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--- abstract: 'We present the IR luminosity function derived from ultra-deep 70$\mu$m imaging of the GOODS-North field. The 70 $\mu$m observations are longward of the PAH and silicate features which complicate work in the MIR. We derive far-infrared luminosities for the 143 sources with $S_{70}> 2$ mJy (S/N $> 3 \sigma$). The majority (81%) of the sources have spectroscopic redshifts, and photometric redshifts are calculated for the remainder. The IR luminosity function at four redshifts ($z \sim$ 0.28, 0.48, 0.78, and 0.97) is derived and compared to the local one. There is considerable degeneracy between luminosity and density evolution. If the evolving luminosity function is described as $\rho(L, z) = (1 + z)^q \rho(L/(1 + z)^p, 0)$, we find $q = -2.19p + 6.09$. In the case of pure luminosity evolution, we find a best fit of $p = 2.78^{+0.34}_{-0.32}$. This is consistent with the results from 24$\mu$m and 1.4GHz studies. Our results confirm the emerging picture of strong evolution in LIRGs and ULIRGs at $0.4 < z < 1.1$, but we find no evidence of significant evolution in the sub-LIRG ($L < 10^{11} L_{\odot}$) population for $z < 0.4$.' author: - 'Minh T. Huynh' - 'David T. Frayer' - Bahram Mobasher - Mark Dickinson - 'Ranga-Ram Chary' - Glenn Morrison bibliography: - 'refs.bib' title: 'The Far-Infrared Luminosity Function from GOODS-N: Constraining the Evolution of Infrared Galaxies for $\lowercase{z} \leq 1$' --- Introduction ============ Deep mid-infrared surveys are revealing a population of mid and far-infrared luminous galaxies out to $z \sim 3$. These luminous (LIRGs, $10^{11} L_\odot < L_{\rm IR} \equiv L_{8-1000\mu{\rm m}} < 10^{12} L_\odot$) and ultraluminous (ULIRGs, $L_{\rm IR} > 10^{12} L_\odot$) infrared galaxies are relatively rare in the local universe, but become increasingly important at high redshift, where dust enshrouded starbursts dominate the total cosmic star formation rate (e.g. [@chary2001], [@blain2002]). The [*Infrared Space Observatory*]{} (ISO) showed that infrared luminous starbursts were much more numerous at $z \sim 1$ than at the present time [@franceschini2001; @elbaz2002]. The ISO results were expanded upon by deep surveys at 24 $\mu$m with the Multiband Imaging Photometer (MIPS) on the [*Spitzer Space Telescope*]{} (e.g. [@chary2004], [@papovich2004]). Using the excellent ancillary data in the Great Observatories Origins Deep Survey (GOODS) South and North fields, 15 $\mu$m and total infrared luminosity functions were derived from thousands of 24 $\mu$m sources [@lefloch2005; @perez2005]. Strong evolution of the IR population was found and the IR luminosity function evolves as $(1 + z)^4$ for $z \lesssim 1$ [@lefloch2005; @perez2005]. The 24 $\mu$m results are dependent on the set of SED templates used to extrapolate the 24 $\mu$m flux densities to 15 $\mu$m and total infrared luminosities. Furthermore, significant variations in the bolometric correction are expected as strong PAH and silicate emission and absorption features are redshifted into the 24 $\mu$m band. Observations with the 70 $\mu$m band of MIPS are closer to the peak in FIR emission and are not affected by PAH or silicate features for $z \lesssim 3$. They should therefore provide more robust estimates of the far-infrared (FIR) luminosities. Studies by ISO in the FIR regime have been limited in sensitivity ($S_{90 \mu{\rm m}} \gtrsim 100$ mJy, $S_{170\mu{\rm m}} > 200$ mJy) and redshift completeness [@serjeant2004; @takeuchi2006]. [@frayer2006] derived a FIR luminosity function (LF) for the Extragalactic First Look Survey (xFLS) from Spitzer 70 $\mu$m data, but this survey had incomplete redshift information at faint fluxes, and it was limited to $z < 0.3$ and bright ($S_{70\mu{\rm m}} \gtrsim 50$ mJy) sources. In this paper we present the infrared luminosity function up to redshift 1 from the ultra-deep 70 $\mu$m survey of GOODS-N. We assume a Hubble constant of $71\,{\rm km}\,{\rm s}^{-1}{\rm Mpc}^{-1}$, and a standard $\Lambda$-CDM cosmology with $\Omega_{\rm M}=0.27$ and $\Omega_{\rm \Lambda}=0.73$ throughout this paper. We define the IR flux as the integrated flux over the wavelength range 8 to 1000 $\mu$m. The Data ======== Ultra-deep 70 $\mu$m Imaging ---------------------------- The GOODS-N field is centered on the Hubble Deep Field North at 12h36m55s, +62$^\circ$14m15s. The MIPS 70$\,\mu$m observations of GOODS-N were carried out during Cycle 1 ([*Spitzer*]{} program ID 3325, [@frayer2006b] and Cycle 3 (January 2006) for the Far Infrared Deep Extragalactic Legacy project (FIDEL, Spitzer PID:30948, PI: Dickinson). Together these data map a region 10$\times$ 18 to a depth of $10.6\,$ksec. The raw data were processed off-line using the Germanium Reprocessing Tools (GeRT), following the techniques described in [@frayer2006b]. We have cataloged 143 sources (over $\sim$$185\,{\rm arcmin}^2$) with $S_{70}\,{\gtrsim}\,2.0\,$mJy (S/N$\,{>}\,3\sigma$) in GOODS-N. The 70$\,\mu$m images have a beam size of 185 FWHM, and in the presence of Gaussian noise the 1$\sigma$ positional error of sources is of the order $\frac{0.5\,\theta_{\rm FWHM}}{{\rm S/N}}$, i.e. 3  for the faintest sources. Redshifts --------- All 70 micron sources were matched to 24 micron and IRAC sources to obtain good positions. The best Spitzer position was then used to search for optical redshifts. About 7% of the 70 micron sources have more than one 24 micron source within the 70 micron beam, and these were deblended individually (e.g. [@huynh2007]). Spectroscopic redshifts are available for 116 of the 143 objects ([@cohen2000]; [@wirth2004]; Stern et al. in prep). Photometric redshifts were derived for 141 of the 143 sources with the extensive photometry available: ACS [*HST*]{} [@giavalisco2004], U- (NOAO), BVRIz- (Subaru-SupremeCam) and JK- (NOAO/KittPeak-Flamingo) imaging. The photometric redshifts were calculated using the $\chi2$ minimization technique as explained in [@mobasher2006]. We have photometric redshifts for 26/27 sources that don’t have a spectroscopic redshift and we therefore have redshift information for 142/143 sources. We quantified the reliability of the photometric redshifts by examining the fractional error, $\Delta \equiv (z_{\rm phot} - z_{\rm spec} / (1 + z_{\rm spec})$. For all 115 70 $\mu$m sources with both photometric and spectroscopic redshifts, we found the median fractional error, $\Delta$, is $0.012 \pm 0.20$. Assuming the 6 cases where the fractional error is greater than 0.2 are outliers, the success rate of the photometric redshift method is 95%. Removing the 6 outliers gives a median fractional error of $0.0014 \pm 0.05$. We therefore conclude that the photometric redshifts are statistically reliable. The 70 micron sources have a median redshift of 0.64 (see Figure 1). The majority (79%) of sources lie at $z < 1$, as expected for the survey sensitivity and steep k-correction that is present at 70 micron. Infrared Luminosities ===================== Many authors argue that the MIR is a good indicator of the bolometric IR luminosity for normal and IR luminous galaxies (e.g. Chary and Elbaz 2001). Based on this, several authors have developed sets of galaxy templates that can be used to estimate the total infrared luminosity ([@chary2001]; [@dh02]; [@lagache2003]). We use the luminosity dependent SED templates based on local galaxies from Chary and Elbaz (2001) to determine the IR luminosities of the 70 $\mu$m galaxies. However it is not clear whether local templates can accurately reproduce the MIR SED of distant galaxies because PAH and silicate absorption features are dependent on complex dust physics, including the intensity of the radiation field, the metallicity of the ISM, and the distribution of grain sizes. For this reason we determine the IR luminosities of the 70 $\mu$m galaxies by fitting templates to the observed 70 $\mu$m flux density only, which is longward of the PAH and silicate features. The IR luminosities as a function of redshift are shown in Figure 1. Most of the sources below redshift $z = 1$ have LIRG-like luminosities. The higher redshift sources are luminous ULIRGs with possibly an embedded AGN. The estimated accuracy of the IR luminosity, from the 70 $\mu$m flux density calibration and PSF fitting errors alone, is 9%. However, the luminosities derived are dependent on the SEDs used. The adopted template SEDs do not reflect the full range of SEDs observed in galaxies, and thus are the main source of systematic errors. For example, the total IR luminosity derived from the MIR regime can vary by a factor of 5 for local galaxies [@dale2005]. We are working longward of the PAHs and silicate features which affected previous work based on the MIR, but, on the other hand, the restframe wavelengths probed at 70 $\mu$m is affected by dust temperatures and emissivity. To test the consistency of our derived IR luminosities and the application of the adopted SEDs, we use the well known FIR-radio correlation. The deep radio image of GOODS-N (5 $\mu$Jy rms at 1.4 GHz, Morrison et al. in preparation) detects 120/143 (84%) of the 70 $\mu$m sources at 3$\sigma$ or above. The FIR-radio correlation, $q = \log ({\rm FIR}/S_{\rm 1.4 GHz})$, where ‘FIR’ here refers to the flux between 40 and 120 $\mu$m (e.g. [@yun2001]), has an observed local value of $q\,{=}\,2.34\pm0.3$ [@yun2001]. Adopting an average factor of 2.0 between IR and FIR (e.g. [@dh02]) and a radio spectral index of $\alpha = -0.8$[^1], we find $q\,{=}\,2.2\pm0.2$ for the radio detected sources. Including the 24 $\mu$m data in the fits to the SEDs gave a slightly larger dispersion in $q$. This suggests that the IR luminosities as estimated from the 70 $\mu$m data alone are reasonable. Infrared Luminosity Functions ============================= In this Section we explore the evolution of the IR luminosity function between redshifts 0 and 1. Methodology ----------- The luminosity functions were derived for 4 redshift bins, $0.2 < z < 0.4$, $0.4 < z < 0.6$, $0.6 < z < 0.9$, and $0.9 < z < 1.1$ using the usual $1/V_{\rm max}$ method (Schmidt 1968). These redshift bins were made wide enough so that there is a reasonable number of sources for calculating the luminosity function. The bins have median redshifts of 0.28, 0.48, 0.78, and 0.97, so a moderate range in redshift is explored. The comoving volume for each source is $V_{\rm max} = V_{z_{\rm max}} - V_{z_{\rm min}} $, where $z_{\rm min}$ is the lower limit of the redshift bin, and $z_{\rm max}$ is the maximum redshift at which the source would be included in the catalog, given the limiting 3$\sigma$ limit, or the maximum redshift of the bin. As mentioned in Section 2.2, we have almost complete redshift information on the 70 $\mu$m sample. A correction factor for each individual source was computed to correct for source detectability across the full image and flux boosting (i.e. the over-estimation of the flux densities of low SN sources). This correction was calculated using the Monte Carlo approach described by [@chary2004] and it is the same correction applied to the source counts [@frayer2006b]. Results and Discussion ---------------------- The luminosity functions were derived from the restframe IR $\mu$m luminosities. In Figure 2 we plot the luminosity functions for the redshift bins explored, and the data is summarized in Table 1. The local IR luminosity function from IRAS sources [@sanders2003] is plotted for comparison. For each luminosity bin the uncertainties, $\sigma_{\rho}$, were estimated using the Poisson statistics on the number of sources, so $\sigma_{\rho} = \left(\sum{\frac{1}{V_{\rm max}^2}}\right)^{1/2}$. Monte Carlo simulations were also performed to disentangle the uncertainties in the derivation of the luminosity function due to photometric errors. Each source was randomly given an IR luminosity within the uncertainty estimates and the luminosity function was re-calculated. We find this adds between 0.03 to 0.09 dex to the luminosity function uncertainty, depending on the bin, but the Poisson statistics dominate the uncertainties. We do not find any significant evolution in the sub-LIRG population ($L < 10^{11} L_{\odot}$) for the lowest redshift bin ($z < 0.4$) (Figure 2). The high redshift LFs show evidence for strong evolution of LIRGs and ULIRGs at $z > 0.4$. The IR LFs derived here are consistent with that derived from 24 micron [@lefloch2005] for the overlapping luminosity bins at $z < 1$. This implies that, on average, similar bolometric luminosities are derived from 24 and 70 micron for moderate luminosity ($L < 10^{11.8} L_{\odot}$) and moderate redshift sources ($z < 1$) sources. We can not say if this is the case for high luminosities and high redshifts (e.g. [@chapman2005; @pope2006]), as those sources are rare in the 70 micron data. Recent 70 micron stacking analysis of galaxies at $z \sim 2$ show that 24 micron observations at high redshift over-estimate LIR in comparison to 70 micron and other LIR indicators [@daddi2007; @papovich2007]. To explore the evolution of IR sources we use the analytical form of the local luminosity function (LF) from [@sanders2003] that comprises of a double power law: $\rho \propto L^{-0.6}$ for $\log (L/L_{\odot}) < 10.5 $, $\rho \propto L^{-2.2}$ for $\log (L/L_{\odot}) > 10.5 $. We assume that the evolving luminosity function can be described by $\rho (L, z) = g(z) \rho[L / f(z), 0]$. In this sense $g(z)$ and $f(z)$ describe the density and luminosity evolution of the LF, respectively. The commonly used form of evolution is to assume $f(z) = (1 + z)^p$ and $g(z) = (1 + z )^q$ (e.g. [@condon1984]; [@haarsma2000]). Using $\chi^2$ minimization, we examine the best fit to the evolution of the IR LF. There is a well known degeneracy between density and luminosity evolution. We find that the best fit evolution parameters follow the relation $q = -2.19p + 6.09$. In the case of pure luminosity evolution ($q = 0$), we find $p = 2.78^{+0.34}_{-0.32}$. These evolution constraints are broadly consistent with 24 $\mu$m studies which found $p = 3.2^{+0.7}_{-0.2}$ and $q = 0.7^{+0.2}_{-0.6}$ for the infrared luminosity function [@lefloch2005]. Our results are also in good agreement with previous studies of IR sources (e.g. [@franceschini2001]). [@hopkins2004] combined star formation rate data with faint radio source counts to find $p = 2.7 \pm 0.6$ and $q = 0.15 \pm 0.60$. If only pure luminosity evolution of radio sources is considered then $p = 2.5 \pm 0.5$ [@Seymour2004] or $p = 2.7$ [@huynh2005]. So our results are consistent with constraints on the evolution of the star forming population from deep radio surveys, indicating that the radio sources overlap with the ultra-deep 70 $\mu$m population, as expected. The constraints on the evolution of the IR LF can be used to determine the cosmic star formation rate (SFR) density. Using the calibration from [@kennicutt1998] and integrating over galaxies with $8.5 < \log(L/L_\odot) < 12.5$, we find the SFR density at $z = 1$ is $0.15^{+0.04}_{-0.03}$ $M_\odot$ yr$^{-1}$ Mpc$^{-3}$ for the best fit pure luminosity evolution case. Here the uncertainties in SFR density do not include the systematics in the FIR/SFR calibration, which add about 0.3 dex to the absolute uncertainty. The SFR density derived here is lower than that estimated by the evolutionary models of [@chary2001] by about a factor of 1.7, but it is consistent with extinction corrected optical measures (e.g. [@kewley2004]) and 24 $\mu$m results [@lefloch2005]. The AGN in our sample can be identified using the deep 2 Ms X-Ray observations of GOODS-N [@alexander2003]. Sources are classed as X-Ray AGN from X-ray band ratios, X-ray luminosity, and X-ray-to-optical flux ratios [@alexander2003; @bauer2004]. At redshifts $z < 0.6$ we find only 7% of the 70 $\mu$m sources are X-Ray AGN, but this fraction increases to 27% for the $0.9 < z < 1.1$ redshift bin. The highest redshift LF in Figure 2 is contaminated by X-Ray AGN but this does not significantly affect the evolution derived in this work. Concluding Remarks ================== Based on ultra-deep 70 $\mu$m observations of GOODS-N, and the spectroscopic and photometric redshifts available of galaxies in this well-studied field, we have derived luminosity functions for $z = 0.3$ to $z = 1.1$. We find strong evolution in galaxies with $L_{\rm IR} > 10^{11} L_{\odot}$ at redshifts $z > 0.4$. Assuming pure luminosity evolution of the form $(1 + z)^p$, we find $p = 2.78^{+0.34}_{-0.32}$. This confirms the strong evolution in LIRGs and ULIRGs between redshift 0 and 1 that has been seen in previous work. The depth of the 70 $\mu$m data allows us to probe sub-LIRG luminosities, and we find little evolution in this population for $z \lesssim 0.4$. In the case of pure luminosity evolution, we find the star formation rate density at $z = 1$ is $0.15^{+0.04}_{-0.03}$ $M_\odot$ yr$^{-1}$ Mpc$^{-3}$. This is the first result from an ultra-deep FIR survey that reaches $z \sim 1$. However, we are limited by poor statistics - the number of bins available for the LF at each redshift is limited by the small number of cataloged sources. The area covered is only 10$\times$ 18 so these results are also affected by cosmic variance. The Far Infrared Deep Extragalactic Legacy (FIDEL) Spitzer legacy project, currently underway, will cover the extended Chandra Deep Field South and the extended Groth Strip at 70 $\mu$m with similar depths to the GOODS-N. The total area covered will be about 9 times that used in this work, and the FIDEL project will detect over 1000 LIRGs at moderate redshift. So in the near future, large ultra-deep FIR surveys such as FIDEL will enable even more detailed studies of the FIR luminosity function and the evolution of infrared galaxies. This work is based on observations made with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. [lllccc]{} log($L_{\rm IR}/L_{\odot}$) & log($L_{\rm IR}/L_{\odot}$) & log($L_{\rm IR}/L_{\odot}$) & N & $\rho ({\rm Mpc}^{-3} {\rm logL}^{-1})$[^2] & Monte Carlo\ low & high & median & & & uncertainty[^3] (dex)\ \ 10.20 & 10.50 & 10.44 & 7 & $3.36 \pm 1.58 \times 10^{-3}$ & 0.04\ 10.50 & 10.70 & 10.63 & 7 & $2.44 \pm 0.94 \times 10^{-3}$ & 0.06\ \ 10.60 & 10.90 & 10.81 & 9 & $1.77 \pm 0.65 \times 10^{-3}$ & 0.07\ 10.90 & 11.10 & 11.00 & 11 & $1.68 \pm 0.51 \times 10^{-3}$ & 0.09\ 11.10 & 11.30 & 11.24 & 9 & $1.24 \pm 0.41 \times 10^{-3}$ & 0.04\ \ 11.10 & 11.40 & 11.25 & 9 & $1.17 \pm 0.47 \times 10^{-3}$ & 0.05\ 11.40 & 11.70 & 11.47 & 13 & $7.90 \pm 2.31 \times 10^{-4}$ & 0.03\ 11.70 & 12.00 & 11.76 & 7 & $2.48 \pm 0.94 \times 10^{-4}$ & 0.07\ \ 11.50 & 11.70 & 11.67 & 8 & $1.44 \pm 0.55 \times 10^{-3}$ & 0.06\ 11.70 & 11.90 & 11.77 & 13 & $1.05 \pm 0.30 \times 10^{-3}$ & 0.07\ \[lf\_table\] ![[*left*]{}: The redshift distribution of the 70 $\mu$m sources. [*right*]{}: The IR luminosity (in $L_{\odot}$) versus redshift for the 70 $\mu$m sources.](f1a.ps "fig:"){width="8.5cm"} ![[*left*]{}: The redshift distribution of the 70 $\mu$m sources. [*right*]{}: The IR luminosity (in $L_{\odot}$) versus redshift for the 70 $\mu$m sources.](f1b.ps "fig:"){width="8.5cm"} {width="14cm"} [^1]: $S_\nu \propto \nu^\alpha$. [^2]: The errors quoted are Poisson uncertainties. [^3]: This is the additional uncertainty to be added to the LF, calculated from Monte Carlo simulations of the uncertainty in the IR luminosity of each source. See Section 4.2.

--- author: - 'P. Papaderos' - 'J.M. Gomes' - 'J.M. Vílchez' - 'C. Kehrig' - 'M.D. Lehnert' - 'B. Ziegler' - 'S. F. Sánchez' - 'B. Husemann' - 'A. Monreal-Ibero' - 'R. Garc[í]{}a-Benito' - 'J. Bland-Hawthorn' - 'C. Coritjo' - 'A. de Lorenzo-C[á]{}ceres' - 'A. del Olmo' - 'J. Falcón-Barroso' - 'L. Galbany' - 'J. Iglesias-Páramo' - 'Á.R. López-Sánchez' - 'I. Marquez' - 'M. Moll[á]{}' - 'D. Mast' - 'G. van de Ven' - 'L. Wisotzki' - the CALIFA collaboration date: 'Received 11 April 2013 / Accepted 2 June 2013' title: 'Nebular emission and the Lyman continuum photon escape fraction in CALIFA early-type galaxies [^1]' --- =1 Introduction \[intro\] ====================== Even though the presence of faint nebular emission () in the nuclei of many early-type galaxies (ETGs) has long been established observationally [e.g., @Phillips1986; @sar06; @sar10; @ani10; @Kehrig2012 hereafter K12], the nature of the dominant excitation mechanism of the warm interstellar medium () in these systems remains uncertain. The [*low-ionization nuclear emission-line region*]{} (LINER) emission-line ratios, as a typical property of ETG nuclei, have prompted various interpretations [see, e.g., K12, @YanBlanton2012], including low-accretion rate active galactic nuclei [AGN; e.g., @Ho1999], fast shocks [e.g. @dop95], and hot, evolved ($\geq 10^8$ yr) post-AGB (pAGB) stars [e.g., @tri91; @bin94; @sta08]. Since each of these mechanisms is tied to distinct and testable expectations on the 2D properties of the , the limited spatial coverage of previous single-aperture and longslit spectroscopic studies has been an important obstacle to any conclusive discrimination between them. Spatially resolved integral field spectroscopy (IFS) over the entire extent of ETGs offers an essential advantage in this respect and promises key observational constraints toward the resolution of this longstanding debate. This Letter gives a brief summary of our results from an ongoing study of 32 ETGs, which were mapped with deep IFS over their entire extent and optical spectral range with the goal of gaining deeper insight into the 2D properties of their . A detailed discussion of individual objects and our methodology will be given in Gomes et al. (2013, in prep.; hereafter G13) and subsequent publications of this series. This study is based on low-spectral-resolution ($R\sim 850$) IFS cubes for 20 E and 12 S0 nearby ($<$150 Mpc) galaxies from the [*Calar Alto Legacy Integral Field Area*]{} (CALIFA) survey [@Sanchez2012 Walcher et al. 2013, in prep.]. These data are being made accessible to the community in a fully reduced and well-documented format [@Husemann2013] through successive data releases. Methodology and results \[meth\] ================================ The CALIFA data cubes were processed with the pipeline (see K12 and G13 for details), which, among various other tasks, permits spaxel-by-spaxel spectral fitting of the stellar component with the population synthesis code [starlight]{} [@cid05] and subsequent determination of emission line fluxes and their uncertainties from the pure emission-line spectrum (i.e. the observed spectrum after subtraction of the best-fitting synthetic stellar model). For each ETG, typically $\sim$1600 to $\sim$3400 individual spectra with a S/N$\geq$30 at 5150 Å  were extracted and modeled in the spectral range 4000–6800 Å using both @bru03 [hereafter BC] and MILES [@san06; @vaz10] simple-stellar population (SSP) libraries, which comprise 34 ages between 5 Myr and 13 Gyr for three metallicities (0.008, 0.019, and 0.03), i.e., 102 elements each. After full analysis and cross-inspection of the relevant output from the BC- and MILES-based models, the emission-line maps for each ETG were error-weighted and averaged spaxel-by-spaxel to reduce uncertainties. An extra module in permits computation of the Lyman continuum () ionizing photon rate corresponding to the best-fitting set of BC SSPs for each spaxel. The  output is then converted into Balmer line luminosities assuming case B recombination for an electron temperature and density of $10^4$ K and 100 cm$^{-3}$, respectively. The same module computes the distance-independent $\tau$ ratio of the  luminosity predicted from pAGB photoionization to the one observed [see @bin94; @cid11 for equivalent quantities]. The latter is optionally corrected for intrinsic extinction, assuming this to be equal to the extinction A$_V$ in the stellar component (cf K12 and G13). Since spectral fits imply a low ($\leq$0.3 mag) A$_V$ in most cases, this correction typically has a weak effect on $\tau$. We preferred to not base corrections of the $\tau$ ratio on nebular extinction estimates since these are consistent with A$_V$ within their uncertainties. We note that state-of-the-art SSP models imply that the  photon rate per unit mass from pAGB stellar populations of nearly-solar metallicity (0.008$\la Z \la$0.03) is almost independent of age, metallicity, and star formation history [e.g. @cid11 G13]. However, substantial uncertainties stem from the fact that existing models differ from one another by a factor $\sim$2 in the mean  output they predict for the pAGB stellar component [@cid11 see also, e.g., Brown et al. 2008 and Woods & Gilfanov 2013 for a discussion related to this subject]. These theoretical uncertainties presumably prevent a determination of the $\tau$ ratio to a precision better than within a factor of $\sim$2 from currently available SSP models. Our analysis in Sects. \[r\_vs\_BPT\] and \[r\_vs\_i\] uses two complementary data sets: i) single-spaxel () determinations from fits with an absolute deviation $\mid\!\!O_{\lambda}-M_{\lambda}\!\!\mid$/$O_{\lambda}$$\leq 2.6$ (cf K12), where $O_{\lambda}$ is the observed spectrum and $M_{\lambda}$ the fit. These are typically restricted to the central, brightest part ($\mu\la$23 $g$ [mag/$\sq\arcsec$]{}) of our sample ETGs. ii) The average of all single-spaxel determinations within isophotal annuli () adapted to the morphology of the (line-free) continuum between 6390 Å and 6490 Å (cf K12). These data, which are to be considered in a *statistical sense*, go $\ga$2 mag fainter, allowing study of the azimuthally averaged properties of the  in the ETG periphery. (8.6,13.8) (0.3,0.4)[![[*From top to bottom:*]{} 3hb, 2ha, $\log$(EW()), and $\log$(\_ext) vs normalized photometric radius /. The gray shaded areas in panels a&b mark the mean and $\pm$1$\sigma$ of the respective quantity, and in panel c the mean EW() for $\geq$ (0.43$\pm$0.65 Å). The light-blue area in panel c depicts the range in EW() that can be accounted for by pAGB photoionization models (0.1–2.4 Å). The color assigned to each ETG is related to its &lt;$\tau$&gt; (cf text and Fig. \[fig:tau2\]) in ascending order, from orange to violet, and is identical in all figures.[]{data-label="fig:r_vs_BPT"}](Fig1.png "fig:"){width="32.60000%"}]{} (16.4,6.4) (0.4,0.4)[{width="32.60000%"}]{} (8.6,6.0) (0.1,0.3)[![Normalized  intensity vs $\log(R^{\star})$ for our sample ETGs, based on  determinations. The diagonal lines correspond to a power-law intensity drop-off of the form $\log(I/I_0) \propto -\alpha\cdot log(R^{\star})$, with $\alpha=1$. The right-hand side table lists the power-law slope $\alpha$ and the radially averaged EW() and $\tau$ for each ETG. []{data-label="fig:tau2"}](Fig3.png "fig:"){width="33.40000%"}]{} Radial behavior of emission-line diagnostics \[r\_vs\_BPT\] ----------------------------------------------------------- Figures \[fig:r\_vs\_BPT\]a&b show the diagnostic 3hb and 2ha line ratios for our sample ETGs as a function of the photometric radius , normalized to the SDSS $r$ band Petrosian\_50 radius . The profiles are based on  determinations, with green error bars illustrating the 1$\sigma$ dispersion (typically $\sim$0.4 dex) of single-spaxel data points within each annulus. All galaxies show LINER-specific @bpt81 [BPT] ratios out to their periphery, with weak (if any) gradients solely within their central part ($\la$). The mean ratios for our sample (shaded regions) were determined to be 0.37$\pm$0.13 for 3hband 0.34$\pm$0.26 for 2ha, with a standard deviation about the mean $\sigma_{\rm N}$ of 0.02 and 0.05. The EW() profiles (panel c) reveal a more complex pattern. For $\ga$, most data points fall between 0.1 Å () and 2.4 Å (), in the range of predictions from  models [e.g., @bin94; @cid11 G13], whereas at smaller radii the sample seems to diverge into a lower ($\la$) and upper ($\ga$) branch. The $\tau$ ratio profiles (panel d) include correction for intrinsic extinction, with vertical bars illustrating the effect that neglecting it would have. The reference line at $\log$(\_ext)=0 corresponds to an equilibrium state where the  photon output from pAGB stars balances the observed  luminosity. Values below ($\log$(\_ext)$<$0) or above ($\log$(\_ext)$>$0) that line imply, in the first case,  photon injection by an additional source (e.g., star formation, AGN, shocks) and, in the second,  photon escape with a [*photon leakage fraction*]{}  = 1-\_ext$^{-1}$. Setting a tentative division line at a radially averaged &lt;$\tau$&gt;=2, we can see that our ETG sample segregates into two groups. In the first one (type i; &lt;$\tau$&gt;$<$2, 14 ETGs), the $\tau$ ratio shows little dependence on radius, with individual data points deviating in most cases by no more than 0.3 dex from the equality line. This suggests a moderate  leakage ($\leq$0.5) and/or dominant contribution of  to the excitation of the . In the second group (type ii; &lt;$\tau$&gt;$\geq$2, 18 ETGs), the  is typically very large ($\ga$0.9) within , and far from negligible ($\ga$0.6) even in the galaxy periphery. As is apparent from panel c, these two groups differ in their EW(), with radially averaged values &lt;EW&gt; of 1.82$\pm$1.04 Å ($\sigma_{\rm N}$=0.28 Å) and 0.41$\pm$0.25 Å ($\sigma_{\rm N}$=0.06 Å). Another salient feature is that EW profiles of type i ETGs are nearly constant beyond $\sim$/2, whereas those of type ii ETGs show a tendency toward a smooth, monotonic increase out to their periphery. Figures \[fig:tau1\]a-c display projections of some quantities of interest onto \_ext. Unsurprisingly, both  and  data delineate a trend toward decreasing EW() with increasing \_ext, with type i and type ii ETGs populating, respectively, the lower and upper parts of a continuous sequence (panel a). This trend is also reflected on a relation log&lt;$\tau$&gt;=(0.23$\pm$0.04)–(1.36$\pm$0.09)$\cdot$log&lt;EW(H$\alpha$)&gt; for our sample (cf right-hand side list in Fig. \[fig:tau2\] for the &lt;$\tau$&gt; and &lt;EW&gt; of individual ETGs). On the 3hb vs log($\tau$) plane (panel b), the two ETG groups differ only marginally from one another (3hb of 0.29$\pm$0.11 and 0.43$\pm$0.12), while a weak trend toward increasing 2ha with log($\tau$) is apparent from panel c (0.11$\pm$0.13 and 0.52$\pm$0.17 for type i and type ii ETGs, respectively). Radial intensity distribution of nebular emission \[r\_vs\_i\] -------------------------------------------------------------- The radial  intensity profiles in Fig. \[fig:tau2\] indicate that faint  is present over nearly the entire optical extent of our sample ETGs. From the Abel integral equation [see, e.g., @P96a for a discussion and solutions for various intensity profiles] it follows that, for an isotropically emitting spheric-symmetric volume, an intrinsic luminosity density distribution $l(r)$ scaling as $\propto r^{-2}$ would be projected onto a power-law intensity profile of the form $\log(I/I_0) \propto -\alpha\log(R^{\star})$ with $\alpha=1$. On the simplifying assumption that the  output from a putative AGN is internally reprocessed into  with a $l(r) \propto r^{-2}$, one can invoke the $\alpha$ inferred from  profile fitting as a minimum consistency check for the AGN illumination hypothesis. The mean $\alpha$ for our sample (1.09), obtained for $\geq$37 (the effective FWHM resolution of CALIFA IFS cubes) is indeed consistent with it and close to the value deduced by @YanBlanton2012 [$\alpha$=1.28] from comparison of two-aperture spectroscopic data. Nevertheless, the large standard deviation in the derived slopes ($\sigma=0.67$) argues against a *universal* power-law index $\alpha\approx1$ for the  intensity drop-off in ETGs. It is interesting though that comparison of Figs. \[fig:r\_vs\_BPT\]&\[fig:tau2\] suggests a tendency for type ii ETGs having shallower  profiles ($\alpha=0.85\pm0.56$; $\sigma_{\rm N}=0.13$) than type i ETGs ($\alpha=1.40\pm0.67$; $\sigma_{\rm N}=0.18$). Discussion and conclusions \[disc\] =================================== Summarizing the evidence from Sect. \[meth\], the ETGs studied here form a broad, continuous sequence with respect to their , EW(), and $\tau$ profiles. Adopting a radially averaged $\tau$ ratio cutoff of &lt;$\tau$&gt;=2, we tentatively subdivide our sample into two groups: Typical properties of type i ETGs are a rather steep  drop-off ($\alpha>1$), nearly constant EWs of $\ga$1 Å beyond , and a &lt;$\tau$&gt; close to unity (0.3…2). Type ii ETGs display shallower  profiles ($\alpha\!<\!1$), overall very low ($\la$…0.5 Å), outwardly increasing EWs, and a large (up to $\sim$20) &lt;$\tau$&gt;. Despite a difference of almost 2 dex in their nuclear $\tau$ ratios, these two groups differ little (by $\la$0.4 dex) in their mean 3hb and 2ha BPT ratios, which in either case are characteristic of LINERs and, within their uncertainties ($\sim$0.4 dex), are radially constant. In our ETG sample, 64% of the S0 galaxies fall into the type i group, and 78% of the E galaxies fall into the type ii group. Clearly, the classification proposed here is only indicative and needs to be refined, both by obtaining better statistics and through a quantitative comparison with other ETG properties: of these, the X-ray luminosity and temperature, the $\alpha_4$ and $(v/\sigma)_{\star}$ parameter, and the star formation history are all being actively investigated. As far as type i ETGs are concerned, various lines of evidence from this study suggest, in line with a substantial body of previous work [e.g. @sar10; @ani10; @YanBlanton2012 K12, among others], that  is the main driver of extended , with nonthermal sources only being potentially important in nuclei: First,  is not confined only to the nuclear regions but is extended out to $\sim$2–4, i.e. is co-spatial with the pAGB stellar background. Second, radial  profiles rule out a power-law intensity drop-off with a *universal* slope $\alpha\approx1$, as a possible signature of a *dominant* AGN contribution to the excitation of the . Third, the EW() is nearly constant beyond $\sim$, pointing to a causal relationship between  and the projected stellar surface density  along the line of sight. This is a plausible expectation from the  scenario, further supported by the quantitative agreement between predicted and observed EWs, as well as the narrow range in $\tau$ ratios ($\simeq1$). These presumably suggest that type i ETGs contain a sufficient amount of  being well mixed with stars, to justify case B recombination as a first-order approximation. Conversely, it is not immediately apparent how AGN or shock excitation alone could lead to the remarkable fine tuning between EW() and   over $\ga$2 dex in . The emerging picture for type ii ETGs appears more complex. The \_ext ratios inferred for these systems imply at face value that the bulk (70%…$\ga$90%) of the  output from hot pAGB stars (*consequently, from any other discrete or diffuse ionizing source*) escapes into the intergalactic space without being reprocessed into . Admittedly,  estimates depend on the pAGB mass inferred from spectral synthesis models. These are known to suffer from degeneracies, the amplitude, topology, and systematics of which remain almost uncharted territory. One might argue that the large number of fits per galaxy (up to $\sim$6800, using two SSP libraries) permits eliminating uncertainties, at least as far as   determinations are concerned. However, this argument would only apply if errors in spectral synthesis were uncorrelated and nonsystematic.\ Nevertheless, in the specific context of type ii ETGs, an essentially model-independent argument for extensive  leakage comes from the virtual absence of , despite a sizeable ionizing photon budget. Quantitatively, for the  escape interpretation to become untenable,  fluxes in type ii ETGs would need be revised upward by up to two orders of magnitude, which can be excluded by any reasonable error budget. The $\tau$ profiles of these ETGs consistently point toward a low, radially dependent volume-filling factor  and/or density for the . A further hint in the same direction comes from their *positive* EW gradients: For a spheric-symmetric volume, these in connection with shallow ($\alpha\!\!<\!\!1$)  profiles are only reproducible when the   luminosity density is monotonically decreasing toward the center. Alternatively, a feature predicted (though not observed) by @sar10 [see their Fig. 10] are positive EW gradients in a spherical stellar host reprocessing its pAGB  output within a *planar* gas configuration. Expanding the considerations by these authors, the high ’s and outwardly increasing EWs of type ii ETGs might be reconcilable for a geometry that involves an oblate distribution of tenuous/clumpy gas within a spherical stellar host. Evidently, such a geometry would *per se* imply pAGB UV photon escape, further reinforcing our interpretation. Regardless of the 3D distribution of the , its high porosity and/or low  call into question the importance of shock excitation in type ii ETGs. If the  is primarily composed of compact cloudlets of radius $r_{\rm c}$, then their large mean-free-path [(2/3)$r_{\rm c}$/, e.g., @JogSolomon1992] would act toward reducing the efficiency of energy dissipation via cloud-cloud collisions and shocks. Given our findings, it is important to ask whether the “weak” AGN interpretation of the optical emission lines is compelling anymore. In the presence of extensive  leakage, emission-line intensities and EWs in type ii ETG nuclei are lowered by factor between $\sim$10 and $\la$100. Consequently, the presence of weak  in these systems is not in itself proof of a “weak” (sub-Eddington accreting) AGN. In fact, the importance of  photon escape, which heretofore has not been investigated in detail, may be a key insight into resolving one of the longstanding enigmas in AGN/LINER research. It offers an *ansatz* for reconciling the fact that many ETGs with prominent signatures of strong AGN activity in radio continuum and/or X-ray wavelengths merely show weak (LINER) optical AGN signatures. In addition, the relative distribution of gas compared to the stars is an important issue. While in a thin, face-on disk, a nuclear EW of, say, $\la$10 Å can safely be regarded as evidence of faint  (and a weak AGN), this is not necessarily the case for a *triaxial* ETG, where the -emitting gas volume may have a more limited extent than the stellar component. In the latter case, nuclear EWs are effectively lowered by the high-surface brightness screen of background and foreground stars along the optical path. In conjunction with the  photon escape, this line-of-sight dilution of the nuclear EWs will conspire to create an observational selection effect, favoring optical detection of AGN activity in oblate, face-on ETGs with atypically low ’s. Arguably, one of the most surprising results of this study are the similar mean BPT ratios of type i and type ii ETGs, despite having substantial differences in their  characteristics. Perhaps the luminosity-weighted emission-line ratios projected along the line of sight “saturate” into the LINER regime for a broad range of distributions and characteristics (i.e. combinations of differing , covering fractions, densities, ionizing photon mean free-path, and ionization parameters), becoming degenerate for ETGs. Circumstantial support for this hypothesis comes from radiation transfer models by [@Ercolano2012], who report that a subset of the *projected* emission line ratio determinations in a 3D model of the [*Pillars of Creation*]{} can mimic LINER characteristics in classical (1D) BPT diagrams. Clearly, detailed 3D radiative transfer modeling of the  in ETGs, including nonequilibrium ionization effects [e.g., @deAvillezBreitschwerdt2012], are important for understanding the nature of  in ETGs. High-quality IFS data, such as those from CALIFA, will no doubt provide crucial observational constraints on these next-generation models. 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This paper is based on data from the Calar Alto Legacy Integral Field Area Survey, CALIFA (http://califa.caha.es), funded by the Spanish Ministery of Science under grant ICTS-2009-10, and the Centro Astronómico Hispano-Alemán. PP is supported by Ciencia 2008 Contract, funded by FCT/MCTES (Portugal) and POPH/FSE (EC), and JMG by a Post-Doctoral grant, funded by FCT/MCTES (Portugal) and POPH/FSE (EC). PP&JMG acknowledge support by the Fundação para a Ciência e a Tecnologia (FCT) under project FCOMP-01-0124-FEDER-029170 (Reference FCT PTDC/FIS-AST/3214/2012), funded by FCT-MEC (PIDDAC) and FEDER (COMPETE). IM acknowledges support from Spanish grant AYA2010-15169 and the Junta de Andalucia through TIC-114 and the Excellence Project P08-TIC-03531. J.F-B. from the Ramón y Cajal Program, grants AYA2010-21322-C03-02 and AIB-2010-DE-00227 from the Spanish Ministry of Economy and Competitiveness (MINECO), as well as from the FP7 Marie Curie Actions of the European Commission, via the Initial Training Network DAGAL under REA grant agreement n$\circ$ 289313. The [starlight]{} project is supported by the Brazilian agencies CNPq, CAPES, and FAPESP. PP has enjoyed inspiring discussions on the effects of  photon leakage with Prof. Nils Bergvall (Uppsala University). We benefited from stimulating discussions with several members of the CALIFA collaboration. This research made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. [^1]: Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck-Institut für Astronomie (MPIA) and the Instituto de Astrofísica de Andalucía (CSIC)

--- author: - 'A. Chanthbouala' - 'R. Matsumoto' - 'J. Grollier' - 'V. Cros' - 'A. Anane' - 'A. Fert' - 'A. V. Khvalkovskiy' - 'K.A. Zvezdin' - 'K. Nishimura' - 'Y. Nagamine' - 'H. Maehara' - 'K. Tsunekawa' - 'A. Fukushima' - 'S. Yuasa' title: 'Vertical current induced domain wall motion in MgO-based magnetic tunnel junction with low current densities' --- **Shifting electrically a magnetic domain wall (DW) by the spin transfer mechanism [@Slonczewski:JMMM:1996; @Berger:PRB:1996; @Grollier:APL:2003; @Klaui:APL:2003] is one of the future ways foreseen for the switching of spintronic memories or registers [@Parkin:Science:2008; @NEC]. The classical geometries where the current is injected in the plane of the magnetic layers suffer from a poor efficiency of the intrinsic torques [@Hayashi:PRL:2007; @Klaui:PRL:2005] acting on the DWs. A way to circumvent this problem is to use vertical current injection [@Ravelosona:PRL:2006; @Boone:PRL:2010; @Rebei:PRB:2006]. In that case, theoretical calculations [@Khvalkovskiy:PRL:2009] attribute the microscopic origin of DW displacements to the out-of-plane (field-like) spin transfer torque [@Slonczewski:PRB:2005; @Theodonis:PRL:2006]. Here we report experiments in which we controllably displace a DW in the planar electrode of a magnetic tunnel junction by vertical current injection. Our measurements confirm the major role of the out-of-plane spin torque for DW motion, and allow to quantify this term precisely. The involved current densities are about 100 times smaller than the one commonly observed with in-plane currents [@Lou:APL:2008]. Step by step resistance switching of the magnetic tunnel junction opens a new way for the realization of spintronic memristive devices [@Strukov:Nature:2008; @Wang:IEEE:2009; @Grollier:patent:2010].** We devise an optimized sample geometry for efficient current DW motion using a magnetic tunnel junction with an MgO barrier sandwiched between two ferromagnetic layers, one free, the other fixed. Such junctions are already the building block of magnetic random-access memories (M-RAMs), which makes our device suitable for memory applications. The large tunnel magnetoresistance [@Yuasa:NatMat:2004; @Parkin:NatMat:2004] allows us to detect clearly DW motions when they propagate in the free layer of the stack [@Kondou:APEX:2008]. The additional advantage of magnetic tunnel junctions is that the out-of-plane field-like torque $\mathbf{T_{OOP}}$ can reach large amplitudes, up to 30$\%$ of the classical in-plane torque $\mathbf{T_{IP}}$ [@Sankey:Nature:2007; @Kubota:Nature:2007], in contrast to metallic spin-valve structures, in which the out-of-plane torque is only a few $\%$ of the in-plane torque [@Stiles:PRB:2002; @Xia:PRB:2002]. This is of fundamental importance since theoretical calculations predict that, when the free and reference layers are based on materials with the same magnetization orientation (either in-plane or perpendicular), the driving torque for steady domain wall motion by vertical current injection is the OOP field-like torque [@Khvalkovskiy:PRL:2009]. Indeed, $\mathbf{T_{OOP}}$ is equivalent to the torque of a magnetic field in the direction of the reference layer, that has the proper symmetry to push the DW along the free layer. On the contrary, the in-plane torque $\mathbf{T_{IP}}$ can only induce a small shift of the DW of a few nm. In magnetic tunnel junctions with the same composition for the top free and bottom reference layers, the OOP field-like torque exhibits a quadratic dependence with bias [@Sankey:Nature:2007; @Kubota:Nature:2007], which could not allow us to reverse the DW motion by current inversion. Therefore we use asymmetric layer composition to obtain an asymmetric OOP field-like torque [@Oh:Nature:2009; @Tang:PRB:2010]. {width=".7\textwidth"} The magnetic stack is sketched in Fig.\[fig1\] (a). The top free layer is (CoFe 1nm/NiFe 4 nm), and the fixed layer is a CoFeB alloy. An S.E.M. top view image of the sample geometry before adding the top contact is shown on Fig.\[fig1\] (b). The half-ring shape was designed for two reasons. First, it facilitates the DW creation [@Saitoh:Nature:2004]. As can be seen from the micromagnetic simulations presented on Fig.\[fig1\] (d), the larger width at the edges stabilizes the DW at an intermediate position in the wire. Secondly, it allows a specific distribution of the Oersted field created by the perpendicular current, as shown by the simulations of Fig.\[fig1\] (c). Thanks to the hollow center, the Oersted field is quasi unidirectional along the wire, and can assist the DW propagation. We first focus on the results obtained with the 210 nm wide wires. A sketch of the sample geometry is given in Fig.\[fig1\] (d), including our convention for the angle of the applied magnetic field. In order to create and pin a DW, we tilt the magnetic field to 75$^{\circ}$. As can be seen in in Fig.\[fig2\] (a), plateaus appear in the resistance vs. field R(H) curve, corresponding to the creation of a magnetic domain wall close to the sample edge (as in the micromagnetic simulation of Fig.\[fig1\] (d)). We chose to work with the plateau obtained at positive fields ($\approx$ + 15 Oe) close to the AP state, which is stable when the field is swept back to zero. This DW creation/pinning process is reproducible, allowing measurements with the same initial state. The strength of the pinning can be evaluated by measuring the corresponding depinning fields. After pinning the DW and coming back to zero field, the R(H) curves have been measured by increasing the field amplitude along 90$^{\circ}$, either to negative or positive values, as shown in Fig.\[fig2\] (b). The positive (resp. negative) depinning fields are $H_{dep}^+$ = +22 Oe and $H_{dep}^-$ = - 43 Oe. This indicates an asymmetry of the potential well which is due to the dipolar field of the synthetic antiferromagnet ($\approx$ + 40 Oe) and also to the asymmetric geometry of the sample close to the edge. {width=".7\textwidth"} In order to study the current induced domain wall depinning, once the domain wall is created, we apply a fixed magnetic field between $H_{dep}^+$ and $H_{dep}^-$, for example - 10 Oe, corresponding to zero effective field, as illustrated by a blue vertical line in Fig.\[fig2\] (b). In our convention, a positive current corresponds to electrons flowing from the synthetic antiferromagnet to the free layer. In Fig.\[fig2\] (c), we show two resistance versus current curves obtained at - 10 Oe, starting always from the same initial DW position (resistance 16.6 $\Omega$). In addition to the expected decrease of the tunnel resistance with bias, we clearly observe irreversible resistance jumps. When the current is swept first to positive values (green curve), the resistance is switched at $I_{dep}^+$ = + 7 mA to a lower resistance state corresponding to another domain wall position, stable at zero current, with a low bias resistance of 16.1 $\Omega$. By resetting the DW position, then applying negative currents (red curve) a resistance jump to a higher resistance state of 17.3 $\Omega$ occurs at $I_{dep}^-$ = -11 mA. We thus demonstrate the possibility to move a domain wall by perpendicular dc current injection in both directions depending on the current sign. The current densities corresponding to the DW motion are lower than 4 10$^6$ A.cm$^{-2}$ (see top x axis of Fig.\[fig2\] (c)). The use of perpendicular current injection therefore allows to reduce the current densities by a factor 100 compared to the classical lateral current injection [@Hayashi:PRL:2007; @Klaui:PRL:2005]. {width=".7\textwidth"} Similar measurements have been performed for several fields between $H_{dep}^+$ and $H_{dep}^-$. As shown on Fig.\[fig2\] (d), the resistance associated with each pinning center changes progressively as a function of the applied magnetic field, which can be ascribed to field-induced DW displacement / deformation within the potential well. The depinning currents strongly depend on the applied magnetic field too. Negative fields favour the domain wall motion in the -90$^{\circ}$ direction, thus reducing the values of $I_{dep}^-$, and increasing $I_{dep}^+$. As expected, the effect is opposite for positive fields. By comparing with the DW motion in applied field, we can define the value of the equivalent field induced by the positive or negative depinning currents : $I_{dep}^{\pm}$ in field $H$ generates an equivalent field of $H_{dep}^{\pm}$-$H$. We therefore can plot the equivalent field generated by the current as a function of the bias voltage, as shown in Fig.\[fig3\] (a) for two samples with the same nominal shape. Additional experiments allow us to discard Joule heating (which could reduce the current-induced depinning fields) as a possible source of measured effective field enhancement at large bias (see methods). For both samples, a positive bias induces an effective field pointing in the direction of the reference layer magnetization and inversely for negative bias. The overall trend is similar, linear at low bias ($<$ 60 mV), with deviations from linearity at large bias. The origins of current-induced DW depinnings are : the two spin-transfer torques (in-plane or out-of-plane) and the Oersted field. Spin diode experiments are a powerful tool to obtain the dc bias dependence of the two spin transfer torques [@Sankey:Nature:2007; @Kubota:Nature:2007; @Wang:PRB:2009]. In order to investigate the physical origin of the perpendicular current-induced domain wall motion in our system, we perform additional spin diode experiments with 70 $\times$ 270 nm$^2$ ellipses patterned in the same stack as the semi-circular wires. The analysis of the resulting rectified voltage vs frequency curves obtained at different bias allow us to plot the two components of the spin transfer torque, the in-plane torque Fig.\[fig3\] (b) and the out-of-plane field-like torque Fig.\[fig3\] (c), expressed in field units, as a function of bias (see methods and supplementary information). As expected, the IP torque is asymmetric with bias. The sign of the OOP torque of our asymmetric structure also changes with the current direction, in agreement with the results of Oh *et al.* obtained by another method in MgO-based tunnel junctions with asymmetric layer compositions. In the low bias region between $\pm$ 60 mV, the OOP field-like torque reaches up to 40 $\%$ of the IP torque. ![Plot of the equivalent field versus dc voltage. Circle symbols : micromagnetic simulations of DW depinning. Red and blue cross symbols : experimental results for the current-induced depinning fields. Green crosses : OOP torque expressed in field units derived from spin diode experiments.[]{data-label="fig4"}](fig4.pdf){width=".45\textwidth"} According to our previous theoretical predictions [@Khvalkovskiy:PRL:2009], only the OOP term can give rise to spin transfer induced steady DW motion. Fig.\[fig3\] (d) directly compares the amplitude of the OOP torque (expressed in field units) derived from spin diode experiments to the effective field determined by domain wall depinning measurements. For both types of experiments, a positive bias induces an equivalent field pointing in the direction of the magnetization of the reference layer, and in the opposite direction for negative bias. Samples 1-A and B have the dimensions corresponding to the S.E.M. picture of Fig.\[fig1\] (b) (width 210 nm), while samples 2-A and B are smaller but with the same aspect ratio (width 120 nm). For the largest samples 1-A and 1-B, the equivalent magnetic field generated by perpendicular injection is clearly increased at high bias. In order to interpret these results, we have performed additional micromagnetic simulations following the experimental procedure of domain wall depinning (supplementary note 3). The spin transfer torques and the Oersted field are introduced in order to determine the equivalent current induced magnetic fields. In the simulations, both spin transfer torques are set linear as a function of current, with $T_{OOP} = 40 \% \:T_{IP}$. The results are summed up on Fig.\[fig4\]. First ignoring the Oersted field, we obtain approximately the same equivalent fields with only $T_{OOP}$ (open green circles) than with both $T_{IP}$ and $T_{OOP}$ (filled green circles), which confirms our predictions *et al.* [@Khvalkovskiy:PRL:2009] that the contribution of $T_{IP}$ to the DW motion is negligible. These simulations are in very good agreement with the OOP torque derived from the spin diode experimental data (green crosses). We find that the contribution of the spin transfer torques to the equivalent field (closed green circles) is not sufficient to account for the experimental equivalent fields for samples 1 and 2 (blue and red crosses), but that a quantitative agreement can be obtained by taking the Oersted field also into account (blue and red filled circles). In particular, this allows us to ascribe unambiguously the larger slope obtained for samples 1-A and B to the increased Oersted field for larger areas. These results also prove the efficiency of our approach combining the actions of the Oersted field and OOP torque to induce DW propagation at current densities lower than 5 10$^6$ A.cm$^{-2}$, thanks to the specific design of our sample. This torque engineering could be of particular interest for the development of the low current density multilevel memory cells proposed by Seagate [@Lou:APL:2008]. From Fig.\[fig3\] (d) and Fig.\[fig4\], it appears that the OOP field-like torque alone can generate an equivalent magnetic field of 10 Oe for 5 10$^6$ A.cm$^{-2}$. Devices using only this mechanism to drive DW motion are therefore possible if the DW is not strongly pinned. This last condition is typically desirable for a DW based spintronic memristor, in which the DW position should be continuously tunable by current injection [@Wang:IEEE:2009; @Grollier:patent:2010]. Memristor devices inherently behave like artificial nano-synapses and have a strong potential for implementation in large-scale neuromophic circuits [@Strukov:Nature:2008]. The more intense is the current through a memristor, and the longer it is injected, the more the resistance changes. Spin-transfer induced DW displacements are precisely proportional to the amplitude of the injected current as well as pulse duration [@Hayashi:PRL:2007]. In addition, by using perpendicularly magnetized layers (domain wall width $<$ 10 nm [@Ravelosona:PRL:2006]), our device could be scaled down below 50x100 nm$^2$. Low current density DW motion by perpendicular current injection in large TMR magnetic tunnel junctions is therefore extremely promising for the future developments of fast and robust spintronic memristors. **Methods** : SAMPLES The magnetic stack was grown by sputtering in a Canon ANELVA chamber. Details of the growth and fabrication process have been presented elsewhere [@Yuasa:JPhysD:2007]. For all samples the TMR is around 65 $\%$, with a low RA product of 3.5 $\Omega.\mu m^2$. Samples 1-A and B have the dimensions corresponding to the S.E.M. picture of Fig.\[fig1\] (b) (width 210 nm, inner diameter 550 nm), while samples 2-A and B are smaller but with the same aspect ratio (width 120 nm, inner diameter 370 nm). TEMPERATURE EVALUATION In order to evaluate the increase of temperature in our samples, we have measured the saturation fields $ H_{SAT} $ of the synthetic antiferromagnet at constant low bias as a function of the temperature, and as a function of bias at constant temperature (RT). By comparing the two sets of measurements, we estimate that the largest temperature increase is $\approx$ 20 K and has a negligible impact on the DW depinning fields ($\approx$ 1 Oe) (see supplementary information). SPIN-TORQUE DIODE MEASUREMENTS The magnetic field is applied along the hard axis of the ellipse, and is chosen large enough to saturate the magnetization of the free layer (experimental applied field range 550 - 650 0e). We inject a constant microwave current of $i_{hf}$ = 40 $\mu A$ and sweep the dc current between -0.9 and +0.9 mA. The microwave current is modulated (on/off 1:1) in order to increase the precision. MICROMAGNETIC SIMULATIONS For the micromagnetic simulations, we use the finite-difference micromagnetic code SpinPM, developed by Istituto P.M. The simulated free layer has the geometry of the S.E.M. images of samples 1 or 2. The mesh cell size is set to 3 $\times$ 3 $\times$ 5 nm$^{3}$. We took the following magnetic parameters: $\alpha$ = 0.01 for the Gilbert damping and $M_{s}(CoFe/NiFe)$ = 1 T for the magnetization of the free layer. The spin polarization has been set to P$_{spin}$ = 0.5 and the current is supposed to be uniform through the structure. **Acknowledgments** : Financial support by the CNRS, RFBR grant (Grant No. 09-02-01423), JSPS Postdoctoral Fellowships for Research Abroad and the European Research Council (Starting Independent Researcher Grant No. ERC 2010 Stg 259068) is acknowledged. Correspondence and requests for materials should be addressed to J.G. **Author Contributions** : J.G., A.C., V.C. and S.Y. conceived the experiments. A.C. and R.M. carried out the measurements and analyzed the data with the help of J.G. and V.C.; A.C. performed the numerical simulations with help from J.G., A.V.K. and K.A.Z.; K.N., Y.N., H.M, K.T. deposited the magnetic stack. A.F. fabricated the samples. J.G. wrote the paper with discussions and comments from A.C., R.M., V.C., A.A., S.Y. and A.F. 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--- author: - 'Roland Diehl[^1]' title: Astrophysics with Radioactive Isotopes --- =@jnl\#1[[\#1]{}]{} å[$\alpha$]{} \[intro\] Origin of Radioactivity ======================= The nineteenth century spawned various efforts to bring order into the elements encountered in nature. Among the most important was an inventory of the [*elements*]{} assembled by the Russian chemist Dimitri Mendeleyev in 1869, which grouped elements according to their chemical properties, their [*valences*]{}, as derived from the compounds they were able to form, at the same time sorting the elements by atomic weight. The genius of Mendeleyev lay in his confidence in these sorting principles, which enforce gaps in his table for expected but then unknown elements, and Mendeleyev was able to predict the physical and chemical properties of such elements-to-be-found. The tabular arrangement invented by Mendeleyev (Fig. \[fig\_1\_periodic\_table\]) still is in use today, and is being populated at the high-mass end by the great experiments in heavy-ion collider laboratories to create the short-lived elements predicted to exist. The second half of the nineteenth century thus saw scientists being all-excited about chemistry and the fascinating discoveries one could make using Mendeleyev’s sorting principles. Note that this was some 30 years before sub-atomic particles and the atom were discovered. Today the existence of 118 elements is firmly established[^2], the latest additions no. 113-118 all discovered in year 2016, which reflects the concerted experimental efforts. ![The periodic table of elements, grouping chemical elements according to their chemical-reaction properties and their atomic weight, after Mendeleyev (1869), in its 2016 version (IUPAC.org)[]{data-label="fig_1_periodic_table"}](Fig_1_IUPAC_PeriodicTable_Nov16.pdf "fig:"){width="\textwidth"}\ In the late nineteenth century, scientists also were excited about new types of penetrating radiation. Conrad Röntgen’s discovery in 1895 of [*X-rays*]{} as a type of electromagnetic radiation is important for understanding the conditions under which Antoine Henri Becquerel discovered radioactivity in 1896. Becquerel also was engaged in chemical experiments, in his research on phosphorescence exploiting the chemistry of photographic-plate materials. At the time, Becquerel had prepared some plates treated with uranium-carrying minerals, but did not get around to make the planned experiment. When he found the plates in their dark storage some time later, he accidentally processed them, and was surprised to find an image of a coin which happened to have been stored with the plates. Excited about X-rays, he believed he had found yet another type of radiation. Within a few years, Becquerel with Marie and Pierre Curie and others recognised that the origin of the observed radiation were elemental transformations of the uranium minerals: The physical process of [*radioactivity*]{} had been found! The revolutionary aspect of elements being able to spontaneously change their nature became masked at the beginning of the twentieth century, when sub-atomic particles and the atom were discovered. But well before atomic and quantum physics began to unfold, the physics of [*weak interactions*]{} had already been discovered in its form of [*radioactivity*]{}. The different characteristics of different chemical elements and the systematics of Mendeleyev’s periodic table were soon understood from the atomic structure of a compact and positively charged nucleus and a number of electrons orbiting the nucleus and neutralising the charge of the atom. Bohr’s atomic model led to the dramatic developments of quantum mechanics and spectroscopy of atomic shell transitions. But already in 1920, Ernest Rutherford proposed that an electrically neutral particle of similar mass as the hydrogen nucleus (proton) was to be part of the compact atomic nucleus. It took more than two decades to verify by experiment the existence of this ’neutron’, by James Chadwick in 1932. The atomic nucleus, too, was seen as a quantum mechanical system composed of a multitude of particles bound by the strong nuclear force. This latter characteristic is common to ’hadrons’, i.e. the electrically charged proton and the neutron, the latter being slightly more massive[^3]. Neutrons remained a mystery for so long, as they are unstable and decay with a mean life of 880 seconds from the weak interaction into a proton, an electron, and an anti-neutrino. This is the origin of radioactivity. The chemical and physical characteristics of an element are dominated by their electron configuration, hence by the number of charges contained in the atomic electron cloud, which again is dictated by the charge of the atomic nucleus, the number of protons. The number of neutrons included in the nucleus are important as they change the mass of the atom, however the electron configuration and hence the properties are hardly affected. Therefore, we distinguish *isotopes* of each particular chemical element, which are different in the number of neutrons included in the nucleus, but carry the same charge of the nucleus. For example, we know of three stable isotopes of oxygen as found in nature, $^{16}$O, $^{17}$O, and $^{18}$O. There are more possible nucleus configurations of oxygen with its eight protons, ranging from $^{13}$O as the lightest and $^{24}$O as the most massive known isotope. An [*isotope*]{} is defined by the number of its two types of nucleons[^4], [*protons*]{} (the number of protons defines the charge number Z) and [*neutrons*]{} (the sum of the numbers of protons and neutrons defines the mass number A), written as $^A$X for an element ’X’. Note that some isotopes may exist in different nuclear quantum states which have significant stability by themselves, so that transitions between these configurations may liberate the binding energy differences; such states of the same isotope are called [*isomers*]{}. The landscape of isotopes is illustrated in Fig. \[fig\_1\_table-of-isotopes\], with black symbols as the naturally-existing stable isotopes, and coloured symbols for unstable isotopes. Unstable isotopes, once produced, will be *radioactive*, i.e. they will transmute to other isotopes through nuclear interactions, until at the end of such a decay chain a stable isotope is produced. Weak interactions will mediate transitions between protons and neutrons and lead to neutrino emission, involvements of atomic-shell electrons will result in X-rays from atomic-shell transitions after electron capture and internal-conversion transitions, and $\gamma$-rays will be emitted in electromagnetic transitions between excitation levels of a nucleus. The production of non-natural isotopes and thus the generation of man-made radioactivity led to the Nobel Prize in Chemistry being awarded to Jean Frédéric Joliot-Curie and his wife Iréne in 1935 – the second Nobel Prize awarded for the subject of radioactivity after the 1903 award jointly to Pierre Curie, Marie Skłodowska Curie, and Henri Becquerel, also in the field of Chemistry. At the time of writing, element 118 called oganesson (Og) is the most massive superheavy element which has been synthesised and found to exist at least for short time intervals, although more massive elements may exist in an island of stability beyond. ![The table of isotopes, showing nuclei in a chart of neutron number (abscissa) versus proton number (ordinate). The stable elements are marked in black. All other isotopes are unstable, or radioactive, and will decay until a stable nucleus is obtained.[]{data-label="fig_1_table-of-isotopes"}](Fig_Table_of_Isotopes.pdf "fig:"){width="\textwidth"}\ Depending on the astrophysical objective, radioactive isotopes may be called *short-lived*, or *long-lived*, depending on how the radioactive lifetime compares to astrophysical time scales of interest. Examples are the utilisation of [$^{26}$Al]{}and [$^{60}$Fe]{}($\tau\sim$My) diagnostics of the early solar system (*short-lived*, Chap. 6) or of nucleosynthesis source types (*long-lived*, Chap. 3-5). Which radioactive decays are to be expected? What are stable configurations of nucleons inside the nuclei involved in a production and decay reaction chain? The answer to this involves an understanding of the nuclear forces and reactions, and the structure of nuclei. This is an area of current research, characterised by combinations of empirical modeling, with some capability of *ab initio* physical descriptions, and far from being fully understood. Nevertheless, a few general ideas appear well established. One of these is recognising a system’s trend towards minimising its total energy, and inspecting herein the concept of *nuclear binding energy*. It can be summarised in the expression for nuclear masses [@Weizsacker:1935]: $$m(Z,A) = Z m_p + (A-Z) m_n - BE$$ with $$BE = a_{volume} A - a_{surface} A^{2/3} - a_{coulomb} {Z^2 \over {A^{1/3}}} - a_{asymmetry} {{{(a-2Z)}^2} \over {4A}} - {\delta \over A^{1/2}}$$ The total *binding energy* (BE) is used as a key parameter for a system of nucleons, and nucleons may thus adopt bound states of lower energy than the sum of the free nucleons, towards a global minimum of system energy. Thus, in a thermal mixture of nucleons, bound nuclei will be formed, and their abundance depends on their composition and shape, and on the overall system temperature, defining how the totally-available phase space of internal and kinetic energy states is populated. The nucleonic systems would thus have local maxima of binding energy from (1) the *odd-even* effect described by the last term, which results in odd-nucleon nuclei being less favored that even-nucleon nuclei, and (2) a general excess of neutrons would be favored by the asymmetry term, which results in heavier nuclei being relatively more neutron rich. The other concept makes use of *entropy*, recognising the relation of this thermodynamic variable to the over-all state of a complex multi-particle and multi-state system. A change in entropy corresponds to a change in the micro-states available to the system. For an infinitesimal change in entropy, we have $$T ds = - \sum_i{\mu_idY_i}$$ where $Y_i$ are the fractional abundances by number of a species $i$, e.g. $i$= $^{12}$C, or $^4$He, or protons $^1$H, and $\mu$ is the thermodynamic potential[^5] of species $i$. Hence, for our application, if the isotopic composition of a nucleonic mixture changes, its entropy will also change. Or, conversely, the entropy, normalised by the number of baryons in the system, will be a characteristic for the composition: $$Y_i \propto {{S}\over {n_b}} = s$$ with the interpretation of entropy related to the (logarithm of) the number $\Gamma$ of micro-states available: $$S=k_b \cdot ln \Gamma$$ This thermodynamic view allows to calculate *equilibrium* compositions, as they depend on the temperature and on the entropy per baryon. With $${{S}\over {n_b}} \propto {{n_\gamma}\over{n_b}}$$ the photon to baryon ratio also serves as a measure of the entropy per baryon. This consideration of thermodynamic equilibrium can be carried through to write down the *nuclear Saha equation* for the composition for an isotope with mass $A$ and charge $Z$: $$\begin{aligned} Y_i=Y(Z_i,A_i)=G(Z_i,A_i) [\zeta(3)^{A_i-1} \pi^{(1-A_i)/2} 2^{(3A_i-5)/2} \cdot \nonumber \\ A_i^{3/2} (k_B T / m_N c^2)^{3(A_i-1)/2} \Phi^{1-A_i} Y_p^{Z_i} Y_n^{A_i-Z_i} exp[BE(Z_i,A_i)/ k_B T ] \end{aligned}$$ Herein, $G(Z_i,A_i)$ is the nuclear partition function giving the number of micro-states for the particular isotope, $\zeta(3)$ is the Riemann function of argument 3, and we find again the *binding energy* $BE$ and also the *thermal energy* $k_BT$. $\Phi$ is defined as ratio of photon number to baryon number, and is proportional to the entropy per baryon, thus including the phase space for the plasma constituents. This equation links the proton and neutron abundances to the abundances of all other isotopes, with the characteristic isotope properties of mass $m_N$, mass and charge numbers $A,Z$, and internal micro-states $G$, using the different forms of energy (rest mass, thermal, and binding), as well as the characteristic entropy. Illustrative examples of how entropy helps to characterise isotopic mixtures are: For high temperatures and entropies, a composition with many nuclei, such as rich in $\alpha$ nuclei would be preferred (e.g. near the big bang in the early universe), while at lower entropy values characteristic for stellar cores a composition of fewer components favouring tightly-bound nucleons in Fe nuclei would be preferred (e.g. in supernova explosions). With such knowledge about nuclear structure in hand, we can look at the possible configurations that may exist: Those with a minimum of total energy will be *stable*, all others *unstable* or *radioactive*. Fig. \[fig\_1\_table-of-isotopes\] shows the table of isotopes, encoded as stable (black) and unstable isotopes, the latter decaying by $\beta^-$-decay (blue) and $\beta^+$-decay (orange). This is an illustration of the general patterns among the available nuclear configurations. The *ragged* structure signifies that there are systematic variations of nuclear stability with nucleon number, some nucleonic numbers allowing for a greater variety of stable configurations of higher binding energy. These are, in particular, [*magic numbers*]{} of protons and neutrons of 2, 8, 20, 28, 50, and 82. We now know approximately 3100 such *isotopes* making up the 118 now-known chemical elements, but only 286 of these isotopes are considered stable. The (7$^{th}$) edition of the Karlsruher Nuklidkarte (2007) [@2007KNucChart..7] lists 2962 experimentally-observed isotopes and 652 isomers, its first edition (1958) included 1297 known isotopes of 102 then-known elements. Theoretical models of atomic nuclei, on the other hand, provide estimates of what might still be open to discovery, in terms of isotopes that might exist but either were not produced in the nearby universe or are too shortlived to be observed. Recent models predict existence of over 9000 nuclei [@Erler:2012; @Xia:2018]. It is the subject of this book to explain in detail the astrophysical implications of this characteristic process of nuclear rearrangements, and what can be learned from measurements of the messengers of radioactive decays. But first we describe the phenomenon of radioactivity in more detail. Processes of Radioactivity {#sec:1_processes} ========================== The number of decays at each time should be proportional to the number of currently-existing radioisotopes: $${{dN}\over{dt}} = -\lambda \cdot N$$ Here $N$ is the number of isotopes, and the [*radioactive-decay constant*]{} $\lambda$ is the characteristic of a particular radioactive species. Therefore, in an ensemble consisting of a large number of identical and unstable isotopes, their number remaining after radioactive decay declines exponentially with time: $$\label{eq_1} \index{decay!exponential} N = N_0 \cdot exp{-t\over\tau}$$ The decay time $\tau$ is the inverse of the radioactive-decay constant, and $\tau$ characterises the time after which the number of isotopes is reduced by decay to $1/e$ of the original number. Correspondingly, the radioactive half-life $T_{1/2}$, is defined as the time after which the number of isotopes is reduced by decay to $1/2$ of the original amount, with $$T_{1/2} = {\tau \over ln(2)}$$ The above exponential decay law is a consequence of a surprisingly simple physical property: The probability per unit time for a single radioactive nucleus to decay is independent of the age of that nucleus. Unlike our common-sense experience with living things, decay does not become more likely as the nucleus ages. Radioactive decay is a nuclear transition from one set of nucleons constituting a nucleus to a different and energetically-favored set with the same number of nucleons. Different types of interactions can mediate such a transition (see below). In *$\beta$-decays* it is mediated by the *weak transition* of a neutron into a proton and vice versa[^6], or more generally, nucleons of one type into the other type[^7]: $$\label{eq_n-decay} n \longrightarrow p \mbox{ } + e^- \mbox{ } + \overline{\nu_e}$$ $$\label{eq_p-decay} p \longrightarrow n \mbox{ } + e^+ \mbox{ } + {\nu_e}$$ If such a process occurs inside an atomic nucleus, the quantum state of the nucleus is altered. Depending on the variety of configurations in which this new state may be realized (i.e. the *phase space* available to the decaying nucleus), this change may be more or less likely, in nature’s attempt to minimize the total energy of a composite system of nucleons. The decay probability $\lambda$ per unit time for a single radioactive nucleus is therefore a property which is specific to each particular type of isotope. It can be estimated by Fermi’s *Golden Rule* formula though time-dependent perturbation theory [e.g. @1962qume.book.....M]. When schematically simplified to convey the main ingredients, the decay probability is: $$\label{eq1} \lambda = \frac{4\pi^2}{h} \mbox{ } V_{fi}^2 \mbox{ } \rho(W) $$ where $\rho(W)$ is the number of final states having suitable energy $W$. The detailed theoretical description involves an integral over the final kinematic states, suppressed here for simplicity. The matrix element $V_{fi}$ is the result of the transition-causing potential between initial and final states. In the general laboratory situation, radioactive decay involves a transition from the ground state of the parent nucleus to the daughter nucleus in an excited state. But in cosmic environments, nuclei may be part of hot plasma, and temperatures exceeding millions of degrees lead to population of excited states of nuclei. Thus, quantum mechanical transition rules may allow and even prefer other initial and final states, and the nuclear reactions involving a radioactive decay become more complex. Excess binding energy will be transferred to the end products, which are the daughter nucleus plus emitted (or absorbed, in the case of electron capture transitions) leptons (electrons, positrons, neutrinos) and $\gamma$-ray photons. The occupancy of nuclear states is mediated by the *thermal* excitation spectrum of the *Boltzmann distribution* of particles, populating states at different energies according to: $${dN \over dE} = G_j \cdot e^{-{{E}\over{k_BT}}}$$ Here $k_B$ is Boltzmann’s constant, $T$ the temperature of the particle population, $E$ the energy, and $G_j$ the statistical weight factor of all different possible states $j$ which correspond to a specific energy $E$[^8]. In natural environments, particles will populate different states as temperature dictates. Transition rates among states thus will depend on temperature. Inside stars, and more so in explosive environments, temperatures can reach ranges which are typical for nuclear energy-level differences. Therefore, in cosmic sites, radioactive decay time scales may be significantly different from what we measure in terrestrial laboratories on *cold* samples (see Section \[sec:1\_processes\] for more detail). Also the atomic-shell environment of a nucleus may modify radioactive decay, if a decay involves *capture or emission of an electron* to transform a proton into a neutron, or vice versa. Electron capture decays are inhibited in fully-ionized plasma, due to the non-availability of electrons. Also $\beta^-$-decays are affected, as the phase space for electrons close to the nucleus is influenced by the population of electron states in the atomic shell. After Becquerel’s discovery of radioactivity in 1896, Rutherford and others found out in the early 20$^{\rm{th}}$ century that there were different types of radioactive decay [@1903PPSL...18..595R]. They called them *$\alpha$ decay, $\beta$ decay* and *$\gamma$ decay*, terms which are still used today. It was soon understood that they are different types of interactions, all causing the same, spontaneous, and time-independent decay of an unstable nucleus into another and more stable nucleus. [*Alpha decay*]{} : This describes the ejection of a $^4$He nucleus from the parent radioactive nucleus upon decay. $^4$He nuclei have since been known also as *alpha particles* for that reason. This decay is intrinsically fast, as it is caused by the *strong* nuclear interaction quickly clustering the nucleus into an alpha particle and the daughter nucleus. Since $\alpha$-nuclei are tighly-bound, they have been found as sub-structures even within nuclei. In the cases of nuclei much heavier than Fe, a nucleus thus consisting of many nucleons and embedded $\alpha$ clusters can find a preferred state for its number of nucleons by separation of such an $\alpha$ cluster, liberating the binding-energy difference[^9]. In such heavy nuclei, Coulomb repulsion helps to overcome the potential barrier which is set up by the strong nuclear force, and decay can occur through emission of an $\alpha$ particle. The $\alpha$ particle *tunnels*, with some calculable probability, through the potential barrier, towards an overall more stable and less-energetic assembly of the nucleons. An example of $\alpha$ decay is $_{88}$Ra$^{226}$ $\Rightarrow$ $_{86}$Rn$^{222}$ + $_2$He$^4$, which is one step in the decay series starting from $^{238}$U. The daughter nucleus , $_{86}$Rn$^{222}$, has charge $Z-2$, where $Z$ is the original charge of the radioactive nucleus ($Z$=88 in this example), because the $\alpha$ particle carried away two charge units from the original radioactive nucleus. Such decay frequently leads to an excited state of the daughter nucleus. Kinetic energy $E_{\alpha}$ for the $\alpha$ particle is made available from the nuclear binding energy liberation expressed by the *Q-value* of the reaction if the mass of the radioactive nucleus exceeds the sum of the masses of the daughter nucleus and of the helium nucleus[^10]: $$\index{isotopes!226Ra} Q_{\alpha} = [M(_{88}\rm{Ra}^{226}) - M(_{86}\rm{Rn}^{222}) - M(_2\rm{He}^4)]\rm{c}^2$$ The range of the $\alpha$ particle (its stopping length) is about 2.7 cm in standard air (for an $\alpha$ particle with E$_{\alpha}$ of 4 MeV), and it will produce about 2$\times$10$^5$ ionizations before being stopped. Even in a molecular cloud, though its range would be perhaps 10$^{14}$ times larger, the $\alpha$ particle would not escape from the cloud. Within small solids (dust grains), the trapping of radioactive energy from $\alpha$ decay provides a source of heat which may result in characteristic melting signatures[^11]. [*Beta decay:*]{} This is the most-peculiar radioactive decay type, as it is caused by the nuclear *weak interaction* which converts neutrons into protons and vice versa. The neutrino $\nu$ carries energy and momentum to balance the dynamic quantities, as Pauli famously proposed in 1930 (Pauli did not publish this conjecture until 1961 in a letter he wrote to colleagues). The $\nu$ was given its name by Fermi, and was discovered experimentally in 1932 by James Chadwick, i.e. *after* Wolfgang Pauli had predicted its existence. Neutrinos from the Sun have been discovered to *oscillate* between flavors. $\beta$ decays are being studied in great detail by modern physics experiments, to understand the nature and mass of the $\nu$. Understanding $\beta$ decay challenges our mind, as it involves several such unfamiliar concepts and particles. There are three types[^12] of $\beta$-decay: $$\label{eq_beta+-decay} ^A_ZX_N\mbox{ } \longrightarrow \mbox{ } ^A_{Z-1}X_{N+1} \mbox{ } + e^+ \mbox{ } + \nu_e$$ $$\label{eq_beta--decay} ^A_ZX_N \mbox{ } \longrightarrow \mbox{ } ^A_{Z+1}X_{N-1} \mbox{ } + e^- \mbox{ } + \overline{\nu_e}$$ $$\label{eq_beta--decay} ^A_ZX_N \mbox{ } + e^- \longrightarrow \mbox{ } ^A_{Z-1}X_{N+1} \mbox{ } + {\nu_e}$$ In addition to eq. \[eq\_n-decay\] (*$\beta^-$ decay*), these are the conversion of a proton into a neutron (*$\beta^+$ decay*), and *electron capture*. The weak interaction itself involves two different aspects with intrinsic and different strength, the vector and axial-vector couplings. The $V_{fi}^2$ term in eq. \[eq1\] thus is composed of two terms. These result in *Fermi* and *Gamow-Teller transitions*, respectively [see @2003RvMP...75..819L for a review of weak-interaction physics in nuclear astrophysics]. An example of $\beta$ decay is $_7^{13}$N $\longrightarrow \mbox{ }_6^{13}$C + e$^{+}$ $+$ $\nu$, having mean lifetime $\tau$ near 10 minutes. The kinetic energy $Q$ of the two leptons, as well as the created electron’s mass, must be provided by the radioactive nucleus having greater mass than the sum of the masses of the daughter nucleus and of an electron (neglecting the comparatively-small neutrino mass). $$Q_{\beta} =[M(_7^{13}\rm{N}) - M(_6^{13}\rm{C})- m_{e}]c^2$$ where these masses are nuclear masses, not atomic masses. A small fraction of the energy release $Q_{\beta}$ appears as the recoil kinetic energy of the daughter nucleus, but the remainder appears as the kinetic energy of electron and of neutrino. ![[$^{26}$Al]{}decay. The [$^{26}$Al]{}nucleus ground state has a long radioactive lifetime, due to the large spin difference of its state to lower-lying states of the daughter nucleus $^{26}$Mg. An isomeric excited state of [$^{26}$Al]{}exists at 228 keV excitation energy. If thermally excited, [$^{26}$Al]{}may decay through this state. Secondary products, lifetime, and radioactive energy available for deposits and observation depend on the environment. []{data-label="fig_1_26Al-decay"}](fig_1_26Al-decay){width="80.00000%"} Capture of an electron is a *two-particle* reaction, the bound atomic electron $e^{-}$ or a free electron in hot plasma being required for this type of $\beta$ decay. Therefore, depending on availability of the electron, electron-capture $\beta$ decay lifetimes can be very different for different environments. In the laboratory case, electron capture usually involves the 1s electrons of the atomic structure surrounding the radioactive nucleus, because those present their largest density at the nucleus. In many cases the electron capture competes with $e^{+}$ $+$ $\nu$ emission. In above example, $^{13}$N can decay not only by emitting $e^{+}$ $+$ $\nu$, but also by capturing an electron: $_7^{13}$N + e$^{-}\longrightarrow_6^{13}$C + $\nu$. In this case the capture of a 1s electron happens to be much slower than the rate of e$^+$ emission. But cases exist for which the mass excess is not large enough to provide for the creation of the $e^{+}$ mass for emission, so that only electron capture remains to the unstable nucleus to decay. Another relevant example is the decay of $^7$Be. Its mass excess over the daugther nucleus $^7$Li is only 0.351 MeV. This excess is insufficient to provide for creation of the rest mass of an emitted $e^{+}$, which is 0.511 MeV. Therefore, the $^7$Be nucleus is stable against $e^{+}$ $+$ $\nu$ emission. However, electron capture adds 0.511 MeV of rest-mass energy to the mass of the $^7$Be nucleus, giving a total 0.862 MeV of energy above the mass of the $^7$Li nucleus. Therefore, the $e^{-}$ capture process (above) emits a monoenergetic neutrino having $E_{\nu}$= 0.862 MeV[^13]. The situation for electron capture processes differs significantly in the interiors of stars and supernovae: Nuclei are ionized in plasma at such high temperature. The capture lifetime of $^7$Be, for example, which is 53 days against 1s electron capture in the laboratory, is lengthened to about 4 months at the solar center [see theory by @1964ApJ...139..318B; @1983NuPhA.404..578T], where the free electron density is less at the nucleus. The range of the $\beta$ particle (its stopping length) in normal terrestrial materials is small, being a charged particle which undergoes Coulomb scattering. An MeV electron has a range of several meters in standard air, during which it loses energy by ionisations and inelastic scattering. In tenuous cosmic plasma such as in supernova remnants, or in interstellar gas, such collisions, however, become rare, and may be unimportant compared to electromagnetic interactions of the magnetic field (*collisionless plasma*). Energy deposit or escape is a major issue in intermediate cases, such as expanding envelopes of stellar explosions, supernovae (positrons from $^{56}$Co and $^{44}$Ti) and novae (many $\beta^+$ decays such as $^{13}$N) (see Chapters 4, 5, and 7 for a discussion of the various astrophysical implications). Even in small solids and dust grains, energy deposition from [$^{26}$Al]{}$\beta$-decay, for example, injects 0.355 W kg$^{-1}$ of heat. This is sufficient to result in melting signatures, which have been used to study condensation sequences of solids in the early solar system (see Chapter 6). [*Gamma decay:*]{} In $\gamma$ decay the radioactive transition to a different and more stable nucleus is mediated by the *electromagnetic interaction*. A nucleus relaxes from its excited configuration of the nucleons to a lower-lying state of the same nucleons. This is intrinsically a fast process; typical lifetimes for excited states of an atomic nucleus are 10$^{-9}$seconds. We denote such electromagnetic transitions of an excited nucleus *radioactive $\gamma$-decay* when the decay time of the excited nucleus is considerably longer and that nucleus thus may be considered a temporarily-stable configuration of its own, a *metastable* nucleus. How is stability, or instability, of a nuclear-excited state effected? In electromagnetic transitions $$\label{eq_photon-decay} A^{\star} \longrightarrow A^{g.s.} + \gamma$$ the spin (angular momentum) is a conserved quantity of the system. The spin of a nuclear state is a property of the nucleus as a whole, and reflects how the states of protons and neutrons are distributed over the quantum-mechanically allowed *shells* or nucleon wave functions (as expressed in the *shell model* view of an atomic nucleus). The photon (*$\gamma$ quantum*) emitted (eq.\[eq\_photon-decay\]) will thus have a *multipolarity* resulting from the spin differences of initial and final states of the nucleus. Dipole radiation is most common and has multipolarity 1, emitted when initial and final state have angular momentum difference $\Delta l=1$. Quadrupole radiation (multipolarity 2, from $\Delta l=2$) is $\sim$6 orders of magnitude more difficult to obtain, and likewise, higher multipolarity transitions are becoming less likely by the similar probability decreases (the *Weisskopf estimates* [see @1951PhRv...83.1073W]). This explains why some excited states in atomic nuclei are much more long-lived (*meta-stable*) than others; their transitions to the ground state are also considered as *radioactivity*, and called *$\gamma$ decay*. The range of a $\gamma$-ray (its stopping length) is typically about 5-10 g cm$^{-2}$ in passing through matter of all types. Hence, except for dense stars and their explosions, radioactive energy from $\gamma$ decay is of astronomical implication only[^14]. An illustrative example of radioactive decay is the [$^{26}$Al]{}nucleus. Its decay scheme is illustrated in Fig. \[fig\_1\_26Al-decay\]. The ground state of [$^{26}$Al]{}is a $5+$ state. Lower-lying states of the neighboring isotope $^{26}$Mg have states $2+$ and $0+$, so that a rather large change of angular momentum $\Delta l$ must be carried by radioactive-decay secondaries. This explains the large $\beta$-decay lifetime of [$^{26}$Al]{}of $\tau\sim$1.04 10$^6$ y. In the level scheme of [$^{26}$Al]{}, excited states exist at energies 228, 417, and 1058 keV. The $0+$ and $3+$ states of these next excited states are more favorable for decay due to their smaller angular momentum differences to the $^{26}$Mg states, although $\Delta l=0$ would not be *allowed* for the 228 keV state to decay to $^{26}$Mg’s ground state. This explains its relatively long lifetime of 9.15 s, and it is a *metastable* state of [$^{26}$Al]{}. If thermally excited, which would occur in nucleosynthesis sites exceeding a few 10$^8$K, [$^{26}$Al]{}may decay through this state without $\gamma$-ray emission as $^{26}\rm{Al}^{g.s.} + \gamma \longrightarrow ^{26}\rm{Al}^{m} \longrightarrow ^{26}\rm{Mg} + e^+$, while the ground state decay is predominantly a *$\beta^+$ decay* through excited $^{26}$Mg states and thus including $\gamma$-ray emission. Secondary products, lifetime, and radioactive energy available for deposits and observation depend on the environment. ![The abundance of elements in the present-day nearby universe. Abundances (by number) are shown in a logarithmic scale, and span 12 orders of magnitude. The interplay of nuclear properties (several are indicated in the graph) with environmental conditions in cosmic nucleosynthesis sites has created this complex abundance pattern during the course of cosmic history.[]{data-label="fig_1_abundances"}](fig_abundances_RD){width="\textwidth"} Radioactivity and Cosmic Nucleosynthesis ======================================== Nuclear reactions in cosmic sites re-arrange the basic constituents of atomic nuclei (neutrons and protons) among the different allowed configurations. Throughout cosmic evolution, such reactions occur in various sites with different characteristic environmental properties. Each reaction environment leads to rearrangements of the relative abundances of cosmic nuclei. The cumulative process is called *cosmic chemical evolution*. [^15] The *cosmic abundance* of a specific isotope is expressed in different ways, depending on the purpose. Counting the atoms of isotope $i$ per unit volume, one obtains $n_i$, the number density of atoms of species $i$ (atoms cm$^{-3}$). The interest of cosmic evolution and nucleosynthesis lies in the fractional abundances of species $i$ related to the total, and how it is altered by cosmic nuclear reactions. Observers count a species $i$ and relate it to the abundance of a reference species. For astronomers this is hydrogen. Hydrogen is the most abundant element throughout the universe, and easily observed through its characteristic atomic transitions in spectroscopic astronomical measurements. Using the definition of Avogadro’s constant $A_{Av}$ as the number of atoms which make up $A$ grams of species $i$ (i.e., one mole), we can obtain abundances *by mass*; $A_{Av}=6.02214~10^{23}$ atoms mole$^{-1}$. The mass contained in a particular species $S$ results from scaling its abundance $N_S$ by its atomic weight $A$. We can get a global measure for cosmic evolution of the composition of matter by tracing how much of the total mass is contained in hydrogen, helium, and the remainder of elements called *metals*[^16], calling these quantities $X$ for hydrogen abundance, $Y$ for helium abundance, and $Z$ for the cumulative abundance of all nuclei heavier than helium. We call these *mass fractions* of hydrogen $X$, helium $Y$, and metals $Z$, with $X+Y+Z=1$. The metalicity $Z$ is a key parameter used to characterise the evolution of elemental and isotopic composition of cosmic matter. The astronomical abundance scale is set from most-abundant cosmic element Hydrogen to $log(X_H)=12$ (Fig. \[fig\_1\_abundances\]), but mineralogists and meteoriticians use $Si$ as their reference element and set $log(X_{Si})=6$. We often relate abundances also to our best-known reference, the solar system, denoting *solar-system* values by the $\odot$ symbol. Abundances of a species $S$ are then expressed in *bracket notation*[^17] as $$\lbrack \frac{ S }{ H } \rbrack \equiv log (\frac{X_S}{X_H})_{\star} - log (\frac{X_S}{X_H})_{\odot} $$ Depending on observational method and precision, our astronomical data are *metalicity*, elemental *enrichments* with respect to solar abundances, or isotopic abundances. Relations to nuclear reactions are therefore often indirect. Understanding the nuclear processing of matter in the universe is a formidable challenge, often listed as one of the *big questions* of science. Big Bang Nucleosynthesis (BBN) about 13.8 Gyrs ago left behind a primordial composition where hydrogen (protons) and helium were the most-abundant species; the total amount of nuclei heavier than He (the *metals*) was less than 10$^{-9}$ (by number, relative to hydrogen [@2007ARNPS..57..463S]).Today, the total mass fraction of metals in [*solar abundances*]{} is $Z=0.0134$ , compared to a hydrogen mass fraction of[^18] $X=0.7381$. This growth of metal abundances by about seven orders of magnitude is the effect of cosmic nucleosynthesis. Nuclear reactions in stars, supernovae, novae, and other places where nuclear reactions may occur, all contribute. But it also is essential that the nuclear-reaction products inside those cosmic objects will eventually be made available to observable cosmic gas and solids, and thus to later-generation stars such as our solar system born 4.6 Gyrs ago. This book will also discuss our observational potential for cosmic isotopes, and we address the constraints and biases which limit our ability to draw far reaching conclusions. The growth of isotopic and elemental abundances from cosmic nucleosynthesis does not occur homogeneously. Rather, the cosmic abundances observed today span a dynamic range of twelve orders of magnitude between abundant hydrogen and rare heavy elements (Fig. \[fig\_1\_abundances\]). Moreover, the elemental abundance pattern already illustrates clearly the prominent effects of nuclear structure (see Fig. \[fig\_1\_abundances\]): Iron elements are among the most-tightly bound nuclei, and locally elements with even numbers of nucleons are more tightly bound than elements with odd numbers of nuclei. The Helium nucleus (*$\alpha$-particle*) also is more tightly bound than its neighbours in the chart of nuclei, hence all elements which are multiples of $\alpha$’s are more abundant than their neighbours. Towards the heavier elements beyond the Fe group, abundances drop by about five orders of magnitude again, signifying a substantially-different production process than the mix of charged-particle nuclear reactions that produced the lighter elements: *neutron capture* on Fe *seed nuclei*. The two abundance peaks seen for heavier elements are the results of different environments for cosmic neutron capture reactions (the *r-process* and *s-process*), both determined by neutron capture probabilities having local extrema near *magic numbers*. The different peaks arise from the particular locations at which the processes’ reaction path encounters these *magic nuclei*, as neutron captures proceed much faster (slower) than $\beta$ decays in the $r$ process ($s$ process).. The subjects of cosmic nucleosynthesis research are complex and diverse, and cover the astrophysics of stars, stellar explosions, nuclear reactions on surfaces of compact stars and in interstellar space. For each of the potential nuclear-reaction sites, we need to understand first how nuclear reactions proceed under the local conditions, and then how material may be ejected into interstellar space from such a source. None of the nucleosynthesis sites is currently understood to a level of detail which would be sufficient to formulate a physical description, sit back and consider cosmic nucleosynthesis *understood*. For example, one might assume we know our Sun as the nearest star in most detail; but solar neutrino measurements have been a puzzle only alleviated in recent years with the revolutionary adoption of non-zero masses for neutrinos, whih allow for flavour oscillations; and even then, the abundances of the solar photosphere, revised by almost a factor two based on three-dimensional models of the solar photosphere [@Asplund:2009], created surprising tension with measurements of helio-seismology and the vibrational behaviour reflected herein, and the physical descriptions in our currently-best solar model are under scrutiny [@Vinyoles:2017]. As another example, there are two types of supernova explosions. Core-collapse supernovae are the presumed outcome of the final gravitational collapse of a massive star once its nuclear fuel is exhausted, and thermonuclear supernovae were thought to originate from detonation of degenerate stars once they exceed a critical threshold for nuclear burning of Carbon, the Chandrasekhar mass limit. The gravitational collapse can not easily be reverted into an explosion, and even the help of neutrinos from the newly-forming neutron star in the center appears only marginally sufficient, so that many massive stars that were thought to explode may collapse to black holes [@Janka:2016]. And the thermonuclear supernova variety appears to require white dwarf collisions as triggering events in some well-constrained cases, while in other cases luminosities deviate by orders of magnitude from the expectation from a Chandrasekhar-mass white dwarf and its nuclear-burning demise that once was thought to be a cosmic standard candle [@Hillebrandt:2013]. For neither of these supernovae, a *physical* model is available, which would allow us to calculate and predict the outcome (energy and nuclear ashes) under given, realistic, initial conditions (see Ch. 4 and 5). Much research remains to be done in cosmic nucleosynthesis. One may consider measurements of cosmic material in all forms to provide a wealth of data, which now has been exploited to understand cosmic nucleosynthesis. Note, however, that cosmic material as observed has gone through a long and ill-determined journey. We need to understand the trajectory in time and space of the progenitors of our observed cosmic-material sample if we want to interpret it in terms of cosmic nucleosynthesis. This is a formidable task, necessary for distant cosmic objects, but here averaging assumptions help to simplify studies. For more nearby cosmic objects where detailed data are obtained, astrophysical models quickly become very complex, and also need simplifying assumptions to operate for what they are needed. It is one of the objectives of cosmic nucleosynthesis studies to contribute to proper models for processes in such evolution, which are sufficiently isolated to allow their separate treatment. Nevertheless, carrying out *well-defined experiments* for a source of cosmic nucleosynthesis remains a challenge, due to this complex flow of cosmic matter (see Ch.’s 6 to 8). The special role of radioactivity in such studies is contributed by the intrinsic decay of such material after it has been produced in cosmic sites. This brings in a new aspect, the clock of the radioactive decay. Technical applications widely known are based on $^{14}$C with its half life of 5700 years, while astrophysical applications extend this to much longer half lives up to Gyrs ($^{235}$U has a decay time of 10$^9$ years). Changes in isotopic abundances with time will occur at such natural and isotope-specific rates, and will leave their imprints in observable isotopic abundance records. For example, the observation of unstable technetium in stellar atmospheres of AGB stars was undisputable proof of synthesis of this element inside the same star, because the evolutionary time of the star exceeds the radioactive lifetime of technetium. Another example, observing radioactive decay $\gamma$-ray lines from short-lived Ni isotopes from a supernova is clear proof of its synthesis in such explosions; measuring its abundance through $\gamma$-ray brightness is a direct *calibration* of processes in the supernova interior. A last example, solar-system meteorites show enrichments in daughter products of characteristic radioactive decays, such as $^{26}$Al and $^{53}$Mn; the fact that these radioactive elements were still alive at the time those solids formed sets important constraints to the time interval between the latest nucleosynthesis event near the forming Sun and the actual condensation of solid bodies in the interstellar gas accumulating to form the young solar system. This book will discuss these examples in detail, and illustrate the contributions of radioactivity studies to the subject of cosmic nucleosynthesis. Observing radioactive Isotopes in the Universe ============================================== ![The electromagnetic spectrum of candidate astronomical measurements ranges across more than twenty orders of magnitude. Not all are easily accessible. Information categories of thermal and non-thermal, and of molecular, atomic, nuclear, and elementary-particle physics origins of cosmic radiation extends over different parts of this broad spectrum. Nuclear physics is directly accessible in a small band (0.1-10 MeV) only. Non-electromagnetic astronomical messengers are indicated at both ends of the electromagnetic spectrum[]{data-label="fig_1_emSpectrum"}](Fig_EM-spectr_physics){width="\textwidth"} Astronomy has expanded beyond the narrow optical band into *new astronomies* in the past decades. By now, we are familiar with telescopes measuring radio and sub-mm through infrared emission towards the long wavelength end, and ultraviolet, X-ray, and $\gamma$-ray emission towards the short wavelength end (see Fig. \[fig\_1\_emSpectrum\]). The physical origins of radiation are different in different bands. Thermal radiation dominates emission from cosmic objects in the middle region of the electromagnetic spectrum, from a few 10K cold molecular clouds at radio wavelengths through dust and stars up to hot interstellar gas radiating X-rays. Non-thermal emission is characteristic for the wavelength extremes, both at radio and $\gamma$-ray energies. Characteristic spectral lines originate from atomic shell electrons over most of the spectrum; nuclear lines are visible only in roughly two decades of the spectrum at 0.1–10 MeV. Few exceptional lines arise at high energy from annihilations of positrons and pions. Cosmic *elements* can be observed in a wide astronomical range. *Isotopes*, however, are observed almost exclusively through $\sim$MeV $\gamma$-rays (see Fig. \[fig\_1\_emSpectrum\]). Note that nucleosynthesis reactions occur among isotopes, so that this is the prime[^19] information of interest when we wish to investigate cosmic nucleosynthesis environment properties. ![*Left:* Example of an isotopic measurement in a stellar atmosphere. Shown is an absorption-line spectrum of a cool star with a present-generation optical telescope, here the Subaru telescope on Hawaii with its IR spectrograph at a resolution of 20000. Molecular lines from the CO molecule isotopologes show isotopic shifts, which can be recognised as changes in line shapes, as resulting from the isotopic abundance ratio. Here the carbon isotopic ratio is determined for the stellar atmosphere of a M dwarf star, comparing the measurement (red dots) with expectations for different ratios $^{12}$C/$^{13}$C [from @Tsuji:2016]. *Right:* The Very Large Telescope (VLT) on Mount Paranal in Chile, with four telescopes (lower right), is one of the modern optical instruments. Equipped with high-resolution spectrographs such as FLAMES (insert lower right), absorption-line spectroscopy of stars provides elemental abundances in stellar atmospheres, even in nearby galaxies. (Figure ESO) []{data-label="fig_1_stellar_spectrosscopy"}](Fig_C-isotopes_Mstar "fig:"){width="45.00000%"} ![*Left:* Example of an isotopic measurement in a stellar atmosphere. Shown is an absorption-line spectrum of a cool star with a present-generation optical telescope, here the Subaru telescope on Hawaii with its IR spectrograph at a resolution of 20000. Molecular lines from the CO molecule isotopologes show isotopic shifts, which can be recognised as changes in line shapes, as resulting from the isotopic abundance ratio. Here the carbon isotopic ratio is determined for the stellar atmosphere of a M dwarf star, comparing the measurement (red dots) with expectations for different ratios $^{12}$C/$^{13}$C [from @Tsuji:2016]. *Right:* The Very Large Telescope (VLT) on Mount Paranal in Chile, with four telescopes (lower right), is one of the modern optical instruments. Equipped with high-resolution spectrographs such as FLAMES (insert lower right), absorption-line spectroscopy of stars provides elemental abundances in stellar atmospheres, even in nearby galaxies. (Figure ESO) []{data-label="fig_1_stellar_spectrosscopy"}](fig_1_VLT.pdf "fig:"){width="55.00000%"} Only few elements such as technetium (Tc) do not have any stable isotope; therefore, elemental photospheric absorption and emission line spectroscopy, the backbone of astronomical studies of cosmic nucleosynthesis, have very limited application in astronomy with radioactivities. This is about to change currently, as spectroscopic devices in the optical and radio/sub-mm regimes advance spectral resolutions. Observational studies of cosmic radioactivities are best performed by techniques which intrinsically obtain isotopic information. These are: - Modern spectrographs on large ground-based telescopes reach R=20000, sufficient to resolve fine structure lines and isotopic features in molecules (see Fig. \[fig\_1\_stellar\_spectrosscopy\]). Radio spectroscopy with CO isotopes has been successfully applied since the 1990ies, and has been used mainly to track the CO molecule at different columns densities, while sub-mm lines from molecules have been demonstrated to observe specific isotopes within molecules such as $^{36}$ArN [@Schilke:2014]. - Precision mass spectroscopy in terrestrial laboratories, which has been combined with sophisticated radiochemistry to extract meteoritic components originating from outside the solar system - Spectroscopy of characteristic $\gamma$-ray lines emitted upon radioactive decay in cosmic environments The two latter [*astronomical disciplines*]{} have a relatively young history. They encounter some limitations due to their basic methods of how astronomical information is obtained, which we therefore discuss in somewhat more detail: - Precision mass spectrometry of meteorites for astronomy with radioactivity began about 1960 with a new discovery of now extinct radioactivity within the young solar system. From heating of samples of bulk meteorite material, the presence of a surprising excess $^{129}$Xe had been puzzling. Through a variety of different chemical processing, this could be tracked to trapped gas enclosures in rather refractory components, which must have been enriched in $^{129}$I at the time of formation of this meteorite. From mineralogical arguments, this component could be associated with the early solar system epoch about 4.6 Gy ago [@PhysRevLett.4.8]. This was the first evidence that the matter from which the solar system formed contained radioactive nuclei whose half-lives are too short to be able to survive from that time until today ($^{129}$I decays to $^{129}$Xe within 1.7 10$^7$y). Another component could be identified from most-refractory Carbon-rich material, and was tentatively identified with dust grains of pre-solar origins. Isotopic anomalies found in such *extra-solar* inclusions, e.g. for C and O isotopes, range over four orders of magnitude for such *star dust* grains as shown in Fig. \[fig\_1\_grain\] [@1998AREPS..26..147Z], while isotopic-composition variations among bulk meteoritic-material samples are a few percent at most. These mass spectroscopy measurements are characterised by an amazing sensitivity and precision, clearly resolving isotopes and counting single atoms at ppb levels to determine isotopic ratios of such rare species with high accuracy, and nowadays even for specific, single dust grains. This may be called an *astronomy in terrestrial laboratories* (see Chapter 11 for instrumental and experimental aspects), and is now an established part of astrophysics and [e.g. @Amari:2014; @Zinner:2014]. --------------------------------------- ------------------- ---------------------------- ------------------------- ------- [**Isotope** ]{} [**Lifetime**]{} [**Presolar Grain** ]{} [**Source**]{} Ref. [**chain** ]{} [**Type**]{} [****]{} $^{49}$V $\longrightarrow$ $^{49}$Ti 330 days SiC, Graphite SNe \[1\] $^{22}$Na $\longrightarrow$ $^{22}$Ne 2.6 years Graphite SNe \[2\] $^{44}$Ti $\longrightarrow$ $^{44}$Ca 60 years SiC, Graphite, Hibonite SNe \[3\] $^{32}$Si $\longrightarrow$ $^{32}$S 153 years SiC SNe, post-AGB stars \[4\] $^{41}$Ca $\longrightarrow$ $^{41}$K 1.02 10$^5$ years SiC, Graphite, Hibonite SNe, RGB, and AGB stars \[5\] $^{99}$Tc $\longrightarrow$ $^{99}$Ru 2.11 10$^5$ years SiC AGB stars \[6\] $^{26}$Al $\longrightarrow$ $^{26}$Mg 7.17 10$^5$ years SiC, Graphite, Corundum, SNe, RGB, and AGB stars \[7\] Spinel, Hibonite, Silicate $^{93}$Zr $\longrightarrow$ $^{93}$Nb 1.61 10$^6$ years SiC AGB stars \[8\] --------------------------------------- ------------------- ---------------------------- ------------------------- ------- ![Meteoritic inclusions such as this SiC grain are recognised as dust formed near a cosmic nucleosynthesis source outside the solar system, from their large isotopic anomalies, which cannot be explained by interstellar nor solar-system processing but are reminiscent of cosmic nucleosynthesis sites. Having condensed in the envelope of a source of new isotopes, laboratory mass spectroscopy can reveal isotopic composition for many elements, thus providing a remote probe of one cosmic nucleosynthesis source.[]{data-label="fig_1_grain"}](fig_1_SiC_Xgrain.pdf){width="60.00000%"} Table 1.1 lists the radioactive isotopes used for studies of pre-solar grains [@Groopman:2015]. Studies of pre-solar dust grain compositions have lead to the distinctions of grain origins from AGB stars, from supernovae, and from novae, all of which are copious producers of dust particles. Formation of stardust occurs in circumstellar environments where temperatures are cool enough [e.g. @Cherchneff:2017 for a recent review of the open issues]. On their journey through the interstellar medium, heating and partial or complete destruction may occur from starlight or even shocks from supernovae [@Zhukovska:2016]. Also a variety chemical and physical reactions may reprocess dust grains [@Dauphas:2016]. Thus, the journey from the stardust source up to inclusion in meteorites which found their way to Earth remains subject to theoretical modelling and much residual uncertainty [@Jones:2009]. Nevertheless, cosmic dust particles are independent astrophysical messengers, and complement studies based on electromagnetic radiation in important ways. Grain composition and morphology from the stardust laboratory measurements are combined with astronomical results such as characteristic spectral lines (e.g. from water ice, or a prominent feature associated with silicate dust), and interpreted through (uncertain) theories of cosmic dust formation and transport [@1998AREPS..26..147Z; @Cherchneff:2016]. Experimental difficulties and limitations arise from sample preparation through a variety of complex chemical methods, and by the extraction techniques evaporising material from the dust grain surfaces for subsequent mass spectrometry (see Chapter 10). ![Example of a $\gamma$-ray line measurement: The characteristic line from $^{26}$Al decay at 1808.63 keV appears Doppler-shifted from large scale galactic rotation, as it is viewed towards different galactic longitudes [left; from @Kretschmer:2013]. This measurement was performed with the SPI spectrometer on INTEGRAL, an example of a present-generation space-borne $\gamma$-ray telescope. The INTEGRAL satellite (artist view picture, ESA) has two main telescopes; the spectrometer SPI, one of them, is shown at the lower-right schematically with its 19-detector Ge camera and the tungsten mask for imaging by casting a shadow onto the camera. Space-based instruments of this kind are required to directly measure the characteristic $\gamma$-ray lines from the decay of unstable isotopes near sites of current-epoch cosmic element formation.[]{data-label="fig_1_gamma"}](Fig_Al-spectra-vs-long.pdf "fig:"){width="32.00000%"} ![Example of a $\gamma$-ray line measurement: The characteristic line from $^{26}$Al decay at 1808.63 keV appears Doppler-shifted from large scale galactic rotation, as it is viewed towards different galactic longitudes [left; from @Kretschmer:2013]. This measurement was performed with the SPI spectrometer on INTEGRAL, an example of a present-generation space-borne $\gamma$-ray telescope. The INTEGRAL satellite (artist view picture, ESA) has two main telescopes; the spectrometer SPI, one of them, is shown at the lower-right schematically with its 19-detector Ge camera and the tungsten mask for imaging by casting a shadow onto the camera. Space-based instruments of this kind are required to directly measure the characteristic $\gamma$-ray lines from the decay of unstable isotopes near sites of current-epoch cosmic element formation.[]{data-label="fig_1_gamma"}](fig_1_INT_SPI.pdf "fig:"){width="67.00000%"} ------------------------------------------------------------------- ------------------ -------------------------------------- ------------------------- ------------------ [**Lifetime**]{} [**$\gamma$-ray Energy**]{} [**Site**]{} [**Process** ]{} \[y\] \[keV\] [**Type** ]{} (branching ratio \[%\]) (detections) $^{7}$Be$\rightarrow ^{7}$Li 0.21 478 (100) Novae explosive H burning $^{56}$Ni$\longrightarrow ^{56}$Co$\longrightarrow^{56}$Fe 0.31 847 (100), 1238 (68) SNe NSE 2598 (17), 1771 (15) (SN1987A, burning (SN1991T, SN2014J) [and 511 from e$^+$]{} $^{57}$Co$\longrightarrow ^{57}$Fe 1.1 122 (86), 136 (11) SNe NSE (SN1987A burning $^{22}$Na$\longrightarrow ^{22}$Ne 3.8 1275 (100) Novae explos. [and 511 from e$^+$]{} H burning $^{44}$Ti$\longrightarrow ^{44}$Sc$\longrightarrow^{44}$Ca 89 68 (95), 78 (96) SNe NSE 1156 (100) (Cas A, SN1987A) $\alpha$ freeze- [and 511 from e$^+$]{} out $^{26}$Al$\longrightarrow ^{26}$Mg 1.04 10$^6$ 1809 (100) ccSNe, WR H burning Novae, AGB (Galaxy) ($\nu$-proc.) [and 511 from e$^+$]{} (Cygnus;Sco-Cen; Orion; Vela) $^{60}$Fe$\longrightarrow ^{60}$Co$\longrightarrow^{60}$Ni 3.8 10$^6$ 1173 (100), 1332 (100) SNe He,C 59 (2) (Galaxy) shell burning e$^{+}\longrightarrow $Ps,..$\longrightarrow\gamma\gamma(\gamma)$ $\sim$10$^7$ 2$\cdot$511 ($\sim$100), cont $<$510 radioactivities $\beta^+$ decay Pulsars, $\mu$QSOs, ... rel. plasma (Galactic bulge; disk) ------------------------------------------------------------------- ------------------ -------------------------------------- ------------------------- ------------------ - Characteristic $\gamma$-ray lines from cosmic sources were not known until the 1960$^{ies}$, when space flight and its investigations of the near-earth space radiation environment had stimulated measurements of $\gamma$-rays. The discovery of a cosmic $\gamma$-ray line feature near 0.5 MeV from the direction towards the center of our Galaxy in 1972 [@1972ApJ...172L...1J] stimulated balloon and satellite experiments for cosmic $\gamma$-ray line spectroscopy. Radioactive isotopes are ejected into the surroundings of their nucleosynthesis sources, and become observable through their gamma-ray line emission once having left dense production sites where not even gamma-rays may escape. Nuclear gamma-rays can penetrate material layers of integrated thickness of a few grams cm$^{-2}$. A typical interstellar cloud would have $\sim$0.1 g cm$^{-2}$, SNIa envelopes are transparent to gamma-rays after 30–100 days, depending on explosion dynamics. Depending on radioactive lifetime, gamma-ray lines measure isotopes which originate from single sources (the short-lived isotopes) or up to thousands of sources as accumulated in interstellar space over the radioactive lifetime of long-lived isotopes (see Table 1.2). Decay of the isotopes $^{26}$Al, $^{60}$Fe, $^{44}$Ti, $^{57}$Ni, and $^{56}$Ni in distant cosmic sites is an established fact, and astrophysical studies make use of such measurements. The downsides of those experiments is the rather poor resolution by astronomy standards (on the order of degrees), and the sensitivity limitations due to large instrumental backgrounds, which effectively only shows the few brightest sources of cosmic $\gamma$-rays until now [see @Diehl:2006g for a discussion of achievements and limitations]. Despite their youth and limitations, both methods to address cosmic radioactivities share a rather direct access to isotopic information, unlike other fields of astronomy. Isotopic abundance studies in the nuclear energy window will be complemented for specific targets and isotopes from the new opportunities in optical and radio/sub-mm spectroscopy. 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[Davis]{} (Elsevier), 181–213 ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_1.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_2.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_3.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_4.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_5.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_6.pdf "fig:"){width="35.00000%"} ![Table of contents: “Astrophysics with radioactive isotopes”, Springer ASSL 453 (2018) []{data-label="fig_TOC"}](TOC_7.pdf "fig:"){width="35.00000%"} [^1]: Max Planck Institut für extraterrestrische Physik, 85748 Garching, Germany [^2]: IUPAC, the international union of chemistry, coordinates definitions, groupings, and naming; see www.IUPAC.org [^3]: The mass difference is [@Patrignani:2016] 1.293332 MeV = 939.565413 - 938.272081 MeV for the mass of neutron and proton, respectively. One may think of the proton as the lowest-energy configuration of a hadron, that is the target of matter in a higher state, such as the combined proton-electron particle, more massive than the proton by the electron mass plus some binding energy of the quark constituents of hadrons. [^4]: The sub-atomic particles in the nucleus are composed of three quarks, and also called *baryons*. Together with the two-quark particles called *mesons*, they form the particles called *hadrons*, which obey the strong nuclear force. [^5]: This is often called *chemical potential*, and describes the energy that is held as internal energy in species $i$, which could potentially be liberated when binding energy per nucleon would change as nucleons would be transferred to different species $j,k,l...$. [^6]: The mass of the neutron exceeds that of the proton by 1.2933 MeV, making the proton the most stable baryon [^7]: In a broader sense, nuclear physics may be considered to be similar to chemistry: elementary building blocks are rearranged to form different species, with macroscopically-emerging properties such as, e.g., characteristic and well-defined energies released in such transitions. [^8]: States may differ in their quantum numbers, such as spin, or orbital-momenta projections; if they obtain the same energy $E$, they are called *degenerate*. [^9]: The binding energy *per nucleon* is maximized for nucleons bound as a Fe nucleus. [^10]: These masses may be either nuclear masses or atomic masses, the electron number is conserved, and their binding energies are negligible, in comparison. [^11]: Within an FeNi meteorite, e.g., an $\alpha$ particle from radioactivity has a range of only $\sim$10 $\mu$m. [^12]: We ignore here two additional $\beta$ decays which are possible from $\nu$ and $\overline{\nu}$ captures, due to their small probabilities. [^13]: This neutrino line has just recently been detected by the Borexino collaboration arriving from the center of the Sun [@2008PhRvL.101i1302A]. [^14]: Gamma-rays from nuclear transitions following $^{56}$Ni decay (though this is a $\beta$ decay by itself) inject radioactive energy through $\gamma$-rays from such nuclear transitions into the supernova envelope, where it is absorbed in scattering collisions and thermalized. This heats the envelope such that thermal and optically bright supernova light is created. Deposition of $\gamma$-rays from nuclear transitions are the engines which make supernovae to be bright light sources out to the distant universe, used in cosmological studies to, e.g., support evidence for *dark energy*. [^15]: We point out that there is no chemistry involved; the term refers to changes in abundances of the chemical elements, which are important for our daily-life experiences. But these are a result of the more-fundamental changes in abundances of isotopes mediated by cosmic nuclear reactions. [^16]: This nomenclature may be misleading, it is used by convenience among astrophysicists. Only a part of these elements are actually metals. [^17]: Deviations from the standard may be small, so that $\lbrack\frac{[S_1}{S_2}\rbrack$ may be expressed in $\delta$ units (parts per mil), or $\epsilon$ units (parts in 10$^4$), or ppm and ppb; $\delta(^{29}Si/^{28}Si)$ thus denotes excess of the $^{29}$Si/$^{28}$Si isotopic ratio above solar values in units of 0.1%. [^18]: This implies a *metalicity* of solar matter of 1.4%. Our local reference for cosmic material composition seems to be remarkably universal. Earlier than $\sim$2005, the commonly-used value for solar metallicity had been 2%. [^19]: Other astronomical windows may also be significantly influenced by biases from other astrophysical and astrochemical processes; an example is the observation of molecular isotopes of CO, where chemical reactions as well as dust formation can lead to significant alterations of the abundance of specific molecular species.

--- abstract: | Because the baryon-to-photon ratio $\eta_{10}$ is in some doubt, we drop nucleosynthetic constraints on $\eta_{10}$ and fit the three cosmological parameters $(h, \Omega_{\mathrm{M}}, \eta_{10})$ to four observational constraints: Hubble parameter $h_{\mathrm{o}} = 0.70 \pm 0.15$, age of the universe $t_{\mathrm{o}} = 14^{+7}_{-2}$ Gyr, cluster gas fraction $f_{\mathrm{o}} \equiv f_{\mathrm{G}}h^{3/2} = 0.060 \pm 0.006$, and effective shape parameter $\Gamma_{\mathrm{o}} = 0.255 \pm 0.017$. Errors quoted are $1\sigma$, and we assume Gaussian statistics. We experiment with a fifth constraint $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$ from clusters. We set the tilt parameter $n = 1$ and the gas enhancement factor $\Upsilon = 0.9$. We consider CDM models (open and $\Omega_{\mathrm{M}} = 1$) and flat $\Lambda$CDM models. We omit HCDM models (to which the $\Gamma_ {\mathrm{o}}$ constraint does not apply). We test goodness of fit and draw confidence regions by the $\Delta\chi^2$ method. CDM models with $\Omega_{\mathrm{M}} =1$ (SCDM models) are accepted only because the large error on $h_{\mathrm{o}}$ allows $h < 0.5$. Baryonic matter plays a significant role in $\Gamma_{\mathrm{o}}$ when $\Omega_{\mathrm{M}} \sim 1$. Open CDM models are accepted only for $\Omega_{\mathrm{M}} \gtrsim 0.4$. The combination of the four other constraints with $\Omega_{\mathrm{o}} \approx 0.2$ is rejected in CDM models with 98% confidence, suggesting that light may not trace mass. $\Lambda$CDM models give similar results. In all of these models, $\eta_{10}$ $\gtrsim 6$ is favored strongly over $\eta_{10}$ $\lesssim 2$. This suggests that reports of low deuterium abundances on QSO lines of sight may be correct, and that observational determinations of primordial $^4$He may have systematic errors. Plausible variations on $n$ and $\Upsilon$ in our models do not change the results much. If we drop or change the crucial $\Gamma_{\mathrm{o}}$ constraint, lower values of $\Omega_{\rm M}$ and $\eta_{10}$ are permitted. The constraint $\Gamma_{\mathrm{o}} = 0.15 \pm 0.04$, derived recently from the IRAS redshift survey, favors $\Omega_{\rm M} \approx 0.3$ and $\eta_{10} \approx 5$ but does not exclude $\eta_{10} \approx 2$. author: - 'Gary Steigman, Naoya Hata, and James E. Felten' title: | Non-Nucleosynthetic Constraints on the Baryon Density\ and Other Cosmological Parameters --- INTRODUCTION {#Sec:Introduction} ============ In a Friedmann-Lemaître big bang cosmology, the universal baryonic mass-density parameter $\Omega_{\mathrm{B}}\; (\,\equiv 8 \pi G \rho_{\mathrm{B}}/3H_0^2\,)$ may be calculated from $$\begin{split} \Omega_{\mathrm{B}}\,h^2 & = 3.675 \times 10^{-3}(T/2.73\,\mathrm{K})^3 \; \eta_{10} \\ & = 3.667 \times 10^{-3} \; \eta_{10}, \label{Eq:Omega_B} \end{split}$$ where $h$ is defined by the present Hubble parameter $H_0 \; [\, h \equiv H_0/(100$ km s$^{-1}$ Mpc$^{-1})\,]$, $T$ is the present microwave background temperature, and $\eta_{10}$ is the baryon-to-photon number ratio in units $10^{-10}$. The last member of equation (\[Eq:Omega\_B\]) is obtained by setting $T = 2.728$ K (Fixsen et al. 1996). In principle, $\eta_{10}$ is well determined (in fact overdetermined) by the observed or inferred primordial abundances of the four light nuclides D, $^3$He, $^4$He, and $^7$Li, if the number of light-neutrino species has its standard value $N_\nu =3$. For some years it has been argued that $\eta_{10}$ is known to be $3.4 \pm 0.3$ (Walker et al. 1991; these error bars are about “1$\sigma$”; cf. Smith, Kawano, & Malaney 1993) or at worst $4.3 \pm 0.9$ (Copi, Schramm, & Turner 1995a; cf. Yang et al. 1984), and that equation (\[Eq:Omega\_B\]) is a powerful constraint on the cosmological parameters $\Omega_{\mathrm{B}}$ and $h$. In practice, it seems recently that $\eta_{10}$ may not be so well determined, and even that the standard theory of big bang nucleosynthesis (BBN) may not give a good fit. With improved abundance data, it appears that the joint fit of the theory to the four nuclide abundances is no longer good for any choice of $\eta_{10}$ (Hata et al. 1995). These authors offer several options for resolving the apparent conflict between theory and observation. Although they suggest that some change in standard physics may be required (e.g., a reduction in the effective value of $N_\nu$ during BBN below its standard value 3), they note that large systematic errors may compromise the abundance data (cf. Copi, Schramm, & Turner 1995b). The nature of such errors is unclear, and this remains controversial. Other authors have reacted to the impending crisis in self-consistency by simply omitting one or more of the four nuclides in making the fit (Dar 1995; Olive & Thomas 1997; Hata et al. 1996, 1997; Fields et al. 1996). This controversy has been sharpened by new observations giving the deuterium abundances on various lines of sight to high-redshift QSOs. These data should yield the primordial D abundance, but current results span an order of magnitude. The low values, D/H by number $\approx 2 \times 10^{-5}$ (Tytler, Fan, & Burles 1996; Burles & Tytler 1996), corresponding to $\eta_{10} \approx 7$ in the standard model, have been revised slightly upward \[D/H $\approx (3-4) \times 10^{-5}$ (Burles & Tytler 1997a,b,c); $\eta_{10} \approx 5$\], but it still seems impossible to reconcile the inferred abundance of $^4$He \[Y$_{\rm P} \approx 0.234$; Olive & Steigman 1995 (OS)\] with standard BBN for this large value of $\eta_{10}$ (which implies Y$_{\rm BBN} \approx 0.247$) unless there are large systematic errors in the $^4$He data (cf. Izotov, Thuan, & Lipovetsky 1994, 1997). Such low D/H values have also been challenged on observational grounds by Wampler (1996) and by Songaila, Wampler, and Cowie (1997), and deuterium abundances nearly an order of magnitude higher, D/H $\approx 2\times10^{-4}$, have been claimed by Carswell et al. (1994), Songaila et al. (1994), and Rugers and Hogan (1996) for other high-redshift systems with metal abundances equally close to primordial. Although some of these claims of high deuterium have been called into question (Tytler, Burles, & Kirkman 1997), Hogan (1997) and Songaila (1997) argue that the spectra of other absorbing systems require high D/H (e.g., Webb et al. 1997). If these higher abundances are correct, then D and $^4$He are consistent with $\eta_{10} \approx 2$, but modellers of Galactic chemical evolution have a major puzzle: How has the Galaxy reduced D from its high primordial value to its present (local) low value without producing too much $^3$He (Steigman & Tosi 1995), without using up too much interstellar gas (Edmunds 1994, Prantzos 1996), and without overproducing heavy elements (cf. Tosi 1996 and references therein)? It appears that $\eta_{10}$, though known to order of magnitude, may be among the less well-known cosmological parameters at present. Despite this, large modern simulations which explore other cosmological parameters are often limited to a single value of $\eta_{10} = 3.4$ (e.g., Borgani et al. 1997). In this situation it may be instructive, as a thought experiment, to abandon nucleosynthetic constraints on $\eta_{10}$ entirely and ask: If we put $\eta_{10}$ onto the same footing as the other cosmological free parameters, and apply joint constraints on all these parameters based on other astronomical observations and on theory and simulation, what values of $\eta_{10}$ and the other parameters are favored? This may indicate the most promising avenue to a resolution of the controversy over $\eta_{10}$. We discuss the following popular CDM models: (1) Open or closed cold dark-matter model with cosmological constant $\Lambda = 0$ (CDM model). The “standard” (flat) CDM model (SCDM), which is an Einstein-de Sitter model, is covered as a special case of this. (2) Flat CDM model with nonzero $\Lambda$ ($\Lambda$CDM model). In a flat model with both hot and cold dark matter, with $\Lambda = 0$ (HCDM model), the constraints will be different; we defer these HCDM models to a later paper. Nonflat models with nonzero $\Lambda$ are not necessarily ruled out by “fine-tuning" arguments and may be of interest (Steigman & Felten 1995), but at the moment we are not compelled to resort to these. Our approach will be to let three parameters range freely, fit the constraints (observables) other than nucleosynthetic constraints, test goodness of fit by $\chi^2$, and draw formal confidence regions for the parameters by the usual $\Delta\chi^2$ method. Because statistical results of this kind are sometimes controversial, we intend to keep the work conceptually simple, review the constraints in a helpful way, and discuss our method carefully. Error bars are $\pm 1\sigma$ unless stated otherwise. The $\Delta\chi^2$ approach is revealing because, in the linear approximation, the confidence regions obtained are rigorous as probability statements and require no “a priori" probability assumptions about the unknown parameters. Most of our results are not surprising, and related work has been done before (Ostriker & Steinhardt 1995, White et al. 1996, Lineweaver et al. 1997, White & Silk 1996, Bludman 1997), but not with these three free variables and the full $\chi^2$ formalism. It is well known that recent cosmological observations and simulations, particularly related to the “shape parameter” $\Gamma$ and the cluster baryon fraction (CBF), pose a challenge to popular models, and that there is some doubt whether any simple model presently fits all data well. Our work, which begins by discarding nucleosynthetic constraints, provides a new way of looking at these problems. The CBF and $\Gamma$ constraints have not been applied jointly in earlier work. We find that, given our conservative (generous) choice of error bar on $h$, the SCDM model is disfavored somewhat but by no means excluded, if we are willing to accept $\eta_{10} \gtrsim 9$. But even with the generous error on h, and allowing $\Omega_{\rm M}$ to range freely, large values ($\gtrsim 5$) of $\eta_{10}$ are favored over small values ($\lesssim 2$). This suggests that the low D abundances measured by Burles and Tytler may be correct, and that the observed (extrapolated) primordial helium-4 mass fraction \[$Y_{\mathrm{P}} \approx 0.23$; cf. OS and Olive, Skillman, & Steigman 1997 (OSS)\], thought to be well determined, may be systematically too low for unknown reasons. CDM MODELS: PARAMETERS AND OBSERVABLES {#Sec:CDM-Models} ====================================== Parameters ---------- We will take the CDM models to be defined by three free parameters: Hubble parameter $h$; mass-density parameter $\Omega_{\mathrm{M}} = 8 \pi\, G\, \rho_{\mathrm{M}}/3H_0^2$; and baryon-to-photon ratio $\eta_{10}$, related to $\Omega_{\mathrm{B}}$ by equation (\[Eq:Omega\_B\]). Here $\Omega_{\mathrm{M}}$ by definition includes all “dynamical mass”: mass which acts dynamically like ordinary matter in the universal expansion. It is not limited to clustered mass only. Other free parameters having to do with structure formation, such as the tilt parameter $n$, could be added (White et al. 1996; Kolatt & Dekel 1997; White & Silk 1996), but we will try in general to avoid introducing many free parameters, so as to avoid generating confidence regions in more than three dimensions. We will, however, show results for two values of $n$ (1 and 0.8), and for a few alternative choices of other parameters affecting some of the observables. Observables ----------- We will consider five observables (constraints) which have measured values and errors which we assume to be normal (Gaussian). The five observables are: (1) measured Hubble parameter $h_{\mathrm{o}}$; (2) age of the universe $t_{\mathrm{o}}$; (3) dynamical mass-density parameter $\Omega_{\mathrm{o}}$ from cluster measurements or from large-scale flows; (4) gas-mass fraction $f_{\mathrm{o}} \equiv f_{\mathrm{G}} h^{3/2}$ in rich clusters; and (5) “shape parameter” $\Gamma_{\mathrm{o}}$ from structure studies. In much of our work we will drop one or another of these constraints. An observable $w_{\mathrm{o}}$ has the central value $\langle w \rangle$ and the standard deviation $\sigma_w$. The theoretical expression for this observable is given by a known function, $w$, of the three free parameters. The $\chi^2$ contribution of this observable is written as $\chi^2 = (\langle w \rangle - w)^2 / \sigma_w^2$. This sets up the usual conditions for the total $\chi^2$ (which, assuming the errors are uncorrelated, is a sum of $\chi^2$ contributions from different observables) to find the confidence regions for the free parameters (Cramer 1946, Bevington & Robinson 1992, Press et al. 1992, Barnett et al. 1996). We state below the theoretical expression $w$ and the observational constraint $w_{\mathrm{o}} = \langle w \rangle \pm \sigma_w$ which we assume. There are other constraints which could be applied, including cluster abundance, the height of the “acoustic peak” in the angular fluctuation spectrum of the cosmic background radiation, the Sunyaev-Zeldovich effect in clusters, the Lyman-alpha forest, and theoretical constraints on $\Lambda$ (White et al. 1996; White & Silk 1996; Lineweaver et al. 1997; Myers et al. 1997; Rauch, Haehnelt, & Steinmetz 1997; Weinberg et al. 1997; Bi & Davidsen 1997; Fan, Bahcall, & Cen 1997; Martel, Shapiro, & Weinberg 1998). We omit these here but intend to pursue them in subsequent work. Observed Hubble Parameter $h_{\mathrm{o}}$ ------------------------------------------ For the Hubble parameter the observable $h_{\mathrm{o}}$ is simply fit with the parameter $h$. Measurements of $h$ still show scatter which is large compared with their formal error estimates (Bureau et al. 1996, Tonry et al. 1997, Kundić et al. 1997, Tammann & Federspiel 1997). This indicates systematic errors. We do not presume to review this subject. To be conservative (permissive), we take $h_{\mathrm{o}} = 0.70 \pm 0.15$. Some may think that a smaller error could be justified. In §3.2 we will experiment with shrinking the error bar. Observed Age of the Universe $t_{\mathrm{o}}$ --------------------------------------------- Pre-Hipparcos observations gave the ages of the oldest globular clusters as $t_{\mathrm{GC}} \approx 14 \pm 2$ Gyr (Bolte & Hogan 1995; Jimenez 1997; D’Antona, Caloi, & Mazzitelli 1997; cf. Cowan et al. 1997, Nittler & Cowsik 1997). Some of the analyses incorporating the new Hipparcos data derive younger ages, $t_{\mathrm{GC}} \approx 12$ Gyr (Chaboyer et al. 1998, Reid 1997, Gratton et al. 1997). However, as Pont et al. (1998) and Pinsonneault (1998) emphasize, there are systematic uncertainties in the main-sequence fitting technique at the 2 Gyr level, and Pont et al. (1998) use Hipparcos data to derive an age of 14 Gyr for M92. The theoretical age for the $\Lambda = 0$ models is given by: $t = 9.78\:h^{-1}f(\Omega_{\mathrm{M}};\Lambda$=0) Gyr \[Weinberg 1972, equations (15.3.11) & (15.3.20)\]. The “observed" age of the universe, $t_{\mathrm{o}}$, should exceed $t_{\mathrm{GC}}$ by some amount $\Delta t$. The best guess for $\Delta t$ might be 0.5 – 1 Gyr. Most theorists believe that $\Delta t$ must be quite small (2 Gyr at most), but we know of no conclusive argument to prove this, and we do not want long-lived models to suffer an excessive $\chi^2$ penalty. We could treat $\Delta t$ as another free parameter, but to avoid this and keep things simple, we introduce asymmetric error bars. We believe that $t_{\mathrm{o}} = 14^{+7}_{-2}$ Gyr is a fair representation of $t_{\mathrm{o}}$ derived from present data on $t_{\mathrm{GC}}$. This allows enough extra parameter space at large ages to accommodate a conservative range of $\Delta t$; extremely large ages will be eliminated by the $h_{\mathrm{o}}$ constraint in any case. The $\chi^2$ analysis will still be valid with the unequal error bars if we assume the $\chi^2$ contribution as $(14 - t)^2/\sigma_t^2$, where $\sigma_t$ takes the value 2 Gyr when $t < 14$ Gyr and the value 7 Gyr when $t > 14$ Gyr. Observed Mass Density $\Omega_{\mathrm{o}}$ ------------------------------------------- The observed $\Omega$ at zero redshift, determined from clusters, has recently been reported as $$\Omega_{\mathrm{CL}} = 0.19 \pm 0.06 {\mbox { (stat)} } \pm 0.04 {\mbox { (sys)} } \label{Eq:Omega_CNOC}$$ (Carlberg, Yee, & Ellingson 1997), where the respective errors are statistical and systematic. This is based on the $M/L$ ratio in clusters and the luminosity density of the universe. If light traces dynamical mass, then we expect that $\Omega_{\mathrm{CL}}$ can be directly fit with $\Omega_{\mathrm{M}}$. There is a difficulty in using equation (\[Eq:Omega\_CNOC\]) as a constraint on the underlying parameter $\Omega_{\mathrm{M}}$. Many consumers of equation (\[Eq:Omega\_CNOC\]) and earlier results (cf. Carlberg et al. 1996) have failed to notice that the result is model-dependent, because the clusters in the sample have substantial redshifts (0.17 $< z <$ 0.55). In their analysis Carlberg et al. (1997) assumed $q_0 = 0.1$ (e.g., $\Omega_{\mathrm{M}} = 0.2$ and $\Lambda = 0$). When the result is $\langle \Omega_{\mathrm{CL}} \rangle= 0.19$, clearly the analysis is approximately self-consistent, but this does not give us sufficient guidance in exploring other values of $\Omega_{\mathrm{M}}$. For example, effects of nonzero $\Lambda$ could be substantial. We believe that if the analysis of Carlberg et al. (1997) were repeated for a $\Lambda$CDM model, the resulting $\langle \Omega_{\mathrm{CL}} \rangle$ might be smaller, perhaps 0.12 rather than 0.19. The parameters $\Omega_{\mathrm{M}}$ and $\Lambda$ need to be incorporated more fully into the analysis. Looking at these and earlier data, we choose, somewhat arbitrarily, to use instead of equation (\[Eq:Omega\_CNOC\]) a somewhat more permissive $\Omega$ constraint from clusters: $$\Omega_{\mathrm{o}} = 0.2 \pm 0.1 \label{Eq:Omega_CL}$$ (Carlberg 1997). If the critical (“closure”) $M/L$ ratio in $B_{\mathrm{T}}$ magnitude is $1500\,h\, (M/L)_{\odot}$ (Efstathiou, Ellis, & Peterson 1988), then equation (\[Eq:Omega\_CL\]) requires that the mean $M/L$ in $B_{\mathrm{T}}$ for galaxies in the local universe be about $(300 \pm 150)\: h$ (cf. Smail et al. 1996). This agrees well with modern reviews (Bahcall, Lubin, & Dorman 1995, Trimble 1987). We will assume in some of our examples that $\Omega_{\mathrm{o}}$ can be directly fit with $\Omega_{\mathrm{M}}$ (light traces mass; “unbiased”), with $\Omega_{\mathrm{o}}$ given by equation (\[Eq:Omega\_CL\]). Obviously, under this assumption, the $\chi^2$ contribution from the observed $\Omega_{\mathrm{o}}$ will rule out the SCDM model ($\Omega_{\mathrm{M}} = 1$) with high confidence. Bias is possible, with the most likely bias being $\Omega_{\mathrm{o}} < \Omega_{\mathrm{M}}$. This would be the case of additional unclustered or weakly clustered dynamical mass. Because such weakly clustered mass is quite possible, we will also do other cases with an alternative to the cluster constraint, as follows. Dekel and Rees (1994; cf. Dekel 1997) studied large-scale flows around voids and concluded that $\Omega_{\mathrm{M}}$ must be quite large: $\Omega_{\mathrm{M}} > (0.4, 0.3, 0.2)$ at confidence levels $(1.6\sigma, 2.4\sigma, 2.9\sigma)$. All values $\Omega_{\mathrm{M}} \ge 0.6$ were permitted. To use this one-way constraint as the third observable in a $\chi^2$ fit, we need a substitute function $\chi^2 (\Omega_{\mathrm{M}})$ having the properties: $\chi^2(0.4) \approx (1.6)^2$, $\chi^2 (0.3) \approx (2.4)^2$, $\chi^2 (0.2) \approx (2.9)^2$. The function $$\chi^2 (\Omega_{\mathrm{M}}) = \left\{ %\begin{cases} \begin{array}{ll} (0.6 - \Omega_{\mathrm{M}})^2/(0.125)^2 & % \text{$(\Omega_{\mathrm{M}} < 0.6)$}, \\ \mbox{$(\Omega_{\mathrm{M}} < 0.6)$}, \\ % 0 & \text{$(\Omega_{\mathrm{M}} \ge 0.6)$} 0 & \mbox{$(\Omega_{\mathrm{M}} \ge 0.6)$} %\end{cases} \end{array} \right. \label{Eq:Omega_DR}$$ is a good approximation. For $\Omega_{\mathrm{M}} \ge 0.6$, this $\chi^2$ implies a “perfect fit” to the $\Omega$ observable. This leaves parameter space open to large $\Omega_{\mathrm{M}}$. We apply the Dekel-Rees constraint instead of the cluster constraint if we just substitute equation (\[Eq:Omega\_DR\]) for the usual $\chi^2$ term arising from $\Omega_{\mathrm{o}}$. Observed Cluster Gas Mass Fraction $f_{\mathrm{o}}$ --------------------------------------------------- Theorists and observers (White & Frenk 1991, Fabian 1991, Briel, Henry, & Böhringer 1992, Mushotzky 1993) have long argued that the large observed gas mass fraction in clusters, $f_{\mathrm{G}}$, is a valuable cosmological datum and poses a serious threat to the SCDM model. This argument was raised to high visibility by the quantitative work of White et al. (1993), and now the problem is sometimes called the “baryon catastrophe” (Carr 1993) or “baryon crisis” (Steigman & Felten 1995). At the risk of boring the experts, we must emphasize that the following argument does not assume that most of the mass in the universe, or any specific fraction of it, is in rich clusters. Rather, we will use $f_{\mathrm{G}}$, not as a constraint on $\Omega_{\mathrm{M}}$, but as a constraint on the universal baryon fraction, the ratio $\Omega_{\mathrm{B}}/\Omega_{\mathrm{M}}$. The idea is that the content of a rich cluster is a fairly unbiased sample of baryonic and dark matter. This is suggested by simulations, which are discussed below. The measurement of $f_{\mathrm{G}}$ poses problems, but this is not the place for a lengthy review. Magnetic pressure (Loeb & Mao 1994) may cause systematic errors, but these are probably not large and do not provide a promising escape hatch for the SCDM model (Felten 1996). Reported values of $f_{\mathrm{G}}$ derived by various methods show quite a wide range from cluster to cluster and also from groups through poor and rich clusters (Steigman & Felten 1995; White & Fabian 1995; Lubin et al. 1996; Mohr, Geller, & Wegner 1996). Loewenstein and Mushotzky (1996) emphasize that the range in $f_{\mathrm{G}}$ is wider than expected from simulations, and they suggest that some significant physics may be missing from the simulations. Cen (1997) argues that the spread may be caused by projection effects in the measurements of $f_{\mathrm{G}}$, arising because of large-scale pancakes and filaments. Evrard, Metzler, and Navarro (1996), using gas-dynamical simulations to model observations, find that the largest error in $f_{\mathrm{G}}$ arises from measurement of the cluster’s [*total*]{} mass, and that this error can be reduced by using an improved estimator and by restricting the measurement to regions of fairly high overdensity. Evrard (1997) applies these methods to data for real clusters and finds $f_{\mathrm{G}}\,h^{3/2} = 0.060 \pm 0.003$. This subject is still controversial so, to be conservative, we will double his error bars and take as our constraint $$f_{\mathrm{o}} \equiv f_{\mathrm{G}}\,h^{3/2} = 0.060 \pm 0.006.$$ Note that this is quite a large gas fraction ($f_{\mathrm{G}} \approx 17\% $ for $h \approx 0.5$), in general agreement with earlier results. The functional dependence of $f_{\mathrm{G}}$ on the cosmological parameters also poses problems. The [*universal*]{} baryonic mass fraction is $\Omega_{\mathrm{B}}/\Omega_{\mathrm{M}}$, but not all baryons are in the form of gas, and furthermore selection factors may operate in bringing baryons and dark matter into clusters. White et al. (1993) introduced a “baryon enhancement factor” $\Upsilon$ to describe these effects as they operate in simulations. $\Upsilon$ may be defined by $$f_{G0} = \Upsilon \, \Omega_{\mathrm{G}}/\Omega_{\mathrm{M}}, \label{Eq:f_G0}$$ where $\Omega_{\mathrm{G}}$ is the initial contribution of [*gas*]{} to $\Omega_{\mathrm{M}}$ (note that $\Omega_{\mathrm{G}} \le \Omega_{\mathrm{B}}$) and $f_{G0}$ is the gas mass fraction in the cluster immediately after formation. We will shortly set $\Upsilon$ equal to some constant. $\Upsilon$ is really the [*gas*]{} enhancement factor, because the simulations do not distinguish between baryonic condensed objects if any (galaxies, stars, machos) and non-baryonic dark-matter particles. All of these are lumped together in the term $(\Omega_{\mathrm{M}} - \Omega_{\mathrm{G}})$ and interact only by gravitation. If all the baryons start out as gas $(\Omega_{\mathrm{G}} = \Omega_{\mathrm{B}})$, and if gas turns into condensed objects only [*after*]{} cluster formation, then equation (\[Eq:f\_G0\]) may be rewritten: $$f_{\mathrm{G}} + f_{\mathrm{GAL}} = \Upsilon \,\Omega_{\mathrm{B}} / \Omega_{\mathrm{M}}, \label{Eq:f_G+f_GAL}$$ where $f_{\mathrm{G}}$ is the present cluster gas-mass fraction and $f_{\mathrm{GAL}}$ the present cluster mass fraction in baryonic condensed objects of all kinds (galaxies, stars, machos). We wish to carry along an estimate of $f_{\mathrm{GAL}}$ to show its effects. White et al. (1993) took some pains to estimate the ratio $f_{\mathrm{G}}/f_{\mathrm{GAL}}$ within the Abell radius of the Coma cluster, counting only galaxies (no stars or machos) in $f_{\mathrm{GAL}}$. They obtained $$f_{\mathrm{G}}/f_{\mathrm{GAL}} = 5.5 \, h^{-3/2}. \label{Eq:f_G/f_GAL}$$ This is large, so unless systematic errors in this estimate are very large, the baryonic content of this cluster (at least) is dominated by the hot gas. Carrying $f_{\mathrm{GAL}}$ along as an indication of the size of the mean correction for all clusters, and solving equations (\[Eq:f\_G+f\_GAL\]) and (\[Eq:f\_G/f\_GAL\]) for $f_{\mathrm{G}} h^{3/2}$, we find $$f_{\mathrm{G}} h^{3/2} = [\,1 + (h^{3/2}/5.5)\,]^{-1} (\Upsilon \, \Omega_{\mathrm{B}}/\Omega_{\mathrm{M}})\,h^{3/2}, \label{Eq:f_Gh}$$ where $\Omega_{\mathrm{B}}$ is given from $\eta_{10}$ and $h$ by equation (\[Eq:Omega\_B\]). This is the appropriate theoretical function of the free parameters to fit to the observation. The second term in brackets in equation (\[Eq:f\_Gh\]) is the small correction term due to $f_{\mathrm{GAL}}$. In deriving this $f_{\mathrm{GAL}}$, given by equation (\[Eq:f\_G/f\_GAL\]), White et al. (1993), using observations in the inner parts of bright ellipticals by van der Marel (1991), assumed that within cluster galaxies the mean ratio of baryonic mass to blue light is $6.4h(M/L)_{\odot}$. We note that this correction term would be larger if cluster galaxies or the cluster as a whole contained baryonic machos amounting to $\sim 20(M/L)_{\odot}$, as suggested for the halo of our Galaxy by theories of observed microlensing events (Chabrier, Segretain, & Méra 1996; Fields, Mathews, & Schramm 1997; Natarajan et al. 1997). Indeed, Gould (1995) has even suggested that the mass in machos could be comparable to that in the gas component, in which case $f_{\mathrm{GAL}} \approx f_{\mathrm{G}}$. What value of the gas enhancement factor $\Upsilon$ should be used in equation (\[Eq:f\_Gh\])? A value $\Upsilon = 3-5$ would do away with the “baryon catastrophe” for the SCDM model. There is no plausible way to obtain an $\Upsilon$ this large. White et al. (1993), when they assumed zero-pressure gas to explore maximizing $\Upsilon$, always found $\Upsilon \le 1.5$ in simulations. More realistic simulations with gas pressure give $\Upsilon \approx 0.9$ (Evrard 1997), or even as small as 2/3 (Cen & Ostriker 1993). The gas preferentially stays out of the clusters to some extent rather than concentrating itself there. Gas can support itself through pressure and shocks, while CDM cannot. We will set $\Upsilon = 0.9$ in most of our examples. This is representative of results from simulations, and it is close to unity, so these cases will also illustrate the approximate consequences if gas is neither enhanced nor excluded in clusters. Cen (1997) finds in simulations that the determination of $f_{\mathrm{G}}$ from X-ray observations is biased toward high $f_{\mathrm{G}}$ by large-scale projection effects; i.e., the calculated $f_{\mathrm{G}}$ exceeds the true $f_{\mathrm{G}}$ present in a cluster. This bias factor can be as large as 1.4. Evrard et al. (1996) and Evrard (1997) have not observed such a bias in their simulations. If Cen is correct, we could explore the effect of such a bias in our statistical tests by using for $\Upsilon$, instead of 0.9, an “effective value” $\Upsilon \approx 0.9 \times 1.4 \approx 1.3$. Since this would also demonstrate the impact of any effect which may cause $\Upsilon$ to exceed unity moderately, we will show results for $\Upsilon = 1.3$ as well as for $\Upsilon = 0.9$. Equations (\[Eq:f\_G+f\_GAL\]) – (\[Eq:f\_Gh\]) above were derived under the assumption that all baryonic condensed objects (galaxies, stars, machos) form from the gas [*after*]{} the collapse of clusters occurs. If, instead, all such objects were formed [*before*]{} collapse, equation (\[Eq:f\_G+f\_GAL\]) should be replaced by: $$f_{\mathrm{G}} + \left ( \frac{\Upsilon - f_{\mathrm{G}}}{1 - f_{\mathrm{G}}} \right ) f_{\mathrm{GAL}} = \Upsilon \, \Omega_{\mathrm{B}} / \Omega_{\mathrm{M}}, \label{Eq:f_Gsum}$$ reflecting the fact that the baryons in condensed objects now escape the gas enhancement occurring during cluster formation. Since these effects are not large for $\Upsilon \approx 1$ and equation (\[Eq:f\_G+f\_GAL\]) is likely to be closer to the true situation than equation (\[Eq:f\_Gsum\]), we will make no further use of equation (\[Eq:f\_Gsum\]). Shape Parameter $\Gamma_{\mathrm{o}}$ from Large-Scale Structure ---------------------------------------------------------------- The last observable we use is the “shape parameter” $\Gamma$, which describes the transfer function relating the initial perturbation spectrum $P_{\mathrm{I}} (k) \propto k^n$ to the present spectrum $P(k)$ of large-scale power fluctuations, as observed, e.g., in the galaxy correlation function. When the spectral index $n$ of $P_{\mathrm{I}} (k)$ has been chosen, $\Gamma$ is determined by fitting the observed $P(k)$. There are some notational differences among papers on this subject. Sometimes $\Gamma$ is used to mean simply the combination $\Omega_{\mathrm{M}}h$. We will avoid this usage here. Results of observations may be cast in terms of an “effective shape parameter” $\Gamma$ (White et al. 1996), which we will take as our observable. Studies show that for the usual range of CDM models, with or without $\Lambda$, the expression for $\Gamma$ is $$\begin{split} \Gamma & \approx \Omega_{\mathrm{M}} h \; \exp \left [ - \Omega_{\mathrm{B}} - (h/ 0.5)^{1/2} (\Omega_{\mathrm{B}} / \Omega_{\mathrm{M}} ) \right ] \\ & \qquad - 0.32 \; (n^{-1} -1) \label{Eq:Gamma_th} \end{split}$$ (Peacock & Dodds 1994; Sugiyama 1995; Liddle et al. 1996a,b; White et al. 1996; Liddle & Viana 1996; Peacock 1997). For $n \approx 1$, if $\Omega_{\mathrm{B}}$ and $\Omega_{\mathrm{B}} /\Omega_{\mathrm{M}}$ are small, we have $\Gamma \approx \Omega_{\mathrm{M}} h$. The Harrison-Zeldovich (scale-invariant, untilted) case is $n = 1$. We will adopt $n = 1$ for our standard case and experiment with different $n$ in §3.3. Approximation (11) has been tested (and, we believe, is valid) only for models in which both $n$ and the exponential term are fairly near unity. This $\Gamma$ is the parameter of the familiar BBKS approximation to the transfer function (Bardeen et al. 1986, Peacock 1997). The BBKS transfer function does not fit the data continuously from long to short wavelengths, and Hu and Sugiyama (1996) have developed a more elaborate approximation for short wavelengths ($\lesssim 3h^{-1}$ Mpc), useful especially in cases with large $\Omega_{\rm B}/\Omega_{\rm M}$. Recent comparisons with data (Webster et al. 1998) still use Sugiyama’s (1995) form for $\Gamma$ as in equation (11) above, and this BBKS approximation is adequate in the regime $(3-100)h^{-1}$ Mpc, where $\Gamma$ is determined. For the observed value of $\Gamma$, we take $$\Gamma_{\mathrm{o}} = 0.255 \pm 0.017 \label{Eq:Gamma_obs}$$ (Peacock & Dodds 1994; cf. Maddox, Efstathiou, & Sutherland 1996). This is based on the galaxy correlation function, and it assumes that light traces mass. The very small errors, from Peacock & Dodds (1994), result from averaging several data sets and may not be realistic. Later we will explore the consequences of inflating these errors and/or moving the central value. Equations (\[Eq:Gamma\_th\]) and (\[Eq:Gamma\_obs\]) imply, very roughly, that $\Omega_{\mathrm{M}} h \approx 0.25$. The shape parameter can be derived from the galaxy peculiar-velocity field instead of the density field. The result from that technique, analogous to $\Omega_{\mathrm{M}} h = 0.25$, is, very roughly and for $n = 1$, $$\Omega_{\mathrm{M}} h^{1.2} = 0.350 \pm 0.087 ~~(90\% \mbox{ CL}) \label{Eq:Omega_Mh_Z}$$ (Zaroubi et al. 1997a), where $\Omega_{\mathrm{B}} h^2 = 0.024$ has been assumed and CL stands for confidence level. Equation (\[Eq:Omega\_Mh\_Z\]) ostensibly includes an estimate of cosmic variance (cf. Kolatt & Dekel 1997, Zaroubi et al. 1997b). Equation (\[Eq:Omega\_Mh\_Z\]) may be used to yield an estimate of $\Gamma_{\mathrm{o}}$ as follows: Adjust the error bar in equation (\[Eq:Omega\_Mh\_Z\]) from $1.65\sigma$ to $1\sigma$. Evaluate $\Omega_{\mathrm{M}}$ from equation (\[Eq:Omega\_Mh\_Z\]) at the midpoint of the “interesting” range of $h$, viz. $h = 0.7$. Substitute the resulting parameters, including $\Omega_{\mathrm{B}}$, into the right-hand side of equation (\[Eq:Gamma\_th\]) and evaluate. The result is $$\Gamma_{\mathrm{o}} = 0.32 \pm 0.05. \label{Eq:Gamma_Z}$$ The independent estimates in equations (\[Eq:Gamma\_obs\]) and (\[Eq:Gamma\_Z\]) agree tolerably within the stated errors. Any difference, if real, could be caused by galaxy bias. The shape-parameter constraint is in a sense the least robust of the constraints we have discussed since it is not part of the basic Friedmann model. Rather, it depends on a theory for the primordial fluctuations and how they evolve. If the Friedmann cosmology were threatened by this constraint, we believe that those who model large-scale structure would find a way to discard it. Therefore we will also explore some consequences of removing this constraint. CDM MODELS: RESULTS {#Sec:CDM-Models-results} =================== CDM with Standard Constraints ----------------------------- We begin the presentation of our results by adopting a standard case with only four constraints, dropping the $\Omega_{\mathrm{o}}$ constraint. For this standard case we assume $n = 1$ and $\Upsilon = 0.9$, and we apply the following “standard constraints”: $h_{\mathrm{o}} = 0.70 \pm 0.15$, $ t_{\mathrm{o}} = 14^{+7}_{-2}$ Gyr, $f_{\mathrm{o}} \equiv f_{\mathrm{G}} h^{3/2} = 0.060 \pm 0.006$, and $\Gamma_{\mathrm{o}} = 0.255 \pm 0.017$. Then $\chi^2$ is the sum of four terms. Results for our standard case are displayed in Figures \[Fig:H-Omega\_M\_L0\]–\[Fig:eta-P\_L0\]. Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\] are a pair which can be understood geometrically. The function $\chi^2 (h, \Omega_{\mathrm{M}}, \eta_{10})$ is computed on the three-dimensional parameter space. It has a minimum $\chi^2_{\mathrm{min}}$ in this space, which in this case is 1.2 for one degree of freedom (DOF) and is located at (0.57, 0.61, 8.7). This value of $\chi^2_{\mathrm{min}}$ is acceptable; it is the 73% point of the distribution. We may draw a closed surface which encloses this point, defined by setting $$\Delta \chi^2 \equiv \chi^2(h, \Omega_{\mathrm{M}}, \eta_{10}) - \chi^2_{\mathrm{min}} = 2.3. \label{Eq:Delta-chi2}$$ The quantity $\Delta\chi^2$ is distributed like a $\chi^2$ variable with 3 DOF (Press et al. 1992, Barnett et al. 1996). Our surface $\Delta\chi^2 = 2.3$ is at the 49% point, so it is a 49% confidence region (“CR”) for the three parameters jointly. Furthermore, its projections on the orthogonal planes ($h, \Omega_{\mathrm{M}}$; Fig. \[Fig:H-Omega\_M\_L0\]) and ($h, \eta_{10}$; Fig. \[Fig:H-eta\_L0\]) give the 68% CRs for the parameters pairwise. These 68% CRs are shown as closed curves in Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\]. Similarly, we construct 95% CRs in these planes by replacing 2.3 by 6.0 in equation (\[Eq:Delta-chi2\]). We also show in Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\] projected CRs obtained by computing $\chi^2$ for single observables alone, or for pairs of observables. They are drawn by setting $\Delta\chi^2 = 1$ and projecting. These regions are not closed. They are merely intended to guide the reader in understanding how the various constraints influence the closed contours which show our quantitative results. One-dimensional confidence intervals (CIs) may similarly be constructed for any parameter by projecting closed surfaces in three-space onto a single axis. These CIs may be described by a likelihood function. Figure \[Fig:eta-P\_L0\] shows the likelihood function $\mathcal{L}$$(\eta_{10})$ for the parameter $\eta_{10}$. Table \[Tab:Table1\] shows the one-parameter CIs for the CDM models. [l l l l l]{} Parameter & &\ $\eta_{10}$ & $8.7\,^{+2.3}_{-1.6}$ & $(>6.1)$& $8.4 \, ^{+2.1}_{-1.5}$ & $(>5.8)$\ $\Omega_{\mathrm{B}} $ & $0.10\,^{+0.08}_{-0.04}$ & $(>0.04)$ & $0.08 \, ^{+0.06}_{-0.03}$ & $(>0.03)$\ $\Omega_{\mathrm{M}} $& $0.61\,^{+0.20}_{-0.14}$ & $(>0.39)$ & $0.53\,^{+0.19}_{-0.11}$ & $(>0.35)$\ $H_0$ (km s$^{-1}$ Mpc$^{-1}$) & $57\,^{+11}_{-10}$ & $(36 - 80)$ & $62\,^{+ 13}_{-11}$ & $(39 - 87)$\ $t_0$ (Gyr) & $12.6\,^{+1.9}_{-1.6}$ & $(9.7 - 16.6)$ & $12.9\,^{+1.5}_{-1.4}$ & $(10.5 - 16.1)$\ CDM: Discussion --------------- It is well known that the condition $\Omega_{\mathrm{M}} h \approx 0.25$ poses some threat to the SCDM $(\Omega_{\mathrm{M}} = 1)$ model. Figure \[Fig:H-Omega\_M\_L0\] shows that this threat is far from acute, with our more accurate form of the $\Gamma$ constraint given in equation (\[Eq:Gamma\_th\]), as long as the error on $h_{\mathrm{o}}$ is large (0.15) and BBN constraints are discarded. (Note again that we have not applied the constraint $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$.) Even our 68% contour extends to $\Omega_{\mathrm{M}}= 1$. With the large error bar, the corresponding value of $H_{0}$, $h = 0.43$, is accepted. The exponential term in equation (\[Eq:Gamma\_th\]) becomes significant because the $f_{\mathrm{G}}$ constraint forces $\Omega_{\mathrm{B}}$ to increase with $\Omega_{\mathrm{M}}$, allowing the product $\Omega_{\mathrm{M}}h$ to exceed 0.25. This has been noted before (White et al. 1996, Lineweaver et al. 1997). We have tested the SCDM model by fixing $\Omega_{\mathrm{M}}$ at unity and fitting the four standard constraints with the remaining two parameters. The CRs for $\eta_{10} - H_0$ are shown in Figure \[Fig:H-eta\_OM1\]. We find $\chi^2_{\mathrm{min}} = 3.4$ for 2 DOF (82% CL), which is acceptable. However, this case encounters severe problems since only $h < 0.48$ and $\eta_{10} > 8$ are accepted at the 95% contour. Indeed, $\eta_{10} \gtrsim 8$ if $h \approx 0.4$ and $\eta_{10} \approx 15$ if $h \approx 0.48$. Such large $\eta$ values pose a serious threat to the consistency between the predictions of BBN and the primordial abundances of the light elements inferred from observations (e.g., Hata et al. 1996). Indeed, this “solution" is only acceptable because of our very generous error bar for $h$ and because we have discarded BBN constraints. When $h$ is better known, the situation for SCDM will change. As an illustration, in Figure \[Fig:H-Omega\_M\_L0\_H70+-7\] we return to our three standard variables but replace our standard constraint on $H_0$ with $h_{\mathrm{o}} = 0.70 \pm 0.07$, assuming, arbitrarily, a 10% error. The $\chi^2_{\mathrm{min}}$ is now 2.2 for 1 DOF (86% CL), so we can still accept the basic Friedmann model, but SCDM is now excluded strongly. In this case the corresponding allowed range of $\eta$, shown in Figure \[Fig:H-eta\_L0\_H70+-7\], is not in strong conflict with BBN although the predicted $^4$He abundance is larger than that inferred from observations of extragalactic regions (OS, OSS). Returning to our CDM case with standard constraints, it is less well known that the $\Gamma$ constraint also poses a threat to [ *low*]{}-density models (Liddle et al. 1996, Kolatt & Dekel 1997, White & Silk 1996). From Figure \[Fig:H-Omega\_M\_L0\] it is apparent that there is tension between our four standard constraints and $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$. Since the 95% contour does not even extend downward to $\Omega_{\mathrm{M}} = 0.3$, we refrained from using this cluster constraint. The $t_{\mathrm{o}}$ and $\Gamma_{\mathrm{o}}$ constraints, combined, force the parameters upward out of the lower part of the figure, and favor $\Omega_{\mathrm{M}} \gtrsim 0.4$. We could nevertheless force a fit to all five constraints and draw CRs. We have done this, and we find that $\chi^2_{\mathrm{min}}$ is 7.8 for 2 DOF (98% CL); i.e., we reject the combined fit with 98% confidence. Among our principal results is that our standard CDM case favors large values of the baryon-to-photon ratio, $\eta_{10} = 8.7^{+2.3}_{-1.6}$ (see Figure \[Fig:eta-P\_L0\] and Table \[Tab:Table1\]). It is the $f_{\mathrm{o}}$ and $\Gamma_{\mathrm{o}}$ constraints which, together, force us to large $\eta_{10}$. Also shown in Figure \[Fig:eta-P\_L0\] are the likelihoods for $\eta$ derived in Hata et al. (1997) for the high deuterium abundance inferred for some QSO absorbers (Songaila et al. 1994, Carswell et al. 1994, Rugers & Hogan 1996) and for the lower D abundance inferred for others (Tytler et al. 1996, Burles & Tytler 1996). It is clear from Figure \[Fig:eta-P\_L0\] that our results here favor the high-$\eta$, low-D choice which is consistent with local deuterium observations (Linsky et al. 1993) and Galactic chemical evolution (Steigman & Tosi 1992, 1995, Edmunds 1994, Tosi 1996). BBN consistency with the observed lithium abundances in very metal-poor halo stars requires that these stars have reduced their surface lithium abundance by a modest factor, $\lesssim 2-3$. However, for consistency with standard BBN predictions for helium, our high value for $\eta$ requires that the abundances inferred from the low-metallicity, extragalactic regions are systematically biased low. This high-$\eta$ range is consistent with estimates of the baryon density derived from observations of the Ly-$\alpha$ forest (Hernquist et al. 1996, Miralda-Escudé et al. 1996, Rauch et al. 1997, Croft et al. 1997, Weinberg et al. 1997, Bi & Davidsen 1997). CDM: More Variations -------------------- Because we dropped the cluster-determined constraint $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$, it is of interest to see how the CRs in Figure \[Fig:H-Omega\_M\_L0\] are affected if we apply instead an alternative constraint to $\Omega_{\mathrm{M}}$. If, for example, we adopt the Dekel-Rees constraint, $\Omega_{\mathrm{DR}}$ \[equation (\[Eq:Omega\_DR\])\], which implies a substantial contribution to $\Omega_{\mathrm{M}}$ arising from mass not traced by light, this does not change Figure \[Fig:H-Omega\_M\_L0\] by much because $\Omega_{\mathrm{M}} > 0.4$ was favored already by our four standard constraints. Small $h$ and large $\Omega_{\mathrm{M}}$ are now favored slightly more. Because this makes little difference, we will proceed in most cases without any constraint $\Omega_{\mathrm{M}}$. The consequences for $\eta$ are found in Table \[Tab:Table2\]. [l l l]{} Variation &\ With $\Omega_{\mathrm{DR}}$ & $9.2\;^{+2.2}_{-1.5}$ & $(>6.5)$\ “Red” tilt $n = 0.8$ & $10.8\;^{+3.5}_{-2.0}$ & $(>7.3)$\ Positive gas bias $\Upsilon = 1.3$ & $5.7\;^{+1.2}_{-0.9}$ & $(>4.0)$\ Without $\Gamma$; With $\Omega_{\mathrm{CL}}$ & $3.1 \pm 1.6$ & $(<6.5)$\ $\Gamma = 0.25 \pm 0.05$ & $8.2\;^{+3.2}_{-2.2}$ & $(>4.2)$\ $\Gamma = 0.15 \pm 0.04$ & $4.6\;^{+1.9}_{-1.2}$ & $(2.1 - 9.2)$\ Figure \[Fig:H-Omega\_M\_L0\_var\], the analog of Figure \[Fig:H-Omega\_M\_L0\], shows the effects of some other variations, taken one at a time. Here we consider only $\Lambda = 0$ models and show only the 95% CRs. The corresponding likelihoods for $\eta$ are shown in Figure \[Fig:eta-P\_L0\_var\]. Tilt in the primordial spectrum, for example, has been investigated in many papers (Liddle et al. 1996a,b, White et al. 1996, Kolatt & Dekel 1997, White & Silk 1996, Liddle & Viana 1996). We show the effect of a moderate “red tilt” ($n = 0.8$ instead of $n = 1$). The $\chi^2_{\mathrm{min}}$ value is 1.5 for 1 DOF (78% CL). The favored likelihood range for $\eta_{10}$ is now higher, though $\eta_{10} \approx 7$ is still allowed (see Figure \[Fig:eta-P\_L0\_var\]). With this tilt the $\Gamma$ constraint favors higher $\Omega_{\mathrm{M}}$, so that the SCDM model is allowed for $h$ up to nearly 0.5. However, as can be seen in Table \[Tab:Table2\], the higher allowed range for $\eta$ threatens the consistency of BBN. Conversely, a “blue” tilt, $n > 1$ (Hancock et al. 1994), would move the CR downward and allow models with $\Omega_{\mathrm{M}} \le 0.3$ at high $h$. Changing to a gas enhancement factor $\Upsilon = 1.3$ (a modest positive enhancement of gas in clusters) instead of 0.9 does not change the contours in Figure \[Fig:H-Omega\_M\_L0\] by much, particularly at the low-$\Omega_{\mathrm{M}}$ end, where the exponential term in $\Gamma$ is close to unity. The $\chi^2_{\mathrm{min}}$ value for this case is 1.1 for 1 DOF (71% CL). Although the acceptable range for $\eta_{10}$ moves downward (see Figure \[Fig:eta-P\_L0\_var\]), $\eta_{10} \le 4$ is still excluded, disfavoring the low D abundance inferred from some QSO absorbers and favoring a higher helium abundance than is revealed by the region data. In §2.6 we mentioned the possibility that the fraction of cluster mass in baryons in galaxies, isolated stars, and machos $(f_{\mathrm{GAL}})$ might be larger – even much larger – than is implied by equation (\[Eq:f\_G/f\_GAL\]). Equations (\[Eq:f\_G/f\_GAL\]) and (\[Eq:f\_Gh\]) show that a large $f_{\mathrm{GAL}}$ would affect the CRs in much the same way as a [ *small*]{} $\Upsilon$, favoring even higher values of $\Omega_{\mathrm{M}}$ and $\eta_{10}$. The $\Gamma$ constraint is crucial for our standard results favoring high $\Omega_{\mathrm{M}}$ and high $\eta$. If, for example, we drop the $\Gamma$ constraint and in its place use the cluster estimate $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$, low $\Omega_{\mathrm{M}}$ and low $\eta$ are now favored (see Figures \[Fig:H-Omega\_M\_L0\_var\] and \[Fig:eta-P\_L0\_var\]). Earlier we mentioned that the Peacock and Dodds (1994) estimate of $\Gamma_{\mathrm{o}}$ may have unrealistically small error bars. Given that the shape parameter plays such an important role in our analysis, we have considered the effects of relaxing the uncertainty in $\Gamma_{\mathrm{o}}$. In Figures \[Fig:H-Omega\_M\_L0\_G+-0.05\] and \[Fig:H-eta\_L0\_G+-0.05\], the analogs of Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\], we show our results for $\Gamma_{\mathrm{o}}$ = 0.25 $\pm$ 0.05. As expected our CRs have expanded and the best-fit values of $\Omega_{\mathrm{M}}$, $h$ and $\eta_{10}$ have shifted: $\Omega_{\mathrm{M}} = 0.48^{+0.22}_{-0.15}$, $h = 0.58 \pm 0.22$ and $\eta_{10} = 8.2^{+3.2}_{-2.2}$. Now the SCDM model with $\Omega_{\mathrm{M}}$ = 1, $h = 0.45$, and $\eta_{10} = 13$ is acceptable (80%). Although the uncertainties are larger, low $\eta_{10}$ is still disfavored. If we add the Dekel-Rees estimate of $\Omega_{\mathrm{M}}$, the five-constraint fit favors somewhat higher values of $\Omega_{\mathrm{M}}$ and $\eta_{10}$ and slightly lower values of $h$. In contrast, if instead we include the cluster estimate, we find a barely acceptable fit ($\chi^2_{\mathrm{min}}$ = 5.0 for 2 DOF, 92% CL), which favors lower values of $\Omega_{\mathrm{M}}$ and $\eta_{10}$ and slightly higher values of $h$. CDM: A Smaller Shape Parameter ------------------------------ The results above show that the Peacock-Dodds shape parameter $\Gamma_{\mathrm{o}} \approx 0.255$, which we have used, clashes with $\Omega_{\mathrm{M}} \approx 0.2$ (the estimate from clusters). The agreement does not become good even if error bars $\pm 0.05$ on $\Gamma_{\mathrm{o}}$ are assumed. A new determination of $\Gamma_{\mathrm{o}}$ from the IRAS redshift survey (Webster et al. 1998) gives $\Gamma_{\mathrm{o}} = 0.15 \pm 0.08$ at 95% confidence. Assuming that the statistics are roughly Gaussian, we can represent this at $\pm 1\sigma$ as $$\Gamma_{\mathrm{o}} = 0.15 \pm 0.04. \label{Eq:newGamma_obs}$$ Combining equation (16) and equation (12) (used earlier) in quadrature would give $\Gamma_{\mathrm{o}} = 0.239 \pm 0.016$ – very close to equation (12). Throwing in equation (14) would bring us even closer to equation (12), which dominates because of its small error. Combining the estimates would be unwise, because they do not agree well. This small value from the IRAS survey is not entirely new (cf. Fisher, Scharf, & Lahav 1994). It has received little attention because the larger value, $\Gamma_{\mathrm{o}} \approx 0.25$, was already seen as a major challenge to the popular SCDM model ($\Omega_{\mathrm{M}} = 1$). The smaller value poses an even more severe challenge to the SCDM model. But it gives more scope to low-density models, which are popular now. Until the discrepant values of $\Gamma_{\mathrm{o}}$ are understood, we think it wise to show joint CRs using separately the larger and the smaller values of $\Gamma_{\mathrm{o}}$. Figures \[Fig:H-Omega\_M\_L0\_G0.15\] and \[Fig:H-eta\_L0\_G0.15\], analogs of Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\], show CRs for our four standard constraints, but with $\Gamma_{\mathrm{o}} = 0.255\pm 0.017$ replaced by $\Gamma_{\mathrm{o}} = 0.15 \pm 0.04$. The $\chi^2_{\mathrm{min}}$ is 0.63 for 1 DOF (good) and is located at ($h$, $\Omega_{\mathrm{M}}$, $\eta_{10}$) = (0.60, 0.30, 4.6). The CRs now exclude the SCDM model strongly and favor low density. The value $\eta_{10} \approx 5$, favored by the Burles and Tytler (1997a,b,c) deuterium abundance determination (see §1), is now near the point of optimum fit. The CRs clearly would accept the added cluster constraint $\Omega_{\mathrm{M}} \approx 0.2$ if we were to apply it. But note that in our three-dimensional CRs, low $\Omega_{\mathrm{M}}$ goes with low $\eta_{10}$, because of the $f_{\mathrm{G}}$ constraint. For example, the combinations (0.7, 0.2, 3) and (0.7, 0.3, 5) give good fits, while (0.7, 0.2, 5) and (0.7, 0.3, 3) give poor fits. In general, the CRs give some preference to $\eta_{10} \approx 5$. But even a value as small as $\eta_{10} \approx 2$ now lies within the 95% CI and cannot be excluded without BBN evidence (see Table \[Tab:Table2\]). $\Lambda$CDM MODELS: RESULTS {#Sec:Lambda-CDM-Models} ============================ Turning to models with nonzero $\Lambda$, we consider here only the popular flat ($k$ = 0) “$\Lambda$CDM” models with $\Omega_\Lambda = 1 - \Omega_{\mathrm{M}}$, where $\Omega_\Lambda \equiv \Lambda /(3H_0^2)$. This means that there are still only three free parameters. The five constraints discussed in §2 are still in force, except that the product of the age and the Hubble parameter is a different function of $\Omega_{\mathrm{M}} = 1 - \Omega_\Lambda$: $ t = 9.78\,h^{-1}f(\Omega_{\mathrm{M}};k=0)$ Gyr \[Carroll, Press, & Turner 1992, equation (17)\]. For a given $\Omega_{\mathrm{M}} < 1$, the age is longer for the flat $(k = 0)$ model than for the $\Lambda = 0$ model. Figures \[Fig:H-Omega\_M\_k0\] and \[Fig:H-eta\_k0\] show the results for our four standard constraints, with no direct $\Omega_{\mathrm{M}}$ constraint. Figures \[Fig:H-Omega\_M\_k0\] and \[Fig:H-eta\_k0\] differ very little from Figures \[Fig:H-Omega\_M\_L0\] and \[Fig:H-eta\_L0\], of which they are the analogs. The longer ages do allow the CRs to slide farther down toward large $h$ and small $\Omega_{\mathrm{M}}$. The $\chi^2_{\mathrm{min}}$ is 0.8 for 1 DOF (good). One-dimensional CIs are listed in Column 3 of Table \[Tab:Table1\]. Because of the longer ages at low $\Omega_{\mathrm{M}}$ (high $\Omega_{\Lambda}$), we can now accept $\Omega_{\mathrm{o}}$ as a fifth constraint \[although we remind the reader that the constraint $\Omega \approx 0.2$ may not be appropriate to a $\Lambda$CDM model (§2.5)\]; the $\chi^2_{\mathrm{min}}$ is 5.4 for 2 DOF (93%, barely acceptable). In this case large $\Omega_{\mathrm{M}}$ and small $h$ are now excluded while $\eta_{10} > 4$ is still favored strongly. We have not imposed any direct constraint on $\Omega_\Lambda$. There are claims that, for a $\Lambda$CDM model, $\Omega_\Lambda < 0.51$ (based on limited statistics of seven supernovae; Perlmutter et al. 1997) and $\Omega_\Lambda < 0.66$ (based on a paucity of lensing events; Kochanek 1996). The lensing constraint has been in dispute because of absorption, but recent work indicates that absorption is probably unimportant (Kochanek 1996; Falco, Kochanek, & Munoz 1997). These $\Omega_\Lambda$ constraints agree in a general way with our result $\Omega_{\mathrm{M}} \gtrsim 0.4$ (Fig. \[Fig:H-Omega\_M\_k0\]). CONCLUSIONS {#Sec:Conclusions} =========== If BBN constraints on the baryon density are removed (or relaxed), the interaction among the shape-parameter $(\Gamma)$ constraint, the $f_{\mathrm{G}}$ (cosmic baryon fraction) constraint, and the value of $\eta_{10}$ assumes critical importance. These constraints still permit a flat CDM model, but only as long as $h < 0.5$ is allowed by observations of $h$. The $f_{\mathrm{G}}$ constraint means that large $\Omega_{\mathrm{M}}$ implies fairly large $\Omega_{\mathrm{B}}$. Therefore the exponential term in $\Gamma$ becomes important, allowing $\Omega_{\mathrm{M}} = 1$ to satisfy the $\Gamma$ constraint. Values of $\eta_{10} \approx 8-15$ are required (Fig. \[Fig:H-eta\_OM1\]). The best-fit SCDM model has $h \approx 0.43$ and $\eta_{10} \approx 13$, which is grossly inconsistent with the predictions of BBN and the observed abundances of D, $^4$He, and $^7$Li. For $h > 0.5$ a fit to SCDM is no longer feasible (Fig. \[Fig:H-eta\_OM1\]). The SCDM model is severely challenged. The $\Gamma$ and age constraints also challenge low-density CDM models. The $\Gamma$ constraint permits $\Omega_{\mathrm{M}} < 0.4$ only for high $h$, while the age constraint forbids high $h$, so $\Omega_{\mathrm{M}} \gtrsim 0.4$ is required. Values $\eta_{10} \gtrsim 6$ are favored strongly over $\eta_{10} \lesssim 2$. The bound $\Omega_{\mathrm{M}} \gtrsim 0.4$ conflicts with the added cluster constraint $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$ at the 98% CL, suggesting strongly that there is additional mass not traced by light. Although a few plausible variations on the CDM models do not affect the constraints very much (Figs. \[Fig:H-Omega\_M\_L0\_var\] – \[Fig:H-eta\_L0\_G+-0.05\]), removing the $\Gamma$ constraint would have a dramatic effect. Both high and low values of $\Omega_{\mathrm{M}}$ would then be permitted. Adopting a smaller observed $\Gamma_{\mathrm{o}} \approx 0.15$ from the IRAS redshift survey also makes a difference. Values $\Omega_{\mathrm{M}} \approx 0.3$ and $\eta_{10} \approx 5$ are then favored, but even $\eta_{10} \approx 2$ is not excluded. At either low or high density, the situation remains about the same for the $\Lambda$CDM models (Figs. \[Fig:H-Omega\_M\_k0\] & \[Fig:H-eta\_k0\]). Because the ages are longer, we can tolerate $\Omega_{\mathrm{M}} \approx 0.3$ for $h = 0.85$. The $\Lambda$CDM model therefore accepts (barely) the added constraint $\Omega_{\mathrm{o}} = 0.2 \pm 0.1$ at the 7% CL, even with the larger $\Gamma_{\mathrm{o}} \approx 0.255$. Improved future constraints on $\Omega_{\Lambda}$ will come into play here. Having bounded the baryon density using data independent of constraints from BBN, we may explore the consequences for the light element abundances. In general, our fits favor large values of $\eta_{10}$ ($\gtrsim 5$) over small values ($\lesssim 2$). While such large values of the baryon density are consistent with estimates from the Ly-$\alpha$ forest, they may create some tension for BBN. For deuterium there is no problem, since for $\eta_{10} \gtrsim 5$ the BBN-predicted abundance, (D/H)$_{\mathrm{P}} \lesssim 4 \times 10^{-5}$ (2$\sigma$), is entirely consistent with the low abundance inferred for some of the observed QSO absorbers (Tytler et al. 1996; Burles & Tytler 1996; Burles & Tytler 1997a,b,c). Similarly, the BBN-predicted lithium abundance, (Li/H)$_{\mathrm{P}} \gtrsim 1.7 \times 10^{-10}$, is consistent with the observed surface lithium abundances in the old, metal-poor stars (including, perhaps, some minimal destruction or dilution of the prestellar lithium). However, the real challenge comes from $^4$He where the BBN prediction for $\eta_{10} \gtrsim 5$, Y$_{\mathrm{P}} \gtrsim 0.246$ (2$\sigma$), is to be contrasted with the region data which suggest Y$_{\mathrm{P}} \lesssim 0.238$ (OS, OSS). We wish to thank Neta Bahcall, Rupert Croft, Eli Dwek, Gus Evrard, Brian Fields, Andrew Gould, David Graff, Craig Hogan, John Huchra, Garth Illingworth, Sasha Kashlinsky, Chris Kochanek, Paul Langacker, Andrew Liddle, Rich Mushotzky, John Peacock, Martin Rees, Caleb Scharf, Allen Sweigart, and David Weinberg for helpful advice. J.E.F. and G.S. worked on this paper during several workshops at the Aspen Center for Physics. The research of G.S. at Ohio State is supported by DOE grant DE-FG02-91ER-40690. N.H. is supported by the National Science Foundation Contract No. NSF PHY-9513835. While preparing the final version of our manuscript, we saw the paper by Lineweaver and Barbosa (1998), which has some overlap with our work. Their analysis in §4 has some similarities to our calculations, but there are some important differences. For example, they have only two free parameters since $\eta_{10}$ is incorporated by a questionable procedure relying on BBN and is not free, and they use a different theoretical expression and different error bars for the shape parameter $\Gamma$. Their §§1-3 are of more interest, since they apply entirely different constraints from the CMB angular power spectrum. It is gratifying that their resulting CRs, based on independent data, are rather similar to ours (e.g., compare their Figure \[Fig:eta-P\_L0\] with our Figure \[Fig:H-Omega\_M\_L0\_G+-0.05\]). Bahcall, N. A., Lubin, L. 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--- author: - | \ ITEP, B.Cheremushkinskaya 25, Moscow, 117259, Russia\ E-mail: - | A.I.Veselov\ ITEP, B.Cheremushkinskaya 25, Moscow, 117259, Russia\ E-mail: title: Upper bound on the cutoff in the Standard Model --- Introduction ============ According to the conventional point of view the upper bound $\Lambda$ on the cutoff in the Electroweak theory (without fermions) depends on the Higgs mass. It is decreased when the Higgs mass is increased. And at the Higgs mass around $1$ Tev $\Lambda$ becomes of the order of $M_H$. At the same time for $M_H \sim 200$ Gev the value of $\Lambda$ can be made almost infinite[^1]. This conclusion is made basing on the perturbation expansion around trivial vacuum. In our presentation we demonstrate that the vacuum of the lattice Weinberg - Salam model is rather complicated, which means that the application of the perturbation expansion around trivial vacuum may be limited. Namely, we investigate the behavior of the topological defects composed of the lattice gauge fields that are to be identified with quantum Nambu monopoles [@Nambu; @BVZ; @Chernodub_Nambu]. We show that their lattice density increases along the lines of constant physics when the ultraviolet cutoff in increased. At sufficiently large values of the cutoff these objects begin to dominate. Moving further along the line of constant physics we reach the point on the phase diagram where the monopole worldlines begin to percolate. This point roughly coincides with the position of the transition between the physical Higgs phase and the unphysical symmetric phase of the lattice model. At infinite bare scalar self coupling $\lambda$ the transition is a crossover and the ultraviolet cutoff achieves its maximal value around $1.4$ Tev at the transition point. At smaller bare values of $\lambda$ correspondent to small Higgs masses the phase transition becomes stronger. Still we do not know the order of the phase transition at small values of $\lambda$. We have estimated the maximal value of the cutoff in the vicinity of the transition point at $\lambda = 0.009$. The obtained value of the cutoff appears to be around $1.4$ Tev. The lattice model under investigation ===================================== The lattice Weinberg - Salam Model without fermions contains gauge field ${\cal U} = (U, \theta)$ (where $ \quad U \in SU(2), \quad e^{i\theta} \in U(1)$ are realized as link variables), and the scalar doublet $ \Phi_{\alpha}, \;(\alpha = 1,2)$ defined on sites. The action is taken in the form $$\begin{aligned} S & = & \beta \!\! \sum_{\rm plaquettes}\!\! ((1-\mbox{${\small \frac{1}{2}}$} \, {\rm Tr}\, U_p ) + \frac{1}{{\rm tg}^2 \theta_W} (1-\cos \theta_p))+\nonumber\\ && - \gamma \sum_{xy} Re(\Phi^+U_{xy} e^{i\theta_{xy}}\Phi) + \sum_x (|\Phi_x|^2 + \lambda(|\Phi_x|^2-1)^2), \label{S}\end{aligned}$$ where the plaquette variables are defined as $U_p = U_{xy} U_{yz} U_{wz}^* U_{xw}^*$, and $\theta_p = \theta_{xy} + \theta_{yz} - \theta_{wz} - \theta_{xw}$ for the plaquette composed of the vertices $x,y,z,w$. Here $\lambda$ is the scalar self coupling, and $\gamma = 2\kappa$, where $\kappa$ corresponds to the constant used in the investigations of the $SU(2)$ gauge Higgs model. $\theta_W$ is the Weinberg angle. Bare fine structure constant $\alpha$ is expressed through $\beta$ and $\theta_W$ as $\alpha = \frac{{\rm tg}^2 \theta_W}{\pi \beta(1+{\rm tg}^2 \theta_W)}$. In our investigation we fix bare Weinberg angle equal to $30^o$. The renormalized fine structure constant can be extracted through the potential for the infinitely heavy external charged particles. Phase diagram ============= The phase diagram at infinite $\lambda$ is represented on Fig.1. The dashed vertical line represents the confinement-deconfinement phase transition corresponding to the $U(1)$ constituent of the model. The continuous horizontal line corresponds to the transition between the broken and the symmetric phases. Real physics is commonly believed to be achieved within the phase of the model situated in the right upper corner of Fig. $1$. The double-dotted-dashed vertical line on the right-hand side of the diagram represents the line, where the renormalized $\alpha$ is constant and is equal to $1/128$. Qualitatively the phase diagram at finite $\lambda$ looks similar to that of infinite $\lambda$. In the three - dimensional ($\beta, \gamma, \lambda$) phase diagram the transition surfaces are two - dimensional. The lines of constant physics on the tree level are the lines ($\frac{\lambda}{\gamma^2} = \frac{1}{8 \beta} \frac{M^2_H}{M^2_W} = {\rm const}$; $\beta = \frac{1}{4\pi \alpha}={\rm const}$). In general the cutoff is increased along the line of constant physics when $\gamma$ is decreased. The maximal value of the cutoff is achieved at the transition point. Nambu monopole density in lattice units is also increased when the ultraviolet cutoff is increased. At $\beta = 12$ the phase diagram is represented on Fig. 2. The physical Higgs phase is situated up to the transition line. The position of the transition is localized at the point where the susceptibility extracted from the Higgs field creation operator achieves its maximum. All simulations were performed on lattices of sizes $8^3\times 16$. Several points were checked using larger lattices up to $16^3\times 24$. At $\lambda = \infty$ we found no significant difference between the results obtained using the mentioned lattices. For small $\lambda$ the careful investigation of the dependence of physical observables on the lattice size has not been performed. Calculation of the cutoff ========================= The following variable is considered as creating the $Z$ boson: $ Z_{xy} = Z^{\mu}_{x} \; = {\rm sin} \,[{\rm Arg} (\Phi_x^+U_{xy} e^{i\theta_{xy}}\Phi_y) ]$. In order to evaluate the masses of the $Z$-boson and the Higgs boson we use the correlators: $$\frac{1}{N^6} \sum_{\bar{x},\bar{y}} \langle \sum_{\mu} Z^{\mu}_{x} Z^{\mu}_{y} \rangle \sim e^{-M_{Z}|x_0-y_0|}+ e^{-M_{Z}(L - |x_0-y_0|)} \label{corZ}$$ and $$\frac{1}{N^6}\sum_{\bar{x},\bar{y}}(\langle H_{x} H_{y}\rangle - \langle H\rangle^2) \sim e^{-M_{H}|x_0-y_0|}+ e^{-M_{H}(L - |x_0-y_0|)}, \label{cor}$$ Here the summation $\sum_{\bar{x},\bar{y}}$ is over the three “space" components of the four - vectors $x$ and $y$ while $x_0, y_0$ denote their “time“ components. $N$ is the lattice length in ”space“ direction. $L$ is the lattice length in the ”time" direction. In lattice calculations we used two different operators that create Higgs bosons: $ H_x = |\Phi|$ and $H_x = \sum_{y} Z^2_{xy}$. In both cases $H_x$ is defined at the site $x$, the sum $\sum_y$ is over its neighboring sites $y$. After fixing the unitary gauge, lattice Electroweak theory becomes a lattice $U(1)$ gauge theory. The $U(1)$ gauge field is $ A_{xy} = A^{\mu}_{x} \; = \,[-{\rm Arg} (\Phi_x^+U_{xy} e^{i\theta_{xy}}\Phi_y) + 2\theta_{xy}] \,{\rm mod} \,2\pi$. The usual Electromagnetic field is $ A_{\rm EM} = A + Z^{\prime} - 2 \,{\rm sin}^2\, \theta_W Z^{\prime}$, where $Z^{\prime} = [ {\rm Arg} (\Phi_x^+U_{xy} e^{i\theta_{xy}}\Phi_y) ]{\rm mod} 2\pi$. The physical scale is given in our lattice theory by the value of the $Z$-boson mass $M^{phys}_Z \sim 91$ GeV. Therefore the lattice spacing is evaluated to be $a \sim [91 {\rm GeV}]^{-1} M_Z$, where $M_Z$ is the $Z$ boson mass in lattice units. At infinite $\lambda$ the real continuum physics should be approached along the the line of constant $\alpha_R = \frac{1}{128}$. The ultraviolet cutoff is $\Lambda = \frac{\pi}{a} = (\pi \times 91~{\rm GeV})/M_Z$. $\Lambda$ is increased slowly along this line with decreasing $\gamma$ and achieves the value around $1.35$ TeV at the transition point between the physical Higgs phase and the symmetric phase. According to our results this value does not depend on the lattice size. In the region of the phase diagram represented on Fig.2 the situation is similar. Our data obtained on the lattice $8^3\times16$ shows that $\Lambda$ is increased slowly with the decrease of $\gamma$ at any fixed $\lambda$. We investigated carefully the vicinity of the transition point at fixed $\lambda = 0.009$ and $\beta = 12$. It has been found that at the transition point the value of $\Lambda$ is equal to $1.4 \pm 0.2$ Tev. The first check of a larger lattice (of size $12^3\times 16$) does not show an increase of this value. However, the careful investigation of the dependence of $\Lambda$ on the lattice size (as well as on $\lambda$) is to be the subject of future investigations. On Fig. 3 the dependence of $M_Z$ in lattice units on $\gamma$ is represented at $\lambda =0.009$ and $\beta = 12$, where $\gamma_c = 0.273 \pm 0.002$. . Unfortunately, we cannot yet estimate the renormalized Higgs boson mass due to the lack of statistics. However, we expect it does not deviate significantly from the tree level estimate $M^0_H = \frac{\sqrt{8\beta \lambda}}{\gamma} \times 80$ Gev. In the vicinity of the phase transition at $\lambda =0.009$, $\beta = 12$ bare value of the Higgs mass is $M^0_H \sim 270$ Gev. The renormalized coupling ========================= The bare constant $\alpha = e^2/4\pi$ (where $e$ is the electric charge) can be easily calculated in our lattice model. It is found to be equal to $1/(4\pi \beta)$. Therefore, its physical value $\alpha(M_Z)\sim 1/128$ could be achieved at the values of $\beta$ in some vicinity of $10$. This naive guess is, however, to be corrected by the calculation of the renormalized coupling constant $\alpha_R$. We perform this calculation using the potential for infinitely heavy external fermions. We consider Wilson loops for the right-handed external leptons: $ {\cal W}^{\rm R}_{\rm lept}(l) = \langle {\rm Re} \,\Pi_{(xy) \in l} e^{2i\theta_{xy}}\rangle. $ Here $l$ denotes a closed contour on the lattice. We consider the following quantity constructed from the rectangular Wilson loop of size $r\times t$: $ {\cal V}(r) = \lim_{t \rightarrow \infty}{ \rm log} \frac{ {\cal W}(r\times t)}{{\cal W}(r\times (t+1))}. $ At large enough distances we expect the appearance of the Coulomb interaction $ {\cal V}(r) = -\frac{\alpha_R}{r} + const. $ The renormalized coupling constant $\alpha$ is found to be close to the realistic value $\alpha(M_Z)=1/128$ along the line represented in Fig. $1$ (at $\lambda \rightarrow \infty$) in the vicinity of $\beta = 15$. We do not observe any dependence of $\alpha_R$ on the lattice size at $\lambda = \infty$. At $\lambda = 0.009$, $\beta = 12$, $ \gamma = \gamma_c(\lambda)\sim 0.273$ the renormalized fine structure constant calculated on the lattice $8^3\times 16$ is $\alpha_R = \frac{1}{98\pm 3}$. The same value has been obtained also on the larger lattice ($12^3\times 16$), which shows that the value of $\alpha_R$ does not depend on the lattice size also for the small values of $\lambda$. Nambu monopole density ======================= According to [@BVZ; @Chernodub_Nambu; @VZ2008] the worldlines of the quantum Nambu monopoles can be extracted from the field configurations as follows: $$j_A = \frac{1}{2\pi} {}^*d([d A]{\rm mod}2\pi)$$ (The notations of differential forms on the lattice [@forms] are used here.) The monopole density is defined as $ \rho = \left\langle \frac{\sum_{\rm links}|j_{\rm link}|}{4L^4} \right\rangle, $ where $L$ is the lattice size. In Fig. $4$ we represent Nambu monopole density as a function of $\gamma$ at $\lambda = 0.009$, $\beta = 12$. The point of the transition is localized as the position of the maximum of the susceptibility $\chi = \langle H^2 \rangle - \langle H\rangle^2$ extracted from $H = \sum_{y} Z^2_{xy}$. The value of monopole density at $\gamma_c = 0.273$, $\beta = 12$, $\lambda = 0.009$ is around $0.1$. At this point the value of the cutoff is $\Lambda \sim 1.4 \pm 0.2$ Tev. The monopole density around $0.1$ means that among $10$ sites there exist $4$ sites that are occupied by the monopole. Average distance between the two monopoles is, therefore, less than $1$ lattice spacing and it is not possible at all to speak of the given configurations as of representing the single Nambu monopole. That’s why these complicated configurations constructed of the gauge field and the scalar field dominate in vacuum in the vicinity of the transition point. This means that the usual perturbation expansion around trivial vacuum (gauge field equal to zero) may not be valid in a vicinity of the phase transition between the physical Higgs phase and the unphysical symmetric phase of the model. This might explain why we do not observe in our numerical simulations the large values of $\Lambda$ predicted by the conventional perturbation theory. Conclusions =========== In Table $1$ we list the values of the lattice spacing used in selected lattice studies of $SU(2)$ Gauge - Higgs Model. From this table it is clear that the correspondent value of the cutoff $\frac{\pi}{a}$ does not exceed $1.5$ Tev. \[tab.01\] [**Reference**]{} [**inverse lattice spacing**]{} $\frac{1}{a}$ (GeV) [**$M_H$**]{} (GeV) ------------------- ----------------------------------------------------- ------------------------- [@1] 140 (space direction) 570 (time direction) 80 [@2] 280 (time direction) 80 [@3] 280 34 [@4] 110 16 [@5] 90 (space direction) 350 (time direction) 34 [@6] 280 48 [@7] 140 35 [@8] 280 20 , 50 [@9] 190 50 [@10] 260 57 - 85 [@11] 200 - 300 47 - 108 [@12] 400 480 [@13] 330 - 470 280 - 720 [@14] 250 - 470 720 ($\lambda =\infty$) Our own numerical data demonstrate that the vacuum structure of the lattice Weinberg - Salam model is rather complicated. Namely, the topological defects identified with quantum Nambu monopoles dominate in vacuum in the vicinity of the phase transition between the symmetric phase and the Higgs phase. This indicates that the usual perturbation expansion around trivial vacuum may not be applied in this region of the phase diagram. As a consequence one cannot apply the conventional perturbation theory to the evaluation of the Ultraviolet cutoff upper bound in this region of the phase diagram. Qualitatively this situation seem to us similar to that of the Ginzburg - Landau theory of superconductivity. Within this theory in a certain vicinity of the phase transition the fluctuations of the order parameter become so strong that the perturbation expansion around the trivial solution of Ginzburg - Landau equations cannot be applied. Thus we conclude that the upper bound on the Ultraviolet cutoff in the lattice Electroweak theory is still not known. The establishing of this upper bound is to be a subject of future investigations. We suppose that the upper bound on the cutoff obtained with the aid of nonperturbative lattice methods may differ from the conventional one obtained via the perturbation expansion around trivial vacuum. This work was partly supported by RFBR grants 09-02-08308, 09-02-00338, 08-02-00661, and 07-02-00237, by Grant for leading scientific schools 679.2008.2,by Federal Program of the Russian Ministry of Industry, Science and Technology No 40.052.1.1.1112. The numerical simulations have been performed using the facilities of Moscow Joint Supercomputer Center. [99]{} Y. Nambu, Nucl.Phys. B [**130**]{}, 505 (1977);\ Ana Achucarro and Tanmay Vachaspati, Phys. Rept. [**327**]{}, 347 (2000); Phys. Rept. [**327**]{}, 427 (2000). B.L.G. Bakker, A.I. Veselov, and M.A. Zubkov, Phys. Lett. B [**583**]{}, 379 (2004); Yad. Fiz. [**68**]{}, 1045 (2005); Phys. Lett. B [**620**]{}, 156 (2005); Phys. Lett. B [**642**]{}, 147 (2006); J. Phys. G: Nucl. Part. Phys. 36 (2009) 075008; arXiv:0708.2864, PoSLAT2007:337,2007 M.N. Chernodub, JETP Lett. [**66**]{}, 605 (1997) A.I. Veselov, and M.A. Zubkov, JHEP 0812:109 (2008) ; M.I. Polikarpov, U.J. Wiese, and M.A. Zubkov, Phys. Lett. B [**309**]{}, 133 (1993). F. Csikor, Z. Fodor, J. Heitger Phys.Rev.Lett. 82 (1999) 21-24 Phys.Rev. D58 (1998) 094504 Nucl.Phys.Proc.Suppl. 63 (1998) 569-571 F. Csikor, Z. Fodor, J. Heitger Phys.Lett. B441 (1998) 354-362 F. Csikor, Z. Fodor, J. Hein, A. Jaster, I. Montvay Nucl.Phys. B474 (1996) 421-445 Joachim Hein (DESY), Jochen Heitger, Phys.Lett. B385 (1996) 242-248 F. Csikor, Z. Fodor, J. Hein, J. Heitger, A. Jaster, I. Montvay Nucl.Phys.Proc.Suppl. 53 (1997) 612-614 Z. Fodor, J. Hein, K. Jansen, A. Jaster, I. Montvay Nucl.Phys. B439 (1995) 147-186 F. Csikor, Z. Fodor, J. Hein, J. Heitger, Phys.Lett. B357 (1995) 156-162 F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Nucl.Phys.Proc.Suppl. 42 (1995) 569-574 F. Csikor, Z. Fodor, J. Hein, K.Jansen, A. Jaster, I. Montvay Phys.Lett. B334 (1994) 405-411 Y. Aoki, F. Csikor, Z. Fodor, A. Ukawa Phys.Rev. D60 (1999) 013001 Nucl.Phys.Proc.Suppl. 73 (1999) 656-658 Y. Aoki Phys.Rev. D56 (1997) 3860-3865 W.Langguth, I.Montvay, P.Weisz Nucl.Phys.B277:11,1986. W. Langguth, I. Montvay (DESY) Z.Phys.C36:725,1987 Anna Hasenfratz, Thomas Neuhaus, Nucl.Phys.B297:205,1988 [^1]: Here we do not consider vacuum stability bound on the Higgs mass related to the fermion loops.

--- abstract: 'We experimentally study the propagation of circularly polarized light in the sub-diffusion regime by exploiting enhanced backscattering (EBS, also known as coherent backscattering) of light under low spatial coherence illumination. We demonstrate for the first time that circular polarization memory effect exists in EBS over a large range of scatterers’ sizes in this regime. We show that EBS measurements under low spatial coherence illumination from the helicity preserving and orthogonal helicity channels cross over as the mean free pathlength of light in media varies, and that the cross point indicates the transition from multiple to double scattering in EBS of light.' author: - 'Young L. Kim' - Prabhakar Pradhan - 'Min H. Kim' - Vadim Backman bibliography: - 'Cir\_pol\_memo.bib' title: | Circular polarization memory effect in enhanced backscattering of light\ under partially coherent illumination --- The circular polarization memory effect is an unexpected preservation of the initial helicity (or handedness) of circular polarization of multiply scattered light in scattering media consisting of large particles. Mackintosh *et al*. \[1\] first observed that the randomization of the helicity required unexpectedly far more scattering events than did the randomization of its propagation in media of large scatterers. Bicout *et al*. \[2\] demonstrated that the memory effect can be shown by measuring the degree of circular polarization of transmitted light in slabs. Using numerical simulations of vector radiative transport equations, Kim and Moscoso \[3\] explained the effect as the result of successive near-forward scattering events in large scatterers. Recently, Xu and Alfano \[4\] derived a characteristic length of the helicity loss in the diffuse regime and showed that this characteristic length was greater than the transport mean free pathlength $l_s^*$ for the scatterers of large sizes. Indeed, the propagation of circularly polarized light in random media has been investigated mainly using either numerical simulations or experiments in the diffusion regime, in part because its experimental investigation in the sub-diffusion regime has been extremely challenging. Therefore, the experimental investigation of circularly polarized light in the low-order scattering (or short traveling photons) regime using enhanced backscattering (EBS, also known as coherent backscattering) of light under low spatial coherence illumination will provide a better understanding of its mechanisms and the polarization properties of EBS as well. EBS is a self-interference effect in elastic light scattering, which gives rise to an enhanced scattered intensity in the backward direction. In our previous publications, \[5-8\] we demonstrated that low spatial coherence illumination (the spatial coherence length of illumination $L_{sc}\!<<l_s^*$) dephases the time-reversed partial waves outside its finite coherence area, rejecting long traveling waves in weakly scattering media. EBS under low spatial coherence illumination ($L_{sc}\!<<l_s^*$) is henceforth referred to as low-coherence EBS (LEBS). The angular profile of LEBS, $I_{LEBS}(\theta)$, can be expressed as an integral transform of the radial probability distribution $P(r)$ of the conjugated time-reversed light paths:\[6-8\] $$I_{LEBS}(\theta)\propto \int^\infty_0 C(r)rP(r)\exp(i2\pi r \theta / \lambda)dr,$$ where $r$ is the radial distance from the first to the last points on a time-reversed light path and $C(r) =|2J_1(r/L_{sc})/(r/L_{sc})|$ is the degree of spatial coherence of illumination with the first order Bessel function $J_1$.\[9\] As $C(r)$ is a decay function of $r$, it acts as a spatial filter, allowing only photons emerging within its coherence areas ($\sim L_{sc}^2$ ) to contribute to $P(r)$. Therefore, LEBS provides the information about $P(r)$ for a small $r$ ($<\sim100~\mu m$) that is on the order of $L_{sc}$ as a tool for the investigation of light propagation in the sub-diffusion regime.  To investigate the helicity preservation of circularly polarized light in the sub-diffusion regime by exploiting LEBS, we used the experimental setup described in detail elsewhere.\[5,6\] In brief, a beam of broadband cw light from a 100 W xenon lamp (Spectra-Physics Oriel) was collimated using a 4-$f$ lens system, polarized, and delivered onto a sample with the illumination diameter of $3~mm$. By changing the size of the aperture in the 4-$f$ lens system, we varied spatial coherence length $L_{sc}$ of the incident light from $35~\mu m$ to $200~\mu m$. The temporal coherence length of illumination was $0.7~\mu m$ with the central wavelength = $520~nm$ and its FWHM = $135~nm$. The circular polarization of LEBS signals was analyzed by means of an achromatic quarter-wavelet plate (Karl Lambrecht) positioned between the beam splitter and the sample. The light backscattered by the sample was collected by a sequence of a lens, a linear analyzer (Lambda Research Optics), and a CCD camera (Princeton Instruments). We collected LEBS signals from two different circular polarization channels: the helicity preserving (h$||$h) channel and the orthogonal helicity (h$\bot$h) channel. In the (h$||$h) channel, the helicity of the detected circular polarization was the same as that of the incident circular polarization. In the (h$\bot$h) channel, the helicity of the detected circular polarization was orthogonal to that of the incident circular polarization. In our experiments, we used media consisting of aqueous suspensions of polystyrene microspheres ($n_{sphere} = 1.599$ and $n_{water} = 1.335$ at $520~nm$) (Duke Scientific) of various radii $a$ = 0.05, 0.10, 0.15, 0.25, and 0.45 $\mu m$ (the size parameter $ka = 0.8 - 7.2$ and the anisotropic factor $g = 0.11- 0.92$). The dimension of the samples was $\pi \times 252~mm^2 \times 50~mm$. Using Mie theory,\[10\] we calculated the optical properties of the samples such as the scattering mean free pathlength of light in the medium $l_{s}$ ($= 1/\mu_s$, where $\mu_s$ is the scattering coefficient), the anisotropy factor $g$ (= the average cosine of the phase function), and the transport mean free pathlength $l_{s}^*$ ($= 1/\mu_s^* = l_{s}/(1 - g)$, where $\mu_s^*$ is the reduced scattering coefficient). We also varied $L_{sc}$ from 40 to 110 $\mu m$. We used $g$ as a metric of the tendency of light to be scattered in the forward direction.  The total experimental backscattered intensity $I_{T}$ can be expressed as $I_T = I_{SS} + I_{MS} + I_{EBS}$, where $I_{SS}$, $I_{MS}$, and $I_{EBS}$ are the contributions from single scattering, multiple scattering, and interference from the time-reserved waves (i.e., EBS), respectively. In media of relatively small particles (radius, $a\leq\lambda$), the angular dependence of $I_T(\theta)$ around the backward direction is primarily due to the interference term, while the multiple and single scattering terms have weaker angular dependence.Thus, $I_{SS} + I_{MS}$ ($=$ the baseline intensity) can be measured at large backscattering angles ($\theta > 3^{\circ}$). Conventionally, the enhancement factor $E = 1 + I_{EBS}(\theta=0^{\circ})/(I_{SS}+I_{MS})$ is commonly used. However, in the studies of circularly polarized light, the enhancement factor should be modified, because the intensity of multiple scattering can be different in the two different channels and because in the (h$||$h) channel, single scattering is suppressed due to the helicity flip. Thus, in our studies, we calculated $I_{EBS}$ by subtracting $I_{SS} + I_{MS}$ from $I_T$. Figure 1 shows representative LEBS intensity profiles $I_{LEBS}(\theta)$ from the suspension of the microspheres with $a$ = 0.15 $\mu m$ ($ka = 2.4$ and $g = 0.73$ at $\lambda = 520~nm$). $I_{LEBS}^{||}$ and $I_{LEBS}^{\bot}$ denote from the (h$||$h) and (h$\bot$h) channels, respectively. We varied $l_{s}^*$ from 67 to 1056 $\mu m$ ($l_s$ from 18 to 285 $\mu m$) with $L_{sc} = 110~ \mu m$. In Fig. 2(a), we plot as a function of $l_s^*$ (the lines are third-degree polynomial fitting), showing two characteristic regimes: (i) the multiply scattering regime ($L_{sc} \gg l_s^*$) and (ii) the minimally scattering regime ($L_{sc} \ll l_s^*$). As expected, in the multiply scattering regime (i), $I_{LEBS}^{||}$ is higher than $I_{LEBS}^{\bot}$ because of the reciprocity principle in the (h$||$h) channel. On the other hand, in the minimal scattering regime (ii), a priori surprisingly, $I_{LEBS}^{||}$ is lower than $I_{LEBS}^{\bot}$ . This is because in this regime, LEBS originates mainly from the time-reversed paths of the minimal number of scattering events in EBS (i.e., mainly double scattering) in a narrow elongated coherence volume.\[8\] In this case, the direction of light scattered by one of the scatterers should be close to the forward direction, while the direction of the light scattered by the other scatterer should be close to the backscattering that flips the helicity of circular polarization. After the cross point, the difference between and remains nearly constant, indicating that LEBS reaches to the asymptotic regime of double scattering. More importantly, Fig. 2(a) shows that and $I_{LEBS}^{||}$ and $I_{LEBS}^{\bot}$ cross over at $l_s^* = 408 \mu m$ ($l_s = 110 \mu m$). The cross point can be understood in the context of the circular polarization memory effect as follows. As shown in the inset of Fig. 2(b), at the cross point,$\int^\infty_0C(r)P^{||}(r)=\int^\infty_0C(r)P^{\bot}(r)$, where $P^{||}(r)$ and $P^{\bot}(r)$ are the radial intensity distributions of the (h${||}$h) and (h${\bot}$h) channels, respectively. Thus, the cross point $R_i$ determines the optical properties ($l_s^*$ or $l_s$) such that $\int^{\sim L_{sc}}_0P^{||}(r)=\int^{\sim L_{sc}}_0P^{\bot}(r)$. In other words, $R_i$ defines $l_s^*$ or $l_s$ such that and are equal within $L_{sc}$ and thus, the degree of circular polarization within $L_{sc}$ becomes zero as well. As shown in Fig. 2(b), the $C(r)rP(r)$, which can be obtained by the inverse Fourier transform of $I_{LEBS}(\theta)$ using Eq. (1), reveals more detailed information about the helicity preservation. For small $r$, $P^{||}(r) < P^{\bot}(r)$. For $r > \sim50~\mu m$, ($\sim l_s/2$), $P^{||}(r) > P^{\bot}(r)$, showing that the initial helicity is preserved. This is because the successive scattering events of the highly forward scatterers direct photons away from the incident point of illumination, while maintaining the initial helicity.  As discussed above, the cross point $R_i$ is determined by both the spatial coherence length of illumination $L_{sc}$ and the optical properties of the media. Thus, we investigated the relationship between $L_{sc}$ and $R_i$ using the fixed scatterer size with $a = 0.25 \mu m$ ($ka = 4.0$, and $g = 0.86$). Fig. 3(a) shows that $R_i$ (in the units of $l_s^*$) is linearly proportional to $L_{sc}$ and that small reduced scattering coefficients $\mu_s^*$ ($= 1/l_s^*$) are necessary to reach a cross point as $L_{sc}$ increases. Because the linear fitting line passes through the origin (the 95% confidence interval of the intercept of the $L_{sc}$ axis is \[$-32~\mu m$, $44~\mu m$\]), $R_i$ can be normalized by $L_{sc}$. Next, in order to elucidate how the tendency of the propagation direction (i.e., $g$) plays a role in the memory effect, we further studied the effect of $g$ on $R_i$ using the various size parameters $ka$ ranging from 0.8 to 7.2 ($g = 0.11 - 0.92$) with the fixed $L_{sc} = 110~\mu m$. In Fig. 3(b), we plot $R_i$ (in the units of $l_s^*$ ) versus $g$. This shows $R_i$ increases dramatically as $g$ increases, which is in good agreement with the conventional notion that a small $\mu_s*$ is required for the memory effect to occur in media of larger particles because of the stronger memory effect in media of larger scatterers. When we plot $R_i$ in the units of $l_s$ versus $g$, as shown in Fig. 3(c), on the other hand, $R_i$ does not depend strongly on $g$. This result shows that when $l_s$ is on the order of $L_{sc}$, the helicity of circular polarization is maintained over a large range of the size parameters. Moreover, Fig. 3(c) demonstrates that the average distance of single scattering events (i.e., $l_s$) is a main characteristic length scale that plays major roles in the memory effect in the sub-diffusion regime. In summary, we experimentally investigated for the first time the circular polarization memory effect in the sub-diffusion regime by taking advantage of LEBS, which suppresses time-reserved waves beyond the spatial coherence area; and thus isolates low-order scattering in weakly scattering media. We reported that LEBS introduces the new length scale (i.e., cross point) at which the degree of circular polarization becomes zero; and the scale is determined by both the spatial coherence length of illumination and the optical properties of the media. Using the cross point of the LEBS measurements from the (h$||$h) and (h$\bot$h) channels, we further elucidated the memory effect in the sub-diffusion regime. Our results demonstrate that the memory effect exists in the EBS phenomenon. Furthermore, we show that the cross point is the transition point from multiple scattering to double scattering events this regime. Finally, our results will further facilitate the understanding of the propagation of circularly polarized light in weakly scattering media such as biological tissue. ———————————————–\ 1. F. C. Mackintosh, J. X. Zhu, D. J. Pine, and D. A. Weitz, “Polarization memory of multiply scattered light,” Phys. Rev. B 40, 9342 (1989).\ 2. D. Bicout, C. Brosseau, A. S. Martinez, and J. M. Schmitt, “Depolarization of Multiply Scattered Waves by Spherical Diffusers - Influence of the Size Parameter,” Phys. Rev. E 49, 1767 (1994).\ 3. A. D. Kim and M. Moscoso, “Backscattering of circularly polarized pulses,” Opt. Lett. 27, 1589 (2002).\ 4. M. Xu and R. R. Alfano, “Circular polarization memory of light,” Phys. Rev. E 72, 065601(R) (2005).\ 5. Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, “Coherent Backscattering Spectroscopy,” Opt. Lett. 29, 1906 (2004).\ 6. Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, and V. Backman, “Low-coherent backscattering spectroscopy for tissue characterization,” Appl. Opt. 44, 366 (2005).\ 7. Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. K. Wali, H. K. Roy, and V. Backman, “Depth-resolved low-coherence enhanced backscattering,” Opt. Lett. 30, 741 (2005).\ 8. Y. L. Kim, P. Pradhan, H. Subramanian, Y. Liu, M. H. Kim, and V. Backman, “Origin of low-coherence enhanced backscattering,” Opt. Lett. 31, 1459 (2006).\ 9. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, Cambridge; New York, 1999).\ 10. H. C. van de Hulst, Light scattering by small particles (Dover Publications, New York, 1995).\

--- abstract: 'The form of the inflationary potential is severely restricted if one requires that it be natural in the technical sense, i.e. terms of unrelated origin are not required to be correlated. We determine the constraints on observables that are implied in such natural inflationary models, in particular on $r$, the ratio of tensor to scalar perturbations. We find that the naturalness constraint does not require $r$ to be lare enough to be detectable by the forthcoming searches for B-mode polarisation in CMB maps. We show also that the value of $r$ is a sensitive discriminator between inflationary models.' author: - | Shaun Hotchkiss, Gabriel Germán,[^1]  Graham G Ross[^2]  and Subir Sarkar\ *Rudolf Peierls Centre for Theoretical Physics,*\ *University of Oxford, 1 Keble Road, Oxford, OX1 3NP, UK* title: | \ Fine tuning and the ratio of tensor to scalar density fluctuations from cosmological inflation --- The nature of the density perturbations originating in the early universe has been of great interest both observationally and theoretically. The hypothesis that they were generated during an early period of inflationary expansion has been shown to be consistent with all present observations. The most discussed mechanism for inflation is the ‘slow roll’ of a weakly coupled ‘inflaton’ field down its potential — the near-constant vacuum energy of the system during the slow-roll evolution drives a period of exponentially fast expansion and the density perturbations have their origin as quantum fluctuations in the inflaton energy density. In such models the detailed structure of the density perturbations which give rise to the large scale structure of the universe observed today depends on the nature of the inflationary potential in the field region where they were generated. Boyle, Steinhardt and Turok [@Boyle:2005ug] have argued that “naturalness” imposes such strong restrictions on the inflationary potential that one may derive interesting constraints on observables today. They concluded that in theories which are “natural” according to their criterion, the spectral index of the scalar density perturbations is bounded as $n_\mathrm{s}<0.98$, and that the ratio of tensor-to-scalar perturbations satisfies $r>0.01$ provided $n_\mathrm{s}>0.95$, in accord with then current measurements [@Spergel:2003cb]. Such a lower limit on the amplitude of gravitational waves is of enormous interest as there is then a realistic possibility of detecting them as ‘B-mode’ polarisation in CMB sky maps (see e.g. [@Efstathiou:2007gz]) and thus verifying a key prediction of inflation. Of course these conclusions are crucially dependent on the definition of naturalness. In this paper we re-examine this important issue and argue that the criterion proposed by Boyle [*et al*]{} does not capture the essential aspects of a [*physically*]{} natural theory. We propose an alternative criterion that correctly reflects the constraints coming from underlying symmetries of the theory and we use this to determine a new bound on $r$ that turns out to quite opposite to the previously inferred one. We emphasise that our result, although superficially similar to the ‘Lyth bound’ [@Lyth:1996im], follows in fact from different considerations and in particular makes no reference to how long inflation lasts. Inflation predicts a near scale-invariant spectrum for the scalar and tensor fluctuations, the former being in reasonable agreement with current observations. Here we explore the predictions for [ *natural*]{} models involving a single inflaton at the time the density perturbations are produced. Models with two or more scalar fields affecting the density perturbations require some measure of fine tuning to relate their contribution to the energy density, whereas the single field models avoid this unnatural aspect. In order to characterize the inflationary possibilities in a model independent way it is convenient to expand the inflationary potential about the value of the field $\phi_\mathrm{H}$ just at the start of the observable inflation era, $\sim 60$ e-folds before the end of inflation when the scalar density perturbation on the scale of our present Hubble radius [^3] was generated, and expand in the field $\phi^\ast \equiv \phi - \phi_\mathrm{H}$ [@German:2001tz]. Since the potential must be very flat to drive inflation, $\phi ^{\ast }$ will necessarily be *small* while the observable density perturbations are produced, so the Taylor expansion of the potential will be dominated by low powers of $\phi^\ast$: $$V (\phi^\ast) = V(0) + V^\prime(0) \phi^\ast + \frac{1}{2}V^{\prime\prime}(0)\phi^{\ast 2} + \ldots \label{expand}$$ The first term $V(0)$ provides the near-constant vacuum energy driving inflation while the $\phi^\ast$-dependent terms are ultimately responsible for ending inflation, driving $\phi^\ast$ large until higher-order terms violate the slow-roll conditions. These terms also determine the nature of the density perturbations produced, in particular the departure from a scale-invariant spectrum. The observable features of the primordial density fluctuations can readily be expressed in terms of the coefficients of the Taylor series [@German:2001tz]. It is customary to use these coefficients first to define the slow-roll parameters $\epsilon$ and $\eta$ [@Liddle:2000cg] which must be small during inflation: $$\epsilon \equiv \frac{M^2}{2}\left(\frac{V^\prime(0)}{V(0)}\right)^2 \ll 1, \qquad |\eta| \equiv M^2 \left\vert \frac{V^{\prime\prime}(0)}{V(0)}\right\vert \ll 1, \label{slowroll}$$ where $M$ is the reduced Planck scale, $M=2.44\times 10^{18}$ GeV. In terms of these the spectral index is given by $$n_\mathrm{s} = 1 + 2\eta - 6\epsilon, \label{spectral}$$ the tensor-to-scalar ratio is $$r = 16\epsilon, \label{indicetensorial}$$ and the density perturbation at wave number $k$ is $$\delta_\mathrm{H}^2 (k) = \frac{1}{150\pi^2}\frac{V(0)}{\epsilon M^4} . \label{densitypert}$$ Finally the ‘running’ of the spectral index is given by $$n_\mathrm{r} \equiv \frac{\mathrm{d}n_\mathrm{s}}{\mathrm{d}\ln k} = 16\epsilon\eta - 24\epsilon^2 - 2\xi , \label{spectraltilt}$$ where $$\xi \equiv M^4 \frac{V^\prime V^{\prime\prime\prime}}{V^2}. \label{xi}$$ At this stage we have four observables, $n_\mathrm{s},$ $n_\mathrm{r}$, $\delta_\mathrm{H}$ and $r$ and four unknown parameters $V(0)$, $V^\prime(0)$, $V^{\prime\prime}(0)$ and $V^{\prime\prime\prime}(0)$ which, for an arbitrary inflation potential, are independent. However for natural potentials these parameters are related, leading to corresponding relations between the observables. Observational confirmation of such relations would provide evidence for the underlying potential, hence crucial clues to the physics behind inflation. As discussed above we are considering the class of natural models in which a single inflaton field dominates when the density perturbations relevant to the large-scale structure of the universe today are being produced.[^4] In classifying “natural” inflation, Boyle [*et al*]{} imposed a set of five conditions [@Boyle:2005ug]: 1. The energy density (scalar) perturbations generated by inflation must have amplitude $\sim 10^{-5}$ on the scales that left the horizon $\approx 60$ e-folds before the end of inflation; 2. The universe undergoes at least $N > 60$ e-folds of inflation; 3. After inflation, the field must evolve smoothly to an analytic minimum with $V = 0$; 4. If the minimum is metastable, then it must be long-lived and $V$ must be bounded from below; 5. Inflation must halt and the universe must reheat without spoiling its large-scale homogeneity and isotropy. They proposed that the level of fine-tuning for potentials satisfying the above conditions should be measured by the integers $Z_{\epsilon, \eta}$ that measure the number of zeros that $\epsilon$ and $\eta$ and their derivatives undergo within the last 60 e-folds of inflation [@Boyle:2005ug]. Here we argue that such a measure does [*not*]{} capture the essential character of physical naturalness. At a purely calculational level this is illustrated by the fact that it is necessary to impose an (arbitrary) cut-off on the number of derivatives included in the criterion.[^5] This is necessary because $\epsilon$ and $\eta$ are defined in terms of the ratio of first or second order derivatives of the potential to the potential itself, so [*all*]{} higher order derivatives must be considered separately when counting the total number of zeros. The difficulty follows from the observation that, as far as naturalness is concerned, it is the inflaton potential that is the primary object, being restricted by the underlying symmetries of the (effective) field theory describing the inflaton dynamics. As stressed by ’t Hooft [@'t; @Hooft:1979bh], a [*natural*]{} theory is one in which all terms in the Lagrangian allowed by the underlying symmetries of the theory are present, with no relations assumed between terms unrelated by the symmetries. It is important however to note that such natural potentials do [ *not*]{} preclude significant contributions from unrelated terms. Indeed such contributions are inevitable if, for example, the inflation field is moving from small to large field values. For small field values the lowest allowed power in the inflaton field is likely to be the most important, but at larger field values higher powers will ultimately dominate. For this reason the last four conditions have a different character to the first in that they involve the end of inflation when naturalness does [*not*]{} require that a single term in the inflation potential should dominate. For example in inflationary models with the inflaton rolling from small to large field values, the higher powers can cause the potential to evolve smoothly to an analytic minimum with $V=0$ or govern the properties of an unstable minimum. Similarly it may be these higher powers that cause inflation to halt and the universe to reheat without affecting the predictions for the observable density perturbation. Given the freedom there is in choosing these higher powers (non-renormalisable terms in the effective field theory description), it is [*always*]{} possible to find a model in which the end of inflation is satisfactory without violating the naturalness constraints [@German:2001tz]. On the other hand the range of $\phi^\ast$ relevant during the production of density perturbations is quite small (corresponding to only 8 of the $\sim 60$ e-folds of inflation) and so it is reasonable to suppose that unrelated terms do not simultaneously contribute significantly to the generation of the observed density perturbation. Although we have made this argument in the context of ‘new inflation’ models where the inflaton field evolves from low to high values, similar considerations apply to the other natural models. For the case where the underlying symmetry is a Goldstone symmetry it is still possible to change the end of inflation in a natural way through the effect of a second ‘hybrid’ field. Given these considerations we do not need to impose the last four conditions when determining the phenomenological implications of natural inflationary models. However we will comment on how these conditions can indeed be satisfied for the various classes of inflation potential. Our definition of naturalness is the standard one in particle physics [@'t; @Hooft:1979bh], viz. pertaining to a potential whose form is guaranteed by a symmetry. This should apply at the time the observable density perturbations are being produced. What form can such natural potentials take? The relevant symmetries that have been identified capable of restricting the scalar inflaton potential are relatively limited. The most direct are Abelian or non-Abelian symmetries, either global or gauge, and continuous or discrete. For a single field inflation model these will either limit the powers of the inflaton field that may appear in the potential or, if the inflaton is a pseudo-Goldstone mode, require a specific form for the potential. Less direct constraints occur in supersymmetric theories where the scalar inflaton field is related to a fermionic partner. In this case chiral symmetries of the associated fermion partner and $R$-symmetries may further restrict the form of the potential.[^6] As observed earlier [@Ross:1995dq], such symmetries are very promising for eliminating the fine-tuning problem in inflationary potentials because they can forbid the large quadratic terms in the inflaton field that, even if absent at tree level, arise in radiative order in non-supersymmetric theories (unless protected by a Goldstone symmetry). We turn now to a discussion of the observable implications of the natural inflation models. In this we find it useful to classify the models into two classes, namely those involving small, sub-Planckian field values only and those that require large, super-Planckian, field values. Here we use the reduced Planck scale, $M$, to define the sub- and super- regimes as this is the scale that orders typical higher order terms in supergravity. In the small-field models we allow for the possibility of higher order terms which can dominate as the vacuum expectation value of the inflaton field becomes large. In the large field models it is necessary to [*forbid*]{} such higher order terms since they would otherwise dominate the potential and there should be an underlying symmetry to enable this to be done. Small field models ================== These potentials are of the ‘new inflation’ form $$V\left( \phi \right) =\Delta ^{4}\left[1-\lambda \left( \frac{\phi }{\Lambda} \right)^p\right], \label{slowrollpot}$$ with a single power of the inflaton field, $\phi$, responsible for the variation of the potential, plus a constant term driving inflation. Such a form does not require fine-tuning as the two terms need not be related and the dominance of a given single power can be guaranteed by a symmetry [@Ross:1995dq]. Since the slow-roll parameters get no contribution from the constant term their main contribution will necessarily come from the leading term involving the inflaton field and the naturalness condition is trivially satisfied because this is dominated by a single power of $\phi$. From Eq.(\[densitypert\]) it is clear that $\delta_\mathrm{H}$ is the only observable that depends on $\Delta$, so one can fit its observed value but cannot predict it without a theory for $\Delta$. The slow-roll parameters are given by $$\begin{aligned} \eta &=&-\lambda p(p-1) \left(\frac{\phi_\mathrm{H}}{\Lambda}\right)^{p-2} \left(\frac{M}{\Lambda }\right)^2,\; \label{eta2} \\ \epsilon &=&\frac{\lambda^2 p^2}{2} \left(\frac{\phi_\mathrm{H}}{\Lambda}\right)^{2p-2} \left(\frac{M}{\Lambda}\right)^2 = \eta^2 \frac{1}{2(p-1)^2}\left(\frac{\phi_\mathrm{H}}{M}\right)^2,\; \label{e2} \\ \xi &=&\lambda^2 p^2 (p-1)(p-2) \left(\frac{\phi_\mathrm{H}}{\Lambda}\right)^{2p-4} \left(\frac{M}{\Lambda}\right)^4 = \eta^2 \frac{\left(p-2\right)}{\left(p-1\right)} . \label{xi2}\end{aligned}$$ Turning to the other observables let us consider first the cases $p \geq 2$. Note that the naturalness arguments apply only if $\phi/\Lambda < 1$ and hence $|\eta| > \epsilon$. In this case $n_\mathrm{s}$ is effectively determined by $\eta$ alone, so the measurement of $n_\mathrm{s}$ does not impose a lower bound on $\epsilon$. Thus the expectation is that $r$ will naturally be small for this class of models [@Ross:1995dq]. To quantify this we note that $\eta \simeq (1-n_\mathrm{s})/2$ hence $$r = 16\epsilon = \eta^2 \frac{8}{(p-1)^2} \left(\frac{\phi_\mathrm{H}}{M}\right)^2.$$ This implies that any value $0.9 \lesssim n_\mathrm{s} \lesssim 1$ can be obtained. Imposing the bound $n_\mathrm{s} > 0.95$ following Boyle [*et al*]{} [@Boyle:2005ug][^7] then requires $r < 0.005$. We emphasise that this makes no explicit reference to the excursion of the field during inflation, as in the ‘Lyth bound’ [@Lyth:1996im]. Note that the precise value for $r$ depends here on the value of $\phi_\mathrm{H}$ which, as discussed earlier, is determined by the higher order terms that may be present in the potential. Specific examples have been constructed [@German:2001tz] showing that $r$ can be much lower than the bound given above, even as small as $10^{-16}$. These results are inconsistent with the [*lower*]{} bound quoted by Boyle [*et al*]{} [@Boyle:2005ug] and reflect our different physical interpretation of naturalness. Finally the prediction for $n_\mathrm{r}$ is $$n_\mathrm{r} \simeq -2\xi \simeq -0.001 \frac{(p-2)}{(p-1)} .$$ The case $p=1$ is special since now $\eta$ and $\xi$ both vanish giving $r=8(1-n_\mathrm{s})/3 < 0.13$ and $n_\mathrm{r} = -2(1-n_\mathrm{s})^2/3\simeq 10^{-3}$. This is the [*only*]{} case of a sub-Planckian model yielding a large tensor amplitude and it has been argued [@Alabidi:2005qi] that this case cannot be realised in a complete model due to the requirement that the universe should undergo at least $\sim 50$ e-folds of inflation. The problem is that for this case $\epsilon = (1 - n_\mathrm{s})/6$ is large, limiting the number of e-foldings, which is given by $$N = \frac{1}{M}\int_{\phi_\mathrm{H}}^{\phi_\mathrm{e}} \frac{1}{\sqrt{2\epsilon}} \mathrm{d}\phi ,$$ where $\phi_\mathrm{e}$ is the field value at the end of inflation. For sub-Planckian models $\phi_\mathrm{e} \leq M$, hence $N < 1/\sqrt{2\epsilon} = \sqrt{3/(1 - n_\mathrm{s})}$. For the case $n_\mathrm{s}=0.95$, which gives the large $r$ value, we have only $N < 8$ e-folds. In this case the effect of higher order terms near the Planck scale does not help as the linear term already contributes too much to the slope of the potential and thus limits the number of e-folds of inflation. The only way out of this is that there should be a subsequent inflationary era which generates $\sim 40-50$ additional e-folds of inflation after the $\phi$ field has settled into its minimum. At first sight this looks like an unnatural requirement. However we have shown elsewhere [@Adams:1997de] that in supergravity models it is natural to expect some $\sim 3\ln(M/\Lambda)$ e-folds of ‘multiple inflation’ due to intermediate scale symmetry breaking along ‘flat directions’, where $\Lambda^4$ is the magnitude of the potential driving this subsequent period of inflation. Taking $\Lambda \sim 10^{11}$ GeV (typical of the supersymmetry breaking scale in supergravity models) one generates $\sim50$ e-folds of inflation. Although this two-stage inflationary model appears complicated, it is still natural in the sense discussed above and should not be ignored as a possibility. Large field models ================== Chaotic inflation ----------------- The simplest potential involves a single power of the inflaton $$V (\phi) = \lambda\frac{\phi^p}{\Lambda^{p-4}}\ , \label{chaotic}$$ where we have allowed for the possibility that the scale, $\Lambda$, relevant for higher dimensional terms in the effective potential need not be the Planck scale but can correspond to the mass of some heavy states that have been integrated out in forming the effective potential. Expanding around $\phi = \phi_\mathrm{H}$ yields $$\begin{aligned} \epsilon &=& \frac{p^2}{2} \left(\frac{M}{\phi_\mathrm{H}}\right)^2 , \\ \eta &=& p(p-1) \left(\frac{M}{\phi_\mathrm{H}}\right)^2 , \\ \xi &=& p^2 (p-1)(p-2) \left(\frac{M}{\phi_\mathrm{H}}\right)^4 . \label{slowrollparams}\end{aligned}$$ The slow-roll conditions, $\epsilon, |\eta| \ll 1$, requires $M/\phi_\mathrm{H} \ll 1$ which means that inflation occurs for $\phi$ above the Planck scale — usually called ‘chaotic inflation’ [@Linde:1983gd].[^8] In this case, in order to explain why ever higher order terms $\phi^m,\;m \rightarrow \infty$, do not dominate, it is necessary to have a symmetry which forbids such terms. One such (Goldstone) symmetry has been invoked in a supergravity context [@Kawasaki:2000yn], although it is not known if this can arise in realistic models. Another recent proposal for a large field potential exploits monodromy in a D-brane setup but contains no Standard Model sector which would give rise to large corrections to the slow-roll parameters [@Silverstein:2008sg]. Thus whether such models can actually be realised remains an open question. What are the observable implications of this potential? As before, $\delta_\mathrm{H}$ is the only observable that depends on $\lambda$ so one can fit its observed value but lacking a theory for $\lambda$ this is not a prediction. The other 3 observables are determined in terms of the parameter $x = M/\phi_\mathrm{H}$ and the power $p$. $$\begin{aligned} n_\mathrm{s} &=& 1-x^2 p(p+2) , \\ r &=& 8 p^2 x^2, \\ n_\mathrm{r} &=& -2x^4 p^2 (p+2) .\end{aligned}$$ From this one sees that the ratio of tensor to scalar fluctuations and the running of the spectral index are tightly constrained by the measurement of $n_\mathrm{s}$ $$\begin{aligned} r &=& \frac{8p}{(p+2)}(1 - n_\mathrm{s}) , \\ n_\mathrm{r} &=& -\frac{2}{(p + 2)}(1 - n_\mathrm{s})^2.\end{aligned}$$ Note that these results are independent of $\phi_\mathrm{H}$ and so, as anticipated above, do not depend on exactly when inflation ends. For the quartic potential $p=4,$ $r=0.27$ and $n_\mathrm{r} \simeq - 8 \times 10^{-4}$. The maximum value of $r$ is $8(1 - n_\mathrm{s}) \simeq 0.4$ with $n_\mathrm{r}=0.$ Natural inflation ----------------- Another class of non-fine-tuned models is based on an approximate Goldstone symmetry [@Freese:1990rb], often called ‘natural inflation’ (although it should now be clear that this is not the [ *only*]{} natural possibility). In this case the potential is not a simple polynomial but has the form $$V(\phi) = \Delta^4 \left(1 + \cos\frac{\phi}{f}\right). \label{freese}$$ The slow-roll parameters are: $$\begin{aligned} \epsilon &=& \frac{1}{2}\left( \frac{M}{f}\right)^2 \frac{\left(\sin\frac{\phi_\mathrm{H}}{f}\right)^2} {\left(1 + \cos\frac{\phi_\mathrm{H}}{f}\right)^2} , \\ \eta &=& -\left(\frac{M}{f}\right)^2 \frac{\cos\frac{\phi_\mathrm{H}}{f}} {\left(1+\cos\frac{\phi_\mathrm{H}}{f}\right)} , \\ \xi &=& -\left(\frac{M}{f}\right)^4 \frac{\left(\sin\frac{\phi_\mathrm{H}}{f}\right)^{2}} {\left(1 + \cos\frac{\phi_\mathrm{H}}{f}\right)^2} .\end{aligned}$$ For these to be small we require $f > M$. Unlike the previous case the predictions now depend sensitively on $\phi_\mathrm{H}$ and hence on the related value of the field at the [*end*]{} of inflation. If the end of inflation is determined, as has usually been [ *assumed*]{}, by the steepening of the above potential then $\phi_\mathrm{H}$ has a value such that $\epsilon$ and $\eta$ are comparable. In this case $\epsilon$ can be close to its slow-roll limit, particularly interesting for tensor fluctuations which can now be large. Imposing the bound $n_\mathrm{s} > 0.95$ [@Hinshaw:2008kr] implies $0.02 < r <0.2$ $\cite{Freese:2008if}$, the range corresponding to the variation of $\phi_\mathrm{H}$ with $f$ for allowed values of $f$. As with the other models, the running is small, $n_\mathrm{r} \sim {\cal O}(10^{-3})$. However it may be more natural for inflation to end much earlier due to a second (hybrid) field. Then $\phi_\mathrm{H}$ is reduced so that $\sin(\phi_\mathrm{H}/f)$ can be small, hence $|\eta| \gg \epsilon$. In this case $r$ will be (arbitrarily) small, being proportional to $\epsilon$, The running is also very small, $n_\mathrm{r} \simeq 12\epsilon\eta$. Conclusions =========== We have discussed natural possibilities for the inflationary potential. From this it is clear that the gravitational wave signal need not be large enough to be observable as argued by Boyle [*et al*]{} [@Boyle:2005ug]. The models considered fall broadly into two classes. The first has $\epsilon$ comparable in magnitude to $\eta$ hence $r$ can saturate the upper bound of 0.4 implied by the slow-roll constraint. A characteristic of these models however is that inflation occurs only at field values higher than the Planck scale and it is not clear if this can be naturally realised. An interesting exception is the ‘new inflation’ model with a leading [*linear*]{} term in the inflaton field which however requires a subsequent period of inflation to create our present Hubble volume. The second class of models has $\eta$ larger than $\epsilon$. These are indeed natural but there is no lower bound to $r$ and the upper bound is (unobservably) small. A characteristic of most such models is that inflation occurs at low field values, much below the Planck scale. Examples of this are provided by ‘new inflation’ where $r$ is bounded from [*above*]{} by 0.005 and is usually much below this bound. A large-feld exception to this is a modified form of ‘natural inflation’ where a hybrid field ends inflation early. To summarise, in models that are not fine-tuned, the amplitude of density perturbations and the spectral index are not predicted, being determined by free parameters of the model. However the tensor-to-scalar ratio, $r$, and the running, $n_\mathrm{r}$, are determined in terms of the spectral index. The ratio $r$ provides a sensitive discriminator of the natural models but there is no requirement that it be greater than 0.01 even if the spectral index is bounded from below $n_\mathrm{s}>0.95$.[^9] All the natural models have the running small, $n_\mathrm{r} \sim (1 - n_\mathrm{s})^2$, so observation of a much larger value would indicate a departure from naturalness, perhaps because more than one inflaton field is active at the time density perturbations are generated. This in turn would suggest there should be a departure from a near-Gaussian distribution of the perturbations. Acknowledgements ================ G.G. acknowledges support from DGAPA, UNAM and the hospitality of the Rudolf Peierls Centre, Oxford. S.S. acknowledges a STFC Senior Fellowship (PP/C506205/1) and the EU network ‘UniverseNet’ (MRTN-CT-2006-035863). We thank Latham Boyle for helpful correspondance. [99]{} L. A. Boyle, P. J. Steinhardt and N. Turok, Phys. Rev. Lett. **96** (2006) 111301. 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K. Freese, W. H. Kinney and C. Savage, arXiv:0802.0227 \[hep-ph\]. W. H. Kinney, E. W. Kolb, A. Melchiorri and A. Riotto, Phys. Rev. D **74** (2006) 023502. [^1]: On sabbatical leave from Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México [^2]: Corresponding author: `g.ross@physics.ox.ac.uk` [^3]: Numerically this is $H_0^{-1} \simeq 3000h^{-1}$ Mpc, where $h \equiv H_0/100$ km s$^{-1}$ Mpc$^{-1} \sim 0.7$ is the Hubble parameter. The density perturbation is measured down to $\sim 1$ Mpc, a spatial range corresponding to $\sim 8$ e-folds of inflation. [^4]: This does not exclude ‘hybrid inflation’ models in which additional fields play a role at the end of inflation. [^5]: Boyle [*et al*]{} imposed the cut-off at 15 derivatives (L. Boyle, private communication). [^6]: In supersymmetric theories it is the superpotential that is constrained by the underlying symmetries. The resulting scalar potential has natural relations between different powers of the inflaton field. [^7]: This is slightly more restrictive than the recent WMAP 5-year result: $n_\mathrm{s} = 0.963_{-0.015}^{+0.014}$ [@Hinshaw:2008kr]. [^8]: In fact “chaotic” actually refers to the initial conditions for inflation and ‘chaotic inflation’ can also be realised in a small-field model [@Linde:1984cd]. [^9]: This conclusion is in agreement with another analysis of specific models [@kinney:2006gm].

--- abstract: 'High-resolution spectroscopic observations were taken of 29 extended main sequence turn-off (eMSTO) stars in the young ($\sim$200 Myr) LMC cluster, NGC 1866 using the Michigan/[*Magellan*]{} Fiber System and MSpec spectrograph on the [*Magellan*]{}-Clay 6.5-m telescope. These spectra reveal the first direct detection of rapidly rotating stars whose presence has only been inferred from photometric studies. The eMSTO stars exhibit H[$\alpha$]{} emission (indicative of Be-star decretion disks), others have shallow broad H[$\alpha$]{} absorption (consistent with rotation $\ga$150 [km s$^{-1}$]{}), or deep H[$\alpha$]{} core absorption signaling lower rotation velocities ($\la$150 [km s$^{-1}$]{}). The spectra appear consistent with two populations of stars - one rapidly rotating, and the other, younger and slowly rotating.' author: - 'A. K. Dupree' - 'A. Dotter' - 'C. I. Johnson' - 'A. F. Marino' - 'A. P. Milone' - 'J. I. Bailey III' - 'J. D. Crane' - 'M. Mateo' - 'E. W. Olszewski' title: | NGC 1866: First Spectroscopic Detection of Fast Rotating Stars\ in a Young LMC Cluster --- Introduction ============ Identification of multiple main sequences in old Milky Way globular clusters from HST photometry (Bedin et al. 2004; Piotto [et al.]{} 2007; Gratton [et al.]{} 2012) created a fundamental change in our concept of their stellar populations for it suggested that cluster stars are neither coeval nor chemically homogeneous. This paradigm shift results from the fact that multiple sequences are visible along the entire color-magnitude diagram (CMD) signaling two or more generations of stars. No completely successful scenario exists to explain multiple populations although many possibilities have been offered. The most popular suggests that a second generation of enriched (polluted) stars forms from gas that was processed at high temperatures in the cores and/or envelopes of intermediate to high mass first generation stars. Each of the many possibilities appears to have at least one fatal flaw (Bastian [et al.]{} 2015; Renzini [et al.]{} 2015; Charbonnel 2016). The situation becomes more complicated when investigating younger clusters, which could reveal the predecessors of the Milky Way clusters. Photometric studies of young and intermediate age clusters (age $<$2 Gyr) in the Large Magellanic Cloud (LMC) support yet another scenario. They have revealed an extended (broadened) main-sequence turnoff (eMSTO) and/or a bimodal main sequence (Mackey [et al.]{} 2008; Milone et al. 2009; Goudfrooij [et al.]{} 2009, 2014). This discovery could imply that a prolonged (100-500 Myr) star-formation history occurred (Mackey [et al.]{} 2008; Conroy & Spergel 2011; Keller [et al.]{} 2011). This could be an attractive simple explanation since there are concerns about the lack of active star-formation in clusters older than 10 Myr (Niederhofer [et al.]{} 2016) and the absence of natal cluster gas after 4 Myr (Hollyhead [et al.]{} 2015) suggesting that multiple stellar generations may not be present. Photometry of young ($\sim$300 Myr) stellar clusters also reveals the eMSTO and a bifurcated main sequence (D’Antona [et al.]{} 2015; Milone [et al.]{}  2016, 2017). A recent claim of detection of young stellar objects in some young clusters in the LMC hints at ongoing star formation (For & Bekki 2017). However, other scenarios have been introduced to explain the eMSTO and bifurcated main sequence including a range of ages (Mackey & Broby Nielsen 2007; Milone [et al.]{} 2009; Keller [et al.]{} 2011; Correnti [et al.]{} 2014; Goudfrooij [et al.]{} 2014), different rotation rates (Bastian & deMink 2009, Bastien [et al.]{} 2016; Niederhofer [et al.]{} 2015; D’Antona [et al.]{} 2015; Milone [et al.]{} 2016, 2017), braking of rapid rotators (D’Antona [et al.]{} 2017), or different metallicities (Milone [et al.]{} 2015). Our target, NGC 1866, a 200 Myr cluster in the LMC, displays the eMSTO and also a bifurcated main sequence (Milone [et al.]{}  2017). These characteristics are not due to photometric errors, field-star contamination, differential reddening, or non-interacting binaries (Milone [et al.]{} 2016, 2017). Comparison with isochrones (Milone [et al.]{} 2017) suggests that the best-fit of the bifurcated main sequence comes from rotating stellar models for the red main sequence and non-rotating models for the blue main sequence. It is believed that abundances are similar among the populations of NGC 1866 (Mucciarelli [et al.]{} 2011), although the ages are not, and may range from 140 Myr to 220 Myr (Milone [et al.]{} 2017). Isochrone modeling provides good agreement with the main-sequence objects but the fit to the eMSTO objects is not as satisfactory. Variable stars, such as $\delta$ Scuti objects might also produce an extended turn off (Salinas [et al.]{} 2016), however these stars become significant in older clusters (1$-$3 Gyr) where the turnoff from the main sequence coincides with the instability strip. Stellar rotation not only affects the colors of the stars but also their lifetimes through rotational mixing. Possibly rotation could cause the observed spreads in the CMD (Bastian & deMink 2009). In fact, narrow and broad-band photometry of bright stars in two young LMC clusters hints at the appearance of H[$\alpha$]{} emission (Bastian [et al.]{} 2017) which is interpreted as signaling the presence of rapidly rotating Be stars. No direct measure of rotation has been carried out for individual stars populating the eMSTO in LMC clusters. In this paper, we report the first high-resolution spectroscopy of the H[$\alpha$]{} line in 29 stars in the extended turnoff region of the LMC cluster NGC 1866. Synthesis of model spectra indicated that narrow photospheric features would be ‘washed out’ and too subtle to detect if the stars are rapidly rotating, making the H[$\alpha$]{} transition a feature of choice to characterize the rotational state of the star. Spectroscopic Material ====================== Stellar spectra were obtained with the Michigan/[*Magellan*]{} Fiber System (M2FS, Mateo [et al.]{} 2012) and the MSpec multi-object spectrograph mounted on the [*Magellan*]{}-Clay 6.5-m telescope at Las Campanas Observatory. The fibers have a diameter of 1.2” and can span a field of view nearly 30 arcminutes in diameter. A 180$\mu$m slit yielded a resolving power $\lambda /\Delta \lambda \sim 28,000$. The spectra were binned by 2 pixels in the spatial direction and remained at 1 pixel along the dispersion. The selected targets, which are likely members of the cluster NGC 1866, were identified by Milone et al. (2017) from the Ultraviolet and Visual Channel of the Wide Field Camera 3 (UVIS/WFC3) of HST. Images taken with the F336W filter and the F814W filter provided the photometry and astrometry. Milone [et al.]{} (2017) noted that the apparent stellar density became constant at radial distances greater than about 3 arcminutes from the cluster center, and concluded that cluster members did not extend beyond that distance. Our targets comply with this criterion. In addition, we selected targets separated by 2.5 arcsec at a minimum from any neighboring stars that are brighter and located away from stars less than 2 magnitudes fainter in the F814W band than the target star. With this selection criterion, coupled with the requirements on fiber placement, very few stars remain within the half-light radius of the cluster (41 arcsec); in fact only two of our targets are located there. The vast majority lie between 41 arcsec and $\sim$180 arcsec from the center. This criterion identified $\sim$ 150 acceptable targets, spanning V = 16.2–20. Positions of the guide and acquisition stars were verified by comparison with the 2MASS catalog and WFI images. The software code for M2FS fiber positioning selected targets according to our priorities. We chose the filter “Bulge-GC1” which spans 6120–6720Å over 6 echelle orders, and allows up to 48 fibers to be placed on our targets. In practice, several fibers are placed on the sky; thus we obtained about 43 stellar targets per configuration. Some targets were “lost” due to low fiber sensitivity, neighboring very bright stars, or possibly inaccurate coordinates. Two configurations - a bright and faint selection – each spanning about 2 magnitudes were implemented. Our principal configuration was observed on 8 December and 12 December 2016 with 7 exposures totaling 5.5 hours varying between 2100s and 2700s each. A fainter target configuration was observed on 11 December and 13 December 2016, but the spectra were severely compromised by the full moon. Standard IRAF procedures performed the bias subtraction, overscan trimming, and combination of the 4 individual CCD quadrants into one monolithic array. The [*dohydra*]{} task was implemented for aperture identification and tracing, scattered-light subtraction, flat-fielding, and line identification for wavelength calibration from the ThAr comparison lamp. Sky emission lines were identified and removed individually and the H$\alpha$ order was continuum normalized with a cubic spline, omitting the H[$\alpha$]{} region. A detailed description of the procedures can be found in Johnson [et al.]{} (2015).[^1] We obtained H[$\alpha$]{} spectra for 29 targets within a 3 arcmin radius of the cluster center. Comparison of stars in the reference field within the color and magnitude boundaries of our sample suggests that $~$10% of our targets (comprising $~$3 targets) in the cluster field might not be cluster members. The solar H[$\alpha$]{} line in absorption frequently appears in the spectra at shorter wavelengths than the LMC spectral features but nevertheless allows definition of the continuum on the short wavelength side of H$\alpha$. Target stars, their positions, magnitudes, and H[$\alpha$]{} characteristics are given in Table 1. Analysis ======== H[$\alpha$]{} spectra of the 29 targets located within 3 arcmin of the cluster center are shown in Fig. 1 where both emission and absorption can be found. The emission features are centered on the velocity of the cluster, $+$298.5 [km s$^{-1}$]{} (Mucciarelli [et al.]{} 2011). The profiles are typical of those found in Be stars in which the emission arises in a Keplerian decretion disk surrounding a rapidly rotating star (Rivinius [et al.]{} 2013; Paul [et al.]{} 2017; Reid & Parker 2012). Differences in the profile shapes result from the angle of observation, from pole-on to equator (Struve 1931). In particular, the narrow ‘wine-bottle’ H[$\alpha$]{} profile of Object 58 suggests it is viewed nearly pole-on; many others (Object 14, 26, 56, 62, 89) exhibit a deep central absorption thought to arise from absorption in the cool circumstellar disk when viewed edge-on (Hummel 1994). The line widths at the continuum level vary as well, from $\pm$110 [km s$^{-1}$]{} in the pole-on object to $> \pm200$ [km s$^{-1}$]{} in stars observed at intermediate angles. Absorption profiles shown in Figure 1 are shallow and broad for many stars. H[$\alpha$]{} absorption wings in B stars are indicative of the stellar gravity, and the core of the line responds to rotation, becoming more shallow with increasing values of $v \sin i$. Several of the stars can be seen by visual inspection to have a deep (narrow) core in the absorption profile. We further examine the absorption profiles in two ways: profile synthesis and broadening assessment. In the first instance, theoretical H[$\alpha$]{} absorption profiles are compared to the observed profiles for three bright targets in Fig. 2a, 2b, 2c. H[$\alpha$]{}  profiles from Castelli and Kurucz LTE models[^2] were broadened using a Gaussian convolution and overlaid on the profiles; also shown are reasonable excursions to the profile with higher and lower rotational velocities. Comparison of LTE vs non-LTE calculations of H[$\alpha$]{} profiles shows that LTE profiles are adequate for stars cooler than $\sim$22,000K (Przybilla & Butler 2004; Nieva & Przybilla 2007). HST colors predicted from the Choi et al. (2016) models suggest these targets have $T_{eff}\sim$15,000K. Synthesis of the spectra for Object 10 and 12 suggests $v \sin i \sim $ 70$-$100 [km s$^{-1}$]{} in contrast to a value $\sim$200 [km s$^{-1}$]{} for Object 30. These values provide a lower limit to the true velocity because the orientations of the stars are unknown. Secondly, we developed a broadening parameter defined as the ratio of the central depth to the line profile depth at a wavelength 4Å longward of line center: $R_{c+4}$. This ratio was measured for the target stars after subtracting the solar scattered continuum. Model profiles demonstrate that this ratio increases with increasing velocity (Fig. 2d). The dependence of the ratio on velocity appears similar for different values of the gravity. The observed profiles map velocities, [*v*]{}, from 50 to 250 [km s$^{-1}$]{}. Inspection of the H[$\alpha$]{} absorption profiles suggests that the majority of the targets with [*v*]{} $\la$ 150 [km s$^{-1}$]{} exhibit deep absorption cores, therefore we denote these stars as ‘slow rotators’ and label the stars with [*v*]{} $\ga$150 [km s$^{-1}$]{} as ‘fast rotators’. Theoretical critical velocities are shown in the HST CMD from MIST isochrones (Dotter 2016; Choi [et al.]{} 2016) for a range of ages corresponding to NGC 1866 (Fig. 3). The isochrones have been shifted by the assumed distance and reddening of NGC 1866 \[($m-M)_0$= 18.31, $E(B-V)$= 0.11, Milone [et al.]{} 2017\]. The MIST isochrones include the effects of rotation; those shown in Fig. 3 are initialized with $\Omega/\Omega_{crit}=0.4$ at the ZAMS. The critical velocities for these targets (350$-$400 [km s$^{-1}$]{}) are larger than the values inferred from Fig. 2d. This may account for the lack of emission in H[$\alpha$]{} as stars have not achieved velocities necessary to shed material producing emission from a surrounding disk. Discussion ========== The majority of eMSTO target stars fall into two categories: fast and slow rotators. Detection of H[$\alpha$]{} emission clearly signals a fast-rotating star with a Keplerian decretion disk - the Be phenomenon (Rivinius [et al.]{} 2013). We do not yet have measurements of the rotational velocity of the emission objects. The H[$\alpha$]{} absorption profiles indicate both rapidly and slowly rotating stars. Isochrone fitting to HST cluster photometry (Milone [et al.]{} 2017) suggested that the blue stars on the bifurcated main sequence are slowly rotating, and represent two stellar generations of 140 Myr and 220 Myr. Red main sequence stars are believed to be rapidly rotating ($\Omega$=0.9$\Omega_{crit}$, a high fraction of the critical rotation rate, $\Omega_{crit}$), and are consistent with an age of 200 Myr. Thus our spectroscopic results confirm the conclusion of Milone [et al.]{} (2017) from photometry that identified fast and slowly rotating populations. Figure 4 shows the HST CMD of NGC 1866 marked with targets and their characteristics. Inspection suggests that two targets, Object 3 and 36, are outliers, and perhaps not cluster members because their colors are $\lesssim -$1 and cluster isochrones (Fig. 3) do not extend to those colors. We exclude them from the calculation of median parameters. Taking the “narrow" absorption targets as those with $R_{c+4} \le 0.90$, corresponding to [*v*]{} $\la$150 [km s$^{-1}$]{}, in comparison to the targets with emission, we find that the median magnitudes, m$_{F814W}$ are essentially identical: 18.01$\pm$0.418 (narrow) and 17.99$\pm$0.377 (emission). Here the dispersion is calculated as the median of the absolute deviations of magnitude about the median magnitude. However, the median colors suggest what is evident from Figure 4, namely $m_{F336W}-m_{F814W}$ equals $-$0.56$\pm$0.10 (narrow) and $-$0.34$\pm$0.12 (emission). Targets exhibiting broad absorption $R_{c+4} > 0.90$ have a median magnitude similar to the others, and a median color lying between the values of the other groups: $-$0.50$\pm$0.10. Additional characteristics of the stars can be compared to the results of the photometric studies of the cluster. Three results derive from the photometry: \(1) [*The red main sequence (rapid rotators) is more centrally concentrated than the blue main sequence (slow rotators).*]{} Milone [et al.]{} (2017) find the fraction of red main sequence stars within 1 arcmin of the cluster core is $\sim$0.68. Excluding outliers, the 5 targets within 1 arcmin include 4 stars which are fast rotators, exhibiting H[$\alpha$]{} emission (corresponding to a fraction of 0.80$\pm$0.53 using Poisson statistics). Thus there appears to be a preponderance of rapidly rotating stars in eMSTO objects located in the core of the cluster. Between 1 and 3 arcmin from the cluster center, our sample of 22 eMSTO stars indicates the fast rotators decrease slightly to 0.55$\pm$0.20 of the targets at this distance - although within the error estimate, the fraction remains comparable to that in the core. \(2) [*The ‘blue component’ comprises about 0.15 of the stars at the top of the main sequence.*]{} We find the fraction of slow rotators (the blue component) to be 0.41 in the total eMSTO sample – a value higher than the photometric results. \(3) [*Isochrone fitting suggests that the fast rotating population has a velocity of 0.9 of critical velocity and corresponds to $\sim$ 200 Myr; the non-rotating isochrones indicate that the blue main sequence may harbor two populations of 140 Myr and 220 Myr.*]{} We find the rapid rotators occur to the red of the slowly rotating stars in the HST CMD (Fig. 4). Inspection of isochrones (Milone [et al.]{} 2017) shown in Fig. 4, suggests that the slowly rotating objects span the non-rotating isochrones between 140 Myr and slightly less than $\sim$220 Myr. The rotating population lies to the ‘red’ of the slower-rotating stars which suggests a population older than 200 Myr if $\Omega \sim 0.9\Omega_{crit}$ and perhaps comparable to the slowly rotating objects. If so, this would remove the uncomfortable problem presented from photometry (Milone [et al.]{} 2017) of three populations harboring slow, fast, and slowly rotating stars formed in sequence. Stars exhibiting H[$\alpha$]{} emission comprise a fraction 0.41 of our total sample. This value is comparable to the fraction (Bastian [et al.]{} 2017) inferred from narrow and broad-band photometry of the eMSTO spanning 0.4-0.62 in the young LMC clusters NGC 1850 (79 Myr) and 0.33 in NGC 1856 (300 Myr). Photometric studies, however, give a lower limit to the emission fraction since only stars with strong emission are detected. Moreover, radial velocity shifts of LMC clusters can be significant and compromise the detection of H[$\alpha$]{}  in the narrow HST filter F656N frequently used as an H[$\alpha$]{} diagnostic. Spectroscopy is advantageous for H[$\alpha$]{} detection because weak emitters can be identified, rapid rotators without H[$\alpha$]{} emission can be detected, and radial velocity shifts are of no consequence. Inclusion of stars with broad H[$\alpha$]{} absorption raises the rapid rotation fraction to 0.61 among the eMSTO population in our total sample and implies the fraction must be higher in other clusters, e.g. NGC 1850 and NGC 1856, as well. Direct spectroscopic measures of the eMSTO stars clearly demonstrates the presence of rapidly rotating stars that are cooler than a population of slowly rotating objects. It is not understood how such conditions were established. If the populations were coeval, slowly rotating stars evolve faster than the rapid rotators and they should have a lower turnoff luminosity. The CMD of the eMSTO objects (Fig. 4) displays the opposite behavior which argues for an actual spread in age: the rapidly rotating population marks the (older) initial burst of star formation, followed by a second generation that is more slowly rotating. Isochrones in Fig. 4 demonstrate the younger non-rotating objects (at 140 Myr) lie to the ’blue’ of an older (200 Myr) isochrone that has a rotation close to the critical velocity, here $\Omega = 0.9 \Omega_{crit}$. Recently, D’Antona et al. (2017) have speculated that rotational braking might mimic an age spread, a conjecture which requires spectroscopic confirmation by abundance measures or detection of a stellar wind. The spatial distribution is also puzzling. Goudfrooij [et al.]{} (2011) find that the upper eMSTO (presumably younger objects) is significantly more centrally concentrated than the lower eMSTO in many massive intermediate age clusters in the LMC. This is in harmony with a second generation of stars formed from material shed by stars of the first generation. Our results might suggest a different scenario. It is the cooler eMSTO objects, spectroscopically determined to be fast rotators, that dominate within 1 arcmin of the cluster core, although we have a small sample. Perhaps this is typical of less massive and/or younger clusters. Yet, it is puzzling that the rapidly rotating stars are concentrated towards the cluster center (Milone et al 2017) where it might be expected that a second stellar generation would form from the material of the first generation. This would appear to suggest that another scenario must be sought for young clusters. Acknowledgments =============== We thank the anonymous referee for thoughtful comments and advice on the manuscript. AFM and APM acknowledge support by the Australian Research Council through Discovery Early Career Researcher Awards DE160100851 and DE150101816. EWO was partially supported by NSF Grant AST1313006. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. And we have used data products from 2MASS, which is a joint project of the University of Massachusetts and IPAC/Caltech, funded by NASA and the NSF. Bastian, N., Cabrera-Ziri, I., Niederhofer, F. [et al.]{}  2017, , 465, 4795 Bastian, N., Cabrera-Ziri, I., & Salaris, M. 2015, , 449, 3333 Bastian, N., & deMink, S. E. 2009, , 398, L11 Bastian, N., Niederhofer, F., Kozhurina-Platais, V. [et al.]{} 2016, , 460, L20 Bedin, L. R., Piotto, G., Anderson, J. [et al.]{} 2004, ApJ, 605, L125 Charbonnel, C. 2016, EAS Publication Series, 80-81, 177 Choi, J., Dotter, A., Conroy, C. 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D., & Broby Nielsen, P. 2007, , 379, 151 Mackey, A. D., Broby Nielsen, P., Ferguson, A. M. N., & Richardon, J. C. 2008, , 681, L17 Mateo, M., Bailey, J. I., Crane, J. [et al.]{} 2012, SPIE, 8446, 84464Y McLaughlin, D. E., & van der Marel, R. P. 2005, , 161, 304 Milone, A. P., Bedin, L. R., Piotto, G., & Anderson, J. 2009, A&A, 497, 755 Milone, A. P., Bedin, L. R., Piotto, G. [et al.]{}  2015, , 450, 3750 Milone, A. P., Marino, A. F., D’Antona, F. [et al.]{} 2016, , 458, 4368 Milone, A. P., Marino, A. F., D’Antona, F. [et al.]{}  2017, , 465, 4363 Mucciarelli, A., Cristallo, S., Brocato, E. [et al.]{} 2011, , 413, 837 Nieva, M. F., & Przybilla, N. 2007, A&A, 467, 295 Niederhofer, F., Bastian, N., Kozhurina-Platais, V. [et al.]{} 2016, A&A, 586, A148 Paul, K. T., Shruthi, S. B., & Subramaniam, A. 2017, JApA, 38, 6 Piotto, G., Bedin, L. R., Anderson, J. [et al.]{} 2007, , 661, L53 Przybilla, N., & Butler, K. 2004, , 609, 1181 Reid, W. A., & Parker, Q. 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[*Panel d:*]{} Models of the relative central depth of H[$\alpha$]{} ($R_{c+4}$) as compared to model fits in panels [*a, b,*]{} and [*c*]{} are denoted by red squares. The measured values of $R_{c+4}$ for the targets are marked by black circles.](fig2a_dupree.ps "fig:") ![[*Panels a, b, c:*]{} The H[$\alpha$]{} region in three objects exhibiting absorption. The spectra are centered on H[$\alpha$]{}. H[$\alpha$]{}  sky contamination is marked in Object 10 and 12. Theoretical profiles are from [*kurucz.harvard.edu*]{} for \[Fe/H\]=$-$0.3, [*log g*]{} values of 2.5 $-$3.5, and broadened by a Gaussian to simulate rotation. The profiles suggest values of [*v sin i*]{} in excess of 50 [km s$^{-1}$]{} and ranging to $\sim$ 250 [km s$^{-1}$]{}. Approximate upper and lower limits of velocities are shown. Object 10 and Object 12, exhibiting deep narrow H[$\alpha$]{} are considered to be slow rotators as compared to Object 30. [*Panel d:*]{} Models of the relative central depth of H[$\alpha$]{} ($R_{c+4}$) as compared to model fits in panels [*a, b,*]{} and [*c*]{} are denoted by red squares. The measured values of $R_{c+4}$ for the targets are marked by black circles.](fig2b_dupree.ps "fig:") ![[*Panels a, b, c:*]{} The H[$\alpha$]{} region in three objects exhibiting absorption. The spectra are centered on H[$\alpha$]{}. H[$\alpha$]{}  sky contamination is marked in Object 10 and 12. Theoretical profiles are from [*kurucz.harvard.edu*]{} for \[Fe/H\]=$-$0.3, [*log g*]{} values of 2.5 $-$3.5, and broadened by a Gaussian to simulate rotation. The profiles suggest values of [*v sin i*]{} in excess of 50 [km s$^{-1}$]{} and ranging to $\sim$ 250 [km s$^{-1}$]{}. Approximate upper and lower limits of velocities are shown. Object 10 and Object 12, exhibiting deep narrow H[$\alpha$]{} are considered to be slow rotators as compared to Object 30. [*Panel d:*]{} Models of the relative central depth of H[$\alpha$]{} ($R_{c+4}$) as compared to model fits in panels [*a, b,*]{} and [*c*]{} are denoted by red squares. The measured values of $R_{c+4}$ for the targets are marked by black circles.](fig2c_dupree.ps "fig:") ![[*Panels a, b, c:*]{} The H[$\alpha$]{} region in three objects exhibiting absorption. The spectra are centered on H[$\alpha$]{}. H[$\alpha$]{}  sky contamination is marked in Object 10 and 12. Theoretical profiles are from [*kurucz.harvard.edu*]{} for \[Fe/H\]=$-$0.3, [*log g*]{} values of 2.5 $-$3.5, and broadened by a Gaussian to simulate rotation. The profiles suggest values of [*v sin i*]{} in excess of 50 [km s$^{-1}$]{} and ranging to $\sim$ 250 [km s$^{-1}$]{}. Approximate upper and lower limits of velocities are shown. Object 10 and Object 12, exhibiting deep narrow H[$\alpha$]{} are considered to be slow rotators as compared to Object 30. [*Panel d:*]{} Models of the relative central depth of H[$\alpha$]{} ($R_{c+4}$) as compared to model fits in panels [*a, b,*]{} and [*c*]{} are denoted by red squares. The measured values of $R_{c+4}$ for the targets are marked by black circles.](fig2d_dupree.ps "fig:")  ![Characteristics of H[$\alpha$]{} profiles in NGC 1866 detected in the targets and displayed in the HST CMD. Isochrones are taken from Georgy [et al.]{} (2013) for non-rotating models ($\Omega = 0$) and ages of 140 Myr and 220 Myr ([*blue curves*]{}) and a rotating model ($\Omega = 0.9\Omega_{crit}$) with an age of 200 Myr ([*red curve*]{}) similar to those shown by Milone [et al.]{} (2017). Stars within 3 arcminutes from cluster center are marked by grey dots. ](fig4_dupree.ps) [lcccrrlrccl]{} 1 & 78.374466 & -65.481554 & 18.048 & 18.670 & 18.468 & 18.506& -0.458 & 1.39& ... & 1\ 3 & 78.431151 & -65.470138 & 15.661 & 16.732 & 16.843 & 16.753 & -1.092 &0.58& ... & 1\ 10 & 78.418615 & -65.475645 & 16.548 & 16.937 & 17.019 & 17.022 & -0.474 &0.69& 0.79 & 2\ 12 & 78.470013 & -65.419725 & 16.428 & 16.881 & 17.038 & 17.095 & -0.667 &3.05& 0.75 & 2\ 14 & 78.425251 & -65.454748 & 16.887 & 22.488 & 15.825 & 17.177 & -0.290 &0.67& ... & 1\ 15 & 78.369840 & -65.465653 & 16.907 & 17.159 & 17.150 & 17.199 & -0.292 &1.06& ... & 1\ 17 & 78.477155 & -65.468106 & 18.134 & 18.679 & 18.723 & 18.705& -0.571 &1.64& 0.87 & 2\ 26 & 78.393400 & -65.455855 & 17.140 & 17.290 & 17.166 & 17.390 & -0.250 &0.69& ... & 1\ 30 & 78.360408 & -65.473644 & 17.046 & 17.423 & 17.437 & 17.493 & -0.447 &1.40& 0.93 & 2\ 31 & 78.447226 & -65.477066 & 17.114 & 17.551 & 17.493 & 17.525 & -0.411 &1.16& 0.73 & 2\ 36 & 78.449744 & -65.441929 & 16.707 & 17.687 & 17.569 & 17.677 & -0.970 &1.65& 0.74 & 2\ 40 & 78.366140 & -65.457688 & 17.164 & 17.633 & 17.639 & 17.727 & -0.563 &1.21& 0.64 & 2\ 48 & 78.459678 & -65.436704 & 17.454 & 17.811 & 17.795 & 17.816 & -0.362 &2.04& 0.93 & 2\ 56 & 78.409850 & -65.475827 & 17.619 & 18.004 & 18.024 & 17.955 & -0.336 &0.69& ... & 1\ 58 & 78.378488 & -65.482908 & 17.389 & 17.864 & 17.910 & 17.968 & -0.579 &1.39& ... & 1\ 61 & 78.354382 & -65.457718 & 17.460 & 17.931 & 17.925 & 17.983 & -0.523 &1.49& 0.76 & 2\ 62 & 78.424431 & -65.442766 & 17.940 & 18.034 & 17.995 & 17.991 & -0.051 &1.34& ... & 1\ 63 & 78.479936 & -65.458286 & 17.272 & 17.911 & 17.987 & 18.010 & -0.738 &1.73& 0.77 & 2\ 65 & 78.398687 & -65.476448 & 17.539 & 17.927 & 17.932 & 18.037 & -0.498 &0.79& ... & 1\ 67 & 78.441385 & -65.452302 & 17.603 & 18.019 & 18.084 & 18.052 & -0.449 &1.03& ... & 1\ 69 & 78.495511 & -65.440840 & 17.503 & 18.006 & 18.030 & 18.096 & -0.593 &2.52& 0.93 & 2\ 71 & 78.463359 & -65.451612 & 17.460 & 18.023 & 18.105 & 18.130 & -0.670 &1.49& 0.84 & 2\ 78 & 78.457044 & -65.466589 & 17.766 & 18.247 & 18.224 & 18.263 & -0.497 &1.13& 0.95 & 2\ 79 & 78.473464 & -65.449753 & 18.127 & 18.306 & 18.314 & 18.290 & -0.163 &1.77& 0.84 & 2\ 83 & 78.441541 & -65.440489 & 18.002 & 18.392 & 18.355 & 18.367 & -0.365 &1.61& ... & 1\ 86 & 78.510130 & -65.437927 & 17.809 & 18.222 & 18.240 & 18.421 & -0.612 &2.92& 0.95 & 2\ 88 & 78.395433 & -65.444091 & 18.291 & 18.899 & 18.898 & 18.428 & -0.137 &1.29& 0.80 & 2\ 89 & 78.452582 & -65.456036 & 18.269 & 18.559 & 18.491 & 18.432 & -0.163 &1.13 & ... & 1\ 91 & 78.443073 & -65.430659 & 17.848 & 18.405 & 18.388 & 18.460 & -0.612 &2.17& 0.80 & 2 [^1]: An outline of procedures is available online ([*https://www.cfa.harvard.edu/oir/m2fsreduction.pdf*]{}). [^2]: Available at [*http://kurucz.harvard.edu.*]{}

--- abstract: | *Octal games* are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, playing an octal game on a path $P_n$ is equivalent to playing the same octal game on a heap of $n$ counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars. address: - 'LIMOS, 1 rue de la Chebarde, 63178 Aubière CEDEX, France.' - 'LAAS-CNRS, Université de Toulouse, CNRS, Université Toulouse 1 Capitole - IUT Rodez, Toulouse, France' - 'Fédération de Recherche Maths à Modeler, Institut Fourier, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères Cedex, France' - 'LAAS-CNRS, Université de Toulouse, CNRS, INSA, Toulouse, France.' - 'Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France.' - 'CNRS/Université Grenoble-Alpes, Institut Fourier/SFR Maths à Modeler, 100 rue des Maths - BP 74, 38402 Saint Martin d’Hères, France.' - 'Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France' author: - Laurent Beaudou - Pierre Coupechoux - Antoine Dailly - Sylvain Gravier - Julien Moncel - Aline Parreau - Éric Sopena title: | Octal Games on Graphs:\ The game 0.33 on subdivided stars and bistars --- Combinatorial Games; Octal Games; Subtraction Games; Graphs Introduction ============ *Combinatorial games* are finite two-player games without chance, with perfect information and such that the last move alone determines which player wins the game. Since the information is perfect and the game finite, there is always a winning strategy for one of the players. A formal definition of combinatorial games and basic results will be given in Section \[sec:def\]. For more details, the interested reader can refer to [@winningways], [@lip] or [@cgt]. A well-known family of combinatorial games is the family of *subtraction games*, which are played on a heap of counters. A subtraction game is defined by a list of positive integers $L$ and is denoted by $Sub(L)$. A player is allowed to remove $k$ counters from the heap if and only if $k \in L$. The first player unable to play loses the game. For example, consider the game $Sub(\{1,2\})$. In this game, both players take turns removing one or two counters from the heap, until the heap is empty. If the initial number of counters is a multiple of 3, then the second player has a winning strategy: by playing in such a way that the first player always gets a multiple of 3, he will take the last counter and win the game. A natural generalization of subtraction games is to allow the players to split a heap into two nonempty heaps after having removed counters. This defines a much larger class of games, called *octal games* [@winningways]. An octal game is represented by an octal code which entirely defines its rules. As an example, $Sub(\{1,2\})$ is defined as [**0.33**]{}. A precise definition will be given in Section \[sec:def\]. Octal games have been extensively studied. One of the most important questions [@Guy96] is the periodicity of these games. Indeed, it seems that all finite octal games have a periodic behaviour in the following sense: the set of initial numbers of counters for which the first player has a winning strategy is ultimately periodic. This is true for all subtraction games and for all finite octal games for which the study has been completed [@althofer; @winningways]. Octal games can also be played by placing counters in a row. Heaps are constituted by consecutive counters and only consecutive counters can be removed. According to this representation, it seems natural to play octal games on more complex structures like graphs. A position of the game is a graph and players remove vertices that induce a connected component which corresponds to consecutive counters. The idea to extend the notion of octal games to graphs was already suggested in [@fleischer]. However, to our knowledge, this idea has not been further developed. With our definition, playing the generalization of an octal game on a path is the same as playing the original octal game. In the special case of subtraction games, players have to keep the graph connected. As an example, playing [**0.33**]{} on a graph consists in removing one vertex or two adjacent vertices from the graph without disconnecting it. This extension of octal games is in line with several take-away games on graphs such as [Arc Kayles]{} [@S78] and [Grim]{} [@adams]. However, it does not describe some other deletion games, such as the vertex and edge versions of the game <span style="font-variant:small-caps;">geography</span> [@S78; @edgegeo], vertex and edge deletion games with parity rules, considered in [@ottaway1] and [@ottaway2], or scoring deletion games such as Le Pic arête [@picarete]. We will first give in Section \[sec:def\] basic definitions from combinatorial game theory as well as a formal definition of octal games on graphs. We then focus on the game [**0.33**]{} which is one of the simplest octal games, and to its study on trees. We first study subdivided stars in Section \[sec:star\]. We prove that paths can be reduced modulo 3 which leads to a complete resolution, in contrast with the related studies on subdivided stars of [Node Kayles]{} [@fleischer] and [Arc Kayles]{} [@H15]. In Section \[sec:bistar\], we extend our results to subdivided bistars (i.e. trees with at most two vertices of degree at least 3) using a game operator similar to the sum of games. Unfortunately, these results cannot be extended to all trees and not even to caterpillars. In a forthcoming paper [@futurpapier], some of our results are generalized to other subtraction games on subdivided stars. Definitions {#sec:def} =========== Basics of Combinatorial Game Theory ----------------------------------- *Combinatorial games* [@winningways] are two-player games such that: 1. The two players play alternately. 2. There is no chance. 3. The game is finite (there are finitely many positions and no position can be encountered twice during the game). 4. The information is perfect. 5. The last move alone determines the winner. In *normal* play, the player who plays the last move wins the game. In *misère* play, the player who plays the last move loses the game. *Impartial games* are combinatorial games where at each turn the moves are the same for both players. Hence the only distinction between the players is who plays the first move. In this paper, we will only consider impartial games in normal play. Positions in impartial games have exactly two possible [*outcomes*]{}: either the first player has a winning strategy, or the second player has a winning strategy. If a game position falls into the first category, it is an *${\mathcal{N}}$-position* (for ${\mathcal{N}}$ext player wins); otherwise, it is a *${\mathcal{P}}$-position* (for ${\mathcal{P}}$revious player wins). From a given position $J$ of the game, the different positions that can be reached by playing a move from $J$ are the *options* of $J$, and the set of options of $J$ is denoted ${\mathrm{opt}}(J)$. If we know the outcomes of the positions in ${\mathrm{opt}}(J)$ we can deduce the outcome of $J$, using the following proposition: \[prop:outcome\] Let $J$ be a position of an impartial combinatorial game in normal play: - If ${\mathrm{opt}}(J)=\emptyset$, then $J$ is a ${\mathcal{P}}$-position. - If there exists a ${\mathcal{P}}$-position $J'$ in ${\mathrm{opt}}(J)$, then $J$ is an ${\mathcal{N}}$-position: a winning move consists in playing from $J$ to $J'$. - If all the options of $J$ are ${\mathcal{N}}$-positions, then $J$ is a ${\mathcal{P}}$-position. Every position $J$ of a combinatorial game can be viewed as a combinatorial game with $J$ as the initial position. We therefore often consider positions as games. Some games can be described as the union of smaller game positions. In order to study them, we define the concept of the sum of games. Given two games $J_1$ and $J_2$, their *disjoint sum*, denoted by $J_1+J_2$, is defined as the game where, at their turn, each player plays a legal move on either $J_1$ or $J_2$. Once $J_1$ (resp. $J_2$) is finished, the two players play exclusively on $J_2$ (resp. $J_1$), until it is over. The player who plays the last move wins the game. The question is now whether we can determine the outcome of a disjoint sum $J_1+J_2$ as a function of the outcomes of $J_1$ and $J_2$. If $J_1$ is a ${\mathcal{P}}$-position, then $J_1+J_2$ has the same outcome as $J_2$: the winning player of $J_2$ applies his strategy on $J_2$, and if the other player plays on $J_1$ then he applies the winning strategy on $J_1$. However, the disjoint sum of two ${\mathcal{N}}$-positions cannot be determined so easily. In order to study the disjoint sum of two ${\mathcal{N}}$-positions, the *equivalence* of two games $J_1$ and $J_2$ is defined as follows: $J_1 \equiv J_2$ if and only if $J_1+J_2$ is a ${\mathcal{P}}$-position. According to this relation, one can attribute to a game a value corresponding to its equivalence class, called the [*Grundy value*]{}. The Grundy value of a game position $P$ for a game $J$, denoted by ${\mathcal{G}}_J(P)$, can be computed from the Grundy value of its options using the following formula: $${\mathcal{G}}_J(P) = \operatorname{mex}({\mathcal{G}}_J(P') | P' \in {\mathrm{opt}}(P))$$ where, for any set of integers $S$, $\operatorname{mex}(S)$ is the smallest nonnegative integer not in $S$. In particular, $P$ is a ${\mathcal{P}}$-position if and only if ${\mathcal{G}}_J(P)=0$. Note that this is consistent with Proposition \[prop:outcome\]. When the context is clear, we will denote ${\mathcal{G}}_J(P)$ as ${\mathcal{G}}(P)$. A fundamental result of Combinatorial Game Theory is the Sprague-Grundy Theorem that gives the Grundy values of the sum of games: \[thm:grundysum\] Let $J_1$ and $J_2$ be two game positions. Then ${\mathcal{G}}(J_1+J_2)={\mathcal{G}}(J_1)\oplus{\mathcal{G}}(J_2)$, where $\oplus$, called the nim-sum, is the bitwise XOR applied to the two values written in base 2. A direct application of this theorem is that for any game position $J$, we have ${\mathcal{G}}(J+J)=0$. Moreover, we can see that two games $J_1$ and $J_2$ have the same Grundy value if and only if their disjoint sum $J_1+J_2$ is a ${\mathcal{P}}$-position. Octal games ----------- A well-known family of impartial games is the family of *octal games*, which are played on heaps of counters. On their turn, each player removes some counters from one heap and may also divide the remaining counters of the heap into two nonempty heaps. The rules of an octal game are encoded according to an octal number as follows: \[def:octalGames\] Let $u_1,u_2,\ldots,u_n,\ldots$ be nonnegative integers such that for all $i$, $u_i \leq 7$. In the octal game ${\bf 0.u_1u_2...u_n...}$, a player can remove $i$ counters from a heap if and only if $u_i \neq 0$. Moreover, if we write $u_i$ as $u_i = b^i_1 + 2 \cdot b^i_2 + 4 \cdot b^i_3$ with $b^i_j \in \{0,1\}$, then, the player can, when removing $i$ counters from a heap: 1. empty the heap if and only if $b^i_1=1$; 2. leave the heap nonempty if and only if $b^i_2=1$; 3. split the remaining heap in two nonempty heaps if and only if $b^i_3=1$. An octal game is [*finite*]{} if it has a finite number of non-zero values. In this case, we stop the code at the last non-zero $u_i$. For example, ${\bf u_i}=3$ means that a player can remove $i$ counters from a heap without splitting it. Octal games with only ${\bf 0}$ and ${\bf 3}$ in their code correspond to [*subtraction games*]{} since the heap is never divided. In particular, the game [**0.33**]{} is the game where one can remove one or two counters from a heap. A value of ${\bf u_i=7}$ means that one can remove $i$ counters from a heap, possibly dividing the heap in two heaps whereas ${\bf u_i=6}$ means that one can remove $i$ counters from a heap except if the heap has exactly $i$ counters, and possibly divide it into two heaps. To study an octal game, it suffices to consider it on a single heap. Indeed, using Theorem \[thm:grundysum\], one can obtain the Grundy value of any octal game by computing the nim-sum of its components. The *Grundy sequence* of an octal game is the sequence of the Grundy values of the game on a heap of $n$ counters with $n=0,1,2,...$. For example, the Grundy sequence of [**0.33**]{} is $0,1,2,0,1,2,...$ since the Grundy value of the game [**0.33**]{} on a heap of size $n$ is $n \bmod 3$. The Grundy sequence of [**0.33**]{} is periodic and one can prove that this is the case for all finite subtraction games [@winningways]. Actually, all the octal games which have been completely studied have an ultimately periodic Grundy sequence[^1]. This led to the following conjecture, proposed by Guy: All finite octal games have ultimately periodic Grundy sequences. Octal games on graphs --------------------- A natural question is whether this periodicity can be extended to more complex structures. A relevant structure is graphs. Indeed, as explained in the introduction, octal games are generally played with counters in a row. Considering a row of counters as a path and replacing the notion of consecutive counters by connected components, we get the following definition of octal games on graphs: \[def:octalGamesOnGraphs\] Let $u_1,u_2,\ldots,u_n,\ldots$ be nonnegative integers such that for all $i$, $u_i \leq 7$. Let $G$ be a graph. In the octal game ${\bf 0.u_1u_2...u_n...}$ played on $G$, a player can remove a set $X_i$ of $i$ vertices of $G$ if and only if $u_i \neq 0$ and $X_i$ induces a connected graph. Moreover, if we write ${ u_i}$ as ${ u_i} = b^i_1 + 2 \cdot b^i_2 + 4 \cdot b^i_3$, with $b^i_j\in\{0,1\}$, and $H$ is the connected component of $G$ containing $X_i$, then: 1. the player can remove $H$ (i.e. $X_i=V(H)$) if and only if $b^i_1=1$; 2. the player can leave $H$ connected with at least one vertex remaining (i.e $H\setminus \{X_i\}$ is nonempty and connected) if and only if $b^i_2=1$; 3. the player can disconnect $H$ if and only if $b^i_3=1$. If $G$ is a path, then the game is equivalent to the corresponding standard octal game of Definition \[def:octalGames\]. We now consider several examples. The game ${\bf 0.33}$ on a connected graph corresponds to the game where one can take one vertex or two adjacent vertices without disconnecting it. The game ${\bf 0.07}$ corresponds to the game where one can remove any two adjacent vertices of the graph. That is exactly the well-known game [Arc Kayles]{} [@S78]. Recently, Adams [*et al.*]{} [@adams] studied the game [Grim]{} that is exactly [**0.6**]{} on some graphs (players are allowed to remove any vertex of the graph, except if it is an isolated vertex). A scoring version of ${\bf 0.6}$ is also currently studied [@DGM]. Hence our definition is relevant with existing work. Note that the well-known game [Node Kayles]{} cannot be seen as such an octal game even though on a path it is equivalent to ${\bf 0.137}$. Indeed, in [Nodes Kayles]{}, when four vertices can be removed, they cannot induce a $P_4$. This cannot match our definition. [In the definition of octal games, if $b^i_3=1$, then the players can split a nonempty heap in exactly two nonempty heaps. Generalizations of octal games may then be defined with $b^i_j$ for $j \geq 4$ in order to allow the splitting of a nonempty heap into more than two nonempty heaps. However, our extension of octal games on graphs do not make this distinction: if $b^i_3=1$, then the players can disconnect the graph and leave as many components as they like. Thus this move is not a move that leaves a given number of components, but one which breaks the connectivity of a graph. This is still coherent with the definition of octal games on a row of counters since the path graph can only be split in two components, and allows us to include previously defined vertex deletion games, such as <span style="font-variant:small-caps;">Arc Kayles</span> and <span style="font-variant:small-caps;">Grim</span>.]{} [We ask for the $i$ removed vertices to form a connected component for two reasons. First, in traditional octal games, the counters are generally taken consecutively. The second reason is that if we remove this condition, then all subtraction games on graphs will be trivial. Indeed, it is always possible to remove a vertex of a connected graph and keep the graph connected. Therefore it is also always possible to remove $i$ vertices of the graph without disconnecting it if the vertices do not need to induce a connected graph. Thus playing a subtraction game on a graph would be equivalent to playing the same game on a path with the same number of vertices and we lose the interest of considering more complex structures. With our definition, subtraction games on graphs are not so straightforward.]{} In the rest of this paper, we focus on one octal game, namely ${\bf 0.33}$, for which we provide a detailed analysis on subdivided stars and bistars: by proving lemmas about reducibility of paths, we provide an equivalence between families of stars and bistars which allows us to determine their Grundy value. A study of the [**0.33**]{} game on subdivided stars {#sec:star} ==================================================== If $n$ is an integer, we define the graph $P_n$ as the path on $n$ vertices, with $n-1$ edges. A subdivided star is the tree obtained by subdividing each edge of a star $K_{1,k}$ (with $k \geq 0$) as many times as we want. Each of the subdivided edges will be called a path. A subdivided star is denoted by [$S_{\ell_1,\ldots,\ell_k}$]{}, where $\ell_i \geq 1$ is the number of vertices of the $i$th path. Figure \[fig:exkpod\] shows an example of such a graph. The standard definition of subdivided stars actually requires $k \geq 3$ and thus excludes the paths, however we will need to consider the paths as base cases for subdivided stars and bistars. This is why we will consider the subdivided star [$S_{\ell_1}$]{} (resp. [$S_{\ell_1,\ell_2}$]{}) which is isomorphic to $P_{\ell_1+1}$ (resp. to $P_{\ell_1+\ell_2+1}$). Note that the star $K_{1,0}$ is isomorphic to $P_1$. For clarity, the notation as paths will be used whenever applicable. \(c) at (1,2) ; \(2) at (0,1) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (7b) at (4,3) ; (8b) at (5,3) ; \(2) to (c); (2b) to (c); (c) to (5) to (6) to (7); (c) to (5b) to (6b) to (7b) to (8b); In the [**0.33**]{} game played on a graph, players can remove a vertex or two adjacent vertices from the graph, provided that they do not disconnect the graph. Figure \[fig:ex033star\] shows the moves that are available for the first player on a subdivided star. Note that in every figure describing moves, the original position will be boxed. (orig) at (0,0) [ ]{}; (playOnP3-1) at (4,0) \(c) at (1,2) ; \(2) at (0,1) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (7b) at (4,3) ; (8b) at (5,3) ; \(2) to (c); (2b) to (c); (c) to (5) to (6); (c) to (5b) to (6b) to (7b) to (8b); ; (playOnP3-2) at (4,-1.5) \(c) at (1,2) ; \(2) at (0,1) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (7b) at (4,3) ; (8b) at (5,3) ; \(2) to (c); (2b) to (c); (c) to (5); (c) to (5b) to (6b) to (7b) to (8b); ; (playOnP4-1) at (4,-3) \(c) at (1,2) ; \(2) at (0,1) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (7b) at (4,3) ; (2) to (c); (2b) to (c); (c) to (5) to (6) to (7); (c) to (5b) to (6b) to (7b); ; (playOnP4-2) at (4,-4.5) \(c) at (1,2) ; \(2) at (0,1) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (2) to (c); (2b) to (c); (c) to (5) to (6) to (7); (c) to (5b) to (6b); ; (playOnP1) at (4,-6) \(c) at (1,2) ; (2b) at (2,1) ; (5) at (0,3) ; (6) at (-1,3) ; (7) at (-2,3) ; (5b) at (2,3) ; (6b) at (3,3) ; (7b) at (4,3) ; (8b) at (5,3) ; (2b) to (c); (c) to (5) to (6) to (7); (c) to (5b) to (6b) to (7b) to (8b); ; at (0,-1) [[$S_{1,1,3,4}$]{}]{}; at (6.5,0) [[$S_{1,1,2,4}$]{}]{}; at (6.5,-1.5) [[$S_{1,1,1,4}$]{}]{}; at (6.5,-3) [[$S_{1,1,3,3}$]{}]{}; at (6.5,-4.5) [[$S_{1,1,2,3}$]{}]{}; at (6.5,-6) [[$S_{1,2,4}$]{}]{}; (orig) to (2,0); (2,0) – (playOnP3-1); (2,0) |- (playOnP3-2); (2,0) |- (playOnP4-1); (2,0) |- (playOnP4-2); (2,0) |- (playOnP1); The [**0.33**]{} game on paths and cycles has the same nim-sequence as the [**0.33**]{} game on heaps of counters: \[prop:033pathsAndCycles\] For any $n \geq 0$, ${\mathcal{G}}(P_n) = {\mathcal{G}}(C_n) = n \bmod 3$. In this section, we will prove a similar result for subdivided stars: every path of length $\ell$ can be reduced to a path of length $\ell \bmod 3$ without changing the Grundy value. \[thm:modkpodes\] For all $\ell_1,\ldots,\ell_k$, we have ${\mathcal{G}}($[$S_{\ell_1,\ldots,\ell_k}$]{}$)={\mathcal{G}}($[$S_{\ell_1 \bmod 3,\ldots,\ell_k \bmod 3}$]{}$)$. To prove this theorem, it suffices to prove that a $P_3$ can be attached to the central vertex or attached to a leaf of a subdivided star without changing the Grundy value. This will follow from a series of lemmas. First we make an observation that will be useful for several proofs. \[lem:keylemma\] Let $P_n$ be a path with $n \geq 4$, and $x$ a vertex of $P_n$. Then a move in $P_n$ that removes $x$ has an equivalent move that does not remove $x$: removing the symmetric of $x$ leads to the same position. In particular, we will use this observation when $x$ is the central vertex of a star with one or two paths. \[lem:grundyStars\] Let $\ell \geq 0$ and $S=$[$S_{1,1,\ell}$]{}. We have ${\mathcal{G}}(S)=|V(S)| \bmod 3=\ell \bmod 3$. We use induction on $\ell$. First, suppose that one can remove the central vertex of $S$. This is only possible if $\ell=0$, thus $S=P_3$ and we are done. Now, if $\ell\geq 1$, then one cannot remove the central vertex of $S$. In this case, up to three moves are available from $S$: - Removing one of the two leaves, leaving $P_{\ell+2}$ whose Grundy value is $(\ell+2) \bmod 3$; - Removing one vertex from the path of length $\ell$, leaving a star whose Grundy value is $(\ell+2) \bmod 3$ by induction hypothesis; - If $\ell \geq 2$, removing two vertices from the path of length $\ell$, leaving a star whose Grundy value is $(\ell+1) \bmod 3$ by induction hypothesis. Thus, we have ${\mathcal{G}}(S)=\operatorname{mex}((\ell+1) \bmod 3, (\ell+2) \bmod 3)=\ell \bmod 3$. Note that if $\ell=1$ then all moves are equivalent and leave $P_3$, thus ${\mathcal{G}}(S)=\operatorname{mex}({\mathcal{G}}(P_3))=\operatorname{mex}(0)=1$. \[lem:modkpodes\] A $P_3$ can be attached to any leaf or to the central vertex of a subdivided star without changing its Grundy value. Let $S$ be a subdivided star, and $S'$ be the subdivided star obtained by attaching a $P_3$ to any leaf or to the central vertex of $S$. We show that $S + S'$ is a ${\mathcal{P}}$-position by proving that the second player can always play to a ${\mathcal{P}}$-position after the first player’s move. The proof is by induction on $|V(S)|$. Suppose first that the first player can remove the central vertex of $S$: - If $S$ is empty (resp. $S=P_1$, $S=P_2$), then $S'=P_3$ (resp. $S'=P_4$, $S'=P_5$), and thus $S+S'$ is a ${\mathcal{P}}$-position since ${\mathcal{G}}(S)={\mathcal{G}}(S')$; - If $S=P_3$, then either $S'=P_6$ and we are done, or $S'=$[$S_{1,1,3}$]{} and the result follows from Lemma \[lem:grundyStars\]. - If $S=S_{\ell}$ with $\ell \geq 4$ or $S=S_{1,\ell}$ with $\ell \geq 2$, then, as stated in Observation \[lem:keylemma\], the second player will always be able to replicate the first player’s move on $S'$, by playing the symmetrical move. By induction hypothesis, the new position will be a ${\mathcal{P}}$-position. Suppose now that the first player cannot remove the central vertex of $S$: - If the first player takes one vertex (resp. two vertices) from the attached $P_3$ in $S'$, then the second player takes two vertices (resp. one vertex) from it, leaving $S + S$ which is a ${\mathcal{P}}$-position. - If the first player plays elsewhere on $S'$, the second player answers by playing the same move on $S$. By induction hypothesis, the new position will be a ${\mathcal{P}}$-position. - If $S \neq P_m$, then the first player cannot remove the central vertex. In this case, for every first player’s move on $S$, the second player can replicate it on $S'$, allowing us to invoke the induction hypothesis. Theorem \[thm:modkpodes\] then directly follows from Lemma \[lem:modkpodes\]. Hence, all paths of length $3p$ can be removed, all paths of length $3p+1$ can be reduced to a single edge, and all paths of length $3p+2$ can be reduced to a path of length 2. If we want to know the Grundy value of a given subdivided star, it then suffices to study the Grundy values of the subdivided stars with paths of length 1 and 2 attached to their central vertex. We are able to build a table of positions and their options: the subdivided star in row $i$ and column $j$, $j \leq i$, has $i$ paths attached to its central vertex, $j$ of them being of length 2. Figure \[fig:tabpos\] shows the first six rows of this table (the first two rows correspond to the empty graph and the subdivided star reduced to its central vertex, respectively): (-1,1.5) – (-1,-5.5); (-1,1.5) – (10.5,1.5); (4.75,2.4) node [Number of paths of length 2 in the subdivided star]{}; (-1.8,-2) node\[rotate=90\] [Number of paths in the subdivided star]{}; (-1.2,0) node [0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{}; (-1.2,-5) node [5]{}; (0,1.8) node [0]{}; (2,1.8) node [1]{}; (4,1.8) node [2]{}; (6,1.8) node [3]{}; (8,1.8) node [4]{}; (10,1.8) node [5]{}; (empty) at (0,1) [$\emptyset$]{}; (p1) at (0,0) [$P_1$]{}; (p2) at (0,-1) [$P_2$]{}; (p3) at (2,-1) [$P_3$]{}; (p3b) at (0,-2) [$P_3$]{}; (p4) at (2,-2) [$P_4$]{}; (p5) at (4,-2) [$P_5$]{}; (s111) at (0,-3) [[$S_{1,1,1}$]{}]{}; (s112) at (2,-3) [[$S_{1,1,2}$]{}]{}; (s122) at (4,-3) [[$S_{1,2,2}$]{}]{}; (s222) at (6,-3) [[$S_{2,2,2}$]{}]{}; (s1111) at (0,-4) [[$S_{1,1,1,1}$]{}]{}; (s1112) at (2,-4) [[$S_{1,1,1,2}$]{}]{}; (s1122) at (4,-4) [[$S_{1,1,2,2}$]{}]{}; (s1222) at (6,-4) [[$S_{1,2,2,2}$]{}]{}; (s2222) at (8,-4) [[$S_{2,2,2,2}$]{}]{}; (s11111) at (0,-5) [[$S_{1,1,1,1,1}$]{}]{}; (s11112) at (2,-5) [[$S_{1,1,1,1,2}$]{}]{}; (s11122) at (4,-5) [[$S_{1,1,1,2,2}$]{}]{}; (s11222) at (6,-5) [[$S_{1,1,2,2,2}$]{}]{}; (s12222) at (8,-5) [[$S_{1,2,2,2,2}$]{}]{}; (s22222) at (10,-5) [[$S_{2,2,2,2,2}$]{}]{}; (p1) to (empty); (p2) to\[out=120, in=-120\] (empty); (p3b) to\[out=120, in=-120\] (p1); (p2) to (p1); (p3) to (p1); (p3) to (p2); (p3b) to (p2); (p4) to (p3b); (p4) to (p2); (p4) to (p3); (p5) to (p3); (p5) to (p4); (s111) to (p3b); (s112) to (p3b); (s112) to (p4); (s112) to (s111); (s122) to (p5); (s122) to (p4); (s122) to (s112); (s222) to (p5); (s222) to (s122); (s1111) to (s111); (s1112) to (s111); (s1112) to (s112); (s1112) to (s1111); (s1122) to (s112); (s1122) to (s122); (s1122) to (s1112); (s1222) to (s122); (s1222) to (s222); (s1222) to (s1122); (s2222) to (s222); (s2222) to (s1222); (s11111) to (s1111); (s11112) to (s1111); (s11112) to (s1112); (s11112) to (s11111); (s11122) to (s1112); (s11122) to (s1122); (s11122) to (s11112); (s11222) to (s1122); (s11222) to (s1222); (s11222) to (s11122); (s12222) to (s1222); (s12222) to (s2222); (s12222) to (s11222); (s22222) to (s2222); (s22222) to (s12222); Since the Grundy value of the empty graph is 0, we can deduce the Grundy value of every star by proceeding inductively from the top lines: \[thm:grunStars\] Figure \[fig:tabgrun\] shows the table of the Grundy values of subdivided stars after reduction of their paths modulo 3. Except for the four first rows, the rows corresponding to stars with an odd number of paths are of the form $1203(12)^*$ whereas the rows corresponding to stars with an even number of paths are of the form $03120(30)^*$. Moreover, except for the four first columns, the columns with an even number of paths of length 2 are of the form $(01)^*$ whereas the columns with an odd number of paths of length 2 are of the form $(23)^*$. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Number of paths of length 2 in the subdivided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number of paths in the subdivided star]{}; (-1.2,0) node [0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{}; (-1.2,-5) node [5]{}; (-1.5,-6) node […]{}; (-1.5,-7) node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) node [2]{}; (3,1.8) node [3]{}; (4,1.8) node [4]{}; (5,1.8) node [5]{}; (6,1.8) node […]{}; (7,1.8) node [$2p$]{}; (8,1.8) node [$2p+1$]{}; (empty) at (0,1) [0]{}; (p1) at (0,0) [1]{}; (p2) at (0,-1) [2]{}; (p3) at (1,-1) [0]{}; (p3b) at (0,-2) [0]{}; (p4) at (1,-2) [1]{}; (p5) at (2,-2) [2]{}; (s111) at (0,-3) [1]{}; (s112) at (1,-3) [2]{}; (s122) at (2,-3) [0]{}; (s222) at (3,-3) [1]{}; (s1111) at (0,-4) [0]{}; (s1112) at (1,-4) [3]{}; (s1122) at (2,-4) [1]{}; (s1222) at (3,-4) [2]{}; (s2222) at (4,-4) [0]{}; (s11111) at (0,-5) [1]{}; (s11112) at (1,-5) [2]{}; (s11122) at (2,-5) [0]{}; (s11222) at (3,-5) [3]{}; (s12222) at (4,-5) [1]{}; (s22222) at (5,-5) [2]{}; (p1) to (empty); (p2) to\[out=120, in=-120\] (empty); (p3b) to\[out=120, in=-120\] (p1); (p2) to (p1); (p3) to (p1); (p3) to (p2); (p3b) to (p2); (p4) to (p3b); (p4) to (p2); (p4) to (p3); (p5) to (p3); (p5) to (p4); (s111) to (p3b); (s112) to (p3b); (s112) to (p4); (s112) to (s111); (s122) to (p5); (s122) to (p4); (s122) to (s112); (s222) to (p5); (s222) to (s122); (s1111) to (s111); (s1112) to (s111); (s1112) to (s112); (s1112) to (s1111); (s1122) to (s112); (s1122) to (s122); (s1122) to (s1112); (s1222) to (s122); (s1222) to (s222); (s1222) to (s1122); (s2222) to (s222); (s2222) to (s1222); (s11111) to (s1111); (s11112) to (s1111); (s11112) to (s1112); (s11112) to (s11111); (s11122) to (s1112); (s11122) to (s1122); (s11122) to (s11112); (s11222) to (s1122); (s11222) to (s1222); (s11222) to (s11122); (s12222) to (s1222); (s12222) to (s2222); (s12222) to (s11222); (s22222) to (s2222); (s22222) to (s12222); (0,-7) node [$0$]{}; (1,-7) node [$3$]{}; (2,-7) node [$1$]{}; (3,-7) node [$2$]{}; (4,-7) node [$0$]{}; (5,-7) node [$3$]{}; (6,-7) node […]{}; (7,-7) node [$0$]{}; (0,-8) node [$1$]{}; (1,-8) node [$2$]{}; (2,-8) node [$0$]{}; (3,-8) node [$3$]{}; (4,-8) node [$1$]{}; (5,-8) node [$2$]{}; (6,-8) node […]{}; (7,-8) node [$1$]{}; (8,-8) node [$2$]{}; (0.2,-7) – (0.8,-7); (1.2,-7) – (1.8,-7); (2.2,-7) – (2.8,-7); (3.2,-7) – (3.8,-7); (4.2,-7) – (4.8,-7); (0.2,-8) – (0.8,-8); (1.2,-8) – (1.8,-8); (2.2,-8) – (2.8,-8); (3.2,-8) – (3.8,-8); (4.2,-8) – (4.8,-8); (7.2,-8) – (7.8,-8); (0,-7.2) – (0,-7.8); (1,-7.2) – (1,-7.8); (2,-7.2) – (2,-7.8); (3,-7.2) – (3,-7.8); (4,-7.2) – (4,-7.8); (5,-7.2) – (5,-7.8); (7,-7.2) – (7,-7.8); (0.2,-7.2) – (0.8,-7.8); (1.2,-7.2) – (1.8,-7.8); (2.2,-7.2) – (2.8,-7.8); (3.2,-7.2) – (3.8,-7.8); (4.2,-7.2) – (4.8,-7.8); (7.2,-7.2) – (7.8,-7.8); The game [**0.33**]{} on subdivided bistars {#sec:bistar} =========================================== Let $S$ and $S'$ be two subdivided stars. The subdivided bistar [S S’]{} is the graph obtained by joining the central vertices of $S$ and $S'$ by a path of $m$ edges. If $m=0$, then the subdivided bistar is a subdivided star. Likewise, if $m \geq 1$, $S=$[$S_{\ell_1,\ldots,\ell_k}$]{} and $S'=\emptyset$, then the subdivided bistar [S S’]{} is the subdivided star [$S_{\ell_1,\ldots,\ell_k,m-1}$]{}. Figure \[fig:exbipod\] shows an example of a subdivided bistar. For the sake of convenience, we will denote [S S’]{} by $S {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S'$. (c1) at (0,0) ; (a11) at (-1,1) ; (a21) at (-1,0) ; (a22) at (-2,0) ; (a23) at (-3,0) ; (a31) at (-1,-1) ; (a32) at (-2,-1) ; (c1) – (a11); (c1) – (a21); (a23) – (a21); (c1) – (a31); (a32) – (a31); (c2) at (2,0) ; (b11) at (3,0) ; (b21) at (3,1) ; (b22) at (4,1) ; (b23) at (5,1) ; (b24) at (6,1) ; (b31) at (3,-1) ; (b32) at (4,-1) ; (b33) at (5,-1) ; (c2) – (b11); (c2) – (b21); (b24) – (b21); (c2) – (b31); (b33) – (b31); \(m) at (1,0) ; (c1) – (m); (m) – (c2); We notice that playing the [**0.33**]{} game on a subdivided bistar is similar to playing the [**0.33**]{} game on the two subdivided stars composing it with an “adjustment” depending on the length of the path linking the two stars, except for some small cases where one of the stars can be emptied so that one can play on the middle path. This section is divided in two parts. In the first part, we will prove that every path of length $\ell$ in a subdivided bistar can be reduced to a path of lenth $\ell \bmod 3$ without changing the Grundy value: \[thm:modbipodes\] For all $\ell_1,\ldots,\ell_k,\ell'_{1},\ldots,\ell'_{k'},m$, we have: $${\mathcal{G}}(S_{\ell_1,\ldots,\ell_k} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m$};\end{tikzpicture} S_{\ell'_{1},\ldots,\ell'_{k'}}) = {\mathcal{G}}(S_{\ell_1 \bmod 3,\ldots,\ell_k \bmod 3} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m\bmod3$};\end{tikzpicture} S_{\ell'_{1} \bmod 3,\ldots,\ell'_{k'} \bmod 3})$$ In the second part, we compute the Grundy value of a subdivided bistar, depending on the Grundy values of each of its two subdivided stars. Reducing the paths of a subdivided bistar ----------------------------------------- In this section, we prove Theorem \[thm:modbipodes\]. We begin by proving the result for the middle path, before proving it for the paths of the two subdivided stars composing the bistar. Note that we allow the length of the middle path to be 0, in which case the subdivided bistar is simply a subdivided star. Thus, if a subdivided bistar has a middle path of $3k$ edges, then it can be reduced to a subdivided star without changing its Grundy value. \[lem:modbipodeschain\] For all $\ell_1,\ldots,\ell_k,\ell'_{1},\ldots,\ell'_{k'},m$, we have: $${\mathcal{G}}(S_{\ell_1,\ldots,\ell_k} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m$};\end{tikzpicture} S_{\ell'_{1},\ldots,\ell'_{k'}}) = {\mathcal{G}}(S_{\ell_1,\ldots,\ell_k} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m\bmod3$};\end{tikzpicture} S_{\ell'_{1},\ldots,\ell'_{k'}})$$ It is enough to prove that adding three edges to the path does not change the Grundy value of the subdivided bistar. Let $S$ and $S'$ be two subdivided stars. Let $B=$[S S’]{} and $B'=$[S S’]{}, $m \geq 0$. We show that $B + B'$ is a ${\mathcal{P}}$-position by proving that for every first player’s move, the second player always has an answer leading to a ${\mathcal{P}}$-position. We use induction on the size of $B$. The first player can play on the middle path if and only if one of the two stars is either empty or reduced to a single vertex or a $P_2$. In this case, $B$ and $B'$ are subdivided stars, and the result follows from Lemma \[lem:modkpodes\]. Assume now that both $S$ and $S'$ have at least two vertices. Hence, the first player is unable to play on the middle path and can play either on $S$ or $S'$. The second player will replicate the same move on the other subdivided bistar. By induction hypothesis, the result follows. In order to prove that the paths of the stars can be reduced, we need a few technical lemmas. \[lem:P3empty\] Let $S$ be a subdivided star, and $B=S$[$S_{1,1}$]{}. We have ${\mathcal{G}}(S)={\mathcal{G}}(B)$. We show that $S + B$ is a ${\mathcal{P}}$-position by proving that for every first player’s move, the second player always has an answer leading to a ${\mathcal{P}}$-position. We use induction on the size of $S$. The cases where $S$ is empty or the first player can remove its central vertex are: - $S$ is empty, thus $B = P_3$, which is a ${\mathcal{P}}$-position; - $S$ is a single vertex, thus $B=$[$S_{1,1,1}$]{}. We know by Lemma \[lem:grundyStars\] that ${\mathcal{G}}(B)=1={\mathcal{G}}(S)$; - $S=P_2$, thus $B=$[$S_{1,1,2}$]{}. Considering Figure \[fig:tabgrun\], we get ${\mathcal{G}}(B)=2={\mathcal{G}}(S)$; - $S=P_3$, and in that case, either $S=$[$S_{1,1}$]{} or $S=S_2$: 1. $B=$[$S_{1,1}$]{}[$S_{1,1}$]{}. $B$ is a ${\mathcal{P}}$-position: the first player has only one available move, and from the resulting graph the second player can play to $P_3$ which is a ${\mathcal{P}}$-position. Both $B$ and $S$ being ${\mathcal{P}}$-positions, we have ${\mathcal{G}}(B)={\mathcal{G}}(S)$. 2. $B=$[$S_{1,1,3}$]{}. By Theorem \[thm:modkpodes\], ${\mathcal{G}}(S)={\mathcal{G}}(B)$. - $S=S_{\ell}$ with $\ell \geq 4$ or $S=S_{1,\ell}$ with $\ell \geq 2$. By Observation \[lem:keylemma\], the second player will always be able to replicate the first player’s move on $B$, by playing the symmetrical move. By induction hypothesis, the new position is a ${\mathcal{P}}$-position. Figure \[fig:P3emptyPROOF\] depicts the cases where the first player does not take the central vertex of $S$, and completes the proof. (orig) at (0,0) [ ]{}; (playOnS1) at (4,0) at (0,0) ; (-0.6,0) node\[scale=0.75\] [$S'$]{}; (0,0) – (-0.75,0.5); (0,0) – (-0.75,-0.5); (-0.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,1) ; at (4,-1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (3,0); (3,0) to (4,1); (3,0) to (4,-1); ; (playOnS2) at (4,-1.5) at (0,0) ; (-0.6,0) node\[scale=0.75\] [$S$]{}; (0,0) – (-0.75,0.5); (0,0) – (-0.75,-0.5); (-0.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,1) ; at (4,-1) ; (1.4,0) node\[scale=0.75\] [$S'$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (3,0); (3,0) to (4,1); (3,0) to (4,-1); ; (playOnS3) at (8,-0.75) at (0,0) ; (-0.6,0) node\[scale=0.75\] [$S'$]{}; (0,0) – (-0.75,0.5); (0,0) – (-0.75,-0.5); (-0.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,1) ; at (4,-1) ; (1.4,0) node\[scale=0.75\] [$S'$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (3,0); (3,0) to (4,1); (3,0) to (4,-1); ; (11,-0.75) node [${\mathcal{P}}$ by induction]{}; (11,-1.25) node [hypothesis]{}; (bli) at (4,-3) at (0,0) ; (-0.6,0) node\[scale=0.75\] [$S$]{}; (0,0) – (-0.75,0.5); (0,0) – (-0.75,-0.5); (-0.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (3,0); (3,0) to (4,1); ; (blir) at (8,-3) at (0,0) ; (-0.6,0) node\[scale=0.75\] [$S$]{}; (0,0) – (-0.75,0.5); (0,0) – (-0.75,-0.5); (-0.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); ; (11,-3) node [${\mathcal{P}}$]{}; (orig) – (2,0); (2,0) – (playOnS1); (2,0) |- (playOnS2); (playOnS1) – (playOnS3); (playOnS2) – (playOnS3); (2,0) |- (bli); (bli) – (blir); Let $S$ be a subdivided star, we denote [S $\emptyset$]{} by $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}$. We then have: \[lem:P3onevertex\] Let $S$ be a subidvided star. We have ${\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})={\mathcal{G}}(S$[$S_{1,1}$]{}$)$. Let $B=S$[$S_{1,1}$]{}. We show that $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}+ B$ is a ${\mathcal{P}}$-position by proving that for every first player’s move, the second player always has an answer leading to a ${\mathcal{P}}$-position. We use induction on the size of $S$. The cases where $S$ is empty, or where the first player can remove either the central vertex of $S$ or both the central vertex of $S$ and the vertex from the middle path of $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}$ are: - $S$ is empty, thus $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}= P_1$ and $B=$[$S_{1,1,1}$]{}. We know by Lemma \[lem:grundyStars\] that ${\mathcal{G}}(B)=1={\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})$ so $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}+B$ is a ${\mathcal{P}}$-position; - $S$ is a single vertex, thus $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}= P_2$ and $B=$[$S_{1,1,2}$]{}. Considering Figure \[fig:tabgrun\], we get ${\mathcal{G}}(B)=2={\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})$ so $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}+B$ is a ${\mathcal{P}}$-position; - $S=P_2$, thus $S {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}= P_3$ and $B=$[$S_{1,1,3}$]{} which by Lemma \[lem:modkpodes\] has the same Grundy value as [$S_{1,1}$]{}, *i.e.* as $P_3$. Thus, ${\mathcal{G}}(B)={\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})$ so $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}+B$ is a ${\mathcal{P}}$-position; - $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}=$[$S_{1,1,1}$]{}, thus $B=$[[$S_{1,1}$]{} [$S_{1,1}$]{}]{}. Considering the table in Figure \[fig:tabgrun\], we get ${\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})=1$. It is easy to see that ${\mathcal{G}}(B)=1$, since only one move is available for the first player (removing one leaf vertex), which leaves [$S_{1,1,3}$]{} which is a ${\mathcal{P}}$-position. Thus ${\mathcal{G}}(S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}})={\mathcal{G}}(B)$ so $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}+B$ is a ${\mathcal{P}}$-position. - $S=S_{\ell}$ with $\ell \geq 4$ or $S=S_{1,\ell}$ with $\ell \geq 2$. By Observation \[lem:keylemma\], the second player will always be able to replicate the first player’s move on $B$, by playing the symmetrical move. By induction hypothesis, the new position is a ${\mathcal{P}}$-position. Figure \[fig:P3onevertexPROOF\] depicts the cases where the first player takes neither the central vertex of $S$ nor both the central vertex of $S$ and the vertex from the middle path of $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}$, and completes the proof. (orig) at (0,0) [ ]{}; (playOnS1) at (5,0) at (-1,0) ; at (0,0) ; (-1.6,0) node\[scale=0.75\] [$S'$]{}; (-1,0) – (0,0); (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); (4,0) to (5,-1); ; (playOnS2) at (5,-1.5) at (-1,0) ; at (0,0) ; (-1.6,0) node\[scale=0.75\] [$S$]{}; (-1,0) – (0,0); (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S'$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); (4,0) to (5,-1); ; (playOnS3) at (10,-0.75) at (-1,0) ; at (0,0) ; (-1.6,0) node\[scale=0.75\] [$S'$]{}; (-1,0) – (0,0); (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S'$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); (4,0) to (5,-1); ; (14,-0.75) node [${\mathcal{P}}$ by induction]{}; (14,-1.25) node [hypothesis]{}; (abli) at (5,-3) at (-1,0) ; (-1.6,0) node\[scale=0.75\] [$S$]{}; (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); (4,0) to (5,-1); ; (bbli) at (5,-4.5) at (-1,0) ; at (0,0) ; (-1.6,0) node\[scale=0.75\] [$S$]{}; (-1,0) – (0,0); (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0.6,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); ; (cbli) at (10,-3.75) at (-1,0) ; (-1.6,0) node\[scale=0.75\] [$S$]{}; (-1,0) – (-1.75,0.5); (-1,0) – (-1.75,-0.5); (-1.75,0.5) arc (135:225:0.7); (0,0) node [+]{}; at (2,0) ; at (3,0) ; at (4,0) ; at (5,1) ; at (5,-1) ; (1.4,0) node\[scale=0.75\] [$S$]{}; (2,0) – (1.25,0.5); (2,0) – (1.25,-0.5); (1.25,0.5) arc (135:225:0.7); (2,0) – (4,0); (4,0) to (5,1); ; (14,-3.5) node [${\mathcal{P}}$ by]{}; (14,-4) node [Lemma \[lem:modkpodes\]]{}; (orig) – (2.5,0); (2.5,0) – (playOnS1); (2.5,0) |- (playOnS2); (playOnS1) – (playOnS3); (playOnS2) – (playOnS3); (2.5,0) |- (abli); (2.5,0) |- (bbli); (abli) – (cbli); (bbli) – (cbli); We are now ready to prove that the paths of the two subdivided stars of a bistar can be reduced: \[lem:modbipodesarms\] For all $\ell_1,\ldots,\ell_k,\ell'_{1},\ldots,\ell'_{k'},m$, we have: $${\mathcal{G}}(S_{\ell_1,\ldots,\ell_k} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m$};\end{tikzpicture} S_{\ell'_{1},\ldots,\ell'_{k'}}) = {\mathcal{G}}(S_{\ell_1 \bmod 3,\ldots,\ell_k \bmod 3} \begin{tikzpicture}[baseline=-4]\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$m$};\end{tikzpicture} S_{\ell'_{1} \bmod 3,\ldots,\ell'_{k'} \bmod 3})$$ Thanks to Lemma \[lem:modbipodeschain\], we only have to prove the result on the subdivided bistars with a middle path of length 1 or 2. Let $S$ and $S'$ be two subdivided stars, $B=$[S S’]{} with $i \in \{1,2\}$, and $B'$ the subdivided bistar obtained by attaching a $P_3$ to a leaf or to the central vertex of (without loss of generality) $S'$. We prove by induction on $|V(B)|$ that ${\mathcal{G}}(B)={\mathcal{G}}(B')$. First, we consider the cases where $S'$ is empty or the first player can remove its central vertex: - If either $S$ or $S'$ is empty (resp. a single vertex), then $B$ is a subdivided star, and the result holds by Lemma \[lem:modkpodes\]; - If $B=S$$P_2$ (resp. $B=S$$P_2$) and the first player empties $S'$ on $B$, then the second player is unable to replicate the move on $B'$. The strategy is then to take two vertices from the attached $P_3$. By Lemma \[lem:P3empty\] (resp. Lemma \[lem:P3onevertex\]), we have ${\mathcal{G}}(B)={\mathcal{G}}(B')$. Now, we consider the cases where the first player cannot take the central vertex of $S'$: - If $S'$ is a path with more than two vertices, and the $P_3$ is attached to its central vertex, then replicating the first player’s move will always be possible and lead to a ${\mathcal{P}}$-position by induction hypothesis. - If the first player takes one vertex (resp. two vertices) from the attached $P_3$ on $B'$, then the second player answers by taking two vertices (resp. one vertex) from it, leaving $B + B$ which is a ${\mathcal{P}}$-position. - Otherwise, the first player plays either on $S$ or on $S'$ in either of the two bistars. Note that the first player cannot remove the central vertex of $S'$, since this case has already been treated above. The second player answers by replicating his move on the other bistar, which is always a legal move, allowing us to invoke the induction hypothesis. Theorem \[thm:modbipodes\] then follows from Lemmas \[lem:modbipodeschain\] and \[lem:modbipodesarms\]. As in the subdivided stars section, we are left with a limited number of bistars to study. The next subsection presents the study of the Grundy value of a subdivided bistar depending on the Grundy values of its subdivided stars. Computing the Grundy value of a subdivided bistar ------------------------------------------------- We will express the Grundy value of a subdivided star as a function of the Grundy values of its two stars. By Lemma \[lem:modbipodeschain\], it is enough to consider bistars whose middle path has length either 1 or 2. We consider these two cases separately. ### When the middle path is of length 1 Playing on a subdivided bistar with a middle path of length 1 is almost equivalent to playing in the disjoint union of the two subdivided stars, except for small cases when some moves are not available in the bistar. We will see in what follows that except for some small cases, the Grundy value of the bistar is indeed the nim-sum of the Grundy values of the two stars. We refine the equivalence relation $\equiv$ for subdivided stars as follows. Let $S$ and $S'$ be two subdivided stars. We say that $S$ and $S'$ are $\sim_1$-equivalent, denoted $S \sim_1 S'$, if and only if for any subdivided star $\hat{S}$, $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv S' {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$. Note that the Grundy value of a bistar $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S'$ only depends of the equivalence class under $\sim_1$ of $S$ and $S'$. The equivalence $\sim_1$ is a refinement of $\equiv$ since taking $\hat{S}=\emptyset$ we have $S\equiv S'$. By Lemma \[lem:P3empty\], we already know that $P_3\sim_1\emptyset$, and thus $S_2 \sim_1 \emptyset$ and [$S_{1,1}$]{}$\sim_1\emptyset$. We will prove that there are actually eight equivalence classes for $\sim_1$: - ${\mathcal C_1^*}=\{P_1,$[$S_{2,1}$]{},[$S_{2,2,2}$]{}$\}$ (these stars have Grundy value 1); - ${\mathcal C_2^*}=\{P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy value 2); - ${\mathcal C_2^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=2$ and $S$ contains one or three paths of length $2$; - ${\mathcal C_3^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=3$ and S contains one or three paths of length $2$; - For $i\in \{0,1,2,3\}$, $\mathcal C_i$: subdivided stars $S$ with ${\mathcal{G}}(S)=i$ and $S$ is not in a previous class. Figure \[fig:tabEquivSim1\] shows the equivalence classes of the subdivided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Number of paths of length 2 in the subdivided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number of paths in the subdivided star]{}; (-1.2,0) node [0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{}; (-1.2,-5) node [5]{}; (-1.5,-6) node […]{}; (-1.5,-7) node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) node [2]{}; (3,1.8) node [3]{}; (4,1.8) node [4]{}; (5,1.8) node [5]{}; (6,1.8) node […]{}; (7,1.8) node [$2p$]{}; (8,1.8) node [$2p+1$]{}; (empty) at (0,1) [0]{}; (p1) at (0,0) [$1^*$]{}; (p2) at (0,-1) [$2^*$]{}; (p3) at (1,-1) [0]{}; (p3b) at (0,-2) [0]{}; (p4) at (1,-2) [$1^*$]{}; (p5) at (2,-2) [$2^*$]{}; (s111) at (0,-3) [1]{}; (s112) at (1,-3) [$2^\Box$]{}; (s122) at (2,-3) [0]{}; (s222) at (3,-3) [$1^*$]{}; (s1111) at (0,-4) [0]{}; (s1112) at (1,-4) [$3^\Box$]{}; (s1122) at (2,-4) [1]{}; (s1222) at (3,-4) [$2^\Box$]{}; (s2222) at (4,-4) [0]{}; (s11111) at (0,-5) [1]{}; (s11112) at (1,-5) [$2^\Box$]{}; (s11122) at (2,-5) [0]{}; (s11222) at (3,-5) [$3^\Box$]{}; (s12222) at (4,-5) [1]{}; (s22222) at (5,-5) [2]{}; (p1) to (empty); (p2) to\[out=120, in=-120\] (empty); (p3b) to\[out=120, in=-120\] (p1); (p2) to (p1); (p3) to (p1); (p3) to (p2); (p3b) to (p2); (p4) to (p3b); (p4) to (p2); (p4) to (p3); (p5) to (p3); (p5) to (p4); (s111) to (p3b); (s112) to (p3b); (s112) to (p4); (s112) to (s111); (s122) to (p5); (s122) to (p4); (s122) to (s112); (s222) to (p5); (s222) to (s122); (s1111) to (s111); (s1112) to (s111); (s1112) to (s112); (s1112) to (s1111); (s1122) to (s112); (s1122) to (s122); (s1122) to (s1112); (s1222) to (s122); (s1222) to (s222); (s1222) to (s1122); (s2222) to (s222); (s2222) to (s1222); (s11111) to (s1111); (s11112) to (s1111); (s11112) to (s1112); (s11112) to (s11111); (s11122) to (s1112); (s11122) to (s1122); (s11122) to (s11112); (s11222) to (s1122); (s11222) to (s1222); (s11222) to (s11122); (s12222) to (s1222); (s12222) to (s2222); (s12222) to (s11222); (s22222) to (s2222); (s22222) to (s12222); (0,-7) node [$0$]{}; (1,-7) node [$3^\Box$]{}; (2,-7) node [$1$]{}; (3,-7) node [$2^\Box$]{}; (4,-7) node [$0$]{}; (5,-7) node [$3$]{}; (6,-7) node […]{}; (7,-7) node [$0$]{}; (0,-8) node [$1$]{}; (1,-8) node [$2^\Box$]{}; (2,-8) node [$0$]{}; (3,-8) node [$3^\Box$]{}; (4,-8) node [$1$]{}; (5,-8) node [$2$]{}; (6,-8) node […]{}; (7,-8) node [$1$]{}; (8,-8) node [$2$]{}; (0.2,-7) – (0.8,-7); (1.2,-7) – (1.8,-7); (2.2,-7) – (2.8,-7); (3.2,-7) – (3.8,-7); (4.2,-7) – (4.8,-7); (0.2,-8) – (0.8,-8); (1.2,-8) – (1.8,-8); (2.2,-8) – (2.8,-8); (3.2,-8) – (3.8,-8); (4.2,-8) – (4.8,-8); (7.2,-8) – (7.8,-8); (0,-7.2) – (0,-7.8); (1,-7.2) – (1,-7.8); (2,-7.2) – (2,-7.8); (3,-7.2) – (3,-7.8); (4,-7.2) – (4,-7.8); (5,-7.2) – (5,-7.8); (7,-7.2) – (7,-7.8); (0.2,-7.2) – (0.8,-7.8); (1.2,-7.2) – (1.8,-7.8); (2.2,-7.2) – (2.8,-7.8); (3.2,-7.2) – (3.8,-7.8); (4.2,-7.2) – (4.8,-7.8); (7.2,-7.2) – (7.8,-7.8); \[thm:equiv1\] The equivalence classes for $\sim_1$ are exactly the sets $\mathcal C_0$, $\mathcal C_1$, ${\mathcal C_1^*}$, $\mathcal C_2$, ${\mathcal C_2^*}$, ${\mathcal C_2^{\Box}}$, $\mathcal C_3$ and ${\mathcal C_3^{\Box}}$. Moreover, Table \[tab:prod1\] describes how the Grundy value of $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S'$ can be computed depending on the equivalence class of $S$ and $S'$. [c|c|c|c|c|c|c|c|c]{} &C\_0 & C\_1 & [C\_1\^\*]{}& C\_2 & [C\_2\^\*]{}& [C\_2\^]{}& C\_3 & [C\_3\^]{}\ C\_0 & & & & & & & &\ C\_1 & & & & & & & &\ [C\_1\^\*]{}& & & 2 & & 0 & & &\ C\_2 & & & & & & & &\ [C\_2\^\*]{}& & & 0 & & 1 & 1 & & 0\ [C\_2\^]{}& & & & & 1 & & &\ C\_3 & & & & & & & &\ [C\_3\^]{}& & & & & 0 & & &\ We will need some technical lemmas before proving the theorem: \[lem:equivstar\] We have: 1. $P_1 \sim_1$ [$S_{2,1}$]{} 2. $P_2 \sim_1$ [$S_{2,2}$]{} 3. [$S_{1,1}$]{} $\sim_1$ [$S_{2,2,1}$]{} 4. [$S_{2,1}$]{} $\sim_1$ [$S_{2,2,2}$]{}. Therefore, any two elements in ${\mathcal C_1^*}$ (resp. ${\mathcal C_2^*}$) are $\sim_1$-equivalent. Each of these equivalences will be proved in the same way: for an equivalence $S \sim_1 S'$, we prove that for every subdivided star $\hat{S}$, ${\mathcal{G}}(\hat{S} {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = {\mathcal{G}}(\hat{S} {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S')$. We will use induction on the size of $\hat{S}$. The base cases will be when $|\hat{S}| \in \{0,1,2\}$, that is to say when the first player is able to take the central vertex of $\hat{S}$. Each of these cases corresponds to a subdivided star, whose Grundy value is given in Figure \[fig:tabgrun\]. In the inductive part, we need to prove that for every move on $\hat{S}{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S + \hat{S} {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S'$ by the first player, the second player has a move leading to a ${\mathcal{P}}$-position. In every case, if the first player plays on $\hat{S}$, then the second player can replicate the move, allowing us to invoke the induction hypothesis. Thus, we will only consider the moves on $S$ or $S'$ in each case. **Case 1 :** $P_1 \sim_1$ [$S_{2,1}$]{} Figure \[fig:pod0EQUIV1pod21\] shows the possible moves on $P_1$ or [$S_{2,1}$]{}, and the answer leading to a ${\mathcal{P}}$-position (for readability, we write $S$ instead of $\hat{S}$ in the figure). (orig) at (0,0) [ ]{}; (removeOnePod0) at (4.5,0) \(2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (-1,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); ; (removeOnePod0B) at (9,0) \(2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (-1,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (6) – (7); ; (13,0.25) node [${\mathcal{P}}$ by]{}; (13,-0.25) node [$\emptyset \sim_1 P_3$]{}; (playToPod2) at (4.5,-1.5) \(1) at (-1,0) ; (2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (1) – (2); (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (6) – (7); ; (playToPod1) at (4.5,-3) \(1) at (-1,0) ; (2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (1) – (2); (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (5); ; (playToPod0) at (9,-2.25) \(1) at (-1,0) ; (2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (1) – (2); (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); ; (13,-2.25) node [${\mathcal{P}}$]{}; (removeOnePod21) at (4.5,-4.5) \(1) at (-1,0) ; (2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (1) – (2); (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); ; (removeOnePod21B) at (9,-4.5) \(2) at (-2,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (-1,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); ; (13,-4.25) node [${\mathcal{P}}$ by]{}; (13,-4.75) node [$\emptyset \sim_1$ [$S_{1,1}$]{}]{}; (playToPm) at (4.5,-6) (2b) at (-2,0) [$P_{m-1}$]{}; (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.4,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); ; (4.5,-7) node [(if the first player can]{}; (4.5,-7.5) node [remove the central vertex,]{}; (4.5,-8) node [then $S=P_m$ with $m \geq 3$)]{}; (playToPmB) at (9,-6) (2b) at (-2,0) [$P_{m-1}$]{}; (-0.75,0) node [+]{}; (3b) at (0.5,0) [$P_{m+2}$]{}; ; (13,-6) node [${\mathcal{P}}$]{}; (orig) – (2.25,0) \[-&gt;\] (2.25,0) – (removeOnePod0); (removeOnePod0) – (removeOnePod0B); (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (playToPod1); (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (playToPod2); (playToPod1) – (playToPod0); (playToPod2) – (playToPod0); (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (removeOnePod21); (removeOnePod21) – (removeOnePod21B); (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (playToPm); (playToPm) – (playToPmB); **Case 2 :** $P_2 \sim_1$ [$S_{2,2}$]{} Figure \[fig:pod1EQUIV1pod22\] shows the possible moves on $P_2$ or [$S_{2,2}$]{}, and the answer leading to a ${\mathcal{P}}$-position (for readability, we write $S$ instead of $\hat{S}$ in the figure). (orig) at (0,0) [ ]{}; (removeTwoPod1) at (5,0) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (-1.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeTwoPod1B) at (10,0) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (-1.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (6) – (7); ; (14,0.25) node [${\mathcal{P}}$ by]{}; (14,-0.25) node [$\emptyset \sim_1$[$S_{2}$]{}]{}; (removeOnePod22) at (5,-1.5) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (5) – (5b); ; (removeTwoPod22) at (5,-3) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (6) – (7); ; (removePod22) at (10,-2.25) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); ; (14,-2.25) node [${\mathcal{P}}$]{}; (removeOnePod1) at (5,-4.5) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (2) – (2b); (-0.9,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeOnePod1B) at (10,-4.5) \(1) at (-1.1,1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (2) – (2b); (-0.9,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); ; (14,-4.25) node [${\mathcal{P}}$ by]{}; (14,-4.75) node [**Case 1**]{}; (orig) – (2.5,0) \[-&gt;\] (2.5,0) – (removeTwoPod1); (removeTwoPod1) – (removeTwoPod1B); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeOnePod22); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeTwoPod22); (removeOnePod22) – (removePod22); (removeTwoPod22) – (removePod22); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeOnePod1); (removeOnePod1) – (removeOnePod1B); **Case 3 :** [$S_{1,1}$]{} $\sim_1$ [$S_{2,2,1}$]{} Figure \[fig:pod11EQUIV1pod221\] shows the possible moves on [$S_{1,1}$]{} or [$S_{2,2,1}$]{}, and the answer leading to a ${\mathcal{P}}$-position (for readability, we write $S$ instead of $\hat{S}$ in the figure). (orig) at (0,0) [ ]{}; (removeTwoPod1) at (5,0) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeTwoPod1B) at (10,0) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (4) to (6); (6) – (7); ; (14,0.25) node [${\mathcal{P}}$ by]{}; (14,-0.25) node [[$S_{1,1}$]{}$\sim_1$[$S_{2}$]{}]{}; (removeOnePod22) at (5,-1.5) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); (4) to (5); (5) – (5b); ; (removeTwoPod22) at (5,-3) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); (6) – (7); ; (removePod22) at (10,-2.25) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); ; (14,-2.25) node [${\mathcal{P}}$]{}; (removeOnePod1) at (5,-4.5) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeOnePod1B) at (10,-4.5) \(1) at (-1.1,1) ; (1b) at (-1.1,-1) ; (2) at (-2.1,0) ; (2b) at (-3.1,0) ; (2c) at (-3.65,0) [$S$]{}; (-3,0) – (-3.75,0.5); (-3,0) – (-3.75,-0.5); (-3.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (14,-4.25) node [${\mathcal{P}}$ by]{}; (14,-4.75) node [**Case 2**]{}; (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeTwoPod1); (removeTwoPod1) – (removeTwoPod1B); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeOnePod22); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeTwoPod22); (removeOnePod22) – (removePod22); (removeTwoPod22) – (removePod22); (orig) – (2.5,0) \[-&gt;\] (2.5,0) |- (removeOnePod1); (removeOnePod1) – (removeOnePod1B); **Case 4 :** [$S_{2,1}$]{} $\sim_1$ [$S_{2,2,2}$]{} Figure \[fig:pod21EQUIV1pod222\] shows the possible moves on [$S_{2,1}$]{} or [$S_{2,2,2}$]{}, and the answer leading to a ${\mathcal{P}}$-position (for readability, we write $S$ instead of $\hat{S}$ in the figure). (orig) at (0,0) [ ]{}; (removeTwoFrom21) at (5.5,0) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (2) – (2b); (2) – (1b); (-1.1,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (8b) at (4,0) ; (3) – (4); (8) – (4); (8) – (8b); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeTwoFrom21B) at (11,0) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (2) – (2b); (2) – (1b); (-1.1,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (14.5,0.25) node [${\mathcal{P}}$ by]{}; (14.5,-0.25) node [**Case 2**]{}; (removeOneFrom222) at (5.5,-1.5) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (1) – (1a); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (8b) at (4,0) ; (3) – (4); (8) – (4); (8) – (8b); (4) to (6); (4) to (5); (6) – (7); ; (removeTwoFrom222) at (5.5,-3) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (1) – (1a); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (removeFrom222) at (11,-2.25) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (1) – (1a); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (4,1) ; (3) – (4); (4) to (6); (4) to (5); (6) – (7); ; (14.5,-2.25) node [${\mathcal{P}}$]{}; (playTo11) at (5.5,-4.5) \(1) at (-2.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.85,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (8b) at (4,0) ; (3) – (4); (8) – (4); (8) – (8b); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (playTo11B) at (11,-4.5) \(1) at (-2.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (2) – (1b); (-0.85,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (14.5,-4.25) node [${\mathcal{P}}$ by]{}; (14.5,-4.75) node [**Case 3**]{}; (playTo2) at (5.5,-6) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (1) – (1a); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (8b) at (4,0) ; (3) – (4); (8) – (4); (8) – (8b); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (playTo2B) at (11,-6) \(1) at (-2.1,1) ; (1a) at (-1.1,1) ; (1b) at (-2.1,-1) ; (2) at (-3.1,0) ; (2b) at (-4.1,0) ; (2c) at (-4.65,0) [$S$]{}; (-4,0) – (-4.75,0.5); (-4,0) – (-4.75,-0.5); (-4.75,0.5) arc (135:225:0.7); (1) – (2); (2) – (2b); (1) – (1a); (-0.6,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (5b) at (4,-1) ; (7) at (4,1) ; (8) at (3,0) ; (3) – (4); (8) – (4); (4) to (6); (4) to (5); (6) – (7); (5) – (5b); ; (14.5,-5.5) node [${\mathcal{P}}$ by]{}; (14.5,-6) node [**Case 3**]{}; (14.5,-6.5) node [since]{}; (14.5,-7) node [[$S_{1,1}$]{}$\sim_1$[$S_{2}$]{}]{}; (orig) – (2.75,0) \[-&gt;\] (2.75,0) – (removeTwoFrom21); (removeTwoFrom21) – (removeTwoFrom21B); (orig) – (2.75,0) \[-&gt;\] (2.75,0) |- (removeOneFrom222); (orig) – (2.75,0) \[-&gt;\] (2.75,0) |- (removeTwoFrom222); (removeOneFrom222) – (removeFrom222); (removeTwoFrom222) – (removeFrom222); (orig) – (2.75,0) \[-&gt;\] (2.75,0) |- (playTo11); (playTo11) – (playTo11B); (orig) – (2.75,0) \[-&gt;\] (2.75,0) |- (playTo2); (playTo2) – (playTo2B);  \ \[lem:onestarone\] Let $S$ be a subdivided star not belonging to ${\mathcal C_1^*}\cup{\mathcal C_2^*}$. Then [$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S \equiv P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S$. We use induction on $|S|$. The base cases are the subdivided stars having an option in ${\mathcal C_1^*}\cup{\mathcal C_2^*}$: 1. $S=\emptyset$. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = {\mathcal{G}}($[$S_{1,1,1}$]{}$) = 1 = {\mathcal{G}}(P_1) = {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. 2. $S=$[$S_{1,1}$]{}. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = {\mathcal{G}}($[$S_{1,1,1}$]{}$)$ (by Lemma \[lem:equivstar\]) $= {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. 3. $S=$[$S_{2,1,1}$]{}. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = 3 = {\mathcal{G}}($[$S_{2,1,1,1}$]{}$) = {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. 4. $S=$[$S_{2,2,1}$]{}. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = {\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}$[$S_{1,1}$]{}$)$ (by Lemma \[lem:equivstar\]) $= 1 = {\mathcal{G}}($[$S_{2,2,1,1}$]{}$) = {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. 5. $S=$[$S_{2,2,2,1}$]{}. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = 3 = {\mathcal{G}}($[$S_{2,2,2,1,1}$]{}$) = {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. 6. $S=$[$S_{2,2,2,2}$]{}. In this case, ${\mathcal{G}}($[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S) = 1 = {\mathcal{G}}($[$S_{2,2,2,2,1}$]{}$) = {\mathcal{G}}(P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. Although tedious, all these values can be computed by considering the Grundy values of the sets ${\mathrm{opt}}(S_{1,1,1}{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$ and ${\mathrm{opt}}(P_1{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S)$. We now prove that if $S$ is a subdivided star not belonging to ${\mathcal C_1^*}\cup{\mathcal C_2^*}$ and not having an option in ${\mathcal C_1^*}\cup{\mathcal C_2^*}$, then [$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S \equiv P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S$. We note that the first player can neither empty $S$ nor take its central vertex. We show that for every first player’s move on [$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S + P_1 {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}S$, the second player can always move to a ${\mathcal{P}}$-position. If the first player plays from $S$ to $S'$, then $S' \not\in {\mathcal C_1^*}\cup{\mathcal C_2^*}$, thus if the second player replicates the move, we can invoke the induction hypothesis. Figure \[fig:onestarone\] shows the case where the first player does not play on $S$, completing the proof. (orig) at (0,0) [ ]{}; (playOnPod0) at (4.5,0) \(2) at (-2.1,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (7) at (3,0) ; (3) to (4); (4) to (5); (4) to (6); (4) to (7); ; (playOnPod111) at (4.5,-1.5) \(1) at (-1.1,0) ; (2) at (-2.1,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (1) – (2); (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (3) to (4); (4) to (5); (4) to (6); ; (playOnPods) at (9,-0.75) \(2) at (-2.1,0) ; (-2,0) – (-2.75,0.5); (-2,0) – (-2.75,-0.5); (-2.75,0.5) arc (135:225:0.7); (2b) at (-2.65,0) [$S$]{}; (-0.4,0) node [+]{}; (1,0) – (0.25,0.5); (1,0) – (0.25,-0.5); (0.25,0.5) arc (135:225:0.7); (3b) at (0.35,0) [$S$]{}; (3) at (1,0) ; (4) at (2,0) ; (6) at (3,1) ; (5) at (3,-1) ; (3) to (4); (4) to (5); (4) to (6); ; (13,-0.5) node [${\mathcal{P}}$ by]{}; (13,-1) node [$\emptyset \sim_1$[$S_{1,1}$]{}]{}; (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (playOnPod0); (orig) – (2.25,0) \[-&gt;\] (2.25,0) |- (playOnPod111); (playOnPod0) – (playOnPods); (playOnPod111) – (playOnPods); We prove by induction on the total number of vertices of $S$ and $S'$ that if $S$ and $S'$ are in the same set $\mathcal C_0$, $\mathcal C_1$, ${\mathcal C_1^*}$, $\mathcal C_2$, ${\mathcal C_2^*}$, ${\mathcal C_2^{\Box}}$, $\mathcal C_3$ or ${\mathcal C_3^{\Box}}$, then they are $\sim_1$-equivalent. By Lemma \[lem:equivstar\], this is true if $S$ and $S'$ are in ${\mathcal C_1^*}$ or in ${\mathcal C_2^*}$. This is also true if $\{S,S'\}=\{\emptyset,$[$S_{1,1}$]{}$\}$ by Lemma \[lem:P3empty\] or if $\{S,S'\}=\{\emptyset,P_3\}$ since it is the same as attaching a $P_3$ to the central vertex of a subdivided star. Furthermore, one can check that the rows and columns for ${\mathcal C_1^*}$ and ${\mathcal C_2^*}$ in Table \[tab:prod1\] are correct. For that, it suffices to prove it for one representant of ${\mathcal C_1^*}$ ($P_1$) and one representant of ${\mathcal C_2^*}$ ($P_2$). Attaching $P_1$ to any subdivided star $\hat{S}$ results in the subdivided star listed directly under $\hat{S}$ in Figure \[fig:tabpos\], while attaching $P_2$ results in the subdivided star listed diagonally to the right and below. For example, for [$S_{1,1,2,2}$]{}, attaching $P_1$ results in [$S_{1,1,1,2,2}$]{}, while attaching $P_2$ results in [$S_{1,1,2,2,2}$]{}. Comparing the Grundy values of the individual stars and the resulting bistar, one can verify that the table columns for ${\mathcal C_1^*}$ and ${\mathcal C_2^*}$ are correct. Identifying the elements of the various sets in Figure \[fig:tabEquivSim1\], we see that below any element of ${\mathcal C_1^*}$ is a bisected star with Grundy value 2, and similarly, below any element of ${\mathcal C_2^*}$ is a bisected star with Grundy value 0. All other stars are either part of a 0-1 pattern (going down the columns), or part of a 2-3 pattern, which fits the computation of the Grundy values via the nim-sum, since $3 \oplus 1 = 2$. This verifies the result for the ${\mathcal C_1^*}$ column. Likewise, one can verify the column ${\mathcal C_2^*}$. Suppose now that $S$ and $S'$ belong to the same set $C$, with $C\neq {\mathcal C_1^*}$ and $C\neq {\mathcal C_2^*}$. Thus both $S$ and $S'$ are either empty or not a path. We prove by induction on the size of $\hat{S}$ that $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ for any subdivided star $\hat{S}$. This is true if $\hat{S}=\emptyset$ (since ${\mathcal{G}}(S)={\mathcal{G}}(S')$) or if $\hat{S}\in {\mathcal C_1^*}\cup {\mathcal C_2^*}$ (as discussed before). Hence we can assume that $\hat{S}\notin {\mathcal C_1^*}\cup{\mathcal C_2^*}$ and $\hat{S}$ is not a path. We will prove that $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} + S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ is a ${\mathcal{P}}$-position. The first player cannot play both in $S$ and $\hat{S}$ nor both in $S'$ and $\hat{S}$ since $\hat{S}$ is not a path. If the first player plays in $\hat{S}$, leading to $\hat{S}'$ in one of the two games, the first player cannot take the central vertex (since $\hat{S}$ is not a path). Hence the second player can reply to $S{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}' + S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}'$ which is a ${\mathcal{P}}$-position by induction hypothesis. Otherwise, the first player plays in $S$ or in $S'$. By symmetry, we can assume that the first player plays in $S$, leading to a game $T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} + S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$. We have to find an answer from that game to a ${\mathcal{P}}$-position. 1. If there is a move from $T$ to $T'$ with $T'$ in the same set as $S$, then the second player plays to $T'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} + S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ (this is always possible since if the move from $T$ to $T'$ is taking the central vertex and $T'$ is not empty, it means that $T$ is a path which is neither $P_3$ nor $P_4$, a contradiction). By induction, $T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} + S'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ is a ${\mathcal{P}}$-position. 2. If there is a move from $S'$ to $T'$ with $T$ and $T'$ in the same set, then the second player plays to $T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}+ T'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ (again, this is always possible since $S'$ is not a path), which is a ${\mathcal{P}}$-position by induction hypothesis. Assume that none of these two cases occurs. If ${\mathcal{G}}(S)=3$ then we are in case [*(ii)*]{}. If ${\mathcal{G}}(S)=0$ then we are in case [*(i)*]{}. Hence we have ${\mathcal{G}}(S)\in \{1,2\}$. If ${\mathcal{G}}(S)=1$, then $S,S'\in \mathcal C_1$. If ${\mathcal{G}}(T)=0$ then we are in case [*(ii)*]{}. Otherwise, ${\mathcal{G}}(T)>1$, and there is always a move from $T$ to $T'\in \mathcal C_1$ and we are in case [*(ii)*]{}. Hence ${\mathcal{G}}(S)=2$. If ${\mathcal{G}}(T)=0$ or if $T\in \mathcal C_1$, then we are in case [*(ii)*]{}. If ${\mathcal{G}}(T)=3$, we are in case [*(i)*]{}. Hence the only remaining case is $T\in {\mathcal C_1^*}$. Then there is a move from $S'$ to $T'$ with $T'\in \mathcal C_1$. By induction, $T'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv $[$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ (indeed, the number of vertices in $S$ and $S'$ is strictly greater than the number of vertices in $T'$ and [$S_{1,1,1}$]{} since $S$ has at least five vertices). By Lemma \[lem:onestarone\], [$S_{1,1,1}$]{}${\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv P_1{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ (since $\hat{S}\notin {\mathcal C_1^*}\cup {\mathcal C_2^*}$). Thus $T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S} \equiv T'{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ and $T{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}+ T' {\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {};}}\hat{S}$ is a ${\mathcal{P}}$-position. To compute Table \[tab:prod1\], it is enough to consider one representant of each class, for instance $\emptyset$, $P_1$, $P_2$, [$S_{1,1,1}$]{}, [$S_{2,1,1}$]{}, [$S_{2,1,1,1}$]{}, [$S_{2,2,2,2,2}$]{}, [$S_{2,2,2,2,2,1}$]{}, respectively, and compute their Grundy value. ### When the middle path is of length 2 The situation in that case will be more complicated than in the previous case. We similarly define an equivalence relation $\sim_2$. Let $S$ and $S'$ be two subdivided stars. We say that $S$ and $S'$ are $\sim_2$-equivalent, denoted $S \sim_2 S'$, if and only if for any subdivided star $\hat{S}$, $S{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\hat{S} \equiv S' {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\hat{S}$. By Lemma \[lem:P3onevertex\], we already know that $P_3\sim_2\emptyset$, and thus $S_2 \sim_2 \emptyset$ and [$S_{1,1}$]{}$\sim_2\emptyset$. We will prove that there are exactly ten equivalence classes for $\sim_2$: - ${\mathcal D_0^*}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=0$ and $S$ contains zero or two paths of length $2$, plus [$S_{2}$]{}; - ${\mathcal D_1^*}=\{P_1,$[$S_{2,1}$]{},[$S_{2,2,2}$]{}$\}$ (these stars have Grundy value 1); - ${\mathcal D_1^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=1$, $S$ contains zero or two paths of length $2$ and $S \neq P_1$; - ${\mathcal D_2^*}=\{P_2$,[$S_{2,2}$]{}$\}$ (these stars have Grundy value 2); - ${\mathcal D_2^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=2$ and $S$ contains one or three paths of length $2$; - ${\mathcal D_3^{\Box}}$: subdivided stars $S$ such that ${\mathcal{G}}(S)=3$ and $S$ contains one or three paths of length $2$; - For $i\in \{0,1,2,3\}$, $\mathcal D_i$: subdivided stars $S$ with ${\mathcal{G}}(S)=i$ and $S$ is not in a previous class. Figure \[fig:tabEquivSim2\] shows the equivalence classes of the subdivided stars. (-1,1.5) – (-1,-8.5); (-1,1.5) – (8.5,1.5); (3.75,2.4) node [Number of paths of length 2 in the subdivided star]{}; (-2.3,-3.5) node\[rotate=90\] [Number of paths in the subdivided star]{}; (-1.2,0) node [0]{}; (-1.2,-1) node [1]{}; (-1.2,-2) node [2]{}; (-1.2,-3) node [3]{}; (-1.2,-4) node [4]{}; (-1.2,-5) node [5]{}; (-1.5,-6) node […]{}; (-1.5,-7) node [$2p$]{}; (-1.5,-8) node [$2p+1$]{}; (0,1.8) node [0]{}; (1,1.8) node [1]{}; (2,1.8) node [2]{}; (3,1.8) node [3]{}; (4,1.8) node [4]{}; (5,1.8) node [5]{}; (6,1.8) node […]{}; (7,1.8) node [$2p$]{}; (8,1.8) node [$2p+1$]{}; (empty) at (0,1) [$0^*$]{}; (p1) at (0,0) [$1^*$]{}; (p2) at (0,-1) [$2^*$]{}; (p3) at (1,-1) [$0^*$]{}; (p3b) at (0,-2) [$0^*$]{}; (p4) at (1,-2) [$1^*$]{}; (p5) at (2,-2) [$2^*$]{}; (s111) at (0,-3) [$1^\Box$]{}; (s112) at (1,-3) [$2^\Box$]{}; (s122) at (2,-3) [$0^*$]{}; (s222) at (3,-3) [$1^*$]{}; (s1111) at (0,-4) [$0^*$]{}; (s1112) at (1,-4) [$3^\Box$]{}; (s1122) at (2,-4) [$1^\Box$]{}; (s1222) at (3,-4) [$2^\Box$]{}; (s2222) at (4,-4) [0]{}; (s11111) at (0,-5) [$1^\Box$]{}; (s11112) at (1,-5) [$2^\Box$]{}; (s11122) at (2,-5) [$0^*$]{}; (s11222) at (3,-5) [$3^\Box$]{}; (s12222) at (4,-5) [1]{}; (s22222) at (5,-5) [2]{}; (p1) to (empty); (p2) to\[out=120, in=-120\] (empty); (p3b) to\[out=120, in=-120\] (p1); (p2) to (p1); (p3) to (p1); (p3) to (p2); (p3b) to (p2); (p4) to (p3b); (p4) to (p2); (p4) to (p3); (p5) to (p3); (p5) to (p4); (s111) to (p3b); (s112) to (p3b); (s112) to (p4); (s112) to (s111); (s122) to (p5); (s122) to (p4); (s122) to (s112); (s222) to (p5); (s222) to (s122); (s1111) to (s111); (s1112) to (s111); (s1112) to (s112); (s1112) to (s1111); (s1122) to (s112); (s1122) to (s122); (s1122) to (s1112); (s1222) to (s122); (s1222) to (s222); (s1222) to (s1122); (s2222) to (s222); (s2222) to (s1222); (s11111) to (s1111); (s11112) to (s1111); (s11112) to (s1112); (s11112) to (s11111); (s11122) to (s1112); (s11122) to (s1122); (s11122) to (s11112); (s11222) to (s1122); (s11222) to (s1222); (s11222) to (s11122); (s12222) to (s1222); (s12222) to (s2222); (s12222) to (s11222); (s22222) to (s2222); (s22222) to (s12222); (0,-7) node [$0^*$]{}; (1,-7) node [$3^\Box$]{}; (2,-7) node [$1^\Box$]{}; (3,-7) node [$2^\Box$]{}; (4,-7) node [$0$]{}; (5,-7) node [$3$]{}; (6,-7) node […]{}; (7,-7) node [$0$]{}; (0,-8) node [$1^\Box$]{}; (1,-8) node [$2^\Box$]{}; (2,-8) node [$0^*$]{}; (3,-8) node [$3^\Box$]{}; (4,-8) node [$1$]{}; (5,-8) node [$2$]{}; (6,-8) node […]{}; (7,-8) node [$1$]{}; (8,-8) node [$2$]{}; (0.2,-7) – (0.8,-7); (1.2,-7) – (1.8,-7); (2.2,-7) – (2.8,-7); (3.2,-7) – (3.8,-7); (4.2,-7) – (4.8,-7); (0.2,-8) – (0.8,-8); (1.2,-8) – (1.8,-8); (2.2,-8) – (2.8,-8); (3.2,-8) – (3.8,-8); (4.2,-8) – (4.8,-8); (7.2,-8) – (7.8,-8); (0,-7.2) – (0,-7.8); (1,-7.2) – (1,-7.8); (2,-7.2) – (2,-7.8); (3,-7.2) – (3,-7.8); (4,-7.2) – (4,-7.8); (5,-7.2) – (5,-7.8); (7,-7.2) – (7,-7.8); (0.2,-7.2) – (0.8,-7.8); (1.2,-7.2) – (1.8,-7.8); (2.2,-7.2) – (2.8,-7.8); (3.2,-7.2) – (3.8,-7.8); (4.2,-7.2) – (4.8,-7.8); (7.2,-7.2) – (7.8,-7.8); \[thm:equiv2\] The equivalence classes for $\sim_2$ are exactly the sets $\mathcal D_0$, ${\mathcal D_0^*}$, $\mathcal D_1$, ${\mathcal D_1^*}$, ${\mathcal D_1^{\Box}}$, $\mathcal D_2$, ${\mathcal D_2^*}$, ${\mathcal D_2^{\Box}}$, $\mathcal D_3$ and ${\mathcal D_3^{\Box}}$. Moreover, Table \[tab:prod2\] describes how the Grundy value of $S{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$ can be computed depending on the equivalence class of $S$ and $S'$. [c|c|c|c|c|c|c|c|c|c|c]{} &D\_0 & [D\_0\^\*]{}& D\_1 & [D\_1\^\*]{}& [D\_1\^]{}& D\_2 & [D\_2\^\*]{}& [D\_2\^]{}& D\_3 & [D\_3\^]{}\ D\_0 & & \_1 & & 2 & \_1 & & 0 & \_1 & & \_1\ [D\_0\^\*]{}& \_1 & \_1 & \_1 & 2 & \_1 & \_1 & 0 & \_1 & \_1 & \_1\ D\_1 & & \_1 & & 3 & \_1 & & 1 & \_1 & & \_1\ [D\_1\^\*]{}& 2 & 2 & 3 & 0 & 3 & 0 & 1 & 1 & 1 & 0\ [D\_1\^]{}& \_1 & \_1 & \_1 & 3 & \_1 & \_1 & 1 & \_1 & \_1 & \_1\ D\_2 & & \_1 & & 0 & \_1 & & 2 & \_1 & & \_1\ [D\_2\^\*]{}& 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3\ [D\_2\^]{}& \_1 & \_1 & \_1 & 1 & \_1 & \_1 & 2 & 0 & \_1 & 1\ D\_3 & & \_1 & & 1 & \_1 & & 3 & \_1 & & \_1\ [D\_3\^]{}& \_1 & \_1 & \_1 & 0 & \_1 & \_1 & 3 & 1 & \_1 & 0\ Notice that when the two subdivided stars are of sufficiently large order, they are in the classes $\mathcal D_0,\mathcal D_1,\mathcal D_2,\mathcal D_3$, and the Grundy value of the bistar is given by the nim-sum of the Grundy values of the two stars. For most of the smallest subdivided stars, ${\mathcal{G}}(S{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S') = {\mathcal{G}}(S) \oplus {\mathcal{G}}(S') \oplus 1$. The following lemma proves the equivalence for ${\mathcal D_1^*}$ and ${\mathcal D_2^*}$. Its proof is not included, since it is similar to the proof of Lemma \[lem:equivstar\]. \[lem:equivstar2\] We have: 1. $P_1 \sim_2$ [$S_{2,1}$]{} 2. $P_2 \sim_2$ [$S_{2,2}$]{} 3. [$S_{1,1}$]{} $\sim_2$ [$S_{2,2,1}$]{} 4. [$S_{2,1}$]{} $\sim_2$ [$S_{2,2,2}$]{}. Therefore, any two elements in ${\mathcal D_1^*}$ (resp. ${\mathcal D_2^*}$) are $\sim_2$-equivalent. We can now prove Theorem \[thm:equiv2\]: Rather than proving the validity of equivalence classes and then deducing the table, we prove by induction on the total number of vertices in $S$ and $S'$ that the Grundy value of $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$ is given by Table \[tab:prod2\]. One can check that the rows and columns for ${\mathcal D_1^*}$ and ${\mathcal D_2^*}$ in Table \[tab:prod2\] are correct: it suffices to prove it for one representant for ${\mathcal D_1^*}$ (say $P_1$) and for ${\mathcal D_2^*}$ (say $P_2$). This is possible since if $S,S' \in {\mathcal D_1^*},{\mathcal D_2^*}$, then they are $\sim_2$-equivalent by Lemma \[lem:equivstar2\]. For any subdivided star $\hat{S}$, $\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_1$ is $\hat{S}$ with a path of length 2 attached to its central vertex. Thus, for every class, we only need to look at the value diagonally to the right and below in Figure \[fig:tabgrun\]. One can check that if ${\mathcal{G}}(\hat{S})=0$, then ${\mathcal{G}}(\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_1)=2$, if $\hat{S} \in {\mathcal D_1^*},\mathcal D_2, {\mathcal D_3^{\Box}}$, then ${\mathcal{G}}(\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_1)=0$, if $\hat{S} \in \mathcal D_1,{\mathcal D_1^{\Box}}$, then ${\mathcal{G}}(\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_1)=3$, if $\hat{S} \in {\mathcal D_2^*},{\mathcal D_2^{\Box}},\mathcal D_3$, then ${\mathcal{G}}(\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_1)=1$. For any subdivided star $\hat{S}$, $\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_2$ is $\hat{S}$ with a path of length 3 attached to its central vertex. Thus, ${\mathcal{G}}(\hat{S} {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}P_2) = {\mathcal{G}}(\hat{S})$. Now we study the Grundy value of $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$ depending on the class of $S$ and $S'$. We can suppose that $S,S' \not\in {\mathcal D_1^*},{\mathcal D_2^*}$, and that neither $S$ nor $S'$ are [$S_{1,1}$]{} or $P_3$ (since, by Lemma \[lem:P3onevertex\], [$S_{1,2}$]{}$\sim_2 \emptyset$; and $P_3 \sim_2 \emptyset$ by Lemma \[lem:modkpodes\]). We can find the Grundy values of the options of $S$ and $S'$ thanks to Figure \[fig:tabEquivSim2\]. None of the options of $S$ and $S'$ involves taking their central vertex. We can verify Table \[tab:prod2\] by computing the Grundy value of $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$ depending on the Grundy values of their options, by using the induction hypothesis: ${\mathcal{G}}(S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S') = \operatorname{mex}( {\mathcal{G}}( T {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S' ) , {\mathcal{G}}( S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}T' ) | T$ option of $S$, $T'$ option of $S' )$ In order to prove that the equivalence classes are correct, we need to check that the Grundy value of $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$ does not change with the classes of the options of $S$ and $S'$. Indeed, two subdivided stars belonging to the same class can have different options. We will prove two cases, the other ones being similar. **Case 1:** $S \in \mathcal D_1$ and $S' \in \mathcal D_3$ In this case, $S$ always has three different options, but these options are not the same depending on $S$. $S$ always has an option in $\mathcal D_0$, and it can have two options either in ${\mathcal D_2^{\Box}}$ and ${\mathcal D_3^{\Box}}$ or in $\mathcal D_2$ and $\mathcal D_3$. $S'$ has three options, which are in $\mathcal D_1,\mathcal D_2$ and $\mathcal D_3$. These possible options of $S$ and $S'$ are shown in Figure \[fig:t1tripat2-c1c3\]. On the left are the possible options of $S$, and on the right are the possible options of $S'$. The notation $\mathcal D_i {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_j$ expresses the fact that the two subdivided stars $T$ and $S'$ (resp. $S$ and $T'$) are in the classes $\mathcal D_i$ and $\mathcal D_j$, and that the subdivided bistar is smaller than $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$, allowing us to invoke the induction hypothesis. (orig) at (0,0) [ ]{}; (right1) at (3,-1.5) (0,0) node [$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_0$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_1$]{}; (2.25,0) node [;]{}; (3,0) node [$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; ; (left1) at (-3,-1) (0,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; (0.75,0) node [;]{}; (1.5,0) node [${\mathcal D_2^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; (2.25,0) node [;]{}; (3,0) node [${\mathcal D_3^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; ; (left2) at (-3,-2) (0,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_2 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; (2.25,0) node [;]{}; (3,0) node [$\mathcal D_3 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; ; (orig) |- (right1); (orig) |- (left1); (orig) |- (left2); Now, we can compute the Grundy value of $S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S'$. First, we compute this value in the case where the options of $S$ are in ${\mathcal D_2^{\Box}}$ and ${\mathcal D_3^{\Box}}$: ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\mathcal{G}}(S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S')$ $=$ $\operatorname{mex}( {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( {\mathcal D_2^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( {\mathcal D_3^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_0 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_1 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2 ) )$ $=$ $\operatorname{mex}( 3,0,1,1,0,3 )$ (by induction hypothesis) $=$ 2 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- 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Now, we compute this value in the case where the options of $S$ are in $\mathcal D_2$ and $\mathcal D_3$: ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- 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${\mathcal{G}}(S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S')$ $=$ $\operatorname{mex}( {\mathcal{G}}( \mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( \mathcal D_2 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( \mathcal D_3 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_0 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_1 ) , {\mathcal{G}}( \mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2 ) )$ $=$ $\operatorname{mex}( 3,1,0,1,0,3 )$ (by induction hypothesis) $=$ 2 ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- 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The Grundy value being the same in both cases, we can conclude that ${\mathcal{G}}(S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S')=2$. **Case 2:** $S \in \mathcal D_0$ and $S' \in \mathcal D_2$ In this case, the possible options of $S$ and $S'$ are shown in Figure \[fig:t1tripat2-c0c2\]. On the left are the options of $S$, and on the right are the options of $S'$. Below each possible bistar is the Grundy value of the bistar, thanks to the induction hypothesis. By computing the $\operatorname{mex}$ value of each of the six sets of options, we always find the value 2. Thus, ${\mathcal{G}}(S {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}S')=2$. (orig) at (0,0) [ ]{}; (right1) at (3,-2) (0,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_0$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_1$]{}; (0,-0.5) node [0]{}; (1.5,-0.5) node [1]{}; ; (right2) at (3,-3) (0,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_0$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_1$]{}; (2.25,0) node [;]{}; (3,0) node [$\mathcal D_0 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_3$]{}; (0,-0.5) node [0]{}; (1.5,-0.5) node [1]{}; (3,-0.5) node [3]{}; ; (left1) at (-3,-1) (0,0) node [${\mathcal D_1^*}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0.75,0) node [;]{}; (1.5,0) node [${\mathcal D_2^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0,-0.5) node [0]{}; (1.5,-0.5) node [1]{}; ; (left2) at (-3,-2) (0,0) node [$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0.75,0) node [;]{}; (1.5,0) node [${\mathcal D_2^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (2.25,0) node [;]{}; (3,0) node [${\mathcal D_3^{\Box}}{{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0,-0.5) node [3]{}; (1.5,-0.5) node [1]{}; (3,-0.5) node [0]{}; ; (left3) at (-3,-3) (0,0) node [$\mathcal D_1 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_2 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (2.25,0) node [;]{}; (3,0) node [$\mathcal D_3 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0,-0.5) node [3]{}; (1.5,-0.5) node [0]{}; (3,-0.5) node [1]{}; ; (left4) at (-3,-4) (0,0) node [$\mathcal D_2 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0.75,0) node [;]{}; (1.5,0) node [$\mathcal D_3 {{\tikz[baseline=-4]{\draw (0,0) node[circle,fill=black,minimum size=0,inner sep=1]{} -- (0.3,0) node[circle,fill=black, minimum size=0, inner sep=1] {} node[midway,above,scale=0.5] {$2$};}}}\mathcal D_2$]{}; (0,-0.5) node [0]{}; (1.5,-0.5) node [1]{}; ; (orig) |- (right1); (orig) |- (right2); (orig) |- (left1); (orig) |- (left2); (orig) |- (left3); (orig) |- (left4);   Overall, there are 36 cases to consider. As they are all similar to the two we already considered, we only present the possible classes of the options of $S$ in Figure \[fig:optionsT\]. The full proof can be found in [@halFullProof]. (c0) at (0,0) \(0) at (0,0) ; (1) at (2.5,0) [${\mathcal D_1^*}$ ; ${\mathcal D_2^{\Box}}$]{}; (2) at (2.5,-0.5) [$\mathcal D_1$ ; ${\mathcal D_2^{\Box}}$ ; ${\mathcal D_3^{\Box}}$]{}; (3) at (2.5,-1) [$\mathcal D_1$ ; $\mathcal D_2$ ; $\mathcal D_3$]{}; (4) at (2.5,-1.5) [$\mathcal D_2$ ; $\mathcal D_3$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); (0) – (1,0) |- (3); (0) – (1,0) |- (4); ; (c0\*) at (5,0) \(0) at (0,0) ; (1) at (2.5,0) [${\mathcal D_1^*}$ ; ${\mathcal D_2^*}$]{}; (2) at (2.5,-0.5) [${\mathcal D_1^{\Box}}$]{}; (3) at (2.5,-1) [${\mathcal D_1^*}$ ; ${\mathcal D_2^*}$ ; ${\mathcal D_2^{\Box}}$]{}; (4) at (2.5,-1.5) [${\mathcal D_1^{\Box}}$ ; ${\mathcal D_2^{\Box}}$ ; ${\mathcal D_3^{\Box}}$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); (0) – (1,0) |- (3); (0) – (1,0) |- (4); ; (c1) at (10,0) \(0) at (0,0) ; (1) at (2.5,0) [$\mathcal D_0$ ; ${\mathcal D_2^{\Box}}$ ; ${\mathcal D_3^{\Box}}$]{}; (2) at (2.5,-0.5) [$\mathcal D_0$ ; $\mathcal D_2$ ; $\mathcal D_3$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); ; (c1b) at (0,-2) \(0) at (0,0) ; (1) at (2.5,0) [${\mathcal D_0^*}$]{}; (2) at (2.5,-0.5) [${\mathcal D_0^*}$ ; ${\mathcal D_2^{\Box}}$ ; ${\mathcal D_3^{\Box}}$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); ; (c2) at (5,-2) \(0) at (0,0) ; (1) at (2.5,0) [$\mathcal D_0$ ; $\mathcal D_1$]{}; (2) at (2.5,-0.5) [$\mathcal D_0$ ; $\mathcal D_1$ ; $\mathcal D_3$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); ; (c2b) at (10,-2) \(0) at (0,0) ; (1) at (2.5,0) [${\mathcal D_0^*}$ ; ${\mathcal D_1^*}$ ; ${\mathcal D_1^{\Box}}$]{}; (2) at (2.5,-0.5) [${\mathcal D_0^*}$ ; ${\mathcal D_1^{\Box}}$ ; ${\mathcal D_3^{\Box}}$]{}; (0) – (1,0) – (1); (0) – (1,0) |- (2); ; (c3) at (2.5,-3.5) \(0) at (0,0) ; (1) at (2.5,0) [$\mathcal D_0$ ; $\mathcal D_1$ ; $\mathcal D_2$]{}; (0) – (1,0) – (1); ; (c3b) at (7.5,-3.5) \(0) at (0,0) ; (1) at (2.5,0) [${\mathcal D_0^*}$ ; ${\mathcal D_1^{\Box}}$ ; ${\mathcal D_2^{\Box}}$]{}; (0) – (1,0) – (1); ; Going through all the cases allows to prove the correctness of Table \[tab:prod2\]. This concludes our study of subdivided bistars. Conclusion ========== In this paper, we introduced a general definition of octal games on graphs, capturing some existing take-away games on graphs. We then focused on one of the simplest octal games, [**0.33**]{}, on some subclasses of trees, namely subdivided stars and bistars. We proved that for subdivided stars and bistars, as in paths, one can reduce the length of the paths to their length modulo 3. Thanks to this result, we have computed the exact Grundy value of any subdivided star, and exihibited a periodic behaviour. We have extended these results to bistars for which one can also reduce the lengths of any path modulo 3. Using operators and equivalence classes similar to the nim-sum and Grundy classes, we could then compute the Grundy value of a subdivided bistar using values of the two stars composing it. However, the reduction of paths modulo 3 cannot be generalized to trees: Attaching a $P_3$ to a vertex of a bistar which is not one of the central vertices of the stars may change the Grundy value (and even the outcome) of the game. Indeed, the bistar of Figure \[fig:contreexemple\] is an ${\mathcal{N}}$-position, but attaching a $P_3$ to $u$ changes it into an ${\mathcal{P}}$-position. at (-1,-1) ; at (-1,1) ; at (0,0) ; at (1,0) ; at (2,0) ; at (3,-1) ; at (3,1) ; (3,1.25) node [$u$]{}; at (4,1) ; (0,0) to (-1,-1); (0,0) to (-1,1); (0,0) – (2,0); (2,0) to (3,-1); (2,0) to (3,1); (4,1) – (3,1); The bistar is an ${\mathcal{N}}$-position: removing $u$ and the leaf attached to it leaves [$S_{1,1,3}$]{} which is equivalent to [$S_{1,1}$]{} by Theorem \[thm:modkpodes\], which is a ${\mathcal{P}}$-position. Attaching a $P_3$ to $u$ changes the outcome: by a straightforward case analysis, one can check that every move leaves a ${\mathcal{N}}$-position. Actually, we conjecture that the Grundy value of trees for the [**0.33**]{} game is not even bounded. For all $n \geq 4$, there exists a tree $T$ such that ${\mathcal{G}}_{{\bf 0.33}}(T)=n$. This conjecture might even be true in the class of caterpillars. A feeble argument to illustrate this intuition comes from our computations. We may provide examples of caterpillars with Grundy values as large as 11. Figure \[fig:large\_grundy\] depicts a caterpillar with a Grundy value of 10 (checked by computer). at (0,0) ; iin [1,...,36]{}[ (i-1,0) – (i,0); at (i,0) ; ]{} iin [ 2,4,6,8,10,12,14,18,20,22,24,26,28,30,34]{}[ at (i,-1) ; (i,0) – (i,-1); ]{} For all $n \geq 4$, there exists a caterpillar $C$ such that ${\mathcal{G}}_{\textbf{0.33}}(C)=n$. However, some of our results can be generalized to other octal games on subdivided stars, see [@futurpapier]. Finally, we would like to mention that it would certainly be interesting to consider the misère version of the 0.33 game on graphs. References {#references .unnumbered} ========== [elsarticle-harv]{} Berkelamp, E. R., Conway, J. H. and Guy, R. K. (2001-2004). [*Winning Ways for Your Mathematical Plays*]{} (2nd ed., 4 volumes). Wellesley, MA: A K Peters. Albert, M., Nowakowski, R. and Wolfe, D. (2007). [*Lessons in Play: An Introduction to Combinatorial Game Theory.*]{} CRC Press. Siegel, A. N. (2013). [*Combinatorial Game Theory.*]{} Graduate Studies in Mathematics (Vol. 146). American Mathematical Society. Guy, R. K. (1996). Unsolved Problems in Combinatorial Games. In R. Nowakowki (Ed.), [*Games of No Chance*]{}. MSRI Book Series (vol. 29). Cambridge: Cambridge University Press.. Althöfer, I. and Bültermann, J. (1995). Superlinear period lengths in some subtraction games. [*Theoretical Computer Science*]{}, 148(1), 111–119. doi:[10.1016/0304-3975(95)00019-S]{} Fleischer, R. and Trippen, G. (2006). Kayles on the way to the stars. In H. Jaap van den Herik et al. (Eds.), [*Computers and Games. 4th International Conference, CG, 2004*]{}. LNCS Book Series (vol. 3846). Berlin Heidelberg: Springer-Verlag. doi:[10.1007/11674399\_16]{} Schaeffer, T. J. (1978). On the complexity of some two-person perfect-information games. [ *J. Comput. System Sci.*]{}, 16(2), 185–225. doi:[10.1016/0022-0000(78)90045-4]{} Adams, R., Dixon, J., Elder, J., Peabody, J., Vega, O. and Will, K. (2015). [*Combinatorial Analysis of a Subtraction Game on Graphs*]{}. Retrieved from arXiv:[1507.05673]{}. Fraenkel, A. S., Scheinerman, E. R. and Ullman, D. (1993). Undirected edge Geography. [*Theoretical Computer Science, 112*]{}(2), 371-381. Nowakowski, R. J. and Ottaway, P. (2005). Vertex deletion games with parity rules. [*Integers: Electronic Journal of Combinatorial Number Theory, 5*]{}(2), A15. Harding, P. and Ottaway, P. (2014). Edge deletion games with parity rules. [*Integers, 14*]{}, G1. Blanc, L., Duchêne, E. and Gravier, S. (2006). A Deletion Game on Graphs: “Le Pic arête”. [*Integers: Electronic Journal of Combinatorial Number Theory, 6*]{}(G02), G02. Huggan, M. (2015). [*Impartial Intersection Restriction Games*]{} (master’s thesis). Retrieved from Carleton University Research Virtual Environment (CURVE) (Record b3819898). Dailly, A., Moncel, J. and Parreau, A. (2018). [*Connected Subtraction Games on Subdivided Stars*]{}. Private communication. Sprague, R. (1935). Über mathematische Kampfspiele. [*Tohoku Mathematical Journal, First Series*]{}, 41, 438–444. Duchêne, É., Gravier, S. and Mhalla, M. (2016). [*Scoring octal games on trees*]{}. Unpublished. Beaudou, L., Coupechoux, P., Dailly, A., Gravier, S., Moncel, J., Parreau, A. and Sopena, É. (2018). [*Octal Games on Graphs: The 0.33 game on subdivided stars and bistars. Full Proof of Theorem 22*]{}. HAL deposit: <https://hal.archives-ouvertes.fr/hal-01807116>. [^1]: For an up-to-date table of octal games, see <http://wwwhomes.uni-bielefeld.de/achim/octal.html>

--- abstract: 'We take a first step towards the solution of QCD in $1+1$ dimensions at nonzero density. We regularize the theory in the UV by using a lattice and in the IR by putting the theory in a box of spatial size $L$. After fixing to axial gauge we use the coherent states approach to obtain the large-$N$ classical Hamiltonian ${\cal H}$ that describes color neutral quark-antiquark pairs interacting with spatial Polyakov loops in the background of baryons. Minimizing ${\cal H}$ we get a regularized form of the ‘t Hooft equation that depends on the expectation values of the Polyakov loops. Analyzing the $L$-dependence of this equation we show how volume independence, à la Eguchi and Kawai, emerges in the large-$N$ limit, and how it depends on the expectation values of the Polyakov loops. We describe how this independence relies on the realization of translation symmetry, in particular when the ground state contains a baryon crystal. Finally, we remark on the implications of our results on studying baryon density in large-$N$ QCD within single-site lattice theories, and on some general lessons concerning the way four-dimensional large-$N$ QCD behaves in the presence of baryons.' --- [**Volume dependence of two-dimensional large-$N$ QCD\ with a nonzero density of baryons.** ]{} Introduction ============ QCD simplifies in the ’t Hooft limit of a large number of colors, and as a result it has been a long-standing goal to understand the properties of the theory in that limit [@largeN], including on the lattice [@lattice-reviews]. One alternative to conventional large volume simulations is the use the large-$N$ equivalence of QCD at large volume to QCD with zero volume [@EK; @BHN1; @MK; @Migdal; @TEK; @AEK; @DW; @GK; @Parisipapers] (see also the related Ref. [@KNN]). These large-$N$ volume reductions allows one, in principle, to study very large values of $N\sim O(100-400)$ with modest resources. Volume reduction holds only if the ground states of the large and zero volume theories respect certain symmetries [@AEK]. Unfortunately, in the most interesting case of QCD in four dimensions these symmetries spontaneously break in the continuum limit when a naive reduction prescription is used [@BHN1; @MK]. An extension of that prescription is thus required and for a recent summary of the literature on this topic we refer to the reviews in the recent Refs. [@QEK; @DEK]. In the case of two space-time dimensions – the ‘t Hooft model – a naive large-$N$ volume reduction is expected to hold and so this theory is generally thought to be completely independent of its volume. In the current paper we analyze this volume dependence. Our motivation is two-fold. Firstly, the ‘t Hooft model is analytically soluble at large-$N$. Thus we can explicitly see how large-$N$ volume reduction works in this case, and what may cause it to fail. This topic was also addressed for zero baryon number by the authors of Ref. [@SchonThies_decompact], and our treatment here differs from that paper by being manifestly gauge invariant, by going beyond zero baryon number, and by using the lattice regularization. Our approach also makes a direct connection with Eguchi-Kawai reduction, and shows how the expectation values of spatial Polyakov loops play a crucial role in the validity of volume independence. Secondly, this paper is a prelude to our companion publication Ref. [@nonzeroBpaper] where we use the formalism presented here to solve the theory in the presence of nonzero baryon density. Considering the current incomplete understanding of the way four-dimensional QCD behaves at low temperatures and large (but not asymptotic) baryon densities, we believe that such a study is useful. Also, there exist certain confusions in the literature about the way large-$N$ gauge theories behave at nonzero baryon number [@Cohen], and seeing how these confusions go away in the soluble two-dimensional case is very helpful. Surprisingly, QCD in two dimensions and nonzero density has not been solved yet : While Ref. [@Salcedo] studied only one and two baryons in an infinite volume, then Ref. [@SchonThies] attempted to extend this but restricted to either (1) translational invariant states which were seen to be inconsistent, or (2) a particular translational non-invariant ansatz for a baryon crystal in the vicinity of the chiral limit. Since $1+1$-dimensional baryons become massless for massless quarks, it is natural to expect that they behave very differently than four-dimensional massive baryons. Furthermore, most of the current literature on the ‘t Hooft model has so far focused on its infinite volume limit where a certain set of gluonic zero modes are irrelevant. With a finite density of baryons, however, these become important and cannot be neglected (at least if the density is increased by fixing the baryon number and decreasing the volume). Thus, given the current status surveyed above, it seems wise to study the dense ‘t Hooft model for arbitrary quark mass, by making as few assumptions on the form of the ground state as possible, and by incorporating correctly the gluonic zero modes. In this paper we develop the machinery to achieve this goal. For the actual solution of the theory for arbitrary baryon numbers we refer to Ref. [@nonzeroBpaper] and for all other discussions on the way nonzero chemical potential affects the system to Ref. [@nonzeroMUpaper]. Former studies of the ‘t Hooft model used a plethora of mathematical methods – for example see [@plethora] for some papers relevant to this work. Common to all these is the need to control the severe IR divergences of this two-dimensional model. A particular clear approach, that we will follow in our study, is the one advocated in the seminal Refs. [@LTYL; @LNT]. There, one works in the Hamiltonian formalism defined in a spatial box of side $L$, and uses the axial gauge to remove all redundant degrees of freedom. This approach is also most suitable for our purpose of investigating the $L$-dependence of this Hamiltonian’s ground state. The outline of the paper is as follows. In Section \[LQCDH\] we present the details of the Hamiltonian approach to lattice QCD. A reader who is familiar with this approach can skip to Section \[Haxial\] where, by generalizing Refs. [@LTYL; @LNT] to the lattice, we show how to fix the axial gauge in the Hamiltonian formalism. Since such gauge fixing is less familiar than the gauge fixing in the Euclidean formalism, we do so in detail. Next, in Section \[GLresolve\], we show how to resolve Gauss law and re-write the electric fields in terms of the fermion color charge densities. This rewriting can be done for all components of the electric field except for those conjugate to the eigenvalues of the spatial Polyakov loops. This set of eigenvalues and their conjugate electric fields is what we refer to above as zero modes, and in Section \[0mode\] we focus on them. Specifically, we show how to represent the zero modes’ electric fields in the Schröedinger picture as differential operators. The end result of Sections \[LQCDH\]-\[Hrecap\] is a Hamiltonian that depends only on the fermions and on the zero-modes, with an overall color neutrality enforced on its Hilbert space. For the convenience of the reader we summarize this emerging structure in Section \[Hrecap\]. We then turn to find the ground state of this Hamiltonian. At large-$N$ this is done in two steps : (1) Solution of the gluon zero modes dynamics - discussed in Section \[SectorG\]. (2) Solution of the fermion sector - Section \[SectorF\]. In the second step we use the coherent state approach of Refs. [@YaffeCoherent] which seems particularly suitable for our problem. The end product is a regularized form of the ‘t Hooft classical Hamiltonian describing color neutral operators that correspond to quark-antiquark pairs and Polyakov-loops (that wrap around the spatial circle), and that interact in the presence of a fixed overall baryon number.[^1] In Section \[otherworks\] we survey other relevant works in the literature that obtain a similar Hamiltonian, pointing out the way they differ from our approach. In Section \[decompact\] we analyze the resulting Hamiltonian and its $L$ dependence for arbitrary baryon number $B$. We show how large-$N$ volume dependence emerges and that for it to hold we need to assume that the ground state has some degree of translation invariance. We also show how it can be violated by giving the Polyakov loops nonzero expectation values.[^2] An interesting phenomena occurs when the ground state contains a baryon crystal and we show how a ‘soft’ form of volume independence emerges. This leads us to remark on the way our results affect studies of nonzero chemical potential that try to rely on large-$N$ volume independence. We conclude in Section \[summary\] by noting some general lessons one can learn about the way large-$N$ QCD behaves in the presence of baryons. Hamiltonian QCD in $1+1$ dimensions : a brief reminder {#LQCDH} ======================================================= In this section we introduce the Hamiltonian formalism of lattice QCD restricted to one spatial dimension and one flavor. A reader familiar with this formalism can skip to Section \[Haxial\]. This Hamiltonian of lattice QCD was first introduced by Kogut and Susskind in 1975 [@Kogut75], shortly after Wilson’s Euclidean formulation. This canonical formalism defines the theory of strong interactions on a spatial lattice with lattice spacing $a$ and continuous time $t$. In one dimension a lattice site is denoted by a single index $x$ taking integer values. We also use $x$ to denote the lattice link that is to the right of the site $x$. Because we define the theory on a finite box we set $x=1,2,\dots,L_s$, where $L_s=L/a$ is the number of lattice sites, and $L$ the physical length of the box. The boundary conditions of the gauge fields are taken to be periodic. In this section and throughout the paper we use standard lattice notations and so the factors of the lattice spacings are implicit and all fields are dimensionless. The quantum fields that describe quarks are the fermion fields $\psi^{a\alpha}_{x}$ that live on the sites $x$ of the lattice. They have color indices $a=1,\dots,N$, and Dirac indices $\alpha=1,2$ (again, recall we are in $1+1$ dimensions). The Fermi fields obey the following anti-commutation relations $$\left\{ \psi^{a\alpha}_x,\psi^{\dag b \beta}_y \right\} = \delta_{xy}\delta_{ab}\delta_{\alpha\beta}. \label{anticomm}$$ Choosing to work in the temporal gauge that fixes $A_0=0$ removes one degree of freedom (and its conjugate momentum). This leaves the spatial gauge field operators $\left(U_x\right)_{ab}$ living on the lattice link between $x$ and $x+1$. For definiteness we note that the operator $\left(U^\dag_{x}\right)_{ab}$ is [*defined*]{} to be equal to $\left(\left(U_{x}\right)_{ba}\right)^\dag$, and so the following holds as a operator identity $$\sum_a\, \left(\left(U_{x}\right)_{ab}\right)^\dag\, \left(U_{x}\right)_{ac} = \sum_a\, \left(U^\dag_{x}\right)_{ba}\, \left(U_{x}\right)_{ac}=\delta_{bc}.$$ The conjugate momenta of the $U$’s also reside on links and are denoted by $E^i_{x}$, $i=1,\dots,N^2-1$. The following are the commutation relations of this set of operators, $$\begin{aligned} \left[ E^i_{x} , \left( U_{x} \right)_{ab} \right] &=& \left( \lambda^i U_{x} \right)_{ab}, \label{comm_plus1} \\ \left[ E^i_{x} , E^j_{x} \right] &=& i f^{ijk} E^k_{x}. \label{comm_plus2}\end{aligned}$$ Here $\lambda^i$ are matrices that represent the traceless generators of $SU(N)$ in the fundamental representation, and $f^{ijk}$ are the structure constants of $SU(N)$. We choose a normalization where $$\begin{aligned} \tr \left(\lambda^i\lambda^j\right) &=& \frac12 \delta^{ij},\label{norm1}\\ \sum_{i=1}^{N^2-1} \, \lambda^i_{ab} \, \lambda^i_{cd} &=& \frac12 \left(\delta_{ad}\, \delta_{cb} - \frac1{N} \, \delta_{ab}\, \delta_{cd}\right).\label{norm2}\end{aligned}$$ Let us now discuss the lattice Hilbert space. A general state $|\Omega \rangle$ is the direct product on all lattice sites, $$|\Omega \rangle = | \Omega \rangle_G \otimes | \Omega \rangle_{\Psi}.$$ Here the first factor is the projection of the state $|\Omega\>$ to the gauge field sector, while the second factor describes the fermionic sector. Any state $|\Omega\>_{\Psi}$ is the following direct product $$|\Omega\>_{\Psi} = \prod_{\otimes x} \left( |\Omega\>_{\Psi} \right)_x.$$ Concentrate on the Hilbert space of each site : the “lowest” state is the no-quantum “drained” state $|\textrm{dr}\rangle_x$, defined as $$\psi^{a\alpha}_x|\textrm{dr}\rangle_x = 0. \label{drained}$$ Applications of various $\psi^{\dag a\alpha}_x$ create the corresponding quarks on that site, $$|a\alpha \rangle_{x\Psi} = \psi^{\dag a\alpha}_x |\textrm{dr}\rangle_x.$$ For the free theory, $\alpha=1$ correspond to creation of positive energy excitations and $\alpha=2$ corresponds to creation of negative energy excitations. To use the usual quark–anti-quark language we write for each site and color $$\psi=\left( \begin{array}{c} b \\ d^{\dag} \end{array} \right).$$ $b^{\dag}$ creates a quark and $d^{\dag}$ an anti-quark. From we see that $b$ and $d^{\dag}$ annihilate the drained state, which means that this state is empty of quarks, and filled with anti-quarks. The local baryon density operator is $$B_x \equiv \frac1{N}\sum_{a=1}^{N} \left[ b^{\dag a} b^{a} - d^{\dag a} d^{a} \right]_x =\frac1{N} \sum_a\psi^{\dag\,a}_x \psi^a_x - 1. \label{B}$$ According to , the baryon number of the drained state is $-1$, corresponding to filling the site with anti-quarks. The vacuum $|0\rangle$ is the state with no quarks and no anti-quarks. This state is the filled Dirac sea on a single site and obeys, $$b|0\rangle=d|0\rangle=0.$$ The baryon number of this state is $B=0$. Because of the Pauli exclusion principle we cannot put too many fermions on a single site. For a single flavor theory the maximum number of local baryon number will be $1$, and is found only in the state $|\textrm{filled} \rangle$ $$b^{\dag}|\textrm{filled}\rangle=d|\textrm{filled}\rangle=0.$$ Below we will see that gauge invariance puts more restrictions on the single site Hilbert space in order that it be color neutral. Moving to the gauge Hilbert space, we also write it as a direct product of the form $$|\Omega\>_G = \prod_{\otimes x} \left( |\Omega\>_G \right)_{x}.$$ Next, we denote the state with no electric field $E$ by $|0\>_G$, $$E^i| 0 \>_G = 0, \qquad \forall i.$$ Any application of the link operators $\left( U_{x} \right)_{ab}$ on $|0\>_G$, creates states which correspond to flux lines on the link $x$. The state $|0 \>_G$ is the only state with no flux at all. Using the fact that the electric field operators generate a $SU(N)$ algebra, one can distinguish between the different quantum states created by the link operators as follows. Define the quadratic Casimir operator $$\vec{E}^2_{x}\equiv\sum_{i=1}^{N^2-1}E^{i2}_{x}. \label{E2}$$ It is clear that the flux-less state $| 0 \>_G$ is an eigenstate of this Casimir, with zero eigenvalue. Next, the commutations of the Casimir with the link operators are verified from to be $$\left[ \vec{E}^2_{x},\left( U_{x} \right)_{ab} \right] = C_F\left( U_{x} \right)_{ab},$$ where $C_F=(N^2-1)/2N$ is the Casimir operator in the fundamental representation. This means that the state $\left[ \left( U_{x} \right)_{ab}| 0 \>_G \right]$ is also an eigenstate of (\[E2\]), with eigenvalue equal to $C_F$. One can now classify the states in $|\Omega \>_G$ according to their $\vec{E}^2$ eigenvalue. The result is a Hilbert space with a ladder-like structure. The lowest state is $|0\>_G$, with zero flux, and is a singlet of $SU(N)$. Repeated applications of the gauge field operators $\left( U_{x} \right)_{ab}$ create states with higher and higher values of flux and the operators $\vec{E}^2_{x}$ measures the flux on the link $x$. Indeed we shall see shortly that it is proportional to the (kinetic) energy of the gauge fields. To complete the picture we now discuss gauge invariance. First recall that the starting point of this formalism was to choose the time-like gauge. This leaves only time-independent gauge transformations as a symmetry. The fermion operators belong to the fundamental representation of the gauge symmetry and transform as $$\psi^a_x \rightarrow \left( V_x \right)^{ab} \psi^b_x, \label{gauge_F1}$$ with $V\in SU(N)$ given in general by $$V_x=\exp \left[ i\sum_{i=1}^{N^2-1} \theta^i_x \lambda^i\right].$$ Using the anti-commutation relations (\[anticomm\]), one can show that the quantum operator ${\cal V}_F$ that realizes in Hilbert space as $$\psi^a_x \rightarrow {\cal V}_F \, \psi^a_x \, {\cal V}^\dag_F,$$ is $${\cal V}_F = \exp \left[ -i\sum_x \sum_{i=1}^{N^2-1} \theta^i_x \left( \psi^{\dag a}_x \lambda^i_{ab} \psi^b_x \right) \right]. \label{gauge_F2}$$ The gauge fields transform according to $$\left( U_{x} \right)_{ab} \rightarrow {\cal V}_G\, \left(U_x\right)_{ab}\, {\cal V}^\dag_G = \left( V_x U_{x} V^\dag_{x+1} \right)_{ab}.$$ Using the commutation relations in , one shows that the quantum operator ${\cal V}_G$ that generates these rotations is given by[^3] $${\cal V}_G = \exp \left[ +i\sum_x \sum_{i=1}^{N^2-1} \theta^i_x \left( E^i_{x} - \left(U^{\rm Adj.}_{x-1}\right)_{ji} \, E^j_{x-1}\right)\right], \label{gauge_G}$$ where here the matrix of operators $\left(U^{\rm Adj.}_{x-1}\right)_{ji}$ is the link matrix in the adjoint representation, i.e.[^4] $$\left(U^{\rm Adj.}_{x}\right)_{ij}= 2 \, \tr \left( \lambda^i \, U_{x} \, \lambda^j \, U^\dag_{x}\right).$$ Putting , and together, we see that the operator that induces gauge transformations is $${\cal V}=\exp \left[ i\sum_x \sum_{i=1}^{N^2-1} \theta^i_x \left( \rho^i_G - \rho^i_F \right)_x\right],\label{gauge_trnsf}$$ where $\rho^i_{Fx}$, and $\rho^i_{Gx}$ are the color charge densities of the fermions and of the gauge fields. These two quantities are given by $$\begin{aligned} \rho^i_{Fx}&=&\psi^{\dag a}_x \lambda^i_{ab} \psi^b_x, \\ \rho^i_{Gx}&=& E^i_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^j_{x-1} \equiv D E^i_x, \label{rhoG} \end{aligned}$$ Since the lattice Hamiltonian is gauge invariant, we know that the generators of the gauge transformations commute with the Hamiltonian $$\left[ H , \rho^i_{Gx} - \rho^i_{Fx} \right]=0, \hskip 1cm \forall i \quad \forall x.$$ This means that we can choose to work with a basis that block diagonalizes $\rho^i_G-\rho^i_F$. This breaks the Hilbert space to separate sectors classified by their eigenvalue $\rho^i_{\rm external,x}\equiv \rho^i_G-\rho^i_F$ on the lattice. Each set of values $\rho^i_{{{\rm}external},x}$ describes a different physical case, with a different [*external*]{} distribution of color charge (that can correspond, for example, to infinitely heavy quarks etc.). To describe the physics of zero external gauge fields, we work with the choice $\rho^i_{{\rm}{external}}=0$. Working in this subspace means that [*all*]{} physical states must be color singlets, since all gauge transformations are trivial. This means that in this sector the following equations hold as operators identities $$E^i_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^j_{x-1} = \rho^i_{Fx}. \label{Glaw}$$ Finally we write the lattice Hamiltonian $H$ of the $1+1$-dimensional $SU(N)$ gauge theory with one flavor of fermions. $H$ is given by (in one dimension there is no magnetic field and so the plaquette term is identically zero) $$H=H_E+H_F. \label{eq:H_initial}$$ Here $H_E$ is the electric term, a sum over links $x$ of the $SU(N)$ Casimir operator $$H_G=\frac{g^2}{2}\sum_{x=1}^{L_s}\left(E^i_{x}\right)^2.$$ Next is the fermion Hamiltonian, $$H_F=-\frac{i}{2}\sum_{x=1}^{L_s}\psi^{\dag a\alpha}_x\, \left(\sigma_3\right)_{\alpha\beta} \, U_{x,ab} \, \psi^{b\beta}_{x+1}+h.c. + m\sum_x \psi^{\dag a\alpha}_x \,\left(\sigma_1\right)_{\alpha\beta} \, \psi^{a\beta}_x. \label{eq:H_F}$$ Here we choose a particular representation of the one-dimensional Dirac matrices using the Pauli matrices $\sigma_{1,3}$ and periodic boundary conditions on the fermions, i.e. $\psi_x=\psi_{x+L_s}$, and $\psi^\dag_x=\psi^\dag_{x+L_s}$. Superficially, the first term in the Hamiltonian is symmetric under a $U(1)_R\times U(1)_L$ chiral symmetry which is explicitly broken by the mass term to the vector $U(1)$. One can, however, spin diagonalize the fermions by writing $\psi^{\alpha}_x\to \left(e^{-i\pi\, \sigma_2/2 } \sigma_3^x\right)_{\alpha\beta} \psi^{\beta}_x$ and see that $$H_F=-\frac{i}2\sum_{x}\psi^{\dag a\alpha}_x \, U_{x,ab} \, \psi^{b\alpha}_{x+1}+h.c. + m\sum_x \, (-1)^x\, \psi^{\dag a\alpha}_x \, \left(\sigma_3\right)_{\alpha\beta} \, \psi^{a\beta}_x. \label{eq:H_F_stag}$$ Here, while the first term is invariant under a $U(2)$ group, the mass term is invariant only under a $U(1)\times U(1)$. In our actual calculations we will drop the second component of this spin-diagonalized Hamiltonian (the $\alpha=2$), and thus work with staggered fermions [@Susskind]. To go back to Dirac fermions is easy and for our purpose it is useful to note that the continuum chiral condensate and baryon number local densities, at position $X$, $\left(\bar \psi\psi\right)(X)$ and $\left(\psi^\dag\psi\right)(X)$, are given, up to overall renormalization factors, by [@Susskind] $$\begin{aligned} \left(\psi^\dag \psi\right)(X)&=&\frac1{\sqrt{2}}\left(\psi^\dag_x \psi_x + \psi^\dag_{x-1} \psi_{x-1} \right),\label{psibarpsi}\\ \left(\bar \psi \psi\right)(X)&=&\frac1{\sqrt{2}}\left(\psi^\dag_x \psi_x - \psi^\dag_{x-1} \psi_{x-1} \right),\label{psidagpsi}\end{aligned}$$ where we take $X/a=x$, and $x$ denotes an even site.[^5] Axial gauge fixing {#Haxial} ================== In this section we show how to fix the axial gauge. Since axial gauge fixing in the Hamiltonian approach is less familiar than it is in the Euclidean approach we begin with an explanation of the general strategy. Temporal gauge fixing left us with a Hamiltonian that is invariant under time-independent gauge transformations discussed in the previous section. With no external charges the generator of such transforms needs to vanish on physical states and this gives rise to a set of local and global Gauss-law constraints that the quantum fields and their conjugate momenta need to obey. Of these constraints, the local ones can be solved (and will be solved in the next section) and consequently a large subset of the gauge fields’ conjugate momenta is written in terms of the fermion color charges. The fact that these momenta become non-dynamical implies that their conjugate gauge fields are not physical and can be removed from the Hamiltonian. These are the fields that, in the action formalism, can be ‘gauged away’. Indeed, from the path integral formalism we know that in a compact system of one spatial dimension almost all gauge fields can be gauged away. The only gluonic modes that remain are those corresponding to a constant spatial gauge field. Furthermore, one can gauge away all but the $N-1$ independent eigenvalues of this zero mode. Anticipating a similar scenario in the Hamiltonian approach, one expects that the following fermionic Hamiltonian will be the remnant of the Hamiltonian after the axial gauge fixing : $$H'_F = -i\sum_{x}\psi^{\dag a}_x \sigma_3 \, e^{i\varphi_a } \, \psi^{a}_{x+1}+h.c. + m\sum_x \psi^{\dag a}_x \sigma_1 \, \psi^{a}_x.\label{H'0}$$ Here we denote by $e^{i\varphi_a}$ the $a^{\rm th}$ eigenvalues of the constant spatial gauge field operator. Clearly, to get from one needs to remove all but the eigenvalues of the spatial Polyakov loops operator from the system. Using a sequence of a change of variables, this is easy to do in the path integral approach. But in the Hamiltonian there seems to be a conceptual difficulty with such a process : the degrees of freedom we wish to gauge away are represented by quantum operators and it is not clear how to ‘gauge them away’. To proceed we choose to generalize, to the lattice, the formalism constructed in the seminal Ref. [@LNT]. The general idea is to find a Hilbert space realization of a unitary operator ${\cal F}$ that will rotate quantum states into a basis where the Hamiltonian looks like . To do so we follow the path integral picture as a guide : first we define the spatial Polyakov loop operator $P$ to be $$P_{ab}= \sum_{c,d,\cdots, z=1}^N U_{1,ac} U_{2,cd} \cdots U_{L_s,zb}\,,$$ and second we define the ‘eigenvalue operators’ $\varphi_a$ through $$P_{ab}= \sum_{c} S^\dag_{ac}(P) \, e^{i\,L_s\,\varphi_c(P)} \, S_{cb}(P).\label{P}$$ Here the matrix of operators $\left( S(P)\right)_{ab}$ is a functional of the operators $P_{ab}$ and is defined implicitly through . Next we ask to find a form of ${\cal F}$ that will induce the following transformations $$\begin{aligned} \psi^a_x &\to& \psi^{'a}_x = {\cal F} \, \psi^a_x \, {\cal F}^{\dag} = \sum_b V_{x,ab} \,\psi^b_x. \label{Psitransform}\\ \psi^{a\dag}_x &\to& \psi^{'\dag a}_x = {\cal F} \, \psi^{\dag a}_x \, {\cal F}^{\dag} = \sum_b \psi^{\dag b}\,V^\dag_{x,ba},\\ U_{x,ab} &\to& U'_{x,ab}={\cal F} \, U_{x,ab} \, {\cal F}^{\dag} = U_{x,ab}, \label{Utransform}\\ U^\dag_{x,ab} &\to& U^{'\dag}_{x,ab}={\cal F} \, U^\dag_{x,ab} \, {\cal F}^{\dag} = U^\dag_{x,ab}.\end{aligned}$$ Crucially, here the operator $\left(V_x\right)_{ab}$ is chosen to be the following functional of the gauge fields operators $\left(U_{x}\right)_{ab}$, $$\begin{aligned} V_{x,ab}&=& \sum_{cde\cdots yz=1}^N U^\dag_{x-1,ac}\, U^\dag_{x-2,cd}\, \dots \, U^\dag_{1,yz} \, S^\dag_{zb}(P) \, e^{i\varphi_b(P) \, x}.\label{Vdef}\end{aligned}$$ This guarantees that $V$ obeys $$\sum_{bc}\left(V^\dag_x\right)_{ab} \,\left(U_{x}\right)_{bc}\,\left(V_{x+1}\right)_{cd} = e^{i\varphi_a }\delta_{ad} \label{Vdef1}$$ as an operator identity. The form of ${\cal F}$ that induces Eqs. (\[Psitransform\])–(\[Utransform\]) is then written as $${\cal F} = \exp \left( -i\sum_{xi} \rho^i_{x,F}\, \Theta^i_x(\left\{U\right\})\right),$$ where $\Theta$ is again an operator in Hilbert space that depends on the gauge field operators $U_{x,ab}$ and that is defined through its following relation to the operator $V$ : $$\begin{aligned} V_{x,ab}&=&\left[\exp \left( i\sum_i \lambda^i \Theta^i_x(\left\{U\right\})\right)\right]_{ab}. \label{Vdef2}\end{aligned}$$ Note that ${\cal F}$ is [*not*]{} a gauge transformation (i.e. its not of the form ). The end result of Eqs. (\[Psitransform\])–(\[Vdef2\]) is that when we rotate the Hilbert space by ${\cal F}$ or, equivalently, conjugate the fermionic part of the Hamiltonian by ${\cal F}$ we indeed get : $$\begin{aligned} H_F &\to & H'_F = {\cal F} \, H_F \, {\cal F}^{\dag} = -i\sum_{x}\psi^{\dag a}_x \sigma_3 \, e^{i\varphi_a } \, \psi^{a}_{x+1}+h.c. + m\sum_x \psi^{\dag a}_x \sigma_1 \, \psi^{a}_x.\label{H'}\end{aligned}$$ This change of quantum basis is how the process of axial gauge fixing works in the Hamiltonian formalism. We now proceed to find the transformed version of $H_E$, of the commutation relations, and of the Gauss-law constraint. We begin by defining the transformed electric fields $E'$ via $$\begin{aligned} E^{'i}_x&\equiv & {\cal F} \, E^{i}_x \, {\cal F}^\dag,\label{E'}\end{aligned}$$ which allows us to write the transformed electric field Hamiltonian as $$\begin{aligned} H_E &\to & H'_E={\cal F} \, H_E \, {\cal F}^\dag = \frac{g^2}{2}\sum_{x,i} \left(E^{'i}_x\right)^2.\end{aligned}$$ It is clear that the commutation relation between the $E'$ fields and the $U$ fields are the same as in and (this is so because of ). Next, we turn to transform the Gauss law constraint. For that we conjugate by ${\cal F}$ $${\cal F} \, \left( E^i_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^j_{x-1} \right) {\cal F}^\dag = {\cal F} \rho^i_{F,x} \, {\cal F}^\dag.$$ This equation, in terms of $E^{'i}_x$, reads $$E^{'i}_x-\left( U^{{\rm}{Adj.}}_{x-1}\right)_{ji} E^{'j}_{x-1} = {\cal F} \rho^i_{F,x} \, {\cal F}^\dag = \sum_{ab}\, \lambda^i_{ab}\, {\cal F} \, \psi^{\dag a}_x \psi^b_x \, {\cal F}^\dag = \sum_{ab \atop a'b'}\, \lambda^i_{ab}\,\, \psi^{\dag a'}_x \, \psi^{b'}_x \, V^\dag_{x,a'a} V_{x,bb'}.$$ Multiplying this equation by $\lambda^i_{cd}$, summing over $i$, and writing it in terms of the adjoint representation of the operator $V$, we get $$\sum_{i}\, \left( V^{\rm Adj.}_x \right)_{ik} \, \left( E^{'i}_{x}-\left( U^{ {\rm}{Adj.}}_{x-1}\right)_{ji} E^{'j}_{x-1} \right) = \rho^k_{F,x}.$$ Finally, if we define $$E^{''k}_x=\left(V^{\rm Adj.}_x\right)_{ik} \, E^{'i}_x, \label{E''def}$$ and $$E^{'j}_x=\left(V^{\rm Adj.}_x\right)_{jk}\, E^{''k}_x,$$ then we get $$E^{''i}_{x}-\left( V^\dag_x \, U^\dag_{x-1}\, V_{x-1}\right)^{\rm Adj.}_{ij} E^{''j}_{x-1} = \rho^i_{F,x}.$$ Using this gives $$E^{''i}_{x}-\left(e^{-i\varphi}\right)^{\rm Adj.}_{ij} E^{''j}_{x-1} = \rho^i_{F,x}. \label{Glaw''}$$ is the starting point to the discussions in Sections \[GLresolve\]–\[0mode\] : first, in Section \[GLresolve\], we solve and write the fields $E^{''i}_x$ in terms of the color charge densities $\rho^i_{F,x}$. This cannot be done to all of the components of $E^{''}$, and a subset of these remains independent of the fermionic charge densities. The way we treat these remaining components is explained in Section \[0mode\]. Resolution of the Gauss law constraint {#GLresolve} ====================================== In this section we resolve the Gauss law constraint of and write the electric field operators $E^{''}$ in terms of the fermion color charge densities $\rho_F$. For that purpose we first specify the basis of the color group generators $\lambda^i$. We choose the $N-1$ traceless generators $\lambda^{i=1,2,\dots,N-1}$ to span the traceless Cartan sub-algebra, and the remaining $N(N-1)$ generators to have no diagonal entries. In this basis we see that $$\left(e^{-i\varphi}\right)^{\rm Adj.}_{ij} \equiv 2\, e^{i(\varphi_a-\varphi_b)}\, \lambda^i_{ab}\, \lambda^j_{ba} = \left[ \begin{array}{cc} \delta_{ij} & {\rm if}\,\,i,j \in [1,N-1],\\ 0 & \hspace{0.5cm} {\rm if}\,\, i\in [1,N-1] {\rm \,\, but\, \, not}\, j, \, \, {\rm and \,vice\,versa} \end{array} \right. .$$ As a result, if we focus on the Cartan sub-algebra (let us denote its generators by $i=I$), and Fourier transform the operators $E''$ and $\rho$ according to $$\begin{aligned} E^{''I}_x &=& \frac1{\sqrt{L_s}} \, \sum_p \,e^{ipn} E^{''I}_p,\label{FTE''}\\ \rho^{I}_x &=& \frac1{\sqrt{L_s}} \, \sum_p \,e^{ipn} \rho^{I}_p,\label{FTrho},\end{aligned}$$ with $p=\frac{2\pi n}{L_s}\, ; \, n=0,1,\dots,L_s-1$, then we can resolve the nonzero momentum components of $E^{''I}_{p\neq 0}$ : $$\begin{aligned} E^{''I}_{p} &=& \frac{\rho^I_{F,p}}{1-e^{-ip}}, \quad {\rm for} \quad p\neq 0. \label{E''p}\end{aligned}$$ For $p=0$ Gauss law cannot be resolved, and instead becomes a constraint on the fermionic global color charges : $$\begin{aligned} \forall I=1,2,\dots,N-1 \qquad :\qquad Q^I&\equiv& \sum_x\, \psi^\dag_x \, \lambda^I \, \psi_x = 0.\label{QI}\end{aligned}$$ The $N-1$ requirements of can be written as $$\forall a=1,\dots,N\quad : \quad\sum_x \, \psi^{\dag a}_x \psi^a_x = {\rm independent \,\, of \,\,} a,$$ and since the baryon number $B$ is equal to $\frac1{N}\sum_{x,a} \psi^{\dag a}_x\psi^a_{x}-L_s$ (see ) then becomes $$\sum_x \, \psi^{\dag a}_x \psi^a_x = B+L_s, \qquad \forall a=1,2,\dots,N.\label{BLs}$$ For the rest of the generators $i\neq I$ we can indeed resolve Gauss law even for zero momentum : multiplying by $\lambda^i_{ab}$ with $a\neq b$, and defining $E^{''}_{x,ab}=\sum_{i\neq I} \, \lambda^i_{ab}\, E^{''i}_x$, and $\rho_{F,x,ab}=\sum_{i\neq I} \, \lambda^i_{ab}\, \rho^{i}_{F,x}$, one finds that the Fourier components of $E^{''}_{x,a\neq b}$ obey $$E^{''}_{p,ab} = \frac{\rho_{F,p,ab}}{1-e^{-i(p+\varphi_a - \varphi_b)}}.\label{Glaw''1}$$ (here the Fourier transformations of $E^{''}_{x,a\neq b}$ and $\rho_{x,a\neq b}$ are defined in a way similar to Eqs. (\[FTE”\]–\[FTrho\])) Note that we work in a Schröedinger picture where $\varphi_a$ is a $c$-number. Also, in we assume the absence of states with $\varphi_a-\varphi_b+p=0$. Since $p$ is quantized in units of $2\pi/L_s$, this assumption will indeed become true – see the discussion below on the importance of the Jacobian of the curvilinear coordinates $\varphi$. We can now use the relations between $E^{''}$ and $\rho_F$, and write the electric Hamiltonian in terms of the fermions and the gluonic zero modes. For that we use the hermiticity of $E'_x$ and get $$\begin{aligned} H'_E &=& \frac{g^2}2\sum_{ix} \left( E^{'i}_x \right)^2=\frac{g^2}2\sum_{ix} E^{'i\dag}_x E^{'i}_x=\frac{g^2}2\sum_{ix} E^{''i\dag}_x E^{''i}_x = \frac{g^2}2\sum_{ip} E^{''i\dag}_p\, E^{''i}_p\nonumber \\ &=&\frac{g^2}2\left( \frac2{L_s}\sum_{ab\atop x,y,p} ^\prime \, \frac{\rho_{x,ab}\, \rho_{y,ba}\, e^{i(y-x)\,p}}{4\sin^2 \left(\frac{p+\varphi_a-\varphi_b}{2}\right)} + \sum_{I=1}^{N-1} E^{''I\dag}_{p=0} E^{''I}_{p=0}\right). \end{aligned}$$ Here, by the primed sum, we mean that one must sum over all $a$, $b$ and $p$ for which the denominator is [*nonzero*]{}. This restriction is a direct consequence of one’s inability to resolve all of the Gauss law constraints and will become the source of the principle value prescription often used in the ‘t Hooft model. Realization of the zero modes in the Schröedinger picture {#0mode} ========================================================= The resolution of Gauss law left us in the glue sector with a restricted set of dynamical degrees of freedom – the $N-1$ eigenvalues of the spatial Polyakov loops, $\varphi_a$, and their $N-1$ conjugate momenta $E^{I''}_{p=0}$. In this section we focus on these operators to which we collectively refer as the zero modes. Specifically, we calculate their commutation relations and also find a simple way to realize $E^{I''}_{p=0}$ in the Schröedinger picture, where $\varphi_a$ are c-numbers. For that purpose we first calculate the commutation relations of $E^{'i}_x$ and the operator $$\left(P_y\right)_{ab} = \left( U_1 \, U_2 \cdots U_{y} \right)_{ab}.$$ The result is easy to obtain and can be written in terms of the operators $S$ and $\varphi$ (see ), and the operator $V$ (), $$\left[E^{'i}_x,\left(P_y\right)_{ab}\right] = \theta(y-x) \left( S^\dag \, e^{i\varphi \, x} V^\dag_x \lambda^i \, V_x e^{-i\varphi \, x } S P_y \right)_{ab}.$$ Here $\theta(x)= 1$ if $x\ge 0$ and zero otherwise. Next we use the definition of $E^{''I}$ in and after some algebra find that (here note that we focus only on the Cartan sub-algebra) $$\left[E^{''I}_x,\left(P_y\right)_{ab}\right] = \theta(y-x) \left( S^\dag \, \lambda^I \, S P_y \right)_{ab}.\label{comm1}$$ Since the dependence on the site index $x$ in the r.h.s. of is trivial, we can easily write the commutation relation of the zero mode $E^{''I}_{p=0}=\frac1{\sqrt{L_s}} \, \sum_x E^{''I}_x$, and we find $$\left[E^{''I}_{p=0},\left(P_y\right)_{ab}\right] = \frac{y}{\sqrt{L_s}} \, \left( S^\dag \, \lambda^I \, S P_y \right)_{ab}.\label{comm2}$$ In particular, for $y=L_s$, we use and get $$\left[E^{''I}_{p=0},P_{ab}\right] = \sqrt{L_s} \, \left( S^\dag \, \lambda^I \, e^{i\varphi L_s} S \right)_{ab}.\label{comm3}$$ If we now define ${\cal E}_a\equiv \sum_I \, \lambda^I_{aa}E^{''I}_{p=0}$, we get $$\begin{aligned} \left[{\cal E}_c,\left(S^\dag e^{i\varphi L_s}S\right)_{ab}\right]&=&\frac12 \, \sqrt{L_s} \left[\, S^{\dag}_{ac} e^{i\varphi_c L_s} S_{cb} -\frac1{N} \, P_{ab}\right].\label{comm4_1}\end{aligned}$$ In Appendix \[gauge\_fixing\] we show that $$\begin{aligned} {\cal E}_a&=&{\cal E}^\dag_a,\label{herm}\\ \left[{\cal E}_a,{\cal E}_b\right]&=&0\label{comm5}.\end{aligned}$$ and we note in passing also that $\sum_a {\cal E}_a=0$. The form of Eqs. (\[comm4\_1\])–(\[comm5\]) leads us to write the following realization of the ${\cal E}$ operators in the Schröedinger picture : $$2\sqrt{L_s}\, {\cal E}_a = - i \left( \frac{\delta}{\delta \varphi_a}-\frac1{N} \sum_c \frac{\delta}{\delta \varphi_c}\right) -\frac{i}2 \left[\frac{\delta \, \log \Delta^2 (\varphi)}{\delta \varphi_a} -\frac1{N}\sum_c\frac{\delta\log \, \Delta^2}{\delta \varphi_c}\right],\label{E_Schroedinger0}$$ where here $\Delta^2(\varphi)$ is the well known Vandermond determinant $$\Delta^2(\{\varphi\})=\prod_{a<b}\, \sin^2 \left(L_s\frac{\varphi_a-\varphi_b}{2}\right). \label{VDM}$$ It is easy to check that the first term in indeed satisfies : the $\varphi_{a=1,\dots,N}$ behave like the coordinates of $N$ particles, while $\frac{\delta}{\delta \varphi_a} - \frac1{N}\sum_b \frac{\delta}{\delta \varphi_b}$ behave like their momenta, relative to the motion of the center of mass coordinate $\varphi_{c.m.}\equiv\sum_a \varphi_a$. This separation into relative coordinates and a center of mass is anticipated in the $SU(N)$ case where $\varphi_{c.m.}$ is not a true degree of freedom. It is also not hard to understand the origin of the second term in : while it trivially obeys , it is necessary for . To see this recall that in the Schröedinger picture the measure of the $\varphi_a$ coordinates is not flat – it is given by the Haar measure $dP$ over the spatial Polyakov loop $P$ (which is the only remnant of the gauge field degrees of freedom). Since the Hamiltonian only depends on the eigenvalues $\varphi_a L_s$ of $P$, then we can replace $$\int_{SU(N)} dP\longrightarrow \int \, \left(\prod_a d\varphi_a\right)\, \,\Delta^2(\left\{\varphi\right\})\, \delta \left(\varphi_{c.m.} \right).$$ Because of the $\Delta^2$ factor in the measure, the simple derivative operator $-i\frac{\delta}{\delta\varphi}$ will not be hermitian (its action to the right will differ from its action to the left by a derivative of $\Delta^2$). It is easy to check the particular choice of in fixes this and makes hermitian.[^6] Our final step will be to simplify even further. For that we define a new wave function $\Psi_{\rm new}$ for the curvilinear coordinates $\varphi$ that will make their measure flat. Specifically, we write $$\Psi(\varphi)\equiv \frac{\Psi_{\rm new}(\varphi)}{\prod_{a<b} \, \sin\left( L_s \frac{\varphi_a-\varphi_b}2 \right)}.\label{change}$$ In terms of the new wave function the Vandermond disappears from the measure and the kinetic term becomes the simple quadratic form (here we rescaled $\varphi \to \varphi/L_s$): $$\begin{aligned} \frac{g^2}{2}\sum_I \left(E^{''I}_{p=0}\right)^2 &=& -\frac{g^2L_s}{4} \sum_{d=1}^N\left( \frac{\delta}{\delta\varphi_d}-\frac1{N}\sum_c \frac{\delta}{\delta\varphi_c} \right)^2.\end{aligned}$$ This will be our final simplified form for the zero mode contribution to $H^{'}_E$. Recap : the Hamiltonian and the restrictions on the Hilbert space. {#Hrecap} =================================================================== To conclude let us write the lattice Hamiltonian $H$ in terms of the operators that cannot be gauged away. It describes the interactions between the lattice fermions $\psi$ and the eigenvalues of the spatial Polyakov loop $\varphi$, and is given by $$\begin{aligned} H&=&H_G + H_K + H_C,\label{Ham}\\ H_G&=&-\frac{g^2L_s}{4} \sum_{d=1}^N\left( \frac{\delta}{\delta\varphi_d}-\frac1{N}\sum_{c=1}^N \frac{\delta}{\delta\varphi_c} \right)^2,\label{HG}\\ H_K & =& -\frac{i}2\sum_{x}\psi^{\dag a}_x \, e^{i\varphi_a/L_s} \, \psi^{a}_{x+1}+h.c. + m\sum_x \, (-1)^x\, \psi^{\dag a}_x \psi^{a}_x,\label{H_F'}\\ H_C&=&\frac{g^2}{L_s}\sum'_{abp\atop}\,\frac{\rho^{ab}_{F,x} \,\rho^{ba}_{F,y}\,e^{ip(y-x)}}{4\sin^2\, \left(\frac{(\varphi_a-\varphi_b)/L_s+p}{2}\right)},\label{HC}\\ \rho^{ab}_{F,x} &\equiv & \frac12 \left(\psi^{\dag b}_x\psi^a_x - \frac{\delta^{ab}}N\sum_c \psi^{\dag c}_x\psi^c_x \right).\end{aligned}$$ Here the primed sum in means we that we should not sum over terms whose denominator is zero (for the derivation of this restriction see section \[GLresolve\]). From here on we discard the lower component of the fermions, and so discuss the staggered fermions formulation of lattice QCD [@Susskind] (this also means that we need to replace the r.h.s. of by $B+L_s/2$, and choose the number of lattice sites, $L_s$, to be even). The reason we choose to work with staggered fermions is two-fold. Firstly, these one-component Grassmann variables are simpler to treat in our formalism and the computational effort involved in the numerical minimization of their classical Hamiltonian (for details see [@nonzeroBpaper]) is a factor of two smaller than for the naive prescription of fermions considered above. Secondly, unlike the situation in four dimensional euclidean calculations, staggered fermions in the one dimensional Hamiltonian approach are ‘un-doubled’ and correspond to a single Dirac fermion in the continuum limit. Finally, the Hilbert space of the system obeys the following : - The operators $\sum_{x}\, \psi^{\dag a}_x \psi^a_x$ are, for each value of the color index $a=1,\dots,N$, constants and equal to $B+L_s/2$. Diagonalization of the Hamiltonian : the coherent states approach {#diagonalize} ================================================================= In this section we use the coherent state approach to study the ground state properties of the Hamiltonian in its large-$N$ limit. This approach is summarized in Refs. [@YaffeCoherent] and we refer to these papers for detailed discussions, while here we only describe its strategy. The main paradigm that underlies the coherent state approach is that QCD, in its large-$N$ limit, becomes a classical theory.[^7] This means that instead of diagonalizing its quantum Hamiltonian, one can instead minimize a corresponding classical function ${\cal H}$, referred to as the ‘classical Hamiltonian’. The minimization is done with respect to a set of coordinates, ${\cal C}$, that corresponds to expectation values of gauge invariant operators. The coherent state approach provides the mathematical prescription for calculating the function ${\cal H}({\cal C})$ and it is shown to be given by $${\cal H}({\cal C}) = \<{\cal C}|H|{\cal C}\>.$$ Here $|{\cal C}\>$ is the so called ‘coherent state’ obtained by applying a unitary color singlet operator, that we denote by ${\cal U(C)}$, on an arbitrary color singlet element $|0\>$ of Hilbert space, the so called ‘reference’ state : $$|{\cal C}\>={\cal U(C)} |0\>. \label{CS}$$ The operator ${\cal U}$ is a functional of gauge invariant operators like spatial Wilson loops of contour $\Gamma$ $$\tr \, W_\Gamma,\label{W}$$ as well as spatial Wilson loops decorated by a single insertion of the electric field matrix operator $\displaystyle{\left(E_x\right)_{ab}\equiv \sum_{i=1}^{N^2-1} E^i\, \lambda_{ab}}$, at point $x$ along the loop, i.e. $$\tr \, E_{x\in \Gamma} \, W_{\Gamma}, \quad {\rm and} \quad \tr W_{\Gamma} \, E_{x\in \Gamma}.\label{Wdecor}$$ (the two operators that appear in are different since the electric fields do not commute with the gauge fields). Also, in the presence of fermions, ${\cal U}$ depends on string-like operators of the form $$\psi^\dag_x \, U_{x\to y} \psi_y,\label{string}$$ where here $U_{x\to y}$ is a string of gauge field operators connecting site $x$ with site $y$. The generic way ${\cal U}({\cal C})$ depends on these operators is $$\begin{aligned} {\cal U}({\cal C}) &\equiv& {\cal U}_F\times {\cal U}_G,\label{Udef1}\\ {\cal U}_F &\equiv& \exp\, \left[ \sum_{xy} {\cal C}^{xy}_F\, \psi^\dag_x \, U_{x\to y} \psi_y \right],\\ {\cal U}_G &\equiv& \exp\, \left[ \sum_\Gamma \, {\cal C}^\Gamma_G\,\tr \, W_\Gamma + \sum_{\Gamma}\sum_{x\in \Gamma} \, \left({\cal C}^{x,\Gamma}_{G,1}\,\tr \, E_{x}\, W_\Gamma + {\cal C}^{x,\Gamma}_{G,2}\,\tr \, W_\Gamma E_{x}\right)\right],\label{Udef3}\end{aligned}$$ where ${\cal C}^{xy}_F,{\cal C}^{\Gamma}_G$, and ${\cal C}^{x\Gamma}_{G,1,2}$ are $c$-numbers. Since all the three type of operators defined in Eqs. (\[W\]–\[string\]) form a closed algebra, the structure of Eqs. (\[Udef1\]–\[Udef3\]) guarantees that ${\cal U}({\cal C})$ is an element of a Lie group referred to as the coherence group. Indeed ${\cal C}^{xy}_F,{\cal C}^{\Gamma}_G$, and ${\cal C}^{x\Gamma}_{G,1,2}$ parameterize this Lie group and furnish coordinates on its manifold. More precisely, the fermionic part of the coherence group is parameterized by the fermionic coordinates ${\cal C}_F$, and the gluonic part by ${\cal C}_{G}$ and ${\cal C}_{G,1,2}$.[^8] The values ${\cal C}_{\rm min}$ that minimize ${\cal H(C)}$ then determine the values of all gauge invariant observables in the ground state of $H$, so for example, the ground state energy $E_{\rm g.s.}$ is given by $$\lim_{N\to \infty} E_{\rm g.s.} = {\cal H(C=C_{\rm min})},$$ etc. In the large-$N$ limit of QCD with the number of flavors, $N_f$, kept fixed, the minimization of ${\cal H}$ process proceeds in two steps : one begins by minimizing the leading $O(N^2)$ contribution of ${\cal H}({\cal C})$. At leading order the latter is equal to the classical function ${\cal H}_G$ given by the classical Hamiltonian of the pure gauge theory $${\cal H}_G = \<0_g|\, {\cal U}^\dag_G\, H_G \, {\cal U}_G\, |0_g\>,$$ where the operator $H_G$ is given in and $|0_g\>$ is a reference state in the pure gauge system. This step sets the value of the expectation values of the gluonic color singlet operators such as the spatial Polyakov loops. For brevity, let us refer to these expectation values by the generic symbol ${\cal P}$, and by ${\cal P}_{\rm min}$ to their value at the minimum of ${\cal H}_G$. Next, one minimizes the $O(N)$ contribution to ${\cal H}$. To leading order it is given by calculating the expectation value of $H_K+H_C$ from Eqs. (\[H\_F’\]) and (\[HC\]) in the subset of coherent states whose gluonic coordinates have already been determined by the gauge dynamics. This contribution is thus given by $${\cal H}_F({\cal C}_F)\equiv \<0_F|\, {\cal U}^\dag_F\, \left(H_K+H_C\right)\, {\cal U}_F\, |0_F\>_{|{\cal P}={\cal P}_{\rm min}},$$ with $|0_F\>$ denoting a fermionic reference state. This two step process reflects the dominance of the gauge field dynamics over the fermion dynamics. Indeed the back-reaction of the fermions on the gauge fields comes from terms which we do not consider in this work and that are subleading in $1/N$. This is true as long as there is no enhancement of these $1/N$ terms by massless modes, which we assume to be the case.[^9] For our purposes it will be easier to simply calculate ${\cal H}_F({\cal C}_F)$ and substitute the set of expectation values ${\cal P}$ by its value in the [*exact*]{} ground state of $H_G$. The latter is known analytically and we repeat its derivation in Section \[SectorG\]. The solution of the fermion sector, however, is done as described above and we calculate the classical function ${\cal H}_{F}({\cal C}_F)$ in Section \[SectorF\]. We now turn to make the following remark on the way ${\cal H}$ is calculated and minimized. As mentioned above, the coordinates ${\cal C}$ corresponds to expectation values of different color singlet operators. More accurately, one can focus ones attentions to expectation values of only ‘non-factorizable’ operators such as the string operators of . Naively, however, one might expect that ${\cal H}$ should also be minimized with respect to coordinates that correspond to expectation values of ‘factorizable’ operators such as $$\frac1{N} \left( \psi^\dag_x \psi_x \right)\,\times\, \left( \psi^\dag_y \psi_y \right), \frac1{N}\left(\psi^\dag_x \, U_{x\to y} \psi_y \right)\,\times \, \left(\psi^\dag_z \,\psi_z\right), \dots.$$ This, however, is incorrect : at large-$N$ the expectation values of such operators factorize, and for example, $$\<\psi^\dag_x \psi_x \,\times \, \psi^\dag_y \psi_y \> \stackrel{N=\infty}{=} \< \psi^\dag_x \psi_x \>\,\times\, \< \psi^\dag_y \psi_y \>.$$ As a result, the expectation values of factorizable operators become determined by the expectation values of the non-factorizable operators and should not be thought of as independent coordinates that parameterize the coherent state manifold or the point within that manifold that represents the ground state. The unique role of non-factorizable operators operators is also reflected by the structure of the unitary operator of Eqs. (\[Udef1\]–\[Udef3\]) which depends only on such operators. Indeed in Ref. [@YaffeCoherent] it is shown that to generate the whole coherence group, it is sufficient to include in its algebra only the non-factorizable operators that appear in Eqs. (\[Udef1\]–\[Udef3\]). Adding other operators greatly complicates the algebra of the coherence group and is unnecessary. An important result of the discussion above, which we wish to emphasize, is that the coherent state $|{\cal C}\>$ defined in is sufficiently general to look for all possible large-$N$ ground states. Put differently, the form in does [*not*]{} correspond to assuming an ansatz for the ground state of the gauge theory. In the next two subsections we show how to implement the program outlined above in practice. We begin in Section \[SectorG\] with the treatment of the pure gauge case that allows us to calculate the expectation values of the color singlet gluonic operators, namely the traces of different powers of spatial Polyakov loops. The pure gauge Hamiltonian in our two-dimensional case is sufficiently simple that we can do so exactly and for any value of $N$. The next step is to apply the coherent state approach to the fermionic part of the Hamiltonian and we do so in Section \[SectorF\]. Solution of the gauge sector {#SectorG} ---------------------------- The solution of the pure gauge sector is well known [@Douglas]. The starting point is to notice that means that the gauge wave functions need to be anti-symmetric to an exchange of two angles, and for odd values of $N$, periodic in $2\pi$: $$\begin{aligned} \Psi(\varphi_1,\varphi_2,\dots,\varphi_k,\dots,\varphi_l,\dots,\varphi_N) &=& -\Psi(\varphi_1,\varphi_2,\dots,\varphi_l,\dots,\varphi_k,\dots,\varphi_N).\\ \Psi(\varphi_1,\varphi_2,\dots,\varphi_k+2\pi,\dots,\varphi_N) &=& +\Psi(\varphi_1,\varphi_2,\dots,\varphi_k,\dots,\varphi_N).\end{aligned}$$ These properties, together with the form of the Hamiltonian, tell us to think of the $\varphi_a$ as the positions of $N$ non-relativistic fermions with mass $2/g^2L_s$, moving on a circle with periodic boundary conditions and a fixed center of mass. The single particle wave functions of such a system are the plane waves $e^{i\varphi n}$ with $n=0,\pm 1,\pm 2,\dots$, and the ground state wave function of this $N$-fermion system is, up to a phase, the $N\times N$ slater determinant obtained by occupying momentum states distributed symmetrically around zero and limited by ‘Fermi momenta’ $n_F=(N-1)/2$: $$\Psi(\{\varphi\}) = \det_{-{n_F} \le a,b \le +n_F} e^{i \, b\,\varphi_a }. \label{gaugeGS}$$ This wave function is an eigenstate of $H_G$ with eigen-energy that has the large-$N$ limit of $\frac{L_sN^2}{48}\times g^2N$. It is easy to show that this determinant differs from the Vandermond determinant by a phase that depends only on the ‘center of mass’ of the fermions, $\sum_c\varphi_c$ , and in $SU(N)$ we set this phase to zero. The result is $$|\Psi(\{\varphi\})|^2 = \Delta^2(\{\varphi\}), \label{gaugeGS1}$$ and so when we calculate expectation values of gauge invariant gluonic operators $O(\left\{\varphi\right\})$ we need to perform the following integral (here we set the normalization such that $\<1\>=1$ and denote $\sum_c \varphi_c$ by $\varphi_{c.m.}$) $$\begin{aligned} \<\hat O\>_{G} &=& \int d\varphi \, \Delta^2(\{\varphi\})\,\delta\left(\varphi_{c.m.} \right)\, \hat O(\left\{\varphi\right\}), \label{vevG}\\ \int d\varphi&\equiv& \frac1{N!}\int \, \prod_{a=1}^N \frac{d\varphi_a}{2\pi}.\end{aligned}$$ Solution of the fermion sector : large-$N$ coherent states {#SectorF} ---------------------------------------------------------- We diagonalize the fermion sector in a variational manner. As discussed above, the most general form of the coherent state $|{\cal C}_F\>$ is given by $$|{\cal C}_F\> \equiv {\cal U}({\cal C}_F) |0\> = \exp \left( -i \sum_{x \in Z_{L_s}}\sum_{ y \in Z} \, {\cal C}^{xy}_F \, \psi^{\dag \, a}_x \, e^{i\varphi_a(y-x)/L_s} \, \psi^a_y\right) \, |0\>, \label{theta}$$ and parameterized by the infinite dimensional hermitian matrix ${\cal C}_F$. The state $|0\>$ is a fermionic reference state which we choose to be a state annihilated by $\psi^\dag_x$ for a subset of lattice sites that are full of fermions and that we denote by $S$. More precisely $|0\>$ is defined to obey $$\psi^\dag_{x,a} |0\> =0 \quad ;\quad \forall x\in S.$$ On the complementary set of sites, $x\in \bar S$, the reference state is annihilated by $\psi_{x}$. This means that we choose $B+L_s/2$ of the lattice sites to be full of baryons and the rest empty of baryons. This choice is convenient for our calculation, but the results are insensitive to it : the only thing that matters is that the overall baryon number that $|0\>$ contains is $B+L_s/2$ (such that the renormalized baryon number is $B$). We emphasize here that in the exponent of , the index $x$ runs over $1,2,\dots, L_s$, while the sum over $y$ is unrestricted : $y=0,\pm 1,\pm 2,\dots, \pm \infty$. The identification $$\psi^a_x=\psi^a_{x+k\,L_s} \quad ;\quad k\in Z,$$ means that ${\cal C}_F$ needs to obey $${\cal C}^{xy}_F={\cal C}^{x+k\,L_s,y+k\, L_s}_F\quad ;\quad k\in Z, \label{thetaBC}$$ in order to have ${\cal U}({\cal C}_F)\,{\cal U}^\dag({\cal C}_F)={\bm 1}$. To find what is ${\cal C}_F$ we minimize the classical Hamiltonian ${\cal H}$ defined by $${\cal H}_F\equiv \int d\varphi\,\, \<{\cal C}_F|H_K+H_C|{\cal C}_F\> \, \Delta^2(\varphi)\times \delta\left(\varphi_{c.m.}\right),\label{Hclassical}$$ in the space of all possible choices of ${\cal C}_F$. The calculation of $\<{\cal C}_F|H_K+H_K|{\cal C}_F\>$ is somewhat lengthy and we postpone it to Appendix \[appHF\]. The resulting expression can be written in terms of the following ‘density matrices’ $\rho^q_{xy}$ and $\bar \rho^q_{xy}$ with $x,y\in [1,L_s]$ and $q\in Z$ $$\begin{aligned} \rho^q_{xy}&\equiv&\sum_{p\in Z\atop z\in S}\, \left(e^{i{\cal C}_F}\right)^{z+pL_s,x}\, \left(e^{-i{\cal C}_F}\right)^{y,z+(q+p)L_s},\label{rho}\\ \bar \rho^q_{xy}&\equiv& \delta_{q,0} \delta_{xy} - \rho^q_{xy},\label{rhobar}\end{aligned}$$ In terms of $\rho^q_{x,y}$ and $\bar \rho^q_{x,y}$ the classical Hamiltonian is $$\begin{aligned} \<{\cal C}_F|H_K+H_C|{\cal C}_F\> &=& \sum_{x\in Z_{L_s}\atop q \in Z} \left[ \left(-\frac{i}2 \rho^q_{x,x+1} + c.c. \right)+ m (-1)^x \rho^q_{xx} \right]\times \sum_a e^{i\varphi_a}\nonumber \\ &+&\frac{g^2}{4L_s} \sum_{xy\in Z_{L_s}}\sum_{p\atop a\neq b} \sum_{qq'\in Z} \frac{\rho^{q'}_{xy}\, \bar \rho^{-q}_{yx} \, e^{-i(x-y)p }\times e^{-i\varphi_a(x-y +q L_s)/L_s + i\varphi_b( x-y + q'L_s)/L_s}}{4\sin^2((\varphi_a-\varphi_b)/L_s + p)/2} + O(1/N). \nonumber \\\label{HKC}\end{aligned}$$ Note that in contrast to the form , here the second term does not contain any contributions from the $a=b$ terms. They are shown to be subleading in Appendix \[appHF\]. To obtain a compact form for the classical Hamiltonian ${\cal H}_C$ we define $P_k$ to be the Polyakov loop operator that winds $k$ times around the torus as $$P_k \equiv \frac1{N} \sum_a e^{i\varphi_a k},$$ and use the identity $$\frac1{4\sin^2\left((k+i\epsilon)/2\right)} = - \frac12 \sum_{Q\in Z} \, |Q|\, e^{-\epsilon |Q| + iQk},\label{FT}$$ with $\epsilon>0$ regularizing the pole of the left hand side, to re-write the second term of as a sum over terms that only contain powers of $e^{i(\varphi_a-\varphi_b)/L_s}$. Indeed, substituting into the second term of , we find that the terms contributing to the sum over the momentum variable $p=2\pi\, l/L_s$ can be isolated and read $$\sum_{l=0}^{L_s-1} e^{-i2\pi l(x-y - Q)/L_s} = L_s \, \sum_{\bar Q \in Z}\delta_{y-x+Q,\bar QL_s}.$$ This allows us to substitute $Q$ in by $x-y+\bar QL_s$. Since the dummy summation variable $\bar Q$ obtains all possible integer values, we drop the ‘bar’ from its notation in the rest of the paper. Thus, the final result of the manipulations in the last paragraph is that gets the following form. $$\begin{aligned} {\cal H}_F(\rho)/N &=& \sum_{x\in Z_{L_s}\atop q \in Z} \left[ \left(-\frac{i}2 \rho^q_{x,x+1} + c.c. \right)+ m(-1)^x \rho^q_{xx} \right] \<P_q\>_G \nonumber \\ &-&\frac{g^2N}{8} \sum_{xy\in Z_{L_s}}\sum_{Qqq'\in Z} \rho^{q'}_{xy}\, \bar \rho^{-q}_{yx} \, |x-y+QL_s| e^{-\epsilon|x-y+QL_s|} \left( \<P_{Q-q} P_{-(Q-q')}\>_G - \frac1N \<P_{q'-q}\>_G\right), \nonumber \\\label{HKC1}\end{aligned}$$ where here we define the gluonic exception values $\<,\>_G$ in . ### Remarks on the form of the classical Hamiltonian {#remarks} above is quite an important ingredient in our paper and so let us now pause here and make the following remark on its form and its implications. What tells us is that the fermionic properties of the system, which are represented by the density matrices $\rho^q_{xy}$, self interact as well as couple to the Polyakov loops $P_q$. The expectation values of the latter are determined by the gauge dynamics and feel no back reaction from the fermions. Thus, performing the minimization in Section \[SectorG\] correctly, and determining the properties of the gluonic vacuum in a consistent way, is crucial to get the correct fermion dynamics. For example, in the Sections \[otherworks\] and \[decompact\] we emphasize that mistreating the glue sector, which is what one does when one ignores the zero modes, leads, through the way couples the glue sector to the fermion sector, to erroneous results for various fermionic expectation values. In particular, one can see that if the expectation values of the different Polyakov loops are incorrectly chosen to be unity, $$P_q =1, \qquad \forall q,$$ then by using large-$N$ factorization of the double trace term $\<P_{Q-q}\,P_{-(Q-q')}\>\stackrel{N=\infty}{=}\<P_{Q-q}\>\,\<P_{-(Q-q')}\>$, one finds that the second term in is strongly dependent on $L_s$ (this is shown explicitly in Section \[symm\]). This volume dependence is contradicting general arguments on large-$N$ gauge theories such as those of Ref. [@EK]. ### Further manipulations of the classical Hamiltonian and preparing for its minimization {#more} In this section we make further important simplifications of that will also allow us to minimize it (see for example Ref. [@nonzeroBpaper] and the next sections). For a $U(N)$ theory one drops the delta functions in , and the gluonic expectation values $\<P_q\>_G$ and $\<P_q\, P_{q'}\>_G$ are calculated in Appendix \[appHF\] (see Eqs. \[Pk\]–\[PkPk’\]). In the $SU(N)$ case the integral in changes only by restricting the sum $\sum \varphi_a$ to be zero, and so we expect the Polyakov loops to be the same in the large-$N$ limit (we show this explicitly for $\<P_q\>_G$ in Appendix \[appHF\]). Using Eqs. (\[Pk\])–(\[PkPk’\]) we see that, as expected, the leading contribution to is from the $q=0$ term in the first line and from the $q=q'=Q$ in the first term of the second line. This gives us[^10] $$\begin{aligned} {\cal H}_F(\rho)/N &=& \sum_{x\in Z_{L_s}} \left[\left(-\frac{i}2 \rho^0_{x,x+1} + c.c. \right)+ m(-1)^x \rho^0_{xx} \right]\nonumber \\ &-&\frac{g^2N}{8} \sum_{xy\in Z_{L_s}}\sum_{Q\in Z} \rho^{Q}_{xy}\, \bar \rho^{-Q}_{yx} \, |x-y+QL_s| e^{-\epsilon|x-y+QL_s|}.\nonumber \\\label{HKC11}\end{aligned}$$ Since the index $Q$ runs over all integers we can write $$\rho^Q_{xy}=\int_{-\pi}^{\pi} \frac{dp}{2\pi} \rho_{xy}(p) e^{ipQ}, \label{FT2}$$ which, re-using the identity gives the form $$\begin{aligned} {\cal H}_F(\rho)/N &=& \int \frac{dp}{2\pi}\sum_{x\in Z_{L_s}} \left\{ \left(-\frac{i}2 \rho_{x,x+1}(p) + c.c. \right)+ m(-1)^x \rho_{xx}(p)\right\} \nonumber \\ &+& \frac{g^2N}{4} \int \int \frac{dp}{2\pi} \frac{dp'}{2\pi} \,\frac1{L_s}\,\sum_{xy\in Z_{L_s}}\sum_{l=1}^{L_s} \frac{\rho_{xy}(p)\, \bar \rho_{yx}(p') \, e^{i2\pi l(x-y)/L_s}}{4\sin^2\left((p-p')/L_s+2\pi l/L_s\right)/2} .\nonumber \\\label{HKC2}\end{aligned}$$ The pole at $p=p'$ and $l=L_s$ in might seem alarming and tracking back its source to one finds that it is the double sum over the color indices $a$ and $b$ that appears in the second term there. Specifically, terms in that sum for which $|a-b|$ is small are causing this divergence. For these terms the argument of the sine in the denominator can be small; for example if $p=2\pi$ and the Polyakov loops have zero expectation value in the gluonic ground state, then heuristically $\varphi_a\sim 2\pi a/N$. (This is not a gauge invariant statement, but for the current discussion this subtlety is not important. In the rest of this paper we take great care to avoid such statements when it is important to do so.) Thus, we see that when $|a-b|\stackrel{<}{_\sim} O(1)$, then the argument of the sine in is very small and this corresponds to the pole in above. This pole is the source of the IR divergence in the usual treatments of the ‘t Hooft model, and that is usually resolved with the [*ad hoc*]{} principle value prescription. In our case this is not neede. The $a=b$ terms were excluded from the sum in . The way this restriction emerges in is through certain conditions obeyed by the density matrices $\rho^Q_{xy}$ and $\bar \rho^Q_{xy}$ which we present in Appendix \[appHF\] (see Eqs. (\[r11\])–(\[r21\])) and that reflect the unitarity of the operator ${\cal U}({\cal C}_F)$. In the language of $\rho_{xy}(p)$ these conditions read $$\sum_{y\in Z_{L_s}} \rho_{xy}(p) \bar \rho_{yz}(p) = 0.$$ Thus the divergence in is removed and we get the principle value prescription. To show that the near vicinity of $p=p'$ (corresponding to $|a-b|$ small but nonzero) is not causing any lower divergences we need to assume a form for the $p$ dependence of $\rho_{xy}(p)$. Instead we have confirmed this numerically whilst minimizing ${\cal H}(\rho)$ with respect to $\rho$ [@nonzeroBpaper]. We conclude this section by writing in a way which is convenient for its minimization. We first solve the constraints Eqs. (\[r11\])–(\[r21\]) on $\rho^Q_{xy}$ by writing $$\begin{aligned} \rho^Q_{xy}&\equiv & \frac1{M} \sum_{a=1}^M \,\sum_{n=1}^{B+L_s/2} \, \phi^n_a(x)\,e^{\frac{2\pi iQa}{N}}\, \phi^{n\star}_a(y), \label{constraints_res1}\\ \bar \rho^Q_{xy}&\equiv & \frac1{M} \sum_{a=1}^M \,\sum_{n=B+L_s/2+1}^{L_s} \, \phi^n_a(x)\,e^{\frac{2\pi iQa}{N}}\, \phi^{n\star}_a(y), \label{constraints_res2}\end{aligned}$$ where for each $a=1,\dots,M$, the single particle wave functions $\phi^n_a(x) \, ; \, n\in [1,L_s]$ span an $L_s\times L_s$ dimensional space, $$\sum_{x\in Z_{L_s}} \phi^n_a(x)\, \phi^{m\star}_a(x) = \delta_{mn}.\label{ortho}$$ is nothing but a discretized way to write the most general expression for $\rho$ and $\bar \rho$, that also obeys Eqs. (\[r11\])–(\[r21\]) and we show that this is correct in Appendix \[app\_rho\_resolve\]. Note that the full space of solutions for $\rho$ and $\bar\rho$ is accessible only if $M=\infty$. In terms of $\phi^n_a(x)$ we get $$\begin{aligned} {\cal H}_F(\phi)/N &=& \frac1{M} \sum_a\sum_{x\in Z_{L_s}}\left\{ \left(-\frac{i}2 \rho^a_{x,x+1} + c.c. \right)+ m (-1)^x \rho^a_{xx}\right\} \nonumber \\ &-& \frac{g^2N}{4} \frac1{L_sM^2} \sum'_{abl} \sum_{xy\in Z_{L_s}} \frac{\rho^a_{xy}\, \rho^b_{yx} \, e^{-i(x-y)\left(\frac{2\pi}{L_s}(\frac{a-b}{M}+l)\right)}}{4\sin^2\left(\frac{2\pi}{L_s}(\frac{a-b}{M}+l)/2\right)}\nonumber \\ \nonumber \\ &+& \frac{g^2N(B+L_s/2)}{4} \frac1{L_sM^2} \sum'_{abl} \frac1{4\sin^2\left(\frac{2\pi}{L_s}(\frac{a-b}{M}+l)/2\right)},\label{HKC4}\label{HF_again} \\ \rho^a_{ab}&\equiv&\sum_{n=1}^{B+L_s/2}\, \phi^n_a(x)\, \phi^{n\star}_a(y).\end{aligned}$$ Here, by the prime on the sums we mean that the terms with $a=b$ and $l=L_s$ are excluded.[^11] We now perform the variation of ${\cal H}$ with respect to the $M$ functions $\phi_a$ $$\frac{\delta}{\delta (\phi^n_a(x))^\star} \left( {\cal H} - \sum_{m,b} \epsilon^b_m \sum_x \phi^m_b(x) \, \phi^{m\star}_b(x) \right)=0, \label{variation}$$ and we find that they must obey the following $M$ coupled nonlinear differential equations (here we use the Lagrange multiplier $\epsilon^a_{n}$ to enforce ) $$\begin{aligned} \sum_{y\in Z_{L_s}} \, h^a_{xy} \ \phi^n_a(y) &=& \epsilon^a_n \, \phi^n_a(x),\label{diagEQ}\end{aligned}$$ with $$\begin{aligned} h^a_{xy}&=& +\frac{i}2 \left(\delta_{y,x+1} - \delta_{y,x-1}\right) + m\, (-1)^x\, \delta_{xy} -g^2N \, v^a_{xy}, \\ v^a_{xy}&=&\frac1{2M}\sum_b \, K_{ab}(y,x)\, \left(\sum_{m=1}^{B+L_s/2}\, \phi^m_b(x)\, \phi^{m\star}_b(y)\right).\end{aligned}$$ and $$\begin{aligned} K_{ab}(y,x) &=& \frac1{L_s}\,\sum_{l\in Z_{L_s}}' \, \frac{e^{\frac{2\pi i(x-y)}{L_s}\left(\frac{a-b}{M}+l\right)}}{4\sin^2\left(\frac12\left(\frac{2\pi (a-b)}{ML_s} + \frac{2\pi l}{L_s}\right)\right)}.\label{Kernel}\end{aligned}$$ Since $K_{ab}$ explicitly depends on $\phi$ then the solution of is a self-consistent process. Within the space of all functions $\phi$ that obey , the correct solution is the one that has the lowest value of ${\cal H}$. The latter [*is not*]{} equal to $\sum_{an}\epsilon^a_n$, since this will count the Coulomb interaction twice. Instead we find $$\begin{aligned} {\cal H}_{\rm solution}/N &=& \frac1{2M}\sum_{n=1}^{B+L_s/2}\,\sum_{a=1}^M\left(\epsilon^n_a + \sum_x \left[{\rm Im}\, \left(\phi^n_a(x) \phi^{n\star}_a(x+1)\right) + m\, (-1)^x\ \phi^n_a(x) \phi^{n\star}_a(x)\right] \right)\nonumber\\ &+&\frac{g^2N(B+L_s/2)}{4} \frac1{L_sM^2} \sum'_{abl} \frac1{4\sin^2\left(\frac{2\pi}{L_s}(\frac{a-b}{M}+l)/2\right)}.\label{Hsol}\end{aligned}$$ Comparison to other relevant works {#otherworks} ================================== In this section we wish to discuss the way that our resulting equations differ from those appearing in other works that also regularize the ‘t Hooft model on a finite circle $L$. 1. Ref. [@LTYL] : this work looked at QCD$_{1+1}$ with the light-cone Hamiltonian, but neglected the curvilinear character of the variables $\varphi$. Thus the ground state with $\varphi_a=0$ was chosen, that in fact has zero measure within the correct solution. This is equivalent to assuming the Polyakov loops are all nonzero, and in Section \[decompact\] we discuss the consequences of such a choice, but in essence it violates volume reduction. In fact, Ref. [@LTYL] showed that this erroneous ground state leads to a phase transition as a function of $L$, which clearly contradicts large-$N$ volume independence. 2. Ref. [@LNT] looked at the Hamiltonian of QCD in axial gauge (in any number of dimensions), and in the continuum. It pointed out to the error made in [@LTYL] but did not discuss the consequences of this on the $1+1$ solution presented in [@LTYL]. In our paper here we generalize the theoretical framework developed in [@LNT] to the lattice regularization. 3. Ref. [@SchonThies_decompact] : This interesting paper formed one of the motivations for our study. Here the authors showed how the phase transition found in [@LTYL] disappears when one chooses an appropriate ansatz for the ground state. The way this choice is made in Ref. [@SchonThies_decompact], however, is not manifestly gauge invariant and breaks a residual gauge symmetry of the Hamiltonian. This has the disadvantage of making it hard to argue that the ansatz used is exact at large-$N$, and to construct a gauge invariant fermionic ground state. We avoid this issue in our paper by working with manifestly gauge invariant operators and states. This is most naturally done with the coherent state approach. Other differences between our paper and [@SchonThies_decompact] is the fact that we use the lattice regularization, we do not restrict to zero baryon number or to a particular ansatz for the ground state, and we analyze the role of translation symmetry for volume reduction. In particular, in the following section, we show how a ‘soft’ form of large-$N$ volume independence works if translation invariance breaks to one of its subgroups. Finally we make the connection with the Eguchi-Kawai volume independence manifest. 4. Ref. [@Salcedo] : To our knowledge this paper is the only one that solves the ‘t Hooft equation at nonzero baryon number for a general ratio $\sqrt{g^2N}/m$.[^12] Unfortunately, the authors restrict to study a single baryon only, and, like Ref. [@LTYL], ignore the curvilinear nature of the $\varphi$ coordinates. Consequently, at short lengths (or large baryon densities) their approach would fail, and exhibit the same phase transition seen in [@LTYL]. Clearly this calls for revisiting of the topic which we aim to perform in Ref. [@nonzeroBpaper]. The list we give above demonstrates the usefulness of our current paper : Firstly, it uses the coherent state approach, which is manifestly gauge invariant throughout. Secondly, it generalize all former studies to the lattice regularization, and extends the study of volume independence to systems with nonzero baryon density and partial translation invariance. Thirdly, it opens the way to study the ‘t Hooft model for arbitrary values of the quark mass, spatial volume, and Baryon number. To our knowledge this is the first time these steps are taken. Large-$N$ volume independence {#decompact} ============================= It is generally expected of large-$N$ QCD in $1+1$ dimensions to be independent of its volume. This equivalence of large-$N$ gauge theories with different volumes was first suggested in [@EK] and caused great excitement since it was seen to be a potentially easy way to solve large-$N$ QCD on the lattice. Shortly after [@EK], the papers [@BHN1; @MK] showed that this equivalence breaks down in the continuum limit for three or more space-time dimensions. This breaking of reduction is signaled by the fact that Polyakov loops that wrap around the volume acquire nonzero expectation values. It is useful to put this large-$N$ equivalence in the more general context of orbifold projections between mother and daughter gauge theories : in our case the mother theory is large volume QCD while the daughter theory is small volume QCD. These projections are expected to become equivalences when the rank of the gauge group, $N$, becomes large. For a review on this topic we refer to Ref. [@AEK]. As shown there, these equivalences hold only between certain sectors of the mother and daughter theories which are defined to be neutral under certain symmetries. In the original Eguchi-Kawai paper it was stressed explicitly that the ground state in both theories needs to be symmetric under the center of the gauge group. For $SU(N)$ gauge theories this means that the global $Z_N$ subgroups of the local $SU(N)$, that correspond to multiplying Polyakov loops in different directions by a $Z_N$ phase $e^{2\pi i/N}$, must be unbroken. Thus these Polyakov loops are the order parameters of these symmetries and must have vanishing expectation values for the equivalence to hold. Another symmetry that the ground states of the mother and daughter theories should respect in the volume projection case is translation symmetry. This is clear intuitively – how can we describe a theory which breaks translations and that as a result has operators with expectation values that depends on the space-time coordinate, by a theory that has no volume ? The requirement of intact translation symmetry can of course be anticipated from the construction of Ref. [@AEK], since this is one of the symmetries that define that neutral sectors of a large-to-small volume mapping. Indeed this was already explicitly pointed out in the first paper in [@YaffeCoherent]. When this symmetry breaks it is no longer true that the physical observables in the large volume theory have a one-to-one mapping to observables in the zero volume theory. If the attempt to map the large and small volume theories fails, then clearly they cannot be large-$N$ equivalent. Unfortunately, we believe that the role of translation symmetry in the Eguchi-Kawai equivalence is not fully stressed in some of the relevant literature. The reason, however, is obvious : the QCD vacuum respects translation symmetry ! But there is at least one physical scenario where one can expect to get broken translation symmetry in QCD and that is at nonzero baryon number or chemical potential, where crystals of different sort can form (for relevant literature on the topic we refer the interested reader to the review in [@MP]). What happens to large-$N$ volume independence in that case ? To answer this question we find it useful to see what happens in a well defined and systematic calculation, and the choice of this paper is the ‘t Hooft model.[^13] Thus our goal in this section is to show how the volume dependent ‘t Hooft classical Hamiltonian ${\cal H}$, derived in , behaves in different cases. We first study the case of a translational invariant ground state (for any value of $B$). We do so for both the original Hamiltonian and also ask what happens if one forces the Polyakov loops to acquire expectation values of different sorts. We then move to discuss what happens when we allow translation symmetry to break. Full translation symmetry {#symm} ------------------------- Since the staggered fermion Hamiltonian is invariant to translations by two lattice sites, it will be easier to discuss the original ‘naive fermions’ case in these sections. These fermions are two-component spinors and it is straight forward to repeat the analysis in Section \[SectorF\] for them. The result is that $\rho^Q_{xy}$ becomes $2\times 2$ matrix and that the classical Hamiltonian in now has the following form (here and below the trace refers to this extra $2$-dimensional Dirac space). $$\begin{aligned} {\cal H}_F/N&=&-\frac{i}2 \sum_x \, \tr \, \sigma_3 \, \rho^{0}_{x,x+1} + c.c. + m \sum_x \tr \sigma_1 \, \rho^{0}_{x,x}\nonumber\\ &+&\frac{g^2N}8\, \sum_{x,y\in Z_{L_s}\atop Q\in Z_N}\,\tr \rho^Q_{xy}\, \rho^{-Q}_{yx} \, |x-y+QL_s| e^{-\epsilon|x-y+QL_s|}.\label{HKC5}\end{aligned}$$ Note that to get we dropped the constant term that appears when one uses to write ${\cal H}_F$ in terms of only $\rho$ (and not $\bar\rho$). Restricting to translation invariant states mean that the coherent states have $${\cal C}^{xy}_F = {\cal C}^{x-y}_F.$$ Using Eqs. (\[rho\]) and (\[thetaBC\]) this means that $\rho^Q_{xy}$ is a function of the combination $x-y+QL_s$ : $$\rho^Q_{xy} \equiv \rho(x-y+QL_s).$$ Combining the sums over $x$ and $Q$ into a single sum over the integers, we get $$\begin{aligned} {\cal H}_F/(NL_s)&=&\tr \left[ \left(-\frac{i}2 \, \rho(-1)\, \sigma_3 + h.c. + m \, \sigma_1\,\rho(0)\right) \right]+\frac{g^2N}8 \sum_{r\in Z}\tr \rho(r)\, \rho(r)\, |r|e^{-\epsilon|r|}. \label{HH}\end{aligned}$$ Since $\rho(r)$ is defined for all integer values $r$, we can write $$\rho(r) = \int \frac{dp}{2\pi} e^{ipr } \rho(p).$$ In terms of $\rho(p)$ the constraint , applied to naive fermions, becomes $$\int \frac{dp}{2\pi} \, \tr \rho(p) = n_B + 1,\label{r311}$$ with $n_B$ equal to the baryon density $B/L_s$, and the classical Hamiltonian has the form $$\begin{aligned} {\cal H}_F/(NL_s) &=& \int_0^{2\pi} \frac{dp}{2\pi}\, \tr \left[\rho(p) \left( -\sigma_3 \sin(p) + m\, \sigma_1 \right) \right] -\frac{g^2 N}{4} {{\displaystyle{-} \hspace{-0.4cm} \int}}{{\displaystyle{-} \hspace{-0.4cm} \int}}\frac{dp}{2\pi} \, \frac{dq}{2\pi}\, \frac{\tr \left(\rho(p)\, \rho(q)\right)}{4\sin^2\left((p-q)/2\right)}.\label{HFrp}\end{aligned}$$ Here by ${{\displaystyle{-} \hspace{-0.4cm} \int}}$ we mean the principle value which now has a precise meaning in the form of the primed sum in . The crucial point that we want to make is that Eqs. (\[r311\])–(\[HFrp\]) are independent of $L_s$ and this is the way volume reduction works in our model.[^14] It is easy to repeat the above steps for a gluonic state that gives nonzero expectation value to some windings of the Polyakov loop. This can be realized by adding a potential for the Polyakov loops, in similar lines to the potential suggested in [@DEK] (although, of course, with an opposite sign since the potential of [@DEK] was devised to null all expectation values of all Polyakov loops). A potential can be chosen such that it induces, in the gluon sector, a spontaneous breaking of the $Z_N$ symmetry of the form $$Z_N\longrightarrow \O.$$ In this case we have $$\<P_q\>_G = 1, \quad \forall q,$$ and it can be easily shown that the only change this causes to is to replace $\rho^Q_{xy}$ by the $Q$-independent function $\tilde \rho_{xy}$, that is given by $$\tilde \rho_{xy} = \sum_{Q\in Z} \rho^Q_{xy}.\label{tilderho}$$ Next, for a translational invariant state we write $$\tilde \rho_{xy} = \frac1{L_s} \sum_{l=1}^{L_s} \tilde \rho(l)\, e^{i2\pi l/L_s (x-y)}.$$ Since $\tilde \rho$ is now independent of $Q$, one can use to perform the sum over $Q$ in and one gets $$\begin{aligned} {\cal H}_F/(NL_s) &=& \frac1{L_s} \sum_{l=1}^{L_s} \tr \left[\tilde \rho(l) ( -\sigma_3 \sin\left(\frac{2\pi l}{L_s}\right) + m\, \sigma_1 ) \right] -\frac{g^2 N}{4} \frac1{L^2_s}\sum_{l,k=1\atop l\neq k}^{L_s} \, \frac{\tr \left(\rho(l)\, \rho(k)\right)}{4\sin^2\left(\pi (l-k)/L_s\right)}.\nonumber \\ \label{HFrp1}\end{aligned}$$ The important point about is that it depends on $L_s$ is a very strong way. Indeed, this is the reason Ref. [@LTYL], which sets all Polyakov loops to be nonzero, saw a strong volume dependence of physical observables which was realized in a phase transition that occurs as a function of $L$. It is also easy to generalize this result to a breaking of $Z_N\to Z_K$. In that case $L_s$ in the l.h.s. of is replaced by $KL_s$, which again means that ${\cal H}_F/L_s$ depends strongly on $L_s$. An important remark noted also in Ref. [@SchonThies_decompact] here is that when we resolved the quantum Gauss law we assumed that the following conditions holds $$\varphi_a - \varphi_b \neq 2\pi n\quad ;\quad n\in Z\label{restrict}$$ This is certainly correct for the correct solution with vanishing Polyakov loops where the field configurations that do not obey have zero Jacobian and thus zero measure (see ). For nonzero Polyakov loops, however, can indeed be violated. This means that more zero modes (except for $E^{''I}$) will be present. Since this is not the main topic of this paper we do not treat these here. To conclude, if we assume translation symmetry and unbroken $Z_N$, then two systems with the same $n_B$, yet different volumes, will be large-$N$ equivalent. Ref. [@SchonThies] showed, in the continuum regularization, that in the translation invariant sector, there is a density $n^c_B$ above which chiral symmetry is restored. Let us denote the dimensionless combination $n^c_B/\sqrt{g^2N}$ by $x^c$.[^15] This can be visualized in a simple phase diagram in the space $B$—$L\sqrt{g^2N}$ that we present in Fig. (\[phaseBL\]). ![Phase diagram for chiral symmetry restoration [*if one assumes translation invariance*]{}. Along fixed lines of fixed $n_B$ there is exact volume independence. See [@SchonThies] for more details on the numeric value of the slope of the phase transition line. []{data-label="phaseBL"}](phase_BL.eps){width="10cm"} Physics along lines of fixed slope is $L$ independent. Nonetheless along horizontal lines, that have a fixed baryon number, one encounters strong $L$ dependence realized as a phase transition. The only case where these two lines coincide is, of course, the $B=0$ case. We note in passing that the authors of [@SchonThies] have shown that the translational invariant ansatz is inconsistent and suggest that the true ground state must break translations. This fits well with the calculation of Ref. [@Salcedo] for the $B=1$ case, and we find preliminary results for $B>1$ which are consistent with this [@nonzeroBpaper]. Broken translation symmetry and a crystal of baryons {#asymm} ---------------------------------------------------- In this section we wish to emphasize that the volume independence obtained above crucially depends on the assumption that $$\rho^Q_{xy} = \rho_{x-y+QL_s},\label{translations}$$ i.e. that the ground state is translation invariant. This condition is anticipated in advance from the point of view of orbifold projections [@AEK] : the symmetry by which one projects QCD at large volume to QCD at small volume is translation invariance. It is useful to make a concrete example to demonstrate this point, and we first consider the case of single baryon in a box. At large-$N$ the baryon has mass of $O(N)$ and is a static particle[^16]. As a result, its presence in the system spontaneously breaks translation invariance. Indeed the calculation in Ref. [@Salcedo] showed that the baryon wave function in the ‘t Hooft model pictorially looks like Fig. \[B1\], where we present a sketch of the baryon density. ![A cartoon of the baryon density of a single baryon at large-$N$ where a complete breakdown of translation invariance takes place and large-$N$ volume reduction does not work at all. []{data-label="B1"}](cartoonB1.eps){width="10cm"} In Ref. [@nonzeroBpaper] we plan to revisit this calculation and to extend it to a finite baryon density. Let us assume that for sufficiently large $B$ a crystal of baryons will form - of the sort sketched in Fig. \[Bcrystal\]. ![A cartoon of the baryon density of three baryons at large-$N$ – a partial breakdown of translation invariance : translations by a third of a unit still leave this state invariant. Thus large-$N$ reduction holds only between this system and a system with a single baryon on a box of size $\Delta$. []{data-label="Bcrystal"}](cartoonB3.eps){width="10cm"} In both the single baryon and the baryon crystal cases the ground state wave function has a characteristic length scale $\Delta$ – it is the baryon width for $B=1$ and the baryon-crystal wave-length for $B\gg 1$. These length scales must depend on $g^2N$, $m$, and $L$, and it is clear that decreasing the box size $L$ will change the ground state wave functions – the baryons will get squashed. This is a result of the compactness of space and the periodic boundary conditions. The only case where the ground state represented by Figs. (\[B1\]-\[Bcrystal\]) is invariant under a change in $L$ is if $\Delta=\infty$ which corresponds to intact translational symmetry. Nonetheless, there is an interesting subtlety to the statement above : in contrast to the single baryon case, the baryon crystal sketched in Fig. \[Bcrystal\] has an unbroken symmetry : shifts by $\Delta$ leave the ground state invariant. Thus the following softer form of large-$N$ volume-independence survives in this case. The meaning of this statement is depicted pictorially in Fig. \[Bcrystal\] : the three baryon system forms a crystal in a box of unit size, and so in these units $\Delta=1/3$. The statement above means we can reproduce the physics of this three-baryon system from a single-baryon system whose box size is $1/3$ – the system bounded between the two dash-dot vertical lines of Fig. \[Bcrystal\]. In these units the baryon number density of both systems is $3$, and this is the only relevant parameter at large-$N$. Let us now show how this happens. The starting point is to modify the ansatz in for the way $\rho^Q_{xy}$ depends on $Q,x$, and $y$. It will be first useful to change coordinates and define $$\begin{aligned} r &\equiv & x + QL -y , \\ s &\equiv& x+ QL +y,\\ \rho_{rs} &\equiv&\rho^Q_{xy}.\end{aligned}$$ Since $x,y \in [1,L_s]$ and $Q\in Z$, we have $r \in Z$. Given one fixes the value of $r$, the values $s$ obtains are $r+2,r+4,r+6,\dots,r+2L_s$. The periodicity in $\Delta$ means that $$\begin{aligned} \rho_{r,s+\Delta} &=& \rho_{r,s}.\end{aligned}$$ One solution to this condition is (here we assume that both $L_s/\Delta$ and $L_s$ are even) $$\rho_{r,s} = \int \frac{dp}{2\pi} \sum_{q\in Z_{2\Delta}} \, \rho_q(p)\, e^{ipr + iq\frac{2\pi}{2\Delta}s}.\label{FTnew}$$ Substituting into the first two terms of we have $$\begin{aligned} 1^{\rm st}-{\rm term} &=& \int\, \frac{dp}{2\pi}\, \tr \left[ -\sigma_3 \sin(p) \left( \rho_{2\Delta}(p) - \rho_{\Delta}(p)\right)\right],\label{H1st}\\ 2^{\rm nd}-{\rm term} &=& \int\, \frac{dp}{2\pi}\, \tr \left[ m \sigma_1\, \left( \rho_{2\Delta}(p) + \rho_{\Delta}(p)\right)\right].\label{H2nd}\end{aligned}$$ Substituting into the third term of ${\cal H}_F$ gives $$\begin{aligned} 3^{\rm rd}-{\rm term}/(NL_s)&=& -\frac{g^2N}{4}\sum_{q=1}^{2\Delta}\, \sum_{k=1}^{4}{{\displaystyle{-} \hspace{-0.4cm} \int}}{{\displaystyle{-} \hspace{-0.4cm} \int}}\frac{dp}{2\pi}\frac{dp'}{2\pi}\, \left[ \frac{\tr (\rho_q(p) \, \rho_{k\Delta-q}(p')}{4\sin^2((p-p')/2 + \pi k/2)} \right], \label{H3rd}\end{aligned}$$ Where we define $\rho_{q<0}(p)=\rho_{q>2\Delta}(p)=0$, and $\rho_{0}(p)\equiv\rho_{2\Delta}(p)$. The sum over $k$ is a result of the summation over $x=(r+s)/2$ and $y=(s-r)/2$ that one performs upon the substitution of into the Coulomb interaction term in the Hamiltonian. Finally the global Gauss Law constraint applied on $\rho_{rs}$ becomes $$\int \frac{dp}{2\pi} \, \tr \left(\rho_\Delta(p) + \rho_{2\Delta}(p)\right) = n_B+1.$$ As it stands, the classical ‘t Hooft Hamiltonian density ${\cal H}_F/(NL_s)$ and the Gauss law constraint, depend only on the parameters $m$, $g^2N$, $n_B$, and $\Delta$ – the volume is irrelevant at large-$N$. Similarly to what we saw in Section \[symm\], this volume independence is a direct result of the ansatz we took in that reflects a partial translation symmetry, and of the fact that all nontrivial winding of the spatial Polyakov loops have a zero expectation value. If we were to neglect the zero modes then all Polyakov loops would effectively be equal and nonzero. This will cause the infinite set of density matrices $\rho^Q_{xy}$ to be replaced by the $Q$-independent quantity $\tilde \rho_{xy}$ (see ) which depends only on the two compact coordinates $x$ and $y$. As a result the coordinate $r=x-y$ would have been as well, and will be replaced by $$\rho_{r,s} = \frac{1}{L_s}\sum_{l=1}^{L_s} \,\sum_{q\in Z_{2\Delta}}\, \rho_{q,l}\, e^{2\pi \,i\, (lr/L_s + qs/2\Delta)}.$$ Plugging this into the classical Hamiltonian will result in a function that strongly depends on $L_s$ and this is wrong. Remark on lattice simulations of volume-reduced large-$N$ QCD at nonzero $B$ or $\mu$ {#remark} ------------------------------------------------------------------------------------- Finally, let us make a remark of relevance to lattice practitioners. Consider large-$N$ QCD defined in a very small box with side $L$ that obeys $L\ll 1/\Lambda_{QCD}$. Let us also assume reduction holds for zero baryon number (this assumption is automatically fulfilled in two dimensions and for higher dimensions is expected to hold in modified versions of QCD – see [@DEK]). Now, force the system to accommodate a single or a few baryons. This can be done by working in the canonical ensemble, or by increasing the chemical potential in the grand canonical ensemble. Because the volume is small, the baryon density will be huge, $n_B\gg (\Lambda_{QCD})^d$ (here $d$ is the number of spatial dimensions), and what we prove in the previous section tells us that at large-$N$ this small volume system is equivalent to a ‘standard’ large-volume system which is extremely dense. For a lattice theory the density will be one in units of the cutoff and the following consequence is immediate : trying to study baryons with single-site reduced models of the Eguchi-Kawai type drives the theory towards the saturation regime of the lattice, where the density is of $O(1/a)$. This regime is dominated by lattice artifacts and is unphysical. This may well be (partly) the reason reason why Ref. [@GHN], which works with a single site model at nonzero chemical potential $\mu$, sees either a ground state that is empty of baryons (for small $\mu$) or a ground state that is full of baryons, with density of $O(1/a)$, and that disappears in the continuum limit. As we discuss above, such behavior is expected from a single-site construction, and cannot be used to study the physical regime where the baryon density is not at the cutoff scale. Conclusions, some remarks, and an outlook {#summary} ========================================= In this paper we study the way large-$N$ QCD depends on its volume. General arguments, such as those found in Ref. [@AEK] and in its references, tell us what are the requirements that the ground state of a large-$N$ gauge theory needs in order to be volume independent. Nonetheless, we find it is useful to see how this phenomenon emerges explicitly in a theory which is exactly soluble. This is the reason we chose to study the ‘t Hooft model in this paper. The formalism we use is the lattice Hamiltonian formalism in axial gauge. Since we are working with a finite box size, the gauge fields cannot be gauged away completely and, in the gluon sector, one is left with a set of $N-1$ curvilinear quantum zero modes. These describe the $N-1$ eigenvalues of the spatial Polyakov loop. Together with their conjugate momenta these zero modes determine the leading $O(N^2)$ dynamics. Thus a systematic large-$N$ treatment of the full Hamiltonian proceeds as follows: 1. Treat the pure gauge Hamiltonian – find the so called ‘large-$N$ master field’. 2. Solve for the dynamics of the fermions. They now interact on the background of this master field. Thus, the back-reaction of the fermions on the gauge fields, which is subleading in $O(1/N)$ compared to the gauge fields, is neglected and this is consistent as long as these $1/N$ effects are not enhanced by any massless modes, which we assume to be the case. To perform steps (1-2) above we choose to use the lattice UV regularization and so generalized the axial gauge fixing of Ref. [@LNT] in the Hamiltonian to the lattice. We solve the fermion sector with the coherent state approach of Ref. [@YaffeCoherent]. The reason we choose this approach is that it is manifestly gauge invariant and easy to justify at large-$N$. We then end up with a regularized form of the ‘t Hooft Hamiltonian that explicitly depends on traces of the Polyakov loops, and describes their interaction with quark-antiquark pairs in the background of baryons. Our next step was to analyze the volume dependence of the emerging ‘t Hooft Hamiltonian. We showed that if translation symmetry is intact then 1. When the $Z_N$ symmetry, whose order parameters are the spatial Polyakov loops, is intact, then the spatial coordinate in the ‘t Hooft Hamiltonian decompactifies and volume independence emerges. 2. In contrast, when the $Z_N$ symmetry breaks to $Z_K$, then the ‘t Hooft Hamiltonian has a strong volume dependence. We emphasize again that this analysis ignores a set of additional zero modes that appear in this case. Since this is not the main topic of this paper we do not study this issue further, but the reader should be aware of this point. (The focus of this paper was the case with unbroken center symmetry, which is free from this subtlety) In our case, the gauge dynamics tell us that the Polyakov loops vanish, the $Z_N$ is intact, and so volume reduction takes place. A important component in the validity of volume reduction is the fact that the ground state is translation invariant. In our calculation we see how this condition arises explicitly. Moreover in the case that translation invariance breaks down by a crystal of baryons, we show that a softer form of volume independence takes place and that, at large-$N$, instead of studying a crystal of $B$ baryons in a volume $L$, one can study a single baryon in a volume $L/B$. In both cases the baryon number density is the same and together with the gauge coupling and the quark mass, these are the only relevant parameters determining the properties of the large-$N$ ground state – the volume is irrelevant. Another aspect of large-$N$ gauge theories which is explicitly exposed in this work is the dominance of the gauge fields dynamics over the fermions dynamics at large-$N$, and that it also happens at nonzero baryon number. This means that the physics of the ground state is planar and that quark loops are suppressed. This is in contrast to the conjecture raised in Ref. [@Cohen], where the author suggests that quark loops are important at nonzero baryon chemical potential, even at $N=\infty$. This conjecture was originally proposed to resolve a subtle apparent confusing contradiction between standard diagrammatic large-$N$ arguments and the phenomenology of QCD. This confusion is absent from our approach to the two-dimensional case, and we see that the conjecture of Ref. [@Cohen] does not hold there. Briefly, the way this contradiction gets resolved is by non-perturbative effects, and so using perturbation theory (even if it is planar) is quite misleading. This means that the ‘contradiction’ of large-$N$ and phenomenology is only apparent also in four dimensions. Further discussion on this point will be given in Ref. [@nonzeroMUpaper]. To conclude, we show how volume independence emerges in a clear and simple way in the Hamiltonian approach to the ‘t Hooft model in the lattice regularization. In the presence of a baryon crystal a partial form of volume independence allows one to substitute the study of the crystal of wave-length $\Delta$, baryon number $B$, and box of size $L$, with a system of a single baryon in a box of size $L=\Delta$. The latter may be useful in our companion study [@nonzeroBpaper] where we aim to solve the ‘t Hooft Hamiltonian of given a baryon number $B\ge 1$. Surprisingly, this has not been done yet for arbitrary quark mass (for the vicinity of the chiral limit where the two-dimensional baryons are nearly massless, see [@SchonThies]). In fact, even in the $B=1$ case, which was studied in Ref. [@Salcedo], the spatial Polyakov loops were set to unity. As discussed above this is inconsistent with the gauge sector dynamics and will give erroneous results at small enough volume (and by the equivalence mention above, at large enough densities, if one increases the density by fixing the baryon number and decreasing the volume). In [@nonzeroBpaper] we also plan to see what will be the effect of correcting this issue. Finally we hope that knowing how two-dimensional QCD behaves at nonzero baryon number will be of value for studies of the physical four-dimensional system. In particular it seems that the partial independence of the QCD ground state on the volume, that we see emerging in the $1+1$ case, is of a general nature and from the orbifold projection point of view can be anticipated on general grounds. This makes the successful modification to the Eguchi-Kawai reduction the four-dimensional case (of the type of [@DEK]) appealing, since it will allow one to study a single baryon in a modestly-sized box (but not of zero size) and conclude on how large-$N$ four-dimensional QCD behaves at moderate/high densities. Acknowledgments {#acknowledgments .unnumbered} =============== I thank R. Narayanan for correspondence during several stages of this work and for discussing his related work [@GHN]. I am grateful to L. Yaffe for numerous enlightening discussions on the coherent state approach and on large-$N$ volume independence. I also thank S. R. Sharpe for comments on this draft and for discussions on staggered fermions, to Carlos Hoyos-Badajoz for discussions on matrix models and related issues, and to V. P. Nair for an interesting discussion on IR divergences. This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-96ER40956. Hermiticity of ${\cal E}_a$ and its commutation relations {#gauge_fixing} ========================================================= In this Appendix we show that the zero mode $E^{''I}_{p=0}$ obeys $$\begin{aligned} \left(E^{''I}_{p=0}\right)^{\dag}&=&E^{''I}_{p=0},\label{zeroherm}\\ \left[E^{''I}_{p=0},E^{''J}_{p=0}\right] &=&0\label{zerocomm}.\end{aligned}$$ To show we use and write the difference between $E^{''I}_{p=0}$ and its hermitian conjugate $$\begin{aligned} \left(E^{''I}_{p=0}\right)^{\dag}-E^{''I}_{p=0} &=& \frac1{\sqrt{L_s}} \sum_x\, \left[ E^{'i}_x , \left(V^{\rm Adj.}_x\right)_{iI} \right] = \frac1{\sqrt{L_s}} \sum_x \left[ E^{'i}_x,\left(U^\dag_{x-1}\cdots U^\dag_1 S^\dag e^{i\varphi x}\right)_{iI}\right]\nonumber\\ &=&\frac1{\sqrt{L_s}} \sum_x \left(U^\dag_{x-1}\cdots U^\dag_1\right)_{ij}\left[E^{'i}_x,\left(S^\dag e^{i\varphi x}\right)_{jI} \right]\nonumber \\ &=&\frac1{\sqrt{L_s}} \sum_x \left(U^\dag_{x-1}\cdots U^\dag_1\right)_{ij}\left(V^{\rm Adj.}_x\right)_{ik}\left[E^{''k}_x,\left(S^\dag e^{i\varphi x}\right)_{jI} \right]\nonumber \\ &=&\frac1{\sqrt{L_s}} \sum_x \left(U^\dag_{x-1}\cdots U^\dag_1\right)_{ij}\left(V^{\rm Adj.}_x\right)_{iK}S^\dag_{jl}\left[E^{''K}_{p=0},\left(e^{i\varphi x}\right)_{lI} \right].\end{aligned}$$ To get the last line we used the fact that, within the physical Hilbert space, and except for $E^{''k}_{p=0}$ with $k\in [1,N-1]$, all other components of the operator $E^{''}$ commute with the gauge fields (see Eqs. (\[E”p\]) and (\[Glaw”1\])), and that $E^{''I}$ commute with the $S$ operators. Finally, it is easy to show that $$\left(e^{i\varphi x}\right)_{lI}=\delta_{lI},$$ which proves that $E^{''I}_{p=0}$ is hermitian within the physical sector of Hilbert space. Since $\lambda^I_{aa}$ is real this also means that ${\cal E}_a$ is hermitian. Let us now show that is obeyed within this subspace. We begin by writing the l.h.s in terms of $E^{'}$. The result is : $$\begin{aligned} \frac1{L_s}\sum_{xy}\left[E^{''I}_x,E^{''J}_y\right] &=& \frac1{L_s}\sum_{xy}\left(V^{\rm Adj.}_x\right)_{iI} \left(V^{\rm Adj.}_y\right)_{jJ} \left[E^{'i}_x,E^{'j}_y \right] \nonumber \\ &+&\frac1{L_s}\sum_{xy}\left\{\left(V^{\rm Adj.}_x\right)_{iI} \left[E^{'i}_x,\left(V^{\rm Adj.}_y\right)_{jJ} \right] E^{'j}_y - \left( I\leftrightarrow J \right) \right\}\nonumber \\ \label{rhs}\end{aligned}$$ Using the commutation relations of $E'$ (which are the same as those of $E$ – see ) it is easy to show that the first term on the r.h.s. is given by $$1^{\rm st}{\rm - term} = \frac1{L_s} \sum_x \sum_k \left(V^{\dag}_x T^k V_x\right)^{\rm Adj.}_{JI} E^{'k}_x.$$ Here the c-number matrix $T_k$ is the $k^{\rm th}$ generator of $SU(N)$ in the adjoint representation. Also, because $I,J \in [1,N-1]$, we can write this term as $$\begin{aligned} 1^{\rm st}{\rm - term} &=& \frac1{L_s} \sum_x \sum_k \left(\Omega^k_x \right)_{JI} E^{'k}_x,\\ \left(\Omega^k_x\right)_{IJ}&\equiv& \left(SU_{1}\cdots U_{x-1}\, T^k \,U^\dag_{x-1}\cdots U^\dag_1 S^\dag\right)_{JI}.\label{Omega}\end{aligned}$$ Next we proceed to the second term and first evaluate the commutation relation of $E^{'i}_x$ with $\left(V^{\rm Adj.}_x\right)_{jJ}$ : because $J\in [1,N-1]$ then $\left(V_y\right)_{jJ}=\left(U^\dag_{y-1} \cdots U^\dag_1 S^\dag \right)_{jJ}$, and we have $$\begin{aligned} \left[E^{'i}_x,\left(V^{\rm Adj.}_y\right)_{jJ}\right]&=&\left(U^\dag_{y-1} \cdots U^\dag_1 \right)_{jl} \left[E^{'i}_x,S^\dag_{lJ}\right] + {\cal C}_F(y-1-x) \left[E^{'i}_x,\left(U_1\cdots U_{y-1}\right)_{lj}\right]S^\dag_{jJ}\nonumber\\ &=&\left(U^\dag_{y-1} \cdots U^\dag_1 \right)_{jl} \left(V^\dag_x\right)_{qi} \left[E^{''q}_x,S^\dag_{lJ}\right] \nonumber\\ &+&{\cal C}_F(y-1-x) \left( U_1 \cdots U_{x-1}\right)_{lf}\left[E^{'i}_x,\left(U_{x}\right)_{fg}\right]\left(U_{x+1}\cdots U_{y-1}\right)_{gj}S^\dag_{jJ}.\label{commE'V}\end{aligned}$$ The first term vanishes since all $E^{''}$ fields commute with the $S$ fields. We now need the commutation relation between $E^{'}$ and $\left(U_x\right)^{\rm Adj.}$. We calculated them explicitly by using , and after some algebra we find an expected result : $$\left[E^{'i}_x,\left(U^{\rm Adj}_x\right)_{fg}\right] = \left(T^i U^{\rm Adj.}_x\right)_{fg}.$$ Using this we obtain the following expression for the $2^{\rm nd}$ term of the r.h.s. of $$2^{\rm nd}{\rm -term} = \frac1{L_s}\sum_{x<y}\left(V_x\right)_{iI} \left(SU_1 \cdots U_{x-1} T^i U_x U_{x+1} \cdots U_{y-1}\right)_{Jj}E^{'j}_y - (I\leftrightarrow J).\label{2nd1}$$ Since $T^{i}_{lk}=-T^{k}_{li}$ we can rearrange to get $$2^{\rm nd}{\rm -term} = -\frac1{L_s}\sum_{x<y} \left[\left(\Omega^k_x \right)_{JI}-\left(\Omega^k_x \right)_{IJ}\right] \left(U_{x}\cdots U_{y-1}\right)_{kj}E^{'j}_y = -\frac2{L_s}\sum_{x<y} \left(\Omega^k_x\right)_{JI}\left(U_x \cdots U_{y-1}\right)_{kj}\, E^{'j}_{y}.\label{2nd2}$$ Here we got the last equality by using the fact that $\Omega_{IJ} = -\left(\Omega\right)_{JI}$ (which is a result of $T^k_{ij}=-T^{j}_{ji}$). The outcome of the above paragraphs is that the commutation relations between $E^{''I}_{p=0}$ and $E^{''J}_{p=0}$ are proportional to a linear combination of $\left(\Omega^k_x\right)_{IJ}$. Let us now show that $(\Omega^k_x)_{ij}=0$ if $i,j\in [1,N-1]$. For that we basically write the definition of $\Omega$ : $$\left(\Omega^k_x\right)_{ij}=\tr \left[\lambda^i \, S U_1 \cdots U_{x-1} \lambda^g U^\dag \cdots U^\dag_1 S^{\dag} \right] \, T^{k}_{gf} \,\tr \left[\lambda^f \, U^\dag_{x-1}\cdots U^\dag_1 S^\dag \, \lambda^j S U_1 \cdots U_{x-1} \right],$$ where here the trace is in the fundamental representation, and we use the fact that $T^k_{gf}=\frac12 \tr \lambda^f\left[\lambda^g,\lambda^k\right]$ and the completeness relation of the $\lambda$ matrices. After some algebra we get $$\left(\Omega^k_x\right)_{ij}=\frac12 \tr \left(\left[\lambda^i , \lambda^j\right] S U_1 \cdots U_{x-1} \lambda^k U^\dag_{x-1} \cdots U^\dag_1 S^\dag\right),$$ which shows that for $i,j$ belonging to the [*abelian*]{} Cartan Sub-algebra, then $\left(\Omega^k\right)_{ij}=0$, and that consequently $$\left[E^{''I}_{p=0},E^{''J}_{p=0}\right]=0.$$ Clearly this also means that $[{\cal E}_a,{\cal E}_b]=0$. Calculation of ${\cal H}_F$. {#appHF} ============================ In this section we present the calculation of ${\cal H}_F({\cal C}_F,{\cal P})$. We begin by evaluating the expectation value of the numerator $\rho_{ab}(x)\rho_{ba}(y)$ of in the fermionic coherent state $|{\cal C}_F\>={\cal U}({\cal C}_F)|0\>$, where the reference state $|0\>$ is defined in Section \[SectorF\]. For that we write $$\<{\cal C}_F|\rho_{ab}(x)\rho_{ba}(y)|{\cal C}_F\> = \<0|\,{\cal U}^\dag({\cal C}_F)\, \rho_{ab}(x)\, {\cal U}({\cal C}_F)\, {\cal U}^\dag({\cal C}_F)\, \rho_{ba}(y) \,{\cal U}({\cal C}_F)\,|0\>.$$ To calculate ${\cal U}^\dag({\cal C}_F)\, \rho_{ab}(x)\, {\cal U}({\cal C}_F)$ we write $$\begin{aligned} {\cal U}^\dag({\cal C}_F)\, \rho_{ab}(x)\, {\cal U}({\cal C}_F) &=& \frac12 \, \left[ {\cal U}^\dag({\cal C}_F)\, \psi^{\dag b}_x \, {\cal U}({\cal C}_F)\, {\cal U}^\dag({\cal C}_F)\, \psi^{a}_x \, {\cal U}({\cal C}_F) \right. \nonumber \\ && \left. -\frac{\delta_{ab}}{N}\sum_c \, {\cal U}^\dag({\cal C}_F)\, \psi^{\dag c}_x \, {\cal U}({\cal C}_F)\, {\cal U}^\dag({\cal C}_F)\, \psi^{c}_x \, {\cal U}({\cal C}_F)\right],\end{aligned}$$ and use the Hadamard lemma to show that $${\cal U}^\dag ({\cal C}_F) \, \psi^a_x \, \,{\cal U}({\cal C}_F) = \sum_{y\in Z} \, e^{i\varphi_a (y-x)/L_s}\, \left(e^{-i{\cal C}_F}\right)^{xy} \, \psi^a_y.\label{UpsiU}$$ This then gives $$\begin{aligned} \<{\cal C}_F|\rho_{ab}(x)\rho_{ba}(y)|{\cal C}_F\> &=& \frac14 \sum_{vw\in Z\atop v'w'\in Z} \left(e^{-i{\cal C}_F}\right)^{xv}\left(e^{i{\cal C}_F}\right)^{wx}\left(e^{-i{\cal C}_F}\right)^{yv'}\left(e^{i{\cal C}_F}\right)^{w'y}\nonumber\\ &&\times \<0|\left[ e^{i\varphi_a(v-x)/L_s + i\varphi_b(x-w)/L_s}\, \psi^{\dag b}_w\psi^a_v - \frac{\delta_{ab}}N\sum_c\,e^{i\varphi_c(v-w)/L_s}\psi^{c\dag}_w\psi^c_v\right]\nonumber\\ &&\times \left[ e^{i\varphi_b(v-y)/L_s + i\varphi_a(y-w')/L_s}\, \psi^{\dag b}_{w'}\psi^a_{v'} - \frac{\delta_{ab}}N\sum_c\,e^{i\varphi_c(v'-w')/L_s}\psi^{c\dag}_{w'}\psi^c_{v'}\right]|0\>.\nonumber \\ \label{rhorho}\end{aligned}$$ We proceed we need to evaluate fermionic contractions of three types: - Terms of type I : for the $a\neq b$ terms of we need to evaluate $\<0|\psi^{\dag b}_w\, \psi^a_v\,\psi^{\dag a}_{w'}\psi^b_{v'}|0\>$. Since $\psi^a_{\tilde z}|0\>=\psi^{\dag a}_z|0\>=0$ for $z\in S$ and $\tilde z\in \bar S$ we get $$\<0|\psi^{\dag b}_w\, \psi^a_v\,\psi^{\dag a}_{w'}\psi^b_{v'}|0\> = \sum_{z\in S\atop \tilde z\in \bar S}\delta_{\bar w z} \delta_{\bar v \tilde z} \delta_{\bar{w}'\tilde z}\delta_{\bar{v}'z}.\label{contract1}$$ Here we use the notation where $\bar{x}=x \,{\rm mod}\,L_s$ (recall that $S$($\bar S$) are the set of sites that are full(empty) of quarks). - Terms of type II : for the $a= b$ terms of we need to evaluate $\<0|\psi^{\dag a}_w\, \psi^a_v\,\psi^{\dag a}_{w'}\psi^a_{v'}|0\>$. Here we have more contractions and we get $$\<0|\psi^{\dag a}_w\, \psi^a_v\,\psi^{\dag a}_{w'}\psi^a_{v'}|0\> = \sum_{z\in S\atop \tilde z\in \bar S}\delta_{\bar w z} \delta_{\bar v \tilde z} \delta_{\bar{w}'\tilde z}\delta_{\bar{v}'z} + \sum_{z_1\in S\atop z_2\in S}\delta_{\bar w z_2} \delta_{\bar v z_2} \delta_{\bar{w'}z_1}\delta_{\bar{v}'z_1}.\label{contract2}$$ - Terms of type III : for the $a= b$ terms of we also need to evaluate $\<0|\psi^{\dag a}_w\, \psi^a_v\,\psi^{\dag c}_{w'}\psi^c_{v'}|0\>$, with $a\neq c$. Here we have only one contractions that gives $$\<0|\psi^{\dag a}_w\, \psi^a_v\,\psi^{\dag c}_{w'}\psi^c_{v'}|0\> = \sum_{z_1\in S\atop z_2\in S}\delta_{\bar w z_2} \delta_{\bar v z_2} \delta_{\bar{w'}z_1}\delta_{\bar{v}'z_1}.\label{contract3}$$ To express in a compact form we define the following ‘matrix densities’ $$\begin{aligned} \rho^q_{xy} &=& \sum_{p\in Z\atop z\in S} \left(e^{i{\cal C}_F}\right)^{z+pL_s,x} \, \left(e^{-i{\cal C}_F}\right)^{y,z+(p+q)L_s},\\ \bar \rho^{q}_{xy} &=& \sum_{p\in Z\atop \tilde z\in \bar S} \left(e^{i{\cal C}_F}\right)^{\tilde z+pL_s,x} \, \left(e^{-i{\cal C}_F}\right)^{y,\tilde z+(p+q)L_s}.\end{aligned}$$ The hermiticity and periodicity of the matrix ${\cal C}_F$ imply that these matrix densities obey $$\begin{aligned} \bar \rho^q_{xy} &=& \delta_{xy}\delta_{q,0} - \rho^q_{xy},\label{r11}\\ \sum_{q\in Z\atop y\in L_s} \rho^q_{xy} \rho^{-q}_{yz} &=& \rho^{q=0}_{xz},\label{r21}\\ \sum_{x\in L_s} \rho^{q}_{xx} &=& (B+L_s/2)\delta_{q,0}.\label{r31}\end{aligned}$$ Substituting Eqs. (\[contract1\])–(\[contract3\]) into we find that $\<{\cal C}_F|H_C|{\cal C}_F\>$ can be brought to the form $$\begin{aligned} \<{\cal C}_F|H_C|{\cal C}_F\> &=&\frac{g^2}{4L_s} \sum_{l=1}^{L_s} \sum_{a\neq b} \sum_{qq'} \frac{\rho^{q'}_{xy}\, \bar \rho^{-q}_{yx} \, e^{-i2\pi l (x-y)/L }}{4\sin^2((\varphi_a-\varphi_b)/L_s + 2\pi l/L_s)/2} \times e^{-i\varphi_a(x-y +q L_s)/L_s + i\varphi_b( x-y + q'L_s)/L_s} \nonumber \\ &+&\frac{g^2}{4L_s}\sum_{l=1}^{L_s-1} \sum_{qq'} \frac{(1-\frac1{N} ) e^{i2\pi l/L (y-x)}}{4\sin^2(\pi l/L_s)} \left[ \rho^q_{xy}\bar \rho^{-q'}_{yx} + \rho^q_{xx} \rho^{-q'}_{yy}\right]\times \sum_a e^{i\varphi (q-q')},\nonumber \\ &-&\frac{g^2}{4L_sN}\sum_{l=1}^{L_s-1}\sum_{qq'}\frac{\rho^q_{xx}\rho^{-q'}_{yy}e^{i2\pi l/L_s(y-x)}}{4\sin^2(\pi l/L_s)} \times \sum_{a\neq c} e^{i\varphi_a q -i\varphi_c q'}. \label{HC1}\end{aligned}$$ To proceed we first show that the terms in the last two lines of are subleading. To see this note that we still need to perform the integral of . This will be done conveniently if we write these terms as $$\begin{aligned} &&\frac{g^2N}{4L_s}\left\{\sum_{l=1}^{L_s-1} \sum_{qq'} \frac{(1-\frac1{N} ) e^{i2\pi l/L (y-x)}}{4\sin^2(\pi l/L_s)} \left[ \rho^q_{xy}\bar \rho^{-q'}_{yx} + \rho^q_{xx} \rho^{-q'}_{yy}\right]\times P^{q-q'}\right. \nonumber \\ &-&\left.\sum_{l=1}^{L_s-1}\sum_{qq'}\frac{\rho^q_{xx}\rho^{-q'}_{yy}e^{i2\pi l/L_s(y-x)}}{4\sin^2(\pi l/L_s)} \times \left(P_q \, P_{-q'} - \frac1{N} P_{q-q'}\right)\right\}. \label{2terms}\end{aligned}$$ Here we have defined the $k$-wound Polyakov loop operator as $$P_k\equiv \frac1{N} \sum_a e^{i\varphi_a k}.$$ The averages over the Haar measure of $P_k$ and $P_k \times P_{k'}$ are known explicitly for the $U(N)$ group [@diaconis; @rains] $$\begin{aligned} \<P_k\> &=& \delta_{k,0},\label{Pk}\\ \<P_k\, P_{-k'}\> &=& \delta_{kk'}\, \left(\delta_{k,0} + \frac{\min(|k|,N)}{N^2}\right).\label{PkPk'}\end{aligned}$$ Using these results we can see that both these terms are at most of $O(1)$. For an $SU(N)$ group these averages are expected to differ by a small amount since the center of mass of the eigenvalues $\sum_c\varphi_c$ is held fixed at zero. For example, the average of $P_k$ can be done explicitly : one can expand the Vandermond as a polynomial of $e^{i\varphi_a}$ and see that $\<P_k\>_{SU(N)}=0$ if $|k| \ge 2N$. Together with the $Z_N$ symmetry of the Vandermond measure and the methods in chapter 8 of [@creutzbook] we get $$\<P_k\> = \delta_{k,0} + \frac1{N} \delta_{|k|,N},\label{Pksun}$$ In contrast to the two subleading terms we discussed above, the first term in includes a double sum over color indices and is thus of $O(g^2N^2)\sim N$. The same is true for the scaling of kinetic contribution of the fermions to the classical Hamiltonian : $\<{\cal C}_F|H_K|{\cal C}_F\>$. Using it is easy to show that it is given by $$\<{\cal C}_F|H_K|{\cal C}_F\> =N \sum_x \left[\left(-\frac{i}2 \rho^q_{x,x+1} + c.c. \right)+ m (-1)^x \rho^q_{xx} \right] \times P_q.$$ which simplifies even further if we use . Resolving the constraints on $\rho^q_{xy}$ {#app_rho_resolve} ========================================== In this appendix we wish to show how the constraints in Eqs. (\[r11\]–\[r31\]) are resolved. Our starting point is to write the Fourier transform $$\rho^q_{xy} = \int_{-\pi}^\pi \, \frac{dp}{2\pi}\, \rho_{xy}(p)\, e^{ip\, q}.$$ This is the most general way to express the dependence of $\rho^q_{xy}$ on the integer $q$ since the range of the latter is the whole integers. In terms of $\rho_{xy}(p)$ the constraints of Eqs. (\[r11\]–\[r31\]) become the following infinite set of equations that hold for any value of $p\in (0,2\pi]$. $$\begin{aligned} \rho_{xy}(p) &=& \rho^\star_{yx}(p),\label{r11p}\\ \sum_{x=1}^{L_s}\, \rho_{xx}(p) &=& \left(B + L_s/2\right),\label{r21p}\\ \sum_y \rho_{xy}(p)\, \rho_{yz}(p)&=&\rho_{xz}(p).\label{r31p}\end{aligned}$$ The equations above can be solved by setting up, for each $p$, an orthogonal basis on the $L_s\times L_s$ space that is furnished by the indices $x$ and $y$. We denote the orthogonal wave functions on that basis by $\phi^n_x(p)$, with $n=1,2,\dots,L_s$. In terms of these wave functions the hermitian matrices $\left(\rho(p)\right)_{xy}$ can be written as $$\rho_{xy}(p) = \sum_{n=1}^{L_s}\, \phi^n_x(p)\, r_n(p)\, \phi^{n\star}_x(p),$$ where $r_n(p)$ are the eigenvalues of the matrix $\rho(p)$ in the basis spanned by $\phi(p)$. Using we see that the eigenvalues $r_n(p)$ are real, while using , we see that they obeys $$r_n(p)\left( r_n(p)-1\right)=0.$$ Thus $r_n(p)$ is either $0$ or $1$. Finally, tells us that $$\sum_{n=1}^{L_s} \, r_n(p) = B + L_s/2, \quad ; \quad \forall p,$$ and by ordering the eigenfunctions $\phi^n(p)$ according to their values of $r_n(p)$ we see that $$r_n(p) = \left[ \begin{array}{lr} 1 & \qquad n \le B+L_s/2,\\ 0 & \qquad n > B+L_s/2. \end{array} \right.$$ The end product of the discussion above is that the most general way to resolve Eqs. (\[r11\]–\[r31\]) is to write $$\rho^q_{xy} = \int_{0}^{2\pi}\, \frac{dp}{2\pi}\, \sum_{n=1}^{B+L_s/2}\, \phi^n_x(p) \, \phi^{n\star}_y(p)\, e^{ipq},\label{rho_resolve_1}$$ with the functions $\phi^n_x(p)$ forming an orthogonal basis for each value of $p$ (note that this basis is not necessarily the same for each $p$ and the precise form of $\phi^n_x(p)$ is determined by the dynamics of the ${\cal H}$ minimization). 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[^4]: Some useful properties of the operator $\left(U^{\rm Adj.}\right)_{ij}$ are that its representation in Hilbert space is Hermitian, $\left(U^{\rm Adj.}\right)_{ij}=\left(\left(U^{\rm Adj.}\right)_{ij}\right)^\dag$, and that $\sum_{k} \left(U^{\rm Adj.}\right)_{ik} \left(U^{\rm Adj.}\right)_{jk}=\delta_{ij}$. The latter relation means that for the operator identity $\left(U^{\rm Adj.}\, U^{\dag \rm Adj.} \right)_{ij}=\delta_{ij}$ to hold we need to define $\left(U^{\dag \rm Adj.}\right)_{ij}\equiv \left(U^{\rm Adj.}\right)_{ji}$. [^5]: Here all fields are still dimensionless. The continuum expectation values $\frac{\<\bar \psi \left({\bm 1},\gamma_0\right)\psi\>^{\rm continuum}}{\sqrt{g^2N}}$ are given by dividing the r.h.s. of Eqs. (\[psibarpsi\])–(\[psidagpsi\]) by $a\sqrt{g^2N}$ ($g$ has dimensions of mass in $1+1$). [^6]: The integral over $\varphi$ can be either restricted to obey $\varphi_a \ge \varphi_{b}$ for $a>b$, or can be simply unrestricted. The difference between these two choices will be reflected by whether the operators $S(P)$ include permutations of the eigenvalues or not. [^7]: Here we implicitly assume that we take the large-$N$ limit when all other parameters are fixed. These includes the volume $L$, the temperature $T$, the mass $m_q$, the ‘t Hooft coupling $g^2N$, and the baryon number $B$. [^8]: To ensure that ${\cal U}(\cal C)$ is unitary, the coordinates ${\cal C}$ need to obey certain conditions and in the next section we make these explicit. [^9]: We thank V. P. Nair for pointing this to us and to L. G. Yaffe for a discussion related to this. [^10]: According to Eqs. (\[Pk\])–(\[PkPk’\]) the terms with $q=q'\neq Q$ are also nonzero, but using the representation in Eqs. (\[constraints\_res1\])–(\[constraints\_res2\]) we verified that they are subleading in $N$ if we take $M\to\infty$ with or after we take $N\to \infty$. [^11]: In our numerical studies we find that, while irrelevant for the minimization of ${\cal H}_F$, the last term in is crucial to include in order to get the right baryon mass. [^12]: Another paper that discusses nonzero baryon number in the ‘t Hooft model is [@SchonThies], but there the authors restrict to translation invariant dense systems, which as they show, is inconsistent. For translation non-invariant states, the authors discuss only the vicinity of the chiral limit. As we already mentioned, the fact that the baryons are massless in this limit makes them very different from the four-dimensional QCD case, and it is important to study the $m\neq 0$ case. [^13]: We note that the following confusion may arise : a tool one can use to make measurements in small volume theories is the Gross-Kitazawa ‘momentum feeding’ trick presented in [@GK]. For example, this was used successfully in [@KNN]. This trick allows one to extract, from a zero volume theory, the meson propagator $G(x)$ for any value of the separation $x$, and to measure the meson mass from the exponential decay in $|x|$. This reflects how large-$N$ projections repackage (but not lose) the large-volume degrees of freedom into the color indices. A natural question now appears : can one also ‘repackage’, in a similar way, a baryon crystal that breaks translations into the color degrees of freedom of a zero volume theory ? As we shall see below the answer to this is no. This confusion arises because $G(x)$, that depends on $x$, [*can*]{} be calculated from zero-volume. This, however, is a direct result of translation invariance. When the latter is broken, the meson propagator depends on two space-time coordinates, and the Gross-Kitazawa trick cannot be used. [^14]: We note in passing that since all physical information on the system is encoded in the classical Hamiltonian, then the theory’s excitation spectrum will also be independent of the volume. For example, the meson spectrum, which is encoded in the $1/N$ fluctuations around the minimum of ${\cal H}$, will also be independent of the volume. For further details on how to extract the spectrum of mesons and glueballs from ${\cal H}$, we refer to Ref. [@YaffeCoherent]. [^15]: Ref. [@SchonThies] get $x^c\simeq 0.0149$. [^16]: We do not consider the chiral limit here, where two dimensional baryons become massless.

--- abstract: 'Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulations to increase their predictive accuracy. While several data-driven turbulence models have been reported, the quantification of the uncertainties introduced has mostly been neglected. Uncertainty quantification for such data-driven models is essential since their predictive capability rapidly declines as they are tested for flow physics that deviate from that in the training data. In this work, we propose a novel data-driven framework that not only improves RANS predictions but also provides probabilistic bounds for fluid quantities such as velocity and pressure. The uncertainties capture both model form uncertainty as well as epistemic uncertainty induced by the limited training data. An invariant Bayesian deep neural network is used to predict the anisotropic tensor component of the Reynolds stress. This model is trained using Stein variational gradient decent algorithm. The computed uncertainty on the Reynolds stress is propagated to the quantities of interest by vanilla Monte Carlo simulation. Results are presented for two test cases that differ geometrically from the training flows at several different Reynolds numbers. The prediction enhancement of the data-driven model is discussed as well as the associated probabilistic bounds for flow properties of interest. Ultimately this framework allows for a quantitative measurement of model confidence and uncertainty quantification for flows in which no high-fidelity observations or prior knowledge is available.' address: 'Center for Informatics and Computational Science, University of Notre Dame, 311 I Cushing Hall, Notre Dame, IN 46556, USA' author: - Nicholas Geneva - Nicholas Zabaras bibliography: - 'mybibfile.bib' title: 'Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks' --- Turbulence ,Reynolds-Averaged Navier-–Stokes Equations (RANS) ,Model Form Uncertainty ,Uncertainty Quantification ,Bayesian ,Deep Neural Networks Introduction ============ Over the past decade, with the exponential power increase of computer hardware, computational fluid dynamics (CFD) has become an ever more predominate tool for fluid flow analysis. The Reynolds-averaged Navier-Stokes (RANS) equation provides an efficient method to compute time-averaged turbulent flow quantities making RANS solvers a frequently selected CFD method. However, it is common knowledge that RANS simulations can be highly inaccurate for a variety of flows due to the modeling of the Reynolds stress term [@pope2001turbulent]. Although over recent years Large Eddy Simulations (LES) or Direct Numerical Simulations (DNS) have become more accessible, these methods still remain out of the scope of practical engineering applications. For example, design and optimization tasks require repeated simulations with rapid turnaround time requirements for which RANS simulations are the choice modeling tool. Thus improving the accuracy of RANS simulations and providing measures of their predictive capability remains essential for the CFD community. Turbulence models seek to resolve the closure problem that is brought about from the time averaging of the Navier-Stokes equations. While CFD and computational technology has made significant strides over the past decade, turbulence models have largely become stagnate with the majority of today’s most popular models being developed over two decades ago. Many of the most widely used turbulence models employ the Boussinesq assumption as the theoretical foundation combined with a set of parameters that are described through one or more transport equations. In general, these turbulence models can be broken down into families based off the number of additional partial differential equations they introduce into the system. For example, the Spalart-Allmaras model [@spalart1992one] belongs to the family of single equation models. While the Spalart-Allmaras model has been proven to be useful for several aerodynamic related flows [@godin1997high], its very general structure severely limits the range of flows that it is applicable. In the two-equation family, models such as the k-$\epsilon$ model [@jones1972prediction; @launder1974application] and the k-$\omega$ model [@wilcox1993turbulence] provide better modeling for a much larger set of flows even though their limitations are well known. In all the aforementioned models, a set of empirically found constants are used for model-calibration thus resulting in potentially poor performance for flows that were not considered in the calibration process. This combined with empirical modeling of specific transport equations, such as the $\epsilon$ equation, result in a significant source of model form uncertainty. While many have proposed more complex approaches such as using different turbulence models for different regions of the flow [@menter1994two] or using a turbulence model with additional transport equations [@walters2008three], these methods still rely heavily on empirical tuning and calibration. Thus model form uncertainty introduced by turbulence models continues to be one of the largest sources of uncertainty in RANS simulations. This work aims to improve turbulence modeling for RANS simulations using machine learning techniques that also allow us to quantify the underlying model error. While the use of machine learning methods in CFD simulations can be traced back to over a decade ago [@milano2002neural], recently there has been a new wave of integrating innovative machine learning algorithms to quantify and improve the accuracy of CFD simulations. Earlier work in quantifying the uncertainty and calibration of turbulence models focused on treating model parameters as random variables and sampling via Monte Carlo to obtain a predictive distribution of outcomes [@cheung2011bayesian; @oliver2011bayesian]. Rather than constraining oneself to a specific model, an alternative approach was to directly perturb components of the anisotropy term of the Reynolds stress [@dow2011uncertainty]. Lately, the use of machine learning models has been shown to provide an efficient alternative to direct sampling. In general, the integration of machine learning with turbulence models can be broken down into three different approaches: modeling the anisotropic term of the Reynolds stress directly, modeling the coefficients of turbulence models and modeling new terms in the turbulence model. Tracey [*et al*.]{} [@tracey2013application] explored the use of kernel regression to model the eigenvalues of the anisotropic term of the Reynolds stress. Later, Tracey [*et al*.]{} [@tracey2015machine] used a single layer neural network to predict a source term in the Spalart-Allmaras turbulence model. Similarly, Signh [*et al*.]{} [@singh2017machine] have used neural networks to introduce a functional corrective term to the source term of the Spalart-Allmaras turbulent model for predicting various quantities over airfoils. Zhang [*et al*.]{} [@zhang2015machine] investigated the use of neural networks and Gaussian processes to model a correction term introduced to the turbulence model. Ling [*et al*.]{} [@ling2016reynolds] considered deep neural networks to predict the anisotropic tensor using a neural network structure with embedded invariance [@ling2016machine]. Ling [*et al*.]{} [@ling2017uncertainty] additionally proposed using random forests to improve RANS predictions for a flow with a jet in a cross flow. While the above works have managed to improve the accuracy of RANS simulations, uncertainty quantification has largely been ignored. Arguably, the integration of black box machine learning models increases the importance of uncertainty quantification in the context of quantifying the error of the improved turbulence model but also quantifying the uncertainty of the machine learning predictions. This is largely due to the significant prediction degradation of these proposed machine learning models for flows that vary from the training data in either fluid properties or geometry [@tracey2013application; @ling2016reynolds]. Past literature has clearly shown that data-driven methods are not exempt from the conflicting objectives of predictive accuracy versus flow versatility seen in traditional turbulence modeling. Several works have taken steps towards using machine learning to provide uncertainty quantification analysis of RANS simulations. For example, Xiao [*et al*.]{} [@xiao2016quantifying] proposed a Bayesian data-driven methodology that uses a set of high-fidelity observations to iteratively tune an ensemble of Reynolds-stress fields and other quantities of interest. While proven to work well for even sparse observational data, this work is limited to a single flow with which the machine learning model was trained explicitly on. Wu [*et al*.]{} [@wu2017priori] used the Mahalanobis distance and kernel density estimation to formulate a method to predict the confidence of a data-driven model for a given flow. While this allows the potential identification of regions of less confidence after training, it is limited to the prediction of the anisotropic stress and fails to provide any true probabilistic bounds. For machine learning methods to be a practical tool for reliably tuning RANS turbulence models, transferability to flows with different geometries and fluid properties is important. Additionally, quantifying the model uncertainty is critical for assessing both the accuracy and confidence of the machine learning model and of the resulting predicted quantities of interest. The novelty of our work is the use of a data-driven model with a Bayesian deep learning framework to provide the means of improving the accuracy of RANS simulations and allow for the quantification of the model form uncertainty arising in the turbulence model. This uncertainty is then propagated to the quantities of interest, such as pressure and velocity. The focus of our work will not be application on flows that are the same or similar to those in the training set, but rather to flows defined by different geometries and fluid properties. We aim to take a much more practical and expansive view of using these innovative machine learning models for improved turbulence modeling. The specific novel contributions of this work are fourfold: (a) the use of a Bayesian deep neural network as a model to predict a tuned Reynolds stress field, (b) introducing a stochastic data-driven RANS algorithm that allows us to calculate probabilistic bounds for any flow field quantity, (c) assessment of the data-driven model on flows that are geometrically different from the training simulations and (d) comparison of both performance and confidence of the data-driven model across several Reynolds numbers. This paper is structured as the following: In Section \[sec:Formulation\], we review the governing equations and motivation for this work. In Section \[sec:Framework\], the proposed data-driven framework is discussed in detail. We discuss the invariant machine learning model in Section \[subsec:invarnn\], its extension to the Bayesian paradigm in Section \[subsec:svgd\] and the stochastic data-driven RANS methodology to propagate uncertainty from the Bayesian data-driven model to quantities of interest in Section \[subsec:uq\]. In Section \[sec:Training\], various implementation details are reviewed including information regarding flow data used, training techniques and integration in the selected CFD solver. Section \[sec:NumericalResults\] details the results of applying this model to two test flows at three different Reynolds numbers. Results for a flow over a backwards step and over a wall mounted cube are presented in Sections \[subsec:backwardsStep\] and \[subsec:wallMountedCube\], respectively. Finally discussion and conclusions are provided in Section \[sec:Conclusions\]. Problem Formulation {#sec:Formulation} =================== Governing Equations ------------------- As previously mentioned, the difficulty of RANS is the fundamental closure problem that is introduced when the Navier-Stokes equations are averaged with respect to time. The RANS momentum equation is as follows: $$\left<u_{j}\right>\frac{\partial \left<u_{i}\right>}{\partial x_{j}} = \frac{\partial}{\partial x_{j}}\left[ -\frac{\left<p\right>}{\rho}\delta_{ij} + \nu\left(\frac{\partial \left<u_{i}\right>}{\partial x_{j}} + \frac{\partial \left<u_{j}\right>}{\partial x_{i}} \right) - \left<u'_{i}u'_{j}\right> \right] + \left<g_{i}\right>.\label{eq:rans}$$ As always, the challenge is to close this equation by approximating the Reynolds stress (R-S) term $\left< u'_{i}u'_{j}\right>$. $u'_{i}$ indicates a fluctuation velocity defined as $u_{i}(x,t)-\left< u_{i}(x,t) \right>$ in which $\left< \cdot \right>$ indicates time-averaged or mean value. The turbulent viscosity theory, originally developed by Boussinesq [@boussinesq1877mem], proposes a form of the R-S that is mathematically analogous to the stress-strain rate of a Newtonian fluid: $$\begin{gathered} \left<u'_{i}u'_{j}\right> = \frac{2}{3}\delta_{ij}k + a_{ij}, \label{eq:rs-1}\\ k = \frac{1}{2}\left<u'_{k}u'_{k}\right>, \label{eq:tke}\\ a_{ij} = -\nu_{t}\left(\frac{\partial\left<u_{i}\right>}{\partial x_{j}}+\frac{\partial \left<u_{j}\right>}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial \left<u_{k}\right>}{\partial x_{k}}\right),\end{gathered}$$ where $k$, $\nu_{t}$ are the turbulent kinetic energy (TKE) and turbulent viscosity, respectively. Assuming that the flow is incompressible results in the following: $$\left<u'_{i}u'_{j}\right>= \frac{2}{3}\delta_{ij}k - \nu_{t}\left(\frac{\partial\left<u_{i}\right>}{\partial x_{j}}+\frac{\partial \left<u_{j}\right>}{\partial x_{i}}\right).$$ This representation is used not because of its accuracy but instead due to the simplifications that result when it is substituted into the RANS equation. This form is known as the *Boussinesq eddy viscosity assumption*. RANS Turbulence Models ---------------------- The context of this work is focused on the $k-\epsilon$ turbulence model [@jones1972prediction; @launder1983numerical; @chien1982predictions] which is the most commonly used closure model for RANS simulations to date [@pope2001turbulent]. Starting with the Boussinesq eddy viscosity assumption, the $k-\epsilon$ model approximates the effective viscosity $\nu_{t}$ in terms of the turbulent kinetic energy $k$ and the turbulent dissipation rate $\epsilon$ with the R-S given as follows: $$\begin{gathered} \left<u'_{i}u'_{j}\right> = -\tau_{ij}= \frac{2}{3}\delta_{ij}k - \nu_{t}\left(\frac{\partial\left<u_{i}\right>}{\partial x_{j}}+\frac{\partial \left<u_{j}\right>}{\partial x_{i}}\right),\\ \nu_{t}=\frac{C_{\mu} k^{2}}{\epsilon},\end{gathered}$$ where $C_{\mu}$ is one of five model constants. Through manipulation of the Navier-Stokes equations, the kinetic energy can be derived precisely for the case of high Reynolds number. On the other hand, the standard transport equation for the turbulent dissipation, $\epsilon$, should be thought of as an empirical fit [@pope2001turbulent]. For this work, we will use the standard $k-\epsilon$ model for fully-turbulent, incompressible flow [@wilcox1993turbulence]: $$\begin{gathered} \frac{\partial k}{\partial t}+\left<u_{i}\right>\frac{\partial k}{\partial x_{i}}=\frac{\partial}{\partial x_{i}}\left[\left(\nu+\frac{\nu_{t}}{\sigma_{k}}\right)\frac{\partial k}{\partial x_{i}}\right]+\tau_{ij}\frac{\partial \left<u_{i}\right>}{\partial x_{j}} - \epsilon,\\ \frac{\partial \epsilon}{\partial t} + \left<u_{i}\right>\frac{\partial\epsilon}{\partial x_{i}} = \frac{\partial}{\partial x_{i}}\left[\left(\nu+\frac{\nu_{t}}{\sigma_{\epsilon}}\right)\frac{\partial \epsilon}{\partial x_{i}}\right] +C_{\epsilon 1}\frac{\epsilon}{k}\tau_{ij}\frac{\partial \left<u_{i}\right>}{\partial x_{j}} - C_{\epsilon 2}\frac{\epsilon^{2}}{k}.\end{gathered}$$ The five constants $ C_{\mu}, C_{\epsilon 1}, C_{\epsilon 2}, \sigma_{k}, \sigma_{\epsilon}$ are tunable parameters whose optimal values depend on the flow under consideration. We use the values originally proposed by Launder [*et al*.]{} [@launder1983numerical] obtained by data fitting over various turbulent flows: $$C_{\mu}=0.09,\quad C_{\epsilon 1}=1.44, \quad C_{\epsilon 2}=1.92, \quad \sigma_{k}=1.0, \quad \sigma_{\epsilon}=1.3.$$ The advantages of the $k-\epsilon$ model are its numerical robustness, computational efficiency, easy implementation and general validity for fully-turbulent flows. However, with this versatility comes some significant drawbacks including poor accuracy for complex fluid flows, and for problems with flow separation and sharp pressure gradients [@menter1994two; @menter1993zonal]. Core assumptions such as the formulation of the turbulent dissipation equations, the turbulent model constants and even the Boussinesq approximation provide large sources of uncertainty for the $k-\epsilon$ model. Converged simulations using the $k-\epsilon$ model with the parameters discussed above will be referred to as *baseline* RANS simulations. Ultimately, we seek to improve the prediction of a baseline simulation through the proposed data-driven framework. Data-Driven Framework {#sec:Framework} ===================== In this work, our goal is to introduce a data-driven model to increase the accuracy of a given RANS simulation and to provide uncertainty bounds for quantities of interest thus capturing the error of the turbulence model. The proposed framework is illustrated in Fig. \[fig:workflow\] which, in a broad sense, shares similar characteristics to earlier works on data-driven turbulence models [@ling2016reynolds; @xiao2016quantifying; @singh2017machine]. However, we introduce several novel modifications to the process. We break this framework down into two key phases: the training of a model using a set of pre-existing flow data and the prediction stage for which the model is sampled to produce fluid flow responses. The training data that is driving our model is a small library of different fluid flows that attempt to capture different fluid physics. Ideally each training flow should bring new information for the model to learn thus increasing its potential predictive capability. For each unique flow, there is a low-fidelity RANS solution and a time-averaged high-fidelity LES solution. The objective of this model is to learn the mapping from some baseline RANS flow input information to a turbulent property yielding an improved R-S field matching that of the corresponding high-fidelity simulation. This turbulent property could be tuned model coefficients, model correction terms or components of the R-S directly. For the scope of this work, we will focus on modeling the R-S tensor directly but this framework can extend to other approaches. An error or loss function that quantifies the discrepancy between the predicted R-S and the true high-fidelity field is used to update the model in an iterative process. We select a Bayesian neural network to serve as this model. Its formulation is discussed in Section \[subsec:invarnn\] with a Bayesian extension presented in Section \[subsec:svgd\]. The methods and techniques used to train the model are outlined in Section \[sec:Training\]. Once the model has been trained, it can be used as a regression model to sample predicted R-S fields for a given reference RANS solution. This process starts with a baseline RANS simulation whose flow field will serve as the input into the calibrated model. From this model, a set of turbulent properties are sampled that correspond to a predicted high-fidelity representation of the R-S field. For each predicted field, an independent forward simulation is completed in which the R-S is held constant and the remaining state variables are relaxed around the predicted field from their baseline values to updated perturbed values. We refer to this process of executing an ensemble of forward simulations as stochastic data-driven RANS (SDD-RANS). The forward simulations for different samples of the R-S can then be used to compute statistical bounds for quantities of interest as discussed in Section \[subsec:uq\]. LES has been chosen as the high-fidelity method for obtaining the training data in the context of this work. However, one could alternatively use higher accuracy methods such as DNS or even a combination of methods assuming that their turbulent statistics are consistent. The use of LES introduces potential physical inconsistencies in the high-fidelity predictions. Namely, LES can yield different results for the same flow depending on various parameters such as the mesh resolution or the subgrid-scale model used. The use of more consistent DNS data will likely make training more efficient and increase predictive accuracy. ![A schematic of the data-driven Bayesian machine learning framework. The top block illustrates the model training using a set of different flows. Once trained, the model is then queried given a baseline RANS flow and a set of Reynolds stress (R-S) field realizations are sampled. Independent RANS simulations are then performed using these predicted fields by stochastic data-driven RANS (SDD-RANS) and statistics for quantities of interest (QoI) are collected.[]{data-label="fig:workflow"}](Fig1.png){width="80.00000%"} Invariant Neural Network {#subsec:invarnn} ------------------------ As previously mentioned, in the scope of this work, we will predict the R-S field directly by the anisotropic component shown in Eq. . This approach has been used by multiple earlier works [@ling2016reynolds; @xiao2016quantifying]. These works consider an explicit representation of the R-S component, specifically in the form of a constant field. The remaining fluid flow quantities (mean velocity and pressure) are then *propagated* forward by solving a numerical system around this constrained R-S field. Such an approach does not constrain our work to model-specific assumptions. We note that explicit R-S approaches can potentially result in significant prediction error of fluid quantities when the Reynolds number approaches $5000$ and above [@thompson2016methodology; @wu2018rans]. Additionally, small errors in the predicted R-S field in an explicit representation can be amplified leading to instabilities. We accept this as an open problem and while alternative implicit approaches have been proposed [@wu2018rans], we leave the discussion of such methods to future works. However, we will show how the proposed framework can reflect such difficulties through the predicted probabilistic bounds on the flow quantities of interest. We select a Bayesian neural network to map the baseline RANS flow to a high-fidelity R-S field due to the impressive performance of neural networks for high-dimensional supervised learning tasks [@lecun2015deep]. For the underlying neural network model, we choose the neural network proposed by Ling [*et al*.]{} [@ling2016reynolds], illustrated in Fig. \[fig:invarnn\]. This neural network predicts the anisotropic tensor of the R-S using the symmetric and antisymmetric tensor components of the velocity gradient tensor. Through use of tensor invariants, the neural network is able to achieve both Galilean invariance as well as invariance to coordinate transformations. This makes such a model attractive for predictions of flows that deviate in geometry from the training data. Here, we briefly review the fundamentals of this neural network for completeness of the presentation. The theoretical foundation of this invariant neural network is the non-linear eddy viscosity model developed by Pope [@pope1975more]. In this model, the normalized anisotropic tensor of the R-S is expressed as a function, $\bm{b}(\bm{s},\bm{\omega})$, of the normalized mean rate-of-strain tensor $\bm{s}$ and rotation tensor $\bm{\omega}$: $$\begin{gathered} \left<u'_{i}u'_{j}\right>= \frac{2}{3}\delta_{ij}k + k b_{ij}(\bm{s},\bm{\omega}),\label{eq:pope-rs}\\ s_{ij} = \frac{1}{2}\frac{k}{\epsilon}\left(\frac{\partial\left<u_{i}\right>}{\partial x_{j}}+\frac{\partial \left<u_{j}\right>}{\partial x_{i}}\right), \quad \omega_{ij} = \frac{1}{2}\frac{k}{\epsilon}\left(\frac{\partial\left<u_{i}\right>}{\partial x_{j}} - \frac{\partial \left<u_{j}\right>}{\partial x_{i}}\right),\end{gathered}$$ where both $\bm{s}$ and $\bm{\omega}$ are scaled by the TKE and turbulent dissipation. For clarity we will refer to the tensor, $\bm{a}=k\cdot\bm{b}(\bm{s},\bm{\omega})$, used when solving the RANS equations as the unnormalized anisotropic tensor. Through application of the Cayley-Hamilton theorem, it can be shown that every second-order anisotropic tensor can be expressed in the following form: $$\begin{gathered} \bm{b}(\bm{s},\bm{\omega})=\sum_{\lambda=1}^{10}G^{\lambda}\left(\mathcal{I}_{1:5}\right)\bm{T}^{\lambda}, \label{eq:geneddypoly} \\ \mathcal{I}_{i} = \left\{Tr(\bm{s}^{2}),\, Tr(\bm{\omega}^{2}),\, Tr(\bm{s}^{3}),\, Tr(\bm{\omega}^{2}\bm{s}),\, Tr(\bm{\omega}^{2}\bm{s}^{2})\right\}, \label{eq:geneddyinvar} \\ \begin{aligned} \bm{T}^{1} & = \bm{s}, & \bm{T}^{2} & = \bm{s}\bm{\omega} - \bm{\omega}\bm{s}, & \bm{T}^{3} &= \bm{s}^{2} - \frac{1}{3}\bm{I}\text{Tr}\left(\bm{s}^{2}\right),\\ \bm{T}^{4} &= \bm{\omega}^{2} - \frac{1}{3}\bm{I}\text{Tr}\left(\bm{\omega}^{2}\right), & \bm{T}^{5} &= \bm{\omega}\bm{s}^{2} - \bm{s}^{2}\bm{\omega}, & \bm{T}^{6} & = \bm{\omega}^{2}\bm{s} + \bm{s}\bm{\omega}^{2} - \frac{2}{3}\bm{I}\text{Tr}(\bm{s}\bm{\omega}^{2}),\\ \bm{T}^{7} &= \bm{\omega}\bm{s}\bm{\omega}^{2} - \bm{\omega}^{2}\bm{s}\bm{\omega}, & \bm{T}^{8} &= \bm{s}\bm{\omega}\bm{s}^{2} - \bm{s}^{2}\bm{\omega}\bm{s}, & \bm{T}^{9} & = \bm{\omega}^{2}\bm{s}^{2} + \bm{s}^{2}\bm{\omega}^{2} - \frac{2}{3}\bm{I}\text{Tr}(\bm{s}^{2}\bm{\omega}^{2}), \\ \bm{T}^{10} & = \bm{\omega}\bm{s}^{2}\bm{\omega}^{2} - \bm{\omega}^{2}\bm{s}^{2}\bm{\omega}, \label{eq:geneddytensor} \end{aligned}\end{gathered}$$ where $\bm{T}^{\lambda}$ is one of $10$ independent, symmetric tensor functions and $G^{\lambda}$ are the respective coefficients in the linear model which can be each expressed as functions of the five invariants $\mathcal{I}_1,\cdots,\mathcal{I}_5$. For complete details on the invariants, tensor functions and the derivation of the representation above, we refer the reader to [@pope1975more]. The neural network model proposed by Ling [*et al*.]{} [@ling2016reynolds] models the anisotropic term by using the linear combination in Eq. . As illustrated in Fig. \[fig:invarnn\], rather than using the components of the symmetric and antisymmetric tensors ($\bm{s}$ and $\bm{\omega}$) directly, the invariants and tensor basis functions in Eqs.  and  are used instead. To enforce invariance to coordinate transformations, the neural network is used to learn the tensor basis coefficients $G^{\lambda}$ which are functions of the five invariants in Eq. . These predicted coefficients, $G^{\lambda}$, can then be used with the tensor basis functions, $\bm{T}^{\lambda}$, to produce the anisotropic tensor $\bm{b}$. Thus while the model predicts the anisotropic tensor given the symmetric and antisymmetric tensors of the velocity gradient, the basis coefficients as functions of the five invariants is what is being learned. If a model uses inputs with specific invariant properties, the model has the same invariance properties as well [@bishop2006machine]. This allows the neural network to be (a) Galilean invariant due to the use of the rate-of-strain and rotation tensors which are functions of the velocity gradient; and (b) invariant to coordinate transformations through the use of the invariant inputs $\mathcal{I}_{i}$. Additionally, since this eddy viscosity model is the most general formulation, this neural network model does not share any of the limitations of other simpler models that place restrictions on the form of the anisotropic term. However, an intrinsic assumption of this model is that the mapping between the RANS and LES physical domains can be thoroughly expressed by the invariants $\mathcal{I}_{i}$. This is clearly not guaranteed, however, the introduction of additional input features would potentially result in loss of coordinate system invariance thus degrading model generalization. This neural network formulation is trained on entirely local (point-wise) information. The key advantage of a spatially local model is that it extends very easily to training flow data provided on non-uniform meshes which are essential in practical CFD simulations. Approaches such as convolution neural networks require training data on a uniform mesh following an image-to-image like regression approach [@zhu2018bayesian]. However, similar to turbulent eddy viscosity models, this approach implies that the R-S mean convection $D\left<u'_{i}u'_{j}\right>/Dt$ is governed entirely by local quantities (*e.g.* $k,\,\epsilon,\, \partial \left<u_i\right>/\partial x_{j}$). This is a questionable assumption for flows that exhibit strong inhomogeneity [@pope2001turbulent]. A model that incorporates spatial correlations would likely be more descriptive, physically robust and potentially easier to train. ![Invariant, fully-connected (some connections are omitted for clarity), neural network architecture proposed by Ling [*et al*.]{} [@ling2016reynolds]. The circles indicate scalar values and the rectangles represent $3 \times 3$ second-order tensors.[]{data-label="fig:invarnn"}](Fig2.png){width="80.00000%"} Bayesian Neural Network ----------------------- \[subsec:svgd\] Traditionally neural networks are not designed to yield predictive statistics, however multiple recent works explore Bayesian reformulations of neural networks. Older techniques for obtaining Bayesian statistics include the placement of distributions over network weights and sampling with Monte Carlo methods to approximate statistical bounds [@mackay1992bayesian; @neal2012bayesian] as well as ensemble methods [@richard1991neural; @barber1998ensemble]. More recently, methods involving stochastic variational inference have brought a new wave a Bayesian neural network techniques [@blundell2015weight; @kingma2015variational; @gal2016dropout; @liu2016stein]. In this work, we choose to use Stein variational gradient decent (SVGD) recently proposed by Liu [*et al*.]{} [@liu2016stein; @liu2017stein] that approximates a variational distribution through a set of particles. SVGD is a non-parametric algorithm of similar form as standard gradient decent. We follow closely the work of Zhu and Zabaras [@zhu2018bayesian] in which SVGD is successfully applied to deep convolutional neural networks used for surrogate modeling. For the invariant neural network architecture discussed previously, we will use the following representation: $$\bm{b} = \bm{f}(\left\{\bm{s}, \bm{\omega}\right\}, \mathbf{w}) = \bm{f}(\bm{x}, \mathbf{w}), \label{eq:nnFunc}$$ where the input $\bm{x}=\{\bm{s},\bm{\omega}\}$ consists of the strain and rotation tensors $\bm{s}$ and $\bm{\omega}$ along with the neural networks parameters $\mathbf{w}$ which include weights and biases. For mathematical convenience, we will represent the anisotropic tensor with a one-dimensional vector $\bm{b}\in\mathbb{R}^{9}$ for the remainder of this section. In the equation above, we have defined the neural network model as a function that has absorbed the calculation of the invariants, tensor basis functions and the linear combination detailed in Eqs. -. Thus we will refer to the function $\bm{f}$ as the invariant neural network model. We wish to treat the neural network’s $K$ learnable parameters as random variables. Due to the potentially large number of weights in a fully-connected neural network, we assume that the weights have a probability density function of a fully-factorizable zero mean Gaussian and Gamma-distributed precision scalar $\alpha$: $$p(\mathbf{w}|\alpha)=\mathcal{N}(\mathbf{w}|0, \alpha^{-1}\bm{I}_{K}), \quad p(\alpha) = Gamma(\alpha|a_{0},b_{0}),$$ where the rate $a_{0}$ and shape parameters $b_{0}$ are taken as $1.0$ and $0.025$, respectively and $\bm{I}_{n}$ denotes the identity matrix in $\mathbb{R}^{n\times n}$. The resulting prior has the density of a narrow Student’s $\mathcal{T}$-distribution centered at zero. This promotes sparsity [@tipping2001sparse] and helps to prevent over-fitting. With the use of a sufficient number of weights and the highly non-linear nature of the neural network, such a prior places little restriction on the network’s final functional form [@neal2012bayesian]. Additionally, output-wise noise is added onto the predicted output to represent inherent uncertainty within the model’s formulation or uncertainty that cannot be reduced with more training data. This results in an additional noise term to the likelihood function of the neural network. We assume that the noise takes the form of a zero mean Gaussian with a learnable precision $\beta$ that is Gamma distributed: $$\begin{gathered} \bm{b} = \bm{f}(\bm{x}, \mathbf{w}) + \bm{\epsilon} \label{eq:nn-noise}, \\ p(\bm{\epsilon}) = \mathcal{N}(\bm{\epsilon} | 0,\beta^{-1}\bm{I}_{9}), \quad p(\bm{b}) = \mathcal{N}(\bm{b}|\bm{f}(\bm{x}, \mathbf{w}),\bm{I}_{9}), \\ p(\beta) = Gamma(\beta| a_{1}, b_{1}).\end{gathered}$$ Since both the LES and RANS simulations are being used for the same flows, we assume that the LES solution will be statistically stationary. We also assume that the LES data has sufficiently converged by averaging over an adequate number of time steps. The output-size noise is assumed to have a small variance and thus we assign in the prior for $\beta$ the shape and rate parameters to be $a_{1}=100$ and $b_{1}=2\cdot10^{-4}$, respectively. This weakly promotes large $\beta$ with an expected value of $5\times10^{5}$ and a variance on the order of $10^{-3}$, which is less than one percent of the scaled training data range. For the sake of brevity, we will drop the notation of the conditional dependence on the hyper-parameters $a_{0}$, $b_{0}$, $a_{1}$ and $b_{1}$ implying that the posterior distribution will be conditionally dependent on these terms. To optimize the parameters in the neural network, SVGD minimizes the Kullback-Leibler (KL) divergence between the true parameter posterior, $p(\mathbf{w},\beta|\mathcal{D})$, given the batch of $M$ i.i.d. training data $\mathcal{D}=\left\{\bm{b}_{i}\right\}_{i=1}^{M}$, with the variational distribution $q(\mathbf{w},\beta)$ that lies in some set of distributions $\mathcal{Q}$: $$q^{*}(\mathbf{w},\beta)=\min\limits_{q\in\mathcal{Q}} \left\{\text{KL}\left(q||p\right)\equiv\mathbb{E}_{q}(\log q(\mathbf{w},\beta)) - \mathbb{E}_{q}(\log \widetilde{p}(\mathbf{w},\beta| \mathcal{D})) + \mathcal{K}\right\},$$ for which $\widetilde{p}(\mathbf{w},\beta|\mathcal{D})$ is the unnormalized posterior and $\mathcal{K}$ is the log normalization constant that is not required to be computed during optimization. For the given neural network, we prescribe a Gaussian likelihood function and the priors discussed previously: $$\begin{gathered} \widetilde{p}(\mathbf{w},\beta| \mathcal{D}) = p(\mathcal{D}|\mathbf{w},\beta)p(\mathbf{w},\beta),\\ \widetilde{p}(\mathbf{w},\beta| \mathcal{D}) = \prod^{M}_{i=1}\left[\mathcal{N}(\bm{b}_{i}|\bm{f}(\bm{x}_{i}, \mathbf{w}), \beta^{-1}\bm{I}_{9})\right] \mathcal{N}(\mathbf{w}|0, \alpha^{-1}\bm{I}_{K})\Gamma(\alpha| a_{0}, b_{0})\Gamma(\beta| a_{1}, b_{1}).\label{eq:likelihood}\end{gathered}$$ Rather than attempting to recover a parametric form of the variational distribution, SVGD describes $q(\mathbf{w}, \beta)$ by a particle approximation. Namely, a set of $N$ deterministic neural networks each representing a particle $\left\{\bm{\theta}_{i}\right\}^{N}_{i=1}, \, \bm{\theta}_{i}=\left\{\mathbf{w}_{i}, \beta_{i}\right\}$, leading to an empirical measure $q_{N}(\mathbf{w}', \beta')=q_{N}(\bm{\theta}')=\frac{1}{N}\sum_{i=1}^{N}\delta(\bm{\theta}_{i}-\bm{\theta}')$. Thus the objective is now for the empirical probability measure, $\mu_{N}$, to converge in distribution towards the true measure of the posterior $\nu$, $$\begin{gathered} \mu_{N}(d\bm{\theta})=\frac{1}{N}\sum_{i=1}^{N}\delta(\bm{\theta}_{i}-\bm{\theta})d\bm{\theta}=\frac{1}{N}\sum_{i=1}^{N}\bm{\theta}_{i}, \\ \nu(d\bm{\theta})=p(\bm{\theta}|\mathcal{D})d\bm{\theta}.\end{gathered}$$ To minimize the KL divergence, we assume that $q(\mathbf{w}, \beta)$ is from a class of distributions that can be obtained through a set of smooth transforms. A small perturbation function, resembling that of standard gradient decent, is used to update the particles: $$\label{eq:steinUpdate} \bm{\theta}^{t+1}_{i} = \bm{T}(\bm{\theta}^{t}_{i}) = \bm{\theta}_{i}^{t}+\eta^{t}\bm{\phi}(\bm{\theta}^{t}_{i}),$$ where $\eta$ is the step size and $\bm{\phi}(\bm{\theta}_{i}^{t})$ is the direction of the update that lies in a function space $\mathcal{F}$ for the $t$-th iteration. It is now a matter of finding the optimal direction to permute the particles which should be chosen such that the KL divergence is maximally reduced, namely, $$\bm{\phi}^{*} = \max_{\bm{\phi}\in\mathcal{F}}\left(-\frac{d}{d\eta}\mathcal{KL}( \bm{T}\mu_{N}||\nu)|_{\eta=0}\right),$$ where $\bm{T}\mu$ denotes the updated empirical measure of the particles. Liu [*et al*.]{} [@liu2016stein] identify connections between the function $\bm{\phi}$ and Stein’s method and show that: $$\frac{\partial}{\partial \eta}KL(\bm{T}\mu_{N}||\nu)|_{\eta = 0} = \mathbb{E}_{\mu}\left(\mathcal{T}_{p}\bm{\phi}\right), \quad \mathcal{T}_{p}\bm{\phi} = \left(\nabla \log p(\bm{\theta}| \mathcal{D})\right)\cdot\bm{\phi} + \nabla\cdot\bm{\phi},$$ in which $\mathcal{T}_{p}$ is known as the Stein’s operator. Assuming that this function space $\mathcal{F}$ is a unit ball in a reproducing kernel Hilbert space $\mathcal{H}$ with positive kernel $k(\bm{\theta}, \bm{\theta}^{'})$, the optimal direction has the closed form: $$\label{eq:steinOpt} \bm{\phi}^{*}(\bm{\theta}) \propto \mathbb{E}_{\bm{\theta}^{\prime}\sim\mu}\left[\left(\nabla_{\bm{\theta}^{\prime}} \log p(\bm{\theta}^{\prime}| \mathcal{D})\right)k(\bm{\theta}, \bm{\theta}^{\prime}) + \nabla_{\bm{\theta}^{\prime}}k(\bm{\theta}, \bm{\theta}^{\prime})\right],$$ where $p(\bm{\theta}^{\prime}| \mathcal{D})$ is given by Eq. . In this work, we choose to use the standard radial basis function kernel for $k(\bm{\theta}, \bm{\theta}^{\prime})$. This formulation results in a simple update procedure in which the optimal decent direction for all particles is calculated with Eq.  and then updated by Eq. . Monte Carlo approximations can then be used to find the predictive mean: $$\begin{aligned} \begin{split} &\mathbb{E}(\bm{b}^{*}|\bm{x}^{*},\mathcal{D})=\mathbb{E}_{p(\mathbf{w},\beta|\mathcal{D})}(\mathbb{E}(\bm{b}^{*}|\bm{x}^{*},\textbf{w},\beta))\\ &\qquad=\mathbb{E}_{p(\mathbf{w}|\mathcal{D})}(\bm{f}(\bm{x}^{*},\mathbf{w}))\approx \frac{1}{N}\sum_{i=1}^{N}\bm{f}(\bm{x}^{*},\mathbf{w}_{i}), \label{eq:svgd-mean} \end{split}\end{aligned}$$ where $\bm{x}^{*}$ and $\bm{b}^{*}$ are the test input and corresponding predictive model output, respectively. The output noise is not present due to its density of a zero mean Gaussian. The approximation of the predictive variance similarly follows: $$\begin{aligned} \begin{split} \text{Cov}(\bm{b}^{*}|\bm{x}^{*},\mathcal{D})&=\mathbb{E}_{p(\mathbf{w},\beta|\mathcal{D})}(\text{Cov}(\bm{b}^{*}|\bm{x}^{*},\textbf{w},\beta)) + \text{Cov}_{p(\mathbf{w},\beta|\mathcal{D})}(\mathbb{E}(\bm{b}^{*}|\bm{x}^{*},\textbf{w},\beta))\\ &=\mathbb{E}_{p(\beta|\mathcal{D})}(\beta^{-1}\bm{I}_{9}) + \text{Cov}_{p(\mathbf{w}|\mathcal{D})}(\bm{f}(\bm{x}^{*},\mathbf{w}))\\ &\begin{aligned} &\approx\frac{1}{N}\sum_{i=1}^{N}\left((\beta_{i})^{-1}\bm{I}_{9}+\bm{f}(\bm{x}^{*},\mathbf{w}_{i})\bm{f}^{T}(\bm{x}^{*},\mathbf{w}_{i})\right)\\ &\qquad\qquad-\mathbb{E}_{p(\mathbf{w}|\mathcal{D})}(\bm{f}(\bm{x}^{*},\mathbf{w}))\mathbb{E}^{T}_{p(\mathbf{w}|\mathcal{D})}(\bm{f}(\bm{x}^{*},\mathbf{w})), \label{eq:svgd-cov} \end{aligned} \end{split}\end{aligned}$$ in which $\mathbb{E}_{p(\mathbf{w}|\mathcal{D})}(\bm{f}(\bm{x}^{*},\mathbf{w}))$ is calculated in Eq. . Although the focus of this paper is to investigate the effect of the model form uncertainty on fluid quantities, the Bayesian neural network also allows for rigorous study of the epistemic uncertainty with Eqs. (\[eq:svgd-mean\]) and (\[eq:svgd-cov\]). Thus one can study the effect of training data, model architecture, and other parameters on predictive confidence. For complete details on SVGD, we direct the reader to the original work by Liu [*et al*.]{} [@liu2016stein; @liu2017stein] along with the work of Zhu and Zabaras [@zhu2018bayesian]. Uncertainty Quantification with SDD-RANS {#subsec:uq} ---------------------------------------- We now wish to propagate this uncertainty obtained for the anisotropic term to the fluid properties such as pressure or velocity. We use a stochastic system approach for which the model parameters in a system of PDEs are considered as random variables. This methodology has been used extensively in the past for model calibration, prediction and selection [@beck1998updating; @beck2002bayesian; @cheung2009bayesian; @cheung2011bayesian]. Consider a dynamical system defined by the model output $h\left(\bm{\phi},\bm{u}(\bm{\phi})\right)$, where $\bm{u}(\bm{\phi})$ are state variables that evolve with the dynamical system and $\bm{\phi}$ represents a set of model parameters with probability density $p(\bm{\phi})$. Traditionally the true form of the distribution $p(\bm{\phi})$ from which the model parameters are sampled from is largely not known. However, under the assumption that samples can be drawn from the parameter distribution, the expected response as well as the respective variance can be approximated by vanilla Monte Carlo simulation (MCS) with $P$ samples of the random model parameters: $$\begin{gathered} \mathbb{E}_{p(\bm{\phi})}(h) \approx \frac{1}{P}\sum_{i=1}^{P}h\left(\bm{\phi}_{i},\bm{u}(\bm{\phi}_{i})\right), \quad \bm{\phi}_{i}\sim p(\bm{\phi}), \\ \text{Var}_{p(\bm{\phi})}(h) \approx \frac{1}{P}\sum_{i=1}^{P}\left[h\left(\bm{\phi}_{i},\bm{u}(\bm{\phi}_{i})\right) - \mathbb{E}_{p(\bm{\phi})}(h)\right]^{2}.\end{gathered}$$ To extend this to the problem of interest and motivate SDD-RANS, let us consider the model output $h$ as the flow field predicted by the RANS equations and the state variables $\bm{u}(\bm{\phi})$ to be the fluid’s velocity, pressure and all other derived properties. As previously discussed, we will be taking an explicit representation of the tuned R-S in which a modified R-S field is predicted and held constant while the other state variables are propagated forward. Thus we are able to view a predicted R-S field as a random model parameter, namely, $p(\bm{\phi}) = p(\bm{b}^{*}|\bm{x}^{*},\mathcal{D})$ where the predictive density of $\bm{b}^{*}$ is given by: $$p(\bm{b}^{*}|\bm{x}^{*},\mathcal{D}) = \int p(\bm{b}^{*}|\bm{x}^{*},\textbf{w},\beta)p(\textbf{w},\beta|\mathcal{D})d\textbf{w}d\beta.$$ Rather than sampling the anisotropic term directly from the predictive distribution, recall the following representation of the likelihood in Eq. : $$\bm{b}^{*} = \bm{f}(\left\{\bm{s}^{*}, \bm{\omega}^{*}\right\}, \mathbf{w}) + \bm{\epsilon}.$$ To sample the predictive distribution, one can first sample the posterior $p(\textbf{w},\beta|\mathcal{D})$ and then execute a forward prediction of the neural network as well as sample the additive output noise yielding the predicted $\bm{b}^{*}$. We can modify the MCS such that we sample the weights of the Bayesian neural network as well as the variance of the additive output-wise noise: $$\begin{gathered} \mathbb{E}_{p(\mathbf{w},\beta|\mathcal{D})}(h) \approx \frac{1}{N}\sum_{i=1}^{N}h\left(\bm{b}^{*}_{i},\bm{u}(\bm{b}^{*}_{i})\right), \\ \text{Var}_{p(\mathbf{w},\beta|\mathcal{D})}(h) \approx \frac{1}{N}\sum_{i=1}^{N}\left[h\left(\bm{b}^{*}_{i},\bm{u}(\bm{b}^{*}_{i})\right) - \mathbb{E}_{p(\mathbf{w},\beta|\mathcal{D})}(h)\right]^{2},\\ \bm{b}^{*}_{i} = \bm{f}(\left\{\bm{s}^{*}, \bm{\omega}^{*}\right\} , \mathbf{w}_{i}) + \bm{\epsilon}_{i}, \quad \bm{\epsilon}_{i} \sim \mathcal{N}(\bm{\epsilon}_{i} | 0,\beta_{i}^{-1}\bm{I}_{9}), \\ \left\{\textbf{w}_{i}, \beta_{i}\right\} \sim p(\mathbf{w}_{i}, \beta_{i}|\mathcal{D}).\end{gathered}$$ The SVGD algorithm provides samples of the posterior $p(\mathbf{w}_{i}, \beta_{i}|\mathcal{D})$. Namely, given that SVGD uses a particle representation, each sample is a particle (or invariant neural network) used during training. In practice, the output-wise noise, $\bm{\epsilon}$, has minimal influence on the predicted values due to the previously made assumptions. Hence, we only take a mean point estimate of the likelihood. In principle, the neural network’s inputs are spatially independent between mesh nodes allowing for each node point in the fluid domain to have independent weight samples resulting in a stochastic field. However, the use of the divergence of the R-S in the RANS equations suggests that the spacial smoothness of the predicted field is of significant importance. Thus, we use a single neural network, $\bm{f}(\left\{\bm{s}^{*}, \bm{\omega}^{*}\right\} , \mathbf{w}_{i})$, to predict the R-S for the entire flow domain. As a result, in the context of the fluid domain, we are in fact sampling a functional representation of the R-S that is dependent on the velocity gradients. This combination of using a Bayesian data-driven model with a stochastic model parameter is why we have named this process stochastic data-driven RANS (SDD-RANS). With SDD-RANS, we have opened up the ability to obtain sample statistics for all flow quantities through traditional MCS. This allows for the quantification of uncertainty regarding our data-driven model beyond the R-S itself. The use of the explicit representation of the R-S and the noisy nature of the neural network’s predictions raise concerns regarding the convergence of the SDD-RANS model. In practice, at higher Reynolds numbers the simulation may fail to converge in some areas of the domain. Due to the nature of SDD-RANS, the statistical averages obtained through MCS accurately reflect the true state of the quantities of interest. The discrepancy from the true solution is reflected by the computed variance or uncertainty estimates. However, while computing each sample, one must still monitor the residuals of the model to ensure the initial transient state has ended. In practice, we run the forward simulation for the same number of iterations as the baseline simulation. Framework Implementation ------------------------ We use this Bayesian framework in the system of RANS equations by setting the R-S term as the stochastic parameter that is sampled from the predictive distribution obtained through the Bayesian neural network. We summarize the offline training process: - The training data consist of both baseline RANS and high-fidelity data for a set of different flows that attempt to capture different flow physics. - The underlying machine learning model is an invariant neural network that uses local fluid quantities to predict the anisotropic term of the R-S. - We extend this invariant model to the Bayesian paradigm by using SVGD in which a set of neural networks approximates the posterior $p(\mathbf{w},\beta|\mathcal{D})$ by a particle representation. - The parameters in each particle (or neural network) are optimized by minimizing the KL divergence between a particle variational approximation and the posterior of the parameters. - An iterative algorithm, resembling the form of standard gradient decent, updates the parameters of each particle until convergence. With the Bayesian neural network trained, one can make predictions for new flows: - For the flow of interest, a baseline RANS solution is obtained and the corresponding invariants and tensor functions at each mesh point are calculated. - Each neural network used during training with SVGD is used to predict a corresponding high-fidelity R-S field. - For each predicted field, the R-S is then constrained to the predicted values and a forward execution of the constrained system updates the remaining state variables. - An equivalent number of state variable samples are then obtained for which probabilistic bounds can be calculated. Numerical Implementation and Training {#sec:Training} ===================================== CFD Methods ----------- For obtaining the training and test flows, the open source CFD platform OpenFOAM (Open source Field Operation And Manipulation) [@weller1998tensorial; @jasak2007openfoam] is used. OpenFOAM is a widely accepted CFD package that contains a vast number of solvers for incompressible, compressible and multi-phase flows along with pre- and post-processing utilities. For the baseline RANS simulations the steady-state, incompressible solver *simpleFoam* was used which employs the semi-implicit method for pressure linked equations (SIMPLE) algorithm [@patankar1983calculation] to solve both the momentum and pressure equations. The high-fidelity LES simulations used the *pimpleFoam* transient solver that combines both the PISO (Pressure Implicit with Split Operator) [@issa1986solution] and SIMPLE algorithms to solve the pressure and momentum equations. The Smagorinsky subgrid-scale model [@smagorinsky1963general] with Van-Driest style damping was used for all LES flows. Both the baseline RANS and LES domains are discretized by second-order methods. Each training and testing flow is outlined in Tables \[tab:cfdflows\] and \[tab:cfd-test-flows\], and all meshes are non-uniform such that the mesh density increases around the feature of interest. All simulations were run with a CFL number below $0.3$ for numerical accuracy. Case Converge Diverge Square Cylinder Periodic Hills Square Duct Tandem Cylinders ------------- ---------------------------------------------------------------- ------------------------------------------- ---------------------------------------------------------------------------- --------------------------------------------- ---------------------------------------------- Reference Schiavo [*et al*.]{} [@schiavo2015large; @langley2018converge] Bosch [*et al*.]{} [@bosch1998simulation] Temmerman [*et al*.]{} [@temmerman2003investigation; @langley2018periodic] Pinelli [*et al*.]{} [@pinelli2010reynolds] Gopalan [*et al*.]{} [@gopalan2015numerical] Mesh RANS $140 \times 50 \times 50$ $100 \times 60 \times 20$ $100 \times 50 \times 50$ $7.5 \pi \times 60 \times 60$ $80 \times 60 \times 50$ Mesh LES $280 \times 100 \times 150$ $280 \times 120 \times 40$ $500 \times 150 \times 250$ $300 \pi \times 150 \times 150$ $470 \times 180 \times 120$ Domain Size $12.56H \times 2H \times 3H$ $20D \times 14D \times 4D$ $9H \times 3.306H \times 4.5H$ $4 \pi H \times 2H \times 2H$ $30D \times 20D \times 3D$ $L_{c}$ Half Channel Height Cylinder Diameter Hill Height Half Channel Width Cylinder Diameter $Re_{b}$ $5000$ $5000$ $6210$ $6680$ $5000$ $\nu$ $2.00$e$-4$ $2.00$e$-4$ $6.07$e$-4$ $2.00$e$-4$ $7.40$e$-4$ : Mesh and CFD parameters for each training flow which includes the respective reference, mesh sizes for both RANS and LES simulations, the domain size, characteristic length $L_{c}$, bulk Reynolds number $Re_{b}$ and kinematic viscosity $\nu$. Streamwise is in the $x-$, wall normal in the $y-$ and spanwise in the $z-$direction.[]{data-label="tab:cfdflows"} Case Backward Step Wall Mounted Cube ------------- ----------------------------------------- -------------------------------------------- -- Reference Gresho [*et al*.]{} [@gresho1993steady] Yakhot [*et al*.]{} [@yakhot2006turbulent] Mesh RANS $220 \times 60 \times 20$ $100 \times 40 \times 80$ Mesh LES $390 \times 100 \times 40$ $200 \times 100 \times 150$ Domain Size $27H \times 2H \times H$ $14H \times 3H \times 7H$ $L_{c}$ Step Height Cube Height $Re_{b}$ $500, 2500, 5000$ $500, 2500, 5000$ $\nu$ $2.00$e$-3$, $4.00$e$-4$, $2.00$e$-4$ $2.00$e$-3$, $4.00$e$-4$, $2.00$e$-4$ : Mesh and CFD parameters for each test flow which includes the respective reference, mesh sizes for both RANS and LES simulations, the domain size, characteristic length $L_{c}$, bulk Reynolds number $Re_{b}$ and kinematic viscosity $\nu$. Streamwise is in the $x-$, wall normal in the $y-$ and spanwise in the $z-$direction.[]{data-label="tab:cfd-test-flows"} Machine Learning Implementation {#sec:TrainingML} ------------------------------- To train the neural network, the Python machine learning library PyTorch [@paszke2017automatic] was used. The software and data used in this work are available at <https://github.com/cics-nd/rans-uncertainty>. The details of the network architecture used are given in Table \[tab:nn\]. The Leaky Rectifier function was used as the activation function as opposed to the standard Rectifier function to prevent too many nodes from becoming zero during training. Additionally the number of nodes in the hidden layers is tapered at the end of the network to prevent weights from being too small, which improved training performance. ----------------- -------------------------------------------------------------------------------------- Architecture $5\rightarrow200\rightarrow200\rightarrow200\rightarrow40\rightarrow20\rightarrow10$ Activation Leaky ReLu Optimizer ADAM [@kingma2014adam] Weight Decay $0.01$ Learning Rate $5$e$-6$, with learning rate decay on plateau Epochs $100$ Training Data $10000$ Mini-batch size $20$ SVGD Particles $20$ ----------------- -------------------------------------------------------------------------------------- : Neural network architecture and training details.[]{data-label="tab:nn"} The network architecture was determined by training an ensemble of neural networks with different number of hidden layers. Other network parameters, such as the taper of the last several layers and learning rate specified in Table \[tab:nn\], were identical between each of the tested architectures. The networks are compared in Fig. \[fig:hiddenLayersMSE\] with the mean negative log likelihood (MNLL) defined by: $$\begin{gathered} \quad {MNLL}=-\frac{1}{T}\sum_{i=1}^{T}\frac{1}{N}\sum_{j=1}^{N}\log{p(\hat{\bm{b}}_{i}|\bm{f}(\hat{\bm{x}}_{i}, \mathbf{w}_{j}), \beta_{j})},\end{gathered}$$ where $T$, $\hat{\bm{x}}$, $\hat{\bm{b}}$ are the number of validation/test data points, the target inputs and target outputs, respectively. We observe little distinguishable difference indicating that training between each architecture is relatively the same. Additionally, the mean squared prediction error of the unnormalized anisotropic tensor $\bm{a}^{*}$ is plotted for a validation set of $1000$ random data points from each of the training flows (i.e. $5000$ total data points). The mean squared prediction error (MSPE) is defined as: $${MSPE}=\frac{1}{T}\sum^{T}_{i=1}{\left\lVert\mathbb{E}(\bm{a}^{*}_{i}|\hat{\bm{x}}_{i},\mathcal{D})-\hat{\bm{a}}_{i}\right\rVert}^{2}_{2}\approx\frac{1}{T}\sum^{T}_{i=1}{\left\lVert\frac{k}{N}\sum_{j=1}^{N}\bm{f}(\hat{\bm{x}}_{i},\mathbf{w}_{j})-\hat{\bm{a}}_{i}\right\rVert}^{2}_{2},$$ where $k$ is the baseline RANS TKE and $\hat{\bm{a}}$ are the target unnormalized anisotropic tensors. For the MSPE, there is a notable difference between the converged accuracy of each network. The networks with above $6$ hidden layers exhibited significant over-fitting of the validation data and are not plotted. One can observe the onset of over-fitting by the noisy MSPE of the $6$ hidden layer neural network. The network architecture with $5$ hidden layers was selected to ensure that over-fitting does not take place. ![(Left) The negative log likelihood (MNLL) and (Right) the mean squared prediction error (MSPE) of the validation data for several neural network architectures.[]{data-label="fig:hiddenLayersMSE"}](Fig3.png){width="100.00000%"} Compared to other potential network models, we found that the model selected originally by Ling [*et al*.]{} [@ling2016reynolds] proved to be exceptionally difficult to train. This is reflected in the original work by the extremely low learning rate used of $2.5 \times 10^{-6}$. During training we also found only very low learning rate could be used for the training process to be stable. To increase training performance and efficiency, we also used the following techniques: - Depending on the size of the fluid domains, the number of training points for a single flow can be large. Thus to increase training efficiency, rather than using every single mesh point, a subset of training points is selected. In this work, we use only $10^4$ total training points that are evenly distributed among all training flows (i.e. $2 \times 10^3$ points for each of the five test flows). Every $10$ epochs these points are then re-sampled at random. This prevents the potential issue of exceeding the available memory on the provided GPU. - Training points are shuffled randomly and mini-batched every epoch such that data from multiple flows can reside in a single mini-batch. This helps prevent the model from over-fitting to a specific flow and improves the quality of predictions. - The invariant inputs to the neural network tended to vary strongly in magnitude including very large values near fixed boundaries. Thus the invariants are re-scaled by a sigmoidal operation that helps to normalize outliers to the range of $+1$ to $-1$ [@li2000fuzzy]. In addition, the tensor basis functions were normalized by the $L_2$ norm of the matrix: $$\hat{\mathcal{I}}_{i} = \frac{1-e^{-\mathcal{I}_{i}}}{1+e^{-\mathcal{I}_{i}}}, \qquad \hat{\bm{T}}^{\lambda} = \frac{\bm{T}^{\lambda}}{{\left\lVert\bm{T}^{\lambda}\right\rVert}_{2}}.$$ To quantify the training quality, the MSPE is calculated for both the validation set along with a test set of $1000$ randomly selected points from each test flow in Table \[tab:cfd-test-flows\]. Additionally, we also plot the MNLL for the training, validation and testing data sets. The results are illustrated in Fig. \[fig:testMSE\]. We note that for both the validation and test datasets the model quickly converges and exhibits minimal over-fitting. The training process on a single NVIDIA P100 GPU took approximately $3.0$ wall-clock hours. ![(Left) The mean squared prediction error (MSPE) of both the validation and test datasets. (Right) The mean negative log likelihood (MNLL) of the training, validation and test datasets.[]{data-label="fig:testMSE"}](Fig4.png){width="100.00000%"} To verify that the trained model has learned a physical interpretation of the training data, we plot the contours of the mixing coefficients $G^{\lambda}$ predicted by the neural network for the square cylinder and periodic hills training flows in Figs. \[fig:squareGCoeff\] and \[fig:periodHillGCoeff\], respectively. While there appears to be some minor over-fitting in front of the square cylinder in Fig. \[fig:squareGCoeff\], for both flows it is clear that the model has indeed identified physical regions of the flow as well as maintained symmetries. ![The learned mixing coefficients of the training neural network for flow around a square cylinder [@bosch1998simulation] at bulk Reynolds number $5000$. No domain reflections were used to artificially impose symmetry.[]{data-label="fig:squareGCoeff"}](Fig5.png){width="100.00000%"} ![The learned mixing coefficients of the training neural network for flow over periodic hills [@temmerman2003investigation; @langley2018periodic] at bulk Reynolds number $6210$.[]{data-label="fig:periodHillGCoeff"}](Fig6.png){width="100.00000%"} Constrained R-S Simulation -------------------------- To integrate the sampled R-S field into OpenFOAM, a small modification is made to the *simpleFoam* solver such that the R-S is now a constant field in the momentum RANS equation. Since the R-S field is held constant, the calculation of the TKE and turbulent dissipation is no longer needed. An important issue is the handling of boundary conditions. This includes the treatment of domain boundaries as well as areas in which wall functions may be used. We address these issues using two different methods. First, the use of the baseline RANS TKE as a scaling factor of the anisotropic term shown in Eq.  allows for many turbulent boundary conditions to be satisfied. Second, to address areas in which wall functions may be used, we take inspiration from hybrid LES/RANS methods and introduce a blending function proposed by Xiao [*et al*.]{} [@xiao2004blending]: $$\bm{b}^{*} = \Gamma \bm{b}_{dd} + (1-\Gamma)\bm{b}_{rans}, \quad \Gamma=\tanh{\left(d/\alpha_{1}\lambda\right)}^{2},$$ where $\bm{b}_{dd}$ is the data-driven prediction of the anisotropic tensor, $\bm{b}_{rans}$ is the baseline RANS anisotropic tensor, $\lambda^{2} = k/\epsilon$, $d$ is the distance from the wall and $\alpha_{1}$ is a tunable parameter. This function allows a smooth transition between the use of the baseline R-S near the wall and the data-driven prediction in the bulk flow. In the original work, it is suggested that the selection of $\alpha_{1}$ be a value that achieves $\Gamma = 0.5$ somewhere is the log region. We found the value of $0.05$ worked well for our test cases. Numerical Results {#sec:NumericalResults} ================= The use of data-driven models for test simulations whose domain is similar or identical to the training data is a frequent occurrence in the literature but does not correctly assess a data-driven model’s performance. Since our selected neural network has already been shown to work adequately for similar flows in [@ling2016reynolds], our test cases are selected to deviate significantly from the training flows in both flow geometry and Reynolds number. We have selected the two test flows detailed in Table \[tab:cfd-test-flows\]: flow over a backwards step and flow around a wall mounted cube both at three different Reynolds numbers. The geometry for each flow can be seen in Fig. \[fig:testFlowSchematic\]. For both test cases, we will study the accuracy and respective uncertainty of both the model’s R-S predictions as well as the predicted fluid quantities of interest. Ultimately, we wish to assess the predictive performance of the data-driven model for these geometrically different flows and use the proposed stochastic data-driven RANS algorithm to calculate probabilistic bounds on flow state variables by conducting uncertainty quantification on the data-driven turbulence model. [0.48]{} ![(a) Flow geometry for the backwards step test flow with height $h$. (b) Flow geometry for the wall mounted cube test flow with height $h$.[]{data-label="fig:testFlowSchematic"}](Fig7a.png "fig:"){width="\textwidth"}   [0.48]{} ![(a) Flow geometry for the backwards step test flow with height $h$. (b) Flow geometry for the wall mounted cube test flow with height $h$.[]{data-label="fig:testFlowSchematic"}](Fig7b.png "fig:"){width="\textwidth"} Backwards Step {#subsec:backwardsStep} -------------- In the first test case, we select a backwards facing step at three different Reynolds numbers. As illustrated in Fig. \[fig:backstepSchematic\], this flow features a constant velocity inlet channel followed by a backwards facing step of height $h$. For this flow, we select the inlet channel to be the same height as the step. In contrast to most backwards step simulations, no-slip walls are on both the top and bottom faces. The $z$ direction is periodic. On the $x-y$ plane, we place the origin at the corner of the step. The flow features of interest are the recirculating regions that appear not only directly after the step but also on the upper channel wall down stream which is seen in the LES simulations in Figs. \[fig:backstepUX500\]-\[fig:backstepUX5000\]. The predicted components of the unnormalized anisotropic term $\bm{a}$, defined by $\bm{a}=k\cdot\bm{b}$ where $k$ is the baseline RANS TKE and $\bm{b}$ is the predicted anisotropic tensor, are shown in Figs. \[fig:backstepDeviatoric500\]-\[fig:backstepDeviatoric5000\] for Reynolds numbers $500$, $2500$ and $5000$. The first trend to notice is the relative consistency of SDD-RANS between all Reynolds numbers in terms of the magnitude of the mean predictions as well as variance. This is clearly an effect of the use of training data that vary little in Reynolds number compared to the test flows. For both $a_{11}$ and $a_{33}$ at Reynolds number $500$ and $2500$, the neural network is able to successfully predict the correct shape of the anisotropic term. This is a notable improvement of the baseline RANS prediction which severally under-predicts all normal components. For Reynolds number $5000$, SDD-RANS favors only a single region for $a_{11}$ and $a_{33}$ as opposed to the two regions seen in lower Reynolds numbers. While these predictions are significant improvements over the baseline RANS solution, there are still key discrepancies including that the anisotropic components are consistently predicted upstream compared to the LES solution. Additionally, for terms such as $a_{11}$, the neural network under-predicts the magnitude for the larger Reynolds numbers as well as consistently under-predicts $a_{22}$. Briefly focusing on the epistemic uncertainty of the model’s predictions, the variance of the neural network’s predictions are relatively small indicating the model is over-confident for this test flow. Also we note that near the inlet (laminar region) there is little variance in the predicted anisotropic term. This shows that the neural network is able to identify regions that are more uncertain than others instead of just placing a uniform variance across the entire field. ![The anisotropic term predictions for the backwards step test flow for Reynolds number $500$. Top to bottom: $a_{11}$, $a_{22}$, $a_{33}$ and $a_{12}$ ($a_{13}$ and $a_{23}$ are omitted due to all fields being zero).[]{data-label="fig:backstepDeviatoric500"}](Fig8.png){width="90.00000%"} ![The anisotropic term predictions for the backwards step test flow for Reynolds number $2500$. Top to bottom: $a_{11}$, $a_{22}$, $a_{33}$ and $a_{12}$ ($a_{13}$ and $a_{23}$ are omitted due to all fields being zero).[]{data-label="fig:backstepDeviatoric2500"}](Fig9.png){width="90.00000%"} ![The anisotropic term predictions for the backwards step test flow for Reynolds number $5000$. Top to bottom: $a_{11}$, $a_{22}$, $a_{33}$ and $a_{12}$ ($a_{13}$ and $a_{23}$ are omitted due to all fields being zero).[]{data-label="fig:backstepDeviatoric5000"}](Fig10.png){width="90.00000%"} The stream-wise velocity contours of the flow for the LES, baseline RANS and the expected velocity prediction of stochastic data-driven RANS (SDD-RANS) are depicted in Figs. \[fig:backstepUX500\]-\[fig:backstepUX5000\]. To keep plot labels uncluttered, we refer to the expected values from the proposed framework as just SDD-RANS. For each Reynolds number, the mean stream-wise velocity profiles are also illustrated with the respective predictive error bars. We look first at the lowest Reynolds number of $500$ for which the model produced the best prediction. Even though this corresponds to a Reynolds number furthest from the training data in Table \[tab:cfdflows\], the stochastic model was able to successfully predict the appearance of the second recirculation region. The baseline RANS simulation only predicted a single eddy behind the step. While the anisotropic predictions are far larger in magnitude compared to LES, the viscous forces are large enough to correct these discrepancies at lower Reynolds numbers. We presume that these upstream over-predictions of the anisotropic terms in magnitude are the reason the second eddy appears closer to the inlet for SDD-RANS compared to the LES solution. Overall, for this lower Reynolds number, the model has little variance in its predictions since the effects of the R-S prediction are dampened. ![Normalized stream-wise mean velocity contours for Reynolds number $500$. The top is the LES solution, below is the baseline RANS prediction followed by the data-driven mean field. Lastly is the stream-wise mean velocity profiles for all simulations shown along with the predictive error bars of the SDD-RANS prediction.[]{data-label="fig:backstepUX500"}](Fig11.png){width="95.00000%"} As the Reynolds number increases, the role of the R-S increases significantly as the viscous forces weaken and a very clear degradation in the predictive performance of SDD-RANS is seen. For Reynolds number $2500$ and $5000$ (see Figs. \[fig:backstepUX2500\]-\[fig:backstepUX5000\]), SDD-RANS does not yield any prediction improvement over the baseline RANS simulation with the exception of the recirculating region near the step ($x/h\leq 7.5$) where SDD-RANS is able to accurately predict the flow. ![Normalized stream-wise mean velocity contours for Reynolds number $2500$. The top is the LES solution, below is the baseline RANS prediction followed by the data-driven mean field. Lastly is the stream-wise mean velocity profiles for all simulations shown along with the predictive error bars of the SDD-RANS prediction.[]{data-label="fig:backstepUX2500"}](Fig12.png){width="95.00000%"} For both higher Reynolds numbers test cases, SDD-RANS fails to predict the upper recirculation region that is present in the LES solution. This is likely due to the under-prediction of the anisotropic components downstream seen in Figs. \[fig:backstepDeviatoric2500\]-\[fig:backstepDeviatoric5000\]. As the R-S is increased, minor deviations in the anisotropic term are amplified resulting in potentially starkly different flow predictions [@wu2018rans]. SDD-RANS accurately captures these phenemona. As the Reynolds number increases, so does the standard deviation indicating a loss of model confidence. In addition, it is clear that the model is extremely confident in its predictions towards the inlet where it is able to match the LES solution. However, this confidence quickly diminishes downstream as flow predictions become increasingly less accurate. With a deterministic data-driven model such indicators would not be present allowing for no interpretable information on prediction confidence without observed high-fidelity data. ![Normalized stream-wise mean velocity contours for Reynolds number $5000$. The top is the LES solution, below is the baseline RANS prediction followed by the data-driven mean field. Lastly is the stream-wise mean velocity profiles for all simulations shown along with the predictive error bars of the SDD-RANS prediction.[]{data-label="fig:backstepUX5000"}](Fig13.png){width="95.00000%"} Wall Mounted Cube {#subsec:wallMountedCube} ----------------- The second test case is flow around a wall mounted cube with height $h$ as shown in Fig. \[fig:wallCubeSchematic\]. Unlike the majority of the flows that have been tested by data-driven models in the literature [@tracey2013application; @singh2017machine; @zhang2015machine; @ling2016reynolds; @xiao2016quantifying] as well as our training flows, this test flow contains an obstacle that is not semi-infinite. This means that flow with this geometry cannot be modeled by a two-dimensional RANS simulation as was the case for all previously considered flows. Additionally, similar to the backwards step, none of our training flows contain a geometry that is similar to this. As a result, we consider this flow an excellent test to investigate the limits of SDD-RANS in generalizing to a true 3D test case. The set-up of this flow consists of an uniform inlet velocity and two channel walls normal to the y-axis. The cube is placed slightly down stream of the inlet. The feature of interest is primarily the recirculation region behind the cube itself. Additionally, as the Reynolds number increases, flow separation occurs on the sides of the cube. As will be shown in the subsequent figures, this flow separation is often non-existent for the baseline RANS predictions. While the mean flow is symmetrical about the $x-y$ plane in the middle of the channel ($z=3.5H$), we simulate the entire cube in order to observe non-symmetrical behavior in predictions. The prediction of the unnormalized anisotropic term along the $x-y$ plane of symmetry is shown in Fig. \[fig:wallRSPred2500\] for Reynolds number $2500$. For brevity, we only show the results for a single Reynolds number since both Reynolds numbers $500$ and $5000$ lead to similar predictions. From this figure, several positive traits are seen for both the mean neural network predictions as well as the associated uncertainty. The bulk region shows little variability within the predictive error bounds. Instead the variance is largely concentrated behind the obstacle. This is a nice attribute because the baseline RANS simulation is accurate in the bulk region, thus perturbing the R-S in the bulk flow would not be of any benefit. Another interesting feature predicted by the neural network is the concentrated region of normal stresses on top of the cube seen in $a_{11}$ and $a_{33}$. Similar regions, yet smaller in magnitude, appear further down-stream in the LES solution as a result of the shear layer that forms between the bulk and recirculation regions. In addition, the SDD-RANS is able to significantly improve the R-S prediction towards the front of the cube where the baseline RANS solution is incorrect. This includes the leading corner of the cube at $x=3$ where SDD-RANS is able to largely correct the baseline RANS solution which has sharp, unphysical predictions near the edge. ![The anisotropic term predictions for the wall mounted cube test flow for Reynolds number $2500$ along the plane of symmetry ($a_{13}$ and $a_{23}$ are omitted due to both fields being zero on the plane of symmetry). From top to bottom: Time-averaged LES solution, baseline RANS prediction, SDD-RANS expected value and the predictive standard deviation error bounds.[]{data-label="fig:wallRSPred2500"}](Fig14.png){width="100.00000%"} Despite the improvements to the upstream edge of the cube, the expected value of SDD-RANS has no marginal improvement on the baseline RANS prediction in the recirculation region. However, as reflected in the predictive error bounds, the variance is larger in the recirculation zone often being able to enclose part of the true LES solution. This is a promising result because, although the mean predictions have not improved, the model is uncertain regarding its predictions in this region. This suggests that this area of the flow could contain physics the model has not seen before in the training data. The stream-wise velocity contours on the plane of symmetry for the LES, baseline RANS and the expected value of SDD-RANS are depicted below in Fig. \[fig:wallUXStreamCont\]. Similar to the backwards step, as the Reynolds number increases, the standard deviation of the SDD-RANS prediction also increases. However, the magnitude of the variance is significantly smaller than that of the backwards step for higher Reynolds numbers reflecting the more accurate predictions for this problem. ![Normalized stream-wise mean velocity contours for Reynolds numbers $500$, $2500$ and $5000$ on the plane of symmetry. The top is the time averaged LES solution, below is the baseline RANS prediction followed by the SDD-RANS expected velocity. The fourth row shows the standard deviation field of the data-driven prediction.[]{data-label="fig:wallUXStreamCont"}](Fig15.png){width="100.00000%"} To take a closer look at the performance of SDD-RANS, stream-wise velocity profiles are plotted for both Reynolds number $500$ and $5000$ in Figs. \[fig:wallUXStreamProfile500\] and \[fig:wallUXStreamProfile5000\], respectively. For each, a plot containing the resulting SDD-RANS velocity field samples are shown as well as the predictive standard deviation error bars. As expected the variance in the velocity samples increases with the increased Reynolds number, however this only occurs in the recirculation region where the instantaneous flow is turbulent. In the bulk region above the recirculation zone, the variance remains small. ![Normalized stream-wise velocity profiles for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re$=500$. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXStreamProfile500"}](Fig16a.png "fig:"){width="48.00000%"}   ![Normalized stream-wise velocity profiles for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re$=500$. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXStreamProfile500"}](Fig16b.png "fig:"){width="48.00000%"} A significant improvement by SDD-RANS for both Reynolds numbers is the prediction of the detached flow on top of the cube ($x/h=4$) which the baseline RANS fails to capture. This is largely due to the large increase of normal stresses from the neural network predictions around the surrounding walls of the cube which results in the shear layer forming above the obstacle. For both Reynolds numbers cases, improvements in the recirculation region prediction are present. However, for the lower Reynolds number case SDD-RANS leads to more accurate predictions. ![Normalized stream-wise velocity profiles for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re=5000. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXStreamProfile5000"}](Fig17a.png "fig:"){width="48.00000%"}   ![Normalized stream-wise velocity profiles for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re=5000. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXStreamProfile5000"}](Fig17b.png "fig:"){width="48.00000%"} Since this obstacle is not semi-infinite, the velocity contours for the horizontal plane at $y=0.5h$ are shown in Fig. \[fig:wallUXSpanCont\]. Similarly, velocity profiles are plotted for Reynolds number $2500$ in Fig. \[fig:wallUXSpanProfile2500\]. In general, we can see the same trends as previous results for which the variance increases with Reynolds number. We note that SDD-RANS is able to predict the presence of the detached flow on the side of the cube. While some asymmetry exists in the SDD-RANS predictions, the predictions overall retain a general symmetric profile. It is likely that increasing the number of samples of R-S fields would further improve the symmetry and smoothness of predictions. ![Normalized stream-wise mean velocity contours for Reynolds numbers $500$, $2500$ and $5000$ on the $y=0.5h$ plane. The top is the time averaged LES solution, below is the baseline RANS prediction followed by the SDD-RANS expected velocity. The fourth row shows the standard deviation field of the data-driven prediction.[]{data-label="fig:wallUXSpanCont"}](Fig18.png){width="100.00000%"} ![Normalized stream-wise velocity on the $y=0.5h$ plane for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re$=2500$. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXSpanProfile2500"}](Fig19a.png "fig:"){width="48.00000%"}   ![Normalized stream-wise velocity on the $y=0.5h$ plane for the baseline RANS, high-fidelity LES, and SDD-RANS predictions on the plane of symmetry at six different locations in the stream-wise direction for Re$=2500$. The left shows the SDD-RANS velocity samples, and the right shows the respective predictive error bars for each profile.[]{data-label="fig:wallUXSpanProfile2500"}](Fig19b.png "fig:"){width="48.00000%"} This framework allows probabilistic bounds to be calculated for other fluid properties such as pressure, drag, shear stress, etc. For example, two pressure profiles along the face of the wall mounted cube are plotted in Figs. \[fig:wallCubePressure1\] and \[fig:wallCubePressure2\]. In general we see that SDD-RANS is able to provide an improved prediction compared to the baseline RANS. Similar to the anisotropic components, SDD-RANS corrects the unphysical pressure drop that occurs on the edge of the leading cube face in the baseline RANS simulation. The uncertainty for the predictive pressure is also very reasonable nearly capturing the true LES prediction for all faces. Similar to the velocity predictions in Figs. \[fig:wallUXStreamCont\] and \[fig:wallUXSpanCont\], the variance of the pressure on the upstream face of the cube is significantly smaller reflecting the model’s confidence in this laminar region. ![Normalized mean surface pressure profile on the plane of symmetry ($z=3.5h$) for Re$=5000$.[]{data-label="fig:wallCubePressure1"}](Fig20a.png "fig:"){width="70.00000%"}   ![Normalized mean surface pressure profile on the plane of symmetry ($z=3.5h$) for Re$=5000$.[]{data-label="fig:wallCubePressure1"}](Fig20b.png "fig:"){width="27.00000%"} ![Normalized mean surface pressure profile on the plane $y=0.5h$ for Re$=5000$.[]{data-label="fig:wallCubePressure2"}](Fig21a.png "fig:"){width="70.00000%"}   ![Normalized mean surface pressure profile on the plane $y=0.5h$ for Re$=5000$.[]{data-label="fig:wallCubePressure2"}](Fig21b.png "fig:"){width="27.00000%"} Conclusions {#sec:Conclusions} =========== As the CFD community continues to investigate the use of machine learning tools for data-driven modeling, the need to accurately quantify the induced uncertainties from the use of such models becomes essential. In this work, we have presented a novel framework that allows for the quantification of such model form uncertainty. To satisfy invariant properties, we use the neural network architecture originally proposed by Ling [*et al*.]{} [@ling2016reynolds]. Using Stein variational gradient decent and following the work of Zhu and Zabaras [@zhu2018bayesian], we extended this invariant neural network model to a Bayesian deep neural network to allow us to compute the distribution of the anisotropic R-S tensor for a given baseline solution. To propagate the uncertainty of this model to fluid flow quantities of interest, a stochastic data-driven RANS algorithm is proposed that utilizes standard Monte Carlo simulation. The integrated framework was rigorously investigated on two flows to observe its generalization property. In the presented implementation of this framework, we found that the invariant neural network used to model the anisotropic tensor proved difficult to train and yield satisfactory predictions for unseen flows and geometries. From our studies, we hypothesize that using just the five invariant inputs does not provide enough descriptive information to accurately map from the coarse to high-fidelity flow physics. Although the network contains desired invariant properties, other flow quantities would most likely need to be used as model inputs to improve the quality of predictions. Thus a critical area to be investigated is the development of more accurate Reynolds stress representations by identifying the important local- and non-local variables that influence its values. The potential use of spatial correlations and information at neighboring nodes (non-local models) may prove to be extremely beneficial. Such an approach, while difficult to implement for non-uniform grids, can be easily applied in the context of convolutional neural networks that are capable of mapping many high-dimensional inputs to multi-outputs of high-dimensionality. While a number of other models in the literature can yield much better training predictions, most of these models remain only useful to a small family of flows resembling those in the training dataset. The generalization property of these models remains an open problem for the data-driven community. With improvements in the representation of the tuned Reynolds stress in the RANS equations, we believe that this framework can provide extremely beneficial information for data-driven models. While the most obvious application is its use to assess a given model’s predictive confidence, the developed framework can also be used to identify locations of potentially lower accuracy. This could be useful for identifying areas that may require finer mesh resolutions or high-fidelity simulations. Additionally, the use of a Bayesian neural network allows us to compute predictive bounds for the quantities of interest. This can be extremely useful in cases where the training data is limited. Even though the use of the Bayesian neural network and SDD-RANS requires more computational time than deterministic data-driven approaches, we found that when compared to high-fidelity simulations the computational cost remains low. The use of a Bayesian neural network opens up the potential of implementing experimental design techniques by investigating the impact of training data on the quality of the model’s predictions. This can range from the assessment of a limited data case or how specific training flows at various Reynolds numbers impact a specific test case prediction. Finally, a detailed analysis of epistemic uncertainty would be beneficial to the data-driven turbulence modeling community. Acknowledgements {#acknowledgements .unnumbered} ================ The authors acknowledge support from the Defense Advanced Research Projects Agency (DARPA) under the Physics of Artificial Intelligence (PAI) program (grant No. HR00111890034). The work of NG is also supported by a National Science Foundation (NSF) Graduate Research Fellowship Program grant No. DGE-$1313583$. The computing was facilitated by the resources of the NSF supported “Extreme Science and Engineering Discovery Environment” (XSEDE) on the Bridges and Bridges-GPU cluster through the startup allocation No. TG-CTS$180011$ and research allocation No. TG-CTS$180038$. Additional computing resources were provided by the University of Notre Dame’s Center for Research Computing (CRC). References {#references .unnumbered} ==========

--- abstract: 'In recent years, the majority of works on deep-learning-based image colorization have focused on how to make a good use of the enormous datasets currently available. What about when the data at disposal are scarce? The main objective of this work is to prove that a network can be trained and can provide excellent colorization results even without a large quantity of data. The adopted approach is a mixed one, which uses an adversarial method for the actual colorization, and a meta-learning technique to enhance the generator model. Also, a clusterization *a-priori* of the training dataset ensures a task-oriented division useful for meta-learning, and at the same time reduces the per-step number of images. This paper describes in detail the method and its main motivations, and a discussion of results and future developments is provided.' author: - Tomaso Fontanini - Eleonora Iotti - Andrea Prati bibliography: - 'mybibliography.bib' title: 'MetalGAN: a Cluster-based Adaptive Training for Few-Shot Adversarial Colorization' --- ![Example images generated using MetalGAN for 100-epochs, and 100-meta-iterations. From left to right: gray scale image, ground truth, output of the network. The example images belong to two different clusters.[]{data-label="fig:meta_net"}](img/visual_abstract_2.png){width="\textwidth"} Introduction ============ The *automatic image colorization* task is an image processing problem that is fundamental and extensively studied in the field of computer vision. The task consists in creating an algorithm that takes as input a gray-scale image and outputs a colorized version of the same image. The challenging part is to colorize it in a plausible and well-looking way. Many systems were developed over the years, exploiting a wide variety of image processing techniques, but recently, the image colorization problem, as many other problems in computer vision, was approached with deep-learning methods. Colorization is a *generative* problem from a machine learning perspective. Generative techniques, such as *Generative Adversarial Networks* (*GANs*) [@goodfellow2014generative], are then suitable to approach such a task. In particular, *conditional GANs* (*cGANs*) models seem especially appropriate to this purpose, since their structure allows the network to learn a mapping from an image $x$ and (only if needed) a random noise vector $z$ to an output generated image $y$. On the contrary, standard GANs only learn the mapping from the noise $z$ to $y$. As many deep-learning techniques, the training of a GAN or a cGAN needs a large amount of images. Large datasets usually grant a great diversity among images, allowing the network to better generalize its results. Nevertheless, having a huge number of images is often not feasible in real-world applications, or simply it requires too much storage space for an average system, and high training computational times. Hence, porting the current deep-learning colorization technologies to a more accessible level and achieving a better understanding of the colorization training process are eased by using a smaller dataset. For these reasons, one of the aims of this work is to achieve good performances in the colorization task using a little number of images compared to standard datasets. In *few-shot learning*, a branch of the deep-learning field, the goal is to learn from a small number of inputs, or from one single input in the ideal case (*one-shot learning*): the network is subject to a low quantity of examples, and it has to be capable to infer something when posed face-to-face to a new example. This problem underpins a high generalization capability of the network, which is a very difficult task and an open challenging problem in deep networks research. Recently, some novel interesting ideas highlight a possible path to reach a better generalization ability of the network. These ideas are based on the concept of learning to learn, i.e., adding a meta-layer of learning information above the usual learning process of the network. The generalization is achieved by introducing the concept of *tasks distribution* instead of a single task, and the concept of *episodes* instead of instances. A tasks’ distribution is the family of those different tasks on which the model has to be adapted to. Each task in the distribution has its own training and test sets, and its own loss function. A meta-training set is composed of training and test images samples, called episodes, belonging to different tasks. During training, these episodes are used to update the initial parameters (weights and bias) of the network, in the direction of the sampled task. Results of meta-learning methods investigated in literature are encouraging and obtain good performances on some few-shot datasets. For this reason and since the goal of this work is to colorize images with a few number of examples, a meta-learning algorithm to tune the network parameters on many different tasks was employed. The chosen algorithm is Reptile [@nichol2018first], and it was combined with an adversarial colorization network composed by a Generator $G$ and a Discriminator $D$. In other words, the proposed method approaches the colorization problem as a meta-learning learning one. Intuitively, Reptile works by randomly selecting tasks, then it trains a fast network on each task, and finally it updates the weights of a slow network. In this proposal, tasks are defined as clusters of the initial dataset. In fact, a typical initial dataset is an unlabeled dataset that contains a wide variety of images, usually photographs. In this setting, for example, a task could be to color all seaside landscape, and another could be to color all cats photos. Those tasks refer to the same problem and use the same dataset, but they are very different at a practical level. A very large amount of images could overwhelm the problem, showing as much seasides and cats as the network needs in order to differentiate between them. The troubles start when only a small dataset is available. As a matter of fact, such a dataset could not have the suitable number of images for making the network learning how to perform both the two example colorizations decently. The idea is to treat different classes of images as different tasks. For dividing tasks, features were extracted from the dataset using a standard approach—e.g., a Convolutional Neural Network (CNN)—and the images were clusterized through K-means. Each cluster is thus considered as a single task. During training, Reptile tunes the network $G$ on the specific task corresponding to an input query image and therefore it adapts the network to a specific colorization class. The problems and main questions that emerge in approaching a few-shot colorization are various. First of all, how the clusterization should be made in order to generate a coherent and meaningful distribution of tasks? Does a task specialization really improve the colorization or the act of automatically coloring a photo is independent from the subject of the photo itself? Second, how the meta-learning algorithm should be combined with cGAN training, also to prevent overfitting the generator on few images? And last, since the purpose of the work is not to propose a solution to the colorization problem in general, but to propose a method that substantially reduce the amount of images involved in training without—or with minor—losses in state-of-the-art results, how to evaluate the actual performance of the network compared to other approaches? In particular, what are the factors that should be taken in account to state an enhancement, not in the proper colorization, but in few-shot colorization? In the light of these considerations, the contributions of this work are summarized as follows: - A new architecture that combines meta-learning techniques and cGAN called *MetalGAN* is proposed, specifying in detail how the generator and the discriminator parameters are updated; - A clusterization and a novel algorithm are described and their ability to tackle image-to-image translation problems is highlighted; - An empirical demonstration that a very good colorization can be achieved even with a small dataset at disposal during training is provided by showing visual results; - A precise comparison between two modalities (i.e. our algorithm and only cGAN training) is performed at experimental time, using the same network model and hyper-parameters. Related Work {#sec:related} ============ ### Image retrieval: Since we need the clusterization to be as accurate as possible we reserved a particular attention to the recent image retrieval techniques that focus on obtaining optimal descriptors. Recently, deep learning allowed to greatly improve the feature extraction phase of image retrieval. Some of the most interesting papers on the subject are [@razavian2016visual; @gong2014multi; @babenko2014neural; @yue2015exploiting; @reddy2015object] and, in particular, MAC descriptors [@tolias2015particular], that we ended up using. ### Conditional GANs: When a GAN generator is not only conditioned with a random noise vector, but also with more complex information like text [@reed2016generative], labels [@mirza2014conditional], and especially images, the model to use is a *conditional* GANs (*cGANs*). cGANs allow a better control over the output of the network and thus are very suitable in a lot of image generation tasks. In particular, cGANs conditioned on images were used both in a paired [@isola2017image] and unpaired [@zhu2017unpaired] way, to produce complex texture [@xian2018texturegan], to colorize sketches [@sangkloy2017scribbler] or images [@cao2017unsupervised] and more recently to produce outstanding image synthesis results [@wang2018high; @park2019semantic]. In this work, the output must be conditioned by the input gray-scale image, in order to train the network at only generating the colors of the image but not shapes, or the image itself. ### Meta-learning: The most relevant meta-learning studies for this work are the Model-Agnostic Meta-Learning (MAML) [@finn2017model] algorithm and Reptile [@nichol2018first] ones. In particular, we incorporate the Reptile algorithm inside the training phase, allowing the parameters of the generator to be updated in the same fashion as Reptile works. A similar work using MAML is MetaGAN [@zhang2018metagan], where a generator is used to enhance classification models in order to discriminate between real and fake data, providing generated samples for a task. The main purpose of MetaGAN is not to improve a generative network, but to perform a better few-shot classification, using generated images to sharpen the decision boundary of the problem. On the contrary, in our approach, the generator is fed with task-related images, and the meta-learner is used to enhance the generator itself, instead of a few-shot classifier. Both MAML and Reptile are based on hyper-parameterized gradient descent, and they learn how to initialize network parameters. Other types of meta-learners work differently. For example, there are many algorithms that learn how to parameterize the optimizer of the network [@hochreiter2001learning; @ravi2016optimization], or in other cases the optimizer itself is a network [@li2017learning; @andrychowicz2016learning; @wichrowska2017learned]. Moreover, one of the most general approach is to use a recurrent neural network trained on the episodes of a set of tasks [@santoro2016meta; @mishra2017simple; @duan2016rl; @wang2016learning]. The most interesting result of these meta-learners is the achievement of high performance on small datasets [@Rezende:2016:OGD:3045390.3045551; @vinyals2016matching; @kiran2018zero], or datasets used for few-shot learning (e.g., Omniglot) [@lake2015human]. Algorithm {#sec:algorithm} ========= This section goes in detail within the algorithm we propose. Therefore, each subsection focuses on a different aspect of the method. Then, the complete architecture is explained. Clusterization of the dataset ----------------------------- In order to exploit Reptile for image colorization we need to treat our image dataset as it would be composed by a series of separate tasks. For this reason we extract features from each image in the dataset using *activation\_43* layer of Resnet50. Then, we calculate MAC descriptors by applying max pooling and L2 normalization on the features. Having these MAC descriptors set $F$, we first apply Principal Componet Analysis (PCA) to reduce features dimension from 2048 to 512 and then apply K-means. K-means produces $k$ clusters, and therefore it divides the dataset in $k$ tasks. ![Some of the results of the clusterization. It is evident how all the images have lots of features in common.[]{data-label="fig:clusters"}](img/klusters.png){width="0.5\linewidth"} Hence, we expect to find, in each of these clusters, images which are similar to each other, accordingly to their features. For example, a cluster could contain images with grass, another one images with pets and so on and so forth. A visual proof of this assumption is showed in Fig. \[fig:clusters\]. cGAN ---- As generator architecture, we choose the U-net [@ronneberger2015u] which is one of the most common for this type of task and we built the discriminator following the classic DCGAN architecture [@radford2015unsupervised], i.e., having each modules composed by Convolutions, Batch Normalization and ReLU layers. Lab is the color space used in this work, because is the one that best approximate human vision and therefore the generator takes as input a grayscale image $x_i$ (the $L$ channel) and outputs the $ab$ channels. Then, we concatenate input and outputs and obtain the final results. We use L1 loss to model the low-frequencies of our output images and adversarial loss to model the high-frequencies in a similar way of the pix2pix architecture proposed by Isola *et al.* [@isola2017image]. Therefore, our objective function became: $$\label{eqn:loss} \mathcal{L} = \textbf{w}_\textbf{adv}\mathcal{L}_\textbf{adv} + \textbf{w}_\textbf{L1}\mathcal{L}_\textbf{L1}$$ where $\textbf{w}_\textbf{adv}$ and $\textbf{w}_\textbf{L1}$ are weights assigned to the different losses, because we want L1 loss to be more effective than adversarial loss during training. Meta-learning ------------- As previously briefly mentioned, we approached the generator training with a Reptile meta-learner. This means that, once a task had been chosen, for a fixed number of meta-iterations, the task is sampled and the gradient of the generator loss function is evaluated to perform a SGD step of optimization. Fixed the initial generator parameters as $\theta_G$, the inner-loop training defines a sequence $\left(\tilde{\theta}_G^{(j)}\right)_{j = 0}^{N_{\mathrm{meta-iter}}}$, where $\tilde{\theta}_G^{(0)} = \theta_G$. Hence it updates the $\tilde{\theta}_G^{(j)}$ parameters in the direction of the task. Once the inner-loop is completed, the parameter are re-aligned with the Reptile rule: $$\theta_G \gets \theta_G + \lambda_{ML}\left(\tilde{\theta}^{(N_{\mathrm{meta-iter}})}_G - \theta_G\right)$$ where $\lambda_{ML}$ is the stepsize hyperparameter of Reptile. Complete architecture of the system ----------------------------------- The *MetalGAN* training process is detailed in Algorithm \[metal\_alg\]. $K(q_i) \gets$ retrieve\_clusters($q_i$) $\tau(q_i) \gets$ get\_task\_from\_cluster($K(q_i)$) sample $\langle \mathrm{input,target} \rangle$ from task $\tau(q_i)$ $\varepsilon_{\mathrm{GAN}} \gets \nabla_{\theta_D}\mathcal{L}_\textbf{adv}$(D(G(input)), label\_real) $\varepsilon_{\mathrm{L1}} \gets \nabla_{\theta_G}\mathcal{L}_\textbf{L1}$(D(G(input)), target) $\varepsilon_G \gets \textbf{w}_\textbf{adv}\varepsilon_{\mathrm{GAN}} + \textbf{w}_\textbf{L1}\varepsilon_{\mathrm{L1}}$ $\tilde{\theta}^{(j)}_G \gets \tilde{\theta}^{(j-1)}_G -\lambda_G\varepsilon_G $ $\theta_G \gets \theta_G + \lambda_{ML}\left(\tilde{\theta}^{(N_{\mathrm{meta-iter}})}_G - \theta_G\right)$ $\varepsilon_{D_{\mathrm{real}}} \gets \nabla_{\theta_D}\mathcal{L}_\textbf{adv}$(D(target), label\_real) $\varepsilon_{D_{\mathrm{fake}}} \gets \nabla_{\theta_D}\mathcal{L}_\textbf{L1}$(D(G(input)), label\_fake) $\varepsilon_D \gets \varepsilon_{D_{\mathrm{real}}} + \varepsilon_{D_{\mathrm{fake}}} $ $\theta_D \gets \theta_D - \lambda_D \varepsilon_D$ \[metal\_alg\] The algorithm is parameterized by the number of epochs $N_{\mathrm{epochs}}$, the number of meta-iterations $N_{\mathrm{meta-iter}}$, the generator and discriminator learning rates $\lambda_G$ and $\lambda_D$, the Reptile stepsize parameter $\lambda_{ML}$, and the loss weights $\textbf{w}_\textbf{adv}$ and $\textbf{w}_\textbf{L1}$. During training, we randomly select a query set $Q = \{q_0,\dots,q_z\}$. Each query $q_i$ corresponds to a single cluster $K(q_i)$. It is worth noting that two queries could point to the same cluster. Having this set, we are able to pick $z$ different images at each epoch by sampling the task $\tau(q_i)$ and to update the generator $G$ as showed in Fig. \[fig:meta\_net\]. ![The MetalGAN architecture: the query $q_i$ points to a cluster $K(q_i)$ that is used as a task to train the Generator $G$ with Reptile.[]{data-label="fig:meta_net"}](img/meta_net.pdf){width="0.8\linewidth"} The generator is updated by evaluating gradients of its loss functions (adversarial loss $\mathcal{L}_{\mathrm{adv}}$ and L1 loss $\mathcal{L}_{\mathrm{L1}}$), and by adding them to obtain the error $\varepsilon_G$. Then, the network parameters obtained in the inner-loop $\tilde{\theta}_G^{N_{\mathrm{meta-iter}}}$ are used to update the outer-loop generator parameters $\theta_G$. In the last step, all images of the task $\tau(q_i)$ are used to train the discriminator, calculating the gradients of the discriminator adversarial and L1 losses, and adding them to obtain the discriminator error $\varepsilon_D$. The discriminator parameters $\theta_D$ are updated consequently. Experimental Results {#sec:results} ==================== For our experiments we choose a slightly modified version of Mini-Imagenet [@ravi2016optimization]. Since our goal is not classification, we create our training and test set using only images from the 64 classes contained in the training section of Mini-Imagenet. The total number of images in the dataset is $38392$. We define two sets of experiments: the first one consists in training the cGAN without the use of Reptile and the second one introduces Reptile and the features clusterization. For both of them we set $\textbf{w}_\textbf{adv} = 1$ and $\textbf{w}_\textbf{L1} = 10^2$. Learning rates of both the generator and the discriminator were set to $\lambda_G = \lambda_D = 10^{-4}$. For K-means clusterization, the parameter $k$ was set to 64 in order to have clusters as much disjoint as possible. For Reptile, we use 100 *meta-iter*, and a stepsize $\lambda_{ML} = 10^{-3}$. The 10% of the dataset images are used as query images. The number of epochs was set to 200. All tests have been executed on a GPU Nvidia 1080 Ti. cGAN results ------------ ![Results obtained using the cGAN only. Each group of three images is composed of the input of the network (grayscale image), the ground truth, and the output of the network.[]{data-label="fig:soloGAN_res"}](img/soloGAN3.png){width="\linewidth"} In Fig. \[fig:soloGAN\_res\] are reported some results produced after the training of the cGAN without the clusterization and without Reptile, i.e., with a standard adversarial algorithm. The training data at disposal are very scarce ($\sim$38k images compared to 1.3M of the whole Imagenet dataset) and, for this reason, the network is not able to produce compelling results. In particular, the network often fails to understand the difference between foreground and background objects and therefore it applies the colors without following edges and borders. In general, for the cGAN is very difficult to propagate the color correctly and is more common the tendency to apply uneven patches of color. Finally, due to the scarcity of data, the network cannot generalize in an acceptable way and hence the colors in the outputs are not sharp, but, on the contrary, the produced results are very blurry and often colors are applied almost randomly. MetalGAN results ---------------- Results of MetalGAN are showed in Fig. \[fig:metalGAN\_res\]. It is immediately evident how Reptile improves the results of the cGAN. In particular, colors are sharper and more bright. The reason is that Reptile tunes the generator on each cluster and therefore allows the network to focus more on the more predominant colors present in each task and, as a consequence, even with few examples the produced results are compelling and plausible. For example, in a task with lots of images containing grass or plants there will be an abundance of different shades of green and thus the network will learn very quickly to reproduce similar colors over the test set. On the contrary, an image that is very different from the majority of images in the rest of its task could be colorized poorly. This problem, however, is not very frequent since the difference has to be very large in order to produce nasty results. Other examples can be found at [implab.ce.unipr.it/?page\_id=1011](implab.ce.unipr.it/?page_id=1011). ![Results of MetalGAN. Each of three images consists of the grayscale input given to the network, the ground truth, and the output of the network. The four represented images belong to different clusters.[]{data-label="fig:metalGAN_res"}](img/MetalGAN3.png){width="\linewidth"} Quantitative evaluation ----------------------- In order to evaluate the quality of the generated samples, we used the Inception Score [@inceptionscore2016], because it is a very good metric to simulate human judgement. We calculated the Inception Score of generated images using both cGAN and MetalGAN (see Table \[tab:res\]). The score also measures the diversity of the generated images, so a high score is better than a lower one. The MetalGAN approach significantly improves standard cGAN score. ----------------------------- -- -- **Dataset & **Mean & **Std\ cGAN & 3.20 & 0.83\ MetalGAN & 9.16 & 1.12\ ****** ----------------------------- -- -- : The Inception Scores are computed on generated images from the MiniImageNet dataset, mean and standard deviation are reported for both cGAN and MetalGAN results.[]{data-label="tab:res"} Conclusions =========== In normal adversarial generative settings, having few images at disposal during training produces a complete failure in the colorization. In this paper, we proposed a novel architecture which mix adversarial training with meta-learning techniques, called MetalGAN. As shown by experimental results, even with few images the network trained with MetalGAN was able to produce a well-looking colorization. The clusterization of the dataset and the use of clusters as tasks help at directing the colorization to the most probable suitable colors for the image, and meta-learning allows to train the network on few examples. As future developments, we plan to include the discriminator in the meta-learning training phase, and to test the method on other small datasets in order to prove the generalization capability of the proposed MetalGAN architecture.

--- abstract: 'A set is effectively chosen in every class of $\fd02$ sets modulo countable.' author: - 'Vladimir Kanovei[^1]' title: 'Definable selector for $\fd02$ sets modulo countable' --- Let $\cnt$ be the equivalence relation of equality modulo countable, that is, $X\cnt Y$ iff the symmetric difference $X\sd Y$ is (at most) countable. Does there exist an , , an effective choice of an element in each equivalence class of sets of certain type? The answer depends on the type of sets considered. For instance, the question answers in the positive for the class of closed sets in Polish spaces by picking the only perfect set in each equivalence class of closed sets. On the other hand, effective selectors for $\cnt$ do not exist in the domain of $\Fs$ sets, , in the Solovay model (in which the axiom of choice AC holds and all ROD sets are LM and have the Baire property) by [@1 Theorem 5.5]. Our goal here is to prove that $\Fs$ is the best possible for such a negative result. There exists a definable selector for $\cnt$ in the domain of $\fd02$ sets in Polish spaces. [($\fd02$ = all sets simultaneously $\Fs$ and $\Gd$.)]{} We’ll make use of the following lemma. If $X$ is a countable $\Gd$ set in a Polish space then the [closure]{} $\clo X$ is countable. Therefore if $X\cnt Y$ are $\fd02$ sets then $\clo X\cnt\clo Y$. Otherwise $X$ is a countable dense $\Gd$ set in an uncountable Polish space $\clo X$, which is not possible. [Difference hierarchy.]{} It is known (see  [@kDST 22.E]) that every $\fd02$ set $A$ in a Polish space $\dX$ admits a representation in the form $A=\bigcup_{\et<\vt}(F_\et\bez H_\et)$, where $\vt<\omi$ and $F_0\qs H_0\qs F_1\qs H_1\qs\dots F_\et\qs H_\et\qs\dots$ is a decreasing sequence of closed sets in $\dX$, defined by induction so that $F_0=\dX$, $H_\et=\clo{F_\et\bez A}$, $F_{\et+1}=H_\et\cap \clo{F_\et\cap A}$, and the intersection on limit steps. The induction stops as soon as $F_\vt=\pu$. The key idea of the proof of Theorem \[mt\] is to show that if $A\cnt B$ are $\fd02$ sets then the corresponding sequences of closed sets $$\left. \bay{l} F_0^A\qs H^A_0\qs F^A_1\qs H^A_1\qs\dots F^A_\et\qs H^A_\et\qs\dots \\[1ex] F_0^B\qs H^B_0\qs F^B_1\qs H^B_1\qs\dots F^B_\et\qs H^B_\et\qs\dots \eay \right\} \quad (\et<\vt=\vt^A=\vt^B),\snos {A shorter sequence is extended to the longer one by empty sets if necessary.}$$ satisfying $A=\bigcup_{\et<\vt}(F^A_\et\bez H^A_\et)$ and $B=\bigcup_{\et<\vt}(F^B_\et\bez H^B_\et)$ as above, also satisfy $F^A_\et\cnt F^B_\et$  and  $H^A_\et\cnt H^B_\et$  —   for all $\et<\vt$. It follows that the perfect kernels $\pk{F^A_\et}$, $\pk{F^B_\et}$ coincide: $\pk{F^A_\et}=\pk{F^B_\et}$, and $\pk{H^A_\et}=\pk{H^B_\et}$ as well. Therefore the sets $\Phi(A)=\bigcup_{\et<\vt}(\pk{F^A_\et}\bez \pk{H^A_\et})$ and $\Phi(B)$ coincide (whenever $A\cnt B$ are $\fd02$ sets), and $A\cnt\Phi(A)$ holds for each $\fd02$ set $A$, so $\Phi$ is a selector required, ending the proof of the theorem. Thus it remains to prove \[\*\]. We argue by induction. We have $F^A_0=F^B_0=\dX$ (the underlying Polish space). Suppose that $F^A_\et\cnt F^B_\et$; prove that $H^A_\et\cnt H^B_\et$. By definition, we have $H^A_\et=\clo{F^A_\et\bez A}$ and $H^B_\et=\clo{F^B_\et\bez B}$, where $(F^A_\et\bez A)\cnt (F^B_\et\bez B)$ (recall that $A\cnt B$ is assumed), hence $H^A_\et\cnt H^B_\et$ holds by Lemma \[fdL\]. It’s pretty similar to show that if $F^A_\et\cnt F^B_\et$ (and then $H^A_\et\cnt H^B_\et$ by the above) then $F^A_{\et+1}\cnt F^B_{\et+1}$ holds. This accomplishes the step $\et\to\et+1$. Finally the limit step is rather obvious. Coming back to the mentioned result of [@1 Theorem 5.5], it is a challenging problem to prove that the equivalence relation $\cnt$ on $\Fs$ sets is not ROD-reducible to the equality of Bodel sets in the Solovay model. As established in [@kl], it is true in some models (including  Cohen and random extensions of $\rL$) that every OD and Borel set is OD-Borel (, has an OD Borel code). In such a model, there is an effective choice of a set and its Borel code, by an OD function, in every class of Borel sets containing an OD set. The author thanks Philipp Schlicht for useful comments. [10]{} V. Kanovei and V. Lyubetsky. 147 (2019), 1277-1282. A. Kechris. . Springer-Verlag, New York, 1995 S. M[ü]{}ller, P. Schlicht, D. Schrittesser, T. Weinert. . , 1811.06489 v4. [^1]: IITP RAS, Bolshoy Karetny, 19, b.1, Moscow 127051, Russia. Partial support of RFBR grant 17-01-00705 acknowledged. [kanovei@googlemail.com]{}.

--- abstract: 'We consider a model of *selective prediction*, where the prediction algorithm is given a data sequence in an online fashion and asked to predict a pre-specified statistic of the upcoming data points. The algorithm is allowed to choose when to make the prediction as well as the length of the prediction window, possibly depending on the observations so far. We prove that, even without *any* distributional assumption on the input data stream, a large family of statistics can be estimated to non-trivial accuracy. To give one concrete example, suppose that we are given access to an arbitrary binary sequence $x_1, \ldots, x_n$ of length $n$. Our goal is to accurately predict the average observation, and we are allowed to choose the window over which the prediction is made: for some $t < n$ and $m \le n - t$, after seeing $t$ observations we predict the average of $x_{t+1}, \ldots, x_{t+m}$. This particular problem was first studied in [@drucker2013high] and referred to as the “density prediction game”. We show that the expected squared error of our prediction can be bounded by $O(\frac{1}{\log n})$ and prove a matching lower bound, which resolves an open question raised in [@drucker2013high]. This result holds for any sequence (that is not adaptive to when the prediction is made, or the predicted value), and the expectation of the error is with respect to the randomness of the prediction algorithm. Our results apply to more general statistics of a sequence of observations, and we highlight several open directions for future work.' author: - | Mingda Qiao\ `mqiao@stanford.edu` - | Gregory Valiant\ `valiant@stanford.edu`[^1] bibliography: - 'main.bib' title: 'A Theory of Selective Prediction[^2]' --- [^1]: This work is supported by NSF awards CCF-1704417 and AF:1813049 and by ONR award N00014-18-1-2295. [^2]: This revised version replaces an older version in which we had missed the closely related work of [@drucker2013high].

--- abstract: 'One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independently of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe basic properties of mutually unbiased bases composed only of product states. The constructions are illustrated with explicit examples in low dimensions. We believe that properties of entanglement in mutually unbiased bases might be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.' address: | $^1$ Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria\ $^2$ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore\ $^3$ Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria author: - 'M Wieśniak$^{1}$[^1], T Paterek$^{2}$[^2], and A Zeilinger$^{1,3}$' title: Entanglement in mutually unbiased bases --- Introduction ============ Quantum complementarity forbids the simultaneous knowledge of almost all pairs of observables. This impossibility is drawn to the extreme in the case of observables described by operators whose eigenstates form mutually unbiased bases (MUBs). Two bases are said to be unbiased if any vector from one basis has an overlap with all vectors from the other basis that is equal in modulo. The definition for a bigger set of MUBs means that the unbiasedness property holds for all pairs of these bases. Accordingly, if we can perfectly predict a measurement result of one such observable corresponding to an eigenstate in one of the bases, then the results of all other observables corresponding to all other basis vectors of all other bases in the set remain completely uncertain. One typical example of a set of three MUBs is the eigenbases of spin-${\mbox{$\textstyle \frac{1}{2}$}}$ projections onto three orthogonal directions: a spin-${\mbox{$\textstyle \frac{1}{2}$}}$ state along one axis leaves us totally uncertain about the results along the orthogonal axes. A spin-${\mbox{$\textstyle \frac{1}{2}$}}$ particle is a two-level quantum system, a qubit, and clearly admits three MUBs. A $d$-level quantum system, a qudit with pure states described in $d$ dimensional Hilbert space, can have at most $d+1$ MUBs [@WF1989], and such a set is referred to as the complete set of MUBs. The first explicit construction of the complete sets of MUBs was presented by Ivanović for $d$ being a prime number [@IVANOVIC]. Subsequently, Wootters and Fields constructed the complete sets for prime-power $d$ [@WF1989]. Since then, many explicit constructions have been derived and they are collected in a recent review [@REVIEW]. If $d$ is not a prime power, the number of MUBs remains unknown although it is considered unlikely that a complete set of MUBs exists in these cases. For example, the works [@BH2007; @BW2008; @arX; @RLE2011] describe failed numerical attempts to find a complete set of MUBs in dimension 6. In addition to this fundamental question, MUBs find applications in quantum tomography [@WF1989], quantum cryptography [@BRUSS; @B-PP; @MOHAMED], the Mean King problem [@VAA1987; @AE2001; @ARAVIND2003; @HHH2005], and other tasks. Here we study the properties of entanglement between subsystems of a global system with a composite (i.e. nonprime) dimension as well as entanglement distribution among the states of MUBs. We show that the amount of entanglement, as measured as a function of the linear entropy of a subsystem, present in states of a complete set of MUBs of a composite dimension always must have a nonzero value that is independent of a chosen set. In other words, entanglement is always present in such a complete set of MUBs and it is always the same independent of the choice of the complete set, being solely a function of dimensions of subsystems. Moreover, for global dimensionality that is big enough, practically all MUBs of a complete set contain entanglement. We then show an experiment-friendly procedure that creates complete sets of MUBs in all dimensions $d=p^2$, which are squares of a prime number. This procedure uses only one entangling operation, which is repeatedly applied to states of product MUBs to give the complete set. Remarkably, the generated set consists of either product states or maximally entangled states. Finally, we discuss the properties of MUBs consisting of product states only. We believe that understanding entanglement in MUBs can lead on the practical side to novel applications and on the conceptual side to an understanding of why complete sets of MUBs can (not) exist for nonprime-power $d$. Conservation of entanglement ============================ Consider a bipartite system composed of subsystems $A$ and $B$, i.e. its global dimension is $d = d_A d_B$. Any (hypothetical) complete set of MUBs allows for efficient quantum tomography as it reveals complete information about an arbitrary quantum state of the system [@WF1989; @LKB2003]. Hence we intuitively expect that the average entanglement over all the states constituting the complete set of MUBs shall be fixed with respect to some measure, independent of the choice of the bases. This intuition is made rigorous in this section. The relevant measure of entanglement is a function of the linear entropy of a reduced density operator. The idea of the proof is to use the property of a complete set of MUBs called a complex projective $2$-design [@BARNUM2002; @KR2005], which here means that the entanglement averaged over a complete set of MUBs is the same as the entanglement averaged over all pure quantum states. The latter is constant due to known results in statistical mechanics [@LUBKIN1978]. The message of this section, namely that the amount of entanglement is the same independent of a choice of the complete set of MUBs, may be well-known to scientists working with designs, but our proof is elementary and has immediate consequences for the distribution of entanglement among the states of MUBs. Complete sets of mutually unbiased bases and designs ---------------------------------------------------- A complete set of MUBs is composed of $d+1$ bases, each basis of $d$ orthonormal vectors. We denote by ${\left | j_m \right\rangle}$ the $j$th state of the $m$th basis, where for convenience we enumerate the states and the bases as $j=0,\dots,d-1$ and $m=0,\dots,d$. To introduce the notion of a $2$-design, one studies polynomials $\mathcal{P}(i) \equiv \mathcal{P}(x_1,x_2,y_1^*,y_2^* | i)$, which are biquadratic in variables $x_1,x_2$ and separately in variables $y_1^*,y_2^*$, where $x_i,y_i$ are any coefficients of arbitrary state $|i\rangle$ with respect to a fixed (say, standard) basis and $^*$ denotes complex conjugation. Any complete set of MUBs is known to be a complex projective $2$-design [@BARNUM2002; @KR2005] because the average of any $\mathcal{P}(j_m)$ over states ${\left | j_m \right\rangle}$ is the same as the average with the Haar measure over all pure states: $$\langle \mathcal{P}(j_m) \rangle_{\mathrm{MUBs}} = \langle \mathcal{P}(i) \rangle_{\mathrm{Haar}}. \label{MUB-HAAR}$$ The conservation law -------------------- In order to utilize the design property of the complete set of MUBs in the studies of entanglement, we characterize the latter by the purity of a reduced density operator, say $\rho_{A|j_m} = \mathrm{Tr}_B({\left | j_m \right\rangle} {\left \langle j_m \right |})$: $$\mathcal{P}(j_m) \equiv \mathrm{Tr}(\rho_{A|j_m}^2).$$ This quantity acquires its minimum of $\frac{1}{d_A}$ for maximally entangled states and its maximum of unity for unentangled product states. By ‘maximally entangled states’ we mean pure states with maximal possible entropy for the smaller of the subsystems. Note that due to the properties of the Schmidt decomposition it does not matter which subsystem is taken into account. Moreover, the assumptions behind Eq. (\[MUB-HAAR\]) are fulfilled and, since per definition $\langle \mathcal{P}(j_m) \rangle_{\mathrm{MUBs}} = \frac{1}{d(d+1)} \sum_{m=0}^{d} \sum_{j=0}^{d-1} \mathrm{Tr}(\rho_{A|j_m}^2)$, using the design property we write $$\mathcal{E} \equiv \sum_{m=0}^{d} \sum_{j=0}^{d-1} \mathrm{Tr}(\rho_{A|j_m}^2) = d(d+1) \langle \mathrm{Tr}(\rho_{A|i}^2) \rangle_{\mathrm{Haar}}.$$ In the last step, we use the result by Lubkin [@LUBKIN1978], who studied how close the average reduced density operator is to a completely mixed state and found that $$\langle \mathrm{Tr}(\rho_{A|i}^2) \rangle_{\mathrm{Haar}} = \frac{d_A + d_B}{d+1}.$$ Therefore, the sum of entanglement over all the states of any complete set of MUBs is fixed and equal to $$\mathcal{E} = d_A d_B (d_A + d_B). \label{ENT_CONST}$$ Note that the right-hand side is symmetric with respect to $d_A$ and $d_B$, which reflects the fact that we can as well study subsystem $B$. Eq. (\[ENT\_CONST\]) has two immediate consequences. The first is that the distribution of entanglement among different states of a complete set of MUBs can be arbitrary as long as there is a proper amount of it. For example, Eq. (\[ENT\_CONST\]) allows a complete set of MUBs to be formed by product and maximally entangled states as well as solely by partially entangled states. The second conclusion is that we cannot have a complete set of MUBs built entirely of product states or entirely of maximally entangled states. Assume that $d_A \le d_B$. In a complete set of MUBs which contains $d_A+1$ product MUBs, all other bases contain only maximally entangled states. \[PROD-ENT\] Proof. The sum of $\mathcal{P}(j_m)$ over the states of product MUBs equals $d_A d_B(d_A + 1)$. The only possibility to obtain the value of (\[ENT\_CONST\]) is when for all the remaining $d_A^2 d_B (d_B-1)$ states, $\mathcal{P}(j_m)$ acquires its minimal value of $\frac{1}{d_A}$. $\Box$ Complete sets of mutually unbiased bases in prime-squared dimension =================================================================== We showed that the complete set of MUBs may be chosen as consisting of product bases and bases containing only maximally entangled states. Here we present a construction of the complete sets with this property in dimension $d = p^2$ , where $p$ is prime. The complete set will be generated from product MUBs with repeated application of a *single* entangling operation, in our case the control-phase operation. This makes our construction experiment-friendly. Explicit examples of MUBs generated by this method together with their factorization into product or maximally entangled bases are presented in the Appendices. Complete sets of mutually unbiased bases in prime dimensions ------------------------------------------------------------ Before we present the new construction, let us briefly recall some of the known ones to which we will refer later on. If $d = p$ is a prime number a complete set of $p+1$ MUBs was first found by Ivanović [@IVANOVIC]. It is convenient to enumerate the bases as $m=0,\dots,p$ with $m=p$ corresponding to a standard basis, i.e. the basis in which the vectors of all other MUBs will be expressed. To simplify the notation and if no confusion arises, we will write the vectors of the standard basis without any index, i.e. ${\left | s \right\rangle} \equiv {\left | s_p \right\rangle}$ enumerates the states of the standard basis. The other $p$ MUBs have the Fourier-Gauss structure, $${\left | j_m \right\rangle} = \frac{1}{\sqrt{p}} \sum_{s=0}^{p-1} \alpha_p^{j s + m s^2} {\left | s \right\rangle} \quad \textrm{for } \quad m=0,\dots,p-1, \quad \textrm{and } \quad p>2, \label{MUB_P}$$ where $\alpha_p = \exp(i 2 \pi / p)$ is the complex $p$th root of unity. The only exception to this formula is the case of $p=2$ where one needs to refer to an imaginary unit $i$, the fourth rather than the square root of unity. For low dimensions, we present these bases explicitly in the Appendices. In odd-prime dimensions, a standard basis and a single MUB are sufficient to generate the complete set of MUBs with an application of a single unitary: $$W = \textrm{diag}[1,\alpha_p,\alpha_p^4,\dots, \alpha_p^{(p-1)^2}], \label{W}$$ which has the standard basis as the eigenbasis and permutes all other MUBs, i.e. $W {\left | j_m \right\rangle} = {\left | j_{m+1} \right\rangle}$ with addition modulo $p$. Alternatively, one can construct complete sets of MUBs using Heisenberg-Weyl operators in prime dimensions, $$\begin{aligned} X=\sum_{s=0}^{p-1}|s+1\rangle\langle s|,\quad Z=\sum_{s=0}^{p-1} \alpha_p^s |s\rangle\langle s|,\end{aligned}$$ with addition inside the kets modulo $p$. These operators span a unitary operator basis with respect to the trace scalar product as $$\mathrm{Tr} \Big[ \left(X^aZ^b \right)^\dagger X^cZ^d \Big] = p\delta_{a,c}\delta_{b,d}.$$ According to the general result of Bandyopadhyay *et al.* [@BANDYOPADHYAY], if one can group elements of the unitary operator basis into disjoint subsets of $d$ commuting operators (unity being the only common element of these sets), the common eigenbases of the commuting operators within each set are mutually unbiased. In the case of a system of a prime dimension, the groups of commuting operators can be chosen as powers of the operators $Z$, $X$, $X Z$, $XZ^2$, $\dots$, $XZ^{p-1}$. Their eigenbases define a complete set of MUBs. It turns out that this set of MUBs is identical to the set of Eq. (\[MUB\_P\]) up to the indexing of bases and states within bases. Bases (\[MUB\_P\]) are the eigenbases of the operators $X$, $X Z$, $\dots$, $XZ^{p-1}$. \[L\_UNIT\_MUB\_P\] Proof. Choosing the standard basis as the eigenbasis of $Z$, the eigenbasis of $X$ is readily the Fourier basis, i.e. $\{{\left | j_0 \right\rangle}\}$. Next note that for $m=1,\dots,p-1$ we have ${\left | j_m \right\rangle} = \frac{1}{\sqrt{p}} \sum_{s=0}^{p-1} \alpha_p^{(j+m) s - 2m \xi_s} {\left | s \right\rangle}$ with $\xi_s = s + \dots + (p-1) = \frac{1}{2}(p-s)(p+s-1)$. The proof that these are exactly the eigenstates of $X Z^{2m}$ is given in Ref. [@BANDYOPADHYAY]. Since $p$ is prime, $2m$ runs through all the powers of $Z$. $\Box$ These two methods of generating complete sets of MUBs in prime dimensions can be generalized to prime-power dimensions. However, these generalizations require a knowledge of elements of finite fields theory; see e.g. [@REVIEW]. We now present our physically motivated construction of the complete set of MUBs in prime-squared dimensions $d=p^2$. Two qubits ---------- We begin with a statement relating the number of MUBs to the possibility of swapping the states of subsystems. The statement itself holds for arbitrary dimension $d=p^2$, but it can be directly used to produce a complete set of MUBs of only two qubits. Assume $d=p^2$ and there exists unitary $U$ that commutes with the swap operation $S$, and such that vectors $\{ U {\left | a_k b_l \right\rangle} \}$, with $k \neq l$, form an MUB with respect to all product symmetric MUBs defined as $\{{\left | a_m b_m \right\rangle}\}$. Then $\{ U {\left | a_l b_k \right\rangle} \}$ is MUB with respect to all the bases mentioned above. \[L\_SWAP\] Proof. The commutativity of $U$ and $S$ and the Hermiticity of $S$ imply $U = S U S$. The assumed unbiasedness is expressed as $|\langle a_m b_m | U | a_k' b_l' \rangle|^2 = \frac{1}{p^2}$ for all bases $m=0,...,p$ and all vectors ${\left | a_m b_m \right\rangle}$ and ${\left | a_k' b_l' \right\rangle}$. The computation of the overlap $$| \langle a_k b_l | U^{\dagger} U | a_l' b_k' \rangle|^2 = | \langle a_k | a_l' \rangle \langle b_l | b_k' \rangle|^2 = \frac{1}{p^2},$$ reveals that the basis from the thesis is unbiased to $\{U {\left | a_k b_l \right\rangle}\}$. The commutativity with the swap operation is used to prove its unbiasedness with respect to all product bases: $$|\langle a_m b_m | U {\left | a_l' b_k' \right\rangle} |^2 = | \langle a_m b_m | S U S | a_l' b_k' \rangle|^2 = | \langle \alpha_m \beta_m | U | \alpha_k' \beta_l' \rangle|^2 = \frac{1}{p^2},$$ where the last equality follows from the assumed unbiasedness and we put $\alpha = b, \beta = a, \alpha' = b',\beta'=a'$. $\Box$ Note that the two bases $\{U {\left | a_k b_l \right\rangle}\}$ and $\{U {\left | a_l b_k \right\rangle}\}$ are simply related by the swap operation because $U {\left | a_l b_k \right\rangle} = S U S {\left | a_l b_k \right\rangle} = S U {\left | \alpha_k \beta_l \right\rangle}$ with $\alpha = b$ and $\beta = a$. In case of $d=4$, this lemma allows us to generate the complete set of MUBs starting with product MUBs. There are three MUBs in dimension $2$ and therefore we begin with the following three product MUBs in dimension $4$: $\{{\left | a_0 b_0 \right\rangle}\}$, $\{{\left | a_1 b_1 \right\rangle}\}$ and $\{{\left | a_2 b_2 \right\rangle}\}$. Consider now application of the control-phase (control-$Z$) operation $$\mathcal{P}_2 = \frac{1}{2}(I \otimes I + I \otimes \sigma_z + \sigma_z \otimes I - \sigma_z \otimes \sigma_z), \label{CPHASE2}$$ where $I$ denotes a single qubit identity operator and $\sigma_z=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$. We apply the control-phase operation on the two qubits prepared in states of the form ${\left | a_0 b_1 \right\rangle}$. The effect is best explained using Pauli operators. For a single qubit, we choose, in accordance with Appendix A, the basis $m=0$ as the eigenbasis of $\sigma_x=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$ and basis $m=1$ as the eigenbasis of $\sigma_y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right)$. Therefore, the basis $\{{\left | a_0 b_1 \right\rangle}\}$ is the eigenbasis of the commuting operators $\sigma_x \otimes I$ and $I \otimes \sigma_y$, and their products. The control-phase operation maps these operators onto $$\begin{aligned} \mathcal{P}_2 (\sigma_x \otimes I) \mathcal{P}_2 & = & \sigma_x \otimes \sigma_z, \\ \mathcal{P}_2 (I \otimes \sigma_y) \mathcal{P}_2 & = & \sigma_z \otimes \sigma_y.\end{aligned}$$ The common eigenstates of these new operators are maximally entangled Bell states. Moreover, such Bell basis is mutually unbiased with respect to all our product MUBs. This can be verified directly or by using, e.g., the result of Bandyopadhyay *et al.* [@BANDYOPADHYAY]. We apply this theorem to tensor products of Pauli operators, eigenbases of which define our MUBs, i.e. the three product MUBs $\{{\left | a_m b_m \right\rangle}\}$ are defined by sets of commuting operators $\{I \otimes I,\! \sigma_x \otimes I,\! I\otimes\sigma_x,\! \sigma_x\otimes\sigma_x\}$, $\{I \otimes I,\! \sigma_y\otimes I,\! I\otimes\sigma_y,\! \sigma_y\otimes\sigma_y\}$ and $\{I \otimes I,\! \sigma_z\otimes I,\! I\otimes\sigma_z,\!\sigma_z\otimes\sigma_z\}$, respectively, whereas the Bell basis $\mathcal{P}_2 {\left | a_0 b_1 \right\rangle}$ is defined by $\{I \otimes I,\! \sigma_x \otimes \sigma_z, \! \sigma_z \otimes\sigma_y,\! \sigma_y \otimes\sigma_x\}$. Each set of four is clearly a set of commuting operators, and according to the mentioned theorem their eigenbases form MUBs. Since the $\mathcal{P}_2$ operation is manifestly invariant under a swap of qubits, according to Lemma \[L\_SWAP\] we obtain the following complete set of MUBs: $\{{\left | a_0 b_0 \right\rangle}\}, \{{\left | a_1 b_1 \right\rangle}\}, \{{\left | a_2 b_2 \right\rangle}\}, \{\mathcal{P}_2 {\left | a_0 b_1 \right\rangle}\}, \{\mathcal{P}_2 {\left | a_1 b_0 \right\rangle}\}$, which is explicitly presented in Appendix C. Note that for this dimension application of Lemma \[L\_SWAP\] has the same effect as the result of Ref. [@IFDTHEND1] stating for general dimension that if there is a set of $d$ MUBs, then there also exists a set of $d+1$ of them. Two qupits ---------- Now we move to a system of a global dimension $d=p^2$ with $p>2$. Two systems, each of prime dimension $p$, admit altogether $p^2+1$ MUBs. We shall show that they all can be generated via the multiple application of a single entangling operation on product bases. For this purpose, we present a lemma which reduces the number of unbiasedness conditions one needs to check. For $p>2$ assume there exists unitary $U$ such that $|{\left \langle a_m b_m \right |} U^n {\left | a_{0}' b_n' \right\rangle}|^2 = \frac{1}{p^2}$ for all $0\leq a,b,a',b'\leq p-1$, $n=1,...,p-1$ and $m=0,...,p$ and that $[U,W \otimes I] = [U,I \otimes W] = 0$, where $W$ is defined in Eq. (\[W\]). Then the bases $\{U^\nu {\left | a_{\mu} b_{\mu+\nu} \right\rangle}\}$, with $\mu,\nu=0,...,p-1$, together with the standard basis $\{{\left | a_p b_p \right\rangle}\}$ form a complete set of MUBs. Addition of indices is modulo $p$. \[L\_TO\_C\_PHASE\] Proof. Consider an overlap between states of two bases of the proclaimed form $$\mathcal{M} \equiv |{\left \langle a_{\mu} b _{\mu + \nu} \right |} (U^{\nu})^{\dagger} U^{\nu'} {\left | a_{\mu'}' b_{\mu'+\nu'}' \right\rangle} |^2 = |{\left \langle a_{\mu } b_{\mu + \nu} \right |} U^{\nu' - \nu} {\left | a_{\mu'}' b_{\mu'+\nu'}' \right\rangle} |^2. \label{OVERLAP_2P}$$ Since $U$ commutes with individual cycling unitary $W \otimes I$ and $I \otimes W$, it also commutes with their products. In particular, we have $U^{\nu' - \nu} =(W^{\mu'}\otimes W^{\mu'+\nu}) U^{\nu' - \nu} (W^{-\mu'} \otimes W^{-mu'-\nu})$. We insert this expression into (\[OVERLAP\_2P\]) and since none of the bases there is the standard basis, the effect is to shift the indices of the local bases and get $$\mathcal{M} = |{\left \langle a_{\mu-\mu'} b_{\mu -\mu'} \right |} U^{\nu' - \nu} {\left | a_0' b_{\nu'-\nu}' \right\rangle} |^2 = \frac{1}{p^2},$$ where the last equality follows from our assumptions. Similarly, overlap with the standard basis equals $|{\left \langle a_{p} b_{p} \right |} U^{\nu} {\left | a_{\mu}' b_{\mu+\nu}' \right\rangle} |^2 = \frac{1}{p^2}$ which follows from our assumptions after noting that the standard basis is not shifted by $W$, whereas the index of the other local bases we shift by $-\mu$. $\Box$ Now we prove that the control-phase operation can be used to generate a complete set of MUBs in all prime-squared dimensions. The control-phase reads $$\label{controlphase} \mathcal{P}_p= \frac{1}{p} \sum_{a,b=0}^{p-1} \alpha_p^{- ab} Z^a \otimes Z^b.$$ In every dimension $d=p^2$ with $p>2$, there exists an integer $\theta$ such that $\mathcal{P}_p^{\theta}$ satisfies requirements of Lemma \[L\_TO\_C\_PHASE\]. Proof. First note that since both $\mathcal{P}_p$ and $W$ are diagonal in the standard basis, $[\mathcal{P}_p^{\theta},W\otimes I]=[\mathcal{P}_p^{\theta},I\otimes W]=0$ is fullfilled for any $\theta$. To prove that the bases $\{ \mathcal{P}_p^{\theta n} {\left | a_0 b_n \right\rangle} \}$ are unbiased to bases $\{ {\left | a_m b_m \right\rangle} \}$, we refer once more to the results of Bandyopadhyay [*et al.*]{} [@BANDYOPADHYAY]. They show that MUBs in prime dimensions $\{ {\left | j_{n} \right\rangle} \}$ may be chosen as eigenstates of sets of commuting operators $X^{\beta} Z^{2 \beta n}$ with $\beta = 0,\dots,p-1$ (see also Lemma \[L\_UNIT\_MUB\_P\]). The idea of the present proof is to show that operators defining bases $\{{\left | a_0 b_n \right\rangle}\}$ are transformed under the application of the control-phase into a new set of distinct operators which are all different from the operators defining bases $\{{\left | a_m b_m \right\rangle}\}$. Since commutation relations are preserved under unitary transformations, the results of [@BANDYOPADHYAY] imply that the new operators define MUBs with respect to $\{{\left | a_m b_m \right\rangle}\}$. The control-phase acts symmetrically on both subsystems and we have up to a global phase: $$\mathcal{P}_p^{\theta n} \left( X^{\alpha} \otimes X^\beta Z^{2 \beta n} \right) \mathcal{P}_p^{-\theta n} = X^{\alpha} Z^{\beta \theta n} \otimes X^\beta Z^{2 \beta n + \alpha \theta n }. \label{TRANSFORMED}$$ Since for different values of $\alpha$ and $\beta$ the initial operators $X^{\alpha} \otimes X^\beta Z^{2 \beta n}$ were orthogonal with respect to the trace scalar product, the final operators are also orthogonal, i.e. we generated a set of trace-orthogonal operators which can be partitioned into proper commuting subsets. We now have to ensure that the generated set does not contain any operators determining product MUBs $\{ {\left | a_m b_m \right\rangle} \}$. Since in the product MUBs the bases of $A$ and $B$ are the same, their defining feature is that operators determining the basis of $A$ commute with the operators determining the basis of $B$. We check whether this commutation condition is satisfied by the operators on the right-hand side of (\[TRANSFORMED\]). The operators of $A$, i.e. $X^{\alpha} Z^{\beta \theta n}$, commute with the operators of $B$, i.e. $X^\beta Z^{2 \beta n + \alpha \theta n}$, if and only if [@BANDYOPADHYAY]: $$n (\alpha^2 \theta + 2 \alpha \beta - \beta^2 \theta) = 0 \textrm{ mod } p.$$ We are interested only in positive $n$ and therefore ask whether the bracket is a multiple of a prime $p$. In other words, we are looking for the solution of the quadratic equation in the prime field $\mathcal{F}_p$. It is well known that for $p>2$ such equations have solutions if and only if there exists a field element $$\Delta = 2 \beta \sqrt{1+\theta^2}.$$ Therefore, we need to choose such a value of $\theta$ that $1+\theta^2$ has no square root in the prime field $\mathcal{F}_p$. Since for any element $x$ in the field $\mathcal{F}_p$ we have $x^2=(p-x)^2$, we have no more than $\frac{1+p}{2}$ elements with square roots. That is, there exist an element $x$ having a square root such that the next element, $1+x$, does not have a square root. Hence there always exists $\theta$ such that $\sqrt{1+ \theta^2}\notin \mathcal{F}_d$. $\Box$ Generally speaking, there is no universal choice of $\theta$ that is independent of $p$. We have neither found a function that for any given $p$ returns $\theta$ such that $1+\theta^2$ does not have a square root in the field, and our construction generates the complete set of MUBs. A good guess of a useful value of $\theta$ is often $1$. Out of the first $1000$ odd prime numbers, the construction with $\theta=1$ fails in $494$ cases, while out of the first $10000$ odd primes it fails in $4988$ cases. The lowest numbers for which this value does not produce the complete set of MUBs are $7$, $17$ and $23$. Three qubits ------------ A similar construction using multiple application of only one entangling operation does not seem to exist for more than two subsystems of prime dimensionality. However, more operations can be used for the task. Here we show that three entangling gates can be used to produce a complete set of nine MUBs for three qubits. We start with the global standard basis $\{{\left | abc \right\rangle}\}$ and eight other bases that do not involve any local standard basis, i.e. ${\left | a_k b_l c_m \right\rangle}$ with $k,l,m=0,1$. We next apply to the basis $\{{\left | a_k b_l c_m \right\rangle}\}$ operation $$\begin{aligned} &\mathcal{G}_{klm} = \frac{1}{2}\left( I \otimes I \otimes I + Z^k \otimes Z^l \otimes Z^m + Z^{1-k} \otimes Z^{1-l} \otimes Z^{1-m} - Z \otimes Z \otimes Z\right).&\nonumber\\\end{aligned}$$ The resulting complete set of MUBs is given in Appendix F. Wocjan-Beth construction ------------------------ We would also like to mention the Wocjan-Beth construction [@WocjanBeth], which is so far the only known construction that gives more MUBs in composite dimensions than there are for the smallest prime-power subsystem. The construction is designed for systems divisible into two identical subsystems. The method utilizes two kinds of vectors. The first kind is the so-called incident vectors, $V$. Exactly $d$ of their $d^2$ entries are equal to $1$; the rest is $0$. The task is to find families of $d$ such vectors that satisfy the following requirements: within each family every pair of vectors is orthogonal, and two vectors from two different families have the scalar product equal to one. For example, for $d=2$ there are only three families of incident vectors, $$\begin{aligned} \left\{\left(\begin{array}{c}1\\1\\0\\0\end{array}\right),\left(\begin{array}{c}0\\0\\1\\1\end{array}\right)\right\},& \left\{\left(\begin{array}{c}1\\0\\1\\0\end{array}\right),\left(\begin{array}{c}0\\1\\0\\1\end{array}\right)\right\},& \left\{\left(\begin{array}{c}1\\0\\0\\1\end{array}\right),\left(\begin{array}{c}0\\1\\1\\0\end{array}\right)\right\}.\end{aligned}$$ There is a one-to-one correspondence between families of incident vectors and mutually orthogonal Latin squares of order $d$. The other type of vectors is phase vectors, $h$, which have $d$ complex entries, each of modulo 1. The two kinds are combined through operation “$\uparrow$”. $h\uparrow V$ shall be understood as $V$ with the first non-zero element multiplied by the first entry of $h$, the next by the second, etc. One needs $d$ orthogonal vectors $h$ and combines every phase vector with the incident vector using $\uparrow$. After normalization we get as many MUBs as the number of incident vector families we found. We would like to mention that when we choose vectors $h$ proportional to the rows of the Fourier matrix and the first two incident families in the most natural way (similarly to the example), two bases generated in this way possess a product structure whereas all others are maximally entangled. The present work suggests that it might be possible to extend this set with the ‘missing’ product bases, which would make the Wocjan-Beth construction even more powerful. Product mutually unbiased bases =============================== Our last topic is limitations on the number of MUBs and their entanglement, which follow from the fact that some bases are formed by product states. We call such bases product MUBs. First we present a straightforward bound on the maximal number of product MUBs; next we discuss classes of product MUBs to show that in every dimension one has two product MUBs such that there is no other product MUB with respect to them. There could still be entangled MUBs and we give an example in which this entanglement does not help us to build a complete set of MUBs. Maximal number -------------- We begin by showing that the only way to construct product MUBs in composite dimension $d_A d_B$ is to build them from MUBs in dimensions $d_A$ and $d_B$ separately. Two product bases $\{{\left | ab \right\rangle}\}$ and $\{{\left | a' b' \right\rangle}\}$ in dimension $d_A d_B$ are mutually unbiased if and only if ${\left | a \right\rangle}$ is mutually unbiased to ${\left | a' \right\rangle}$ in dimension $d_A$ and ${\left | b \right\rangle}$ is mutually unbiased to ${\left | b' \right\rangle}$ in dimension $d_B$. \[L\_PROD\_MUB\] Proof. If local bases are mutually unbiased, then clearly their product bases are also mutually unbiased. Conversely, assume the product bases are MUBs, i.e. $|\langle a | a' \rangle |^2 |\langle b | b' \rangle |^2 = \frac{1}{d_A d_B}$ for all $a,a',b,b'$. Since the right-hand side is positive, neither of the scalar products of the left-hand side is zero. In particular, this implies that keeping $a,a',b$ fixed we have for all values of $b'$ that $|\langle b | b' \rangle |^2 = 1/ d_A d_B |\langle a | a' \rangle |^2$. Since the squared moduli are the quantum probabilities they sum up to $\sum_{b'} |\langle b | b' \rangle |^2 = 1$, which implies that $|\langle a | a' \rangle |^2 = \frac{1}{d_A}$ and hence $|\langle b | b' \rangle |^2 = \frac{1}{d_B}$. $\Box$ The maximal number of product MUBs follows as a corollary. In a general dimension $d = d_1 \dots d_n$ there are at most $\min_j \mathcal{M}_{j}$ product MUBs, where $\mathcal{M}_{j}$ is the maximal number of MUBs in dimension $d_j$. Note that the maximal number of product MUBs is also a corollary to Lemma \[PROD-ENT\]. Direct and indirect bases ------------------------- Not every set of product bases can be of the cardinality described below Lemma \[L\_PROD\_MUB\]. The crucial distinction between the product bases is whether their states can be distinguished with (i) local measurements only or with (ii) additional classical communication [@PATER_PLA]. The bases (i) are of form $\{{\left | a \right\rangle}{\left | b \right\rangle}\}$ having all states ${\left | a \right\rangle}$ orthogonal in the first subspace and all states ${\left | b \right\rangle}$ orthogonal in the second subspace. We shall call them *direct* product bases because the matrix $P_{\mathrm{direct}}$ having vectors ${\left | a \right\rangle}{\left | b \right\rangle}$ as columns is a tensor product of matrices $A$ and $B$ having as columns vectors ${\left | a \right\rangle}$ and ${\left | b \right\rangle}$, respectively: $$P_{\mathrm{direct}} = A \otimes B.$$ An example of a direct product basis is the tensor product of standard bases. Product bases (ii) can be written as ${\left | a \right\rangle}{\left | b(a) \right\rangle}$, i.e. for every fixed vector ${\left | a \right\rangle}$ orthogonality of the product basis requires states of the second subsystem ${\left | b(a) \right\rangle}$ to be orthogonal, but importantly for different states ${\left | a \right\rangle}$ the orthogonal bases of the second subsystem may be different. The measurement in such product basis requires classical communication: after measuring the first subsystem the result needs to be fed-forward to a device measuring second subsystem in order to adapt its setting to a suitable basis. We shall call such bases *indirect* product bases. In matrix notation, the matrix of an indirect product basis $P_{\mathrm{indirect}}$ cannot be written as a tensor product of matrices of local bases, but rather is of the form $$P_{\mathrm{indirect}} = \sum_{a} {\left | a \right\rangle} {\left \langle a \right |} \otimes B(a),$$ where the columns of matrix $B(a)$ are vectors ${\left | b(a) \right\rangle}$. An example of an indirect product basis of two qubits is $$\begin{aligned} {\left | 0_0 \right\rangle} {\left | 0_0 \right\rangle}, & \qquad & {\left | 1_0 \right\rangle} {\left | 0_1 \right\rangle}, \nonumber \\ {\left | 0_0 \right\rangle} {\left | 1_0 \right\rangle}, & \qquad & {\left | 1_0 \right\rangle} {\left | 1_1 \right\rangle}. \label{FOURIER4}\end{aligned}$$ Blocking product mutually unbiased bases ---------------------------------------- A set of product MUBs is *blocked* if there exists no other mutually unbiased product basis with respect to this set. Indirect product bases lead to the minimal blocked set of product MUBs, and they have consequences for completeness of sets containing them. In every composite dimension, there is a blocked set of two product MUBs. \[L\_UNEXT\] Proof. The first basis is a standard basis: direct product basis. The second basis is an indirect product basis exhausting all possible MUBs for at least one subsystem. We order the subsystems such that $d_1 \ge d_2 \ge \dots \ge d_n$. In dimension $d_2$, there are at most $d_2$ MUBs with respect to the local standard basis. Since $d_1 \ge d_2$, for every orthogonal vector in dimension $d_1$ one can have different orthogonal bases in dimension $d_2$ which exhaust the whole set of local MUBs. According to Lemma \[L\_PROD\_MUB\] there is no other product MUB. $\Box$ A compact explicit example of an indirect product basis that together with the standard basis forms blocking product MUBs can be given in dimension being a power of a prime $d = p^r$, which is regarded as the dimension of a Hilbert space of a set of $r$ elementary $p$-level systems. For every subsystem there are exactly $p$ MUBs with respect to the local standard basis and so is the number of distinguishable local states. Therefore, the basis $\{{\left | (j_1)_0 \right\rangle} {\left | (j_2)_{j_1} \right\rangle} {\left | (j_3)_{j_1} \right\rangle} \dots {\left | (j_r)_{j_{r-1}} \right\rangle}\}$ exhausts all allowed MUBs for all but the first subsystem. Here we denoted by ${\left | (j_n)_m \right\rangle}$ the state of the $n$th elementary subsystem in the $m$th MUB. The indirect product bases can block extendibility not only of a set of product MUBs but also of MUBs in general with no restriction to product bases. For example, in dimension $4$ the set of three MUBs composed of the standard basis, the indirect product basis of Eq. (\[FOURIER4\]) and the Fourier basis cannot be extended by any other MUB [@GRASSL_P; @MUB_OLS]. Conclusions =========== We were studying aspects of entanglement in states of MUBs in composite dimensions. Independently of the way a global system is split into subsystems, there is no complete set of MUBs that does not contain entanglement. In contrast, practically all MUBs are entangled as the dimension of at least one of the subsystems grows to infinity. The higher the dimension of the total system, the smaller the ratio of the number of product MUBs to the cardinality of the complete set of MUBs. This cardinality is proportional to the total dimension $d$, whereas the largest number of product MUBs is of the order of the smallest prime-power factor of $d$. Therefore, the ratio is the highest if $d = p^2$ is a square of a prime, and even in this case the cardinality of the complete set is a square of the cardinality of the product MUBs and the ratio vanishes in the limit $d \to \infty$. We showed that entanglement of states of *any* complete set of MUBs is fixed. This has consequences for the distribution of entanglement among the states of a complete set and might be a useful hint for a search of the complete sets or one of the ingredients to (dis)prove their existence. This conservation law holds true independent of a division into subsystems and therefore perhaps an argument could be found that there is a finite set of divisions under which the entanglement cannot simultaneously match the proper value. Another route to follow is to begin with a set of states with a proper amount of entanglement and apply local operations and classical communication in order to search for a complete set of MUBs. We also considered practical implementations of complete sets of MUBs and showed that for two subsystems, each with the same prime number of orthogonal states, the complete set can be generated via the multiple application of a single entangling operation on product states. The outcomes of this construction together with other examples of MUBs are explicitly presented in the Appendices for low dimensions (see also Ref. [@SMALL_MUB]). Turning to possible experiments, we note that there are various avenues for implementing quantum states in higher dimensions. For photons these include multiports and spatial-mode superpositions [@RECK; @WEIHS; @MULTIPORT; @OBRIEN; @LANGFORD; @OBRIEN_MULTIPORT] or Hermite-Gauss and Laguerre-Gauss modes, most notably orbital angular momentum states [@OAM_NATURE; @PADGETT_OAM; @PADGETT_REVIEW; @NL_OAM; @NL_GROUP]. We acknowledge discussions with Markus Grassl and Huangjun Zhu. This research is supported by ERC Advanced Grant QIT4QAD, FWF SFB-grant F4007 of the Austrian Science Fund, and the National Research Foundation and Ministry of Education in Singapore. Appendix {#appendix .unnumbered} ======== We present here explicit examples of complete sets of MUBs and for composite dimensions we emphasize division into entangled and product states. The notation used is explained on the example of a qubit ($d=2$). d=2 === The symbol ${\left | j_m \right\rangle}$ denotes the $j$th vector of the $m$th MUB. The standard basis is either denoted with subscript $d$ or has no subscript at all: $$B_2 = \left(\begin{array}{cc} 1&0\\ 0&1 \end{array}\right) = \left\{\begin{array}{l} |0 \rangle \\ |1 \rangle \end{array}\right\} = \left\{\begin{array}{l} |0_2 \rangle \\ |1_2 \rangle \end{array}\right\}.$$ Note that when we write a basis as a matrix, we can freely permute columns, since it only changes the order of the vectors in the basis. $$\label{dim2} B_0 = \frac{1}{\sqrt{2}} \left(\begin{array}{ccc} 1&1\\ 1&-1 \end{array}\right) = \left\{\begin{array}{l} |0_0\rangle\\ |1_0\rangle \end{array}\right\}, \quad B_1 = \frac{1}{\sqrt{2}} \left(\begin{array}{ccc} 1&1\\ i & -i \end{array} \right) = \left\{\begin{array}{l} |0_1\rangle\\ |1_1\rangle \end{array}\right\}.$$ d=3 === $$\label{dim3} B_3 =\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right) = \left\{ \begin{array}{l}|0 \rangle\\|1 \rangle \\ |2 \rangle \end{array}\right\},$$ $$B_0=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1&1&1\\1&\alpha_3&\alpha_3^2\\1&\alpha_3^2&\alpha_3\end{array}\right) = \left\{ \begin{array}{l}|0_0\rangle\\|1_0\rangle \\ |2_0\rangle \end{array}\right\},$$ $$B_1= \frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1&1&1\\\alpha_3&\alpha_3^2&1\\\alpha_3&1&\alpha_3^2\end{array}\right) = \left\{\begin{array}{l}|0_1\rangle \\|1_1\rangle \\ |2_1\rangle \end{array}\right\},$$ $$B_2=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}1&1&1\\\alpha_3^2&1&\alpha_3\\\alpha_3^2&\alpha_3&1\end{array}\right) = \left\{\begin{array}{l}|0_2\rangle\\|1_2\rangle \\ |2_2\rangle \end{array}\right\},$$ where $\alpha_d = \exp{2 \pi / d}$ is the complex $d$th root of unity. d=4 === The bases of this Appendix present explicitly the result of construction described in section 3.2 of the main text. $$\label{dim4a} B_4 = \left(\begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{array} \right) = \left\{\begin{array}{c} |0\rangle\\ |1 \rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0\rangle\\ |1\rangle \end{array} \right\},$$ $$B_0 = \frac{1}{2} \left(\begin{array}{cccc} 1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{array} \right) = \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle \end{array} \right\},$$ $$B_1 = \frac{1}{2} \left( \begin{array}{cccc} 1&1&1&1\\ i&-i&i&-i\\ i&i&-i&-i\\ -1&1&1&-1 \end{array}\right) = \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle \end{array} \right\},$$ $$\begin{aligned} B_2 & = & \frac{1}{2} \left( \begin{array}{cccc} 1&1&1&1\\ i&-i&i&-i\\ 1&1&-1&-1\\ -i&i&i&-i \end{array}\right) = \frac{1}{\sqrt{2}} \left\{ \begin{array}{c} |0_1 \rangle |0_0 \rangle + i |1_1 \rangle |1_0 \rangle \\ |0_1 \rangle |0_0 \rangle - i |1_1 \rangle |1_0 \rangle \\ |1_1 \rangle |0_0 \rangle + i |0_1 \rangle |1_0 \rangle \\ |1_1 \rangle |0_0 \rangle - i |0_1 \rangle |1_0 \rangle \end{array}\right\} \\ & = & \mathcal{P}_2 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_3 &=& \frac{1}{2} \left( \begin{array}{cccc} 1&1&1&1\\ 1&-1&1&-1\\ i&i&-i&-i \\-i&i&i&-i \end{array} \right) = \frac{1}{\sqrt{2}} \left\{ \begin{array}{c} |0_0 \rangle |0_1 \rangle + i |1_0 \rangle |1_1 \rangle \\ |0_0 \rangle |1_1 \rangle + i |1_0 \rangle |0_1 \rangle \\ |0_0 \rangle |0_1 \rangle - i |1_0 \rangle |1_1 \rangle \\ |0_0 \rangle |1_1 \rangle - i |1_0 \rangle |0_1 \rangle \end{array}\right\} \\ & = & \mathcal{P}_2 \left[ \left\{\begin{array}{c} |0_1 \rangle\\ |1_1 \rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle \end{array} \right\} \right],\end{aligned}$$ where the kets refer to MUBs for the two-level system (Appendix A) and $\mathcal{P}_2$ is the control-phase operation between two qubits defined in Eq. (\[CPHASE2\]) of the main text. d=5 === $$\label{dim5} B_5 = \left(\begin{array}{ccccc} 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array}\right) = \left\{\begin{array}{c} |0\rangle\\ |1\rangle\\ |2\rangle\\ |3\rangle\\ |4\rangle \end{array}\right\},$$ $$\begin{aligned} B_0 &=& \frac{1}{\sqrt{5}} \left(\begin{array}{ccccc} 1&1&1&1&1\\ 1&\alpha_5&\alpha_5^2&\alpha_5^3&\alpha_5^4\\ 1&\alpha_5^2&\alpha_5^4&\alpha_5&\alpha_5^3\\ 1&\alpha_5^3&\alpha_5&\alpha_5^4&\alpha_5^2\\ 1&\alpha_5^4&\alpha_5^3&\alpha_5^2&\alpha_5 \end{array}\right) = \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle\\ |3_0\rangle\\ |4_0\rangle \end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_1 & = & \frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1&1&1&1&1\\ \alpha_5&\alpha_5^2&\alpha_5^3&\alpha_5^4&1\\ \alpha_5^4&\alpha_5&\alpha_5^3&1&\alpha_5^2\\ \alpha_5^4&\alpha_5^2&1&\alpha_5^3&\alpha_5\\ \alpha_5&1&\alpha_5^4&\alpha_5^3&\alpha_5^2 \end{array}\right) = \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle\\ |3_1\rangle\\ |4_1\rangle \end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_2 &=& \frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1&1&1&1&1\\ \alpha_5^2&\alpha_5^3&\alpha_5^4&1&\alpha_5\\ \alpha_5^3&1&\alpha_5^2&\alpha_5^4&\alpha_5\\ \alpha_5^3&\alpha_5&\alpha_5^4&\alpha_5^2&1\\ \alpha_5^2&\alpha_5&1&\alpha_5^4&\alpha_5^3 \end{array}\right) = \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle\\ |3_2\rangle\\ |4_2\rangle \end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_3 &=& \frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1&1&1&1&1\\ \alpha_5^3&\alpha_5^4&1&\alpha_5&\alpha_5^2\\ \alpha_5^2&\alpha_5^4&\alpha_5&\alpha_5^3&1\\ \alpha_5^2&1&\alpha_5^3&\alpha_5&\alpha_5^4\\ \alpha_5^3&\alpha_5^2&\alpha_5&1&\alpha_5^4 \end{array}\right) = \left\{\begin{array}{c} |0_3\rangle\\ |1_3\rangle\\ |2_3\rangle\\ |3_3\rangle\\ |4_3\rangle \end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_4&=&\frac{1}{\sqrt{5}}\left(\begin{array}{ccccc} 1&1&1&1&1\\ \alpha_5^4&1&\alpha_5&\alpha_5^2&\alpha_5^3\\ \alpha_5&\alpha_5^3&1&\alpha_5^2&\alpha_5^4\\ \alpha_5&\alpha_5^4&\alpha_5^2&1&\alpha_5^3\\ \alpha_5^4&\alpha_5^3&\alpha_5^2&\alpha_5&1 \end{array}\right) = \left\{\begin{array}{c} |0_4\rangle\\ |1_4\rangle\\ |2_4\rangle\\ |3_4\rangle\\ |4_4\rangle \end{array}\right\}.\end{aligned}$$ d=6 === In this dimension it is not known if there exist more than three MUBs. A possible choice of three is to take the products $$\label{dim6} B_6 = \left(\begin{array}{cccccc} 1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{array}\right) = \left\{\begin{array}{c} |0\rangle\\ |1\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0\rangle\\ |1\rangle\\ |2\rangle \end{array} \right\},$$ $$B_0 =\frac{1}{\sqrt{6}}\left(\begin{array}{cccccc} 1&1&1&1&1&1\\ 1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3\\ 1&1&1&-1&-1&-1\\ 1&\alpha_3&\alpha_3^2&-1&-\alpha_3&-\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&-1&-\alpha_3^2&-\alpha_3 \end{array}\right) = \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\},$$ $$B_1 = \frac{1}{\sqrt{6}}\left(\begin{array}{cccccc} 1&1&1&1&1&1\\ \alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2\\ i&i&i&-i&-i&-i\\ i \alpha_3&i \alpha_3^2&i&-i \alpha_3&-i \alpha_3^2&-i\\ i \alpha_3^2&i&i \alpha_3^2 &-i \alpha_3&-i&-i \alpha_3^2 \end{array}\right) = \left\{\begin{array}{l} |0_1\rangle\\ |1_1\rangle \end{array} \right\} \otimes \left\{\begin{array}{l} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\},$$ where the kets in two-dimensional vectors refer to qubit MUBs (Appendix A) and the kets in three-dimensional vectors refer to qutrit MUBs (Appendix B). d=8 === The bases of this Appendix present explicitly the result of construction described in section 3.4 of the main text. $$B_8 =\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&0&0&0&0&0&0&10\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&1&0&0&1&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1 \end{array}\right)=\left\{\begin{array}{c} |0 0 0\rangle\\ |0 0 1\rangle\\ |0 1 0\rangle\\ |0 1 1\rangle\\ |1 0 0\rangle\\ |1 0 1\rangle\\ |1 1 0\rangle\\ |1 1 1\rangle \end{array}\right\},$$ $$B_0=\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ i&-i&1&-i&i&-i&i&-i\\ i&i&-i&-i&i&i&-i&-i\\ -1&1&1&-1&-1&1&1&-1\\ i&i&i&i&-i&-i&-i&-i\\ -1&1&-1&1&1&-1&1&-1\\ -1&-1&1&1&1&1&-1&-1\\ -i&i&i&-i&i&-i&-i&i \end{array}\right)=\left\{\begin{array}{c} |0_00_00_0\rangle\\ |0_00_01_0\rangle\\ |0_01_00_0\rangle\\ |0_01_01_0\rangle\\ |1_00_00_0\rangle\\ |1_00_01_0\rangle\\ |1_01_00_0\rangle\\ |1_01_01_0\rangle \end{array}\right\},$$ $$\begin{aligned} B_1&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ i&i&-i&-i&i&i&-i&-i\\ -i&i&i&-i&-i&i&i&-i\\ i&i&i&i&-i&-i&-i&-i\\ i&-i&i&-i&-i&i&-i&i\\ 1&1&-1&-1&-1&-1&1&1\\ -1&1&1&-1&1&-1&-1&1 \end{array}\right) \\ & = & \frac{1}{\sqrt{2}}\left\{\begin{array}{c} |0_0 0 0_1\rangle+|1_01 1_1\rangle\\ |0_00 1_1\rangle+|1_01 0_1\rangle\\ |0_01 0_1\rangle+|1_00 1_1\rangle\\ |0_01 1_1\rangle+|1_00 0_1\rangle\\ |0_00 0_1\rangle-|1_01 1_1\rangle\\ |0_00 1_1\rangle-|1_01 0_1\rangle\\ |0_01 0_1\rangle-|1_00 1_1\rangle\\ |0_01 1_1\rangle-|1_00 0_1\rangle\end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_2&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ i&-i&i&-i&i&-i&i&-i\\ 1&1&-1&-1&1&1&-1&-1\\ -i&i&i&-i&-i&i&i&-i\\ i&i&i&i&-i&-i&-i&-i\\ 1&-1&1&-1&-1&1&-1&1\\ -i&-i&i&i&i&i&-i&-i\\ -1&1&1&-1&1&-1&-1&1 \end{array}\right) \\ & = & \frac{1}{\sqrt{2}}\left\{\begin{array}{c} |0 0_10_0\rangle+i|1 1_11_0\rangle\\ |0 0_11_0\rangle+i|1 1_10_0\rangle\\ |0 1_10_0\rangle+i|1 0_11_0\rangle\\ |0 1_11_0\rangle+i|1 0_10_0\rangle\\ |0 0_10_0\rangle-i|1 1_11_0\rangle\\ |0 0_11_0\rangle-i|1 1_10_0\rangle\\ |0 1_10_0\rangle-i|1 0_11_0\rangle\\ |0 1_11_0\rangle-i|1 0_10_0\rangle\end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_3&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&1&-1&-1&1&1&-1&-1\\ -1&1&1&-1&-1&1&1&-1\\ i&i&i&i&-i&-i&-i&-i\\ -i&i&-i&i&i&-i&i&-i\\ i&i&-i&-i&-i&-i&i&i\\ i&-i&-i&i&-i&i&i&-i \end{array}\right) \\ & = & \frac{1}{\sqrt{2}}\left\{\begin{array}{c} |0_00_10\rangle+|1_01_11\rangle\\ |0_00_11\rangle+|1_01_10\rangle\\ |0_01_10\rangle+|1_00_11\rangle\\ |0_01_11\rangle+|1_00_10\rangle\\ |0_00_10\rangle-|1_01_11\rangle\\ |0_00_11\rangle-|1_01_10\rangle\\ |0_01_10\rangle-|1_00_11\rangle\\ |0_01_11\rangle-|1_00_10\rangle\end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_4&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ i&-i&i&-i&i&-i&i&-i\\ i&i&-i&-i&i&i&-i&-i\\ 1&-1&-1&1&1&-1&-1&1\\ 1&1&1&1&-1&-1&-1&-1\\ -i&i&-i&i&i&-i&i&-i\\ i&i&-i&-i&-i&-i&i&i\\ -1&1&1&-1&1&-1&-1&1 \end{array}\right) \\ & = & \left\{\begin{array}{c} |0_10_00\rangle+i|1_11_01\rangle\\ |0_10_01\rangle+i|1_11_00\rangle\\ |0_11_00\rangle+i|1_10_01\rangle\\ |0_11_01\rangle+i|1_10_00\rangle\\ |0_10_00\rangle-i|1_11_01\rangle\\ |0_10_01\rangle-i|1_11_00\rangle\\ |0_11_00\rangle-i|1_10_01\rangle\\ |0_11_01\rangle-i|1_10_00\rangle\end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_5&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ i&i&-i&-i&i&i&-i&-i\\ i&-i&-i&i&i&-i&-i&i\\ 1&1&1&1&-1&-1&-1&-1\\ -1&1&-1&1&1&-1&1&-1\\ -i&-i&i&i&i&i&-i&-i\\ i&-i&-i&i&-i&i&i&-i \end{array}\right) \\ & = & \frac{1}{\sqrt{2}}\left\{\begin{array}{c} {\left | 0 0_00_1 \right\rangle}+{\left | 1 1_01_1 \right\rangle}\\ {\left | 0 0_01_1 \right\rangle}+{\left | 1 1_00_1 \right\rangle}\\ {\left | 0 1_00_1 \right\rangle}+{\left | 1 0_01_1 \right\rangle}\\ {\left | 0 1_01_1 \right\rangle}+{\left | 1 1_01_1 \right\rangle}\\ {\left | 0 0_00_1 \right\rangle}-{\left | 1 1_01_1 \right\rangle}\\ {\left | 0 0_01_1 \right\rangle}-{\left | 1 1_00_1 \right\rangle}\\ {\left | 0 1_00_1 \right\rangle}-{\left | 1 0_01_1 \right\rangle}\\ {\left | 0 1_01_1 \right\rangle}-{\left | 1 1_01_1 \right\rangle} \end{array}\right\},\end{aligned}$$ $$\begin{aligned} B_6&=&\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ i&-i&i&-i&i&-i&i&-i\\ 1&1&-1&-1&1&1&-1&-1\\ -i&i&i&-i&-i&i&i&-i\\ 1&1&1&1&-1&-1&-1&-1\\ -i&i&-i&i&i&-i&i&-i\\ -1&-1&1&1&1&1&-1&-1\\ i&-i&-i&i&-i&i&i&-i \end{array}\right) \\ & = & \frac{1}{\sqrt{2}}\left\{\begin{array}{c} |0 0_00_1\rangle+i|1 1_01_1\rangle\\ |0 0_01_1\rangle+i|1 1_00_1\rangle\\ |0 1_00_1\rangle+i|1 0_01_1\rangle\\ |0 1_01_1\rangle+i|1 0_00_1\rangle\\ |0 0_00_1\rangle-i|1 1_01_1\rangle\\ |0 0_01_1\rangle-i|1 1_00_1\rangle\\ |0 1_00_1\rangle-i|1 0_01_1\rangle\\ |0 1_01_1\rangle-i|1 0_00_1\rangle\end{array}\right\},\end{aligned}$$ $$B_7=\frac{1}{2\sqrt{2}}\left(\begin{array}{cccccccc} 1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&-1&1&-1&-1&1&-1&1\\ 1&1&-1&-1&-1&-1&1&1\\ -1&1&1&-1&1&-1&-1&1 \end{array}\right)=\left\{\begin{array}{c} |0_10_10_1\rangle\\ |0_10_11_1\rangle\\ |0_11_10_1\rangle\\ |0_11_11_1\rangle\\ |1_10_10_1\rangle\\ |1_10_11_1\rangle\\ |1_11_10_1\rangle\\ |1_11_11_1\rangle \end{array}\right\}$$ $d=9$ ===== The bases of this Appendix present explicitly the result of construction described in section 3.3 of the main text for $d=3^2=9$. $$B_9 = \left(\begin{array}{ccccccccc} 1&0&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&1 \end{array}\right) = \left\{\begin{array}{c} |0\rangle\\ |1\rangle\\ |2\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0\rangle\\ |1\rangle\\ |2\rangle \end{array} \right\},$$ $$B_0 = \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ 1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3\\ 1&1&1&\alpha_3&\alpha_3&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ 1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3\\ 1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1\\ 1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3\\ 1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1\\ 1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2 \end{array}\right) = \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\},$$ $$B_1 = \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2\\ \alpha_3&\alpha_3&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1\\ \alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1\\ \alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2\\ \alpha_3&\alpha_3&\alpha_3&1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ \alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2\\ \alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3 \end{array}\right) = \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\},$$ $$B_2 = \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3\\ \alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1&\alpha_3&\alpha_3&\alpha_3\\ \alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2\\ \alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3&1&1&1\\ \alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3\\ \alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1 \end{array}\right) = \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\},$$ $$\begin{aligned} B_3 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3&\alpha_3^2&1&\alpha_3&1&1&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3&1&\alpha_3^2\\ 1&1&1&\alpha_3&1&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ 1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3\\ \alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2\\ 1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3\\ \alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_0 \rangle |0_9 \rangle +\alpha_3 |1_0 \rangle |1_9 \rangle + \alpha_3 |2_0 \rangle | 2_9 \rangle\\ |0_0 \rangle |0_9 \rangle +\alpha_3^2 |1_0 \rangle |1_9 \rangle + |2_0 \rangle | 2_9 \rangle\\ |0_0 \rangle |0_9 \rangle + |1_0 \rangle |1_9 \rangle + \alpha_3^2 |2_0 \rangle | 2_9 \rangle\\ |1_0 \rangle |0_9 \rangle +\alpha_3 |2_0 \rangle |1_9 \rangle + \alpha_3 |0_0 \rangle | 2_9 \rangle \\ |1_0 \rangle |0_9 \rangle +\alpha_3^2 |2_0 \rangle |1_9 \rangle + |0_0 \rangle | 2_9 \rangle\\ |1_0 \rangle |0_9 \rangle + |1_0 \rangle |2_9 \rangle + \alpha_3^2 |0_0 \rangle | 2_9 \rangle \\ |2_0 \rangle |0_9 \rangle +\alpha_3 |0_0 \rangle |1_9 \rangle + \alpha_3 |1_0 \rangle | 2_9 \rangle \\ |2_0 \rangle |0_9 \rangle +\alpha_3^2 |0_0 \rangle |1_9 \rangle + |1_0 \rangle | 2_9 \rangle \\ |2_0 \rangle |0_9 \rangle + |1_0 \rangle |0_9 \rangle + \alpha_3^2 |1_0 \rangle | 2_9 \rangle \end{array}\right\} = \mathcal{P}_3^2 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_4 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3\\ \alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1\\ 1&1&1&\alpha_3&\alpha_3&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ 1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3\\ \alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3\\ 1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3\\ \alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3\\ 1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_0 \rangle |0_9 \rangle +\alpha_3^2 |1_0 \rangle |2_9 \rangle + \alpha_3^2 |2_0 \rangle | 1_9 \rangle\\ |0_0 \rangle |0_9 \rangle +\alpha_3 |1_0 \rangle |2_9 \rangle + |2_0 \rangle | 1_9 \rangle\\ |0_0 \rangle |0_9 \rangle + |1_0 \rangle |2_9 \rangle + \alpha_3 |2_0 \rangle | 1_9 \rangle \\ |1_0 \rangle |0_9 \rangle +\alpha_3^2 |2_0 \rangle |2_9 \rangle + \alpha_3^2 |0_0 \rangle | 1_9 \rangle\\ |1_0 \rangle |0_9 \rangle +\alpha_3 |2_0 \rangle |2_9 \rangle + |0_0 \rangle | 1_9 \rangle \\ |1_0 \rangle |0_9 \rangle + |2_0 \rangle |2_9 \rangle + \alpha_3 |0_0 \rangle | 1_9 \rangle \\ |2_0 \rangle |0_9 \rangle +\alpha_3^2 |0_0 \rangle |2_9 \rangle + \alpha_3^2 |1_0 \rangle | 1_9 \rangle \\ |2_0 \rangle |0_9 \rangle +\alpha_3 |0_0 \rangle |2_9 \rangle + |1_0 \rangle | 1_9 \rangle \\ |2_0 \rangle |0_9 \rangle + |0_0 \rangle |2_9 \rangle + \alpha_3 |1_0 \rangle | 1_9 \rangle \end{array}\right) = \mathcal{P}_3 \left[ \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_5 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ 1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3\\ \alpha_3&\alpha_3&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1\\ \alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1\\ 1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1\\ \alpha_3&\alpha_3&\alpha_3&1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ 1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1\\ \alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_1 \rangle |0_9 \rangle +|1_1 \rangle |2_9 \rangle + |2_1 \rangle | 1_9 \rangle\\ |0_1 \rangle |0_9 \rangle +\alpha_3^2 |1_1 \rangle |2_9 \rangle + \alpha_3 |2_1 \rangle | 1_9 \rangle\\ |0_1 \rangle |0_9 \rangle + \alpha_3 |1_1 \rangle |2_9 \rangle + \alpha_3^2 |2_1 \rangle | 1_9 \rangle\\ |1_1 \rangle |0_9 \rangle +|2_1 \rangle |2_9 \rangle + |0_1 \rangle | 1_9 \rangle\\ |1_1 \rangle |0_9 \rangle +\alpha_3^2 |2_1 \rangle |2_9 \rangle + \alpha_3 |0_1 \rangle | 1_9 \rangle \\ |1_1 \rangle |0_9 \rangle + \alpha_3 |2_1 \rangle |2_9 \rangle + \alpha_3^2 |0_1 \rangle | 1_9 \rangle \\ |2_1 \rangle |0_9 \rangle +|0_1 \rangle |2_9 \rangle + |1_1 \rangle | 1_9 \rangle \\ |2_1 \rangle |0_9 \rangle +\alpha_3^2 |0_1 \rangle |2_9 \rangle + \alpha_3 |1_1 \rangle | 1_9 \rangle\\ |2_1 \rangle |0_9 \rangle + \alpha_3 |0_1 \rangle |2_9 \rangle + \alpha_3^2 |1_1 \rangle | 1_9 \rangle \end{array}\right\} = \mathcal{P}_3 \left[ \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_6 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3\\ \alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1\\ \alpha_3&\alpha_3&\alpha_3&\alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1\\ \alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3\\ \alpha_3&\alpha_3&\alpha_3&1&1&1&\alpha_3^2&\alpha_3^2&\alpha_3^2\\ \alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3\\ \alpha_3^2&\alpha_3&1&\alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_1 \rangle |0_9 \rangle + \alpha_3^2 |1_1 \rangle |1_9 \rangle + \alpha_3^2 |2_1 \rangle | 2_9 \rangle\\ |0_1 \rangle |0_9 \rangle + |1_1 \rangle |1_9 \rangle + \alpha_3 |2_1 \rangle | 2_9 \rangle \\ |0_1 \rangle |0_9 \rangle + \alpha_3 |1_1 \rangle |1_9 \rangle +|2_1 \rangle |2_9 \rangle\\ |1_1 \rangle |0_9 \rangle + \alpha_3^2 |2_1 \rangle |1_9 \rangle + \alpha_3^2 |0_1 \rangle | 2_9 \rangle\\ |1_1 \rangle |0_9 \rangle + |2_1 \rangle |1_9 \rangle + \alpha_3 |0_1 \rangle | 2_9 \rangle \\ |1_1 \rangle |0_9 \rangle + \alpha_3 |2_1 \rangle |1_9 \rangle +|0_1 \rangle |2_9 \rangle \\ |2_1 \rangle |0_9 \rangle + \alpha_3^2 |0_1 \rangle |1_9 \rangle + \alpha_3^2 |1_1 \rangle | 2_9 \rangle \\ |2_1 \rangle |0_9 \rangle + |0_1 \rangle |1_9 \rangle + \alpha_3 |1_1 \rangle | 2_9 \rangle \\ |2_1 \rangle |0_9 \rangle + \alpha_3 |0_1 \rangle |1_9 \rangle +|1_1 \rangle |2_9 \rangle \end{array}\right\} = \mathcal{P}_3^2 \left[ \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_7 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ 1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1&\alpha_3&\alpha_3&\alpha_3\\ \alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2\\ 1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2&\alpha_3^2&\alpha_3&1\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3&1&1&1\\ 1&\alpha_3&\alpha_3^2&\alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_2 \rangle |0_9 \rangle + |1_2 \rangle |1_9 \rangle + |2_2 \rangle | 2_9 \rangle \\ |0_2 \rangle |0_9 \rangle + \alpha_3 |1_2 \rangle |1_9 \rangle + \alpha_3^2 |2_2 \rangle | 2_9 \rangle \\ |0_2 \rangle |0_9 \rangle + \alpha_3^2 |1_2 \rangle |1_9 \rangle + \alpha_3 |2_2 \rangle |2_9 \rangle \\ |1_2 \rangle |0_9 \rangle + |2_2 \rangle |1_9 \rangle + |0_2 \rangle | 2_9 \rangle\\ |1_2 \rangle |0_9 \rangle + \alpha_3 |2_2 \rangle |1_9 \rangle + \alpha_3^2 |0_2 \rangle | 2_9 \rangle \\ |1_2 \rangle |0_9 \rangle + \alpha_3^2 |2_2 \rangle |1_9 \rangle + \alpha_3 |0_2 \rangle |2_9 \rangle \\ |2_2 \rangle |0_9 \rangle + |0_2 \rangle |1_9 \rangle + |1_2 \rangle | 2_9 \rangle\\ |2_2 \rangle |0_9 \rangle + \alpha_3 |0_2 \rangle |1_9 \rangle + \alpha_3^2 |1_2 \rangle | 2_9 \rangle\\ |2_2 \rangle |0_9 \rangle + \alpha_3^2 |0_2 \rangle |1_9 \rangle + \alpha_3 |1_2 \rangle |2_9 \rangle \end{array}\right\} = \mathcal{P}_3^2 \left[ \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_0\rangle\\ |1_0\rangle\\ |2_0\rangle \end{array} \right\} \right],\end{aligned}$$ $$\begin{aligned} B_8 & = & \frac{1}{3} \left(\begin{array}{ccccccccc} 1&1&1&1&1&1&1&1&1\\ \alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1&\alpha_3&\alpha_3^2&1\\ \alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2&\alpha_3&1&\alpha_3^2\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&1&1&1&\alpha_3&\alpha_3&\alpha_3\\ \alpha_3&\alpha_3^2&1&\alpha_3^2&1&\alpha_3&1&\alpha_3&\alpha_3^2\\ \alpha_3^2&\alpha_3&1&1&\alpha_3^2&\alpha_3&\alpha_3&1&\alpha_3^2\\ \alpha_3^2&\alpha_3^2&\alpha_3^2&\alpha_3&\alpha_3&\alpha_3&1&1&1\\ \alpha_3^2&1&\alpha_3&\alpha_3&\alpha_3^2&1&1&\alpha_3&\alpha_3^2\\ \alpha_3&1&\alpha_3^2&1&\alpha_3^2&\alpha_3&\alpha_3^2&\alpha_3&1 \end{array}\right) \\ & = & \frac{1}{\sqrt{3}} \left\{ \begin{array}{l} |0_2 \rangle |0_9 \rangle + \alpha_3 |1_2 \rangle |2_9 \rangle + \alpha_3 |2_2 \rangle | 1_9 \rangle \\ |0_2 \rangle |0_9 \rangle + |1_2 \rangle |2_9 \rangle + \alpha_3^2 |2_2 \rangle |1_9 \rangle \\ |0_2 \rangle |0_9 \rangle + \alpha_3^2 |1_2 \rangle |2_9 \rangle + |2_2 \rangle |1_9 \rangle \\ |1_2 \rangle |0_9 \rangle + \alpha_3 |2_2 \rangle |2_9 \rangle + \alpha_3 |0_2 \rangle | 1_9 \rangle \\ |1_2 \rangle |0_9 \rangle + |2_2 \rangle |2_9 \rangle + \alpha_3^2 |0_2 \rangle |1_9 \rangle \\ |1_2 \rangle |0_9 \rangle + \alpha_3^2 |2_2 \rangle |2_9 \rangle + |0_2 \rangle |1_9 \rangle\\ |2_2 \rangle |0_9 \rangle + \alpha_3 |0_2 \rangle |2_9 \rangle + \alpha_3 |1_2 \rangle | 1_9 \rangle \\ |2_2 \rangle |0_9 \rangle + |0_2 \rangle |2_9 \rangle + \alpha_3^2 |1_2 \rangle |1_9 \rangle \\ |2_2 \rangle |0_9 \rangle + \alpha_3^2 |0_2 \rangle |2_9 \rangle + |1_2 \rangle |1_9 \rangle \end{array}\right\} = \mathcal{P}_3 \left[ \left\{\begin{array}{c} |0_2\rangle\\ |1_2\rangle\\ |2_2\rangle \end{array} \right\} \otimes \left\{\begin{array}{c} |0_1\rangle\\ |1_1\rangle\\ |2_1\rangle \end{array} \right\} \right],\end{aligned}$$ where the kets refer to MUBs for a qutrit (Appendix B), and $\mathcal{P}_3$ is the control-phase gate for two qutrits as given in Eq. (\[controlphase\]) of the main text. References {#references .unnumbered} ========== [99]{} Wootters W K and Fields B D 1989 *Ann. Phys. (N.Y.)* [**191**]{} 363 Ivanović I D 1981 *J. Phys. A* [**14**]{} 3241 Durt T, Englert B-G, Bengtsson I and Życzkowski K 2010 *Int. J. Quant. Inf.* [**8**]{} 535 Butterley P and Hall W 2007 *Phys. Lett. A* [**369**]{} 5 Brierley S and Weigert S 2008 *Phys. Rev. A* [**78**]{} 042312 Thiang G C 2010 arXiv:1012.3147 Raynal P, Lü X and Englert B-G 2011 arXiv:1103.1025 Bru[ß]{} D 1998 *Phys. Rev. Lett.* [**81**]{} 3018 Bechmann-Pasquinucci H and Peres A 2000 *Phys. Rev. Lett.* [**85**]{} 3313 Cerf N J, Bourennane M, Karlsson A and Gisin N 2002 *Phys. Rev. Lett.* [**88**]{} 127902 Vaidman L, Aharonov Y and Albert D Z 1987 *Phys. Rev. Lett.* [**58**]{} 1385 Aharonov Y and Englert B-G 2001 *Z. Naturforsch. A* [**56**]{} 16 Aravind P K 2003 *Z. Naturforsch. A* [**58**]{} 2212 Hayashi A, Horibe M and Hashimoto T 2005 *Phys. Rev. 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A* [**55**]{} 2564 Politi A, Cryan M J, Rarity J G, Yu S and O’Brien J L 2008 *Science* [**320**]{} 646 Lanyon, B P, Weinhold T J, Langford N K, O’Brein J L, Resch K J, Gilchrist A, and White A G 2008 [*Phys. Rev. Lett.*]{} [**100**]{}, 060504 Peruzzo A, Laing A, Politi A, Rudolph T and O’Brien J L 2008 2011 arXiv:1007.1372 Mair A, Vaziri A, Weihs G and Zeilinger A 2001 *Nature* [**412**]{} 313 Leach J, Padgett M J, Barnett S M, Franke-Arnold S and Courtial J 2002 *Phys. Rev. Lett.* [**88**]{} 257901 Franke-Arnold S, Allen S and Padgett M J 2008 *Laser and Photon. Rev.* [**2**]{} 299 Berkhout G C G and Beijersbergen 2008 *Phys. Rev. Lett.* [**101**]{} 100801 Pors J B, Oemrawsingh S S R, Aiello A, van Exter M P, Eliel E R, ’t Hooft G W and Woerdman J P 2008 *Phys. Rev. Lett.* [**101**]{} 120502 [^1]: Present address: Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland. [^2]: Author to whom any correspondence should be addressed. Email: tomasz.paterek@nus.edu.sg

--- abstract: 'Suppose we have identified three clusters of galaxies as being topological copies of the same object. How does this information constrain the possible models for the shape of our Universe? It is shown here that, if our Universe has flat spatial sections, these multiple images can be accommodated within any of the six classes of compact orientable 3-dimensional flat space forms. Moreover, the discovery of two more triples of multiple images in the neighbourhood of the first one, would allow the determination of the topology of the Universe, and in most cases the determination of its size.' author: - | G.I. Gomero[^1],\ \ Instituto de Física Teórica,\ Universidade Estadual Paulista,\ Rua Pamplona, 145\ São Paulo, SP 01405–900, Brazil title: '**Determining the shape of the Universe using discrete sources**' --- Introduction ============ The last two decades have seen a continuously increasing interest in studying cosmological models with multiply connected spatial sections (see [@Review] and references therein). Since observational cosmology is becoming an increasingly high precision science, it would be of wide interest to develop methods to systematically construct specific candidates for the shape of our Universe in order to analyse whether these models are consistent with observational data. Since one of the simplest predictions of cosmological models with multiply connected spatial sections is the existence of multiple images of discrete cosmic objects, such as clusters of galaxies,[^2] the following question immediately arises: Suppose we have identified three clusters of galaxies as being different topological copies of the same object, how does this information constrain the possible models for the shape of our Universe? The initial motivation for this work was the suggestion of Roukema and Edge that the X–ray clusters RXJ 1347.5–1145 and CL 09104+4109 may be topological images of the Coma cluster [@RE97]. Even if these particular clusters turn out not to be topological copies of the same object, the suggestion of Roukema and Edge raises an interesting challenge. *What if* one day a clever astrophysicist discovers three topological copies of the same object? It is shown here that these (would be) multiple images could be accommodated within any of the six classes of compact orientable 3-dimensional flat space forms. Moreover, and this is the main result of this paper, the discovery of two more triples of multiple images in the neighbourhood of the first one, would be enough to determine the topology of the Universe, and in most cases even its size. Thus, two interesting problems appear now, (i) does our present knowledge of the physics of clusters of galaxies (or alternatively, of quasars) may allow the identification of a triple of multiple images if they actually exist?, and (ii) given that such an identification has been achieved, how easy can other triples of topological copies near the first one be identified? The present paper does not deal with these two problems, however it should be noticed that a recent method proposed by A. Bernui and me in [@BerGo] (see also [@Gomero]) could be used to test, in a purely geometrical way, the hypothesis that any two given clusters of galaxies are topological copies. The model building procedure is explained in the next section, while section \[Examples\] presents some numerical examples illustrating specific candidates for the shape of our Universe, under the pressumed validity of the Roukema–Edge hypothesis. In section \[Decide\] it is discussed the main result of this paper: how the topology of space could be determined with the observation of just two more triples of images; and how, in most cases, one could even determine the size of our Universe. Finally, section \[Concl\] consists of discussions of the results presented in this letter and suggestions for further research. Model Building {#ModBuild} ============== Suppose that three topological copies of the same cluster of galaxies have been identified. Let $C_0$ be the nearest copy from us, $C_1$ and $C_2$ the two other copies, $d_1$ and $d_2$ the distances from $C_0$ to $C_1$ and $C_2$ respectively, and $\theta$ the angle between the geodesic segments $\overline{C_0C_1}$ and $\overline{C_0C_2}$. Roukema and Edge [@RE97] have suggested an example of this configuration, the Coma cluster being $C_0$ and the clusters RXJ 1347.5–1145 and CL 09104+4109 being $C_1$ and $C_2$ (or vice versa). The distances of these clusters to Coma are 970 and 960$h^{-1}$ $Mpc$ respectively (for $\Omega_0=1$ and $\Lambda=0$), and the angle between them, with the Coma cluster at the vertex, is $\approx \! 88^o$. Under the assumption that these multiplicity of images were due to two translations of equal length and in orthogonal directions, they constructed FL cosmological models whose compact flat spatial sections of constant time were (i) 3-torii, (ii) manifolds of class ${\mathcal{G}_2}$, or (iii) manifolds of class ${\mathcal{G}_4}$, all of them with square cross sections, and scale along the third direction larger than the depth of the catalogue of X-ray clusters used in the analysis. Let us consider the possibility that at least one of the clusters $C_i$ is an image of $C_0$ by a screw motion, and do not assume that the distances from $C_0$ to $C_1$ and $C_2$ are equal, nor that they form a right angle (with $C_0$ at the vertex). It is shown in this section that one can accommodate this generic configuration of clusters within any of the six classes of compact orientable 3-dimensional flat space forms, thus providing a plethora of models for the shape of our Universe consistent with the (would be) observational fact that these clusters are in fact the same cluster. Moreover, one could also consider the possibility that one of the clusters $C_i$ is an image of $C_0$ by a glide reflection, thus giving rise to non–orientable manifolds as models for the shape of space. However, these cases will not be considered here since they do not give qualitatively different results, and the corresponding calculations can be done whenever needed. The diffeomorphic and isometric classifications of 3-dimensional Euclidean space forms given by Wolf in [@Wolf] were described in detail by Gomero and Rebouças in [@GR02]. The generators of the six diffeomorphic compact orientable classes are given in Table \[Tb:OESF\], where an isometry in Euclidean 3-space is denoted by $(A,a)$, $a$ is a vector and $A$ is an orthogonal transformation, and the action is given by $$\label{action} (A,a) : p \mapsto Ap + a \; ,$$ for any point $p$. The orientation preserving orthogonal transformations that appear in the classification of the Euclidean space forms take the matrix forms $$\begin{aligned} \label{Rot3} A_1 = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right) \; , & A_2 = \left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right) \; , & A_3 = \left( \begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array} \right) \; , \nonumber \\ \\ B = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{array} \right) \; , & C = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array} \right) \quad\mbox{and} & D = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 1 \end{array} \right) \; , \nonumber\end{aligned}$$ in the basis formed by the set $\{a,b,c\}$ of linearly independent vectors that appear in Table 1. We will fit the set of multiple images $\{C_0,C_1,C_2\}$ within manifolds of classes ${\mathcal{G}_2}-{\mathcal{G}_6}$, since the class ${\mathcal{G}_1}$ (the 3–torus) is trivial. [|c|\*[6]{}[|c]{}|]{} Class & ${\mathcal{G}_1}$ & ${\mathcal{G}_2}$ & ${\mathcal{G}_3}$ & ${\mathcal{G}_4}$ & ${\mathcal{G}_5}$ & ${\mathcal{G}_6}$\ & & & & & & ($A_1$,$a$)\ Generators & $a$, $b$, $c$ & ($A_1$,$a$), $b$, $c$ & ($B$,$a$), $b$, $c$ & ($C$,$a$), $b$, $c$ & ($D$,$a$), $b$, $c$ & ($A_2$,$b+c$)\ & & & & & & ($A_2$,$b-c$)\ Let us first deal with the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$. The generators for the corresponding covering groups are $\alpha = (A,a)$, $\beta=(I,b)$ and $\gamma=(I,c)$, with $A=A_1,B,C$ and $D$ for the classes ${\mathcal{G}_2}$, ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ and ${\mathcal{G}_5}$ respectively, and $I$ is the identity transformation. For these classes we will consider the following non-trivial configuration: denoting the position of $C_0$ by $p$, $C_1$ is located at $\alpha(p)$ and $C_2$ at $\beta(p)$. The configuration in which $C_2$ is located at $\gamma(p)$ is equivalent to the former, while the configuration in which $C_1$ and $C_2$ are images of $C_0$ by pure translations (strictly possible only in ${\mathcal{G}_2}$, and a convenient approximation in ${\mathcal{G}_4}$ if $\theta \approx 90^o$, and the distances of $C_1$ and $C_2$ to $C_0$ are almost equal, as is the case in the Roukema–Edge hypothesis) is equivalent to that of a torus. For space forms of the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$ the following facts are easily derivable from the generators of their corresponding covering groups (see [@GR02] for details): 1. The vector $a$ is orthogonal to both $b$ and $c$. 2. \[angle\] The angle between $b$ and $c$ is a free parameter for the class ${\mathcal{G}_2}$, while its value is fixed to be $120^o$, $90^o$ and $60^o$ for the classes ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ and ${\mathcal{G}_5}$ respectively. 3. Denoting by $|a|$ the length of the vector $a$, and similarly for any other vector, one has that $|b| = |c|$ for the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$, while both lengths are independent free parameters in the class ${\mathcal{G}_2}$. Moreover, in all classes ${\mathcal{G}_2}-{\mathcal{G}_5}$, $|a|$ is an independent free parameter. 4. Denoting the canonical unitary basis vectors in Euclidean space by $\{\hat{\imath},\hat{\jmath},\hat{k}\}$, one can always write $a = |a| \hat{\imath}$, $b = |b| \hat{\jmath}$ and $c = |c| \cos \varphi \hat{\jmath} + |c| \sin \varphi \hat{k}$, for the basis $\{a,b,c\}$, where $\varphi$ is the angle between $b$ and $c$, as established in the item \[angle\]. [\*[4]{}[|c]{}|]{} & & &\ Class & $\alpha(p)$ & $\delta_{\alpha}(p)$ & $\delta_{\alpha}(p) \cos(\alpha,\beta)$\ & & &\ & & &\ ${\mathcal{G}_2}$ & $(x+|a|,-y, -z)$ & $\sqrt{|a|^2+4(y^2+z^2)}$ & $-2y$\ & & &\ & & &\ ${\mathcal{G}_3}$ & $(x+|a|, -\frac{1}{2} y - \! \frac{\sqrt{3}}{2} z, \, \frac{\sqrt{3}}{2} y - \! \frac{1}{2} z)$ & $\sqrt{|a|^2 + 3(y^2+z^2)}$ & $- \frac{\sqrt{3}}{2}(\sqrt{3}y + z)$\ & & &\ & & &\ ${\mathcal{G}_4}$ & $(x+|a|,-z,y)$ & $\sqrt{|a|^2+2(y^2+z^2)}$ & $-(y+z)$\ & & &\ & & &\ ${\mathcal{G}_5}$ & $(x+|a|, \frac{1}{2} y - \! \frac{\sqrt{3}}{2} z, \, \frac{\sqrt{3}}{2} y + \frac{1}{2} z)$ & $\sqrt{|a|^2+(y^2+z^2)}$ & $-\frac{1}{2}(y + \sqrt{3}z)$\ & & &\ Writing $p=(x,y,z)$ for the components of the position of $C_0$ in the basis $\{\hat{\imath},\hat{\jmath},\hat{k}\}$,[^3] one can easily work out the expressions for the components of the position of $C_1$, $\alpha(p)$, the distance function $\delta_{\alpha}(p)$, and the cosine of the angle between $\overline{C_0C_1}$ and $\overline{C_0C_2}$, $\cos(\alpha,\beta)$. The resulting expressions are shown in Table \[Tb:Express\]. For the configuration we are dealing with, one trivially has $d_2=\delta_{\beta}(p)=|b|$, since $\beta$ is a pure translation. More interestingly, from $\delta_{\alpha}(p) = d_1$ and $\cos(\alpha,\beta)=\cos\theta$, one can partially solve the equations for the components of the position of $C_0$. The resulting expressions are shown in Table \[Tb:Positions\]. Observe that for each class we have two solutions in terms of the free parameter $|a|$. For the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$ the two solutions are those for which $d_1 \!\cos\theta$ is given by the fourth column in Table \[Tb:Express\]. Two remarks are in order here. First, it is convenient to write down the components of the position of $C_0$ in terms of the parameter $|a|$, because this parameter can be easily determined once two more triples of multiple images, say $\{D_0,D_1,D_2\}$ and $\{E_0,E_1,E_2\}$, in the neighbourhood of $\{C_0,C_1,C_2\}$ have been identified, as shown in Section \[Decide\].[^4] Once this has been done, the positions of $C_0$, $D_0$ and $E_0$ can be used to predict multiple images of them due to the inverse isometry $\alpha^{-1}$, thus yielding a definitive observational test for the hypothesis of the multiply connectedness of our Universe. Second, note that the $x-\,$coordinate is not constrained by this configuration of topological images. This freedom of the $x-\,$coordinate is a consequence of homogeneity of manifolds of classes ${\mathcal{G}_2}-{\mathcal{G}_5}$ along the $x-\,$axis. This *partial* homogeneity is due to the fact that the orthogonal transformations involved in the corresponding covering groups have the $x-\,$axis as their axis of rotation. [\*[3]{}[|c]{}|]{} & &\ Class & $y$ & $z$\ & &\ & &\ ${\mathcal{G}_2}$ & $-\frac{1}{2} \, d_1 \cos\theta$ & $\pm \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2}$\ & &\ & &\ ${\mathcal{G}_3}$ & $\pm \frac{\sqrt{3}}{6} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{\sqrt{3}}{6} \, d_1\cos\theta$\ & &\ & &\ ${\mathcal{G}_4}$ & $\pm \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{1}{2} \, d_1\cos\theta$\ & &\ & &\ ${\mathcal{G}_5}$ & $\pm \frac{\sqrt{3}}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{\sqrt{3}}{2} \, d_1\cos\theta$\ & &\ We now fit the multiple images $\{C_0,C_1,C_2\}$ within manifolds of class ${\mathcal{G}_6}$. The generators for the covering group of a manifold of this class are $\alpha=(A_1,a)$, $\beta=(A_2,b+c)$ and $\mu=(A_2,b-c)$. The vectors $\{a,b,c\}$ are mutually orthogonal but their lengths are free parameters. For manifolds of class ${\mathcal{G}_6}$ we have two possible configurations, both of them with $C_0$ located at $p$, 1. $C_1$ located at $\alpha(p)$ and $C_2$ at $\beta(p)$, and 2. $C_1$ located at $\beta(p)$ and $C_2$ at $\mu(p)$. The case in which $C_1$ is at $\alpha(p)$ and $C_2$ at $\mu(p)$ is equivalent to the first configuration. The expressions for the distances $\delta_{\alpha}(p)$, $\delta_{\beta}(p)$ and $\delta_{\mu}(p)$, and angles $\cos(\alpha,\beta)$ and $\cos(\beta,\mu)$ are $$\begin{aligned} \delta_{\alpha}(p) & = & \sqrt{|a|^2 + 4(y^2+z^2)} \nonumber \\ \label{DistFunc} \delta_{\beta}(p) & = & \sqrt{|b|^2 + 4x^2 + (2z-|c|)^2} \\ \delta_{\mu}(p) & = & \sqrt{|b|^2 + 4x^2 + (2z+|c|)^2} \nonumber\end{aligned}$$ and $$\begin{aligned} \cos(\alpha,\beta) & = & \frac{4z^2 - 2(|a|x + |b|y + |c|z)}{\delta_{\alpha}(p) \delta_{\beta}(p)} \nonumber \\ \label{CosAngle} & & \\ \cos(\beta,\mu) & = & \frac{4x^2 + 4z^2 +|b|^2 - |c|^2}{\delta_{\beta}(p) \delta_{\mu}(p)} \nonumber\end{aligned}$$ For the first configuration one has $\delta_{\alpha}(p) = d_1$, $\delta_{\beta}(p) = d_2$ and $\cos(\alpha,\beta) = \cos\theta$, thus yielding the equations $$\begin{aligned} y^2 + z^2 & = & \frac{1}{4} (d_1^2 - |a|^2) \nonumber \\ \label{EqFirstG6} 4x^2 + (2z - |c|)^2 & = & d_2^2 - |b|^2 \\ 4z^2 -2(|a|x + |b|y + |c|z) & = & d_1d_2\cos\theta \; . \nonumber\end{aligned}$$ This is a system of three quadratic equations with six unknowns, the three coordinates $(x,y,z)$ of the point $p$, and the three coordinates $(|a|,|b|,|c|)$ in the parameter space of the ${\mathcal{G}_6}$ manifold (see [@GR02]). An algebraic solution of these equations for $(x,y,z)$ in terms of $(|a|,|b|,|c|)$, or vice versa, would in general yield higher degree (decoupled) equations for each variable, and thus are not so illuminating. Particular solutions can be obtained by (i) assuming specific values for the parameters $(|a|,|b|,|c|)$, and then calculating numerically the position of $C_0$, or (ii) assuming some particular position for $C_0$, and then calculating the parameters $(|a|,|b|,|c|)$. This second method does not follow the strategy of determining the parameters of the manifold using two more triples of clusters of galaxies (see Section \[Decide\]), thus it will not be pursued here. The next section presents examples of application of the first method. Finally, let us examine the second configuration which is simpler. One has $\delta_{\beta}(p)=d_1$, $\delta_{\mu}(p)=d_2$ and $\cos(\beta,\mu) = \cos\theta$, thus yielding the equations $$\begin{aligned} 4x^2 + (2z - |c|)^2 & = & d_1^2 - |b|^2 \nonumber \\ \label{EqSecondG6} 4x^2 + (2z + |c|)^2 & = & d_2^2 - |b|^2 \\ 4x^2 + 4z^2 + |b|^2 -|c|^2 & = & d_1d_2\cos\theta \; . \nonumber \end{aligned}$$ These equations can be partially solved giving $$\begin{aligned} z & = & \frac{1}{8|c|} \, (d_2^2 - d_1^2) \nonumber \\ \label{ModelSecondG6} |c| & = & \frac{1}{2} \sqrt{d_1^2 + d_2^2 - 2d_1d_2\cos\theta} \\ x^2 + z^2 & = & \frac{1}{16} \, (d_1^2 + d_2^2 + 2 d_1d_2\cos\theta) - \frac{1}{4} \, |b|^2 \; . \nonumber\end{aligned}$$ In this case the $y-\,$coordinate is not constrained by the configuration of topological images, since the only orthogonal transformation involved in the calculations has the $y-\,$axis as its axis of rotation. Numerical Examples {#Examples} ================== Let us now apply the results obtained in the previous section to the proposed multiple images of Roukema and Edge [@RE97], in a FL universe whose matter components are pressureless dust and a cosmological constant. The models presented below are small universes with compactification scales much smaller than the horizon radius, so they may seem to be in conflict with constraints on the topology coming from observations of the CMBR. However, it must be recalled that all current constraints for flat universes hold exclusively for models with (i) toroidal spatial sections [@Torus; @Inoue], or (ii) any flat (compact and orientable) spatial section, but in cosmological models without a dark energy component, and moreover, with the observer located on the axis of rotation of a screw motion of the corresponding covering group [@FlatCMB]. As has been shown by Inoue [@Inoue], the addition of a cosmological constant term makes the constraints less stringent, whereas the effect of considering the observer out of an axis of rotation is totally unknown. Since the models presented below consider both, a cosmological constant term, and the observer off an axis of rotation, they can not be considered as being ruled out by current observational data. The models constructed here consider $C_1$ as being the cluster RXJ 1347.5–1145 and $C_2$ the cluster CL 09104+4109. Then for the values $\Omega_{m0} = 0.3$ and $\Omega_{\Lambda 0} = 0.7$, one has $d_1 = 1158h^{-1} \, Mpc$, $d_2 = 1142h^{-1} \, Mpc$ and $\theta = 87^o$. Other examples can be built by simply reversing these identifications, i.e. by considering $C_1$ as being the cluster CL 09104+4109 and $C_2$ the cluster RXJ 1347.5–1145. As before, let us first examine the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$. One has $|b| = 1142h^{-1} \, Mpc$, and because of the expression $\sqrt{d_1^2\sin^2 \theta - |a|^2}$ in Table \[Tb:Positions\] one also has the constraint $$\label{|a|-Max} |a| \leq 1156.4h^{-1} \, Mpc \; .$$ The models within class ${\mathcal{G}_2}$ are special because they have a fixed value of $y$, say $y = - 30.3h^{-1} \, Mpc$; however the $z-\,$coordinate depends on the parameter $|a|$, and remarkably is sensible to this value as can be seen with the following two examples. 1. First consider the case when $|a|$ is slightly lower than the maximum value allowed by (\[|a|-Max\]), say $|a|=1156h^{-1} \, Mpc$. Then $z = \pm 15.5h^{-1} \, Mpc$. 2. Second consider the symmetric case when $|a| = |b| = 1142h^{-1} \, Mpc$. In this case one has $z = \pm \, 91.0h^{-1} \, Mpc$. [\*[6]{}[|c]{}|]{} & & &\ Class & $|a| \, (h^{-1} \, Mpc)$ & &\ & & &\ ${\mathcal{G}_3}$ & $1156$ & $-21.4$ & $-39.2$ & $-32.9$ & $-2.0$\ & $1142$ & $22.2$ & $82.8$ & $108.5$ & $73.5$\ ${\mathcal{G}_4}$ & $1156$ & $-14.9$ & $-45.8$ & $-45.8$ & $-14.9$\ & $1142$ & $60.7$ & $-121.3$ & $-121.3$ & $60.7$\ ${\mathcal{G}_5}$ & $1156$ & $-3.5$ & $-57.1$ & $-67.9$ & $-37.0$\ & $1142$ & $127.3$ & $-187.9$ & $143.5$ & $38.5$\ Note that the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$ do not yield models with a fixed value of $y$; instead, both $y$ and $z$ depend on the parameter $|a|$. In Table \[Tb:Examples\] we show the values of $y$ and $z$ calculated from Table \[Tb:Positions\] for $|a| = 1156$ and $1142h^{-1} \, Mpc$. In this table the first column for each coordinate corresponds to the first solution of Table \[Tb:Positions\], and the second column for the second solution. Now we deal with models within class ${\mathcal{G}_6}$. For the first configuration one obtains from eqs. (\[EqFirstG6\]) $$(2x + |a|)^2 + (2y + |b|)^2 = d_1^2 + d_2^2 - 2d_1d_2\cos\theta - |c|^2 \; ,$$ which implies that $$|c|^2 \leq d_1^2 + d_2^2 - 2d_1d_2\cos\theta \; .$$ Furthermore, from the first and third equations in (\[EqFirstG6\]) one also has $$|a| \leq d_1 \qquad\mbox{and}\qquad |b| \leq d_2 \; .$$ A family of simple examples are obtained by taking $|a| = d_1$. In fact, in this case one has $$y=z=0 \qquad , \qquad x = - \frac{1}{2} d_2\cos\theta \qquad , \qquad |b|^2 + |c|^2 = d_2^2\sin^2\theta \; .$$ Thus, taking $|b|=|c|$, one model of a universe with spatial sections of class ${\mathcal{G}_6}$ that fits the first configuration with the Roukema–Edge hypothesis is $$|a| = 1158h^{-1} \, Mpc \qquad\mbox{and}\qquad |b| = |c| = 1140.4h^{-1} \, Mpc \; ,$$ with Coma located at $$x = - 29.9h^{-1} \, Mpc \qquad\mbox{and}\qquad y=z=0 \; .$$ On the other side, for the second configuration one has $$\label{ExampleSecondG6} |c| = 791.6h^{-1} \, Mpc \quad\mbox{,}\quad z = -5.8h^{-1} \, Mpc \quad\mbox{,}\quad |b| \leq 834.1h^{-1} \, Mpc \; ,$$ the last inequality being obtained from the last equation in (\[ModelSecondG6\]). It is illustrative to give two specific examples as done with the ${\mathcal{G}_2}$ models. 1. First consider the case when $|b|$ is slightly lower than its maximum value allowed by (\[ExampleSecondG6\]), say $|b| = 834h^{-1} \, Mpc$, then one has $x = \pm 7h^{-1} \, Mpc$. 2. Second consider the symmetric case when $|b| = |c| = 791.6h^{-1} \, Mpc$. In this case $x = \pm 131.5h^{-1} \, Mpc$. The case of three triples of images {#Decide} =================================== In this section it is shown that the discovery of two additional triples of clusters of galaxies close to $\{C_0,C_1,C_2\}$ would allow the determination of the topology of the universe, and in most cases the determination of its size. Let us denote by $\{D_0,D_1,D_2\}$ and $\{E_0,E_1,E_2\}$ these two additional triples of topological images. Mathematically, to characterize the *closeness* relation between two triples $\{C_i\}$ and $\{D_i\}$ it suffices the lengths of the geodesic segments $\overline{C_iD_i}$ ($i=0,1,2$) to be the same and smaller than the injectivity radius. Observationally, it is enough that $C_0$ and $D_0$ are two nearby clusters of galaxies, while the distances between $C_i$ and $D_i$ ($i=1,2$) are equal (within the observational error bounds) to the distance between $C_0$ and $D_0$.[^5] By parallel transporting the triangle $C_1D_1E_1$ along the geodesic segment $\overline{C_0C_1}$, one obtains two triangles with a common vertex, namely the triangle of nearby clusters $C_0D_0E_0$, and that of *transported* clusters of $C_1D_1E_1$. It is just a matter of elementary analytic geometry to determine the unique rotation that takes one triangle to the other. Note however that one can easily find also the unique reflection that takes one triangle to the other, if it exists. If the angle of rotation is different from $\pi$, $2 \pi/3$, $\pi/2$ or $\pi/3$, then the isometry that takes $C_0$ to $C_1$ is not a screw motion, but a reflection, and the Universe would be spatially non-orientable. On the contrary, if the angle of rotation is either $\pi$, $2 \pi/3$, $\pi/2$ or $\pi/3$, then one can think this is not by coincidence, so the Universe would be spatially orientable. In such a case, if the angle of rotation is different from $\pi$, it uniquely determines to which class the topology of the Universe belongs, namely ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$ respectively.[^6] Let us restrict our analysis to the orientable case in order to be specific. The determination of the rotation taking $C_0D_0E_0$ to the parallel transportation of $C_1D_1E_1$ provides also the direction of the axis of rotation of the screw motion linking $C_0$ with $C_1$. If the Universe has a topology of class ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$, the translation vector is parallel to this axis, so elementary geometry can be used to determine the parameter $|a|$ and the position of the axis. Moreover, the isometry linking $C_0$ with $C_2$ has to be a translation, and a parallel transport of the triangle $C_2,D_2,E_2$ to $C_0D_0E_0$ would confirm it. A remarkable fact is that, if the topology of the Universe has been identified to be of class ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ or ${\mathcal{G}_5}$, the vector $c$ is automatically fixed, and observational searches can be performed to find the topological images of $C_0$, $D_0$ and $E_0$ due to the isometries $\gamma$ and $\gamma^{-1}$ for validation of the model. Let us now consider the case when the angle of rotation taking $C_0D_0E_0$ to the parallel transport of $C_1D_1E_1$ is $\pi$. In this case the topology of the Universe has to be of class ${\mathcal{G}_2}$ or ${\mathcal{G}_6}$. One can decide between these two possibilities by parallel transporting $C_2D_2E_2$ to $C_0D_0E_0$. If the angle of rotation between these triangles is null, then the isometry linking $C_0$ to $C_2$ is a translation, and the Universe has topology of class ${\mathcal{G}_2}$. On the other hand, if the angle of rotation is $\pi$, the Universe has topology of class ${\mathcal{G}_6}$. In the former case one can proceed as before and determine the length $|a|$ and the position of the axis of the screw motion. However, since for the class ${\mathcal{G}_2}$, the vector $c$ is a free parameter, its modulus and direction remain undetermined. If the topology of the Universe turns out to be of class ${\mathcal{G}_6}$, the multiple images can be fitted within the two inequivalent configurations described in Section \[ModBuild\]. One can decide between both configurations by just looking at the directions of the axes of rotation, for if they are orthogonal the first configuration would be the correct one, while if they are parallel the correct one is the second. Using elementary geometry one can completely determine the three axes of rotation and translations (thus determining the global shape of space) if the multiple images fit with the first configuration. On the other hand, with the second configuration one can determine the vectors $b$ and $c$, and thus the direction of $a$, but it is impossible to determine the length $|a|$, as could have been anticipated from eqs. (\[EqSecondG6\]). However, in this latter case, one can design effective search procedures to look for multiple images due to the isometries $\alpha$ and $\alpha^{-1}$, thus providing at least robust constraints for the parameter $|a|$ (see [@BerGo]). Discussion and Further Remarks {#Concl} ============================== The work presented in this paper originated with the following problem in the context of cosmological models with flat spatial sections: Suppose we have identified three clusters of galaxies as being different topological images of the same object. How do these multiple images constrain the possible models for the shape of our Universe? A natural extension of this work would be the study of this problem in the context of universes with non–flat spatial sections, specifically those with positive curvature (since multiply connected spaces of negative curvature are very unlikely to have a detectable topology [@Detect]). It has been shown here that one can accommodate any of the six classes of compact orientable 3–dimensional flat space forms to fit with any configuration of three topological images of a cosmic object. It can be seen from the construction of the models that one could also easily fit any of the non–orientable flat manifolds. Moreover, the main result in this paper is that the identification of two more triples of multiple images of clusters of galaxies, in the neighbourhood of the first one, is enough to completely determine the topology of space, as well as its size in most of the cases. Even if the primary goal of this paper is not to construct specific candidates for the shape of our universe, but to present a systematic procedure for building such models, it turns out that the illustrative examples constructed by using the Roukema–Edge hypothesis are not in contradiction with current observational data. In view of these results, it seems of primary importance to state and test hypotheses like that of Roukema and Edge, i.e. that the clusters RXJ 1347.5–1145 and CL 09104+4109 are topological images of the Coma cluster, since the identification of a very small quantity of multiple images is, as has been shown here, enough to determine (or almost determine) the global shape of the universe. The problem of testing this kind of hypothesis can be solved by the Local Noise Correlations (LNC) method proposed in [@BerGo]. The problem of generating such kind of hypotheses seems to be much harder, although current efforts are being done to find multiple images of our Galaxy [@Galaxy], clusters of galaxies [@Rou] and radio-loud AGNs [@RMBS]. To close this paper, let us stress that there have only been considered here models in which the topological images are related by the generators of the covering groups of the corresponding manifolds. This needs not be the case, for one could also consider other isometries (compositions of the generators) as being the responsible for the multiple images. Thus the list of possible models presented here is not exhaustive. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank CLAF/CNPq and FAPESP (contract 02/12328-6) for the grants under which this work was carried out, and to the CBPF for kind hospitality. I also thank to Bruno Mota, Marcelo Rebouças and Armando Bernui for critical reading of previous versions of this work and for their valuable suggestions. [99]{} M. Lachièze-Rey & J.-P. Luminet, *Phys. Rep.* [**254**]{}, 135 (1995).\ B. Roukema, *Méthodes aux grandes échelles pour mesurer la topologie (globale) de l’Universe*, astro-ph/9801225 (1998).\ G.D. Starkman, *Class. Quantum Grav.* [**15**]{}, 2529 (1998). See also the other articles in this special issue featuring invited papers from the Topology of the Universe Conference, Cleveland, Ohio, October 1997. Guest editor: Glenn D. Starkman.\ R. Lehoucq, J.-P. Uzan & J.-P. Luminet, *Astron. Astrophys.* [**363**]{}, 1–8 (2000).\ J. Levin, *Phys. Rep.* [**365**]{}, 251 (2002).\ K.T. Inoue, *Exploring Topology of the Universe in the Cosmic Microwave Background*, PhD Thesis, Kyoto University, astro-ph/0103158 (2001). G.I. Gomero, M.J. Rebouças & R. Tavakol, *Class. Quantum Grav.* [**18**]{}, 4461–4476 (2001).\ G.I. Gomero, M.J. Rebouças & R. Tavakol, *Class. Quantum Grav.* [**18**]{}, L145–L150 (2001).\ E. Gausmann, R. Lehoucq, J.-P. Luminet, J.-P. Uzan & J. Weeks, *Class. Quantum Grav.* [**18**]{}, 5155– (2001).\ G.I. Gomero, M.J. Reboucas & R. Tavakol, *Int. J. Mod. Phys. A* [**17**]{}, 4261–4272 (2002).\ J.R. Weeks, *Detecting topology in a nearly flat hyperbolic universe*, astro-ph/0212006 (2002). G.I. Gomero & M.J. Rebouças, *Phys. Lett. A* [**311**]{}, 319–330 (2003). B.F. Roukema & A. Edge, *Mon. Not. R. Astron. Soc.* [292]{}, 105–112 (1997). A. Bernui & G.I. Gomero, *Local Noise Correlations in Cosmic Topology*, in preparation. G.I. Gomero, *Int. J. Mod. Phys. A* [**17**]{}, 4281–4286 (2002). J.A. Wolf, *Spaces of Constant Curvature*, fifth ed., Publish or Perish Inc., Delaware (1984). D. Stevens, D. Scott & J. Silk, *Phys. Rev. Lett.* [**71**]{}, 20–23 (1993).\ A. de Oliveira–Costa & G.F. Smoot, *Ap. J.* [**448**]{}, 477–481 (1993).\ A. de Oliveira–Costa, G.F. Smoot & A.A. Starobinsky, *Ap. J.* [**468**]{}, 457–461 (1993).\ B.F. Roukema, *MNRAS* [**312**]{}, 712–723 (2000). K.T. Inoue, *Class. Quantum Grav.* [**18**]{}, 1967–1978 (2001).\ K.T. Inoue, *Phys. Rev.* [**D67**]{}, 043003 (2003). J. Levin, E. Scannapieco & J. Silk, *Phys. Rev.* [**D58**]{}, 103516 (1998).\ G. Rocha et al., [**astro-ph/0205155**]{}, submitted to *MNRAS*. V. Blanl[œ]{}il & B.F. Roukema, Editors of the electronic proceedings of the *Cosmological Topology in Paris 1998*, astro-ph/0010170 (2000). See the article of U.F. Wichoski, *Topological Images of the Galaxy*.\ B.F. Roukema, *The Milky Way as the Ultimate Extragalactic Source*, astro-ph/0106272 (2001). B.F. Roukema, *Clusters of Galaxies as Standard Candles for Global Observational Cosmology*, astro-ph/0105548 (2001). B.F. Roukema, A. Marecki, S. Bajtlik & R.E. Spencer, *Topological Lensing: Is the matter both here and there?*, astro-ph/ 0111052 (2001). [^1]: german@ift.unesp.br [^2]: Provided that the scale of compactification is small enough (see [@Detect] and [@GR02]). [^3]: Note that the origin of a coordinate system is implicitly determined by the axes of rotation of the orthogonal transformations in (\[Rot3\]), and can be taken as the centre of the fundamental polyhedron for the corresponding manifold. Moreover, this origin does not necessarily coincide with the position of our galaxy. [^4]: Actually, it can be done much more than that. If the topology of the Universe turns out to be of any of the classes ${\mathcal{G}_2}-{\mathcal{G}_6}$, the triples $\{D_0,D_1,D_2\}$ and $\{E_0,E_1,E_2\}$ would be enough to decide which topology our Universe has, and except in the case of ${\mathcal{G}_2}$ and a configuration in ${\mathcal{G}_6}$, it would be possible to specify completely the parameters of the manifold that models the spatial sections of the spacetime. [^5]: Strictly speaking, this *closeness* relation is not a necessary condition, but observationally it would be simpler to look for other triples of images in the neighborhood of the first one. [^6]: Note however that if there exists a reflection taking one triangle to the other, in order to settle definitely the orientability of space, it would be necessary to identify a fourth triple of multiple images.

--- abstract: 'We study the interaction between the microstructures of regular AdS Hayward black hole using Ruppeiner geometry. Our investigation shows that the dominant interaction between the black hole molecules is attractive in most of the parametric space, as in van der Waals system. However, in contrast to the van der Waals fluid, there exists a weak dominant repulsive interaction for small black hole phase in some parameter domain. This result clearly distinguishes the interactions in a magnetically charged black hole from that of van der Waals fluid. However, the interactions are universal for charged black holes since they do not dependent on magnetic charge or temperature.' author: - 'Naveena Kumara A.' - 'Ahmed Rizwan C.L.' - Kartheek Hegde - 'Ajith K.M.' bibliography: - 'BibTex.bib' title: 'Repulsive Interactions in the Microstructure of Regular Hayward Black Hole in Anti-de Sitter Spacetime' --- [ ]{} [ ]{} [ ]{} [ ]{} Introduction ============ In recent years the subject of black hole chemistry has become an attractive window to probe the properties of AdS black holes. In black hole chemistry, the negative cosmological constant of the AdS spacetime is identified as the thermodynamic variable pressure to study the phase transition of the AdS black holes [@Kastor:2009wy; @Dolan:2011xt]. Interestingly the phase transition of certain AdS black hole analytically resembles that of the van der Waals system [@Kubiznak2012; @Gunasekaran2012; @Kubiznak:2016qmn]. Recently by studying the phase transitions, attempts were made to investigate of the underlying microscopic properties of the AdS black holes [@Wei2015; @Wei2019a; @Wei2019b; @Guo2019; @Miao2017; @Zangeneh2017; @Wei:2019ctz; @Kumara:2019xgt; @Kumara:2020mvo; @Xu:2019nnp; @Chabab2018; @Deng2017; @Miao2019a; @Chen2019; @Du2019; @Dehyadegari2017]. In these researches, the geometric methods were the key tools in probing the microscopic details of the black holes. Contrast to the statistical investigation in ordinary thermodynamics the approach here is upside down, the macroscopic thermodynamic details are ingredients for the microscopic study [@Ruppeinerb2008]. The technique is inspired by the applications of thermodynamic geometry in ordinary thermodynamic systems [@Ruppeiner95; @Janyszek_1990; @Oshima_1999x; @Mirza2008; @PhysRevE.88.032123]. Recently, a general Ruppeiner geometry framework is developed from the Boltzmann entropy formula, to study the black hole microstructure [@Wei2019a]. The fluctuation coordinates are taken as the temperature and volume, and a universal metric was constructed in that scheme. When this methodology is applied to the van der Waals fluid only a dominant attractive interaction was observed, as it should be. However, when the same methodology is used for the RN AdS black hole, a different result was obtained. In a small parameter range, the repulsive interaction is also found in addition to the dominant attractive interaction between the black hole molecules [@Wei2019a; @Wei2019b]. Even so, in the case of five-dimensional neutral Gauss-Bonnet black hole only a dominant attractive interaction was discovered, which is similar to van der Waals fluid [@Wei:2019ctz]. Therefore, in general, the nature of the black hole molecular interactions are not universal. In our recent work [@Kumara:2020mvo], we have observed that there exists a repulsive interaction in regular Hayward black hole, like that of RN AdS case. In the present work, we will make a detailed study of the previously observed repulsive interaction. The primary motivation for our research is due to the great interest on the regular black holes in black hole physics since they do not possess singularities. Wide variety of regular black holes exists, ranging from the first solution given by Bardeen [@Bardeen1973], the later versions [@AyonBeato:1998ub; @AyonBeato:2000zs], to the one on which we are interested, the Hayward black hole [@Hayward:2005gi]. (We suggest the readers to go through our previous article [@Kumara:2020mvo] for the chronological discussion on this). Hayward black hole is the solution to Einstein gravity non-linearly coupled to an electromagnetic field, which carries a magnetic charge. In this article, we probe the phase structure and repulsive interactions in the microstructure of this magnetically charged AdS black hole. The article is organised as follows. After a brief introduction, we discuss the phase structure of the black hole in section \[secone\]. Then the Ruppeiner geometry for the black hole is constructed for microstructure scrutiny (section \[sectwo\]). Then we present our findings in section \[secthree\]. Phase structure of the Hayward AdS Black Hole {#secone} ============================================= The Hayward black hole solution in the four dimensional AdS background is given by [@Fan:2016hvf; @Fan:2016rih] (see [@Kumara:2020mvo] for a brief explanation), $$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega ^2,$$ where $d\Omega ^2=d\theta ^2+\sin \theta ^2d\phi ^2$ and the metric function, $$f(r)=1-\frac{2 M r^2}{g^3+r^3}+\frac{8}{3} \pi P r^2.$$ We study the phase structure in the extended phase space where the cosmological constant $\Lambda$ gives the pressure term $P=-\Lambda /8\pi$. The parameter $g$ is related to the total magnetic charge of the black hole $Q_m$ as, $$Q_m=\frac{g^2}{\sqrt{2\alpha}},$$ where $\alpha$ a free integration constant. The thermodynamic quantities temperature, volume and entropy of the black hole are easily obtained to be, $$T=\frac{f'(r_+)}{4\pi}=\frac{2 P r^4}{g^3+r^3}-\frac{g^3}{2 \pi r \left(g^3+r^3\right)}+\frac{r^2}{4 \pi \left(g^3+r^3\right)}; \label{temperature}$$ $$V=\frac{4}{3} \pi \left(g^3+r^3\right) \quad \text{and} \quad S=2 \pi \left(\frac{r^2}{2}-\frac{g^3}{r}\right).$$ These results are consistent with the first law $$dM=TdS+\Psi dQ_m+VdP+\Pi d \alpha,$$ and the Smarr relation, $$M=2(TS-VP+\Pi \alpha)+\Psi Q_m.$$ The heat capacity of the black hole system at constant volume is, $$C_V=T\left( \frac{\partial S}{\partial T}\right)_V=0. \label{cv}$$ Inverting the expression for the Hawking temperature (\[temperature\]) we get the equation of state, $$P=\frac{g^3}{4 \pi r^5}+\frac{g^3 T}{2 r^4}-\frac{1}{8 \pi r^2}+\frac{T}{2 r}.$$ From the state equation one can see that the black hole shows critical behaviour similar to van der Waals system. This often interpreted as the transition between a small black hole and a large black hole phases. In our earlier studies [@Kumara:2020mvo], we have shown that an alternate interpretation is possible, using Landau theory of continuous phase transition, where the phase transition is between the black hole phases at different potentials. In this alternate view the black hole phases, namely high potential, intermediate potential and low potential phases, are determined by the magnetic charge. In either of these interpretations the phase transition can be studied by choosing a pair of conjugate variables like $(P-V)$ or $(T-S)$. With the conjugate pair $(P,V)$, the Maxwell’s equal area law has the form, $$P_0(V_2-V_1)=\int _{V_1}^{V_2}PdV. \label{equalarea}$$ Since there exists no analytical expression for the coexistence curve for Hayward AdS black hole we seek numerical solutions most of the time. For that, we obtain the key ingredient from the Maxwell’s equal area law. Using the equation (\[equalarea\]) and expressions for $P_0(V_1)$ and $P_0(V_2)$ from equation of state we get, $$r_2=g\left[ \frac{x \left(x^3+6 x^2+6 x+1\right)+\sqrt{y}}{x^4}\right]^{1/3}, \label{r2eqn}$$ $$P_0=\frac{3 \left[\frac{\sqrt{y}+ x \left(x^3+6 x^2+6 x+1\right)}{x^4}\right]^{1/3} \left[\left(-2 x^4-11 x^3-20 x^2-11 x-2\right) \sqrt{y}+ z\right]}{16 \pi g^2 x \left(x^2+4 x+1\right) \left(3 x^2+4 x+3\right)^2}, \label{peqn}$$ $$T_0=\frac{\left[\frac{\sqrt{y}+x \left(x^3+6 x^2+6 x+1\right)}{x^4}\right]^{2/3} \left[u-\left(x^3+4 x^2+4 x+1\right) \sqrt{y}\right]}{4 \pi g x \left(3 x^4+16 x^3+22 x^2+16 x+3\right)}. \label{teqn}$$ Where $$y=x^2 \left(x^6+12 x^5+54 x^4+82 x^3+54 x^2+12 x+1\right),$$ $$z=x \left(2 x^7+23 x^6+104 x^5+213 x^4+213 x^3+104 x^2+23 x+2\right),$$ $$u=x \left(x^6+10 x^5+37 x^4+54 x^3+37 x^2+10 x+1\right).$$ We have taken $x=r_1/r_2$, where $r_1$ and $r_2$ are the radii of black holes for first order phase transition points. The critical values are readily obtained by setting $x=1$, $$T_{c}=\frac{\left(5 \sqrt{2}-4 \sqrt{3}\right) \left(3 \sqrt{6}+7\right)^{2/3}}{4\ 2^{5/6} \pi g},$$ $$P_{c}=\frac{3 \left(\sqrt{6}+3\right)}{16\ 2^{2/3} \left(3 \sqrt{6}+7\right)^{5/3} \pi g^2},$$ and $$V_c=4 \left(2 \sqrt{6}+5\right) \pi g^3.$$ The reduced thermodynamic variables are defined as, $$T_r=\frac{T}{T_c},\quad P_r=\frac{P}{P_c}, \quad V_r=\frac{V}{V_c}.$$ Using these we can write the equation of state in the reduced parameter space, $$P_r=\frac{2^{2/3} \left(3 \sqrt{6}+7\right)^{5/3} \left[V_r \left(\left(6 \sqrt{6}+14\right)^{2/3} T_r \left(3 \left(2 \sqrt{6}+5\right) V_r-1\right)^{1/3}-4 \sqrt{6}-10\right)+2\right]}{\left(\sqrt{6}+3\right) \left[3 \left(2 \sqrt{6}+5\right) V_r-1\right]^{5/3}}. \label{reducedeos}$$ The reduced equation of state is independent of the magnetic charge parameter $g$. From the reduced state equation we obtain the spinodal curve, which separates metastable phases from the unstable phase, using the condition, $$\left( \partial _{V_r} P_r \right)_{T_r}=0.$$ The explicit form of spinodal curve is, $$T_{rsp}=\frac{ 2^{4/3} \left(2 \sqrt{6}+5\right) \left[\left(2 \sqrt{6}+5\right) V_r-2\right]}{\left(3 \sqrt{6}+7\right)^{2/3} \left[\left(2 \sqrt{6}+5\right) V_r+1\right] \left[3 \left(2 \sqrt{6}+5\right) V_r-1\right]^{1/3}}.$$ Solving this for $V_r$ and substituting in equation (\[reducedeos\]) we obtain the curve in $P-V$ plane. The spinodal curve along with the coexistence curve display the stable, unstable and metastable phases of the black hole. The coexistence curve is obtained numerically using the equations (\[r2eqn\]), (\[peqn\]) and (\[teqn\]). The spinodal and coexistence curves are shown together in fig \[fig1\]. Actually, by fitting the coexistence curve in $P-T$ plane we have obtained the following expression, $$\begin{aligned} P_r=&5.622 \times 10^{-7}-5.539\times 10^{-5} T_r+0.693 T_r^2+0.1365 T_r^3+0.1966 T_r^4\nonumber \\ &-0.4255 T_r^5+1.134 T_r^6-1.698 T_r^7+1.621 T_r^8-0.8651 T_r^9+0.2085 T_r^{10}.\end{aligned}$$ The small black hole, large black hole and supercritical black hole phases are depicted in fig \[HPT\]. The coexistence curve separates the small black hole and large black hole phases. It terminates at the critical point, after which the distinction between the SBH and LBH states is not possible, hence corresponds to the supercritical black holes. The region between the coexistence curve and spinodal curve corresponds to the metastable states, namely the supercooled LBH and the superheated SBH, which are shown in the $T-V$ diagram. An observable feature in these diagrams is that the spinodal and coexistence curves meet each other at the critical point. Another important property associated with the spinodal curve is that the Ruppeiner scalar curvature diverges at that curve (section \[sectwo\]). Now, we would like to study the change in volume at the black hole phase transition as a function of temperature and pressure. Using equation (\[r2eqn\]), we make the functional change $V(r)\rightarrow V(x)$ to obtain a parametric expression for $\Delta V_r$. The parametric expression of $\Delta V_r$ along with that of $T_r$ and $P_r$ (equations \[teqn\] and \[peqn\]) are used to plot fig \[Deltav\], which gives the behaviour of $\Delta V_r$. From the figure \[Deltav\] it is clear that, $\Delta V_r$ decreases with increase in both temperature and pressure. It approaches zero at the critical point ($T_r=1$ and $P_r=1$). The behaviour near the critical point is, $$\Delta V_r \sim (1-T_r)^{1/2} \sim (1-P_r)^{1/2}.$$ This suggests that the change in volume $\Delta V_r$ can serve as the order parameter to characterise the black hole phase transition, with the universal critical exponent $1/2$. Microstructure of the Hayward AdS Black Hole {#sectwo} ============================================ In this section we examine the microstructure of the black hole using Ruppeiner geometry in which $T$ and $V$ are taken as fluctuation coordinates. The line element in these parameter space has the form [@Wei2019a], $$dl^2=\frac{C_V}{T^2}dT^2-\frac{\left( \partial _V P\right)_T }{T}dV^2. \label{line}$$ The heat capacity $C_V$ vanishes for the Hayward AdS black hole (equation \[cv\]). This makes the line element (\[line\]) singular, and hence the corresponding geometry will not give the information regarding the microstructure of the black hole. Therefore the normalised scalar curvature is used for studying the microscopic interactions, $$R_N=C_V R.$$ From a straightforward calculation, for the Hayward AdS black hole we obtain, $$R_N=\frac{\left(8 \pi g^3-V\right) \left\{ 8 g^3 \left[\pi ^{5/3} T \left(6 V-8 \pi g^3\right)^{1/3}+\pi \right]+V \left[2 \pi ^{2/3} T \left(6 V-8 \pi g^3\right)^{1/3}-1\right]\right\}}{2 \left\{4 \pi g^3 \left[\pi ^{2/3} T \left(6 V-8 \pi g^3\right)^{1/3}+2\right]+V \left[\pi ^{2/3} T \left(6 V-8 \pi g^3\right)^{1/3}-1\right]\right\}^2}.$$ In terms of the reduced parameters, $$R_N= \frac{4 \left[\left(2 \sqrt{6}+5\right) V_r-2\right]\left[-A(T_r,V_r)+2 \left(2 \sqrt{6}+5\right) V_r-4\right]}{\left[A-4 \left(2 \sqrt{6}+5\right) V_r+8\right]^2},$$ where, $$A(T_r,V_r)=2^{1/6} \left(3 \sqrt{6}+7\right)^{2/3} T_r \left(\sqrt{2} V_r+5 \sqrt{2}-4 \sqrt{3}\right) \left(3 \left(2 \sqrt{6}+5\right) V_r-1\right)^{1/3}.$$ Similar to the case of charged AdS black hole and Gauss Bonnet black hole $R_N$ is independent of $g$. The behaviour of $R_N$ with reduced volume $V_r$ for fixed temperature is studied in fig (\[RN\]). For $T_r<1$, below critical temperature, $R_N$ has two negative divergence points. They come nearer as the temperature increases and merge together at $V_r=1$ for $T_r=1$. These divergences do not exist for temperatures greater than the critical value. We see that always there exist small regions where the curvature scalar is positive (shown in inlets). We need to examine whether these regions are thermodynamically stable. Setting $R_N=0$ we get, $$T_0=\frac{T_{rsp}}{2}=\frac{2^{5/6} \left[\left(2 \sqrt{6}+5\right) V_r-2\right]}{\left(3 \sqrt{6}+7\right)^{2/3} \left(\sqrt{2} V_r+5 \sqrt{2}-4 \sqrt{3}\right) \left[3 \left(2 \sqrt{6}+5\right) V_r-1\right]}.$$ This is the sign-changing temperature, which is half of the spinodal curve temperature as in vdW system, RN AdS and Gauss-Bonnet black holes. Another solution for this is, $$V_r=\frac{2}{2 \sqrt{6}+5} \equiv V_0.$$ The normalised curvature scalar $R_N$ diverges along the spinodal curve. The regions under the spinodal curve for $V_r>V_0$, $R_N$ is positive. This region corresponds to the coexistence phase of SBH and LBH, similar to van der Waals fluid’s coexistence phase. Everywhere below $V_0$, $R_N$ is positive, including region below and above the coexistence curve. The region under the coexistence curve is the same as the previous case, a coexistence phase. However in the region above the curve, small black hole phase, we can safely say that the black hole molecules possess repulsive interaction. Therefore in Hayward black hole for a small parameter range there exist dominant repulsive interaction. This result is similar to RN-AdS black hole and in contrast to five-dimensional neutral Gauss-Bonnet, where there is no repulsive interaction in the latter case. Finally, we consider the behaviour of the scalar curvature $R_N$ along the coexistence curve. Since there exists no analytical expression for the coexistence curve, the analytic study of the curvature scalar behaviour near the critical point is not possible. The numerical solution is obtained and shown in fig \[HRT\]. Both the SBH branch and LBH branch of $R_N$ have the divergence near the critical point. For a large black hole, the sign of $R_N$ is always negative and hence the microscopic interaction is always attractive. Interestingly, for the small black hole, there is a lower temperature range where $R_N$ is positive (zoomed-in in the inlet). This indicates a repulsive interaction between the black hole molecules. From this, we can conclude that in the low-temperature regime the microstructure, as well as microscopic interaction of the black hole, changes drastically during the phase transition. Whereas in the high-temperature range only microstructure changes and the nature of interaction remains attractive in both phases. These results are strikingly different from that of van der Waals fluid, where the dominant interaction among the molecules is always attractive. Discussions {#secthree} =========== In this paper, we have studied the phase transitions and microstructure of the Hayward AdS black hole. The microscopic properties are analysed from the behaviour of Ruppeiner curvature scalar along the coexistence curve. Since an analytical expression for the coexistence curve is not feasible we have carried out our investigation numerically. In the first part of the paper, we probed the phase structure of the black hole using the coexistence curve in $P_r-T_r$ and $T_r-V_r$ planes. Along with this the spinodal curve also displayed, which enable us to identify the metastable phases of the black holes, namely the superheated small black hole and the supercooled large black hole. It is shown that the change in volume $\Delta V_r$ during the small black hole - large black hole phase transition can serve an order parameter to describe the same. The behaviour of $\Delta V_r$ has a critical exponent $1/2$ which is universal. In the second part of this article, we have focused on the Ruppeiner geometry of the black hole. We have adopted the definition of curvature scalar given in the ref. [@Wei2019a], where the fluctuation coordinates are temperature and volume. The normalised curvature scalar diverges to the negative infinity at the critical point. Even though the black hole shows van der Waals like phase transition, the microstructure properties differ in some aspects. In van der Waals fluid the dominant interaction among the constituent molecules is always attractive, which does not change during the phase transition. The change in microstructure does not lead to any change in the nature of microscopic interaction. However, in Hayward black hole there exists a domain, low-temperature range for the small black hole, where the dominant interaction between the black hole molecules is repulsive. This is inferred from the positive sign of the normalised curvature scalar. During the phase transition, in this temperature range, the microscopic interaction of the black hole changes significantly. This result is similar to what is observed in RN AdS black hole and in contrast to the five-dimensional neutral Gauss-Bonnet black hole, where the interaction is always attractive like van der Waals fluid. To conclude, the magnetic charge in the Hayward black hole plays a similar role as the electric charge in RN AdS black hole in contributing to the microstructure. We believe that this is another significant step in understanding the black hole microstructure properties. Author N.K.A. and A.R.C.L. would like to thank U.G.C. Govt. of India for financial assistance under UGC-NET-SRF scheme.

--- abstract: 'A key property of Majorana zero modes is their protection against local perturbations. In the standard picture this is the result of a high degree of spatial wavefunction non-locality. A careful quantitative definition of non-locality in relation to topological protection goes beyond purely spatial separation, and must also take into account the projection of wavefunction spin. Its form should be physically motivated from the susceptibility of the Majorana mode to different local perturbations. Non-locality can then be expressed as one of various wavefunction overlaps depending on the type of perturbation. We quantify Majorana non-locality using this approach and study its dependence with Majorana nanowire parameters in several classes of experimentally relevant configurations. These include inhomogeneous nanowires with sharp and smooth depletion and induced pairing, barriers and quantum dots. Smooth inhomogeneities have been shown to produce near-zero modes below the critical Zeeman field in the bulk. We study how accurately their non-locality can be estimated using a purely local measurement on one end of the nanowire, accessible through conventional transport experiments. In uniform nanowires the local estimator quantifies non-locality with remarkable accuracy, but less so in nanowires with inhomogeneities greater than the superconducting gap. We further analyse the Majorana wavefunction structure, spin texture and the spectral features associated with each type of inhomogeneity. Our results highlight the strong connection between internal wavefunction degrees of freedom, non-locality and protection in smoothly inhomogeneous nanowires.' author: - 'Fernando Peñaranda$^1$, Ramón Aguado$^2$, Pablo San-Jose$^2$, Elsa Prada$^1$' bibliography: - 'biblio.bib' title: 'Quantifying wavefunction non-locality in inhomogeneous Majorana nanowires' --- Introduction ============ A unique electronic state by the name of Majorana zero mode (MZM)[@Kitaev:PU01] associated with topological superconductivity has been the subject of intense research recently. The pace picked up after the first experimental hints of its existence were reported six years ago [@Mourik:S12] in so-called Majorana nanowires, i.e. semiconducting nanowires with induced superconductivity and spin-orbit coupling subjected to a Zeeman field above a critical value $B>B_c$. These pioneering experiments were quickly followed by others [@Deng:NL12; @Das:NP12; @Churchill:PRB13; @Lee:NN14; @Deng:S16; @Zhang:N18; @Grivnin:A18], mostly revolving around robust zero energy midgap states in tunneling spectroscopy. ![**Inhomogeneous nanowires.** Sketch of an inhomogeneous nanowire, hosting Majorana zero modes of overlap $\Omega_s$. The overlap may be estimated by a local quantity $\eta$ measured by a local probe. Five types of inhomogeneous profiles of the electrostatic potential $\phi(x)$ and pairing $\Delta(x)$ are considered: uniform, S’S, NS, Barrier-S and Dot-S. The latter two are subtypes of the general NS case. []{data-label="fig:sketch"}](sketch.pdf){width="\columnwidth"} The reason for all the excitement is manyfold. From a technological perspective, MZM are viewed by many as a possible foundation of a new form of quantum computer architecture that could achieve topologically protection against some forms of logic errors [@Nayak:RMP08; @Cheng:PRB12a; @Sarma:NQI15]. From the point of view of fundamental physics, standard theory predicts, moreover, that a MZM should exhibit some truly exotic properties. It is a zero energy state inside a superconducting gap that is typically localised in space [@Kitaev:PU01; @Oreg:PRL10; @Lutchyn:PRL10]. The most common place to find a MZM is at boundaries between regions of distinct electronic topology [@Qi:RMP11; @Aguado:RNC17; @Lutchyn:NRM18]. A MZM behaves in many respects as half an electron, with each MZM emerging simultaneously with a second Majorana partner located at some other position in the system. Two such “electron-halves” form a rather unusual, spatially non-local fermion [@Jackiw:PS12; @Fu:PRL10; @Semenoff:C16]. The non-local nature of this fermion pins it to zero energy regardless of any local perturbations performed on either of the MZMs. This is often called topological protection [@Cheng:PRB12a], although protection through non-locality is perhaps a better description, as will be argued here. Each MZM also exhibits non-Abelian braiding statistics upon exchange [@Nayak:RMP08]. They are hence sometimes called fractionalised non-Abelian Ising anyons [@Bonderson:PRB13; @Aasen:PRX16]. All these exotic properties are expected to be remarkably robust, and to not require any fine tuning of the system’s state. The reason is that, at least within standard theory, they are a consequence of the different band topology at either side of the boundary they inhabit, which does not depend on microscopic details. However, while the MZM analysis in terms of band-topology can be made rigorous for boundaries between semi-infinite systems, it becomes problematic when applied to closed, finite-sized systems, for example. It also fails to account for the properties of so-called trivial Andreev zero modes, also known as pseudo-MZMs or quasi-MZMs, predicted to appear in smoothly inhomogeneous nanowires without any obvious form of band-topological order [@Kells:PRB12; @Prada:PRB12; @Liu:PRB17; @Moore:PRB18; @Setiawan:PRB17; @Moore:18; @Liu:18; @Vuik:18]. An example of such states relevant to the present work arises in fully trivial nanowires ($B<B_c$) hosting a sufficiently smooth normal-superconductor interface, wherein modes of arbitrarily small energy localise. All the experimental evidence so far of conventional topological MZMs can be mimicked by pseudo-MZMs. This realisation has given rise to an intense debate regarding possible loopholes in the interpretation of the experimental observations, and on the protection, or lack thereof, of the observed zero modes. Intriguingly, these states share most properties with MZMs at the end of a uniform $B>B_c$ topological nanowire, except in one crucial aspect: they may be highly local in space. Since spatial non-locality is conventionally associated to the resilience of MZMs against error-inducing local perturbations, it is often argued that pseudo-MZMs, unlike MZMs, would not be useful for topological quantum computation. Perhaps for this reason the idea that two distinct types of zero modes, trivial and non-trival, can exist in real samples has taken hold in recent literature. Instead of classifying states into trivial and non-trivial [we will characterise them in terms of their resilience against perturbations]{} [@Cheng:PRB12a; @Aseev:A18; @Knapp:PRB18]. The associated susceptibility is expressed as different spatial overlap integrals $\Omega$ of the Majorana Nambu-spinorial wavefunctions, depending on the type of perturbation [@Penaranda:18]. Despite all of these integrals expressing non-locality, the way the internal spin degrees of freedom combine in the overlap integral is different. This leads to several [measures]{} of non-locality $0\leq 1-\Omega\leq 1$ that go beyond purely spatial separation, and are directly connected to protection [against]{} perturbations. This quantity $1-\Omega$ provides an extension of the concept of *topological* non-triviality, but in contrast to the latter it is no longer an all-or-nothing proposition, but a matter of degree. It has also the distinct advantage of being generally applicable to zero modes in arbitrary isolated systems, [not only semi-infinite ones]{}. From this point of view, $1-\Omega$ [is proposed here as]{} the essential figure of merit of a given MZM, ultimately associated to its resilience against decoherence [@Cheng:PRB12a; @Penaranda:18]. Beyond this, the distinction between ‘proper’ MZMs and pseudo-MZMs ceases to make sense. As an aside, we note that an alternative theoretical framework has been recently proposed that allows to recover a well-defined and unambiguous trivial/non-trivial classification within this continuum of MZMs of isolated systems. It defines the topological nature of these zero modes in more general terms by considering the exceptional-point topology of the non-hermitian Hamiltonians that describe the system when it is coupled to a reservoir [@Avila:A18; @Pikulin:JL12; @Pikulin:PRB13]. In essence, the coupling to the reservoir makes the system infinite, so that it is once more amenable to a rigorous topological classification. This approach is related to band topology, but is more general, and in it the degree of non-locality of the isolated states studied here plays a crucial role. In this work we further consider the practical problem of quantifying and detecting the degree of non-locality of a given zero mode using purely local measurements by local spectroscopic probes. These include e.g. a tunnel contact or a quantum dot coupled to a certain point in a Majorana nanowire, a setup routinely used today in the lab to perform tunnelling spectroscopy, see Fig. \[fig:sketch\]. This challenge seems a priori hopeless since quantifying non-locality involves knowing the distribution of the zero mode throughout an extended region in space, not just at one point. We show, however, that the spatial distribution of subgap states is not completely arbitrary in realistic systems, but span a finite volume in the space of all possible wavefunctions. Within this constraint, local measurements remain highly correlated with the actual Majorana wavefunction overlap throughout the system. This work is organised as follows. Section \[sec:nonlocality\] presents the basic concepts and definitions of overlaps and local estimators, and the five types of nanowire configurations to be studied, see Fig. \[fig:sketch\]. Section \[sec:uniform\] is devoted to uniform nanowires. The basic Lutchyn-Oreg model is presented, together with its phenomenology regarding spectrum, zero-mode overlaps and their correlation with local estimators. Sections \[sec:S’S\] and \[sec:NS\] present the corresponding analysis in inhomogeneous superconductor-superconductor and normal-superconductor nanowires, respectively. In the latter case we also analyse specific barrier-superconductor and quantum dot-superconductor configurations, of relevance to many experimental devices. We finally present, in Sec. \[sec:spin\], a discussion of the spatial spin density of the Majorana wavefunction in the various types of inhomogeneous nanowires as a function of their smoothness. This will allow us to distinguish between two characteristic types of wavefunctions, that of conventional abrupt Majoranas at sharp insulating interfaces, and that of Gaussian-like smooth Majoranas that develop at smooth superconductor interfaces. Finally, in Sec. \[sec:conclusion\] we conclude. Non-locality and local estimator {#sec:nonlocality} ================================ Majorana basis -------------- Consider a generic subgap eigenstate $c$ in a quasi-1D superconductor $$c = \int dx \sum_\sigma u_\sigma(x)\psi_\sigma(x) + v_\sigma(x)\psi^\dagger_\sigma(x).$$ Here $\sigma$ denotes spin projections on a given axis, chosen in this work as the $x$ axis along which a Zeeman field will later be applied. This fermionic state can be decomposed into two Majorana components $\gamma_{L}$ and $\gamma_{R}$ $$\begin{aligned} c &=& \frac{\gamma_L+i\gamma_R}{\sqrt{2}},\nonumber\\ c^\dagger &=& \frac{\gamma_L-i\gamma_R}{\sqrt{2}}, \label{decomposition}\end{aligned}$$ so that $$\gamma_{L,R} = \int dx \sum_\sigma u^{L,R}_\sigma(x)\psi_\sigma(x) + \left[u^{L,R}_\sigma(x)\right]^*\psi^\dagger_\sigma(x).\nonumber$$ By definition, $\gamma_{L,R}$ are self-conjugate $\gamma_{L,R}=\gamma^\dagger_{L,R}$ (Majorana reality condition). Their wavefunctions $u^{L,R}_\sigma(x)$ can be expressed in terms of the particle and hole wavefunction components $u_\sigma(x), v_\sigma(x)$ of eigenstate $c$ as $$\begin{aligned} u^{L}_\sigma(x) &=& \frac{u_\sigma(x) + v_\sigma(x)}{\sqrt{2}}, \nonumber\\ u^{R}_\sigma(x) &=& \frac{u_\sigma(x) - v_\sigma(x)}{i\sqrt{2}} . \label{mwf}\end{aligned}$$ All wavefunctions $u^{L,R}_\sigma(x), u_\sigma(x), v_\sigma(x)$ are normalised. In particular $$\int dx \|\mathbf{u}^{L,R}(x)\|^2=1. \label{normalisation}$$ where $\mathbf{u}^{L,R}$ denotes the spinor of $u_\sigma^{L,R}$ components, and the $\|\dots\|$ denotes its norm. Note that the Majorana decomposition of Eqs. (\[decomposition\], \[mwf\]) is possible for *any* Andreev state with finite energy $E_0$, not only for those with zero energy. Only if the subgap eigenstate $c$ has zero energy, the $\gamma_{L,R}$ will themselves be zero energy eigenstates. This is true *regardless* of their spatial non-locality or their topological/trivial origin. In this work we will always call these self-conjugate $\gamma_{L,R}$ zero modes, *Majorana* zero modes (MZMs), since they satisfy the Majorana reality condition. We thus refer to MZMs independently of whether the system has a trivial or non-trivial band topology. Protection and wavefunction non-locality ---------------------------------------- The standard definition of topological protection [@Sarma:NQI15] relies on Majorana midgap states with a sufficiently large gap to higher excitations and an exponentially suppressed energy, $E_0\sim e^{-L/\xi}$ resulting from spatial separation of Majorana wavefunctions, with $L$ the wire’s length and $\xi$ the Majorana coherence length. This allows in practice to achieve a degenerate [ground state. The spatial]{} non-locality of the Majoranas leads also to an exponentially suppressed sensitivity to arbitrary local perturbations. This is ultimately the reason why non-locality is such a key Majorana property. This picture has to be extended in more general situations, wherein zero modes arise due to smooth inhomogeneities. In such case, the connection between protected ground state degeneracy and non-locality becomes less obvious, [as one can have stable zero modes with highly]{} overlapping wavefunctions. The question remains as to whether [these smoothly confined MZMs]{} are susceptible to local perturbations. The specific type of perturbation and the spinorial internal structure $\mathbf{u}^{L,R}(x)$ become crucial in this regard. They lead to different forms of the susceptibility, expressed as overlap integrals that give a sense of non-locality, but in which internal degrees of freedom combine in different ways. In this subsection we analyse these different forms. To make connection with published literature, we start by considering the response to global perturbations to the chemical potential $\mu$ in the nanowire. In Ref. [@Ben-Shach:PRB15] it was shown that the zero-temperature change in the energy $E_0$ of a subgap Andreev state $c$ in response to such change is the state’s dimensionless charge $\partial E_0/\partial\mu = \delta N =|Q|/e=\langle c^\dagger c-c c^\dagger\rangle$ which in the Majorana basis takes the form of a wavefunction overlap, $$\delta N = \left|\int dx\,\mathbf{u}^L(x)\cdot\mathbf{u}^R(x)\right|.$$ Here $|\dots|$ denotes the absolute value [and $\cdot$ the scalar product]{}. It becomes clear that for well separated, exponentially decaying Majorana wavefunction $\mathbf{u}^{L,R}(x)$ separated by a distance $L$, as corresponds to the MZMs of uniform topological nanowires, this susceptibility $\delta N$ is exponentially suppressed [@Ben-Shach:PRB15], $\delta N\sim e^{-L/\xi}$, just like $E_0$. We next consider the susceptibility to local noise, such as spatially-uncorrelated changes of electrostatic potential or Zeeman field perturbations. For completely arbitrary local *spin-uncorrelated* noise $\langle V_\sigma(x)V_{\sigma'}(x')\rangle = a_0V_0^2\delta(x-x')\delta_{\sigma\sigma'}$, where $a_0$ is some short lengthscale, one obtains a susceptibility $\Omega_s=\partial E_0/\partial V_0$ of the form $$\Omega_s = \int dx \sum_\sigma\left|u_\sigma^L(x)u_\sigma^R(x)\right|. \label{OmegaS}$$ Despite this again taking the form of a spatial overlap integral, [the absolute value]{} and the spin degree of freedom enter differently as compared to $\delta N$. We note that this expression is not SU(2) symmetric, as assuming uncorrelated spin fluctuations requires one to define a preferred spin quantization axis. It is the relevant susceptibility when the system is subject to a Zeeman field that breaks SU(2) symmetry, such as a Majorana nanowire, and one does not know anything about the noise produced by the environment. It is therefore the ‘pessimistic’ form of the susceptibility, which again takes the form of a measure of non-locality. This measure is stricter than $\delta N$, since $\delta N\leq \Omega_s$. If we consider only *spin-independent* local noise, such as electrostatic potential fluctuations $\langle V_\sigma(x)V_{\sigma'}(x')\rangle = a_0V_0^2\delta(x-x')$, we have the following $\Omega_0=\partial E_0/\partial V_0$ susceptibility instead, $$\Omega_0 = \int dx \left|\mathbf{u}^L(x)\cdot\mathbf{u}^R(x)\right|.$$ Since the noise is more restricted, it is natural that $\Omega_0\leq\Omega_s$. It is thus a less strict measure of non-locality than $\Omega_s$. It is also SU(2) symmetric. Finally, we can define a degree of non-locality that is purely spatial, and independent of the spin degree of freedom, defined as $1-\Omega_\mathrm{max}$, with $$\Omega_\mathrm{max} \equiv \int dx \|\mathbf{u}^L(x)\|\, \|\mathbf{u}^R(x)\|. \label{Omegamax}$$ This Majorana overlap [completely discards spin information]{} and does not seem to be directly related with a linear-response susceptibility to any type of noise, but has the interesting property of being a strict upper bound to all other definitions. Taking into account Eq. (\[normalisation\]), we have $$0\leq\delta N\leq\Omega_0\leq\Omega_s\leq\Omega_\mathrm{max}\leq 1.$$ Remarkably, we will show that all these measures of non-locality become essentially equal for zero modes in globally topological nanowires $B>B_c$, but not so for zero modes below the $B_c$ in inhomogeneous nanowires. This reflects the intricate relation between spin, non-locality and protection derived from spin-orbit-induced spin textures. Throughout this work we will deal in particular with the problem of estimating the most pessimistic susceptibility $\Omega_s$ with a local probe. This was the subject of theoretical and experimental studies in quantum dot-nanowire setups, see Refs. [@Prada:PRB17; @Deng:A17]. In this context, $\Omega_s$ is the pertinent measure of non-locality, and given its meaning as a susceptibility to unrestricted local perturbations, it is also the most conservative measure of Majorana zero mode protection, beyond topological considerations. A trivial example of a MZM is a highly local Yu-Shiba-Rusinov state [@Yu:APS65; @Shiba:POTP68; @Rusinov:JL69; @Lee:NN14] tuned to zero energy. In general, such a fully local Andreev state will have equal Majorana components $|u_\sigma^L(x)| = |u_\sigma^R(x)|$, and hence $\Omega_s=1$. A MZM of topological origin will in contrast have exponentially small overlap $\Omega_s\approx 0$, as e.g. the conventional $B>B_c$ topological zero modes in very long Majorana nanowires. As will become apparent in the course of this work, a continuum of ABSs are possible under broken time reversal symmetry with any value of $\Omega_s$ between zero and one. The so-called pseudo-MZMs at smooth interfaces will be shown to lie at any point within this spectrum depending on nanowire parameters. Local estimator $\eta$ ---------------------- It was proposed [@Prada:PRB17] to estimate $\Omega_s$ by relating it to a quantity $\eta$ that can be extracted by a local measurement performed at a given point $x=0$. This point is chosen here as the left end of the nanowire, of total length $L$. Other choices for $x$ can be considered, but in our models at least one of the zero modes will often be concentrated around said end, so this becomes the optimal choice for our purposes, and the one relevant for most experiments currently. The local quantity to be measured is given in terms of the ratio of the norms of the two Majorana components at $x=0$, $$\eta \equiv \sqrt{\frac{\|\mathbf{u}^R(x=0)\|}{\|\mathbf{u}^L(x=0)\|}}. \label{eta}$$ The original observation that this quantity is correlated with Majorana non-locality was made in Ref. [@Prada:PRB17], where it was shown that in $B>B_c$ uniform nanowires $$\Omega_s\approx \eta.$$ In the rest of this work we will quantitatively evaluate how well the approximation holds in more general situations, including inhomogeneous samples with smooth interfaces hosting near-zero modes at $B<B_c$. At least one scheme has recently been proposed [@Prada:PRB17; @Clarke:PRB17] and demonstrated experimentally [@Deng:A17] to access this local quantity $\eta$. It consists in measuring, using tunnelling spectroscopy, the splitting of the zero mode as it is tuned to resonance with a quantum dot state coupled in series to the end of the nanowire (Fig. \[fig:sketch\]). This scheme, applied to two subsequent dot-wire resonances, offers enough information to obtain not only $\eta$, but also the spin-canting angles of the zero modes at $x=0$ [@Prada:PRB17; @Schuray:A18]. We will return to the spin of the MZMs in Sec. \[sec:spin\], after analysing in detail $\eta$ as an estimator of $\Omega_s$. We first note that a MZM of topological origin, with $\Omega_s\approx 0$, is guaranteed to have $\eta=0$, as $\Omega_s \approx 0 \Rightarrow u_\sigma^R(0)\approx 0$ (though the converse is not true). Likewise, a perfectly local ABS with $\Omega_s=1$ requires $u_\sigma^L(x) = u_\sigma^R(x)$ for all $x$, so $\eta=1$ in that case. The correlation between $\Omega_s$ and $\eta$ in intermediate situations is not so simple, and depends on the specific microscopic configuration of the Majorana nanowire. {width="90.00000%"} We will explore three such types of configurations: (1) a uniform Lutchyn-Oreg nanowire, (2) a superconducting nanowire with a smooth step in the electronic density (S’S), and (3) a nanowire with a step both in charge density and induced pairing (NS). Within this latter class we further specialise in two specific cases of particular experimental relevance, a superconducting nanowire with a narrow normal barrier (4) or a quantum dot (5) at the left end of the nanowire (see sketch in Fig. \[fig:sketch\]). These five setups, each corresponding to a different device design, play an important role in the ongoing discussion around the interpretation of recent experimental observations and even of theoretical results themselves. They will now be discussed in turn. Before proceeding, it is interesting to make the connection between the local estimator $\eta$ and the non-Hermitian topological classification theory mentioned in the introduction. Zero modes with intermediate $\Omega_s$ acquire a distinct topological classification when coupled to a reservoir at $x=0$ [@Avila:A18]. Within this theory, any deviation from perfect locality $\eta<1$ translates into a dimensionless asymmetry in the couplings of each Majorana component to the reservoir, denoted as $\gamma_0/\Gamma_0$, where $\gamma_0$ and $\Gamma_0$ are, respectively, the half-difference and average of the escape rates of the two Majoranas into the reservoir. The connection with $\eta$ is simply $\eta^4=(1-\gamma_0/\Gamma_0)/(1+\gamma_0/\Gamma_0)$. A finite asymmetry, in turn, stabilises the zero mode through an exceptional point bifurcation. This happens if the coupling asymmetry exceeds the Majorana splitting $\gamma_0>|E_0|$. Hence, a mode with $\eta<1$ *exactly* at zero energy will be rendered non-trivial when coupled to a reservoir. Uniform Majorana nanowires {#sec:uniform} ========================== The established standard to describe Majorana nanowires is the Lutchyn-Oreg model [@Lutchyn:PRL10; @Oreg:PRL10]. It consists in a modification of an original proposal by Fu and Kane [@Fu:PRL08] to engineer one-dimensional topological superconductivity by proximity to a conventional superconductor. The ingredients of the Lutchyn-Oreg model are a one dimensional semiconducting nanowire with Fermi energy $\mu$ and spin-orbit coupling $\alpha$, a Zeeman field $B$ applied parallel to it, and an s-wave superconductor to induce a pairing $\Delta$ on the nanowire by proximity effect. The Bogoliubov-de Gennes Hamiltonian of the model reads $$H = \left(\frac{p_x^2}{2m^*} - \mu\right) \tau_z + B\sigma_x\tau_z - \frac{\alpha}{\hbar} p_x \sigma_y \tau_z + \Delta\tau_x. \label{H}$$ Here $\tau_i$ and $\sigma_i$ are Pauli matrices in the $(c^\dagger, c)$ particle-hole and $(\uparrow, \downarrow)$ spin sectors. Only one spinful nanowire subband is included. We use, for concreteness, the effective mass of InSb, $m^*=0.015m_e$. The model describes a trivial superconducting phase for $B<B_c\equiv \sqrt{\mu^2+\Delta^2}$, while for $B>B_c$ it develops a topological p-wave gap with Majorana zero modes at either end of a long nanowire. For smaller lengths $L$ around one micron, Majoranas start to develop a finite overlap $\Omega_s$ and hibridise away from zero energy $E_0>0$. This model has been extensively used to characterise the basic physical regimes of Majorana nanowires. The correlation between $\Omega_s$ and the estimator $\eta$ established in Ref. referred mainly to this model. Here we use it as a starting point for the more complicated inhomogeneous nanowire models discussed later. We computed its associated phenomenology using the MathQ package [@MathQ]. Figure \[fig:uniform\] summarises the results. Panel (a) shows, for a $L=1.2$ $\mu$m nanowire, the $B$ dependence of the Bogoliubov spectrum, the typical wavefunctions, their hybridisation energy $E_0$, their overlaps $\delta N$, $\Omega_0$, $\Omega_s$ and $\Omega_\mathrm{max}$, and the overlap estimator $\eta$. Panel (b) shows the equivalent results for a longer $L=3$ $\mu$m nanowire. The spectra in panels (a,b) illustrate the well-known band inversion at a finite $B_c\approx 0.36$ meV (dotted vertical line), after which a zero mode emerges that oscillates as a function of $B$ due to the small, but finite, spatial overlap of its Majorana components. The corresponding Majorana wavefunctions $\|\mathbf{u}^{L,R}(x)\|$ for the lowest eigenstate are depicted for three values of $B$ (numbered circles), one below $B_c$ and two above. Atop the wavefunctions we represent the corresponding band-topological phase along the nanowire, with trivial in yellow (S) and topological in red (TS). Note that for field $B_1=0.25\mathrm{meV}<B_c$, although the topological transition has not yet taken place, the Majorana components of the lowest eigenstate appear as precursors to the Majorana zero modes at the higher $B_2=1.0 \mathrm{meV}>B_c$ and $B_3=2.0 \mathrm{meV}>B_c$. Their overlap ceiling $\Omega_\mathrm{max}$ (red shaded region) starts from $\Omega_\mathrm{max}=1$ ([strictly local after neglecting spin]{}) at $B=0$, and decreases as $B$ increases. We see that the wavefunction overlap between MZMs is a continuous non-monotonic quantity as a function of Zeeman field (this will also be the case in inhomogeneous nanowires). Thus, it is incorrect to think about fully local/non-local MZMs, before/after a topological transition, [in nanowires of realistic length]{}. Interestingly, in uniform nanowires the spin-independent [susceptibilities]{} coincide for $B<B_c$, $\delta N\approx\Omega_0$, while the overlap corresponding to spin-uncorrelated noise essentially follows the purely spatial overlap, $\Omega_s\sim\Omega_\mathrm{max}$. All forms of the overlap reach a common minimum just beyond the critical field $B\gtrsim B_c$, and then *grow* together in an oscillatory fashion, out of phase with the splitting (blue curve). Namely, MZMs deep in the topological regime [enjoy *less protection*]{} than the ones near $B_c$ (compare wavefunctions at $B_2$ and $B_3$). In the two uniform nanowires simulated in panels (a,b) we see how the estimator $\eta$ (black curve) roughly traces the overlap $\Omega_s$, particularly around the $B\gtrsim B_c$ region of minimal overlap. This good correlation corresponds to two particular configurations of the uniform Lutchyn-Oreg model. To fully assess the overall correlation between $\eta$ and $\Omega_s$ for arbitrary configurations, we simulate an ensemble of $\sim 2\cdot 10^4$ uniform nanowires with varying model parameters, including $B$, $\alpha$, $L$, $\mu$ and $\Delta$, distributed uniformly within realistic ranges (see caption). Amongst all configurations, we select those with a near zero mode (splitting $E_0<10\mu$eV) well separated from higher excitations (second eigenvalue greater than $50\mu$eV). This preselection is experimentally feasible at temperatures below $\sim 100$mK using local tunnel spectroscopy. In the uniform case it excludes in particular any field in the trivial regime $B<B_c$. We then compute the $(\eta, \Omega_s)$ pair for the near-zero modes in the ensemble and collect all these points (see Fig. \[fig:uniform\]c) to build the probability density $P(\eta,\Omega_s)$ that a given zero mode with a measured $\eta$ have a given $\Omega_s$. The probability $P(\eta,\Omega_s)$ is shown in Fig. \[fig:uniform\]d. Its profile gives an accurate account of the quality of $\eta$ as an estimator of $\Omega_s$ within the whole space of uniform Lutchyn-Oreg nanowire models with near zero modes. A perfect correlation would appear as a straight, thin $P(\eta,\Omega_s)$ along the diagonal. We see that while the actual dependence of the typical $\Omega_s$ with $\eta$ is non-linear, the probability distribution is rather narrow and close to the diagonal, which reveals the high quality of the estimator within this model space. The Pearson correlation coefficient between $\eta$ and $\Omega_s$ is $r=0.98$. The small panels in red, green and blue dissect the ensemble according to their $\mu$ (see legend, low densities in red, higher densities in blue). We find that the precision of $\eta$ is greater if the nanowire is known to have low electronic density. The same statistical analysis with $\Omega_0$ instead of $\Omega_s$ yields very similar results. Smooth S’S nanowires {#sec:S'S} ==================== In the remaining sections we consider inhomogeneous nanowires, described by a generalised Lutchyn-Oreg model with position dependent pairing $\Delta(x)$ and electrostatic potential $\phi(x)$, $$H = \left(\frac{p_x^2}{2m^*} -\mu+\phi(x)\right) \tau_z + B\sigma_x\tau_z - \frac{\alpha}{\hbar} p_x \sigma_y \tau_z + \Delta(x)\tau_x. \label{H}$$ (In what follows we reabsorb $\mu$ into $\phi(x)$ for simplicity.) Much of the current debate as to the potential non-triviality of transport signatures in Majorana nanowires revolves around the possibility that near zero modes may arise as the result of smooth spatial variations in $\Delta(x)$ and/or $\phi(x)$, independently of a band-topological phase transition. The debate has thus [far]{} centered mostly on distinguishing between topological MZMs and such pseudo-MZMs in this system. In our view, as summarised in the introduction, this formulation of the question is missing the broader point that topological protection and non-triviality in finite or inhomogeneous systems is a matter of degree, connected to wavefunction overlaps (see Sec. \[sec:nonlocality\]) and protection to perturbations. Regardless of their relation to band-topology, smoothly-confined pseudo-MZMs with a sufficiently small overlap $\Omega$ will be, for all purposes, genuine MZMs protected against the corresponding type of local perturbation, exactly like the topological $B>B_c$ MZMs in finite-length nanowires. The debate is thus reduced to clarifying whether smoothly confined near-zero modes can have significantly suppressed overlaps or not. Two gneric types of smooth variations are possible within the Lutchyn-Oreg model, smooth S’S and smooth NS boundaries. We first concentrate on the S’S case, wherein $\Delta$ is uniform along the nanowire but $\phi(x)$ is position dependent. It can be positive (insulating regions) or negative (higher density regions). The spatial variation may arise due to e.g. non-uniform screening from contacts or gates. We model $\phi(x)$ in a nanowire spanning $0<x< L_{S'}+L_S$ as $$\phi(x) = \phi_{S'} + (\phi_{S}-\phi_{S'})\theta_\zeta(x-L_{S'})$$ where $\theta_\zeta(x) = \frac{1}{2}[1+\tanh(x/\zeta)]$ is a smooth step function of width $\zeta$, see Fig. \[fig:sketch\]. This length controls the smoothness of the boundary between the left $S'$ side, of length $L_{S'}$ and the right $S$ side, of length $L_S$. {width="90.00000%"} Similarly to the uniform nanowire, a non-uniform S’S system with sufficiently long $L_{S,S'}$ may still be analysed from the conventional point of view of band topology of the two sides. The two $\phi_{S'}$ and $\phi_S$ now define two critical fields $B_c^{S', S} = \sqrt{\Delta^2+\phi_{S', S}^2}$. For a given $B$, we can have all possible combinations S’S, TS’S, S’TS, and TS’TS, where S stands for a trivial superconductor, and TS a topological superconductor, depending on whether $B<B_c^{S',S}$ or $B>B_c^{S',S}$. Whenever the [band]{} topology of the left and right sides is different (*locally* topological nanowire) and the corresponding halves are long enough, a pseudo-MZM will be localised somewhere in the smooth junction, regardless of $\zeta$. This state is actually a consequence of the bulk boundary correspondence. There is therefore nothing ‘pseudo’ about it. Crucially, moreover, we will show in the next section that this state is essentially identical to the so-called pseudo-MZMs of smooth NS junctions, where band-topological arguments do not apply. We thus argue that it is incorrect to distinguish between MZMs and pseudo-MZM in general isolated systems, as the two types of states are ultimately connected. The discussion, once more, should focus instead on the overlap $\Omega_s$, not on artificial distinctions between classes of zero modes. Figure \[fig:S’S\], analogous to Fig. \[fig:uniform\], shows the overlaps, $\eta$ and spectral phenomenology of a smooth S’S nanowire as depicted in case (2) of Fig. \[fig:sketch\]. When the junction is sufficiently smooth, the S’S to TS’S transition at $B=B_c^{S'}$ manifests as a single subgap state dropping into the gap towards zero energy. A lone subgap level detaching from the quasicontinuum of levels is a recurrent and distinct feature of smooth configurations that replaces the band inversion [pattern]{} of uniform wires. It is clearly visible in the spectrum of panels (a) and (b), blue curve, where parameters are chosen so that $B_c^{S'} < B_c^S$ (the two critical fields are shown as dotted vertical lines). The two panels (a,b) correspond, respectively, to S’S nanowires with weaker and stronger inhomogeneity $\Delta\mu=0.5$ meV and $\Delta\mu=1.8$ meV, where $\Delta\mu\equiv\max_x(\phi(x))-\min_x(\phi(x))$ is the maximum variation of the Fermi energy in the nanowire. We again show the Majorana component wavefunctions of the lowest eigenstate at three fixed fields (numbered circles). At $B_1=0.1\mathrm{meV}<B_c^{S'}$ the nanowire is in an S’S configuration (trivial-trivial), and the finite energy state dropping into the gap is merely a precursor of the Majorana zero modes at larger fields, concentrated on the less dense $S'$ side. It already exhibits a slightly suppressed overlap $\Omega_s<1$, with its two Majorana components starting to separate \[wavefunction (1)\]. As the nanowire enters the TS’S configuration \[$B_c^{S'}<B_2 = 0.6 \mathrm{meV}<B_c^S$, wavefunction (2)\] the MZMs at the smooth junction (blue) moves away from the left Majorana at $x=0$ (red). The distance between the two is $B$-dependent, since the TS length of the nanowire that satisfies $B>\sqrt{\Delta^2+\phi(x)^2}$ \[see colored bar atop wavefunction (2)\] grows with $B$ due to the smooth $\phi(x)$ profile. The spatial decoupling suppresses $\Omega_s$ (solid red curve in bottom panel)\] until $B_c^S$ is reached, wherein the type of $\Omega_s$ oscillations we observed in the uniform case appear, and the Majorana wavefunctions become [standard]{}, confined to the ends of the nanowire \[wavefunction (3)\]. Contrary to conventional lore, the Majorana overlap in the globally topological TS’TS phase at $B>B_c^{S,S'}$ is not necessarily smaller than in the locally topological TS’S case with a Majorana within the bulk of the nanowire. For example, $\delta N$ and $\Omega_0$ can be substantially suppressed for $B<B_c$, which suggests a strong resilience of locally topological Majorana zero modes against electrostatic potential fluctuations, see Figs. \[fig:S’S\](a,b), bottom panels. Actually, as was noted also for the uniform case, in typical S’S nanowires shorter than around $L\sim 3\mu$m all forms of the Majorana overlap reach their minimum within the TS’S regime, $B_c^{S'}<B<B_c^{S}$, and begin to *increase* into the TS’TS phase. The same will be noted in Sec. \[sec:NS\] for smooth NS nanowires. This behavior is due to the different (faster) decay profile of $u_\sigma^R(x)$ when it lies at the smooth TS’S junction than when it shifts to the abrupt right boundary of the nanowire. It is important to appreciate the difference between these two types of MZMs. The MZM at a smooth boundary is also spatially smooth, with a [highly confined]{} Gaussian-like profile [@Kells:PRB12; @Fleckenstein:PRB18], while the MZM at an abrupt boundary has fast $\sim k_F$ spatial harmonics and a double-exponential decay [@Klinovaja:PRB12]. We will analyse in more detail the profiles and spin densities of these two types of MZMs in Sec. \[sec:spin\]. The faster spatial decay of smooth Majoranas suggests that the accuracy of the local estimator $\eta$ should be worse in this case, as compared to the case of uniform nanowires and abrupt MZMs. Indeed, the estimator may become suppressed as the smooth Majorana moves away from $x=0$ at a faster rate (Gaussian) than the overlap (exponential). We find that in realistic nanowires (see parameter ranges in the caption to Fig. \[fig:S’S\]) the accuracy of $\eta$ is indeed reduced, particularly under strong inhomogeneities $\Delta\mu\gg\Delta$. This is shown in Fig. \[fig:S’S\](c,d). Here we have [again performed a uniform sampling over all nanowire parameters, this time including also $\phi_{S'}$, $\phi_S$, $L_{S'}$, $L_S$ and $\zeta$ in $\sim 10^5$ configurations]{} \[see Figs. \[fig:S’S\](c,d)\]. The resulting $P(\eta,\Omega_s)$ is similar to that of the uniform case, albeit with a slightly reduced Pearson coefficient $r=0.91$. This [reduction]{} is precisely the result of the Gaussian profile of smooth MZMs, which translates into a slight ‘bulge’ above the origin and another one to its right. In the subpanels to the right we disect $P(\eta,\Omega_s)$ into partial probability densities for increasing degree of Fermi energy inhomogeneity $\Delta\mu$. We find that for inhomogeneities $\Delta\mu<1$ meV, the estimator preserves a high $r=0.95$ correlation with $\Omega_s$ (red subpanel), but increasing $\Delta\mu$ (green, blue subpanels) suppresses $r$, though the effect is not drastic, with $r\approx 0.9$ still. This remains true regardless of the maximum nanowire density considered. {width="90.00000%"} ![**Barrier-Superconductor nanowires**. Specific Barrier-S configuration of the NS model, similar to the one discussed in Ref. [@Vuik:18], with positive $\phi_{N}=2$meV and short $L_{N}=\zeta=0.1\mu$m, which forms a smooth, Zeeman-polarised insulating barrier around $x=0$. Other parameters: $\phi_S=-1.8$ meV, $\Delta_S=0.3$ meV, $L_S=2\mu$m and $\alpha=0.4$eV$\mathrm{\AA}$.[]{data-label="fig:BS"}](bs.pdf){width="0.95\columnwidth"} ![**Dot-Superconductor nanowires**. Specific Dot-S configuration of the NS model, relevant to a number of experiments [@Deng:S16; @Deng:A17], with negative $\phi_{N}=-14$meV and short $L_{N}=0.15\mu$m, which confines states in a quantum-dot region around $x=0$. Other parameters: $\phi_S=-0.8$ meV, $\Delta_S=0.5$ meV, $\zeta=0$, $L_S=2\mu$m and $\alpha=0.2$eV$\mathrm{\AA}$.[]{data-label="fig:DS"}](ds.pdf){width="0.95\columnwidth"} Smooth NS nanowires {#sec:NS} =================== We now [study]{} the second type of inhomogeneous nanowire, wherein the [pairing $\Delta(x)$ is also position dependent, like $\phi(x)$]{}. We again consider a simple profile that interpolates between a left side and a right side. The left side is always normal in this case, with $\Delta_N=0$, so that the nanowire contains a smooth NS interface centered at $x=L_N$, $$\begin{aligned} \phi(x) &=& \phi_{N} + (\phi_{S}-\phi_{N})\theta_{\zeta}(x-L_{N}),\nonumber\\ \Delta(x) &=& \Delta_S \theta_{\zeta}(x-L_{N}).\end{aligned}$$ This model is relevant to many devices explored in recent experiments. Nanowires are often proximitised by epitaxial growth of a superconductor on the surface of the nanowire. Often, the epitaxial coverage of the nanowire is incomplete, so it is natural to assume a suppressed pairing in the exposed portions. Like in the S’S nanowire, a thorough microscopic validation of this model would require a detailed characterisation of the device in question. The fundamental interest of the Lutchyn-Oreg model with a smooth NS interface is particularly high due to the fact that, perhaps surprisingly, it can also host near-zero modes at finite Zeeman field B, much like the smooth S’S, despite not developing a topological gap on the normal side. This is shown in Fig. \[fig:NS\], which is the NS version of Fig. \[fig:S’S\]. The suppressed pairing gives rise to Andreev levels in the normal region. Depending on the normal length $L_N$, their level spacing $\delta\epsilon$ can be much smaller than the induced gap $\Delta$, which results is many subgap levels (unlike the S’S case, where only a lone level, detached from the quasicontinuum appears). A finite $B$ field Zeeman-splits all these subgap levels, that evolve avoiding each other due to spin-orbit coupling. This is true for all except the lowest two [eigenstates]{} (blue), which converge to zero energy with a *finite* slope at low $B$-fields [@Vaitiekenas:PRL18] (this is unlike in the S’S case, where the lone detached level starts off *flat* at $B=0$) [@Moor:18]. Despite the superficial resemblance to Zeeman-induced parity crossings in quantum dots [@Lee:NN14; @Moor:18], see Fig. \[fig:DS\], [almost]{} perfect Andreev reflection of N electrons on the smooth NS interface stabilises this low-lying subgap level near zero energy for $B>\delta\epsilon$, but still well before $B_c^S$. From the point of view of its Majorana components, this near-zero mode is remarkably similar to the corresponding zero mode at the TS’S junction. Comparing the wavefunctions at field $B_2$ and $B_3$ in Figs. \[fig:S’S\] and \[fig:NS\], we see that the essential difference between the S’S and NS cases lies in the Majorana component $\mathbf{u}^L (x)$ on the left side (red wavefunction). In the NS case it is delocalised throughout the N region of length $L_N$, whilst in the TS’S case it is confined within a coherence length of the $x=0$ boundary. The smooth Majorana at the junction, however, is very similar and, remarkably, remains confined at the junction instead of decaying into the N side. We note that in the NS case this confinement is not the result of a bulk-boundary correspondence, as the left side is not gapped in the [$L_N\to\infty$]{} limit, but of the high smoothness $\zeta$ of the boundary which enhances Andreev reflection. This observation hints at a deeper reason for the zero energy of smoothly confined states beyond topology, connected instead to the exact charge-conjugate symmetry of quasiparticles that undergo perfect Andreev reflection at an adiabatically smooth S boundary. The similar wavefunction phenomenology produces a similar behaviour also for $\Omega_s$ and $E_0$ as a function of $B$. Three regimes are visible, with crossovers at Zeeman $B\approx\delta\epsilon$ and $B\approx B^S_c=\sqrt{\Delta_S^2+\phi_S^2}$, see Fig. \[fig:NS\](a,b). In the regime $\delta\epsilon<B<B_c^S$ with smooth Majoranas, we see that $\eta$ underestimates the overlap $\Omega_s$. The general $(\eta,\Omega_s)$ correlation analysis is shown in Fig. \[fig:NS\](c,d). The overall correlation, with identical sampling and zero-mode preselection scheme as in the S’S, is now $r=0.93$. This reduced value respect to the uniform case is once more due to the effect of the smooth MZMs. Due to their faster decay, their overlap with the [left Majorana components delocalised within the N segment]{} is greater than what $\eta$ would estimate. [This enhances the ’bulge’ at $(\eta,\Omega_s)\approx (0.2,0.4)$ in the probability distribution as compared to the S’S case]{}. The states in this region of underestimated overlap, however, come from highly inhomogeneous samples, as can be seen from the small subpanel decomposition. If the nanowire Fermi energy inhomogeneity $\Delta\mu$ is known to be low enough ($\Delta\mu<1\mathrm{meV}\approx 3\Delta_S$, red subpanel), the correlation remains strong at $r=0.96$. Smooth Barrier-S and Dot-S nanowires {#sec:BDS} ------------------------------------ A particular case of the NS nanowires in the preceding section that is of relevance to many devices is the limit in which $L_N$ is small. Nanowires designed to be probed by tunneling spectroscopy are often left uncovered by the superconductor at $x=0$ in order to allow efficient gating of the contact to the metallic reservoir. A finger gate under $x=0$ can then, thanks to the reduced screening by the superconductor shell, tune the transparency of the contact by inducing a positive $\phi_N$. This defines a barrier of finite smoothness $\zeta$, see case (4) in Fig. \[fig:sketch\]. Such a setup was recently discussed in Ref. , where the smoothness allowed for the development of a stable $B<B_c$ near-zero mode, with a different coupling of its Majorana components [to an outside reservoir]{} across the Zeeman-polarised barrier by virtue of their opposite spin orientation at the smooth contact. The phenomenology of a such a Barrier-S configuration is shown in Fig. \[fig:BS\]. We see that the near-zero modes at the barrier for $B<B_c$ are characterised by a high overlap $\Omega_s$ but a reduced charge $e\delta N$ due to Andreev processes. An opposite voltage of the finger gate can make $\phi_N$ strongly negative. This may trap discrete states around $x=0$ in an effective quantum dot-superconductor configuration. Additionally, screening effects in the nanowire may produce, in a mean-field approximation, a quantum dot-superconductor profile spontaneously [@Dominguez:NQM17; @Escribano:A17] that can also trap states. To gain insight on these cases we simulate nanowires with short, normal dot regions abruptly connected to the nanowire ($\zeta=0$) without an additional intervening barrier, so the confinement is merely the result of the potential and pairing mismatch at $L_N$. This is a likely situation in [some]{} experiments. Its associated phenomenology is shown in Fig. \[fig:DS\]. The trapped states are Zeeman-split as $B$ is increased, and can cross zero energy at specific values of $B=B_1<B_c$ [@Lee:NN14; @Moor:18], analogous to Shiba state parity crossings. The crossings are considerably flattened due to the effect of Andreev reflections from the nanowire, which are enhanced by the lack of a confining dot-nanowire barrier. The near-zero mode is not completely stabilised at zero, unlike in Fig. \[fig:NS\], because Andreev reflection is however not perfect (that requires a smooth dot-S contact). The state remains very concentrated within the quantum dot region, and is therefore considerably local, with $\Omega_s$ and $\eta$ both close to one. Its charge $e\delta N$ and susceptibility $\Omega_0$ to local potential fluctuations are comparatively suppressed, again due to Andreev processes. This once more showcases the fact that seemingly trivial, spatially overlapping near-zero modes are not necessarily fragile, and may exhibit, due to Andreev particle-hole mixing, a highly non-trivial response to certain perturbations.[^1] It is important to note that these Barrier-S and Dot-S types of configurations of the generic NS model are included in the NS sampling of Figs. \[fig:NS\](c,d), which therefore remains representative of the quality of the $\eta$ estimator expected in these cases. We have performed samplings of purely Dot-S and Barrier-S configuration ensembles, and found similar $P(\eta,\Omega_s)$ distributions as for the NS case, including the $(\eta,\Omega_s)\approx (0.2,0.4)$ bulge. The general conclusions on the NS model class can [therefore also be applied specifically]{} to Barrier-S and Dot-S models. Spin texture and smoothness {#sec:spin} =========================== ![**Majorana spin in S’S junctions.** (a) Wavefunction $|\mathbf{u}^L|$ and $|\mathbf{u}^R|$ of the lowest eigenstate in an TS’S junction of increasing smoothness $\zeta=(0.0,0.1,1.0)\mu$m. (b) The corresponding spin density $\langle\sigma_x\rangle$ along the Zeeman field. The shading under the spin density curves encodes the Majorana canting angle $\theta$ relative to the Zeeman field along $x$. Parameters: $\phi_{S'}=0$, $\phi_{S}=-1$ meV, $L_{S'}=L_S=1.8\mu$m, $\Delta_{S'}=\Delta_S=0.4$ meV, $\alpha=0.4$eV$\mathrm{\AA}$.[]{data-label="fig:szss"}](sz_ss.png){width="\columnwidth"} ![**Majorana spin in NS junctions.** Same as Fig.\[fig:szss\] for an NS nanowire ($\Delta_N=0$).[]{data-label="fig:szns"}](sz_ns.png){width="\columnwidth"} ![**Majorana spin in Barrier-S junctions.** Same as Fig.\[fig:szss\] for an Barrier-S nanowire ($\Delta_N=0$, $\phi_N=1$ meV).[]{data-label="fig:szbs"}](sz_bs.png){width="\columnwidth"} In this final section we analyse the spin structure of MZMs associated to different types of interfaces as a function of their smoothness. This aspect of the MZM wavefunction is relevant, since current experiments that extract $\eta$ to estimate the Majorana overlap use spin-polarised quantum dots coupled to the nanowire. The hybridisation of the dot levels and the MZMs at resonance depends strongly on the spin orientation of the latter. Furthermore, MZM spin is important in view of a recent arguments [@Vuik:18] that relate potential smoothness and [Majorana]{} spin polarisation. [As mentioned in Sec. \[sec:BDS\]]{}, this work points out that at a smooth barrier, highly local MZMs acquire opposite spin polarisation, which may result in a highly asymmetric coupling to a reservoir due to the Zeeman-polarisation of the barrier. We show here that such an effect is a particular manifestation of the MZM non-locality produced by the barrier smoothness. We once more analyse different nanowire configurations separately. Fig. \[fig:szss\] shows, for an S’S nanowire of increasing smoothness $\zeta$ at $B_c^{S'}<B<B_c^S$, the Majorana wavefunctions of the lowest energy mode (a) and their spin polarisation $\langle \sigma_x\rangle$ along the nanowire (b). The first row shows a completely abrupt TS’S junction. The left Majorana $\mathbf{u}^L(x)$ centered at the abrupt $x=0$ boundary to vacuum (red curve) exhibits rapid oscillations associated to the Fermi wavevector $k_F$. We call this an *abrupt* Majorana. Its fast spatial harmonics are the result of perfect $k_F\to -k_F$ normal reflection at the $x=0$ boundary. Its spin density [$\langle\sigma_x(x)\rangle=\mathbf{u}^{L,R}(x)^\dagger\sigma_x\mathbf{u}^{L,R}(x)$]{} likewise oscillates spatially within the $z-x$ plane. The angle $\theta$ in this plane (with $\theta=0$ for spin along $x$) is known as the canting angle, and is color coded in a gray-orange scale. The right Majorana, in blue, lies at the sharp TS’S interface where Andreev reflection processes are possible. It has a different profile from the abrupt Majorana, but still shows considerable density and spin oscillations. As the junction smoothness $\zeta$ increases (second and third row), the left Majorana remains unchanged, but the right Majorana at the junction becomes increasingly smooth, loosing the fast spatial harmonics both in $\mathbf{u}^R(x)$ and $\langle\sigma_x(x)\rangle$. Thus, a Majorana of Gaussian-like profile emerges, which we call here *smooth* Majorana. Its spin becomes well defined, with canting angle converging to $\theta=0$ (orange) along the Zeeman field direction. The equivalent smoothness phenomenology for the NS nanowire is shown in Fig. \[fig:szns\]. In this case, the abrupt Majorana takes the form of a standing wave in the N region, with oscillatory density and spin. Its spin, however, is predominantly aligned along $-x$ (i.e. $\theta=\pi$, gray). The smooth Majorana, as remarked in Sec. \[sec:NS\], bears a strong resemblance to the one in smooth S’S junctions. It does not leak into the N side, even for moderate smoothness $\zeta\sim 0.1\mu$m, and acquires a well-defined spin polarisation along $x$ (i.e. $\theta=0$, orange). Again, the difference in density and spin texture of abrupt and smooth Majoranas in smooth nanowires is stark. Finally, we present in Fig. \[fig:szbs\] the results for a Barrier-S nanowire (insulating left side, $\phi_N>0$), with a barrier of increasing smoothness. For a sharp barrier (top row), the two Majoranas are very similar to the abrupt Majorana [at the $x=0$ boundary to vacuum]{} in the S’S case. The only difference is that the barrier side has a finite potential, and a slight leakage of the two Majoranas is possible. The leakage, as pointed out in Ref. [@Vuik:18], depends on the spin density of each Majorana, as the barrier height is different for the two spin orientations due to the uniform Zeeman field in the whole system, barrier included. Said spin orientation for the abrupt junction is rapidly varying, as corresponds to abrupt Majoranas. The difference in leakage becomes more pronounced as the barrier smoothness increases (middle and bottom rows). The spin of the two Majoranas in this case becomes increasingly well defined, and opposite, so that one Majorana penetrates more and more into the barrier as it becomes smoother. This leads to a simultaneous *spatial and spin decoupling* (suppression of $\Omega_s$) of the two Majoranas at smooth barriers. We thus see that smoothness-induced non-locality and spin-induced decoupling of Majoranas are one and the same. We conclude that, in the context of nanowires coupled to external reservoir [@Avila:A18], a different decay of MZMs into the outside world can always be traced back to a finite degree of non-locality. Conclusion {#sec:conclusion} ========== To summarise, in this work we have studied the properties of inhomogeneous Majorana nanowires. We have considered Majorana zero modes emerging before and after the band-topological transition, and analysed their wavefunction profiles. This allows us to distinguish between two distinct types, the smooth and abrupt Majoranas, each with characteristic spin textures. We also showed that the nanowire spectrum is a rich fingerprint of the nanowire inhomogeneities. From the spectrum it is possible to extract information about the type of pairing and potential inhomogeneities in the nanowire. For example, a Zeeman splitting that starts with zero or finite slope at $B=0$ can distinguish between uniform and non-uniform pairing in the nanowire. Similarly, a lone Andreev level detaching into the gap as a function of $B$ reveals non-uniform and smooth electrostatic potentials. We have finally studied in depth the protection to local perturbations of Majorana zero modes, and its relation to wavefunction overlaps and non-locality. As a result, we obtain several expressions for the degree of non-locality, differing in the role of internal degrees of freedom of the spinorial wavefunction. We study their evolution with nanowire parameters and Zeeman field. The different susceptibilities $\delta N$, $\Omega_0$ and $\Omega_s$ essentially coincide for globally topological nanowires, and match the purely spatial definition $\Omega_\mathrm{max}$, but significantly differ in nanowires with non-uniform topology. The $\Omega$’s can be minimised in smooth NS or S’S junctions before even crossing into a topological superconductor phase. Once established, and regardless of the underlying mechanism, a small $\Omega$ protects states at zero energy, and suppresses their decoherence due to a noisy environment [@Cheng:PRB12a; @Penaranda:18]. Thus, the wavefunction overlap emerges as the only relevant figure of merit of Majorana zero modes in isolated inhomogeneous nanowires. Spatial non-locality is intrinsically difficult to measure. The local-detection scheme proposed in Refs. [@Prada:PRB17; @Clarke:PRB17] and analysed in detail here is much simpler than alternative schemes based on interferometry [@Hell:PRB18] or spatially correlated measurements [@Li:SR14; @Moore:PRB18]. Unlike the latter, however, the predictive power of the local detection scheme is merely statistical. In this work we have assessed the accuracy, in a statistical sense, of local quantity $\eta$ as an estimator of the spin-uncorrelated susceptibility $\Omega_s$, as the most conservative, physically motivated measure of Majorana non-locality. Its accuracy is rather high, particularly in the case of uniform nanowires. The significance of this for current experiments is large, as it quantifies the likelihood that a zero bias anomaly observed in transport is connected to a non-local Majorana zero mode. We have also analysed carefully the extent to which the estimator $\eta$ remains valid in the presence of smooth inhomogeneities. We found that for large smooth inhomogeneities with $\Delta\mu >1$ meV (of the order or greater than the superconducting gap) its accuracy is lessened, although only weakly, statistically speaking. Even if $\Delta\mu$ is very large, however, $\eta$ can still provide an upper bound for $\Omega_s$. A small $\eta\lesssim 0.2$ is a statistical guarantee that the overlap should remain bounded to $\Omega_s\lesssim 0.4$. We have finally considered the effect of smoothness in inhomogeneous nanowires in connection to the wavefunction and spin density of Majorana zero modes. A smooth interface NS or S’S interface creates smooth Majoranas with uniform spin. These remain confined at the interface regardless of whether one of its two sides is ungapped (NS) or not (S’S). We also note that at a smooth insulating barrier, the uniform spin-polarisation of smooth Majoranas leads to their spatial separation due to a spin-dependent barrier penetration, and a suppression of their overlap as the smoothness increases. Likewise, near-perfect Andreev reflection at smooth [NS and Dot-S]{} interfaces leads to near-equal particle and hole amplitudes, suppressed charge and a correspondingly small sensitivity to electrostatic perturbations, despite their apparently local wavefunctions. This highlights the strong connection between internal spin and particle/hole degrees of freedom, non-locality and protection in smoothly inhomogeneous nanowires. We acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Grant Nos. FIS2015-65706-P, FIS2015-64654-P and FIS2016-80434-P (AEI/FEDER, EU), the Ramón y Cajal programme, Grant Nos. RYC-2011-09345 and RYC-2013-14645 and the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0377) [^1]: We note that introducing a barrier between dot and nanowire progressively suppresses these non-trivial Andreev effects until one reaches, for high barriers, a standard quantum dot behaviour with unitary charge and a conventional unprotected response to electrostatic and Zeeman noise.

--- abstract: | Non-standard sandwich gravitational waves are constructed from the homogeneous $pp$ vacuum solution and the motions of free test particles in the space-times are calculated explicitly. They demonstrate the caustic property of sandwich waves. By performing limits to impulsive gravitational wave it is demonstrated that the resulting particle motions are identical regardless of the “initial” sandwich. PACS number(s): 04.30.-w, 04.20.Jb, 98.80.Hw --- 6.6in 21.7cm J. Podolsk' y, K. Veselý ** Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic [ Electronic address: podolsky@mbox.troja.mff.cuni.cz]{} **1 Introduction** Plane-fronted gravitational waves with parallel rays ([*pp*]{} waves) are characterized by the existence of a quadruple Debever-Penrose null vector field which is covariantly constant. In suitable coordinates (cf. [@KSMH]) the metric of vacuum [*pp*]{} waves can be written as $${{\rm d}}s^2=2\,{{\rm d}}\zeta {{\rm d}}\bar\zeta-2\,{{\rm d}}u{{\rm d}}v-(f+\bar f)\,{{\rm d}}u^2\ , \label{E1}$$ where $f(u,\zeta)$ is an arbitrary function of $u$, analytic in $\zeta$. The only non-trivial components of the curvature tensor are proportional to $f_{,\zeta\zeta}$ so that (\[E1\]) represents flat Minkowski space-time when $f$ is linear in $\zeta$. The simplest case for which the metric describe gravitational waves arise when $f$ is of the form $$f(u,\zeta)=d(u)\zeta^2\ , \label{E2}$$ where $d(u)$ is an [*arbitrary*]{} function of $u$; such solutions are called homogeneous [*pp*]{} waves (or “plane” gravitational waves). Performing the transformation (cf. [@Penrose]) $$\begin{aligned} \zeta&=&\frac{1}{\sqrt{2}}\,(Px+iQy) \ ,\nonumber\\ v &=&\frac{1}{2}\,(t+z+PP'x^2+QQ'y^2) \ ,\label{E3}\\ u &=&t-z \ ,\nonumber\end{aligned}$$ where real functions $P(u)\equiv P(t-z)$, $Q(u)\equiv Q(t-z)$ are solutions of differential equations $$P''+d(u)\,P=0\ ,\qquad Q''-d(u)\,Q=0\ , \label{E4}$$ (here prime denotes the derivation with respect to $u$) the metric can simply be written as $${{\rm d}}s^2 = - {{\rm d}}t^2 + P^2 {{\rm d}}x^2 + Q^2 {{\rm d}}y^2 + {{\rm d}}z^2\ . \label{E5}$$ This form of the homogeneous [*pp*]{} waves is suitable for physical interpretation. Considering two free test particles standing at fixed $x$, $y$ and $z$, their relative motion in the $x$-direction is given by the function $P(u)$ while it is given by $Q(u)$ in the $y$-direction. The motions are unaffected in the $z$-direction which demonstrate transversality of gravitational waves. The coordinate $u=t-z$ can now be understood as a “retarded time” and the function $d(u)$ as a “profile” of the wave. Note also that functions $P, Q$ may have a higher degree of smoothness than the function $d$ so that relative motions of particles are continuous even in the case of a shock wave (with a step-function profile, $d(u)\sim\Theta(u)$), or an impulsive wave (with a distributional profile, $d(u)\sim\delta(u)$). **2 Standard sandwich wave** A sandwich gravitational wave [@BPR; @BP] is constructed from the homogeneous [*pp*]{} solution (\[E1\]), (\[E2\]) if the function $d(u)$ is non-vanishing only on some finite interval of $u$, say $u\in[u_1, u_2]$. In such a case the space-time splits into three regions: a flat region $u<u_1$ (“Beforezone”), a curved region $u_1<u<u_2$ (“Wavezone”), and another flat region $u_2<u$ (“Afterzone”). In the region $u<u_1$ where $d(u)=0$ it is natural to choose solutions of Eqs. (\[E4\]) such that $P=1$ and $Q=1$ so that the metric (\[E5\]) is explicitly written in Minkowski form. The form of the metric (\[E5\]) for $u>u_1$ is then given by solutions of Eqs. (\[E4\]) where the function $P, Q$ are chosen to be continuous up to the first derivatives at $u_1$ and $u_2$. A standard example of a sandwich wave can be found in textbooks (cf. [@Rindler]). The “square” profile function $d(u)$ is given simply by $$d(u)=\left\{ \begin{array}{l} 0, \qquad u<0 \\ a^{-2}, \quad 0\le u\le a^2 \\ 0, \qquad a^2<u \end{array}\right. \label{E6}$$ where $a$ is a constant. It is easy to show that the corresponding functions $P, Q$ are given by $$\begin{aligned} P(u)&=&\left\{ \begin{array}{l} 1, \hskip 57mm u\le0 \\ \cos(u/a), \hskip44mm 0\le u\le a^2 \\ -u \sin a/a+\cos a+a\sin a, \hskip11mm a^2\le u \end{array}\right. \label{E7}\\ Q(u)&=&\left\{ \begin{array}{l} 1, \hskip 57mm u\le 0 \\ \cosh(u/a), \hskip42mm 0\le u\le a^2 \\ u \sinh a/a+\cosh a-a\sinh a, \qquad a^2\le u \end{array}\right. \label{E8}\end{aligned}$$ Therefore, particles which were in rest initially accelerate within the wave in such a way that they approach in $x$-direction and move apart in $y$-direction. Behind the wave they move uniformly (see Fig. 1a). **3 Non-standard sandwich waves** Now we construct some other (non-trivial) sandwich waves. Our work is motivated primarily by the possibility of obtaining impulsive gravitational waves by performing appropriate limits starting from [*different*]{} sandwich waves (see next Section). This also enables us to study particle motions in such radiative space-times. Moreover, the standard sandwich wave given by (\[E6\]) is very special and “peculiar” since it represents a [*radiative*]{} space-time containing [*stationary*]{} regions. Indeed, for $d(u)$ being a positive constant, the Killing vector $\partial_u$ is timelike where $|Re\,\zeta|>|Im\,\zeta|$. This “strange” property remained unnoticed in literature so far. It may be useful to introduce more general sandwich waves which are [*not*]{} stationary. [**a) Sandwich wave with “$\bigwedge$” profile**]{} Let us consider a solution (\[E1\]), (\[E2\]) for which the function $d(u)$ takes the form $$d(u)=\left\{ \begin{array}{l} 0, \hskip27mm u\le -a \\ b\,(a+u)/a, \hskip10mm -a\le u\le0 \\ b\,(a-u)/a, \hskip10mm 0\le u\le a \\ 0, \hskip27mm a\le u \end{array}\right. \label{E9}$$ where $a, b$ are arbitrary real (positive) constants. The wave has a “wedge” profile illustrated in Fig. 1b. Straightforward but somewhat lengthy calculations give the following form of the functions $P(u), Q(u)$ (continuous up to the second derivatives everywhere including the points $u=-a$, $u=0$ and $u=a$): $$\begin{aligned} P(u)&=&\left\{ \begin{array}{l} 1, \hskip 70mm u\le -a \\ c\,\sqrt{u_1}\,J_{-\frac{1}{3}}(\frac{2}{3}u_1^{3/2}), \hskip40mm -a\le u\le 0 \\ \sqrt{u_2}\,\left[A\,J_{ \frac{1}{3}}(\frac{2}{3}u_2^{3/2}) +B\,J_{-\frac{1}{3}}(\frac{2}{3}u_2^{3/2})\right], \hskip10mm 0\le u\le a \\ C\,u+D, \hskip57mm a\le u \end{array}\right. \label{E10}\\ Q(u)&=&\left\{ \begin{array}{l} 1, \hskip 70mm u\le -a \\ c\sqrt{u_1}\,I_{-\frac{1}{3}}(\frac{2}{3}u_1^{3/2}), \hskip40mm -a\le u\le 0 \\ \sqrt{u_2}\,\left[E\,I_{ \frac{1}{3}}(\frac{2}{3}u_2^{3/2}) +F\,I_{-\frac{1}{3}}(\frac{2}{3}u_2^{3/2})\right], \hskip10mm 0\le u\le a \\ G\,u+H, \hskip57mm a\le u \end{array}\right. \label{E11}\end{aligned}$$ where $c=3^{-1/3}\Gamma(\frac{2}{3})$ ($\Gamma$ being the gamma function), $J_n$ is the Bessel function, $I_n$ is the modified Bessel function (cf. [@Abram]), $$u_1=\sqrt[3]{b/a}\,(a+u)\ ,\qquad u_2=\sqrt[3]{b/a}\,(a-u)\ ,\qquad \label{E12}$$ and $A, B, C, D, E, F, G, H$ are real constants given by the relations $$\begin{aligned} &&A=-2cZ\beta\delta \ ,\hskip25mm B= cZ(\beta\gamma+\alpha\delta) \ ,\nonumber\\ &&C=-A \sqrt[3]{9b/a}\,/\,\Gamma(1/3) \ ,\hskip6.3mm D= B/c-Ca \ ,\label{E13}\\ &&E=-2cZ\nu\sigma \ ,\hskip25mm F= cZ(\nu\rho+\mu\sigma) \ ,\nonumber\\ &&G=-E \sqrt[3]{9b/a}\,/\,\Gamma(1/3) \ ,\hskip6.3mm H= F/c-Ga \ ,\nonumber\end{aligned}$$ with $Z=\frac{2\pi}{3\sqrt3}(a\sqrt b)^{1/3}$, $$\begin{aligned} &&\alpha=J_{\frac{1}{3}}(\kappa)\ ,\hskip27mm \beta=J_{-\frac{1}{3}}(\kappa) \ ,\nonumber\\ &&\gamma= (a\sqrt b)^{2/3}\,J_{-\frac{2}{3}}(\kappa)\ ,\hskip9.5mm \delta=-(a\sqrt b)^{2/3}\,J_{\frac{2}{3}}(\kappa) \ ,\label{E14}\\ &&\mu=I_{\frac{1}{3}}(\kappa) \ ,\hskip28mm \nu=I_{-\frac{1}{3}}(\kappa) \ ,\nonumber\\ &&\rho =(a\sqrt b)^{2/3}\,I_{-\frac{2}{3}}(\kappa) \ ,\hskip10mm \sigma=(a\sqrt b)^{2/3}\,I_{\frac{2}{3}}(\kappa) \ ,\nonumber\end{aligned}$$ $\kappa=\frac{2}{3}a\sqrt b$ (note that $\beta\gamma-\alpha\delta=1/Z=\nu\rho-\mu\sigma$). Typical behaviour of the particles in the above sandwich space-time is shown in Fig. 1b. [**b) Sandwich wave with “$/\!\!\hskip1pt|$” profile**]{} Another sandwich wave can be obtained using the function $d(u)$ such that $$d(u)=\left\{ \begin{array}{l} 0, \hskip27mm u\le -a \\ b\,(a+u)/a, \hskip10mm -a\le u<0 \\ 0, \hskip27mm 0<u \end{array}\right. \label{E15}$$ where $a, b$ are again constants. In fact, it has a “saw” profile (see Fig. 1c) which is one “half” of the sandwich discussed above. It is non-symmetric and contains two discontinuities of different types. The functions $d(u)$ defined by Eq. (\[E9\]) and (\[E15\]) coincides for $u\le 0$ so that the functions $P, Q$ are identical in both cases. It is only necessary to join the solution at $u=0$ differently: $$\begin{aligned} P(u)&=&\left\{ \begin{array}{l} 1, \hskip 70mm u\le -a \\ c\,\sqrt{u_1}\,J_{-\frac{1}{3}}(\frac{2}{3}u_1^{3/2}), \hskip40mm -a\le u\le 0 \\ K\,u+L, \hskip57mm 0\le u \end{array}\right. \label{E16}\\ Q(u)&=&\left\{ \begin{array}{l} 1, \hskip 70mm u\le -a \\ c\,\sqrt{u_1}\,I_{-\frac{1}{3}}(\frac{2}{3}u_1^{3/2}), \hskip40mm -a\le u\le 0 \\ M\,u+N, \hskip57mm 0\le u \end{array}\right. \label{E17}\end{aligned}$$ where $$K = c\delta\,\sqrt[3]{b/a}\ ,\qquad L = c\beta\,\sqrt[6]{a^2 b}\ ,\qquad M = c\sigma\,\sqrt[3]{b/a}\ ,\qquad N = c\nu\,\sqrt[6]{a^2 b} \ .\label{E18}$$ Relative motions of test particles are illustrated in Fig. 1c. Qualitatively, they resemble motions in both previous cases (cf. Fig. 1a and Fig. 1b), only the relative velocities of particles in the region behind the wave (given in $x$-direction by $\ -\sin a/a$, $C$, and $K$ , respectively; in $y$-direction by $\ \sinh a/a$, $G$, and $M$) depend differently on particular choice of the parameters $a$ and $b$. [**c) Asymptotic sandwich wave**]{} Let us also consider the function $d(u)$ of the form $$d(u)=\frac{n}{2} \exp(-n|u|) \ , \label{E19}$$ (shown in Fig. 1d) where $n$ is an arbitrary real positive constant . Now there are [*no flat regions*]{} in front of the wave and behind it. The space-time is curved everywhere (it is of Petrov type N and therefore radiative), it becomes flat only asymptotically as $u\to\pm\infty$. For this reason we choose the functions $P, Q$ such that $P(u\to-\infty)\to 1$, $P'(u\to-\infty)\to 0$ and similarly $Q(u\to-\infty)\to 1$, $Q'(u\to-\infty)\to 0$. Then it can be shown that the functions are given by $$\begin{aligned} P(u)&=&\left\{ \begin{array}{l} J_0\left(\sqrt{\frac{2}{n}}\exp(\frac{n}{2}u)\right), \hskip63mm u\le 0 \\ A_1\, J_0\left(\sqrt{\frac{2}{n}}\exp(-\frac{n}{2}u)\right) +A_2\, Y_0\left(\sqrt{\frac{2}{n}}\exp(-\frac{n}{2}u)\right), \hskip10mm 0\le u \end{array}\right. \label{E20}\\ Q(u)&=&\left\{ \begin{array}{l} I_0\left(\sqrt{\frac{2}{n}}\exp(\frac{n}{2}u)\right), \hskip63.5mm u\le 0 \\ B_1\, I_0\left(\sqrt{\frac{2}{n}}\exp(-\frac{n}{2}u)\right) +B_2\, K_0\left(\sqrt{\frac{2}{n}}\exp(-\frac{n}{2}u)\right), \hskip10mm 0\le u \end{array}\right. \label{E21}\end{aligned}$$ where $$\begin{aligned} &&A_1=-\frac{\pi}{2}\lambda(\tilde\alpha\tilde\delta+\tilde\beta\tilde\gamma) \ ,\hskip15.0mm A_2=\pi\lambda\tilde\alpha\tilde\gamma \ ,\label{E22}\\ &&B_1=\lambda(\tilde\mu\tilde\sigma+\tilde\nu\tilde\rho) \ ,\hskip21.5mm B_2=2\lambda\tilde\mu\tilde\rho \ ,\nonumber\end{aligned}$$ with $$\begin{aligned} &&\tilde\alpha=J_0(\lambda)\ ,\hskip8mm \tilde\beta =Y_0(\lambda) \ ,\hskip8mm \tilde\gamma=J_1(\lambda)\ ,\hskip8mm \tilde\delta=Y_1(\lambda) \ ,\label{E23}\\ &&\tilde\mu=I_0(\lambda) \ ,\hskip9mm \tilde\nu=-K_0(\lambda) \ ,\hskip4mm \tilde\rho =I_1(\lambda) \ ,\hskip8mm \tilde\sigma=K_1(\lambda) \ ,\nonumber\end{aligned}$$ $\lambda=\sqrt{2/n}$ (note also that $\tilde\alpha\tilde\delta-\tilde\beta\tilde\gamma= -\frac{2}{\pi\lambda}$ and $\tilde\mu\tilde\sigma-\tilde\nu\tilde\rho=-\frac{1}{\lambda}$). Typical behaviour of the functions $P, Q$ is shown in Fig. 1d. It can be observed that in both asymptotic regions $u\to\pm\infty$ the particles move uniformly. **4 Impulsive limit** Now we can use the above results to construct impulsive gravitational waves. For standard sandwich wave (\[E6\])-(\[E8\]) it is easy to perform the limit $a\to0$. Then the profile function $d(u)$ approaches the $\delta$ function distribution and, using $\sin a/a\to 1$, $\sinh a/a\to 1$ we get $$\begin{aligned} P(u)&=&1-u\,\Theta(u)\ ,\nonumber\\ Q(u)&=&1+u\,\Theta(u)\ ,\label{E24}\end{aligned}$$ where $\Theta$ is the Heaviside step function ($\Theta=0$ for $u<0$, $\Theta=1$ for $u>0$). For non-standard sandwiches introduced in previous section one has to perform similar limits more carefully. It is well known that the sequence of “$\wedge$” functions given by (\[E9\]) approach the $\delta$ function (in a distributional sense) as $a\to 0$ if the second parameter is $b=1/a$ (so that the normalization condition $\int_{-\infty}^{+\infty} d(u)\,{{\rm d}}u=1$ holds for arbitrary $a$). Considering this limit, $\kappa=\frac{2}{3}\sqrt{a}$, the parameters (\[E14\]) are $\alpha\sim\mu \sim a^{1/6}\, 3^{-1/3} /\Gamma(4/3)$, $\beta \sim\nu \sim a^{-1/6}\, 3^{1/3} /\Gamma(2/3)$, $\gamma\sim\rho\sim 3^{2/3}/\Gamma(1/3)$, $\delta\sim-\sigma\sim-a^{2/3}\,3^{1/3}/(2\Gamma(2/3))$, and (\[E13\]) gives $$C\to-1 \ ,\quad D\to 1 \ ,\quad G\to 1 \ ,\quad H\to 1 \ .\label{E25}$$ Therefore, the functions $P, Q$ describing relative motions of test particles in the corresponding impulsive wave are again given by (\[E24\]). Analogously, the limit $a\to 0$ of “$/\!\!\hskip1pt|$” sandwiches given by (\[E15\]) with $b=2/a$ gives $\beta \sim\nu\sim (2a)^{-1/6}\, 3^{1/3} /\Gamma(2/3) $, $\delta\sim-\sigma\sim -(2a)^{2/3}\, 3^{1/3}/(2\Gamma(2/3))$, so that the parameters (\[E18\]) are $$K\to-1 \ ,\quad L\to 1 \ ,\quad M\to 1 \ ,\quad N\to 1 \ .\label{E26}$$ Again, the resulting functions $P, Q$ can be written in the form (\[E24\]). Finally, we can perform a limit $n\to\infty$ of “asymptotic sandwich waves” given by profile functions (\[E19\]). For $u\le0$ it follows immediately from (\[E20\]), (\[E21\]) that $P\to1$ and $Q\to1$. For $u\ge0$ calculations are more complicated: for $n\to\infty$ we get $\tilde\alpha\sim\tilde\mu\sim 1$, $\tilde\beta \sim -\frac{1}{\pi}\ln n$, $\tilde\gamma\sim\tilde\rho\sim (2n)^{-1/2}$, $\tilde\delta\sim -\frac{1}{\pi}[(2n)^{-1/2}\ln n+\sqrt{2n}\,]$, $\tilde\nu\sim -\frac{1}{2}\ln n$, $\tilde\sigma\sim \frac{1}{2} [\sqrt{2n}-(2n)^{-1/2}\ln n\,]$, so that $A_1\sim B_1\sim 1$, $A_2\sim \pi/n$ and $B_2\sim 2/n$. Since $J_0\sim I_0\sim1 $, $Y_0\sim-\frac{1}{\pi}[n u+\ln n\,]$ and $K_0\sim \frac{1}{2}[n u+\ln n\,]$ as $n\to\infty$, in the limit we obtain $$P\sim 1-u\ ,\qquad\qquad Q\sim 1+u \ ,\label{E27}$$ i.e. the relations (\[E24\]) are revealed once more. Note finally that using the transformation (\[E3\]) with the functions $P$ and $Q$ given by (\[E24\]) the metric of the impulsive homogeneous [*pp*]{} wave $${{\rm d}}s^2 = - {{\rm d}}t^2 + (1-u\Theta(u))^2 {{\rm d}}x^2 + (1+u\Theta(u))^2 {{\rm d}}y^2 + {{\rm d}}z^2\ , \label{E28}$$ goes over to $${{\rm d}}s^2=2\,{{\rm d}}\zeta {{\rm d}}\bar\zeta-2\,{{\rm d}}u{{\rm d}}v -\delta(u)(\zeta^2+\bar \zeta^2)\,{{\rm d}}u^2\ , \label{E29}$$ i.e. the metric (\[E1\]) with $f(u,\zeta)=\delta(u)\zeta^2$. Although this form of the impulsive wave is illustrative with the pulse evidently localized along the hyperplane $u=0$, the metric (\[E28\]) is more convenient in the sense that the metric system is continuous, $\delta$ function appearing only in the components of the curvature tensor. The transformation (\[E3\]) with (\[E24\]) also relates to the “scissors-and-paste” approach to the construction of impulsive solutions [@Penrose] which recently enabled new impulsive gravitational waves of somewhat different type [@Nutku; @Hogan] to be found. **5 Concluding remarks** We constructed three new types of non-standard sandwich [*pp*]{} waves with “wedge” (\[E9\]), “saw” (\[E15\]) and “asymptotic” (\[E19\]) profiles. Contrary to the standard sandwich wave (\[E6\]) they do not contain stationary regions. Particle motions were calculated explicitly and the corresponding limits to impulsive gravitational wave were performed. It was shown that all these limits give the same result (\[E24\]). Moreover, for sandwich waves presented above there exist critical values of $u$ for which the function $P(u)$ vanishes so that all the particles initially at rest on the $x$-axis collide. This demonstrates the caustic property of (plane) sandwich waves [@BP]. **Acknowledgments** We acknowledge the support of grants No. GACR-202/96/0206 and No. GAUK-230/1996 from the Czech Republic and Charles University. [99]{} Kramer D., Stephani H., MacCallum M.A.H., and Herlt H.: Exact Solutions of the Einstein’s Field Equations, Cambridge University Press, Cambridge, 1980. Penrose R.: in General Relativity (Ed. L.O’Raifeartaigh), Clarendon Press, Oxford, 1972. Bondi H., Pirani F.A.E., and Robinson I.: Proc. Roy. Soc [*A 251*]{} (1959) 519. Bondi H. and Pirani F.A.E.: Proc. Roy. Soc [*A 421*]{}, (1989) 395. Rindler W.: Essential Relativity, Springer, New York, 1977. Abramowitz M. and Stegun I.A.: Handbook of Mathematical Functions, Dover Publications, New York, 1964. Nutku Y.: Phys. Rev. D [*44*]{} (1991) 3164. Hogan P.A.: Phys. Lett. A [*171*]{} (1992) 21; Phys. Rev. Lett. [*70*]{} (1993) 117. Fig. 1. Typical exact behaviour of functions $P(u)$ and $Q(u)$ determining relative motions of free test particles (initially at rest) in $x$ and $y$-directions, respectively, caused by sandwich gravitational waves of various profiles: a) standard “square”, b) “wedge”, c) “saw”, d) “asymptotic” sandwich wave. In the last case also the limiting procedure $n\to\infty$ leading to an impulsive wave is indicated by corresponding dashed lines.

--- author: - 'Jun-ichi [Igarashi]{}[^1] and Tatsuya [Nagao]{}$^{1}$' title: 'Lattice Distortion and Resonant X-Ray Scattering in DyB$_{2}$C$_2$' --- Introduction ============ Resonant x-ray scattering has recently attracted much interest, since the resonant enhancement for the prohibited Bragg reflection corresponding to the orbital order has been observed in several transition-metal compounds by using synchrotron radiation with photon energy around the $K$ absorption edge. [@Murakami98a; @Murakami98b; @Murakami99c; @Murakami00b] For such $K$-edge resonances, $4p$ states of transition metals are involved in the intermediate state in the electric dipolar ($E_1$) process, and they have to be modulated in accordance with the orbital order for the signal to be observed. This modulation was first considered to come from the anisotropic term of the $4p$-$3d$ intra-atomic Coulomb interaction, [@Ishihara1] but subsequent studies based on the band structure calculation [@Elfimov; @Benfatto; @Takahashi1; @Takahashi2] have revealed that the modulation comes mainly from the crystal distortion via the oxygen potential on the neighboring sites. This is because $4p$ states are so extending in space that they are very sensitive to the electronic structure at neighboring sites. Rare-earth compounds also show the orbital order (usually an ordering of quadrupole moments). In CeB$_6$, RXS experiments were carried out around the Ce $L_{\rm III}$ absorption edge, and resonant enhancements have been found on quadrupolar ordering superlattice spots. [@Nakao01] Only one peak appeared as a function of photon energy, which was assigned to the $E_1$ process. In the $E_1$ process, $5d$ states of Ce in the intermediate state are to be modulated in accordance with the superlattice spots. Since the lattice distortion seems extremely small and the $5d$ states are less extending than the $4p$ states in transition-metal compounds, it is highly possible that the modulation is mainly caused by the Coulomb interaction between the $5d$ states and the orbital ordering $4f$ states. In our previous papers,[@Nagao; @Igarashi] we demonstrated this scenario by calculating the RXS spectra on the basis of the effective Hamiltonian of Shiina et al.[@Shiina; @Sakai; @Shiba] Without the help of lattice distortion, we obtained sufficient intensities of the spectra, and reproduced well the temperature and magnetic field dependences. This situation contrasts with those in transition-metal compounds. ![ (a) Sketch of the crystal structure of DyB$_2$C$_2$ ($P4/mbm$: $a=5.341$ ${\rm \AA}$, $c=3.547$ ${\rm \AA}$ at $30$ K). Gray large circles are Dy atoms. Solid and open small circles are B and C atoms, respectively. (b) Local coordinate frames attached to each sublattice. \[fig.cryst\]](fig.print.1.eps){width="8.0cm"} Another example for rare-earth compounds is RXS experiments on DyB$_2$C$_2$, where the intensity is resonantly enhanced near the Dy $L_{\rm III}$ absorption edge. [@Tanaka; @Hirota; @Matsumura] This material takes a tetragonal form at high temperatures as shown in Fig. \[fig.cryst\](a), and undergoes two phase transitions with decreasing temperatures in the absence of the magnetic field: a quadrupole order below $T_{\rm Q}$ ($=24.7$ K) (Phase II) and a magnetic order below $T_{\rm C}$ ($=15.3$ K) (Phase III).[@Yamauchi] Corresponding to the transition at $T_{\rm Q}$, a large non-resonant intensity is found in the $\sigma\to\sigma'$ channel on the $(h0\frac{\ell}{2})$ spot ($h$ and $\ell$ are odd integers).[@Matsumura] This suggests that some structural change takes place at $T=T_{\rm Q}$ from the tetragonal phase at high temperatures.[@Tanaka; @Hirota] A buckling of sheets of B and C atoms was proposed,[@Tanaka] and the non-resonant intensities by the buckling has recently been evaluated; about $0.01$ ${\rm \AA}$ shift of B and/or C atoms may be sufficient to give rise to such large intensities.[@Adachi] It is not clear in experiments whether the intensity on this spot is resonantly enhanced at the $L_{\rm III}$ edge, since the non-resonant part is so large that it may mask the resonant behavior. On the other hand, the resonant enhancement of RXS intensities has clearly been observed on the superlattice spot $(00\frac{\ell}{2})$. In this paper, we study the mechanism of the RXS spectra at the $L_{\rm III}$ edge in Phase II of DyB$_2$C$_2$. Since the $5d$ states are so extended in space that they are sensitive to lattice distortion caused by the buckling of sheets of B and C atoms. Then the question arises whether the direct influence of the lattice distortion on the $5d$ states is larger than the influence of the anisotropic $4f$ charge distribution associated with the quadrupole order through the $5d$-$4f$ Coulomb interaction. Lovesey and Knight[@Lovesey] have discussed the mechanism from the symmetry viewpoint, and have pointed out that the RXS intensities on $(00\frac{\ell}{2})$ and $(h0\frac{\ell}{2})$ spots come from lowering the local symmetry probably due to lattice distortion. The argument based on symmetry alone is powerful in some respect, but does not shed light on this issue. In the transition-metal compounds, the corresponding question has already been answered by [*a*b initio]{} calculations as mentioned above. However, such [*a*b initio]{} calculations are difficult in rare-earth compounds. We resort to a model calculation by treating the $5d$ states as a band and the $4f$ states as localized states. The buckling of sheets of B and C atoms causes modulations of the $5d$ bands and of the $4f$ states. We analyze such effects of lattice distortion on the basis of the point charge model,[@Hutchings] which leads to four inequivalent Dy sites with principal axes of the crystal field shown in Fig. \[fig.cryst\](b). These principal axes seem to correspond well to the direction of magnetic moments in the magnetic phase.[@Yamauchi] Of course, the point charge model is not good in quantitative viewpoint. Nonetheless, we construct an effective model that the $5d$ and $4f$ states are under the crystal field of the same form and with the same principal axes as the above analysis. The crystal field modulates the $5d$ states. Although the actual effect may come from hybridizations to $2p$, $3s$ states of B and C, it can be included into a form of the crystal field. The crystal field also makes the quadrupole moment of the $4f$ states align along the principal axes, establishing a quadrupole order. A molecular field caused by the Dy-Dy interaction may also act on the $4f$ states in Phase II in addition to the crystal field. This interaction may be mediated by the RKKY interaction, but the explicit form has not been derived yet. Note that the Ce-Ce interaction in CeB$_6$ has been extensively studied, describing well the phase diagram under the magnetic field. [@Shiina; @Sakai; @Shiba] But the molecular field may change little and even stabilize the quadrupole order. Therefore, we need not explicitly consider the molecular field by regarding the crystal field as including the effect. The charge anisotropy associated with the quadrupole order modulates the $5d$ states through the intra-atomic $5d$-$4f$ Coulomb interaction. We calculate the RXS intensity within the $E_1$ transition. We take account of the above two processes, direct and indirect ones, of modulating the $5d$ states. Both processes give rise to the RXS intensities on the $(00\frac{\ell}{2})$ and on the $(h0\frac{\ell}{2})$ spots. Both give similar photon-energy dependences and the same azimuthal-angle dependence in agreement with the experiment. However, the mechanism of direct modulation of the $5d$ band gives rise to the intensities much larger than the mechanism of indirect modulation through the $5d$-$4f$ Coulomb interaction in a wide parameter range of the crystal field. This suggests that the RXS intensities are mainly controlled by the lattice distortion. This paper is organized as follows. In § 2, we analyze the buckling of sheets of B and C atoms. In § 3, we briefly summarize the formulae used in the calculation of the RXS spectra. In § 4, we calculate the RXS spectra on two mechanisms. Section 5 is devoted to concluding remarks. Lattice Distortion ================== ![ Sketch of a B$_2$C$_2$ sheet ($z=c/2$). Open circles represent B and C atoms; big and small circles move to positive and negative directions along the $z$ axis, respectively. The directions are reversed on the plane of $z=-c/2$. Hatched circles represent Dy atoms at the plane of $z=0$. \[fig.distortion\]](fig.print.2.eps){width="8.0cm"} For making clear the effect of lattice distortion on electronic states, we first calculate the electrostatic potential on the basis of a point charge model. [@Hutchings] Point charges $q_{Dy}$, $q_B$, and $q_C$ are placed on Dy, B, and C sites, respectively. Figure \[fig.distortion\] shows the positions of B and C atoms on the plane of $z=c/2$ and those of Dy atoms on the plane of $z=0$. For the buckling of sheets of B and C atoms (up- and down-movements along the $c$ axis) specified in Fig. \[fig.distortion\], the electrostatic potential is evaluated within the second order of coordinates around Dy sites. As shown in Fig. \[fig.cryst\](b), four inequivalent sites arises. For sites $j$ ($=1\cdots 4$), the electrostatic potential is obtained as $$V_{\rm crys}(j) = A_2^0 Q_2^0 + A_2^2(j) Q_2^2 + A_{xy}(j)Q_{xy} , \label{eq.Vcry}$$ with $$\begin{aligned} Q_2^0 &= \frac{1}{2}(3z^2-r^2), \\ Q_2^2 &= \frac{\sqrt{3}}{2}(x^2-y^2), \\ Q_{xy}&= \sqrt{3}xy,\end{aligned}$$ where the coefficients are given by $$\begin{aligned} A_{2}^{0} & = \frac{2q_D}{a^3} \left[ 1 -2 \frac{|q_B|}{q_D} \left(\frac{a}{R_{B}} \right)^3 \left\{ 1 - 3 \left( \frac{c}{2 R_{B}} \right)^2 \right\} \right. \nonumber \\ & \left. -2 \frac{|q_C|}{q_D} \left(\frac{a}{R_{C}} \right)^3 \left\{ 1 - 3 \left( \frac{c}{2 R_{C}} \right)^2 \right\} \right] \nonumber \\ &= \frac{2q_D}{a^3}\left[ 1 + 1.40\frac{|q_B|}{q_D} + 1.79\frac{|q_C|}{q_D} \right], \\ A_{2}^{2}(j) &= \Lambda_{B} \cos ( 2 \theta_{B}^{(j)} ) + \Lambda_{C} \cos ( 2 \theta_{C}^{(j)} ), \label{eq.defA22} \\ A_{xy}(j) &= \Lambda_{B} \sin ( 2 \theta_{B}^{(j)} ) + \Lambda_{C} \sin ( 2 \theta_{C}^{(j)} ), \label{eq.defAxy}\end{aligned}$$ with $$\begin{aligned} \Lambda_{B} & = - \frac{|q_B|}{a^3} \frac{60}{\sqrt{3}} \left(\frac{r_B}{a} \right)^2 \left(\frac{a}{R_{B}}\right)^5 \left( \frac{c}{2 R_{B}} \right)^2 \frac{d_B}{\frac{c}{2}} \nonumber \\ &= -22.3 \frac{|q_B|}{a^3} \frac{d_B}{\frac{c}{2}}, \\ \Lambda_{C} & = - \frac{|q_C|}{a^3} \frac{60}{\sqrt{3}} \left(\frac{r_C}{a} \right)^2 \left(\frac{a}{R_{C}}\right)^5 \left( \frac{c}{2 R_{C}} \right)^2 \frac{d_C}{\frac{c}{2}} \nonumber \\ &= -24.0 \frac{|q_C|}{a^3} \frac{d_C}{\frac{c}{2}}.\end{aligned}$$ The first term in eq. (\[eq.Vcry\]) represents the crystal field without lattice distortion, while the second and third terms arise from the buckling. The $R_B$ ($=2.732$Å) and $R_C$ ($=2.676$Å) are distances from the origin to B and C sites, respectively. The $d_B$ and $d_C$ represent the absolute values of shifts along the $c$ axis from the $z=\pm c/2$ planes, respectively. Angles $\theta_{B(C)}^{(j)}$ and $\theta_{B(C)}^{(j)}$ in eqs. (\[eq.defA22\]) and (\[eq.defAxy\]) are given by $$\begin{aligned} \theta_B^{(1)} &= 65.6^{\circ}, \quad \theta_C^{(1)} = 19.1^{\circ},\nonumber\\ \theta_B^{(2)} &= 180^{\circ}-65.6^{\circ}, \quad \theta_C^{(2)} = 180^{\circ}-19.1^{\circ},\nonumber\\ \theta_B^{(3)} &= 90^{\circ}+\theta_B^{(1)}, \quad \theta_C^{(3)} = 90^{\circ}+\theta_C^{(1)},\nonumber\\ \theta_B^{(4)} &= 90^{\circ}+\theta_B^{(2)}, \quad \theta_C^{(4)} = 90^{\circ}+\theta_C^{(2)}.\end{aligned}$$ Now we search for the local coordinate frames in which the third term in eq. (\[eq.Vcry\]) is eliminated. Rotating the original coordinate frame by angle $\phi_j$ around the $c$ axis for each sublattice, we have the operators transformed as $$\begin{aligned} Q_2^2 &= \cos(2\phi_j)\tilde Q_2^2(j) - \sin(2\phi_j)\tilde Q_{xy}(j), \nonumber\\ Q_{xy} &= \sin(2\phi_j)\tilde Q_2^2(j) + \cos(2\phi_j)\tilde Q_{xy}(j), \label{eq.rotQ}\end{aligned}$$ where tilde operators $\tilde Q(j)$’s are represented with respect to the local coordinate frames. $Q_2^0$ is unchanged. Inserting eq. (\[eq.rotQ\]) into eq. (\[eq.Vcry\]), we have $$V_{crys}(j) = A_2^0 \tilde Q_2^0(j) + \tilde A_2^2(j)\tilde Q_2^2(j) + \tilde A_{xy}(j)\tilde Q_{xy}(j),$$ with $$\begin{aligned} \tilde A_{2}^{2}(j) &= \Lambda_{B} \cos [2(\theta_{B}^{(j)}-\phi_j)] + \Lambda_{C} \cos [ 2 (\theta_{C}^{(j)}-\phi_j)], \nonumber \\ \\ \tilde A_{xy}(j) &= \Lambda_{B} \sin [2(\theta_{B}^{(j)}-\phi_j)] + \Lambda_{C} \sin [2(\theta_{C}^{(j)}-\phi_j)]. \nonumber \\\end{aligned}$$ Condition $\tilde A_{xy}(j)=0$ determines $\phi_j$’s, which take values between $\theta_{B}^{(j)}$ and $\theta_{C}^{(j)}$. For example, assuming $q_B=q_C=-(3/4)e$, $q_{Dy}=3e$ (e: proton charge), $d_B=0.01$Å, $d_C=0.02$Å, we have $\phi_1=31.8^{\circ}$, $\phi_2=180^{\circ}-\phi_1$, $\phi_3=90^{\circ}+\phi_1$, $\phi_4=90^{\circ}+\phi_2$. The principal axes thus estimated correspond well with the directions of the ordered magnetic moments in Phase III.[@Yamauchi] The equivalent operator method allows us to write the crystal field energy $H_{crys}(j)$ at site $j$ within the subspace of angular momentum $J$ as $$H_{\rm crys}(j) = D_{J}[3\tilde J_z^2(j)-J(J+1)]+E_{J}[\tilde J^2_x(j) - \tilde J^2_y(j)]. \label{eq.crystal}$$ The $5d$ states are forming an energy band with width $\sim 15$ eV through a hybridization with $s$ and $p$ states of neighboring B and C atoms as well as $5d$ states of neighboring Dy atoms. We need the density of states (DOS) for calculating the RXS intensity. We assume a Lorentzian with full width of half maximum 5 eV for the DOS’s projected onto symmetries $xy$, $x^2-y^2$, $yz$, $zx$, and $3z^2-r^2$. The center of each component of the DOS is separate to each other in accordance with the first term of eq. (\[eq.crystal\]), although the first term need not be respected so much because of the large band effect. Explicitly they are assumed to be $(1/\pi)\Delta/((\epsilon-2.5)^2+\Delta^2)$ for $xy$, $x^2-y^2$, $(1/\pi)\Delta/((\epsilon-7)^2+\Delta^2)$ for $yz$, $zx$, and $(1/\pi)\Delta/((\epsilon-8.5)^2+\Delta^2)$ for $3z^2-r^2$, with energies in units of eV and $\Delta=2.5$ eV. This arbitrary assumption for the DOS form may be justified by the fact that the RXS spectra is not sensitive to the assumption. The second term of eq. (\[eq.crystal\]), which arises from the buckling of sheets of B and C atoms, gives rise to a small modification on the $5d$ band. Although the actual modulation of the $5d$ states may come through the hybridization to the $s$ and $p$ states of B and C atoms, such effects can be included into the second term. This term makes the local symmetry twofold. Dy$^{3+}$ ion is approximately in the $4f^9$-configuration ($^6$H$_{15/2}$). Equation (\[eq.crystal\]) is now applied to the subspace of $J=15/2$. Since the $4f$ states are much localized than the $5d$ states, the crystal field is much smaller here than that on the $5d$ states. The coefficient $D_f$ of the first term is expected to be positive from the analysis of magnetic susceptibility.[@Yamauchi] This leads to the lowest energy states $|\pm\frac{1}{2}\rangle$ and the next energy states $|\pm\frac{3}{2}\rangle$, both of which form Kramers’ doublets ($|M\rangle$ represents the state of $J_z=M$). The axial symmetry is kept instead of fourfold symmetry without the lattice distortion. Terms of $O_4^4\equiv \frac{1}{2}(J_+^4 + J_-^4)$, which admixes the states $|M\rangle$ with $|M\pm 4\rangle$, come from the higher order expansion to make the local symmetry fourfold. The detailed study along this line, however, is beyond the scope of the present study. In any event, the second term makes the local symmetry twofold. The lowest energy state is admixed by $|M\rangle$ with $|M|>1/2$. The quadrupole moment is ordered; $\langle\tilde O_{x^2-y^2}\rangle$ ($\equiv\frac{\sqrt{3}}{2}\langle\tilde J_x^2-\tilde J_y^2\rangle$) takes a finite value with $\langle\tilde O_{xy}\rangle$ ($\equiv\frac{\sqrt{3}}{2}\langle \tilde J_x\tilde J_y +\tilde J_y\tilde J_x\rangle$)$=0$ in the local coordinate frame for each sublattice ($\langle\cdots\rangle$ means the average over the lowest Kramers doublet). With increasing values of $|E_f|/D_f$, the average quadrupole moment increases, as shown in Fig. \[fig.parameter\](b). It becomes largest, $\langle \tilde{O}_{x^2-y^2}\rangle=46.1$ at $D_{f}=0.1$. Cross Section of Resonant X-Ray Scattering ========================================== We briefly summarize here the formulae used in the calculation of the RXS spectra in the next section. The conventional RXS geometry is shown in Fig. \[fig.geom\]; photon with frequency $\omega$, momentum ${\textbf k}_i$ and polarization $\mu$ ($=\sigma$ or $\pi$) is scattered into the state with momentum ${\textbf k}_f$ and polarization $\mu'$ ($=\sigma'$ or $\pi'$). The scattering vector is defined as ${\textbf G}\equiv{\textbf k}_f-{\textbf k}_i$. Near the Dy $L_{\rm III}$ absorption edge, a $2p$ core electron is virtually excited to $5d$ states in the $E_1$ process. Subsequently it recombines with the core hole. Since the $2p$ states are well localized around Dy sites, it is a good approximation to describe the scattering tensor as a sum of contributions from each site of the core hole. Therefore, the cross section in the $E_1$ process is given by $$I_{\mu\to\mu'}({\textbf G},\omega) \propto | \sum_{\alpha\alpha'}P'^{\mu'}_{\alpha} M_{\alpha\alpha'}({\textbf G},\omega)P^{\mu}_{\alpha'} |^2, \label{eq.cross}$$ where $$M_{\alpha\alpha'}({\textbf G},\omega) = \frac{1}{\sqrt{N}} \sum_j M_{\alpha\alpha'}(j,\omega) \exp(-i{\textbf G}\cdot{\textbf r}_j), \label{eq.scatensor}$$ with $$M_{\alpha\alpha'}(j,\omega) = \sum_{\Lambda} \frac{\langle\psi_n|x_\alpha(j)|\Lambda\rangle \langle \Lambda|x_{\alpha'}(j)|\psi_j\rangle} {\hbar\omega-(E_{\Lambda}-E_j)+i\Gamma}, \label{eq.dipole}$$ $\alpha$ $(P^\sigma)_\alpha$ $(P'^{\sigma'})_\alpha$ $(P'^{\pi'})_\alpha$ ---------- ---------------------- ------------------------- ---------------------------------------------------- 1 $\cos\beta\cos\psi$ $\cos\beta\cos\psi$ $-\sin\theta\cos\beta\sin\psi+\cos\theta\sin\beta$ 2 $- \sin\psi$ $- \sin\psi$ $-\sin\theta\cos\psi+\cos\theta\sin\beta$ 3 $-\sin\beta\cos\psi$ $-\sin\beta\cos\psi$ $\cos\theta\cos\beta$ : Geometrical factors \[tab.azim\] where $N$ is the number of Dy sites. Note that the cross section is an order of $N$. The $P^\mu$ and $P'^{\mu}$ are geometrical factors for the incident and scattered photons, respectively. Their explicit forms are given in Table \[tab.azim\]. The $|\psi_j\rangle$ represents the initial state with energy $E_j$. The intermediate state $|\Lambda\rangle$ consists of an excited electron on $5d$ states and a hole on $2p$ states with energy $E_{\Lambda}$. The $\Gamma$ is the life-time broadening width of the core hole. The dipole operators $x_\alpha(j)$’s are defined as $x_1(j)=x$, $x_2(j)=y$, and $x_3(j)=z$ in the coordinate frame fixed to the crystal axes with the origin located at the center of site $j$. The scattering amplitude $M_{\alpha\alpha'}({\textbf G},\omega)$ contains the square of the dipole matrix element $A_{dp}=\langle 5d|r|2p\rangle = \int_0^{\infty} R_{5d}(r)rR_{2p}(r)r^2{\rm d}r $ with $R_{5d}(r)$ and $R_{2p}(r)$ being the radial wavefunctions for the $5d$ and $2p$ states. For Dy$^{3+}$ atom, it is estimated as $2.97\times 10^{-11}$ cm in the $4f^9$ configuration within the HF approximation.[@Cowan] ![ Scattering geometry. Incident photon with wave vector ${\textbf k}_i$ and polarization $\sigma$ or $\pi$ is scattered into the state with wave vector ${\textbf k}_f$ and polarization $\sigma'$ or $\pi'$ at Bragg angle $\theta$. The sample crystal is rotated by azimuthal angle $\psi$ around the scattering vector ${\textbf G}={\textbf k}_f-{\textbf k}_i$. \[fig.geom\]](fig.print.3.eps){width="8.0cm"} Calculation of RXS spectra ========================== In the $E_1$ transition, an electron is excited from $2p$ states to $5d$ states at a Dy site. A $2p$ core hole is left behind, and its state is split into the states of $j_p=3/2$ and $j_p=1/2$ ($j_p$ is the total angular momentum) due to the strong spin-orbit interaction. We consider only the $j_p=3/2$ states ($L_{\rm III}$ edge). We describe the photoexcited $5d$ electron in the band by introducing a local Green’s function, $$G^{5d}_{m^d}(\hbar\omega) = \int_{\epsilon_F}^{\infty} \frac{\rho^{5d}_{m^d}(\epsilon)}{\hbar\omega-\epsilon+i\delta} {\rm d}{\epsilon}, \label{eq.5dgreen}$$ where $\rho^{5d}_{m^d}(\epsilon)$ is the $m^d$ component of the $d$ DOS defined in §2. The Fermi energy $\epsilon_F$ is set to be zero so that the $5d$ band is almost empty. Finite RXS intensities on superlattice spots arise from modulating the $5d$ states with wave vectors of superlattice spots. There are two origins to giving rise to such modulation. One is a direct influence from the buckling of sheets of B and C atoms, which is represented by the second term of eq. (\[eq.crystal\]). Another is the charge anisotropy of the $4f$ states in the quadrupole ordering phase. We discuss both origins separately. In the actual calculation, we specify the local coordinate frames with $\phi_1=28^{\circ}$, $\phi_2=180^{\circ}-28^{\circ}$, $\phi_3=90^{\circ}+28^{\circ}$, $\phi_4=90^{\circ}+152^{\circ}$ for four sublattices in accordance with the experiment.[@Yamauchi] Direct influence of lattice distortion -------------------------------------- Let the $E_1$ transition take place at a particular site (called as “origin"). The excited $5d$ electron is attracted by the core hole potential at the origin. What is more important is that the $5d$ electron is under the influence of the second term of eq. (\[eq.crystal\]). Taking account of the multiple scattering from these terms at the origin, we evaluate the resolvent $1/(\hbar\omega-H_{\rm int})$ with respect to the intermediate-state Hamiltonian $H_{\rm int}$: $$\begin{aligned} & \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_ {m^ds^d\lambda;m'^ds'^d\lambda'} \nonumber \\ = &[G^{5d}_{m^d}(\hbar\omega+i\Gamma-\epsilon_\lambda)^{-1} \delta_{\lambda\lambda'}\delta_{m^dm'^d}\delta_{s^ds'^d} \nonumber\\ &- V_{m^ds^d\lambda;m'^ds'^d\lambda'}]^{-1}, \label{eq.matrix1}\end{aligned}$$ where $m^d$ and $s^d$ specify the orbital and spin of the $d$ electron, respectively. The $\epsilon_{\lambda}$ represents the energy of the core hole with $\lambda$ in the $j_p=3/2$ subspace. The scattering potential $V_{m^ds^d\lambda;m'^ds'^d\lambda'}$ includes the second term of eq. (\[eq.crystal\]) and the Coulomb interaction between the $5d$ electron and the $2p$ hole. The latter quantity is expressed in terms of the Slater integrals, which are evaluated within the HF approximation in a Dy$^{3+}$ atom (see Table \[tab.slater\]).[@Com1] The core hole life-time width is set to be $\Gamma=2.5$ eV. Equation (\[eq.matrix1\]) is numerically evaluated. --------------------- --------------------- ---------------- ---------------- $F^{k}(4f,4f)$ $F^{k}(2p,5d)$ $F^{k}(2p,4f)$ $F^{k}(4f,5d)$ $F^0$ 32.19 $F^0$ 16.13 $F^0$ 44.62 $F^0$ 14.70 $F^2$ 15.31 $F^2$ 0.489 $F^2$ 1.982 $F^2$ 3.614 $F^4$ 9.607 $F^4$ 1.741 $F^6$ 6.911 $G^{k}(2p,5d)$ $G^{k}(2p,4f)$ $G^{k}(4f,5d)$ $G^1$ 0.414 $G^2$ 0.207 $G^1$ 1.615 $G^3$ 0.245 $G^4$ 0.133 $G^3$ 1.321 $G^5$ 1.009 $\zeta_{4f}=$ 0.273 $\zeta_{5d}=$ 0.181 --------------------- --------------------- ---------------- ---------------- : Slater integrals and the spin-orbit interaction for Dy$^{3+}$ atoms in the Hartree-Fock approximation (in units of eV). $\ast$In the RXS calculation, the above values of the anisotropic terms are reduced by multiplying a factor 0.8, while the values for $F^{(0)}(nl,n'l')$ are replaced by much smaller values, $F^{(0)}(4f,5d) = 3.0$, $F^{(0)}(4f,4f) = 7.0$, $F^{(0)}(2p,5d) = 4.0$, $F^{(0)}(2p,4f) = 12.0$. \[tab.slater\] Before calculating the RXS spectra, we touch on the absorption coefficient. We calculate the absorption coefficient $A(\omega)$ in the $E_1$ process from the resolvent by using the relation, $$\begin{aligned} A(\omega) & \propto \sum_{j} \sum_{\alpha} \langle\psi_j|x^\alpha(j)|m^ds^d\lambda\rangle \nonumber \\ & \times \left(-\frac{1}{\pi}\right) {\rm Im}\left(\frac{1}{\hbar\omega - H_{\rm int} + i\delta}\right)_ {m^ds^d\lambda;m'^ds'^d\lambda'} \nonumber\\ & \times\langle m'^ds'^d\lambda'|x^\alpha(j)|\psi_j\rangle . \label{eq.absorp}\end{aligned}$$ Here ${\rm Im}X$ indicates the imaginary part of the quantity $X$. Figure \[fig.absorp\] shows the calculated $A(\omega)$ in comparison with the fluorescence experiment.[@Hirota] We adjust the core hole energy such that the calculated peak coincides with the experimental one. The calculated curve reproduces the experimental one. Since the spectra is proportional to the $d$ DOS when the $5d$-$2p$ Coulomb interaction is neglected, the assumed DOS seems reasonable. The intensity seen in the high energy region of the experiment may come from the $d$ symmetric states mixing with $3s$, $3p$ states of B and C atoms, which is outside our interest. ![ Absorption coefficient $A(\omega)$ (lower panel) in comparison with the $L_{\rm III}$-edge fluorescence spectra (upper panel).$^{17)}$ \[fig.absorp\]](fig.print.4.eps){width="8.0cm"} Now we discuss the RXS spectra. The resolvent above calculated is used to calculate the scattering amplitude at the origin with the help of the relation, $$\begin{aligned} & \sum_{\Lambda}\frac{|\Lambda\rangle\langle\Lambda|} {\hbar\omega-(E_{\Lambda}-E_j)+i\Gamma} \nonumber \\ &= \sum_{m^ds^d\lambda}\sum_{m'^ds'^d\lambda'} |m^ds^d\lambda\rangle \nonumber \\ & \times \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_ {m^ds^d\lambda;m'^ds'^d\lambda'} \langle m'^ds'^d\lambda'| . \label{eq.intermed}\end{aligned}$$ This expression is independent of the quadrupole ordering $4f$ states. It is inserted into eq. (\[eq.dipole\]) to calculate an RXS amplitude. The extension to general site $j$ is straightforward. In the coordinate frame fixed to crystal (not in the local coordinate frames), their forms are given by $$\begin{aligned} \hat M(1,\omega) &= \left( \begin{array}{ccc} \xi(\omega) & \eta(\omega) & 0 \\ \eta(\omega) & \zeta(\omega) & 0 \\ 0 & 0 & \gamma(\omega) \end{array} \right), \nonumber \\ % \label{eq.Mamp1} %\end{equation} %\begin{equation} \hat M(2,\omega) &= \left( \begin{array}{ccc} \zeta(\omega) & \eta(\omega) & 0 \\ \eta(\omega) & \xi(\omega) & 0 \\ 0 & 0 & \gamma(\omega) \end{array} \right), \nonumber \\ %\label{eq.Mamp2} %\end{equation} %\begin{equation} \hat M(3,\omega) &= \left( \begin{array}{ccc} \zeta(\omega) & -\delta(\omega) & 0 \\ -\delta(\omega) & \xi(\omega) & 0 \\ 0 & 0 & \gamma(\omega) \end{array} \right), \nonumber \\ %\label{eq.Mamp3} %\end{equation} %\begin{equation} \hat M(4,\omega) &= \left( \begin{array}{ccc} \xi(\omega) & -\delta(\omega) & 0 \\ -\delta(\omega) & \zeta(\omega) & 0 \\ 0 & 0 & \gamma(\omega) \end{array} \right). \label{eq.Mamp4}\end{aligned}$$ Owing to the factor $\exp(-i{\textbf G}\cdot{\textbf r}_j)$ in eq. (\[eq.scatensor\]), the total amplitude is given by a combination of $\hat M(1,\omega)+\hat M(2,\omega)-\hat M(3,\omega)-\hat M(4,\omega)$. Thus, we have the final form, $$\frac{\hat M({\textbf G},\omega)}{\sqrt{N}} = \left( \begin{array}{ccc} 0 & \eta(\omega) & 0 \\ \eta(\omega) & 0 & 0 \\ 0 & 0 & 0 \end{array} \right). \label{eq.final1}$$ The geometrical factors are given by setting $\beta=0$ in Table \[tab.azim\], which are combined to eq. (\[eq.final1\]) to calculate the scattering intensity. We have the RXS intensity as a function of azimuthal angle $\psi$ as $$\begin{aligned} I_{\sigma\to\sigma'}({\textbf G},\omega) &\propto |\eta(\omega)|^2\sin^2 2\psi, \nonumber \\ I_{\sigma\to\pi'}({\textbf G},\omega) &\propto |\eta(\omega)|^2\sin^2\theta\cos^2 2\psi, \label{eq.azim1}\end{aligned}$$ with $\theta$ the Bragg angle. Here $\psi$ is defined such that $\psi=0$ corresponds to the scattering plane containing the $b$ axis. Figure \[fig.spec\] shows the calculated RXS spectra as a function of photon energy in comparison with the experiment ($\psi=-45^\circ$).[@Hirota] The crystal field parameter is set to be $E_d=-0.1$ eV in eq. (\[eq.crystal\]). As shown in the middle panel, the calculated spectra show a single-peak in agreement with the experiment. (Only the $\sigma\to\sigma'$ channel gives finite intensity for $\psi=-45^\circ$.) The photon energy dependence in the $\sigma\to\pi'$ channel is found to be the same as in the $\sigma\to\sigma'$ channel in the calculation. On the other hand, an extra peak has been observed in the $\sigma\to\pi'$ channel at $\hbar\omega=7782$ eV (pre-edge peak) for $\psi=0$.[@Hirota] This peak may come from the electric quadrupole ($E_2$) transition. ![ RXS spectra for ${\textbf G}=(00\frac{5}{2})$ at $\psi=-45^\circ$ in the $\sigma\to\sigma'$ channel, as a function of photon energy. Top: experimental spectra at $T=20$ K (Phase II).$^{17)}$ Middle: Calculated spectra by taking account of the direct influence of lattice distortion with $E_d=-0.1$ eV. Bottom: Calculated spectra by taking account of the influence of quadrupole ordering $4f$ states with $E_f/D_f=-0.2$. \[fig.spec\] ](fig.print.5.eps){width="8.0cm"} ![ (a) Main peak intensity as a function of the crystal field parameter $|E_d|$ on the $5d$ states, which is calculated by taking account of the direct influence of lattice distortion. (b) Main peak intensity as a function of the crystal field parameter $|E_f|/D_f$ on the $4f$ states, which is calculated by taking account of the influence of quadrupole ordering $4f$ states (open circles). Crosses represent the quadrupole moments $\langle \tilde{O}_{x^2-y^2} \rangle$ in the local coordinate frame for each sublattice. \[fig.parameter\]](fig.print.6.eps){width="8.0cm"} Figure \[fig.parameter\](a) plots the peak intensity as a function of $|E_d|$ (at $\psi=-45^\circ$ in the $\sigma\to\sigma'$ channel). The intensity of the “main" peak increases with increasing values of $|E_d|$. It is nearly proportional to $|E_d|^2$. Figure \[fig.azim1\] shows the azimuthal angle dependence of the main peak intensity for ${\textbf G}=(00\frac{5}{2})$, in good agreement with the experiment. The same dependence as eq. (\[eq.azim1\]) has been proposed on the basis of the symmetry of the scattering tensor.[@Matsumura] Note that the intensity ratio between the $\sigma\to\sigma'$ channel and the $\sigma\to\pi'$ channel is determined by a geometrical factor; the oscillation amplitude of intensity in the $\sigma\to\pi'$ channel is the factor $\sin^2\theta=0.312$ smaller than that in the $\sigma\to\sigma'$ channel. ![ Azimuthal angle dependence of the RXS intensity of the main peak on ${\textbf G}=(00\frac{5}{2})$, in comparison with the experiment at $T=20$ K.$^{17)}$ \[fig.azim1\]](fig.print.7.eps){width="8.0cm"} The total amplitude is given by a combination of $\hat M(1,\omega)-\hat M(2,\omega)-\hat M(3,\omega)+\hat M(4,\omega)$. Thus, we have the final form, $$\frac{\hat M({\textbf G},\omega)}{\sqrt{N}} = \left( \begin{array}{ccc} \frac{1}{2}(\xi(\omega)-\zeta(\omega)) & 0 & 0 \\ 0 & \frac{1}{2}(\zeta(\omega)-\xi(\omega)) & 0 \\ 0 & 0 & 0 \end{array} \right). \label{eq.final2}$$ Combining the geometrical factor in Table I to eq. (\[eq.final2\]), we obtain the scattering intensity. The spectral shape as a function of photon energy is found almost the same as that for the $(00\frac{\ell}{2})$ spot, so that we omit the corresponding figure. The azimuthal angle dependence is given by $$\begin{aligned} I_{\sigma\to\sigma'}({\textbf G},\omega) &\propto \frac{1}{4}|\xi(\omega)-\zeta(\omega)|^2 (\cos^2\beta\cos^2\psi-\sin^2\psi)^2, \nonumber \\ I_{\sigma\to\pi'}({\textbf G},\omega) &\propto \frac{1}{4}|\xi(\omega)-\zeta(\omega)|^2 \nonumber\\ &\times [\cos\theta\sin\beta(\cos\beta\cos\psi+\sin\psi) \nonumber \\ & -\sin\theta(1+\cos^2\beta)\sin\psi\cos\psi)]^2, \label{eq.azim2}\end{aligned}$$ where $\beta$ is determined from $\tan\beta=(2hc)/(\ell a)$. Figure \[fig.azim2\] shows the azimuthal dependence of the peak intensity for the $(30\frac{3}{2})$ spot. A large non-resonant intensity has been observed in the $\sigma \rightarrow \sigma'$ channel, and the resonant behavior is not clear in the experiment.[@Tanaka; @Hirota] The non-resonant intensity may come from the Thomson scattering due to the lattice distortion. We hope that the resonant behavior discussed here is observed in the $\sigma \rightarrow \pi'$ channel, since the non-resonant intensity is expected to disappear in this channel. ![ Azimuthal angle dependence of the RXS intensity of the main peak on ${\textbf G}=(30\frac{3}{2})$. \[fig.azim2\]](fig.print.8.eps){width="8.0cm"} Influence of quadrupole ordering 4f states ------------------------------------------ The initial state is evaluated in § 2, where the $4f$ quadrupole moment is ordered. The intermediate state is evaluated by the following steps. Let the $E_1$ transition take place at the origin. The complex of $4f$ electrons and the $2p$ hole is assigned as the eigenstate $|\nu\rangle$ with energy $E_{\nu}$, which is calculated by diagonalizing the matrix of the intra-atomic Coulomb interaction. To keep the matrix size manageable, the space of $4f$ states is restricted within the space of $J=15/2$. This restriction causes only minor errors in the RXS spectra, since the RXS amplitude contains the overlap between the $4f$ states in the intermediate state and that in the initial state, which becomes very small for the $4f$ states outside the $J=15/2$ subspace. Since the photoexcited $5d$ electron interacts with the $2p$ hole and with $4f$ electrons, the complex of $4f$ electrons and the $2p$ hole serves as a scatterer to the $5d$ electron. Thus we have the resolvent at the origin, $$\begin{aligned} & \left(\frac{1}{\hbar\omega - H_{\rm int} + i\Gamma}\right)_ {m^ds^d\nu;m'^ds'^d\nu'} \nonumber \\ & = [G^{5d}_{m^d}(\hbar\omega+i\Gamma-E_\nu)^{-1} \delta_{\nu\nu'}\delta_{m^dm'^d}\delta_{s^ds'^d} \nonumber\\ & - U_{m^ds^d\nu;m'^ds'^d\nu'}]^{-1}, \label{eq.matrix2}\end{aligned}$$ where $m^d$ and $s^d$ specify $5d$ states. The potential $U_{m^ds^d\nu;m'^ds'^d\nu'}$ includes the $5d$-$4f$ Coulomb interaction as well as the $5d$-$2p$ Coulomb interaction. They are expressed in terms of the Slater integrals, which are given in Table II.[@Cowan; @Com1] Since the $5d$-$4f$ Coulomb interaction is implicitly included in the $5d$-band energy, we eliminate the average of the $5d$-$4f$ interaction from the potential $U$. The energies of the $4f$ states coming from the crystal field much smaller than other energies in the intermediate state, and thus can be neglected. We do not include the crystal field on the $5d$ states due to the lattice distortion (the second term of eq. (\[eq.crystal\])). Within the present approximation, the right hand side of eq. (\[eq.matrix2\]) becomes a matrix with dimensions $640\times 640$ ($640=10\times 64$), which are numerically inverted. The resolvent thus obtained is the same on four sublattices. The scattering amplitudes become different on different sublattices after the transition matrix elements from the initial state are taken into account. The scattering amplitude is found to take the same forms as eqs. (\[eq.Mamp4\]). Therefore, the azimuthal-angle dependence is the same as eqs. (\[eq.azim1\]) and (\[eq.azim2\]). As regards the photon-energy dependence, the bottom panel in Fig. \[fig.spec\] shows the RXS spectra at $\psi=-45^{\circ}$ in the $\sigma\to\sigma'$ channel for ${\textbf G}=(00\frac{5}{2})$. We put $E_f/D_f=-0.2$. The spectral shape is not so different from the curve given in the preceding subsection, although a small hump appears at the low energy side. Difference is the magnitude of the intensity. Figure \[fig.parameter\](b) shows the main-peak intensity as a function of $|E_f|/D_f$. It increases with increasing values of $|E_f|/D_f$. In the same figure, we have also plotted the sublattice quadrupole moment as a function of $|E_{f}|/D_{f}$. It also increases with increasing values of $|E_{f}|/D_{f}$. This coincidence is plausible, since the charge anisotropy in the $4f$ states increases with increasing the ordered quadrupole moment. However, its magnitude remains much smaller than those given by the direct influence of lattice distortion in a wide parameter range of crystal field by comparison with the curve in Fig. \[fig.parameter\](a). Concluding Remarks ================== We have studied the mechanism of RXS at the $L_{\rm III}$ edge in the quadrupole ordering phase of DyB$_2$C$_2$. Having analyzed the effect of the bucking of sheets of B and C atoms on the $5d$ and $4f$ states, we have constructed an effective model that the crystal field is acting on the $5d$ and $4f$ states with the principal axes different for different sublattices. We have calculated the RXS spectra in the $E_1$ process by treating the $5d$ states as a band and the $4f$ states as localized states. We have considered two mechanisms separately that the lattice distortion directly modulates the $5d$ band and that the charge anisotropy of the quadrupole ordering $4f$ states modulate the $5d$ band through the $5d$-$4f$ intra-atomic Coulomb interaction. We have found that both mechanisms give rise to the RXS intensities on $(00\frac{\ell}{2})$ and $(h0\frac{\ell}{2})$ spots with similar photon-energy dependences and the same azimuthal angle dependence. Both explain well the experimental RXS spectra. However, it is shown that the former mechanism gives rise to the intensity much larger than the latter one for a wide parameter range of crystal field. This suggests that the main-peak of the RXS spectra is not a direct reflection of the quadrupole order but mainly controlled by the lattice distortion. To confirm this observation more quantitatively, band structure calculations may be useful since the $5d$ states are considerably extended in space. This study is left in the future. As regards the pre-edge peak, we have estimated its intensity within the $E_2$ transition. In that estimate, we have used the same initial state as discussed above and have taken account of the full multiplets of the $f^{10}$-configuration for the intermediate state. The transition matrix element has been evaluated by the atomic Hartree-Fock wave function.[@Cowan] The pre-edge peak intensity thus evaluated is found to be more than three-order of magnitude smaller than the main-peak intensity evaluated by the mechanism of the charge anisotropy of the quadrupole ordering $4f$ states. This is inconsistent with the experiments, where the pre-edge peak intensity is the same order of magnitude to the main peak intensity. Clarifying this point is also left in the future. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank S. W. Lovesey, Y. Tanaka, and T. Inami for valuable discussions. This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture. (\#1,\#2,\#3)[[\#1]{} (\#2) \#3]{} [99]{} Y. Murakami, H. Kawata, M. Tanaka, T. Arima, Y. Moritomo and Y. Tokura: Phys. Rev. Lett. (80,1998,1932). Y. Murakami, J. P. Hill, D. Gibbs, M. Blume, I. Koyama, M. Tanaka, H. Kawata, T. Arima, Y. Tokura, K. Hirota and Y. Endoh: Phys. Rev. Lett. (81,1998,582). M. von Zimmermann, J.P. Hill, D. Gibbs, M. Blume, D. Casa, B. Keimer, Y. Murakami, Y. Tomioka and Y. Tokura: Phys. Rev. Lett. (83,1999,4872). M. Noguchi, A. Nakazawa, T. Arima, Y. Wakabayashi, H. Nakao and Y. Murakami: Phys. Rev. B (62,2000,R9271). S. Ishihara and S. Maekawa: Phys. Rev. Lett. (80,1998,3799). I. S. Elfimov, V. I. Anisimov and G. Sawatzky: Phys. Rev. Lett. (82,1999,4264). M. Benfatto, Y. Joly and C. R. Natoli: Phys. Rev. Lett. (83,1999,636). M. Takahashi, J. Igarashi and P. Fulde: J. Phys. Soc. Jpn. (68,1999,2530). M. Takahashi, J. Igarashi and P. Fulde: J. Phys. Soc. Jpn. (69,2000,1614). H. Nakao, K. Magishi, Y. Wakabayashi, Y. Murakami, K. Koyama, K. Hirota, Y. Endoh and S. Kunii: J. Phys. Soc. Jpn. (70,2001,1857). T. Nagao and J. Igarashi: J. Phys. Soc. Jpn. (70,2001,2892). J. Igarashi and T. Nagao: J. Phys. Soc. Jpn. (71,2002,1771). R. Shiina, H. Shiba and P. Thalmeier: J. Phys. Soc. Jpn. (66,1997,1741). O. Sakai, R. Shiina, H. Shiba and P. Thalmeier: J. Phys. Soc. Jpn. (66,1997,3005). H. Shiba, O. Sakai and R. Shiina: J. Phys. Soc. Jpn. (68,1999,1988). Y. Tanaka, T. Inami, T. Nakamura, H. Yamauchi, H. Onodera, K. Ohoyama and Y. Yamaguchi: J. Phys. Condens. Matter (11,1999,L505). K. Hirota, N. Oumi, T. Matsumura, H. Nakao, Y. Wakabayashi, Y. Murakami and Y. Endoh: Phys. Rev. Lett. (84,2000,2706). T. Matsumura, N. Oumi, K. Hirota, H. Nakao, Y. Murakami, Y. Wakabayashi, T. Arima, S. Ishihara and Y. Endoh: Phys. Rev. B (65,2002,094420). H. Yamauchi, H. Onodera, K. Ohoyama, T. Onimaru, M. Kosaka, M. Ohashi and Y. Yamaguchi: J. Phys. Soc. Jpn. (68,1999,2057). H. Adachi, H. Kawata, M. Mizumaki, T. Akao, M. Sato, N. Ikeda, Y. Tanaka and H. Miwa: Phys. Rev. Lett. (89,2002,206401). S. W. Lovesey and K. S. Knight: Phys. Rev. B (64,2001,094401). M. T. Hutchings: Solid State Physics, (16,1964,227). R. Cowan: [*T*he Theory of Atomic Structure and Spectra]{} (University of California Press, Berkeley, 1981). The anisotropic terms of the Coulomb interaction are slightly reduced in solids; we use the atomic values in Table \[tab.slater\] by reducing them with multiplying a factor 0.8. On the other hand, the values of $F^0(n\ell,n'\ell')$ are considerably screened in solids, so that they are replaced by much smaller values. [^1]: E-mail: jigarash@spring8.or.jp

--- abstract: 'Uncertainty quantification plays an important role in biomedical engineering as measurement data is often unavailable and literature data shows a wide variability. Using state-of-the-art methods one encounters difficulties when the number of random inputs is large. This is the case, e.g., when using composite Cole-Cole equations to model random electrical properties. It is shown how the number of parameters can be significantly reduced by the Karhunen-Loève expansion. The low-dimensional random model is used to quantify uncertainties in the axon activation during deep brain stimulation. Numerical results for a Medtronic 3387 electrode design are given.' author: - '[^1]' title: 'Low-Dimensional Stochastic Modeling of the Electrical Properties of Biological Tissues' --- Uncertainty, random processes, principal component analysis, biomedical engineering. Introduction ============ electrical properties of biological tissue are based on experimental data and are subject to large variability in literature [@gabriel2009; @schmidtieee2013], which arises from difficulties associated with the measuring process. Their properties vary over frequency and exhibit a non-symmetrical distribution of relaxation times, which can be described by composite Cole-Cole equations. Randomness in the material can be accounted for by modeling the parameters in the Cole-Cole equations as random variables. This gives rise to random material laws which are physically motivated but contain a large number of random parameters. Hence, they are not well suited for the majority of uncertainty quantification methods that scale unfavorably with the dimension of the parameter space. In this study we exploit correlation in the random Cole-Cole equation to substantially reduce the number of parameters. In particular, we use an eigendecomposition of the covariance matrix to derive a low-rank approximation. The truncated Karhunen-Loève (KL) expansion [@loeve1978; @ghanem1991] of the random material is then spanned in direction of the dominant eigenfunctions. This procedure is closely related to principal component analysis and proper orthogonal decomposition. The final computational goal is to quantify uncertainties in the axon activation during Deep Brain Stimulation (DBS) [@schmidtieee2013]. To this end, the stimulation electrode and the surrounding brain tissue are modeled as a volume conductor, see Figure \[fig:axons\] (left). A numerical approximation of the electric potential is obtained by the finite element method. The quantity of interest is the minimal electrode current to be applied in order to activate a particular axon in the electrode’s vicinity. This optimization is formulated as a root-finding problem for a function obtained from post-processing the solution of the volume conductor problem. Brent’s method is applied for its numerical solution. at (0,0) [![Axons aligned perpendicular to electrode (left) and computational domain with mesh using rotational symmetry (right)[]{data-label="fig:axons"}](fig01.png "fig:"){width="0.13\columnwidth"}]{}; (0.4,0.1) circle (.4ex); (0.6,0.1) circle (.4ex); (0.8,0.1) circle (.4ex); (1,0.1) circle (.4ex); (1.2,0.1) circle (.4ex); (1.4,0.1) circle (.4ex); (1.6,0.1) circle (.4ex); (1.8,0.1) circle (.4ex); (2,0.1) circle (.4ex); (2.2,0.1) circle (.4ex); at (1,1) [axons]{}; (1,0.8) – (1,0.4); at (0,0) [![Axons aligned perpendicular to electrode (left) and computational domain with mesh using rotational symmetry (right)[]{data-label="fig:axons"}](fig02.png "fig:"){width="0.54\columnwidth"}]{}; at (1.3,0.1) [boundary $\Gamma$]{}; In the presence of randomness in the electric coefficients, uncertainty quantification techniques are required. We use a stochastic quadrature on sparse grids [@xiu2005; @babuska2010] to efficiently compute the mean value and standard deviation of the axon activation. The method is non-intrusive as it only requires repetitive runs of the volume conductor model and the activation potential post-processing routine. The paper is organized as follows: Sections \[sec:cole\] and \[sec:KL\] contain the random Cole-Cole equation together with the KL expansion. Section \[sec:problem\] briefly summarizes the main equations needed for modeling DBS. Section \[sec:uq\] introduces a stochastic setting together with the stochastic quadrature. Finally, numerical results for a Medtronic 3387 electrode design are given in Section \[sec:num\]. Random Cole-Cole Equation {#sec:cole} ========================= Electrical properties of biological tissues can be modeled by the Cole-Cole equation $$f(\omega) = \epsilon_\infty + \frac{\varkappa_i}{j \omega \epsilon_0} + \sum_{i=1}^4 \frac{\Delta \epsilon_n}{1+(j \omega \tau_n)^{1-\alpha_n}}, \label{eq:cole_cole}$$ where $\omega$ denotes frequency, $j$ the imaginary unit, $\varkappa_i$ the static ionic conductivity and $\tau_n$ represents relaxation time constants. Also, $\epsilon_\infty$ and $\Delta \epsilon_n$ denote the high frequency and difference of the low to high frequency relative permittivitiy, respectively. From the permittivity and electric conductivity are inferred as $\epsilon(\omega) = {\mathrm{Re}(f(\omega))}$ and $\varkappa(\omega) = -{\mathrm{Im}(\epsilon_0 \omega f(\omega))}$, with $\mathrm{Re}$ and $\mathrm{Im}$ referring to the real part and imaginary part, respectively. In , $\epsilon_\infty,\varkappa_i,\Delta \epsilon_n,\tau_n$ and $\alpha_n$ are parameters that need to be inferred from measurements. As uncertainties are inevitably connected to this process we consider these parameters to be random variables $Y_i: \Theta \rightarrow \mathbb{R}$, $i=1,\ldots,14$, where $\Theta$ refers to a set of random outcomes. Then, with $\theta \in \Theta$ denoting a random event, the random Cole-Cole equation reads $$f(\theta,\omega) = Y_1(\theta) + \frac{Y_2(\theta)}{j \omega \epsilon_0} + \sum_{i=1}^4 \frac{Y_{3 i}(\theta)}{1+(j \omega Y_{3 i + 1}(\theta))^{1-Y_{3 i + 2}(\theta)}}. \label{eq:random_cole_cole}$$ In view of , both the electric permittivity and the conductivity are random. Since the following derivation is identical for both $\epsilon$ and $\varkappa$, we use the function $g$ referring to either of them. Important measures of the random field $g$ are the expected value and the covariance, given as $$\begin{aligned} {\mathrm{E}}_g(\omega) &= \int_{\Theta} g(\theta,\omega) \ \mathrm{d} P(\theta), \\ {\mathrm{Cov}}_g(\omega,\omega') &= \int_{\Theta} (g(\theta,\omega)-{\mathrm{E}}[g](\omega)) \notag \\ & \hspace*{3em} \cdot (g(\theta,\omega')-{\mathrm{E}}[g](\omega')) \ \mathrm{d} P(\theta),\end{aligned}$$ where $P$ refers to a probability measure. Discrete Karhunen-Loève Expansion {#sec:KL} ================================= When is used within simulations, both the large number of random variables and their possible correlation pose difficulties. The former results in a high computational complexity, whereas a possible correlation of the inputs cannot be handled by many state-of-the-art uncertainty quantification methods. In the following we apply the discrete Karhunen-Loève expansion (KLE) to reduce the number of random variables in . Although, the KLE is readily applicable to random fields such as , we consider its discrete variant, also referred to as principal component analysis. The exposition thereby follows [@elia2013coarse]. Given a set of frequency points $\{\omega_n\}_{n=1}^N$, chosen equidistantly over a fixed interval on a logarithmic scale, we consider the covariance matrix ${\mathbf{C}}$ with entries $$C_{n_1,n_2} = {\mathrm{Cov}}_g(\omega_{n_1},\omega_{n_2}), \ n_1,n_2=1,\ldots,N \label{eq:cov}$$ and denote its eigenvectors and eigenvalues with $\mathbf{b}_n$ and $\lambda_n$, respectively. Then, ${\mathbf{C}}$ can be decomposed as $$\mathbf{C} = \mathbf{V} \mathbf{E} \mathbf{V}^\top,$$ with $\mathbf{V}$ storing the eigenvectors $\mathbf{b}_n$ column-wise and $\mathbf{E}$ containing the eigenvalues $\lambda_n$ in decreasing order on its diagonal. As ${\mathbf{C}}$ is symmetric positive definite, the $\lambda_n$ are real and positive. Moreover, given a strongly correlated random field $g$, the eigenvalues decrease rapidly [@elia2013coarse]. Hence, we only consider the $M$ largest eigenvalues by introducing $$\mathbf{C} \approx \mathbf{C}_M = \mathbf{V}_M \mathbf{E}_M \mathbf{V}_M^\top. \label{eq:rank_M}$$ Numerical examples discussing the error committed by this low-rank approximation are given in Section \[sec:num\]. As only the largest eigenvalues and eigenfunctions are required a Krylov subspace method, such as the Lanczos algorithm can be used. Moreover, the underlying random field has only one dimension (frequency). This results in a moderate size of the covariance matrix and acceleration techniques for the matrix-vector product can be omitted. Numerical techniques and properties of the Karhunen-Loève expansion for one-dimensional random fields were also investigated in [@roemer2016] in the context of nonlinear magnetic material properties. Based on a new discrete random field is defined as $$\mathbf{g}_{M}(\theta) = {\mathrm{E}}_{\mathbf{g}} + \mathbf{V}_M \mathbf{E}_M^{1/2} {\mathbf{Y}}_M(\theta), \label{eq:KL}$$ where $({\mathrm{E}}_{\mathbf{g}})_n = {\mathrm{E}}_g(\omega_n)$. A frequency dependent random field $g_M$ is recovered from $\mathbf{g}_{M}$ by spline interpolation. The new random variables are uncorrelated and can be inferred from $$Y_{M;m}(\theta) = \left((\mathbf{g}(\theta) - {\mathrm{E}}_{\mathbf{g}} )^\top \mathbf{b}_m \right )/\sqrt{\lambda_m}, \ m=1,\dots,M,$$ based on observations $(\mathbf{g}(\theta))_n= g(\theta,\omega_n)$. It should be noted that the variables ${\mathbf{Y}}_M$ are also independent in the case of a Gaussian random field. In general independence needs to be assured by introducing a transformation to another set of random variables. Here, we simply assume independence. Problem Description {#sec:problem} =================== The computational model to estimate the activation during DBS is based on a 2D rotational symmetric finite element volume conductor model of the stimulation electrode and surrounding brain tissue coupled to axons in the target area. The electric potential in the tissue is computed by solving the Laplace equation for complex material properties $$\nabla \cdot \left[\left(\varkappa(\omega, \boldsymbol r)+\mathsf{j}\omega\epsilon_0\epsilon_r(\omega, \boldsymbol r)\right) \nabla \phi_{\mathrm{e}}(\omega, \boldsymbol r)\right] = 0 \label{eq:potential}$$ with the electric conductivity $\varkappa(\omega, \boldsymbol r)$ and relative permittivity $\epsilon_r(\omega, \boldsymbol r)$ of the encapsulation layer and brain tissue. Following the approach in [@schmidtieee2013], a current-controlled stimulation pulse $I(\omega)$ is introduced to one electrode contact, while the other boundaries of the electrode are modeled as insulation. The boundary of the surrounding tissue is set to ground, i.e., $\phi_\mathrm{e} |_{\Gamma} = 0$, see Figure \[fig:axons\] (right). The time-dependent electric potential resulting from the applied stimulus is computed using the Fourier Finite Element Method (FFEM) [@butson2006], for which the Laplace equation (\[eq:potential\]) is solved in the frequency-domain for $N$ logarithmically distributed frequency nodes and interpolated for the Fourier components of the stimulation signal in the considered frequency range. In order to investigate the activation of neuronal tissue during DBS, a number of axon cable models are positioned perpendicular to the electrode contact where the stimulus is applied. Each axon cable model consists of a number of compartments, for which the inner potential in each compartment is defined by the following equation [@mcintyre2002]: $$\begin{aligned} \begin{split} g_\mathrm{A}(\boldsymbol r)\Delta^2\phi_\mathrm{e}(\boldsymbol r, t)&=c(\boldsymbol r) \frac{\mathrm{d}\phi_\mathrm{m}(\boldsymbol r, t)}{dt}+\\ &+i_\mathrm{ion}(\phi_\mathrm{m}(\boldsymbol r, t),\boldsymbol r)-\\ &-g_\mathrm{A}(\boldsymbol r)\Delta^2\phi_\mathrm{m}(\boldsymbol r, t) \end{split} \label{eq:neuron}\end{aligned}$$ with the membrane capacitance $c$, the ionic current $i_\mathrm{ion}$, the axial conductance $g_\mathrm{A}$, the membrane potential $\phi_\mathrm{m}(\boldsymbol r, t)$, and second spatial difference $\Delta^2$ in direction of the axon. The membrane potential is defined by $$\phi_\mathrm{m}(\boldsymbol r, t)=\phi_\mathrm{i}(\boldsymbol r,t)-\phi_\mathrm{e}(\boldsymbol r,t)+\phi_\mathrm{r}(\boldsymbol r,t) \label{eq:membranepotential}$$ with the innercellular potential $\phi_\mathrm{i}(\boldsymbol r,t)$, the resting potential $\phi_\mathrm{r}(\boldsymbol r,t)$, and the extracellular potential $\phi_\mathrm{e}(\boldsymbol r,t)$. The time-dependent electric potential at each compartment center provided by the volume conductor model is applied as extracellular potential $\phi_\mathrm{e}(\boldsymbol r, t)$ to the compartment equation (\[eq:neuron\]). The computational goal is to determine the minimum stimulation amplitude, required to excite an action potential in a specific axon. This can be expressed as a root-finding problem as follows: for a given current stimulus, problem is solved repeatedly as outlined above to obtain a time-dependent potential $\phi_\mathrm{e}$. Then, for the axon under investigation, for each compartment, is solved. The axon is activated, if the inner potential at the outer compartment $\phi_\mathrm{i}^{\mathrm{out}}$ is larger than zero. Hence, to determine the required stimulation amplitude $I$, a root of $\phi_\mathrm{i}^{\mathrm{out}}(I)$ needs to be found. This root is found numerically with Brent’s method here. Uncertainty Quantification {#sec:uq} ========================== Randomness in the electric conductivity and relative permittivity gives rise to a stochastic volume conductor model. This stochastic model in turn can be used to compute statistics of the axon activation current $I$. The methods presented in this section assume independence of the inputs. Unfortunately, the KLE as presented in Section \[sec:KL\] applied to both the permittivity and the conductivity does not ensure independence of the parameters, as both are modeled by the same random process . An extension to obtain uncorrelated and independent electrical parameters at the same time is possible but beyond the scope of this paper. We have observed that the conductivity is the parameter with a larger sensitivity and hence, in the following, only $\varkappa$ is subject to uncertainty. We obtain the parametric equation $$\begin{aligned} \!\!\nabla \cdot \left[\left(\varkappa({\mathbf{y}}_{M},\omega,\boldsymbol r)+\mathsf{j}\omega\epsilon_0\epsilon_r(\omega,\boldsymbol r)\right) \nabla \phi_{\mathrm{e}}({\mathbf{y}}_M,\omega, \boldsymbol r)\right] = 0, \label{eq:vol_cond_parametric}\end{aligned}$$ where lowercase symbols are used for the realization of a random variable, i.e., ${\mathbf{y}}_M ={\mathbf{Y}}_M(\theta)$. The parameter dependency is inherited by the activation potential and the minimum current required for activation. In particular, a current is associated to each ${\mathbf{y}}_M$ through $$\phi_\mathrm{i}^{\mathrm{out}}({\mathbf{y}}_M,I) = 0. \label{eq:root_para}$$ We denote with $\rho$ and $\Gamma$ the joint probability density function and the image of ${\mathbf{Y}}_M$, respectively. Then the expected value and variance can be rewritten as $$\begin{aligned} {\mathrm{E}}[I] &= \int_{\Gamma} I({\mathbf{y}}_M) \ \rho({\mathbf{y}}_M) \mathrm{d} {\mathbf{y}}_M, \\ {\mathrm{Var}}[I] &= \int_{\Gamma} (I({\mathbf{y}}_M) - {\mathrm{E}}[I])^2 \ \rho({\mathbf{y}}_M) \mathrm{d} {\mathbf{y}}_M.\end{aligned}$$ \[eq:moments\] The aim is to find an efficient numerical approximation of . A state-of-the art technique for uncertainty quantification is the stochastic collocation method [@xiu2005; @babuska2010] based on tensor or sparse grids. The procedure is summarized as follows: given a set of collocation points $({\mathbf{y}}_M^{(k)})_{k=1}^K$, is solved for each ${\mathbf{y}}_M^{(k)}$ to obtain $I({\mathbf{y}}_M^{(k)})$. This involves solutions of the volume conductor model and post-processing to obtain the respective action potentials. We emphasize that no modification of the code is required as simulations are simply repeated with different conductivities $\varkappa({\mathbf{y}}_M^{(k)})$. In this sense, the method is non-intrusive. The collocation points are given by tensor grid or sparse grid constructions. Given $(I({\mathbf{y}}_M^{(k)}))_{k=1}^K$, a polynomial approximation of the output quantity can be computed by enforcing the collocation conditions. Here, we are mainly interested in the approximation of the expected value and variance which can be directly obtained using a dedicated numerical quadrature. The knots and weights of univariate quadrature rules are given for instance by the Gauss or Clenshaw-Curtis abscissas. Then, tensor product formulas of different degree in different directions are combined to obtain efficient quadrature rules, see, e.g., [@novak1999], [@babuska2010]. Numerical Example {#sec:num} ================= In a first step, the KLE was applied to the random conductivity given by . The random vector ${\mathbf{Y}}$ was modeled with a mean value $$\begin{gathered} {\mathrm{E}}[{\mathbf{Y}}]= {\scriptstyle(4,0.02,45,7.96 \times10^{-12},0.1,400,15.92\times 10^{-9},0.15, }\\ {\scriptstyle2 \times 10^5, 106.10 \times 10^{-6},0.22,4.5 \times 10^{7},5.31 \times 10^{-3},0)^\top}\end{gathered}$$ according to values given in literature. Due to the lack of further data, the vector was assumed to be uniformly distributed on a interval of 10$\%$ deviation around the mean value. Equidistant points with a stepsize of $0.004$ on the logarithmic scale of the interval $2 \pi [130, 5 \cdot 10^5]$ Hz were considered. The eigenvalues and eigenfunctions were computed with the eigs function of MATLAB. Figure \[fig:cov\](a) depicts the covariance sampled $10^{3}$ times. The relative error for the KL expansion and $M=4$ is shown in Figure \[fig:cov\](b). As it is in the order of $10^{-6}$ the low-rank approximation is justified. This is further illustrated in Figure \[fig:cov\](c) showing the fast (exponential) decay of the eigenvalues. The low-dimensional stochastic model for the conductivity was then used for the simulation of the required axon activation current. Simulation details are summarized as follows: the electrode model, which represents the Medtronic 3387 electrode design commonly used in human DBS [@schmidtieee2013], was encapsulated by a $0.2\,\mathrm{mm}$ thick tissue layer, which is formed due to body reactions at the interface between the electrode and the brain tissue [@grant2010]. Cathodal current-controlled square-wave stimulation pulses with a frequency of $130\,\mathrm{Hz}$ and a pulse duration of $60\,\mathrm{\mu s}$, as used in clinical practice [@schmidtieee2013], were applied. The model equation was discretized with $27{,}000$ elements and solved using the software COMSOL Multiphysics$\ $ at $N=3846$ frequency points in the considered frequency range. By subsequent refinement of the frequency interval an accuracy of 1$\%$ was ensured. Ten axon cable models, each of them with 221 compartments, were positioned perpendicular to the second electrode contact in a distance between $1\,\mathrm{mm}$ to $10\,\mathrm{mm}$ to the electrode center. The axon activation was obtained by solving with the backward Euler method. A time step of $10\,\mathrm{\mu s}$ was employed. The minimal current required for axon activation was computed with Brent’s method with an absolute tolerance of $1\cdot10^{-5}$. Table \[tab:uq\] gives the mean value and standard deviation of the axon activation obtained with a stochastic quadrature. Clenshaw-Curtis abscissas were used for the univariate quadrature rules. Quadrature points and weights for the multivariate case were chosen to exactly integrate total degree polynomials of level three, which corresponds to 137 quadrature points and weights in total. Details can be found in [@babuska2010]. The values in Table \[tab:uq\] are rounded to significant figures, estimated with a higher order quadrature. It is observed that the standard deviation is in the order of 10$\%$ of the mean value for each axon. This reflects a moderate sensitivity of the goal fucntion with respect to the variable conductivity input parameters. \[tab:uq\] Axon mean value \[mA\] standard deviation \[mA\] ------ ------------------- --------------------------- 1 0.14 0.01 2 0.56 0.06 3 1.44 0.15 4 2.97 0.31 5 5.30 0.56 6 8.65 0.91 7 13.20 1.39 8 19.15 2.02 9 26.67 2.81 10 36.06 3.80 : Expected value and standard deviation of axon activation using stochastic quadrature Conclusion ========== Uncertainties in the axon activation in deep brain stimulation have been quantified using a stochastic quadrature and a volume conductor model for the electric potential distribution in the brain tissue. The activation was found to be moderately sensitive to variations in the conductivity parameters. Hence, the problem is well conditioned with respect to deviations in the electrical input parameters. A crucial ingredient for the efficiency of the scheme is a random model of the electric conductivity based on the Karhunen-Loève expansion. It requires the computation of the eigenvalues and eigenfunctions of the covariance matrix at a discrete set of frequency points. An exponential decay of the eigenvalues was observed, allowing for a significant reduction of the number of random parameters. Additionally, the new parameters are uncorrelated. Considering uncertainties in the conductivity and permittivity at the same time requires the study of cross-correlation effects which is the subject of ongoing work. Acknowledgment {#acknowledgment .unnumbered} ============== U. Römer and S. Schöps acknowledge the support of the DFG through the Graduate School of Computational Engineering in Darmstadt. [1]{} C. Gabriel, A. Peyman, and E. H. Grant, “Electrical conductivity of tissue at frequencies below 1 MHz,” *Phys. Med. Biol.*, vol. 54, pp. 4863–4878, 2009. C. Schmidt, P. Grant, M. Lowery, U. van Rienen, “Influence of Uncertainties in the Material Properties of Brain Tissue on the Probabilistic Volume of Tissue Activated,” *IEEE Trans. Biomed. Eng.*, vol. 60, pp. 1378-1387, 2013. C. McIntyre, A. Richardson, W. Grill, “Modeling the excitability of mammalian nerve fibers: influence of afterpotentials on the recovery cycle.,” *J. Neurophysiol*, vol. 87, pp. 995–1006, 2002. M. Loève, *Probability Theory*, Springer, New York, 1978. R. Ghanem, P. D. Spanos, *Stochastic Finite Elements: a Spectral Approach*, Springer, New York, 1991. D. Xiu, and Jan S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” *SIAM Journal on Scientific Computing*, vol. 27.3, pp. 1118–1139, 2005. I. Babuška, F. Nobile, and R. Tempone, “A stochastic collocation method for elliptic partial differential equations with random input data,” *SIAM review*, vol. 52.2, pp. 317–355, 2010. M. D’Elia, and M. Gunzburger, “Coarse-Grid Sampling Interpolatory Methods for Approximating Gaussian Random Fields,” *SIAM/ASA J. Uncertainty Quantification*, vol. 1, pp. 270–296, 2013. U. Römer, S. Schöps, T. Weiland, “Stochastic Modeling and Regularity of the Nonlinear Elliptic curl–curl Equation,” *SIAM/ASA Journal on Uncertainty Quantification*, vol. 4, pp. 952-979, 2016. C. R. Butson, and C. C. McIntyre, “Tissue and electrode capacitance reduce neural activation volumes during deep brain stimulation,” *Clinical neurophysiology*, vol. 116, pp. 2490–2500, 2005. E. Novak, and K. Ritter, “Simple cubature formulas with high polynomial exactness,” *Constructive approximation*, vol. 15.4, pp. 499–522, 1999. P. F. Grant, and M. M. Lowery, “Effect of Dispersive Conductivity and Permittivity in Volume Conductor Models of Deep Brain Stimulation,” *IEEE Transactions on Biomedical Engineering*, vol. 57, pp. 2386–2393, 2010. [^1]: Manuscript received xxx; revised xxx. Corresponding author: U. Römer (email: roemer@temf.tu-darmstadt.de).

--- abstract: 'We prove asymptotic formulas for the density of integral points taking coprime polynomial values on the affine quadrics defined by $Q(X_1,\cdots,X_n)=m$, where $Q$ is a non-degenerate quadratic form in $n\geqslant 3$ variables and $m$ a non-zero integer. This is a quantitative version of the arithmetic purity of strong approximation off infinity for affine quadrics, a property that has been established in our previous work, and may also be viewed as a refined version of the Hardy-Littlewood property in the sense of Borovoi-Rudnick’s work.' address: - 'Yang CAO Lebniz Universität Hannover Welfengarten 1, 30167 Hannover, Germany' - 'Zhizhong HUANG Lebniz Universität Hannover Welfengarten 1, 30167 Hannover, Germany' author: - Yang Cao - Zhizhong Huang title: | Arithmetic purity, geometric sieve\ and counting integral points on affine quadrics --- [UTF8]{}[gkai]{} 岂曰无衣，与子同裳。\ 同气连枝，共盼春来。 Introduction ============ #### **Background and empiricism** The behavior of integral points on affine varieties defined over a number field is sometimes more subtle than rational points. Studying integral points on an open part of a variety naturally involves infinitely many congruence conditions, a problem to overcome when trying to move integral points around in showing approximation results in adelic topology. When the complementary of the open set has codimension at least two, no cohomological or topological obstructions to the local-to-global principle for this open set can arise, which serves as positive evidence for the following question first raised by Wittenberg (c.f. [@Wittenberg §2.7 Question 2.11]). \[q:purity\] Let $X$ be a smooth variety over a number field satisfying strong approximation (off a finite set of places). Does any open subset $U\subset X$ satisfy also this property, whenever ${{\mathrm{codim}}}(X\setminus U,X)\geqslant 2$? We shall say that such $X$ verifies the *arithmetic purity (of strong approximation)* (c.f. [@Cao-Huang Definition 1.2]). Recently the authors in [@Cao-Huang §1.3] settled this question in the affirmative for a wide class of semisimple simply connected linear algebraic groups and consequently for their homogeneous spaces (with connected stabilizers). We refer to the references therein for an account of known results towards Question \[q:purity\]. The purpose of this article is to address an effective or statistic aspect of Question \[q:purity\] concerning the arithmetic purity off the real place ${{\mathbb {R}}}$. Let $X$ and $U$ be as before defined over ${{\mathbb {Q}}}$, with a fixed integral model for each (still denoted by $X,U$ by abuse of notation). Assume that $X$ is quasi-affine. Embed $X$ into an affine space ${{\mathbb {A}}}^n$ equipped with an archimedean height function $\|\cdot\|$. We ask the following: \[q:countingpurity\] Does there exist an asymptotic formula for $$\label{eq:NUT} N_U(T):=\#\{{\underline{\mathbf{X}}}\in U({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T\},\quad T\to\infty?$$ When $U=X$, this question is identical to the usual one of counting integral points on varieties. The situation for symmetric varieties is relatively well-understood and different methods can be applied. See notably the work [@Duke-Rudnick-Sarnak] and the more recent one [@Browning-Gorodnik]. When such an asymptotic formula exists (and assume that $X$ satisfies strong approximation), then one expects that the order of growth of $N_X(T)$ should be $T^{n-\deg X}$ (depending on the embedding $X\hookrightarrow {{\mathbb {A}}}^n$), and the leading constant should be the product of local densities. Varieties of this type are called *(strongly) Hardy-Littlewood* after Borovoi-Rudnick [@Borovoi-Rudnick Definition 2.2] (see also [@Duke-Rudnick-Sarnak p. 143]), and they satisfy $$\label{eq:strongHL} N_X(T)\sim\tau_{\infty}(X;T)\prod_{p<\infty}\hat{\tau}_p(X),$$ where $\hat{\tau}_p(X)$ are the $p$-adic local factors (c.f. [@Borovoi-Rudnick (0.0.3)]) of $X$: $$\label{eq:HLpadic} \hat{\tau}_p(X):=\lim_{t\to\infty} \frac{\#X({{\mathbb {Z}}}/p^t{{\mathbb {Z}}})}{p^{t\dim X}},$$ and if $X$ is cut off in ${{\mathbb {A}}}^n$ by polynomials $f_1,\cdots,f_r\in{{\mathbb {Q}}}[X_1,\cdots,X_n]$, then $$\label{eq:realHL} \tau_{\infty}(X;T):=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^r}{{\mathrm{vol}}}_{{{\mathbb {R}}}^n}\{{\mathbf{x}}\in{{\mathbb {R}}}^n:\|{\mathbf{x}}\|\leqslant T,|f_i({\mathbf{x}})|<\frac{\varepsilon}{2},\forall 1\leqslant i\leqslant r\}$$ is the *real Hardy-Littlewood density* (for the embedding $X\hookrightarrow{{\mathbb {A}}}^n$) ([@Borovoi-Rudnick (0.0.4)]) (or *singular integral*). The infinite product $$\label{eq:singularseries} \mathfrak{G}(X)=\prod_{p<\infty}\hat{\tau}_p(X)$$ is called the *singular series* of $X$. It happens that an asymptotic formula exists for $N_X(T)$, even if $X$ fails the integral Hasse principle and strong approximation (this failure is explained by Brauer-Manin obstruction). Such $X$ is called *relatively Hardy-Littlewood* ([@Borovoi-Rudnick Definition 2.3]), for which a certain density function is included in describing $N_X(T)$. A discussion on affine quadrics in three variables, amongst varieties of this type, is given in §\[se:quadricarith\]. Handling the cases $U\subsetneq X, {{\mathrm{codim}}}_X(X\setminus U)\geqslant 2 $ requires special care regarding certain infinite congruence conditions. For instance, if $Z=X\setminus U$ is defined by two regular functions $f,g\in{{\mathbb {Q}}}[X]$ with integral coefficients, the estimation of $N_U(T)$ in now boils down to $$\label{eq:gcd1} \#\{{\underline{\mathbf{X}}}\in X({{\mathbb {Z}}}): \|{\underline{\mathbf{X}}}\|\leqslant T,\gcd(f({\underline{\mathbf{X}}}),g({\underline{\mathbf{X}}}))=1\}.$$ Here an “infinite” congruence condition comes in, since $$\gcd(f({\underline{\mathbf{X}}}),g({\underline{\mathbf{X}}}))=1\Leftrightarrow {\underline{\mathbf{X}}}{\ \mathrm{mod}\ }p\not\in Z,\forall p,$$ although it is actually a condition about finitely many primes if we bound the height of ${\underline{\mathbf{X}}}$. A *geometric sieve* was inaugurated by Ekedahl [@Ekedahl] when dealing with $X={{\mathbb {A}}}^n$. Further pursued by Poonen [@Poonen Theorem 3.1] and Bhargava [@Bhargava], this sieve method has demonstrated surprising applications on the density of square-free polynomial values in various circumstances. Their results provide $$N_U(T)\sim N_{{{\mathbb {A}}}^n}(T) \prod_{p<\infty}\left(1-\frac{\#Z({{\mathbb {F}}}_p)}{\#{{\mathbb {A}}}^n({{\mathbb {F}}}_p)}\right) .$$ The Lang-Weil estimate (c.f. [@Lang-Weil], [@Cao-Huang Corollary 3.5]) shows that $$\frac{\#Z({{\mathbb {F}}}_p)}{\#{{\mathbb {A}}}^n({{\mathbb {F}}}_p)}=\frac{c_p}{p^2},$$ where $c_p\geqslant 0$ is uniformly bounded for any prime $p$. Hence the above infinite product is absolutely convergent. This motivates, at least for $X$ being strongly Hardy-Littlewood, that if Question \[q:purity\] has a positive answer to $U$ (which implies that $N_U(T)\to\infty$), and if an asymptotic formula should exist, then we expect $$N_U(T)\sim N_X(T)\left(\prod_{p<\infty }\tau_p(Z;X)\right),$$ where for almost all prime $p$ (usually when $X{\ \mathrm{mod}\ }p$ is smooth), $$\tau_p(Z;X)=1-\frac{\#Z({{\mathbb {F}}}_p)}{\# X({{\mathbb {F}}}_p)}.$$ The quantity $\tau_p(Z;X)$ signifies the proportion of integral points lying in $U$ (or outside $Z$) modulo $p$. This empiricism also suggests that we do not expect, among other reasons (e.g., existence of non-constant invertible functions, c.f. [@Borovoi-Rudnick Lemma 1.5.2]), that the quantity $N_U(T)$ could grow in the same magnitude as $N_X(T)$ when ${{\mathrm{codim}}}_X(X\setminus U)=1$, as the infinite product would diverge to $0$. #### **Results on affine quadrics** The goal of this article is to provide further positive answers towards Question \[q:countingpurity\] beyond affine spaces, namely affine quadrics, as motivated by our recent progress [@Cao-Huang Theorem 1.5] on Question \[q:purity\] for homogeneous spaces under spin groups. The authors also learn of forthcoming work of Browning and Heath-Brown [@Browning-HB] dealing with projective quadratic hypersurfaces. Let $Q({\underline{\mathbf{X}}})$ be an integral quadratic form in $n$ variables ${\underline{\mathbf{X}}}=(X_1,\cdots,X_n)$. Suppose that $Q$ is indefinite (non-compact as a real form). For $m\in{{\mathbb {Z}}}_{\neq 0}$, let us define the affine quadric $$\label{eq:Vm} V_m:=(Q({\underline{\mathbf{X}}})=m)\subset {{\mathbb {A}}}^n,$$ also serving as an integral model. Then $V_m$ is a regular affine symmetric homogeneous space under the group $G_Q={\mathsf{Spin}}_Q$. We fix once and for all a codimension two subset $Z\subset V_m$ and $U=V_m\setminus Z$, and we assume throughout this article that $V_m({{\mathbb {Z}}})\neq 0$. We define the local $p$-adic factor for the open set $U$ as follows. But to emphasize its complementary $Z$ we denote it by $\hat{\tau}_p(Z;V_m)$ by abuse of notation: $$\label{eq:deltap} \begin{split} \hat{\tau}_p(Z;V_m)=\lim_{t\to\infty}\frac{\#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi{\ \mathrm{mod}\ }p\not\in Z\}}{p^{t(n-1)}}. \end{split}$$ Our main result provides an asymptotic formula for with leading constant in accordance with expectation. \[thm:mainthmgcd=1\] Let $V_m$ be the affine quadric with $n\geqslant 4$. Then $$N_U(T)\sim \left(\prod_{p} \tau_p(Z;V_m) \right)N_{V_m}(T),$$ where $$\label{eq:taup} \tau_p(Z;V_m)=\frac{\hat{\tau}_p(Z;V_m)}{\hat{\tau}_p(V_m)}.$$ For all sufficiently large prime numbers $p$ (such that $V_m$ is smooth modulo $p$), $$\label{eq:taupexpect} \tau_p(Z;V_m)=1-\frac{\# Z({{\mathbb {F}}}_p)}{\# V_m(\mathbb{F}_p)}.$$ The affine quadric $V_m$ satisfying strong approximation off infinity ${{\mathbb {R}}}$, the set $V_m({{\mathbb {Z}}})$ is dense in any arbitrarily small adelic neighborhood of $$V_m(\widehat{{{\mathbb {Z}}}}):=\prod_{p<\infty} V_m({{\mathbb {Z}}}_p).$$ Fix $S$ a finite set of places including ${{\mathbb {R}}}$, such that $V_m$ is smooth over ${{\mathbb {Z}}}_S$, and that for each $p\in S$, fix an integer $n_p\in{{\mathbb {N}}}_{\neq 0}$ and a residue $\xi_p\in V_m({{\mathbb {Z}}}/p^{n_p}{{\mathbb {Z}}})$. They form an adelic neighborhood $$\label{eq:WS} W_S:=\left(\prod_{p\in S} \{{\mathbf{x}}\in V_m({{\mathbb {Z}}}_p):{\mathbf{x}}\equiv \xi_p{\ \mathrm{mod}\ }p^{n_p}{{\mathbb {Z}}}_p\}\right)\times \left(\prod_{p\not\in S}V_m({{\mathbb {Z}}}_p)\right),$$ as part of the adelic topological basis. From the counting viewpoint, we seek an asymptotic formula for the counting function $$\begin{aligned} N_U^S(T)=\#\{{\underline{\mathbf{X}}}\in W_S\cap U({{\mathbb {Z}}}_S):\|{\underline{\mathbf{X}}}\|\leqslant T\}\end{aligned}$$ describing the density of integral points in the set above, generalizing and refining . To state the result, we define for each $\xi_p\in V_m({{\mathbb {Z}}}/p^{n_p}{{\mathbb {Z}}})$, the $p$-adic density with residue $\xi_p$: $$\label{eq:HLpadicxi} \hat{\tau}_p(\xi_p;V_m)=\lim_{t\to\infty}\frac{\#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi\equiv\xi_p{\ \mathrm{mod}\ }p^{n_p}\}}{p^{t(n-1)}}.$$ \[th:mainthm\] Under the same assumptions in Theorem \[thm:mainthmgcd=1\], we have $$N_U^S(T)\sim\left(\prod_{p\in S}\tau_p(\xi_p;V_m)\right) \left(\prod_{p\not\in S} \tau_p(Z;V_m)\right) N_{V_m}(T),$$ where $\tau_p(Z;V_m)$ is given by and $$\label{eq:iotap} \tau_p(\xi_p;V_m)=\frac{\hat{\tau}_p(\xi_p;V_m)}{\hat{\tau}_p(V_m)}.$$ Moreover $\tau_p(\xi_p;V_m)=\left(\#V_m({{\mathbb {Z}}}/p^{n_p}{{\mathbb {Z}}})\right)^{-1}$ if $V_m$ is smooth modulo $p$. The quantity $\tau_p(\xi_p;V_m)$ signifies the probability of $p$-adic integral points of $V_m$ taking the residue $\xi_p$ modulo $p^{n_p}$. So the above formula is coherent with expectation. To see that agrees with whenever $V_m$ is smooth over ${{\mathbb {F}}}_p$, let us first consider $p\not\in S$. Then by Hensel’s lemma, for any $t\geqslant 1$, $$\# V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}})=p^{(t-1)(n-1)}\# V_m({{\mathbb {F}}}_p),$$ $$\label{eq:HenseltauZ} \#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi\in Z{\ \mathrm{mod}\ }p\}=p^{(t-1)(n-1)}\#Z({{\mathbb {Z}}}/p{{\mathbb {Z}}}),$$ so that $$\hat{\tau}_p(Z;V_m)=\frac{\#V_m({{\mathbb {F}}}_p)-\#Z({{\mathbb {F}}}_p)}{p^{n-1}}.$$ Thus for any such $p$, $$\tau_p(Z;V_m)=\frac{\hat{\tau}_p(Z;V_m)}{ \hat{\tau}_p(V_m)}=1-\frac{\#Z({{\mathbb {F}}}_p)}{\#V_m({{\mathbb {F}}}_p)}.$$ Now assume $p\in S$. Hensel’s lemma tells that for any $t>n_p$, $$\#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi\equiv \xi_p{\ \mathrm{mod}\ }p^{n_p}\}=p^{(t-n_p)(n-1)},$$ so that $$\tau_p(\xi_p;V_m)=\frac{1}{p^{n_p(n-1)}}\left(\frac{\# V_m({{\mathbb {Z}}}/p^{n_p}{{\mathbb {Z}}})}{p^{n_p(n-1)}}\right)^{-1}=\frac{1}{\# V_m({{\mathbb {Z}}}/p^{n_p}{{\mathbb {Z}}})}.$$ The distribution of integral points on affine quadrics in three variables are more subtle. They are divided into finitely many orbits under the spin group $G_Q$. One key feature is that some orbits may fail to verify the integral local-to-global principle and strong approximation. Each adelic orbit $\mathfrak{O}_{{\mathbf{A}}}:=G_Q({\mathbf{A}}_{{\mathbb {Q}}})\cdot(x_p)_{p\leqslant \infty}$ with $(x_p)_{p\leqslant \infty}\in V_m({\mathbf{A}}_{{\mathbb {Q}}})$ is the restricted product of the orbits $\mathfrak{O}_p:=G_Q({{\mathbb {Q}}}_p)\cdot x_p$ with respect to $V_m({{\mathbb {Z}}}_p)$. We associate to ${\mathfrak{O}_{{\mathbf{A}}}}$ a counting function $$\label{eq:NOT} N_{\mathfrak{O}_{{\mathbf{A}}}}(T):=\#\{{\mathbf{x}}\in\mathfrak{O}_{{\mathbf{A}}}\cap {{\mathbb {Z}}}^n:\|{\mathbf{x}}\|\leqslant T\},$$ and we define its $p$-adic density $$\label{eq:HLadelic} \hat{\tau}_p({\mathfrak{O}_{{\mathbf{A}}}})=\lim_{t\to\infty}\frac{\#(\mathfrak{O}_p\cap V_m({{\mathbb {Z}}}_p){\ \mathrm{mod}\ }p^t)}{p^{t(n-1)}}.$$ For the closed subset $Z$ and the residue $\xi_p$, we also define (again by abuse of notation) $$\label{eq:HLpadicorbit} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)=\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z;V_m)=\lim_{t\to\infty}\frac{\#\{\xi\in\mathfrak{O}_p\cap V_m({{\mathbb {Z}}}_p) {\ \mathrm{mod}\ }p^t:\xi{\ \mathrm{mod}\ }p\not\in Z\}}{p^{t(n-1)}},$$ $$\label{eq:HLpadicorbitresidue} \begin{split} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)=\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p;V_m)= \lim_{t\to\infty}\frac{\#\{\xi\in\mathfrak{O}_p\cap V_m({{\mathbb {Z}}}_p) {\ \mathrm{mod}\ }p^t:\xi\equiv \xi_p{\ \mathrm{mod}\ }p^{n_p}\}}{p^{t(n-1)}}. \end{split}$$ \[thm:mainthmn=3\] Let $n=3$. Assume moreover that the form $Q$ is anisotropic over ${{\mathbb {Q}}}$ and that the integer $-m\det Q$ is not a square. Then $$N_U^S(T)\sim\sum_{\substack{{\mathfrak{O}_{{\mathbf{A}}}}(*)\\\mathfrak{O}_{{\mathbf{A}}} \cap V_m({{\mathbb {Q}}})\neq\varnothing }}\left(\left(\prod_{p\in S}\tau_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)\right) \left(\prod_{p\not\in S} \tau_p(\mathfrak{O}_{{\mathbf{A}}};Z)\right) N_{\mathfrak{O}_{{\mathbf{A}}}}(T)\right),$$ where (\*) means that we sum over all (finitely many) orbits intersecting non-trivially with $V_m({{\mathbb {R}}})\times V_m(\widehat{{{\mathbb {Z}}}})$, and $$\label{eq:tauorbits} \tau_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)=\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})},\quad \tau_p(\mathfrak{O}_{{\mathbf{A}}};Z)=\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}.$$ For almost all $p$, the quantity $\tau_p(\mathfrak{O}_{{\mathbf{A}}};Z)$ agrees with for any $\mathfrak{O}_{{\mathbf{A}}}$ verifying (\*). #### **Strategies** Our approach of deriving the main term builds on the finer equidistribution result due to Browning and Gorodnik [@Browning-Gorodnik Theorem 2.1], generalizing a result of Nevo and Sarnak [@Nevo-Sarnak Theorem 3.1]. It furnishes asymptotic formulas about the growth of integral points of bounded height in certain affine symmetric homogeneous spaces (including affine quadrics) with certain prescribed congruence residue, in which the main terms are given by Tamagawa measures of the adelic set defining this congruence condition. Several main characteristics of their results, based on mixing properties of algebraic groups, are the polynomial growth of error terms, and the uniformity of dependence on the level of congruence. For simplicity let us ignore the condition on finitely many residues $(\xi_p)_{p\in S}$. Any condition imposing points of $V_m({{\mathbb {Z}}})$ to avoid the closed subset $Z$ for a finite set ${{\mathcal {P}}}$ of primes (of small moduli) forms an adelic subset ${{\mathcal {C}}}\subset V_m({{\mathbb {R}}})\times V_m(\widehat{{{\mathbb {Z}}}})$. We show in §\[se:mainterm\] that the density of such points has the expected main term, i.e., the Tagamawa measure of ${{\mathcal {C}}}$. In doing so we consider alternating sums over the density of integral points with fixed residue lying in $Z$ modulo integers whose prime divisors are in ${{\mathcal {P}}}$. For instance, if $Z$ is the locus of two integral regular functions $f,g\in{{\mathbb {Z}}}[V_m]$, this procedure is nothing but a Möbius inversion resulting from the inclusion-exclusion principle: $$\begin{aligned} &\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,\forall p,p\mid f({\underline{\mathbf{X}}}),p\mid g({\underline{\mathbf{X}}})\Rightarrow p\geqslant M\}\\ =&\sum_{\substack{D\in{{\mathbb {N}}}_{\neq 0}\\ p\mid D\Rightarrow p<M}}\mu(D)\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,D\mid f({\underline{\mathbf{X}}}),D\mid g({\underline{\mathbf{X}}})\}\\ =&\sum_{\substack{D\in{{\mathbb {N}}}_{\neq 0}\\ p\mid D\Rightarrow p<M}}\mu(D)\sum_{\xi:f(\xi)\equiv g(\xi)\equiv 0{\ \mathrm{mod}\ }D}\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,{\underline{\mathbf{X}}}\equiv \xi{\ \mathrm{mod}\ }D\}.\end{aligned}$$ Then one applies the equidistribution results for the inner expression, each one being dominated by the Tamagawa measure of corresponding adelic neighborhood defined in terms of $\xi$ and the level of congruence $D$. If one chooses the moduli $M=M(T)$ so that it grows as $O(\log T)$ (as to be determined in the last part) as $T\to\infty$, the so obtained main term, ignoring congruence conditions at large primes, approximates the Tamagawa measure of $V_m({{\mathbb {R}}})\times U(\widehat{{{\mathbb {Z}}}})$, which finally contributes to the leading constant with infinite product in our main theorems. Handling residues arising from large primes, which are expected to be negligible, necessitates radically different ideas. To deal with prime moduli growing as $T^\alpha$ with $\alpha>0$ small enough to be determined later (and no slower than $\log T$), the above equidistribution result can be applied, again thanks to the polynomial growth of the error term. Prime moduli larger than a power of $T$ diverges into two parts, named respectively “intermediate primes” and “very large primes”: $$T^\alpha<p< T\quad \text{ and }\quad p\geqslant T.$$ For the first case, we appeal to uniform estimates for integral points on quadrics in [@Browning-Gorodnik §4 §5] developed originally aiming at studying power-free polynomial values on symmetric spaces. This slicing argument, applied straightforward to the cases $n\geqslant 4$, however *a priori* does not provide desired power saving for the case $n=3$, as was also encountered in [@Browning-Gorodnik p. 1078]. To overcome this difficulty, we make essential use of the assumption that $-m\det Q$ is non-squared and that the form $Q$ being ${{\mathbb {Q}}}$-anisotropic in our argument. All these guarantee that no sliced piece contains any line or conic and thus contributes few integral points. can Note that these conditions also appear in results of Liu-Sarnak [@Liu-Sarnak]. One technical core of this paper is the treatment of very large primes, which combines the Ekedahl-type geometric sieve [@Ekedahl] with the half-dimensional sieve due to Friedlander-Iwaniec [@Iwaniec] [@Iwaniec-Friedlander], both developed for affine quadrics. The latter result can be rephrased as the density of quadratic polynomial values represented by a binary quadratic form, which may be of independent interest. \[thm:halfdimsieve\] Let $Q_1({\mathbf{x}})$ be a quadratic polynomial in $M\geqslant 1$ variables, $Q_2(u,v)$ be a binary positive-definite quadratic form, and $m\in{{\mathbb {Z}}}_{\neq 0}$. Assume that, - if $M\geqslant 2$, then the affine quadric $(Q_1({\mathbf{x}})=0)\subset {{\mathbb {A}}}^M$ is smooth; - if $M=1$, then the affine quadric $$\tag{$**$} (Q_1({\mathbf{x}})=Q_2(u,v))\subset {{\mathbb {A}}}^{3}$$ is smooth and has anisotropic stabilizer. Then $$\#\{{\mathbf{x}}\in{{\mathbb {Z}}}^{M}:\|{\mathbf{x}}\|\leqslant T,\exists (u,v)\in{{\mathbb {Z}}}^2,(**) \text{ holds}\}=O\left(\frac{T^{n-2}}{\sqrt{\log T}}\right),$$ where the implicit constant may depend on $Q_1,Q_2$. \[rmk:notsquare\] Without the assumption that the stabilizer being anisotropic, the estimate in Theorem \[thm:halfdimsieve\] is false, as clearly seen from the example $x^2+1=u^2+v^2$. It would be interesting to investigate whether Theorem \[thm:mainthmn=3\] remains valid for affine quadrics of dimension two without assuming that the form $Q$ being anisotropic, or even for those with isotropic stabilizers. A different feather is that the singular series diverges and the order of magnitude of the main term would be $T\log T$ instead of $T$ (c.f. [@Duke-Rudnick-Sarnak p. 146]). #### **Structure of the paper** In Section \[se:quadricarith\] we recall Tamagawa measures on affine quadrics, and we discuss the convergence of the infinite products in Theorems \[thm:mainthmgcd=1\], \[th:mainthm\], \[thm:mainthmn=3\]. In Section \[se:mainterm\], the main terms are derived. Section \[se:errorterms\] is entirely devoted to the treatment of error terms. In Section \[se:final\] we assemble all afore-obtained estimates and finish the proof. More layouts are sketched at the beginning of each section. #### **Notations and conventions** Unless otherwise specified, the implicit constant is only allowed to depend on the affine quadric $V_m\hookrightarrow {{\mathbb {A}}}^n$, the closed subset $Z$ and the residues $(\xi_p)_{p\in S}$. Any extra dependence of the arguments and results that we use of in course of the proof will be explicitly stated. The letter $p$ is always reserved for prime numbers. We write $p^k\| n$ for certain $k\in{{\mathbb {N}}},n\in{{\mathbb {N}}}_{\neq 0}$ if $p^{k}\mid n$ and $p^{k+1}\nmid n$. We write $\mu(\cdot)$ for the Möbius function, and $\Omega(n)$ for the number of prime divisors of $n\in{{\mathbb {N}}}_{\neq 0}$. Tamagawa measures on affine quadrics {#se:quadricarith} ==================================== The goal of this section is to recall the definition of *Tamagawa measures*. We refer to [@Borovoi-Rudnick §1.6] and [@Weil §2] for details. Let $L$ be a connected unimodular (i.e., equipped with a (both left and right) invariant gauge form) group over ${{\mathbb {Q}}}$. Let $\varrho_L$ be the Galois representation on the space of characters $\mathbf{X}^*(L)\otimes_{{\mathbb {Z}}}{{\mathbb {Q}}}$ and let $t_L$ be the multiplicity of the trivial representation in $\varrho_L$. Let $$L(s,\varrho_L)=\prod_{p<\infty}L_p(s,\varrho_L)$$ be the associated Artin $L$-function. The function $L(s,\varrho_L)$ has a pole of order $t_L$ at $s=1$. Define the *convergence factors* $$r_L=\lim_{s\to 1}(s-1)^{t_L}L(s,\varrho_L),\quad \lambda_{p,L}=L_p(1,\varrho_L)^{-1}.$$ Then the Tamagawa measure (with respect to the convergence factors $(r_L,(\lambda_{p,L})_{p<\infty})$) is the product measure on $L(\mathbf{A}_{{\mathbb {Q}}})$ $$m_L=r_L^{-1}m_{\infty,L}\prod_{p<\infty}\lambda_{p,L}^{-1}m_{p,L},$$ where $m_{\infty,L},m_{p,L}~(p<\infty)$ are local measures induced by the fixed gauge form, and $m_L$ is independent of the choice of gauge form. This product measure is *absolutely convergent*, meaning that applying to any adelic neighborhood the resulting infinite product is absolutely convergent, by Ono’s theorem [@Weil Appendix 2]. The choice of convergence factors is such that the additive group and the multiplicative group both have Tamagawa number equal to one. If $L$ is moreover semisimple and simply connected without non-trivial ${{\mathbb {Q}}}$-characters, the convergence factors above are not necessary and can be sorted out, and $m_L=m_{\infty,L}\prod_{p<\infty} m_{p,L}$ remains absolutely convergent. Returning to the affine quadric $V_m$ . We suppose throughout that $V_m({{\mathbb {Q}}})\neq\varnothing$ and that $V_m({{\mathbb {R}}})$ has no compact components. Then $V_m$ is a homogeneous space under $G_Q={\mathsf{Spin}}_Q$, the double cover of ${\mathsf{SO}}_Q$, with stabilizer $H_Q\simeq {\mathsf{Spin}}_Q|_{P^\perp}$, where $P\in V_m({{\mathbb {Q}}})$ and $P^\perp$ is the orthogonal complement of $P$. We have $$\dim G_Q=\frac{n(n-1)}{2},\quad \dim H_Q=\frac{(n-1)(n-2)}{2}.$$ Since $n\geqslant 3$, the group $G_Q$ is semisimple and simply connected, so is $H_Q$ if $n\geqslant 4$. Special attention is paid to affine quadrics in three variables because $H_Q$ is isomorphic to a torus. It is anisotropic over ${{\mathbb {Q}}}$ (hence has no ${{\mathbb {Q}}}$-characters) precisely when $-m\det(Q)$ is not a square, a condition that we shall always assume and denoted by $-m\det(Q)\neq \square$ in the sequel. To ease notations, we write $G=G_Q, H=H_Q$. Both $G,H$ are unimodular, so upon fixing an invariant gauge form for each, then on $V_m$ there exists an invariant gauge form matching together with those of $G,H$. Let $m_{\infty,V_m},m_{p,V_m}~ (p<\infty)$ be the corresponding local measures. Having defined the convergence factors $ r_G,r_H,(\lambda_{p,G})_{p<\infty}, (\lambda_{p,H})_{p<\infty}$ as above, the product measure $$m_{V_m}=r_{V_m}^{-1}m_{\infty,V_m}\prod_{p<\infty}\lambda_{p,V_m}^{-1}m_{p,V_m},$$ where $$r_{V_m}=\frac{r_G}{r_H},\quad \lambda_{p,V_m}=\frac{\lambda_{p,G}}{\lambda_{p,H}},$$ is the induced Tagamawa measure on $V_m({\mathbf{A}}_{{\mathbb {Q}}})$, which is also absolutely convergent (c.f. [@Borovoi-Rudnick Lemma 1.6.5]). As $G$ is semisimple and simply connected, the above formula for $m_{V_m}$ can be simplified to $$\label{eq:TamagawaX} m_{V_m}=r_Hm_{\infty,V_m}\prod_{p<\infty }\lambda_{p,H}m_{p,V_m}.$$ Note that for any $p$-adic neighborhood ${{\mathcal {Y}}}\subset V_m({{\mathbb {Z}}}_p)$, one has (c.f. [@Borovoi-Rudnick Lemma 1.8.1]) $$m_{p,V_m}({{\mathcal {Y}}})=\lim_{t\to\infty}\frac{\#({{\mathcal {Y}}}{\ \mathrm{mod}\ }p^t)}{p^{t(n-1)}}.$$ We first discuss the case $n\geqslant 4$, in which $H_Q$ is simply connected. So $V_m$ is strongly Hardy-Littlewood by [@Borovoi-Rudnick Proposition 2.9], and orbits of rational points are in bijection with those of adelic points (c.f. [@Borovoi-Rudnick Theorem 3.2]). So now writes $$\label{eq:globallocal} N_{V_m}(T)\sim m_{V_m}(V_m({{\mathbb {R}}})\times V_m(\widehat{{{\mathbb {Z}}}}))=\tau_{\infty}(V_m;T)\prod_{p<\infty}\hat{\tau}_p(V_m),$$ with real Hardy-Littlewood density (c.f. and [@Borovoi-Rudnick Lemma 1.8.2]) $$\label{eq:HLinfinity} m_{\infty,V_m}(V_m({{\mathbb {R}}}))=\tau_{\infty}(V_m;T) $$ and absolutely convergent singular series (c.f. ) $$\mathfrak{G}(V_m)=\prod_{p<\infty} \hat{\tau}_p(V_m),$$ where the local factors $\hat{\tau}_p(V_m)$ are given by . If $V_m({{\mathbb {Z}}}_p)\neq \varnothing$ for all prime $p$, then $\hat{\tau}_p(V_m)\neq 0$ for any prime $p$ and (c.f. [@Duke-Rudnick-Sarnak Example 1.5]) $$\label{eq:globalgrowth} N_{V_m}(T)\sim c_{V_m}T^{n-2}$$ for certain $ c_{V_m}>0$ (and *a fortiori* $V_m({{\mathbb {Z}}})\neq\varnothing$). Thus a quantitative local-to-global principle holds with the leading constant expressed as product of local factors. We will need the following (recall ). \[le:infprod1\] The infinite product $$\label{eq:infiniteprod1} \prod_{p<\infty} \tau_p(Z;V_m)=\prod_{p<\infty}\frac{\hat{\tau}_p(Z;V_m)}{\hat{\tau}_p(V_m)}$$ is absolutely convergent. The Lang-Weil estimate [@Cao-Huang Corollary 3.5] and Hensel’s lemma gives $$\label{eq:LangWeiltauZ} 1-\hat{\tau}_p(Z;V_m)=\frac{\#Z({{\mathbb {Z}}}/p{{\mathbb {Z}}})}{p^{n-1}}=O\left(\frac{1}{p^2}\right),$$ uniformly for any $p$ such that $V_m$ is smooth over ${{\mathbb {F}}}_p$. Thus the infinite product $$\prod_{p<\infty} \hat{\tau}_p(Z;V_m)$$ and the singular series $\mathfrak{G}(V_m)$ are both absolutely convergent, and from which we deduce the absolute convergence of . Secondly we discuss the case $n=3$ where the stabilizer $H$ is a torus and fails to be simply connected, so that $V_m$ is not strongly but only relatively Hardy-Littlewood, with a locally constant density function $\delta(\cdot)$ on $V_m({\mathbf{A}}_{{\mathbb {Q}}})$ with values in $\{0,2\}$ (c.f. [@Browning-Gorodnik p. 1047-1048], [@Borovoi-Rudnick §3]) characterizing orbits intersecting $V_m({{\mathbb {Q}}})$: $$\delta(\mathfrak{O}_{{\mathbf{A}}})>0\Leftrightarrow \mathfrak{O}_{{\mathbf{A}}}\cap V_m({{\mathbb {Q}}})\neq\varnothing.$$ As we assume that $H$ is anisotropic, then $t_H=0$ (in other words, ${\operatorname{rk}}_{{{\mathbb {Z}}}}\mathbf{X}^*(H)^{{{\mathrm{Gal}}}(\bar{{{\mathbb {Q}}}}/{{\mathbb {Q}}})}=0$) and $r_H=L(1,\varrho_H)$ in . The action of $G({\mathbf{A}}_{{\mathbb {Q}}})$ on $V_m({\mathbf{A}}_{{\mathbb {Q}}})$ is open by Lang’s theorem (c.f. [@Borovoi-Rudnick Lemma 1.6.4]), but not transitive in general. So there are finitely many (open) orbits $\mathfrak{O}_{{\mathbf{A}},i},i\in I$ of $V_m({\mathbf{A}}_{{\mathbb {Q}}})$ under $G({\mathbf{A}}_{{\mathbb {Q}}})$ intersecting non-trivially with $V_m({{\mathbb {R}}})\times V_m(\widehat{{{\mathbb {Z}}}})$, so that $$N_{V_m}(T)=\sum_{i\in I}N_{\mathfrak{O}_{{\mathbf{A}}},i}(T).$$ For any orbit ${\mathfrak{O}_{{\mathbf{A}}}}$ among them, we have $V_m({{\mathbb {Z}}}_p)\subset \mathfrak{O}_p$ for almost all $p$, and so (recall ) $$\label{eq:hattauporbit} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})=\hat{\tau}_p(V_m),\quad \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)=\hat{\tau}_p(Z;V_m).$$ To interpret the leading constant as product of local densities, we define analogously the real Hardy-Littlewood density $\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}};T)$ for each (fixed connected component of) orbit ${\mathfrak{O}_{{\mathbf{A}}}}$ (c.f. [@Browning-Gorodnik p. 1048 (1.11)]). Then by [@Borovoi-Rudnick Theorem 5.3], the quantity is asymptotically equivalent (up to the constant $\delta(\mathfrak{O}_{{\mathbf{A}}})$) to the Tamagawa measure (restricted to $V_m({{\mathbb {R}}})\times V_m(\widehat{{{\mathbb {Z}}}})$) of the orbit: $$\label{eq:globallocalorbit} N_{\mathfrak{O}_{{\mathbf{A}}}}(T)\sim \delta(\mathfrak{O}_{{\mathbf{A}}})L(1,\varrho_H)\left(\prod_{p<\infty}L_p(1,\varrho_H)^{-1}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})\right)\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}};T).$$ The infinite product in is absolutely convergent. The same argument as proving yields: \[le:infprod2\] The infinite product $$\label{eq:infiniteprod2} \prod_{p<\infty}\tau_p(\mathfrak{O}_{{\mathbf{A}}};Z)=\prod_{p<\infty}\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}$$ is absolutely convergent. The density function $\delta(\cdot)$ can equivalently be defined in terms of orthogonal locus of the Brauer group $\operatorname{Br}(V_m)$, as the failure of integral Hasse principle for $V_m$ can be explained by integral Brauer-Manin obstruction (c.f. [@CT-Xu Theorem 3.7, §5.6, §5.8]). See the work of Wei-Xu [@Wei-Xu] for various explicit formulas in this spirit. Upon choosing “good” integral models, the finiteness of orbits of $V_m({{\mathbb {Z}}})$ under ${\mathsf{Spin}}_Q({{\mathbb {Z}}})$ is guaranteed by Borel-Harish–Chandra theory, which also allows to work with every single orbit. See [@Liu-Sarnak]. Towards the main term – primes of small moduli {#se:mainterm} ============================================== The goal of this section is to derive asymptotic formulas for the density of integral points in a fixed adelic neighborhood of $V_m(\widehat{{{\mathbb {Z}}}})$ (which we choose to be $W_S$ ) lying in the open set $U$ modulo any sufficiently small prime with explicit error terms, and prove that they contribute to the main term. For the sake of simplicity, we shall only state our results in terms of affine quadrics: Theorem \[thm:maintermsmallp\] for $n\geqslant 4$ and Theorem \[thm:maintermsmallporbit\] for $n=3$. Our proof can be extended *verbatim* to any affine symmetric spaces verifying the conditions (i)–(iv) in [@Browning-Gorodnik p. 1045], in particular to any principal homogeneous spaces under semisimple simply connected groups. #### **Ingredients** Results in what follows are recorded from [@Browning-Gorodnik §3]. They show that the growth of integral points with prescribed residue modulo any non-zero integer is in accordance with the Tagamawa measure of the corresponding adelic neighborhood. Let us define for $l\in{{\mathbb {N}}}_{\neq 0},\xi\in ({{\mathbb {Z}}}/l{{\mathbb {Z}}})^n$, $$\label{eq:NlT} \begin{split} V_l(T;\xi)&:=\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,{\underline{\mathbf{X}}}\equiv \xi{\ \mathrm{mod}\ }l\}. \end{split}$$ \[prop:equidistresidue\] Assume that $n\geqslant 4$. Then there exists $0<\delta_{V_m}<1$, depending only on $V_m$, such that, uniformly for any $l\in{{\mathbb {N}}}_{\neq 0},\xi\in V_m({{\mathbb {Z}}}/l{{\mathbb {Z}}})$, $$V_l(T;\xi)=\tau_{\infty}(V_m;T)\prod_{p<\infty} \hat{\tau}_p(V_m;\xi,l)+O(l^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(V_m;T)^{1-\delta_{V_m}}),$$ where $\tau_{\infty}(V_m;T)$ is the real Hardy-Littlewood density , and $$\hat{\tau}_p(V_m;\xi,l):=\lim_{t\to\infty}\frac{\#\{{\mathbf{x}}\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):{\mathbf{x}}\equiv\xi{\ \mathrm{mod}\ }p^{{{\mathrm{ord}}}_p (l)}\}}{p^{t(n-1)}}.$$ The group ${\mathsf{Spin}}_Q$ is always ${{\mathbb {Q}}}$-simple if $n\geqslant 5$ or $n=3$, whereas when $n=4$, it can happen that ${\mathsf{Spin}}_Q\simeq J\times J$ where $J$ is a ${{\mathbb {Q}}}$-form of ${\operatorname{SL}}_{2,{{\mathbb {Q}}}}$. As explained in [@Browning-Gorodnik Remark 2.4], the above asymptotic formula remains true for affine quadrics in four variables, whose spin group ${\mathsf{Spin}}_Q$ can possibly be not ${{\mathbb {Q}}}$-simple. We state correspondingly the result for adelic orbits for the treatment of the case $n=3$. \[prop:equidistresidue2\] There exists $0<\delta_{V_m}<1$, depending only on $V_m$, such that, uniformly for every orbit $\mathfrak{O}_{{\mathbf{A}}}$ of $V_m({\mathbf{A}}_{{\mathbb {Q}}})$ under $G({\mathbf{A}}_{{\mathbb {Q}}})$ and for every $l\in{{\mathbb {N}}}_{\neq 0}$ and $\xi\in V_m({{\mathbb {Z}}}/l{{\mathbb {Z}}})$, we have $$\begin{aligned} V_l(\mathfrak{O}_{{\mathbf{A}}};T;\xi)&:=\#\{{\underline{\mathbf{X}}}\in \mathfrak{O}_{{\mathbf{A}}}\cap V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,{\underline{\mathbf{X}}}\equiv \xi{\ \mathrm{mod}\ }l\}\\ &=\delta(\mathfrak{O}_{{\mathbf{A}}})\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}};T)L(1,\varrho_H)\prod_{p<\infty} L_p(1,\varrho_H)^{-1}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi,l)\\ &\quad\quad +O(l^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}};T)^{(1-\delta_{V_m})}), \end{aligned}$$ where $\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}};T)$ is the real Hardy-Littlewood density of the orbit ${\mathfrak{O}_{{\mathbf{A}}}}$ [@Browning-Gorodnik (1.11)] and $$\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi,l):= \lim_{t\to\infty}\frac{\#\{\xi\in\mathfrak{O}_{{\mathbf{A}}}\cap V_m({{\mathbb {Z}}}_p) {\ \mathrm{mod}\ }p^t:\xi\equiv \xi_p{\ \mathrm{mod}\ }p^{{{\mathrm{ord}}}_p(l)}\}}{p^{t(n-1)}}.$$ #### **Affine quadrics of dimension greater than three** Let us recall the adelic neighborhood $W_S$ and define $$\label{eq:NSMT} \begin{split} N^{S,M}_{U}(T):=\#\{{\underline{\mathbf{X}}}\in W_S\cap V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,\forall p\not\in S,p<M,{\underline{\mathbf{X}}}{\ \mathrm{mod}\ }p\not\in Z\}. \end{split}$$ \[thm:maintermsmallp\] Assume that $n\geqslant 4$. Then for any $M$ sufficiently large, $$N^{S,M}_{U}(T)= \tau_{\infty}(V_m,T)\left(\prod_{p\in S} \hat{\tau}_p(\xi_p;V_m)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(Z;V_m)\right) +\operatorname{Er}(T;M),$$ where $$\label{eq:prodNS} \operatorname{Er}(T;M) =O_\varepsilon\left(2^{\Omega(\mathfrak{P}_{M,S})}\mathfrak{P}_{M,S}^{\frac{(3n-2)(n-1)}{2}+n-3+\varepsilon}T^{(n-2)(1-\delta_{V_m})}+\frac{T^{n-2}}{M}\right),\quad \mathfrak{P}_{M,S}:=\prod_{p\not\in S, p<M} p.$$ To simply notations and to keep summation formulas from being overloaded, let $P_S=\prod_{p\in S}p^{n_p}$, and by Chinese remainder theorem, we extract $\xi_0\in V_m({{\mathbb {Z}}}/P_S{{\mathbb {Z}}})$ the unique element such that $\xi_0\equiv\xi_p{\ \mathrm{mod}\ }p^{n_p}$, so that the conditions ${\underline{\mathbf{X}}}\equiv\xi_p{\ \mathrm{mod}\ }p^{n_p},\forall p\in S$ integrate into only one condition ${\underline{\mathbf{X}}}\equiv \xi_0{\ \mathrm{mod}\ }P_S$. Next for any $D\in{{\mathbb {N}}}_{\neq 0}$ square-free, such that $\gcd(D,P_S)=1$, we define the quantity $$\label{eq:NSTD} \begin{split} V_{S,D}(T;Z) :=\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,{\underline{\mathbf{X}}}\equiv \xi_0{\ \mathrm{mod}\ }P_S,{\underline{\mathbf{X}}}{\ \mathrm{mod}\ }D\in Z\}. \end{split}$$ By the Lang-Weil estimate, for any $l\in{{\mathbb {N}}}_{\neq 0}$, we have $$\#Z({{\mathbb {Z}}}/l{{\mathbb {Z}}})\ll_{\varepsilon} l^{n-3+\varepsilon}.$$ On invoking Proposition \[prop:equidistresidue\] for $l=D^\prime=DP_S$, we estimate $V_{S,D}(T;Z)$ as follows. $$\label{eq:mainterm1} \begin{split} &V_{S,D}(T;Z)=\sum_{\substack{\xi\in V_m({{\mathbb {Z}}}/D^\prime {{\mathbb {Z}}})\\\xi{\ \mathrm{mod}\ }D\in Z\\\xi\equiv\xi_0{\ \mathrm{mod}\ }P_S}}V_{D^\prime}(T;\xi)\\&=\sum_{\substack{\xi\in V_m({{\mathbb {Z}}}/D^\prime {{\mathbb {Z}}})\\\xi{\ \mathrm{mod}\ }D\in Z\\\xi\equiv\xi_0{\ \mathrm{mod}\ }P_S}}\left(\tau_{\infty}(V_m;T)\prod_{p<\infty} \hat{\tau}_p(V_m;\xi,D^\prime)+ O(D^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(V_m;T)^{1-\delta_{V_m}})\right)\\ &=\tau_{\infty}(V_m;T)\left(\prod_{p\in S}\hat{\tau}_p(V_m;\xi_0,P_S)\right) \left(\sum_{\xi\in Z({{\mathbb {Z}}}/D{{\mathbb {Z}}})}\prod_{p\mid D}\hat{\tau}_p(V_m;\xi,D)\right)\left(\prod_{p\nmid D^\prime} \hat{\tau}_p(V_m;\xi,D^\prime)\right)\\ &\quad \quad +O(\#Z({{\mathbb {Z}}}/D{{\mathbb {Z}}})D^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(V_m;T)^{1-\delta_{V_m}})\\ &=\tau_{\infty}(V_m;T)\left(\prod_{p\in S}\hat{\tau}_p(\xi_p;V_m)\right)\eta_D(Z)\left(\prod_{p\nmid D} \hat{\tau}_p(V_m)\right) +O_\varepsilon(D^{\frac{(3n-2)(n-1)}{2}+n-3+\varepsilon}T^{(n-2)(1-\delta_{V_m})}), \end{split}$$ where $$\begin{aligned} \eta_D(Z)&:=\prod_{p\mid D}\lim_{t\to\infty}\frac{\#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi{\ \mathrm{mod}\ }p\in Z\}}{p^{t(n-1)}}\\ &=\sum_{\xi\in Z({{\mathbb {Z}}}/D{{\mathbb {Z}}})}\prod_{p\mid D}\hat{\tau}_p(V_m;\xi,D),\end{aligned}$$ and because $$\begin{aligned} \forall p\in S,&\quad \hat{\tau}_p(V_m;\xi_0,P_S)=\hat{\tau}_p(V_m;\xi_p,p^{n_p})=\hat{\tau}_p(\xi_p;V_m),\\ \forall p\nmid D^\prime,&\quad \hat{\tau}_p(V_m;\xi,D^\prime)=\hat{\tau}_p(V_m).\end{aligned}$$ By means of the inclusion-exclusion principle and the Chinese remainder theorem, $$\begin{aligned} \sum_{\substack{D\in{{\mathbb {N}}}_{\neq 0} \text{ square-free}\\D\mid\mathfrak{P}_{M,S}}}\mu(D)\eta_D(Z)&=\prod_{\substack{p<M,p\not\in S}}\lim_{t\to\infty}\frac{\#\{\xi\in V_m({{\mathbb {Z}}}/p^t{{\mathbb {Z}}}):\xi{\ \mathrm{mod}\ }p\not\in Z\}}{p^{t(n-1)}}\\ &=\prod_{p<M,p\not\in S}\hat{\tau}_p(Z;V_m).\end{aligned}$$ Using the above formulae of $V_{S,D}(T;Z)$ , we can estimate the quantity $N^{S,M}_{U}(T)$ . $$\begin{aligned} N^{S,M}_{U}(T)&=\sum_{\substack{D\in{{\mathbb {N}}}_{\neq 0} \text{ square-free}\\D\mid\mathfrak{P}_{M,S}}}\mu(D) V_{S,D}(T;Z)\\ &=\tau_{\infty}(V_m;T)\left(\prod_{p\in S}\hat{\tau}_p(\xi_p;V_m)\right)\ \left(\prod_{\substack{p<M,p\not\in S}}\hat{\tau}_p(Z;V_m)\right)\left(\prod_{p\geqslant M}\hat{\tau}_p(V_m)\right)\\ &\quad\quad+O(2^{\Omega(\mathfrak{P}_{M,S})}\mathfrak{P}_{M,S}^{\frac{(3n-2)(n-1)}{2}+n-3+\varepsilon}T^{(n-2)(1-\delta_{V_m})}).\end{aligned}$$ We see that the error term contributes to the first part of $\operatorname{Er}(T;M)$. To get the desired main term, it remains to extend the finite product $\prod_{\substack{p<M,p\not\in S}}\hat{\tau}_p(Z;V_m)$ to infinite product. Thanks to the Lang-Weil estimate ([@Lang-Weil], [@Cao-Huang Corollary 3.5]) $$c_p:=\frac{\# Z({{\mathbb {F}}}_p)}{\# V_m(\mathbb{F}_p)}=O\left(\frac{1}{p^2}\right).$$ Then by Lemma \[le:infprod1\], $$\begin{aligned} \left(\prod_{p\geqslant M} \frac{\hat{\tau}_p(V_m)}{\hat{\tau}_p(Z;V_m)}\right) -1&=\left(\prod_{p\geqslant M}\frac{\# V_m(\mathbb{F}_p)}{\# V_m(\mathbb{F}_p)-\# Z({{\mathbb {F}}}_p)}\right)-1\\&=\exp\left(-\sum_{p\geqslant M}\log\left(1-\frac{c_p}{p^2}\right)\right)-1\\ &= O\left( \sum_{p\geqslant M}\frac{1}{p^2}\right)=O\left(\frac{1}{M}\right).\end{aligned}$$ We recall that $$\tau_{\infty}(V_m;T)=O(T^{n-2})$$ by . Therefore this contributes to second term of $\operatorname{Er}(T;M)$, and finishes the proof. #### **Two-dimensional quadrics** Similarly to , let us define for each orbit ${\mathfrak{O}_{{\mathbf{A}}}}$, $$\begin{aligned} N_{U,\mathfrak{O}_{{\mathbf{A}}}}^{S,M}(T):=\#\{{\underline{\mathbf{X}}}\in{\mathfrak{O}_{{\mathbf{A}}}}\cap W_S\cap V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,\forall p\not\in S,p<M,{\underline{\mathbf{X}}}{\ \mathrm{mod}\ }p\not\in Z\}.\end{aligned}$$ Employing an argument akin to the one in proving Theorem \[thm:maintermsmallp\] immediately yields: \[thm:maintermsmallporbit\] For any orbit $\mathfrak{O}_{{\mathbf{A}}}$ satisfying (\*) in Theorem \[thm:mainthmn=3\], $$\begin{aligned} &N_{U,\mathfrak{O}_{{\mathbf{A}}}}^{S,M}(T)\\ =&\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}},T)L(1,\varrho_H)\delta(\mathfrak{O}_{{\mathbf{A}}})\left(\prod_{p\in S}L_p(1,\varrho_H)^{-1} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)\right)\left(\prod_{p\not \in S,p<\infty} L_p(1,\varrho_H)^{-1}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)\right) +\operatorname{Er}(T;M), \end{aligned}$$ where $\operatorname{Er}(T;M)$ is identical to the one in Theorem \[thm:maintermsmallp\]. The work [@Liu-Sarnak] deals with two-dimensional quadrics defined by anisotropic forms, using a hyperbolic lattice point counting approach totally different from the works [@Browning-Gorodnik] [@Nevo-Sarnak]. It also provides estimates similar to Proposition \[prop:equidistresidue2\] (aiming at sieve problems about almost primes solutions), whose error terms also have polynomial growth and are related to the Selberg eigenvalue conjecture. Treatment of error terms {#se:errorterms} ======================== The whole section is devoted to showing that residues arising from other prime moduli are negligible. For any $0<N_1<N_2\leqslant \infty$ sufficiently large, let us define $$\label{eq:VN} V(T;N_1,N_2):=\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,\exists p\in [N_1,N_2],{\underline{\mathbf{X}}}{\ \mathrm{mod}\ }p\in Z\}.$$ According to the range of $N_1,N_2$, we separate our discussion into three parts, in which is treated using different methods. In §\[se:er:primeextpoly\] we derive an upper bound for valid for arbitrary $N_1,N_2$, which is satisfactory if we take $N_1$ going to infinity as $T$ grows and $N_2=T^\alpha$ with $\alpha>0$ sufficiently small. In §\[se:er:primeinter\], on taking $N_1=T^\alpha,N_2=T$, we deal with the residues coming from intermediate primes, by employing various Serre-type uniform bounds for integral points on quadrics. In §\[se:er:geometricsieve\] and §\[se:polyresp\] we match together a generalised geometric sieve *à la Ekedahl* and a half-dimensional sieve *à la Friedlander-Iwaniec* for affine quadrics so as to derive a satisfactory upper bound for $V(T;T,\infty)$. We shall always throughout assume that $V_m({{\mathbb {Q}}})\neq \varnothing$ and $-m\det Q\neq\square$ if $n=3$. And we shall emphasise the uniformity of dependence on variables in the assumption of all statements. Unless otherwise specified, all implicit constants are only allowed to depend on $V_m$ and $Z$. Extension to prime moduli of polynomial growth {#se:er:primeextpoly} ---------------------------------------------- The purpose of section is to prove \[thm:primepoly\] Suppose that $N_2<\infty$. Then $$\begin{aligned} V(T;N_1,N_2)=O\left(\frac{T^{n-2}}{N_1}+N_2^{\frac{(3n-2)(n-1)}{2}+n-2}T^{(n-2)(1-\delta_{V_m})}\right). \end{aligned}$$ We shall only deal with the case $n\geqslant 4$, and that of $n=3$ is reduced to considering every single orbit and applying Proposition \[prop:equidistresidue2\]. We now rerun the argument in proving Theorem \[thm:maintermsmallp\]. Analogous to , for $p\not\in S$, we define the quantity $V_p(T;Z)$ by forgetting the congruent conditions for primes in $S$: $$\label{eq:NpT} V_p(T;Z):=\#\{{\underline{\mathbf{X}}}\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,{\underline{\mathbf{X}}}{\ \mathrm{mod}\ }p\in Z\}.$$ For any $p_0$ sufficiently large and $\xi\in Z({{\mathbb {F}}}_{p_0})$, by Hensel’s lemma in a manner similar to , $$\hat{\tau}_{p_0}(V_m;\xi,p_0)=p_0^{-(n-1)},$$ and for any $p\neq p_0$, $$\hat{\tau}_p(V_m;\xi,p)=\hat{\tau}_p(V_m).$$ By Lang-Weil estimate (c.f. [@Cao-Huang Corollary 3.5]), we have $$\label{eq:LangWeilZFp} Z({{\mathbb {F}}}_p)\ll p^{n-3}$$ for any prime $p$. Applying Proposition \[prop:equidistresidue\] by taking $l=p_0>N_1$, we get: $$\begin{aligned} V_{p_0}(T;Z)&=\sum_{\xi\in Z({{\mathbb {F}}}_{p_0})}V_{p_0}(T;\xi)\\ &=\sum_{\xi\in Z({{\mathbb {F}}}_{p_0})}\left(\tau_{\infty}(V_m;T)\hat{\tau}_{p_0}(V_m;\xi,p_0)\prod_{p\neq p_0}\hat{\tau}_p(V_m)+ O(p_0^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(V_m;T)^{1-\delta_{V_m}})\right)\\ &=\tau_{\infty}(V_m;T)\mathfrak{G}(V_m)\frac{\#Z({{\mathbb {F}}}_{p_0})}{\#V_m({{\mathbb {F}}}_{p_0})}+O(\#Z({{\mathbb {F}}}_{p_0})p_0^{\frac{(3n-2)(n-1)}{2}}\tau_{\infty}(V_m;T)^{1-\delta_{V_m}})\\ &=O\left(\frac{T^{n-2}}{p_0^2}+p_0^{\frac{(3n-2)(n-1)}{2}+n-3}T^{(n-2)(1-\delta_{V_m})}\right), \end{aligned}$$ where $\mathfrak{G}(V_m)$ is the singular series . Summing over all such $p_0$’s, we get an the following upper bound for $V(T;N_1,N_2)$: $$\begin{aligned} V(T;N_1,N_2)&\leqslant \sum_{N_1\leqslant p_0\leqslant N_2}V_{p_0}(T;Z)\\ &=\sum_{N_1\leqslant p_0\leqslant N_2}O\left(\frac{T^{n-2}}{p_0^2}+p_0^{\frac{(3n-2)(n-1)}{2}+n-3}T^{(n-2)(1-\delta_{V_m})}\right)\\ &\ll\frac{T^{n-2}}{N_1}+N_2^{\frac{(3n-2)(n-1)}{2}+n-2}T^{(n-2)(1-\delta_{V_m})}. \end{aligned}$$ This gives the desired expression for the bound. Intermediate primes of polynomial range {#se:er:primeinter} --------------------------------------- The goal of this section is to show: \[th:intermediateprimes\] Assume moreover that the quadratic form $Q$ is anisotropic over ${{\mathbb {Q}}}$ if $n=3$. Then for any $\alpha>0$, $$\begin{aligned} V(T;T^\alpha, T)=O_{\alpha}\left( \frac{T^{n-2}}{\log T}\right). \end{aligned}$$ Compared to the order of growth $T^{n-2}$ of the main term, this is satisfactory. We shall separate our discussion into the cases $n=3$ and $n\geqslant 4$. While dealing with the latter is relatively straightforward, the case $n=3$ requires some slightly more involved analysis. It turns out that we need satisfactory bounds for the quantity , which are uniform respect to $p$ and $\mathbf{\xi}\in {{\mathbb {F}}}_p^n$, and we need to prove that the contribution from summing over all $p$ in range is still satisfactory. Our argument is inspired by [@Browning-Gorodnik §5]. ### Ingredients We record here a uniform estimate of integral points on affine quadrics due to Browning-Gorodnik, which is also useful in §\[se:polyresp\]. For $q\in {{\mathbb {Z}}}[x_1,\cdots,x_m]$ a polynomial of degree two, the *quadratic part* $q_0$ of $q$ is the homogeneous degree two part of $q$. Let ${\operatorname{rk}}(q_0)$ denote the rank of the quadratic form $q_0$. \[thm:BrowningGorodnikaffinequadric\] Let $q\in{{\mathbb {Z}}}[x_1,\cdots,x_n]$ be an irreducible polynomial of degree two. Assume that ${\operatorname{rk}}(q_0)\geqslant 2$. Then we have for any $\varepsilon>0$, $$\#\{{\mathbf{x}}\in{{\mathbb {Z}}}^n:\|{\mathbf{x}}\|\leqslant T,q({\mathbf{x}})=0\}=O_{\varepsilon} (T^{n-2+\varepsilon}).$$ We emphasis that the implicit constant is independent of the polynomial $q$. The case of $n=3$ is a direct consequence of dimensional growth bounds obtained by Browning, Heath-Brown and Salberger (c.f. [@Browning-Gorodnik Lemma 4.1]). ### Two-dimensional affine quadrics We first recall the following basic fact. \[le:notsquare\] Assume that $V_m({{\mathbb {Q}}})\neq \varnothing$. Then the condition that $-m\det Q\neq\square$ is equivalent to the one that $V_m$ does not contain any line over ${{\mathbb {Q}}}$. We first note that ${\mathrm{Pic}}((V_{m})_{\overline{{{\mathbb {Q}}}}})={{\mathbb {Z}}}$. To see how ${\mathrm{Pic}}((V_{m})_{\overline{{{\mathbb {Q}}}}})$ is generated, we recall that a projective quadric surface $S$ in ${{\mathbb {P}}}^3$ over $\overline{{{\mathbb {Q}}}}$ has Picard group ${\mathrm{Pic}}(S)\simeq {{\mathbb {Z}}}^2$. Any hyperplane section $H$ intersects $S$ into a conic curve $C$ of divisor type $(1,1)$. So that ${\mathrm{Pic}}(S\setminus H)\simeq {{\mathbb {Z}}}^2/{{\mathbb {Z}}}(1,1)\simeq {{\mathbb {Z}}}$. In particular, if the conic curve $C$ splits into two lines, then any of them generates ${\mathrm{Pic}}(S\setminus H)$. So if we compactify $V_m$ into $\overline{V_m}\subset{{\mathbb {P}}}^3$ and view $V_m=\overline{V_m}\setminus H$ for some hyperplane section $H$, then the class of any ${{\mathbb {Q}}}$-line on $V_m$ generates ${\mathrm{Pic}}(V_m)$. Let $d=-m\det Q$ and fix $P\in V_m({{\mathbb {Q}}})$. Let $H\simeq (x^2-dy^2=1)$ be the stabilizer of ${\mathsf{Spin}}_Q$ acting on $P$. The group $G={{\mathrm{Gal}}}({{\mathbb {Q}}}(\sqrt{d}/{{\mathbb {Q}}}))$ operates on $\hat{H}\simeq {\mathrm{Pic}}((V_{m})_{\overline{{{\mathbb {Q}}}}})$, so that $${\mathrm{Pic}}(V_m)\simeq \hat{H}^G.$$ On the one hand, if $d=\square$, then the tangent plane at $P$ intersects $V_m$ into two lines over ${{\mathbb {Q}}}$ (c.f. [@CT-Xu p. 333]). On the other hand, if $d\neq\square$, by [@CT-Xu p. 331], we have ${\mathrm{Pic}}(V_m)=0$. So we conclude that $V_m$ cannot contain a ${{\mathbb {Q}}}$-line when $d\neq\square$. With this at hand, we now show: \[prop:intern3\] Assume that the quadric $(Q=0)$ is anisotropic over ${{\mathbb {Q}}}$ and that $-m\det Q\neq\square$. Then we have, uniformly for any $1\ll p<T$ and $\mathbf{\xi}\in {{\mathbb {F}}}_p^3$, $$V_p(T;\xi)\ll_{\varepsilon} \frac{T^\varepsilon}{p^\varepsilon}+\frac{T^{1+\varepsilon}}{p^\frac{4}{3}}.$$ Either $V_p(T;\xi)=0$, for which the desired estimate is evident, or we can find ${\underline{\mathbf{X}}}_0\in V_m({{\mathbb {Z}}})$ such that, $\|{\underline{\mathbf{X}}}_0\|\leqslant T,{\underline{\mathbf{X}}}_0\equiv\xi{\ \mathrm{mod}\ }p$. By making the change of variables ${\underline{\mathbf{X}}}={\underline{\mathbf{X}}}_0+p\mathbf{y}$, the new variable $\mathbf{y}\in{{\mathbb {Z}}}^3$ verify the following equations: $$\label{eq:changevar1} \|{\mathbf{y}}\|\leqslant \frac{2T}{p},\quad \mathbf{y}\cdot \nabla Q({\underline{\mathbf{X}}}_0)+pQ(\mathbf{y})=0.$$ Since ${\underline{\mathbf{X}}}_0$ is a smooth point of $V_m$, we have $\nabla Q({\underline{\mathbf{X}}}_0)\neq \mathbf{0}$. Moreover $p\nmid \gcd({\underline{\mathbf{X}}}_0)$, since otherwise $p\mid m$, which cannot happen for $p$ large enough. Since ${\underline{\mathbf{X}}}_0\equiv\xi{\ \mathrm{mod}\ }p$, the second equation implies that ${\mathbf{y}}$ lies in the lattice $$\Gamma_{\xi}:=\{{\mathbf{x}}\in{{\mathbb {Z}}}^3:p\mid \mathbf{x}\cdot \nabla Q(\xi)\}$$ of determinant $\gg\ll p$, with implicit constants depending only on $Q$ (c.f. [@Browning-Gorodnik p. 1076]). Thus $$\label{eq:AM} V_p(T;\xi)\leqslant M_{{\underline{\mathbf{X}}}_0,\xi}(T),$$ where $$M_{{\underline{\mathbf{X}}}_0,\xi}(T):=\#\{{\mathbf{y}}\in\Gamma_{\xi}:\|{\mathbf{y}}\|\leqslant \frac{2T}{p},\mathbf{y}\cdot \nabla Q({\underline{\mathbf{X}}}_0)+pQ(\mathbf{y})=0\}.$$ We are led to bounding $M_{{\underline{\mathbf{X}}}_0,\xi}(T)$. Choose a minimal basis $\mathbf{L}=(\mathbf{l}_1,\mathbf{l}_2,\mathbf{l}_3)$ of $\Gamma_{\xi}$ such that (c.f. [@Browning-Gorodnik (5.3)]) $$\|\mathbf{l}_1\|\leqslant \|\mathbf{l}_2\|\leqslant \|\mathbf{l}_3\|, \quad \|\mathbf{l}_1\|\|\mathbf{l}_2\|\|\mathbf{l}_3\|\gg\ll p,$$ so that, by making the non-singular change of variables ${\mathbf{y}}\mapsto \mathbf{L}\cdot{\mathbf{z}}$, the new variable ${\mathbf{z}}=(z_1,z_2,z_3)\in{{\mathbb {Z}}}^3$ verifies, by and [@Browning-Gorodnik p. 1075], $$\label{eq:changevar2} |z_i|\ll \frac{T}{\|\mathbf{l}_i\|p},1\leqslant i\leqslant 3,\quad \widetilde{Q}({\mathbf{z}})+{\mathbf{z}}\cdot\mathbf{Y}_0=0,$$ where $$\widetilde{Q}({\mathbf{z}})=Q(\mathbf{L}{\mathbf{z}}),\quad \mathbf{Y}_0=p^{-1}\mathbf{L}\cdot \nabla Q({\underline{\mathbf{X}}}_0).$$ Next we slice the second equation of into the variable $z_3$ and get for each fixed integer $z_3$ a resulting polynomial $q_{z_3}\in {{\mathbb {Z}}}[z_1,z_2]$. By , the total number of $z_3$ is $$\ll 1+\frac{T}{p\|\mathbf{l_3}\|}.$$ Observe that for any $z_3\in {{\mathbb {Q}}}$, $$(q_{z_3})_0=\widetilde{Q}(z_1,z_2,0).$$ The latter, viewed as quadratic form in two varieties, has rank $1$ (that is, ${\operatorname{rk}}((q_{z_3})_0)=1$) if and only if $z_3=0$ is the tangent plane at certain point defined over ${{\mathbb {Q}}}$ of the projective quadric $(Q=0)\subset {{\mathbb {P}}}^2$ with homogeneous coordinates $[z_1:z_2:0]$. By assumption that $(Q=0)$ is ${{\mathbb {Q}}}$-anisotropic, we conclude that this is impossible, and hence for any $z_3\in {{\mathbb {Q}}}$, ${\operatorname{rk}}((q_{z_3})_0)=2$. Next, if for certain $z_3=\kappa$, the polynomial $q_{\kappa}$ is reducible over ${{\mathbb {Q}}}$, that is, it splits into two polynomials $f_1,f_2$ of degree one, then $(z_3=\kappa)\cap (f_i=0)$ defines a ${{\mathbb {Q}}}$-line on $V_m$ for $i=1,2$ (since the change of variables at each step is non-singular). This is absurd by Lemma \[le:notsquare\] as we always assume that $-m\det Q\neq\square$. So for any $z_3\in{{\mathbb {Q}}}$, the polynomial $q_{z_3}$ is irreducible over ${{\mathbb {Q}}}$. Theorem \[thm:BrowningGorodnikaffinequadric\] shows that the contribution from integral points on the quadric $q_{z_3}=0$ is (as we assume $p\leqslant T$) $$\begin{aligned} A_{z_3}(T)&:=\#\{(z_1,z_2)\in{{\mathbb {Z}}}^2:\max_{i=1,2}|z_i|\leqslant T,q_{z_3}(z_1,z_2)=0\}\\ &=O_\varepsilon\left(1+\left(\frac{T}{p\|\mathbf{l}_1\|}\right)^\varepsilon\right)=O_\varepsilon\left(\frac{T^\varepsilon}{p^{\varepsilon}}\right). \end{aligned}$$ Therefore we obtain an upper bound for $M_{{\underline{\mathbf{X}}}_0,\xi}(T)$ as follows: $$\begin{aligned} M_{{\underline{\mathbf{X}}}_0,\xi}(T)&\leqslant \sum_{|z_3|\ll 1+\frac{T}{p\|\mathbf{l_3}\|}}A_{z_3}(T)\\ &\ll_{\varepsilon}\left( 1+\frac{T}{p\|\mathbf{l_3}\|}\right)\times \frac{T^\varepsilon}{p^\varepsilon}=O_{\varepsilon}\left(\frac{T^\varepsilon}{p^\varepsilon}+\frac{T^{1+\varepsilon}}{p^\frac{4}{3}}\right), \end{aligned}$$ because $\|\mathbf{l_3}\|\gg p^\frac{1}{3}$. This finishes the proof, thanks to . Since $\dim Z=0$, we have $\#Z({{\mathbb {F}}}_p)\leqslant \deg Z$ for every prime $p$. Employing the estimate (c.f. [@Axler Proposition 10]) $$\label{eq:primesumaxler} \sum_{p\leqslant X} \frac{1}{p^\sigma}\ll_\sigma \frac{X^{1-\sigma}}{\log X},\quad 0<\sigma<1,$$ and Proposition \[prop:intern3\], we sum over all primes in the interval $[T^{\alpha},T]$ and we get, for any $0<\varepsilon<\min(\frac{1}{2},\frac{1}{3}\alpha)$, $$\begin{aligned} V(T;T^\alpha, T) \leqslant &\sum_{T^\alpha\leqslant p\leqslant T}\sum_{\xi\in Z({{\mathbb {F}}}_p) }V_p(T;\xi)\\ \ll_{\varepsilon} &(\deg Z)\sum_{T^\alpha\leqslant p\leqslant T}\left(\frac{T^\varepsilon}{p^\varepsilon}+\frac{T^{1+\varepsilon}}{p^\frac{4}{3}}\right)\\ \ll_\varepsilon &\frac{T^{\varepsilon}\times T^{1-\varepsilon}}{\log T}+T^{1+\varepsilon-\frac{1}{3}\alpha}=O_\varepsilon\left(\frac{T}{\log T}\right). \end{aligned}$$ We fix $\varepsilon>0$ small enough in terms of $\alpha$ so that the implicit constant above depends only on $\alpha$. This gives the desired upper-bound. ### Affine quadrics of dimension $\geqslant 3$ Now we turn to the case $n\geqslant 4$. The key input for us is the following estimate obtained by Browning and Gorodnik. \[prop:intern4\] Assume $n\geqslant 4$. Then $$V_p(T;\xi)\ll_{\varepsilon}\left(\frac{T}{p}\right)^{n-3+\varepsilon}\left(1+\frac{T}{p^{\frac{n}{n-1}}}\right)$$ holds uniformly for any $\mathbf{\xi}\in {{\mathbb {F}}}_p^n$ and for any prime $p\leqslant T$. On recalling the Lang-Weil estimate , we infer that from Proposition \[prop:intern3\] and that $$\begin{aligned} V(T;T^\alpha, T)\leqslant &\sum_{T^{\alpha}\leqslant p\leqslant T}\sum_{\xi\in Z({{\mathbb {F}}}_p) }V_p(T;\xi)\\ \ll_{\varepsilon}&\sum_{T^{\alpha}\leqslant p\leqslant T} p^{n-3}\times \left(\frac{T}{p}\right)^{n-3+\varepsilon}\left(1+\frac{T}{p^{\frac{n}{n-1}}}\right)\\ \ll_\varepsilon &\sum_{T^{\alpha}\leqslant p\leqslant T} \left(\frac{T^{n-3+\varepsilon}}{p^\varepsilon}+\frac{T^{n-2+\varepsilon}}{p^{1+\frac{1}{n-1}+\varepsilon}}\right)\\ \ll_\varepsilon & T^{n-3+\varepsilon}\times \frac{T^{1-\varepsilon}}{\log T}+T^{n-2+\varepsilon}\times T^{-\frac{\alpha}{n-1}} \ll_{\varepsilon} \frac{T^{n-2}}{\log T}, \end{aligned}$$ for any $0<\varepsilon<\min(\frac{1}{2},\frac{\alpha}{n-1})$. It remains to fix $\varepsilon>0$ small enough depending only on $\alpha$ to get the desired dependence for the implicit constant. A geometric sieve for affine quadrics {#se:er:geometricsieve} ------------------------------------- The Ekedahl sieve [@Ekedahl] (or the geometric sieve) has been generalised and applied by several authors to similar counting problems for the affine space ${{\mathbb {A}}}^n$ [@Poonen Theorem 3.1] (also over some other base fields), and in some more general setting [@Bhargava]. The goal of this section is to generalise this sieve method to affine quadrics. The idea is inspired by a discussion with Tim Browning, to whom we express our gratitude. \[th:largeprimes\] Let $N_1=T,N_2=\infty$ in . Then $$\begin{aligned} V(T;T,\infty)=O\left(\frac{T^{n-2}}{(\log T)^{\frac{1}{2}}}\right) \end{aligned}$$ ### Proof of Theorem \[th:largeprimes\] Let us first fix a diagonal integral model for the affine quadratic $V_m$: $$\label{eq:vm} \sum_{i=1}^{n} a_ix_i^2=m.$$ This may affect via a different choice of equivalent height functions in terms of the equation of $Q$, which is clearly negligible. We may assume that $a_{n-1}\cdot a_n>0$. By multiplying $-1$ to the equation if necessary we can assume that both of them are $>0$. We can furthermore assume that $a_n=1$, and all other $a_i$’s are integers and square-free, and we write from now on $a_{n-1}=a>0$. Next, upon enlarging the codimension two subvariety $Z$, we can assume that $Z$ is the locus of together with two polynomials $f,g$ satisfying $f \in {{\mathbb {Z}}}[x_1,\cdots,x_{n-1}],g\in {{\mathbb {Z}}}[x_1,\cdots,x_{n-2}]$. In order to do so, let us consider the maps ${\operatorname{pr}}_1:{{\mathbb {A}}}^n\to{{\mathbb {A}}}^{n-1},{\operatorname{pr}}_2:{{\mathbb {A}}}^{n-1}\to{{\mathbb {A}}}^{n-2}$, and ${\operatorname{pr}}_3={\operatorname{pr}}_2\circ{\operatorname{pr}}_1$, the first (resp. second) being the projection onto the first $(n-1)$ (resp. $(n-2)$) coordinates. Since $\dim(Z)= n-3$, its image ${\operatorname{pr}}_3(Z)$ has codimension at least one in ${{\mathbb {A}}}^{n-2}$. Therefore we can choose $g\in {{\mathbb {Z}}}[x_1,\cdots,x_{n-2}]$ such that ${\operatorname{pr}}_3(Z)\subseteq Z^\prime:=(g=0)\subset {{\mathbb {A}}}^{n-2}$. On the other hand, since the map ${\operatorname{pr}}_1|_{V_m}$ is affine and finite, the closed subset ${\operatorname{pr}}_1(Z)\subset{{\mathbb {A}}}^{n-1}$ has codimension at least two, and ${\operatorname{pr}}_1(Z)\subset {\operatorname{pr}}_2^{-1}(Z^\prime)=Z^\prime\times{{\mathbb {A}}}^1$, the latter being of codimension one in ${{\mathbb {A}}}^{n-1}$. We can choose $f \in {{\mathbb {Z}}}[x_1,\cdots,x_{n-1}]$ such that ${\operatorname{pr}}_1(Z)\subset (f=0)\cap{\operatorname{pr}}_2^{-1}(Z^\prime)$. This gives $$Z\subset V_m\cap (f=g=0)\subset{{\mathbb {A}}}^n.$$ Note that this above procedure may be reformulated using the elimination theory. **Case 1.** First of all we recall a quantitative version of Ekedahl’s geometric sieve [@Ekedahl], due to Bhargava [@Bhargava]. \[le:Ekedahl\] Let $Y\subset {{\mathbb {A}}}^N_{{\mathbb {Z}}}$ be a subvariety of codimension $k$. Then for any $M\gg_Y 1$, 1. $\#\{{\mathbf{x}}\in Y({{\mathbb {Z}}}):\|{\mathbf{x}}\|\leqslant T\}=O_Y(T^{N-k});$ 2. $\#\{{\mathbf{x}}\in{{\mathbb {Z}}}^N:\exists p>M,{\mathbf{x}}{\ \mathrm{mod}\ }p\in Y\}=O_Y\left(\frac{T^N}{M^{k-1}\log M}+T^{N-k+1}\right)$. We are going to consider three conditions in which we can reduce the problem to the affine space ${{\mathbb {A}}}^{n-2}$ via the fibration $\operatorname{pr}_3|_{V_m}$ and adopt Lemma \[le:Ekedahl\] to get satisfactory upper bounds. If $f$ is constant in the variable $x_{n-1}$, then $Z=(Y_1\times {{\mathbb {A}}}^2)\cap V_m$ where $Y_1=(f=g=0)\subset{{\mathbb {A}}}^{n-2}$ is of codimension two. Otherwise, if $f$ is non-constant in $x_{n-1}$, then by an induction on the degree of $f$ in $x_{n-1}$ plus the above elimination process, we may assume that the leading coefficient of $f$ in $x_{n-1}$, say $h\in{{\mathbb {Z}}}[x_1,\cdots,x_{n-2}]$, together with $g$, defines a codimension two subvariety $Y_2\subset {{\mathbb {A}}}^{n-2}$. We now extract the set ${{\mathcal {B}}}_1$ consisting of ${\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}$ verifying at least one of the following conditions: - $g({\mathbf{x}})=0$; - the polynomial $f$ is constant in $x_{n-1}$ and $\exists p\geqslant T,{\mathbf{x}}{\ \mathrm{mod}\ }p\in Y_1$; - the polynomial $f$ is non-constant in $x_{n-1}$, and $\exists p\geqslant T$ such that $p\mid g({\mathbf{x}})$, and $f({\mathbf{x}},x_{n-1})$, as a polynomial in $x_{n-1}$, is $\in p{{\mathbb {Z}}}[x_{n-1}]$. In particular this implies $p\mid h({\mathbf{x}})$ and thus ${\mathbf{x}}{\ \mathrm{mod}\ }p\in Y_2$. On rearranging the contribution from $V(T;T,\infty)$ by fixing the first $n-2$ variables and summing over integral points on the fibres of $\operatorname{pr}_3|_{V_m}$, we introduce $V_1(T;T,\infty)$ to bound the overall contribution in the subcases above: $$\begin{aligned} &V_1(T;T,\infty):=\#\{{\underline{\mathbf{X}}}=(x_1,\cdots,x_n)\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,(x_1,\cdots,x_{n-2})\in {{\mathcal {B}}}_1\}\\ &\leqslant\sum_{\substack{\|{\mathbf{x}}\|\leqslant T\\{\mathbf{x}}\in{{\mathcal {B}}}_1}} H_1({\mathbf{x}})\leqslant \left(\sum_{\substack{{\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}:\|{\mathbf{x}}\|\leqslant T\\g({\mathbf{x}})=0}} 1+ \sum_{\substack{(\text {if }f \text{ is constant in } x_{n-1})\\{\mathbf{x}}\in {{\mathbb {Z}}}^{n-2}:\|{\mathbf{x}}\|\leqslant T\\\exists p\geqslant T, {\mathbf{x}}{\ \mathrm{mod}\ }p\in Y_1}}1 + \sum_{\substack{(\text {if }f \text{ is non-constant in } x_{n-1})\\ {\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}:\|{\mathbf{x}}\|\leqslant T\\\exists p\geqslant T, {\mathbf{x}}{\ \mathrm{mod}\ }p\in Y_2}} 1\right)H_1({\mathbf{x}}), $$ where for any ${\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}$, $$H_1({\mathbf{x}}):=\#\{(u,v)\in {{\mathbb {Z}}}^2:u^2+av^2=m-\sum_{i=1}^{n-2}a_ix_i^2\}.$$ We have clearly $H_1({\mathbf{x}})=O_\varepsilon(T^\varepsilon)$ for any $\|{\mathbf{x}}\|\leqslant T$. As for the three sums in bracket, one uses Lemma \[le:Ekedahl\] ((1) applied to the codimension one subvariety $(g=0)\subset{{\mathbb {A}}}^{n-2}$ for the first sum, and (2) applied to the codimension two varieties $Y_1,Y_2\subset{{\mathbb {A}}}^{n-2}$ in the remaining two sums), and gets $$\label{eq:V1infty} V_1(T;T,\infty)= O\left(T^{n-3}+\frac{T^{n-2}}{T\log T}+T^{n-3}\right)\times O_\varepsilon(T^\varepsilon)=O_\varepsilon(T^{n-3+\varepsilon}).$$ **Case 2.** Let us consider from now on the set ${{\mathcal {B}}}_2$ of ${\mathbf{x}}\in {{\mathbb {Z}}}^{n-2}$ verifying all of the conditions below: - $g({\mathbf{x}})\neq 0$; - for any $p\geqslant T$, $f({\mathbf{x}},x_{n-1}){\ \mathrm{mod}\ }p$ is a non-zero polynomial in $x_{n-1}$. To give an upper bound for the overall contribution in Case 2, we consider $$\begin{aligned} &V_2(T;T,\infty)\\ &:=\#\{{\underline{\mathbf{X}}}=(x_1,\cdots,x_n)\in V_m({{\mathbb {Z}}}):\|{\underline{\mathbf{X}}}\|\leqslant T,\exists p\geqslant T,p\mid g({\underline{\mathbf{X}}}),p\mid f({\underline{\mathbf{X}}}),(x_1,\cdots,x_{n-2})\in {{\mathcal {B}}}_2\}\\ &\leqslant \sum_{\substack{{\mathbf{x}}\in{{\mathcal {B}}}_2,\|{\mathbf{x}}\|\leqslant T\\\exists u,v\in{{\mathbb {Z}}},m-\sum_{i=1}^{n-2}a_ix_i^2=u^2+av^2}} H_2({\mathbf{x}}),\end{aligned}$$ where for any ${\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}$, $$H_2({\mathbf{x}}):=\sum_{\substack{p:p\geqslant T\\p\mid g(\mathbf{x})}}\sum_{\substack{y\in{{\mathbb {Z}}},|y|\leqslant T\\ p\mid f(\mathbf{x},y)}}\#\{z\in{{\mathbb {Z}}}:z^2=m-ay^2-\sum_{i=1}^{n-2}a_ix_i^2\}.$$ Under the assumption that ${\mathbf{x}}\in{{\mathcal {B}}}_2$, we have, $$\sum_{\substack{p:p\geqslant T\\p\mid g(\mathbf{x})\neq 0}}1=O(\deg g),$$ because $g({\mathbf{x}})\neq 0$, and $g({\mathbf{x}})\ll T^{\deg g}$, so the number of primes $\geqslant T$ dividing $g({\mathbf{x}})$ is $\ll \deg g$. Moreover, for any $p\geqslant T$, $$\sum_{\substack{y\in{{\mathbb {Z}}},|y|\leqslant T\\ p\mid f(\mathbf{x},y)}}1\ll(\deg f) \left(\frac{T}{p}+1\right)=O(\deg f),$$ because $f({\mathbf{x}},x_{n-1}) {\ \mathrm{mod}\ }p$ is a non-zero polynomial in $x_{n-1}$. Hence all implicit constants above depend only on the polynomials $f,g$, that is, the variety $Z$. So $$\begin{aligned} H_2({\mathbf{x}})\ll O(1)\times O(1)\times O(1)=O(1).\end{aligned}$$ Returning to the error term $V_2(T;T,\infty)$. The bound for $H_2({\mathbf{x}})$ results in $$V_2(T;T,\infty)\ll\#C(T) \times O(1),$$ where $$C(T):=\{{\mathbf{x}}\in{{\mathbb {Z}}}^{n-2}:\|{\mathbf{x}}\|\leqslant T,\exists u,v\in{{\mathbb {Z}}},m-\sum_{i=1}^{n-2}a_ix_i^2=u^2+av^2\}.$$ We are reduced to bounding $\#C(T)$. For this we appeal to Theorem \[thm:halfdimsieve\], whose proof will be given in the next section, by setting $Q_1({\mathbf{x}})=m-\sum_{i=1}^{n-2}a_ix_i^2$ and $Q_2(u,v)=u^2+av^2$. If $n\geqslant 4$, then the affine quadric $(Q_1({\mathbf{x}})=0)\subset{{\mathbb {A}}}^{n-2}$ is clearly smooth. When $n=3$, the condition $-m\det Q\neq \square $ is equivalent to the stated one in Theorem \[thm:halfdimsieve\]. So all assumptions of Theorem \[thm:halfdimsieve\] are satisfied. We thus obtain $$\label{eq:V2infty} V_2(T;T,\infty)\ll \frac{T^{n-2}}{(\log T)^{\frac{1}{2}}}.$$ Finally, thanks to $$V(T;T,\infty)\leqslant V_1(T;T,\infty)+V_2(T;T,\infty),$$ the bounds obtained in **Case 1** and in **Case 2** complete the proof. A half-dimensional sieve for affine quadrics {#se:polyresp} -------------------------------------------- The goal of this section is devoted to proving Theorem \[thm:halfdimsieve\]. Our strategy is based on an upper-bound version of the combinatorial sieve in dimension one half, developed in works [@Iwaniec] [@Iwaniec-Friedlander]. Keeping the notations in Theorem \[thm:halfdimsieve\], let us fix throughout this section a quadratic polynomial $Q_1({\mathbf{x}})\in{{\mathbb {Z}}}[x_1,\cdots,x_M]$, and a primitive positive-definite integral quadratic form $Q_2(u,v)$. Let $D_{Q_2}$ be the discriminant of the form $Q_2$. We have $D_{Q_2}\leqslant -3$. We shall give full details for the case $M\geqslant 2$ in §\[se:n4resp\], and we indicate necessary modifications in §\[se:n3resp\] for the case $M=1$, as is already inexplicit in [@Iwaniec-Friedlander Remarks 2 p.2]. ### Representation by binary quadratic forms {#se:resp} To start, let us define two arithmetic functions ${\mathfrak{b}}(\cdot)$ and ${\mathfrak{b}^*}(\cdot)$, characterizing integers represented by the form $Q_2(u,v)$. First define $$\label{eq:legendresym} {{\mathcal {P}}}_{Q_2}=\left\{p:\left(\frac{D_{Q_2}}{p}\right)=-1\right\},$$ where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol of modulus $p$. For $n\in{{\mathbb {Z}}}$, let $${\mathfrak{b}}(n)=\begin{cases} 1 &\text{ if } \exists u,v\in{{\mathbb {Z}}},n=Q_2(u,v);\\ 0 &\text{ otherwise}, \end{cases}$$ and $${\mathfrak{b}^*}(n)=\begin{cases} 1 &\text{ if } \forall p\mid n,p\not\in{{\mathcal {P}}}_{Q_2};\\ 0 &\text{ if } \exists p\in{{\mathcal {P}}}_{Q_2},p\mid n. \end{cases}$$ We now collect some well-known facts about representation of integers by primitive binary quadratic forms of negative discriminant. See for example [@James Lemmas 1 & 2]. If there exists $p\in {{\mathcal {P}}}_{Q_2}$ such that $n=p^{2k+1} m$, with $k\in{{\mathbb {N}}},\gcd(p,m)=1$, then ${\mathfrak{b}}(n)=0$. So, if ${\mathfrak{b}}(n)=1$, then for any $p\in{{\mathcal {P}}}_{Q_2},p\mid n$, there exists $k\in{{\mathbb {N}}}$, such that $p^{2k}\|n$. We conclude from this that if ${\mathfrak{b}}(n)=1$ then ${\mathfrak{b}^*}(n/r)=1$ with certain $r\mid n$ square-full, whose prime divisors (if any) are all in ${{\mathcal {P}}}_{Q_2}$. The function ${\mathfrak{b}^*}$ is clearly multiplicative, however it is not the case for ${\mathfrak{b}}$ in general. ### The half-dimensional sieve Let ${{\mathcal {A}}}=(a_i)_{i\in I}$ be a finite sequence of integers indexed by $I$. For any $r\in{{\mathbb {N}}}_{\neq 0}$, let ${{\mathcal {A}}}_r$ be the subsequence consisting of elements of ${{\mathcal {A}}}$ divisible by $r$. That is, ${{\mathcal {A}}}_r=(a_i)_{i\in I_r}$ with $I_r=\{i\in I:r\mid a_i\}$. Let ${{\mathcal {P}}}$ be a subset of prime numbers. For any $z>1$, define the sifting function $$S({{\mathcal {A}}},{{\mathcal {P}}},z):=\#\{i\in I:\gcd(a_i,\prod_{\substack{p:p\in{{\mathcal {P}}}\\ p<z}}p)=1\}.$$ We will employ the following version of the half dimensional sieve due to Friedlander and Iwaniec. \[thm:I-F\] There exists a continuous function $G:\mathopen]0,\infty[\to{{\mathbb {R}}}_{>0}$ such that, for any arithmetic function $\varrho$ verifying $\varrho(p)\neq p$ for any $p\in{{\mathcal {P}}}$, and $$\label{eq:half-dim} \left|\sum_{\substack{p:p\in{{\mathcal {P}}}\\ p<z}}\frac{\varrho(p)}{p-\varrho(p)}\log p-\frac{1}{2}\log z\right|\leqslant K,$$ for certain $K>0$, we have, for any $y,z\geqslant 2$, $$S({{\mathcal {A}}},{{\mathcal {P}}},z)\leqslant X\left(G\left(\frac{\log y}{\log z}\right)+O_K((\log y)^{- \frac{1}{5}})\right)\prod_{\substack{p:p\in{{\mathcal {P}}}\\p<z}}\left(1-\frac{\varrho(p)}{p}\right)+\sum_{\substack{d<y\\p\mid d\Rightarrow p<z,p\in{{\mathcal {P}}}}}|R(d)|,$$ where $X=\#{{\mathcal {A}}}(=\# I)$ and $$R(d)=\# {{\mathcal {A}}}_d-\frac{\varrho(d)}{d}X.$$ ### Proof of Theorem \[thm:halfdimsieve\] for the case $M\geqslant 2$ {#se:n4resp} We consider $${{\mathcal {A}}}(T)=\{Q_1({\mathbf{x}})\}_{{\mathbf{x}}\in{{\mathbb {Z}}}^M:\|{\mathbf{x}}\|\leqslant T}.$$ For $r\in{{\mathbb {N}}}_{\neq 0}$, we define the subsequence $${{\mathcal {A}}}(T)_r=\{Q_1({\mathbf{x}})/r\}_{\substack{{\mathbf{x}}\in{{\mathbb {Z}}}^M:\|{\mathbf{x}}\|\leqslant T\\r\mid Q_1({\mathbf{x}})}}.$$ Consider the arithmetic multiplicative functions $\omega,\varrho$, defined for $N\in{{\mathbb {N}}}_{\neq 0}$, $$\omega(N):=\#\{\xi\in ({{\mathbb {Z}}}/N{{\mathbb {Z}}})^M:Q_1(\xi)\equiv 0{\ \mathrm{mod}\ }N\},\quad \varrho(N):=\frac{\omega(N)}{N^{M-1}}.$$ Then $\omega(N)\ll_{\varepsilon} N^{M-1+\varepsilon}$. We have, by Chinese remainder theorem, for any $N\in{{\mathbb {N}}}_{\neq 0}$, $$\label{eq:omegaN} \begin{split} \#{{\mathcal {A}}}(T)_{N}&=\sum_{\substack{\xi\in({{\mathbb {Z}}}/N{{\mathbb {Z}}})^M\\ Q_1(\xi)\equiv 0{\ \mathrm{mod}\ }N}}\#\{\mathbf{y}\in (N{{\mathbb {Z}}})^M:\|y+\xi\|\leqslant T\}\\ &=\omega(N)\left(\frac{T}{N}+O(1)\right)^M. \end{split}$$ Our ultimate goal is to give non-trivial upper bounds for $$\label{eq:goalhalfsieve} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}}(Q_1({\mathbf{x}}))$$ via $S({{\mathcal {A}}},{{\mathcal {P}}},z)$ and to apply the Theorem \[thm:I-F\] with appropriately chosen ${{\mathcal {A}}},{{\mathcal {P}}},z$. Our discussion in §\[se:resp\] shows that $$\label{eq:step1} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}}(Q_1({\mathbf{x}}))\leqslant\sum^*_{r} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}^*}(Q_1({\mathbf{x}})/r),$$ where the sum with superscript $*$ means that $r$ is restricted to square-full integers whose prime divisors are all in ${{\mathcal {P}}}_{Q_2}$ . By assumption, the affine variety $(Q_1=0)$ is smooth modulo any sufficient large $p$, we have by Lang-Weil estimate [@Lang-Weil], $$\varrho(p)=1+O(p^{-\frac{1}{2}}).$$ Let us define $$\label{eq:CP2} {{\mathcal {P}}}_{Q_2}^\prime={{\mathcal {P}}}_{Q_2}\setminus \{p:\varrho(p)=p\}.$$ Since there are at most finitely primes $p$ verifying $\varrho(p)=p$, the primes in the set ${{\mathcal {P}}}_{Q_2}$, and hence ${{\mathcal {P}}}_{Q_2}^\prime$, have density one half amongst the prime residues modulo $D_{Q_2}$. With these notions, for any $r\in{{\mathbb {N}}}_{\neq 0},\lambda>0$, one has $$\label{eq:step1prime} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}^*}(Q_1({\mathbf{x}})/r)\leqslant S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2},T^\lambda)\leqslant S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2}^\prime,T^\lambda).$$ We observe that it suffices to deal with sufficiently small $r$’s, more precisely $r<T^\gamma$ for certain $0<\gamma<\Delta_M:=\frac{1}{4M}$. Because once a square-full $r=q^2\mid Q_1({\mathbf{x}})$ for some $\|{\mathbf{x}}\|\leqslant T$, we have $q\leqslant\sqrt{|Q_1({\mathbf{x}})|}\ll T$, and so by , $$\begin{aligned} \#{{\mathcal {A}}}(T)_{q^2}&=\#\{{\mathbf{x}}\in{{\mathbb {Z}}}^M:\|{\mathbf{x}}\|\leqslant T,q^2\mid Q_1({\mathbf{x}})\}\\ &\ll \omega(q^2)\times \left( \left(\frac{T}{q^2}\right)^M+1\right)\\ &\ll_\varepsilon (q^2)^{M-1+\varepsilon}\times\left( \left(\frac{T}{q^2}\right)^M+1\right)\\ &\ll_{\varepsilon}\frac{T^M}{q^{2-\varepsilon}}+q^{2M-2+\varepsilon}.\end{aligned}$$ Then the contribution from all such $q\in[T^\frac{\gamma}{2},T^{\frac{1}{2}+\Delta_M}]$ is $$\sum_{q\in[T^\frac{\gamma}{2},T^{\frac{1}{2}+\Delta_M}]}\#{{\mathcal {A}}}(T)_{q^2}\ll_\varepsilon T^{M-\frac{\gamma}{2}+\varepsilon}+T^{M-\Delta_M+\varepsilon}\ll T^{M-\frac{\gamma}{2}+\varepsilon}.$$ This is satisfactory compared to the expected leading term $\frac{T^M}{\sqrt{\log T}}$. Next, for $$\label{eq:bdq} T^{\frac{1}{2}+\Delta_M}<q\ll T,$$ we regard the $(M+1)$-tuple $({\mathbf{x}},q)=(x_1,\cdots,x_M,q)$ as an integral point on the affine quadric $$Q_{s}({\mathbf{x}},q):Q_1({\mathbf{x}})-sq^2=0,$$ where $s$ is an auxiliary integer parameter satisfying $s\ll T^{1-2\Delta_M}$, thanks to the preassigned bound for $q$. We want to insert the uniform upper bound in Theorem \[thm:BrowningGorodnikaffinequadric\] for the quadrics $Q_{s}({\mathbf{x}},q)$ with $s\neq 0$. Recall that we assume $M\geqslant 2$, so ${\operatorname{rk}}(Q_{s})_0\geqslant 2$ whenever $s\neq 0$, and moreover the quadratic polynomial $Q_1({\mathbf{x}})-sq^2$ is irreducible. Since otherwise $Q_1({\mathbf{x}})-sq^2=(s_1q+A_1({\mathbf{x}}))(s_2q+A_2({\mathbf{x}}))$ and this would imply that $Q_1({\mathbf{x}})=A_1({\mathbf{x}})A_2({\mathbf{x}})$, a contradiction to the assumption that $(Q_1({\mathbf{x}})=0)$ is smooth. So the hypotheses of Theorem \[thm:BrowningGorodnikaffinequadric\] are verified. As for $s=0$, we use the evident upper bound (Lemma \[le:Ekedahl\] (1)) $$B(T):=\#\{{\mathbf{x}}\in{{\mathbb {Z}}}^M:\|{\mathbf{x}}\|\leqslant T,Q_1({\mathbf{x}})=0\}\ll T^{M-1}.$$ We conclude that the contribution of such $q$ in this case is $$\begin{aligned} \sum_{T^{\frac{1}{2}+\Delta_M}<q\ll T}\#{{\mathcal {A}}}(T)_{q^2}&\ll B(T)+\sum_{0\neq s\ll T^{1-2\Delta_M}}\#\{({\mathbf{x}},q)\in{{\mathbb {Z}}}^{M+1}:\|({\mathbf{x}},q)\|\ll T,Q_s({\mathbf{x}},q)=0\} \\ &\ll_{\varepsilon}T^{M-1}+\sum_{0\leqslant s\ll T^{1-2\Delta_M}} T^{M+1-2+\varepsilon}\ll T^{M-2\Delta_M+\varepsilon}.\end{aligned}$$ This is also satisfactory and proves our claim. Gathering together , equation now writes $$\label{eq:step2} \begin{split} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}}(Q_1({\mathbf{x}}))&\leqslant \sum^*_{r<T^\gamma} \sum_{{\mathbf{x}}\in{{\mathbb {Z}}}^M,\|{\mathbf{x}}\|\leqslant T} {\mathfrak{b}^*}(Q_1({\mathbf{x}})/r)+\sum_{T^\frac{\gamma}{2}\leqslant q\ll T}\#{{\mathcal {A}}}(T)_{q^2}\\ &\leqslant\sum^*_{r<T^\gamma} S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2}^\prime,T^\lambda)+O_\varepsilon(T^{M-\frac{\gamma}{2}+\varepsilon}), \end{split}$$ where $0<\gamma<\Delta_M,\lambda>0$ are to be chosen later. Everything now boils down to the estimation of $S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2}^\prime,T^\lambda)$. Our task is to apply Theorem \[thm:I-F\] to each ${{\mathcal {A}}}(T)_r$. By taking a crude estimate on the error term of , we get $$\label{eq:omegaNr} X_r:=\#{{\mathcal {A}}}(T)_r=\frac{\omega(r)}{r^M}T^M+O(r^{M-1+\varepsilon}T^{M-1}).$$ We define arithmetic functions $$\omega_r(N):=\frac{\omega(rN)}{\omega(r)},\quad \varrho_r(N)=\frac{\omega_r(N)}{N^{M-1}}=\frac{\varrho(rN)}{\varrho(r)},\quad N\in{{\mathbb {N}}}_{\neq 0},$$ for $\omega(r)\neq 0$. Otherwise we simply put $\omega_r=\varrho_r=0$. Let us now verify that whenever $\omega(r)\neq 0$, the function $\omega_r$, and hence $\varrho_r$, is multiplicative. Indeed, for any $N\in{{\mathbb {N}}}_{\neq 0}$, we factorize $N=N_1N_2,r=r_1r_2$ such that $\gcd(r_1N_1,r_2N_2)=1$. Then $$\begin{aligned} \omega_r(N)=\frac{\omega(r_1r_2N_1N_2)}{\omega(r_1r_2)}&=\frac{\omega(r_1N_1)}{\omega(r_1)}\frac{\omega(r_2N_2)}{\omega(r_2)}\\ &=\frac{\omega(r_1N_1)\omega(r_2)}{\omega(r_1)\omega(r_2)}\frac{\omega(r_2N_2)\omega(r_1)}{\omega(r_2)\omega(r_1)}\\ &=\frac{\omega(rN_1)}{\omega(r)}\frac{\omega(rN_2)}{\omega(r)}=\omega_r(N_1)\omega_r(N_2).\end{aligned}$$ By Hensel’s lemma, for any $p\gg 1$ (such that the variety $(Q_1=0) {\ \mathrm{mod}\ }p$ is smooth) and $p\mid r$, one has $\omega(pr)=\omega(r)p^{M-1}$, and $\omega(rp)=\omega(r)\omega(p)$ for $p\nmid r$ by Chinese remainder theorem, and hence $$\label{eq:varrp} \varrho_r(p)=\begin{cases} \varrho(p) & \text{ if } p\nmid r;\\ 1 &\text{ if } p\mid r. \end{cases}$$ This implies $\varrho_r(p)=1+O(p^{-\frac{1}{2}})$ uniformly for any $p\gg 1$ and any $r\in{{\mathbb {N}}}_{\neq 0}$, so by Mertens’ first theorem on arithmetic progressions (c.f. [@Iwaniec-Kolwalski Theorem 2.2]), the hypothesis is verified for the arithmetic function $\varrho_r$ and the set ${{\mathcal {P}}}_{Q_2}^\prime$ uniformly for any $r$ (that is, the remainder term $K$ in depends only on the polynomial $Q_1$ and is independent of $r$). Next we need to evaluate for each $d\in{{\mathbb {N}}}_{\neq 0}$ square-free, the cardinality of the subsequence ${{\mathcal {A}}}(T)_{rd}$, using and the definition of $\omega_r,\varrho_r$. $$\begin{aligned} \#{{\mathcal {A}}}(T)_{rd}&=\frac{\omega(dr)}{(dr)^M}T^M+O(\omega(dr)T^{M-1})\\ &=\frac{\omega_r(d)}{d^M}\frac{\omega(r)}{r^{M}}T^{M}+O((dr)^{M-1+\varepsilon}T^{M-1})\\ &=\frac{\varrho_r(d)}{d}X_r+O((dr)^{M-1+\varepsilon}T^{M-1}).\end{aligned}$$ Define $$R_r(d):=\#{{\mathcal {A}}}(T)_{rd}-\frac{\varrho_r(d)}{d}X_r.$$ The above computation shows that $$R_r(d)=O((dr)^{M-1+\varepsilon}T^{M-1}).$$ On applying Theorem \[thm:I-F\] with $z=T^\lambda,y=T^\beta$, for $\lambda,\beta>0$ small enough, we get, for each $r$ square-full with all prime divisors in ${{\mathcal {P}}}_{Q_2}$, $$\begin{aligned} &S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2}^\prime,T^\lambda)\\ \leqslant & \left(G(\beta/\lambda)+O((\log T)^{-\frac{1}{5}})\right)X_r\prod_{\substack{p:p\in{{\mathcal {P}}}_{Q_2}^\prime\\p<T^\lambda}}\left(1-\frac{\varrho_r(p)}{p}\right)+O\left(\sum_{d<T^{\beta}}R_r(d)\right)\\ \leqslant & \left(G(\beta/\lambda)+O((\log T)^{-\frac{1}{5}})\right)\left(\frac{\omega(r)}{r^M} \prod_{\substack{p:p\in{{\mathcal {P}}}_{Q_2}^\prime\\p<T^\lambda}}\left(1-\frac{\varrho_r(p)}{p}\right)\right)T^M+O\left(\sum_{d<T^{\beta}}(dr)^{M-1+\varepsilon}T^{M-1}\right).\end{aligned}$$ Thanks to , the leading term in the last expression, up to the factor $\left(G(\beta/\lambda)+O((\log T)^{-\frac{1}{5}})\right)T^M$, can be written as $$\left( \prod_{\substack{p:p<T^\lambda\\ p\in {{\mathcal {P}}}_{Q_2}^\prime}}\left(1-\frac{\varrho(p)}{p}\right)\right) \times\left(\frac{\omega(r)}{r^M}\prod_{\substack{p:p\mid r\\p\in {{\mathcal {P}}}_{Q_2}^\prime}}\left(\left(1-\frac{1}{p}\right)\left(1-\frac{\varrho(p)}{p}\right)^{-1}\right)\right).$$ The series $\sum_{r=\square} c_r$, formed by $$c_r:=\frac{\omega(r)}{r^M}\prod_{\substack{p:p\mid r\\p\in {{\mathcal {P}}}_{Q_2}^\prime}}\left(\left(1-\frac{1}{p}\right)\left(1-\frac{\varrho(p)}{p}\right)^{-1}\right)=\frac{\omega(r)}{r^M}\prod_{\substack{p:p\mid r\\p\in {{\mathcal {P}}}_{Q_2}^\prime}}\frac{p-1}{p-\varrho(p)},$$ converges, because $$c_r\ll_{\varepsilon} \frac{r^{M-1+\varepsilon}}{r^{M}}\times r^\varepsilon\ll_{\varepsilon}\frac{1}{r^{1-\varepsilon}}.$$ We therefore conclude $$\label{eq:step3} \begin{split} &\sum_{\substack{r=\square, r<T^\gamma\\ p\mid r\Rightarrow p\in \mathcal{P}_{Q_2}}}S({{\mathcal {A}}}(T)_r,{{\mathcal {P}}}_{Q_2}^\prime,T^\lambda)\\ &\leqslant \left(\left(G(\beta/\lambda)+O((\log T)^{-\frac{1}{5}})\right)T^M\prod_{\substack{p:p<T^{\lambda}\\ p\in {{\mathcal {P}}}_{Q_2}^\prime}}\left(1-\frac{\varrho(p)}{p}\right)\right)\left(\sum_{\substack{r=\square\\ p\mid r\Rightarrow p\in \mathcal{P}_{Q_2}}}c_r\right) +O_\varepsilon\left(\sum_{\substack{r,d\in{{\mathbb {N}}}_{\neq 0}\\ r<T^\gamma,d<T^{\beta }}} (dr)^{M-1+\varepsilon}T^{M-1}\right)\\ &\asymp_{\beta,\lambda} \frac{T^M}{\sqrt{(\log T)}}+O\left(\frac{T^M}{(\log T)^{\frac{1}{2}+\frac{1}{5}}}\right)+O_\varepsilon(T^{M-1+(\gamma+\beta) M+\varepsilon}). \end{split}$$ On taking $\gamma,\beta>0$ small enough, and on taking into account, we finish the proof of Theorem \[thm:halfdimsieve\]. It would be interesting to ask whether a lower bound of expected magnitude $\frac{T^M}{\sqrt{\log T}}$ exists for , just as was established in [@Iwaniec-Friedlander Theorem 1] for the case $M=1$ under some mild assumptions. ### Sketch of proof of Theorem \[thm:halfdimsieve\] for the case $M=1$ {#se:n3resp} In this case it is equivalent to showing: For $b_1,b_2\in{{\mathbb {Z}}}_{\neq 0}$, assume that $D_{Q_2}b_1b_2\neq\square$. Then $$\#\{x\in {{\mathbb {Z}}}:|x|\leqslant T:\exists u,v\in{{\mathbb {Z}}},b_1x^2+b_2=Q_2(u,v)\}\ll \frac{T}{\sqrt{\log T}}.$$ Most reasoning (especially the treatment of error terms) is akin to the one in §\[se:n4resp\]. We choose to only outline how the dominant term comes out, which is the major difference between these two cases. Let $\mathfrak{D}$ be the square-free fundamental discriminant of the polynomial $b_1x^2+b_2$. We may assume that $b_1>0$, as the case $b_1<0$ is trivial the set above has finite cardinality. Let us keep the notations defined in §\[se:n4resp\]. Then for any $p\nmid 2b_1b_2$, $$\varrho(p)=\begin{cases} 2 &\text{ if } \left(\frac{\mathfrak{D}}{p}\right)=1;\\ 0 &\text{ otherwise}. \end{cases}$$ As we assume $D_{Q_2}b_1b_2\neq\square$, then the Legendre symbols of $D_{Q_2}$ and $\mathfrak{D}$ are “independent”. More precisely, let $D_0=\prod_{p\mid D_{Q_2}\mathfrak{D}}p$, then if we break all primes into residues modulo $D_0$, then exactly $\phi(D_0)/4$ (where $\phi$ is the Euler totient function), that is one quarter of the residue classes verify $$\left(\frac{D_{Q_2}}{p}\right)=-\left(\frac{\mathfrak{D}}{p}\right)=-1.$$ So by Mertens’ theorem regarding primes in arithmetic progressions, one deduces that $$\sum_{\substack{p:p\in{{\mathcal {P}}}_{Q_2}\\ p<z}}\frac{\varrho(p)}{p-\varrho(p)}\log p\sim \sum_{\substack{p:\left(\frac{D}{p}\right)=-\left(\frac{\mathfrak{D}}{p}\right)=-1\\ p<z}}\frac{2\log p}{p}\sim 2\times \frac{1}{\phi(D_0)}\frac{\phi(D_0)}{4}\log z=\frac{1}{2}\log z,$$ thereby verifying the condition in the half-dimensional sieve (Theorem \[thm:I-F\]). So the dominant term in takes the desired form: $$T\prod_{\substack{p\in{{\mathcal {P}}}_{Q_2}\\p<T^\lambda}}\left(1-\frac{\varrho(p)}{p}\right)=T\prod_{\substack{p:\left(\frac{D_{Q_2}}{p}\right)=-\left(\frac{\mathfrak{D}}{p}\right)=1\\p<T^\lambda}}\left(1-\frac{2}{p}\right)\asymp \frac{T}{\sqrt{\log T}}.$$ To further clarify Remark \[rmk:notsquare\] at this point, note that if $D_{Q_2}b_1b_2=\square$, which is equivalent to $D_{Q_2}\mathfrak{D}=\square$, then $$\left(\frac{D_{Q_2}}{p}\right)=-1\Leftrightarrow \left(\frac{\mathfrak{D}}{p}\right)=-1.$$ The main term in the sifting function now goes like $$T\prod_{\substack{p\in{{\mathcal {P}}}_{Q_2}\\p<T^\lambda}}\left(1-\frac{\varrho(p)}{p}\right)=T\prod_{\substack{p\in{{\mathcal {P}}}_{Q_2}\\p<T^\lambda}}\left(1-\frac{2}{p}\right)\asymp T,$$ which does not give the log saving. Dénouement {#se:final} ========== We are finally in a position to gather together fragmentary estimates obtained in preceding sections to prove our main theorems. #### **Proof of Theorems \[thm:mainthmgcd=1\] and \[th:mainthm\]** Keeping the notations in Theorems \[thm:maintermsmallp\], \[thm:primepoly\], \[th:intermediateprimes\], \[th:largeprimes\], we begin by choosing appropriate parameters to make the error term satisfactory. Let $\alpha>0$ satisfy $$0<\alpha<\frac{\delta_{V_m}(n-2)}{\frac{(3n-2)(n-1)}{2}+n-2}.$$ Fix $\varepsilon=1$ in the error term $\operatorname{Er}(T,M)$ in Theorem \[thm:maintermsmallp\]. We want take sufficiently large $M$ satisfying $$M\log 2+\left(\frac{(3n-2)(n-1)}{2}+n-3+\varepsilon\right)\sum_{p<M}\log p<\frac{\alpha}{2} \log T,$$ so that $$2^{\Omega(\mathfrak{P}_{M,S})}\mathfrak{P}_{M,S}^{\frac{(3n-2)(n-1)}{2}+n-2}< T^\frac{\alpha}{2}.$$ It suffices to take $$M\asymp_{\alpha}\log T,$$ since (c.f. [@Axler Proposition 10]) $$\sum_{p<X} \log p=O(X),$$ and $$\Omega(\mathfrak{P}_{M,S})\sim \frac{M}{\log M}<M.$$ Therefore $$\operatorname{Er}(T,M)=O_\alpha\left(T^{n-2-\frac{\alpha}{2}}+\frac{T^{n-2}}{\log T}\right).$$ This is satisfactory, as the choice of $\alpha$ depends only on $V_m$. We thus have, with the choice $N_1=N_1(T)=M<N_2=N_2(T)=T^\alpha$, $$\begin{aligned} N_U^S(T)&=N_U^{S,M}(T)+O\left(V(T;M,T^\alpha)+V(T;T^\alpha, T)+V(T; T,\infty)\right)\\ &= \tau_{\infty}(V_m,T)\left(\prod_{p\in S} \hat{\tau}_p(\xi_p;V_m)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(Z;V_m)\right)\\ &\quad\quad+O\left(\operatorname{Er}(T;M)+\frac{T^{n-2}}{M}+\frac{T^{n-2}}{\log T}+\frac{T^{n-2}}{(\log T)^{\frac{1}{2}}}\right)\\ &=\tau_{\infty}(V_m,T)\left(\prod_{p\in S} \hat{\tau}_p(\xi_p;V_m)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(Z;V_m)\right)+O\left(\frac{T^{n-2}}{\sqrt{\log T}}\right). \end{aligned}$$ We next recover the infinite product of local factors in Theorem \[thm:maintermsmallp\]. A comparison of leading terms from and Theorem \[thm:maintermsmallp\] gives $$\begin{aligned} &\tau_{\infty}(V_m,T)\left(\prod_{p\in S} \hat{\tau}_p(\xi_p;V_m)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(Z;V_m)\right)\\ \sim & \frac{N_{V_m}(T)}{\prod_{p<\infty} \hat{\tau}_p(V_m)}\left(\prod_{p\in S} \hat{\tau}_p(\xi_p;V_m)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(Z;V_m)\right)\\ =&N_{V_m}(T)\left(\prod_{p\in S}\frac{ \hat{\tau}_p(\xi_p;V_m)}{ \hat{\tau}_p(V_m)}\right)\left(\prod_{p\not \in S,p<\infty}\frac{\hat{\tau}_p(Z;V_m)}{ \hat{\tau}_p(V_m)}\right),\end{aligned}$$ because are infinite products are absolutely convergent by Lemma \[le:infprod1\]. (Recall that we always assume that $V_m({{\mathbb {Z}}})\neq\varnothing$, so all the denominators above are non-zero.) So the proof of Theorem \[th:mainthm\] is thus achieved. #### **Proof of Theorem \[thm:mainthmn=3\]** With exactly the same argument as before, we appeal correspondingly to Theorems \[thm:maintermsmallporbit\], \[thm:primepoly\], \[th:intermediateprimes\] and \[th:largeprimes\] to get an asymptotic formula for $N_U^S(T)$. The verification for the leading term in Theorem \[thm:mainthmn=3\] is similar. Indeed, it suffices to care about every single orbit $\mathfrak{O}_{{\mathbf{A}}}$ satisfying (\*) and $\delta({\mathfrak{O}_{{\mathbf{A}}}})>0$. In particular $\hat{\tau}({\mathfrak{O}_{{\mathbf{A}}}})\neq 0$ for any prime $p$. Then substituting the main term of into that of Theorem \[thm:maintermsmallporbit\] gives $$\begin{aligned} &\tau_{\infty}(\mathfrak{O}_{{\mathbf{A}}},T)\left(\prod_{p\in S} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)\right)\left(\prod_{p\not \in S,p<\infty} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)\right)\\ \sim & N_{{\mathfrak{O}_{{\mathbf{A}}}}}(T)\frac{L(1,\varrho_H)\delta(\mathfrak{O}_{{\mathbf{A}}})\left(\prod_{p\in S}L_p(1,\varrho_H)^{-1} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)\right)\left(\prod_{p\not \in S,p<\infty} L_p(1,\varrho_H)^{-1}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)\right)}{L(1,\varrho_H)\delta(\mathfrak{O}_{{\mathbf{A}}})\left(\prod_{p<\infty}L_p(1,\varrho_H)^{-1}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})\right)}\\ =&N_{{\mathfrak{O}_{{\mathbf{A}}}}}(T)\left(\prod_{p\in S}\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}\right)\left(\prod_{p\not \in S,p<\infty}\frac{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)}{\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}})}\right).\end{aligned}$$ Again because the infinite products in Lemma \[le:infprod2\] and in are absolutely convergent. To see that $p$-adic local factors are the expected ones, we recall the discussion in §\[se:quadricarith\] that $V_m({{\mathbb {Z}}}_p)\subset \mathfrak{O}_p$ for almost all $p$ and holds. So we are reduced to the previous proof. Acknowledgment {#acknowledgment .unnumbered} ============== The first author is supported by a Humboldt-Forschungsstipendiaten. The second author is supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft. We are in debt to Tim Browning for an invitation to IST Austria. The enlightening discussion with him yields overall improvements of a preliminary version of this paper. We would like also to address our gratitude to Ulrich Derenthal for his generous support. Part of this work was carried out and reported during a visit to University of Science and Technology of China. We thank Yongqi Liang for offering financial support and warm hospitality. [99]{} C. Axler, *On a family of functions defined over sums of primes*. J. 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--- abstract: 'We study the location and field distribution of zero-energy corner states in a non-Hermitian quadrupole insulator (QI) and discover an unexpected splitting of the parameter space into three distinct regimes: near-Hermitian QI, intermediate phase, and trivial insulator. In the newly discovered intermediate phase, the Hamiltonian becomes defective, and our analysis using Jordan decomposition reveals the existence of a new corner state without a Hermitian counterpart. Resonant excitation of corner states in this region is found to be highly counter-intuitive owing to disparity of field profiles between left Jordan basis states and the corresponding right states: the most efficient excitation corresponds to placing the source as far as possible from the corner state’s location.' author: - Yang Yu - Minwoo Jung - Gennady Shvets bibliography: - 'nhqi.bib' title: 'Zero-energy Corner States in a Non-Hermitian Quadrupole Insulator' --- *Introduction.*—Higher-order topological insulators (HOTIs) are characterized by exotic topological signatures with dimensionality that is lower by at least two than that of the protecting bulk. One such signature is fractionally quantized corner charges in two-dimensional (2D) crystals with $C_n$ symmetry [@benalcazar2019quantization]. In the presence of an additional chiral (sublattice) symmetry, $e/2$ corner charges become associated with mid-gap (“zero-energy") corner-localized states [@benalcazar2019quantization]. Similar fractionalized vortex states can also exist [*inside*]{} a 2D lattice with an appropriate order parameter twists [@hou2007electron]. While the fractional nature of the topological charge is of particular significance for fermionic systems, the localized nature and robust spectral pinning of such corner/vortex states is of great practical importance for bosonic (e.g., acoustic, photonic, and radio-frequency) lattices [@serra2018observation; @peterson2018quantized; @imhof2018topolectrical; @ni2019observation]. Among many types of HOTIs supporting zero-energy corner states, the quadrupole insulator (QI) is a particularly interesting one because its lowest non-vanishing bulk polarization moment is quadrupolar [@benalcazar2017quantized; @benalcazar2017electric], i.e., its dipole polarization moment strictly vanishes. QI is the first type of HOTI to be theoretically predicted [@benalcazar2017quantized] and experimentally implemented [@serra2018observation; @peterson2018quantized]. Non-Hermitian physics also attracted considerable interest in recent years because of its relevance to non-equilibrium (e.g., undergoing photo-ionization) systems [@baker_pra84; @lopata_jctc13]. Some of its notable phenomena include “exceptional points" (EPs) [@heiss2004exceptional; @berry2004physics; @moiseyev2011non] and real-valued spectra despite non-Hermiticity. At the EP, both the complex-valued eigenvalues of two bands as well as their corresponding eigenvectors coalesce [@liang_feng_nphot17; @el2018non]. In other words, the matrix corresponding to the Hamiltonian at the EP becomes [*defective*]{} [@golub2013matrix; @lee2016anomalous]. The completely real spectrum of some non-Hermitian systems can be related to parity-time (PT) symmetry [@bender2007making; @PhysRevLett.104.054102; @PhysRevA.88.062111] or pseudo-Hermiticity [@mostafazadeh2002pseudo], though in general it is hard to assert a real spectrum without directly calculating the eigenvalues. Extending the rich and rapidly growing field of topological physics to non-Hermitian systems has been of great interest [@shen2018topological; @yao2018edge; @liu2019second] because of their relevance to non-equilibrium topological systems [@sobota_prl12; @marsi_pss18; @shen_fu_prl18]. However, some of the earlier obtained results must be reconsidered using the appropriate mathematical formalism and modern computational techniques, and considerable gaps remain in the parameter space studied so far. In this Letter, we concentrate on a non-Hermitian version of a QI model proposed in Ref. [@benalcazar2017quantized]. We pay special attention to the locations and field profiles of the zero-energy corner states, and to exotic behaviors without Hermitian counterparts in some regions of the parameter space when the Hamiltonian becomes defective. We also discuss the excitation of the corner states by external drives. *Tight-Binding Model*—The non-Hermitian QI model studied in this Letter is schematically shown in Fig. \[fig:nhqi\](a), where the intra/inter-cell hopping amplitudes $t\pm\gamma$ and $\lambda$ are all taken to be real. It is a natural non-Hermitian generalization of the QI model described in Ref. [@benalcazar2017quantized], with the intracell hopping strength becoming asymmetric, characterized by a finite $\gamma$, while maintaining the chiral symmetry $\Sigma H\Sigma^{-1}=-H$. Here the chiral operator $\Sigma=P_1-P_2-P_3+P_4$, where $P_j=\sum_{x,y}|x,y,j\rangle\langle x,y,j|$ are the sublattice projection operators, and $|x,y,j\rangle$ are the tight-binding states, where $x$ and $y$ are integer-valued coordinates of the unit cells as defined in Fig. \[fig:nhqi\](a), and $j=1,\dots,4$ denote four sub-lattice sites of each unit cell. This model can also be viewed as a two-dimensional (2D) generalization of the non-Hermitian Su-Schrieffer-Heeger (SSH) model [@lieu2018topological; @yin2018geometrical; @yao2018edge]. An earlier study [@liu2019second] of this 2D non-Hermitian HOTI model did not identify an important parameter regime (the cyan region in Fig. \[fig:nhqi\](b)) and incorrectly reported the numbers and spatial locations of the corner states in other parameter regimes. Below we rigorously resolve these issues using a mathematical technique of “partial Jordan decomposition", which is critical when the Hamiltonian matrix is close to defective. The significance of the defectiveness of the Hamiltonian was raised in the study of edge states in a non-Hermitian linear chain [@lee2016anomalous]. As we demonstrate below, our deceptively simple model supports rich physics with novel non-Hermtian phenomena. ![\[fig:nhqi\] (a) Tight binding model of a non-Hermitian QI on a square lattice. Grey dashed line: boundary of unit cell with four (sublattice) sites (numbered 1 to 4). Red and blue lines with arrows: asymmetric intra-cell hopping amplitudes $\pm t \pm \gamma$, green lines: symmetric inter-cell hopping amplitudes $\pm \lambda$. Dashed lines: negative hopping terms. All four sublattices have the same on-site potentials (set to $\epsilon_j \equiv 0$). (b) The phase diagram of a large non-Hermitian QI with open boundary condition. Green region ($|\lambda| > |t| + |\gamma|$): near-Hermitian regime with $4$ zero-energy corner states, each localized at a separate corner. Cyan region ($\sqrt{|t^2-\gamma^2|} <|\lambda| < |t|+|\gamma|$): intermediate regime with $2$ zero-energy corner states at the top-left corner. White region ($|\lambda| < \sqrt{|t^2-\gamma^2|}$): no corner states. Bandgap vanishes along solid black lines. The spectrum is complex-valued between the two dashed orange lines, real-valued elsewhere.](nH-QI.eps){width="\linewidth"} *Non-Bloch bulk continuum*—As was pointed in the context of the non-Hermitian SSH system [@yao2018edge], the open-boundary spectrum can significantly differ from that of the periodic-boundary system described by the Bloch Hamiltonian $H(\vec{k})$. That is because the usual Bloch phase-shift factor $e^{ik}$ for bulk eigenstates (i.e., eigenstates in the continuum spectrum) of an open-boundary system needs to be modified to $\beta\equiv \beta_0e^{ik}$, where $\beta_0$ can be non-unity (i.e., the wavevector acquires an imaginary part: $k\to k-i\ln \beta_0$). This extra *bulk localization factor* $\beta_0$ must be taken into account when calculating the spectrum of the open-boundary system. The same argument applies to our 2D non-Hermitian QI system, where $\vec{k}\equiv(k_x,k_y)\to(k_x-i\ln \beta_0,k_y-i\ln \beta_0)$, and $\beta_0=\sqrt{|(t-\gamma)/(t+\gamma)|}$[@liu2019second]. With this substitution, the corrected Bloch Hamiltonian shows (see the Supplemental Material) agreement with numerical simulations of an open-boundary system, that a finite bulk bandgap exists for all values of the hopping amplitudes except at $t^2 = \gamma^2 \pm \lambda^2$. The zero-gap condition is represented in Fig. \[fig:nhqi\](b) by the solid black lines. Another important consequence of this extra factor $\beta_0$ is that the bulk spectrum is real-valued for $|t| > |\gamma|$. While there are also edge and corner states, our numerical results show that the entire spectrum is real for arrays of any size whenever $|t| > |\gamma|$. This fact can be related to the pseudo-Hermiticity of the Hamiltonian [@liu2019second]. *Zero-energy corner states.*—Having established the bulk properties of non-Hermitian QIs, we now proceed with investigating the existence conditions and spatial properties of zero-energy corner states supported by a large ($N\times N$ array, $N\gg1$) non-Hermitian QI with open boundary conditions. In what follows, we focus on the systems with entirely real-valued spectrum: $t > \gamma >0$ and $\lambda > 0$. When the inter-cell hopping strength dominates over the intra-cell one, i.e. $\lambda > t + \gamma$, it can be shown that the four corner states identified in Hermitian QIs [@benalcazar2017quantized] still persist in the thermodynamic limit $N\gg1$ (where the coupling between different corners of the domain is negligible), albeit with modified field distributions: $$\begin{aligned} &|\psi_1\rangle=\sum_{x,y}(-\frac{t-\gamma}{\lambda})^{x+y}|x,y,1\rangle,\label{eq:supra}\\ &|\psi_2\rangle=\sum_{x,y}(-\frac{t+\gamma}{\lambda})^{-x}(-\frac{t-\gamma}{\lambda})^y|x,y,2\rangle,\\ &|\psi_3\rangle=\sum_{x,y}(-\frac{t-\gamma}{\lambda})^{x}(-\frac{t+\gamma}{\lambda})^{-y}|x,y,3\rangle,\\ &|\psi_4\rangle=\sum_{x,y}(-\frac{t+\gamma}{\lambda})^{-x-y}|x,y,4\rangle.\end{aligned}$$ We verify $H\psi_i\approx0$ in the thermodynamic limit in the Supplemental Material. Just as in the case of a Hermitian QI, each corner state is localized at one corner of the array, and has support on only one sublattice. The asymmetric intracell coupling is the reason for the different states to have different spatial localization lengths, and for those lengths to be different in the $x$ and $y$ directions. Therefore, we refer to this parameter regime as “near-Hermitian". Figure \[fig:sp1\] presents the field distributions of the four corner states (see Fig. S1(a) for the full spectrum). ![\[fig:sp1\] Field distribution of the four zero-energy (mid-gap) corner states of a large square domain of a non-Hermitian QI in the “near-Hermitian" regime $t=0.6, \gamma=0.4, \lambda=1.5$. Domain size: $20 \times 20$ unit cells, near-identical on-site potentials: $\epsilon_j = 10^{-3} \times j$.](sp1.eps){width="\linewidth"} An earlier work has incorrectly concluded that all four corner eigenstates are localized in the upper-left corner [@liu2019second] as shown in Fig. \[fig:sp1\](a). The reason for this numerical artifact is the finite (albeit exponentially small in the system size) coupling between different corner states. This coupling is asymmetric because of the non-Hermiticity of the Hamiltonian, resulting in one of the corner states dominating the others in the coupled eigenstates. This artifact can be overcome by adding a small (but larger than the exponential coupling) energy offset (“on-site potential") to one sublattice with respect to the others. An on-site potential of order $\epsilon_j \sim 10^{-3}$ is used in obtaining Fig. \[fig:sp1\]. As we enter the *intermediate* regime $\sqrt{t^2-\gamma^2} <\lambda < t+\gamma$ range, see Fig. \[fig:nhqi\](b), only the first of the above four corner states survives, see Fig. \[fig:sp2\](a). Additionally, a new corner state – also localized at the top-left corner, but having support on two ($2$ and $3$) sublattices – emerges. It has the following field distribution: $$\label{eq:sub} |\phi\rangle=\sum_{x,y}(r_1^x-r_2^x)r_1^y(|x,y,2\rangle-|y,x,3\rangle),$$ where $r_1=-(t-\gamma)/\lambda, r_2=-\lambda/(t+\gamma)$, see Fig. \[fig:sp2\](b). We verify $H\phi\approx0$ in the thermodynamic limit in the Supplemental Material. We refer to the surviving $\psi_1$ as “mono-sublattice", and the less localized (since $|r_2|>|r_1|$) $\phi$ as “multi-sublattice". This contrast of localization length is evident in Fig. \[fig:sp2\] (see Fig. S1(b) for the full spectrum). Both states are corner states since they have a different localization factor than that of bulk states, $\beta_0$. The change of the location of corner states has been observed in the non-Hermitian SSH model as well [@yao2018edge]. Although the numerical eigenvalue calculation shows zero eigenenergy of multiplicity four, these two corner states are the only two linearly independent eigenstates. This implies that the Hamiltonian is defective at zero energy – a common feature of non-Hermitian systems [@lee2016anomalous]. Remarkably, the Hamiltonian is not defective at zero energy in the near-Hermitian regime. Thus, the transition between these two regimes is not induced via a bulk bandgap closure. ![\[fig:sp2\] Corner states of a non-Hermitian QI in the intermediate regime $t=0.6, \gamma=0.4, \lambda=0.7$. (a) Field distribution of the mono-sublattice state: similar to Fig. \[fig:sp1\](a). (b) The emerging multi-sublattice state: also localized at the top-left corner, but supported on the sub-lattices $2$ and $3$. Domain size: $20 \times 20$ unit cells.](sp2.eps){width="\linewidth"} When the inter-cell hopping amplitude is further reduced to $\lambda < \sqrt{t^2-\gamma^2}$, zero-energy corner states disappear (trivial regime). The three regimes of a square finite-sized non-Hermitian QI with open boundary conditions are summarized by a phase diagram shown in Fig. \[fig:nhqi\](b). Only trivial and near-Hermitian regimes have been previously identified [@liu2019second]. Below we demonstrate that the neglected intermediate regime exhibits highly counter-intuitive behaviors such as non-local excitation and unidirectional amplification of corner states. *Excitation of corner states.*—Having classified the number and properties of zero-energy corner states in 2D non-Hermitian QI, we now discuss how to observe them. In bosonic systems, a (periodic) drive corresponds to adding a source term $\xi$ to the equation of motion of the system: $i{d\psi}/{dt}=H\psi+\xi$. Because the spectrum of the system is purely real when $|t|>|\gamma|$, adding an overall small loss to the system ensures that all transients eventually decay. Therefore, only the driven equation $(E-H)\psi=\xi$ needs to be solved, where $E$ is the driving frequency. If $H$ is not defective ($E_n$’s are eigenvalues), one can still obtain an expression similar to the one in the Hermitian case: $(E-H)^{-1} = \sum_n|\eta_n^R\rangle \langle \eta_n^L|/(E-E_n)$, provided that the left and right eigenvectors of $H$ are normalized according to the bi-orthogonality condition: $\langle\eta_m^L|\eta_n^R\rangle = \delta_{mn}$ [@datta2016matrix]. Not surprisingly, in the near-Hermitian regime, the most efficient excitation of a corner state occurs when the source is localized in the same corner (see the Supplemental Material). This behavior is expected based on our intuition derived from the property of the eigenstates $\langle x|\eta_n^R\rangle = \langle \eta_n^L |x\rangle^{\ast}$ of the fully-Hermitian systems [@serra2018observation; @peterson2018quantized; @imhof2018topolectrical; @ni2019observation]. The situation changes dramatically when the Hamiltonian matrix $H$ becomes defective, as is the case in the intermediate regime of our non-Hermitian QI. First, we present the results of driven simulations with localized sources, and then interpret the results based on the spectral properties of defective matrices. The responses of the system introduced in Fig. \[fig:sp2\] (see the caption for the lattice parameters) to external sources localized at different sub-lattice sites are shown in Fig. \[fig:resp\](a-c). For this numerical study we have chosen $E=0.01i$, a small uniform on-site loss. Surprisingly, our simulations reveal that placing the source at the bottom-right corner gives the strongest excitation of the top-left corner states. This contradicts our intuition developed by studying Hermitian systems, where one finds it most efficient to place the source in close proximity of the targeted state’s maximum. This contradiction is resolved by the noted difference between the left and right eigenstates of a non-Hermitian systems. Moreover, we find that the mono-sublattice state is predominantly excited by placing the source on the sublattices $1$, $2$, or $3$. On the other hand, the multi-sublattice state is predominantly excited when the source is on the sublattice $4$. Finally, the response in the intermediate regime is much larger compared to that of near-Hermitian regime (at least $4$ orders of magnitude: compare Fig. S2 and Fig. \[fig:resp\]). ![\[fig:resp\] The response of a non-Hermitian QI in the intermediate regime to external sources placed at different sub-lattice sites in the lower-right corner of the domain. The source sublattice sites are $1$ (left), $2$ (middle), and $4$ (right). Color: magnitude of the complex field $\psi$. mono-sublattice (left and middle) and multi-sublattice (right) corner states are predominantly excited (cf. Fig. \[fig:sp2\]). Source frequency: $E=0.01i$, a small uniform on-site loss. Other lattice parameters (domain size and hopping amplitudes) of the tight-binding model: same as in Fig. \[fig:sp2\].](response.eps){width="\linewidth"} *Partial Jordan decomposition of Hamilton*—In this section we explain why the system has such a non-local response in the intermediate regime, i.e. the source and the excited zero-energy state must be counter-located for most efficient excitation. We also prove that there are only two linearly independent corner states in the intermediate regime. Since we already know that the Hamiltonian matrix might be defective (or, in the case of a numerical solution, nearly-defective), we need to examine its Jordan decomposition $H=PJP^{-1}$ (where $J$ is no longer diagonal) instead of its eigenvalue decomposition. Even though the complete Jordan decomposition might be hard to obtain in general, we simplify the problem by focusing on the Jordan blocks for $E=0$ that are relevant to zero-energy corner states. Numerical results for Fig. \[fig:sp2\] show that $E=0$ is an eigenvalue of $H$ with algebraic multiplicity $4$. From the experience of obtaining Eq. (\[eq:sub\]), it is not too difficult to see that the following four vectors can serve as the four columns of the Jordan basis matrix $P$ corresponding to the $E=0$ Jordan blocks (these states can also be obtained numerically using the Schur decomposition [@golub2013matrix]): $$\begin{aligned} &|\eta_1^R\rangle=\sum_{x,y}r_1^{x+y}|x,y,1\rangle,\nonumber\\ &|\eta_2^R\rangle=\sum_{x,y}(r_1^x-r_2^x)r_1^y(|x,y,2\rangle+|y,x,3\rangle),\nonumber\\ &|\eta_3^R\rangle=\sum_{x,y}(r_1^x-r_2^x)r_1^y(|x,y,2\rangle-|y,x,3\rangle),\nonumber\\ &|\eta_4^R\rangle=\sum_{x,y}(r_1^x-r_2^x)(r_1^y-r_2^y)|x,y,4\rangle,\end{aligned}$$ and the Jordan blocks for $E=0$ are $$J_0= \begin{pmatrix} 0 & 2\kappa & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \kappa \\ 0 & 0 & 0 & 0 \end{pmatrix},$$ where $\kappa=t+\gamma-\lambda^2/(t-\gamma)$. We observe from $J_0$ that the geometric multiplicity of the $E=0$ eigenvalue is $2$, indicating that the $E=0$ subspace is defective. Note that $|\eta_1^R\rangle$ is the mono-sublattice state given by Eq. , and $|\eta_3^R\rangle$ is the multi-sublattice state given by Eq. . Next, the corresponding four rows of $P^{-1}$ must be determined. This can be done by repeating the above analysis for $H^T$. It turns out they are localized at the bottom-right corner: $$\begin{aligned} &\langle{\eta}_1^L|=A_1\sum_{x,y}(r_1^{\bar{x}}-r_2^{\bar{x}})(r_1^{\bar{y}}-r_2^{\bar{y}})\langle x,y,1|,\nonumber\\ &\langle{\eta}_2^L|=A_2\sum_{x,y}r_1^{\bar{x}}(r_1^{\bar{y}}-r_2^{\bar{y}})(\langle x,y,2|+\langle y,x,3|),\nonumber\\ &\langle{\eta}_3^L|=A_3\sum_{x,y}r_1^{\bar{x}}(r_1^{\bar{y}}-r_2^{\bar{y}})(\langle x,y,2|-\langle y,x,3|),\nonumber\\ &\langle{\eta}_4^L|=A_4\sum_{x,y}r_1^{\bar{x}+\bar{y}}\langle x,y,4|, \end{aligned}$$ where $\bar{x}=N+1-x,\bar{y}=N+1-y$. It can be directly verified that $\langle{\eta}_m^L|{\eta}_n^R\rangle=0$ for $m\neq n$ as required. Normalization constants $A_n\sim r_2^{-2N}$ so that $\langle{\eta}_n^L|{\eta}_n^R\rangle=1$. The normalization constants are huge simply because left and right states are both well-localized and spatially far away. Now we are ready to calculate the driven response of the Hamiltonian, or equivalently, the Green’s function of the system near zero energy. The benefit of finding the Jordan normal decomposition is that in order to solve the driven equation $(E-H)\psi=\xi$, we instead need to solve a much simpler equation $(E-J)\psi'=\xi'$, where $\psi'=P^{-1}\psi, \xi'=P^{-1}\xi$, and $(E-J)^{-1}$ is easy to compute. To understand the behavior of $H$ near $E=0$, we only need to work in the above mentioned four-dimensional subspace because only the vectors in this subspace can diverge as $1/E$ or faster. Therefore, below we appropriate the notations $\xi'$ and $\psi'$ to just represent the four dimensional vectors. As mentioned, $(E-J_0)^{-1}$ is easy to compute: $$(E-J_0)^{-1}=\begin{pmatrix} 1/E & 2\kappa/E^2 & 0 & 0 \\ 0 & 1/E& 0 & 0 \\ 0 & 0 & 1/E & \kappa/E^2 \\ 0 & 0 & 0 & 1/E \end{pmatrix}$$ Because of the form of ${\eta}_n^L$, placing a source on sublattice $1$ gives $\xi'\propto(1,0,0,0)^T$. By calculating $\psi'=(E-J_0)^{-1}\xi'$ we see that the mono-sublattice state is excited. Likewise, placing a source on sublattice $4$ induces $\xi'\propto(0,0,0,1)^T$, so the multi-sublattice state is excited. Note that placing a source on either sublattice $2$ or $3$ induces $\xi'\propto(0,1,\pm1,0)^T$, but the mono-sublattice state still dominates due to its faster divergence rate $1/E^2$. This is clearly observed in Fig. \[fig:resp\], where the response to the sources placed on sublattices $2$ and $4$ (middle and left figures) is stronger than that to the source placed on sublattice $1$ (left figure). Remarkably, placing the source as far away as possible from the corner states leads to stronger excitation of the latter because the localization of the ${\eta}_n^L$ at the bottom-right corner maximizes the overlap. The huge amplitude of the response $\psi$ (see Fig. \[fig:resp\]) is mainly due to the exponentially large normalization constant $A_n$. Such non-local response in the intermediate regime presents a remarkable opportunity for [*unidirectional amplification*]{} of corner states. Specifically, placing a source at the bottom-right corner will lead to huge response at the top-left corner, but a source at the top-left corner will only lead to weak response throughout the system in comparison. Compared to the response of an isolated site to the same source, whose amplitude would simply be $|1/E|$, the amplitude of the response of an array is amplified by roughly $|A_n|$ (Fig. \[fig:resp\](a)) or $|A_n\kappa/E|$ (Fig. \[fig:resp\](b-c)). Such behavior is absent in the near-Hermitian regime (see the Supplemental Material for demonstration). While this has not been previously recognized, unidirectional amplification can also be realized for the non-Hermitian SSH model because the latter possesses a similarly defined intermediate regime. An important advantage of the non-Hermitian QI is that we can selectively excite two distinct corner states, whereas only one edge state is supported by a 1D chain in the intermediate regime of the non-Hermitian SSH model. *Conclusions.*—A non-Hermitian quadrupole insulator with asymmetric intracell coupling strengths has been investigated, with the focus on zero-energy corner states it supports. We identified a previously unknown “intermediate regime" in the parameter space, where a new type of a corner state without counterpart in Hermitian QIs exists. The peculiarity of this regime arises from the defective nature of its Hamiltonian matrix at zero energy. We also used partial Jordan decomposition of the Hamiltonian matrix to explain the response of the system to external sources. The techniques used in this Letter are applicable to other non-Hermitian systems. This work was supported by the Office of Naval Research (ONR) under Grant No. N00014-17-1-2161, by the National Science Foundation (NSF) under Grant No. DMR-1741788, and by the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1719875). M. J. was supported in part by Kwanjeong Educational Foundation.

--- abstract: 'In this paper, we show that SVRG and SARAH can be modified to be fundamentally faster than all of the other standard algorithms that minimize the sum of $n$ smooth functions, such as SAGA, SAG, SDCA, and SDCA without duality. Most finite sum algorithms follow what we call the “span assumption”: Their updates are in the span of a sequence of component gradients chosen in a random IID fashion. In the big data regime, where the condition number $\kappa=\mathcal{O}(n)$, the span assumption prevents algorithms from converging to an approximate solution of accuracy $\epsilon$ in less than $n\ln(1/\epsilon)$ iterations. SVRG and SARAH do not follow the span assumption since they are updated with a hybrid of full-gradient and component-gradient information. We show that because of this, they can be up to $\Omega(1+(\ln(n/\kappa))_+)$ times faster. In particular, to obtain an accuracy $\epsilon = 1/n^\alpha$ for $\kappa=n^\beta$ and $\alpha,\beta\in(0,1)$, modified SVRG requires $\mathcal{O}(n)$ iterations, whereas algorithms that follow the span assumption require $\mathcal{O}(n\ln(n))$ iterations. Moreover, we present lower bound results that show this speedup is optimal, and provide analysis to help explain why this speedup exists. With the understanding that the span assumption is a point of weakness of finite sum algorithms, future work may purposefully exploit this to yield even faster algorithms in the big data regime.' author: - 'Robert Hannah[^1]' - 'Yanli Liu[^2]' - 'Daniel O’Connor[^3]' - 'Wotao Yin[^4]' bibliography: - 'Master\_Bibliography.bib' title: 'Breaking the Span Assumption Yields Fast Finite-Sum Minimization' --- Introduction ============ Finite sum minimization is an important class of optimization problem that appears in many applications in machine learning and other areas. We consider the problem of finding an approximation $\hat{x}$ to the minimizer $x^{*}$ of functions $F:\RR^{d}\to\RR$ of the form: $$\begin{aligned} F\p x & =f(x)+\psi(x)=\frac{1}{n}\sum_{i=1}^{n}f_{i}\p x +\psi\p{x}.\label{eq:Average-of-fi}\end{aligned}$$ We assume each function $f_{i}$ is smooth[^5], and possibly nonconvex; $\psi$ is proper, closed, and convex; and the sum $F$ is strongly convex and smooth. It has become well-known that under a variety of assumptions, functions of this form can be minimized much faster with variance reduction (VR) algorithms that specifically exploit the finite-sum structure. When each $f_{i}$ is $\mu$-strongly convex and $L$-smooth, and $\psi=0$, SAGA [@DefazioBachLacoste-Julien2014_saga], SAG [@RouxSchmidtBach2012_stochastic], Finito/Miso [@DefazioDomkeCaetano2014_finito; @Mairal2013_optimization], SVRG [@JohnsonZhang2013_accelerating], SARAH [@NguyenLiuScheinbergTakac2017_sarah], SDCA [@Shalev-ShwartzZhang2013_stochastic], and SDCA without duality [@Shalev-Shwartz2016_sdca] can find a vector $\hat{x}$ with expected suboptimality $\EE\p{f\p{\hat{x}}-f\p{x^{*}}}=\cO\p{{\epsilon}}$ with only $\cO\p{\p{n+L/\mu}\ln\p{1/{\epsilon}}}$ calculations of component gradients $\nabla f_{i}\p x$. This can be up to $n$ times faster than (full) gradient descent, which takes $\cO\p{n L/\mu \ln\p{1/{\epsilon}}}$ gradients. These algorithms exhibit sublinear convergence for non-strongly convex problems[^6]. Various results also exist for nonzero convex $\psi$. Accelerated VR algorithms have also been proposed. Katyusha [@Allen-Zhu2017_katyusha] is a primal-only Nesterov-accelerated VR algorithm that uses only component gradients. It is based on SVRG and has complexity $\cO\p{\p{n+\sqrt{n\kappa}}\ln\p{1/{\epsilon}}}$) for condition number $\kappa$ which is defined as $L/\mu$. In [@Defazio2016_simple], the author devises an accelerated SAGA algorithm that attains the same complexity using component proximal steps. In [@LanZhou2017_optimal], the author devises an accelerated primal-dual VR algorithm. There also exist “catalyst” [@LinMairalHarchaoui2015_universala] accelerated methods [@LinLuXiao2014_accelerateda; @Shalev-ShwartzZhang2016_accelerated]. However, catalyst methods appear to have a logarithmic complexity penalty over Nesterov-accelerated methods. In [@LanZhou2017_optimal], authors show that a class of algorithms that includes SAGA, SAG, Finito (with replacement), Miso, SDCA without duality, etc. have complexity $K(\epsilon)$ lower bounded by $\Omega\p{\p{n+\sqrt{n\kappa}}\ln\p{1/{\epsilon}}}$ for problem dimension $d\geq 2K(\epsilon)$. More precisely, the lower bound applies to algorithms that satisfy what we will call the **span condition**. That is $$\begin{aligned} x^{k+1} & \in x^{0}+\text{span}\cp{\nabla f_{i_{0}}\p{x^{0}},\nabla f_{i_{1}}\p{x^{1}},\ldots,\nabla f_{i_{k}}\p{x^{k}}}\label{eq:SpanCondition}\end{aligned}$$ for some fixed IID random variable $i_k$ over the indices $\cp{1,\ldots,n}$. Later, [@WoodworthSrebro2016_tight] and [@ArjevaniShamir2016_dimensionfreea] extend lower bound results to algorithms that do not follow the span assumption: SDCA, SVRG, SARAH, accelerated SAGA, etc.; but with a smaller lower bound of $\Omega\p{n+\sqrt{n\kappa}\ln\p{1/{\epsilon}}}$. The difference in these two expressions was thought to be a proof artifact that would later be fixed. However we show a surprising result in Section \[sec:OptimalSVRG\], that SVRG, and SARAH can be fundamentally faster than methods that satisfy the span assumption, with the full gradient steps playing a critical role in their speedup. More precisely, for $\kappa=\cO\p n$, SVRG and SARAH can be modified to reach an accuracy of ${\epsilon}$ in $\cO((\frac{n}{1+(\ln\p{n/\kappa})_+} )\ln\p{1/{\epsilon}})$ gradient calculations[^7], instead of the $\Theta(n\ln(1/\epsilon))$ iterations required for algorithms that follow the span condition. We also improve the lower bound of [@ArjevaniShamir2016_dimensionfreea] to $\Omega(n+(\frac{n}{1+\p{\ln\p{n/\kappa}}_+}+\sqrt{n\kappa})\ln\p{1/{\epsilon}})$ in Section \[sec:Optimality\]. That is, the complexity $K(\epsilon)$ of a very general class of algorithm that includes all of the above satisfies the lower bound: $$\begin{aligned} K\p{\epsilon} & =\begin{cases} \Omega(n+\sqrt{n\kappa}\ln\p{1/{\epsilon}}), &\text{ for } n=\cO\p{\kappa},\\ \Omega(n+\frac{n}{1+\p{\ln\p{n/\kappa}}_+}\ln\p{1/{\epsilon}}), &\text{ for } \kappa=\cO\p{n}. \end{cases}\end{aligned}$$ Hence when $\kappa=\cO\p n$ our modified SVRG has optimal complexity, and when $n=\cO\p{\kappa}$, Katyusha is optimal. SDCA doesn’t quite follow the span assumption. Also the dimension $n$ of the dual space on which the algorithm runs is inherently small in comparison to $k$, the number of iterations. We complete the picture using different arguments, by showing that its complexity is greater than $\Omega(n\ln(1/\epsilon))$ in Section \[sec:SDCA\], and hence SDCA doesn’t attain this logarithmic speedup. We leave the analysis of accelerated SAGA and accelerated SDCA to future work. Our results identify a significant obstacle to high performance when $n\gg\kappa$. The speedup that SVRG and SARAH can be modified to attain in this scenario is somewhat accidental since their original purpose was to minimize memory overhead. However, with the knowledge that this assumption is a point of weakness for VR algorithms, future work may more purposefully exploit this to yield better speedups than SVRG and SARAH can currently attain. Though the complexity of SVRG and SARAH can be made optimal to within a constant factor, this factor is somewhat large, and could potentially be reduced substantially. Having $n\gg\kappa$, which has been referred to as the “big data condition”, is rather common: For instance [@RouxSchmidtBach2012_stochastic] remarks that $\kappa=\sqrt{n}$ is a nearly optimal choice for regularization for empirical risk minimization in some scenarios, [@SridharanShalev-shwartzSrebro2009_fast] considers $\kappa=\sqrt{n}$, and [@EbertsSteinwart2011_optimal] considers $\kappa=n^{\beta}$ for $\beta<1$. So for instance, we now have the following corollary (which will follow from Corollary \[Cor:OptimalUpperBound\] ahead): \[cor:nlogn-to-log\] To obtain accuracy $\epsilon = 1/n^\alpha$ for $\kappa=n^\beta$ and $\alpha,\beta\in\p{0,1}$, modified SVRG requires $\cO\p{n}$ iterations, whereas algorithms that follow the span assumption require $\cO\p{n\ln\p{n}}$ iterations [@LanZhou2017_optimal] for sufficiently large $d$. For large-scale problems, this $\ln\p n$ factor can be rather large: For instance in the KDD Cup 2012 dataset ($n=149,639,105$ and $\ln\p n\approx18$), Criteo’s Terabyte Click Logs ($n=4,195,197,692$ and $\ln\p n\approx22$), etc. Non-public internal company datasets can be far larger, with $n$ potentially larger than $10^{15}$. Hence for large-scale problems in this setting, SVRG, SARAH, and future algorithms designed for the big-data regime can be expected to have much better performance than algorithms following the span condition. We also analyze Prox-SVRG in the case where $f_{i}$ are smooth and potentially nonconvex, but the sum $F$ is strongly convex. We build on the work of [@Allen-Zhu2018_katyusha], which proves state-of-the-art complexity bounds for this setting, and show that we can attain a similar logarithmic speedup without modification. Lower bounds for this context are lacking, so it is unclear if this result can be further improved. Optimal Convex SVRG {#sec:OptimalSVRG} =================== In this section, we show that the Prox-SVRG algorithm proposed in [@XiaoZhang2014_proximal] for problem can be sped up by a factor of $\Omega(1+(\ln(n/\kappa))_+)$ when $\kappa=\cO(n)$. A similar speedup is clearly possible for vanilla SVRG and SARAH, which have similar rate expressions. We then refine the lower bound analysis of [@ArjevaniShamir2016_dimensionfreea] to show that the complexity is optimal[^8] when $\kappa=\cO(n)$. Katyusha is optimal in the other scenario when $n=\cO(\kappa)$ by [@ArjevaniShamir2016_iterationa]. \[assumption 1\] $f_i$ is $L_i-$Lipschitz differentiable for $i=1,2,...,n$. That is, $$\|\nabla f_i(x)-\nabla f_i(y)\|\leq L_i\|x-y\|\quad \text{for all} \,\, x,y\in\mathbb{R}^d.$$ $f$ is $L-$Lipschitz differentiable. $F$ is $\mu-$strongly convex. That is, $$F(y)\geq F(x)+\langle \tilde{\nabla}F(x), y-x\rangle+\frac{\mu}{2}\|y-x\|^2 \quad \text{for all}\,\,x,y\in\mathbb{R}^d\,\,\text{and}\,\, \tilde{\nabla}F(x)\in\partial F(x).$$ \[assumption 2\] $f_i$ is convex for $i=1,2,...,n$; and $\psi$ is proper, closed, and convex. **Input:** $F(x)=\psi(x)+\frac{1}{n}\sum_{i=1}^{n} f_i(x)$, initial vector $x^0$, step size $\eta>0$, number of epochs $K$, probability distribution $P=\cp{p_1,\ldots,p_n}$\ **Output:** vector $x^K$ \[alg\_1\] We make Assumption \[assumption 1\] throughout the paper, and Assumption \[assumption 2\] in this section. Recall the Prox-SVRG algorithm of [@XiaoZhang2014_proximal], which we reproduce in Algorithm \[alg\_1\]. The algorithm is organized into a series of $K$ **epochs** of size $M^k$, where $M^k$ is a geometric random variable with success probability $1/m$. Hence epochs have an expected length of $m$. At the start of each epoch, a snapshot $\mu = \nabla f(x^k)$ of the gradient is taken. Then for $M^k$ steps, a random component gradient $\nabla_{i_t}f(w_t)$ is calculated, for an IID random variable $i_t$ with fixed distribution $P$ given by $\PP[i_t=i]=p_i$. This component gradient is used to calculate an unbiased estimate $\tilde{\nabla}_t$ of the true gradient $\nabla f(w_t)$. Each time, this estimate is then used to perform a proximal-gradient-like step with step size $\eta$. At the end of these $M^k$ steps, a new epoch of size $M^{k+1}$ is started, and the process continues. We first recall a modified Theorem 1 from [@XiaoZhang2014_proximal]. The difference is that in [@XiaoZhang2014_proximal], the authors used a epoch length of $m$, whereas we use a random epoch length $M^k$ with expectation $\EE M^k = m$. The proof and theorem statement only require only trivial modifications to account for this. This modification is only to unify the different version of SVRG in [@XiaoZhang2014_proximal] and [@Allen-Zhu2018_katyusha], and makes no difference to the result. It becomes useful to define the **effective Lipschitz constant** $L_Q = \text{max}_i L_i/(p_in)$, and the **effective condition number** $\kappa_Q=L_Q/\mu$ for this algorithm. These reduce to the standard Lipschitz constant $L$, and the standard condition number $\kappa$ in the standard uniform scenario where $L_i=L, \forall i$, and $P$ is uniform. \[upper complexity\] Let Assumptions \[assumption 1\] and \[assumption 2\] hold. Then Prox-SVRG defined in Algorithm \[alg\_1\] satisfies: $$\begin{aligned} \label{linear convergence} \mathbb{E}[F(x^{k})-F(x^*)]&\leq \rho^{k}[F(x^0)-F(x^*)]\\ \text{ for } \rho&=\frac{1+\mu\eta\p{1+4mL_Q\eta}}{\mu \eta m\p{1-4L_Q\eta}}\label{linear rate}\end{aligned}$$ In previous work, the optimal parameters were not really explored in much detail. In the original paper [@JohnsonZhang2013_accelerating], the author suggest $\eta=0.1/L$, which results in linear convergence rate $1/4\leq \rho \leq 1/2$ for $m\geq50\kappa$. In [@XiaoZhang2014_proximal], authors also suggest $\eta=0.1/L$ for $m=100\kappa$, which yields $\rho\approx 5/6$. However, they observe that $\eta=0.01/L$ works nearly as well. In [@NguyenLiuScheinbergTakac2017_sarah], authors obtain a similar rate expression for SARAH and suggest $\eta=0.5/L$ and $m=4.5\kappa$ which yields $\rho\approx 7/9$. In the following corollary, we propose a choice of $\eta$ and $m$ that leads to an optimal complexity to within a constant factor for $\kappa=\cO (n)$. This result helps explain why the optimal step size observed in prior work appears to be much smaller than the “standard” gradient descent step of $1/L$. \[Cor:OptimalUpperBound\] Let the conditions of Theorem \[upper complexity\] hold, and let $m=n+121 \kappa_Q$, and $\eta=\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}/(2L_Q)$. The Prox-SVRG in Algorithm \[alg\_1\] has convergence rate $\rho \leq\sqrt{\frac{100}{121+(n/\kappa_Q)}}$, and hence it needs: $$\begin{aligned} K(\epsilon)=\cO\p{\p{\frac{n}{1+(\ln(\frac{n}{\kappa_Q}))_+}+\kappa_Q}\ln{\frac{1}{\epsilon}}+n+\kappa_Q} \label{upper complexity of Prox-SVRG}\end{aligned}$$ iterations in expectation to obtain a point $x^{K(\epsilon)}$ such that $\EE\sp{f\p{x^{K(\epsilon)}}-f\p{x^*}}<\epsilon$. This result is proven in Appendix \[App:UpperComplexity\]. The $n+\kappa_Q$ term is needed because we assume that at least one epoch is completed. For $n=\cO(\kappa_Q)$, we have a similar convergence rate ($\rho\approx\frac{10}{11}$) and complexity to algorithms that follow the span assumption. For $n\gg\kappa_Q$, we have a convergence rate $\rho\to 0$, and complexity $\cO\p{\frac{n}{1+(\ln(n/\kappa))}\ln(1/\epsilon}$, which can can be much better than $n\ln(1/\epsilon)$. See also Corollary \[cor:nlogn-to-log\]. In Theorem \[upper complexity\] and Corollary \[Cor:OptimalUpperBound\], the optimal choice of the probability distribution $P=\{p_1,p_2,...,p_n\}$ on $\{1,2,...,n\}$ is $p_i=\frac{L_i}{\sum_{i=1}^n L_j}$ for $i=1,2,...,n$, and $L_Q=\frac{\sum_{i=1}^n L_i}{n}$. Optimality {#sec:Optimality} ---------- The major difference between SAGA, SAG, Miso/Finito, and SDCA without duality, and SVRG and SARAH, is that the former satisfy what we call the **span condition** . SVRG, and SARAH, do not, since they also involve full-gradient steps. We refer to SVRG, and SARAH as **hybrid methods**, since they use full-gradient and partial gradient information to calculate their iterations. We assume for simplicity that $L_i=L$, for all $i$, and that $\psi = 0$. We now present a rewording of Corollary 3 from [@LanZhou2017_optimal]. For every ${\epsilon}$ and randomized algorithm on that follows the span assumption, there are a dimension $d$, and $L$-smooth, $\mu$-strongly convex functions $f_i$ on $\RR^d$ such that the algorithm takes at least $\Omega\p{\p{n+\sqrt{\kappa n}}\ln\p{1/\epsilon}}$ steps to reach sub-optimality $\EE f\p{x^k}-f\p{x^*}<{\epsilon}$. The above algorithms that satisfy the span condition all have known upper complexity bound $\cO\p{\p{n+\kappa}\ln\p{1/{\epsilon}}}$, and hence for $\kappa =\cO\p{n}$ we have a sharp convergence rate. However, it turns out that the span assumption is an obstacle to fast convergence when $n\gg\kappa$ (for sufficiently high dimension). In the following theorem, we improve[^9] the analysis of [@ArjevaniShamir2016_dimensionfreea], to show that the complexity of SVRG obtained in Corollary \[Cor:OptimalUpperBound\] is optimal to within a constant factor without fundamentally different assumptions on the class of algorithms that are allowed. Clearly this also applies to SARAH. The theorem is actually far more general, and applies to a general class of algorithms called $p-$*CLI oblivious* algorithms introduced in [@ArjevaniShamir2016_dimensionfreea]. This class contains all VR algorithms mentioned in this paper. In Appendix \[App:LowerBound\], we give the definition of $p-$CLI oblivious algorithms, as well as the proof of a more general version of Theorem \[lower complexity of SVRG, SARAH\]. \[lower complexity of SVRG, SARAH\] For all $\mu,L$, there exist $L$-smooth, and $\mu$-strongly convex functions $f_i$ such that at least[^10] $$\begin{aligned} K\p{{\epsilon}} &=\tilde{\Omega}\p{\p{\frac{n}{1+(\ln(\frac{n}{\kappa}))_+}+\sqrt{n\kappa}}\ln{\frac{1}{\epsilon}}+n} \label{Lower Bound Oblivious PCLI}\end{aligned}$$ iterations are needed for SVRG or SARAH to obtain expected suboptimality $\EE\sp{f\p{K\p{{\epsilon}}}-f\p{X^*}}<{\epsilon}$. SDCA {#sec:SDCA} ---- To complete the picture, in the following proposition, which we prove in Appendix \[App:lower complexity of SDCA\], we show that SDCA has a complexity lower bound of $\Omega(n\ln(1/{\epsilon}))$, and hence attains no logarithmic speedup. SDCA aims to solve the following problem: $$\min_{x\in \mathbb{R}^d} F(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)= \frac{1}{n}\sum_{i=1}^n \big(\phi_i(x^Ty_i)+\frac{\lambda}{2}\|x\|^2\big),$$ where each $y_i\in\mathbb{R}^d$, $\phi_i:\mathbb{R}\rightarrow \mathbb{R}$ is convex and smooth. It does so with coordinate minimization steps on the corresponding dual problem: $$\min_{\alpha\in \mathbb{R}^n} D(\alpha)\coloneqq \frac{1}{n}\sum_{i=1}^n \phi^*_i(-\alpha_i)+\frac{\lambda}{2}\|\frac{1}{\lambda n}\sum_{i=1}^n\alpha_i y_i\|^2,$$ Here $\phi^*_i(u)\coloneqq \max_z\big(zu-\phi_i(z)\big)$ is the convex conjugate of $\phi_i$. Let $i_k$ be an IID sequence of uniform random variables on $\{1,...,n\}$. SDCA updates a dual point $\alpha^k$, while maintaining a corresponding primal vector $x^k$. SDCA can be written as: $$\begin{aligned} \label{eq:SDCA} \alpha^{k+1}_i &= \begin{cases} \alpha^k_i, &\text{if}\,\,i\neq i_k,\\ \argmin_{z} D(\alpha^k_1,...,\alpha^k_{i-1}, z, \alpha^k_{i+1},...,\alpha^k_{n}), &\text{if}\,\,i=i_k, \end{cases}\\ x^{k+1}&=\frac{1}{n\lambda}\sum_{i=1}^n\alpha^{k+1}_iy_i,\end{aligned}$$ Since SDCA doesn’t follow the span assumption, and the number of iterations $k$ is much greater than the dual problem dimension $n$, different arguments to the ones used in [@LanZhou2017_optimal] must be used. Motivated by the analysis in [@ArjevaniShalev-ShwartzShamir2016_lower], which only proves a lower bound for dual suboptimality, we have the following lower complexity bound, which matches the upper complexity bound given in [@Shalev-ShwartzZhang2013_stochastic] for $\kappa=\cO(n)$. \[lower complexity of SDCA\] For all $\mu,L,n>2$, there exist $n$ functions $f_i$ that are $L-$smooth, and $\mu-$strongly convex such that $$\begin{aligned} K(\epsilon)=\Omega\big(n\ln\frac{1}{\epsilon}\big) \label{lower bound of SDCA}\end{aligned}$$ iterations are needed for SDCA to obtain expected suboptimality $\mathbb{E}[F(K(\epsilon))-F(x^*)]\leq \epsilon$. Why are hybrid methods faster? ============================== In this section, we explain why SVRG and SARAH, which are a hybrid between full-gradient and VR methods, are fundamentally faster than other VR algorithms. We consider the performance of these algorithms on a variation of the adversarial function example from [@LanZhou2017_optimal; @Nesterov2013_introductory]. The key insight is that the span condition makes this adversarial example hard to minimize, but that the full gradient steps of SVRG and SARAH make it easy when $n\gg\kappa$. We conduct the analysis in $\ell^{2}$, for simplicity[^11], since the argument readily applies to $\RR^{d}$. Consider the function introduced in [@Nesterov2013_introductory] that we introduce for the case $n=1$: $$\begin{aligned} \phi\p x & =\frac{L-\sigma}{4}\p{\frac{1}{2}\dotp{x,Ax}-\dotp{e_{1},x}}\text{, for }A=\p{\begin{array}{cccc} 2 & -1\\ -1 & 2 & -1\\ & -1 & 2 & \ddots\\ & & \ddots & \ddots \end{array}}\end{aligned}$$ The function $\phi\p x+\frac{1}{2}\sigma\n x^{2}$ is $L$-smooth and $\sigma$-strongly convex. Its minimizer $x^{*}$ is given by $\p{q_{1},q_{1}^{2},q_{1}^{3},\ldots}$ for $q_{1}=\p{\kappa^{1/2}-1}/\p{\kappa^{1/2}+1}$. We assume that $x^{0}=0$ with no loss in generality. Let $N\p x$ be position of the last nonzero in the vector. E.g. $N\p{0,2,3,0,4,0,0,0,\ldots}=5$. $N\p x$ is a control on how close $x$ can be to the solution. If $N\p x=N$, then clearly: $$\begin{aligned} \n{x-x^{*}}^{2} & \geq\min_{y\text{ s.t. }N\p y=N}\n{y-x^{*}}^{2}=\n{\p{0,\ldots,0,q_{1}^{N+1},q_{1}^{N+2},\ldots}}^{2}=q_{1}^{2N+2}/\p{1-q_{1}^{2}}\end{aligned}$$ Because of the tridiagonal pattern of nonzeros in the Hessian $\nabla_{x}^{2}\p{\phi\p x+\frac{1}{2}\sigma\n x^{2}}\p y=\frac{L-\sigma}{4}A+\sigma I$, the last nonzero $N\p{x^{k}}$ of $x^{k}$ can only increase by $1$ per iteration *by any algorithm that satisfies that span condition* (e.g. gradient descent, accelerated gradient descent, etc.). Hence since we have $N\p{x^{0}}=0,$ we have $\n{x^{k}-x^{*}}^{2}/\n{x^{0}-x^{*}}^{2}\geq q_{1}^{2k}$. For the case $n>1$, let the solution vector $x=\p{x_{1},\ldots,x_{n}}$ be split into $n$ coordinate blocks, and hence define: $$\begin{aligned} f\p x & =\sum_{i=1}^n\big(\phi\p{x_{i}}+\frac{1}{2}\sigma\n x^{2}\big)\\ &=\sum_{i=1}^n\p{\frac{L-\sigma}{4}\p{\frac{1}{2}\dotp{x_{i},Ax_{i}}-\dotp{e_{1},x_{i}}}+\frac{1}{2}\p{\sigma n}\n{x_{i}}^{2}}\nonumber\\ & =\sum_{i=1}^n\p{\frac{\p{L-\sigma+\sigma n}-\sigma n}{4}\p{\frac{1}{2}\dotp{x_{i},Ax_{i}}-\dotp{e_{1},x_{i}}}+\frac{1}{2}\p{\sigma n}\n{x_{i}}^{2}}. \label{eq:Adversarial-f-rep-2}\end{aligned}$$ $f$ is clearly the sum of $n$ convex $L$-smooth functions $\phi\p{x_{i}}+\frac{1}{2}\sigma\n x^{2}$, that are $\sigma$-strongly convex. shows it is $\sigma n$-strongly convex and $L-\sigma+\sigma n$-smooth with respect to coordinate $x_{i}$. Hence the minimizer is given by $x_{i}=\p{q_{n},q_{n}^{2},q_{n}^{3},\ldots}$ for $q_{n}=\p{\p{\frac{\kappa-1}{n}+1}^{1/2}-1}/\p{\p{\frac{\kappa-1}{n}+1}^{1/2}+1}$ for all $i$. Similar to before, $\p{N\p{x_{1}},\ldots,N\p{x_{n}}}$ controls how close $x$ can be to $x^{*}$: $$\begin{aligned} \frac{\n{x-x^{*}}^{2}}{\n{x^{*}}^{2}} & =\frac{\sum_{i=1}^{n}\n{x_{i}-\p{q_{n},q_{n}^{2},\ldots}}^{2}}{nq_{n}^{2}/\p{1-q_{n}^{2}}}\geq\sum_{i=1}^{n}q_{n}^{2N\p{x_{i}}}/n\end{aligned}$$ Let $I_{K,i}$ be the number of times that $i_{k}=i$ for $k=0,1,\ldots,K-1$. For algorithms that satisfy the span assumption, we have $N\p{x_{i}^{k}}\leq I_{k,i}$. If we assume that $i_{k}$ is uniform, then $I_{K,i}$ is a binomial random variable of probability $1/n$ and size $k$. Hence: $$\begin{aligned} \EE\n{x^{k}-x^{*}}^{2}/\n{x^{0}-x^{*}}^{2} & \geq\EE\sum_{i=1}^{n}q_{n}^{2N\p{x_{i}^{k}}}/n\geq\EE\sum_{i=1}^{n}q_{n}^{2I_{k,i}}/n\nonumber\\ & =\EE q_{n}^{2I_{k,i}}=\p{1-n^{-1}\p{1-q_{n}^{2}}}^{k}\label{eq:Binomial-RV}\\ & \geq\p{1-4n^{-1}/\p{\p{\frac{\kappa-1}{n}+1}^{1/2}+1}}^{k}\nonumber\\ & \geq\p{1-2n^{-1}}^{k}\nonumber\end{aligned}$$ for $n\geq\kappa$. the second equality in follows from the factor that $I_{i,k}$ is a binomial random variable. Hence after 1 epoch, $\EE\n{x^{k}-x^{*}}^{2}$ decreases by a factor of at most $\approx e^{2}$, whereas for SVRG it decreases by at least a factor of $\sim\p{n/\kappa}^{1/2}$, which is $\gg e^{2}$ for $n\gg\kappa$. To help understand why, consider trying the above analysis on SVRG for 1 epoch of size $n$. Because of the full-gradient step, we actually have $N\p{w_{i}^{n}}\leq1+I_{n,i}$, and hence: $$\begin{aligned} \EE\n{w^{n}-x^{*}}^{2}/\n{x^{0}-x^{*}}^{2} & \geq\EE\sum_{i=1}^{n}q_{n}^{2\p{I_{n,1}+1}}\geq q_{n}^{2}\p{1-2n^{-1}}^{n} \approx\p{\frac{1}{4}\frac{\kappa-1}{n}}^{2}e^{-2}\end{aligned}$$ Hence attempting the above results in a much smaller lower bound. What it comes down to is that when $n\gg\kappa$, we have $\EE\sum_{i=1}^{n}q_{n}^{2I_{i,k}}/n\gg\EE\sum_{i=1}^{n}q_{n}^{2\EE I_{i,k}}/n$. The interpretation is that for this objective, the progress towards a solution is limited by the component function $f_{i}$ that is minimized the least. The full gradient step ensures that at least some progress is made toward minimizing every $f_{i}$. For algorithms that follow the span assumption, there will invariably be many indices $i$ for which no gradient $\nabla f_{i}$ is calculated, and hence $x_{i}^{k}$ can make no progress towards the minimum. This may be related to the observation that sampling without replacement can often speed up randomized algorithms. However, on the other hand, it is well known that full gradient methods fail to achieve a good convergence rate for other objectives with the same parameters $\mu,L,n$ (e.g. $f\p x=\frac{1}{n}\sum_{i=1}^n\phi\p x+\frac{1}{2}\mu\n x^{2}$). Hence we conclude that it is because SVRG combines both full-gradient and VR elements that it is able to outperform both VR and full-gradient algorithms. Nonconvex Prox-SVRG =================== In this section, we show that when $f_i$ is merely assumed to be $L_i$ smooth and possibly nonconvex, there is also a logarithmic speedup. This is based on the analysis of Prox-SVRG found in [@Allen-Zhu2018_katyusha]. The proof of Theorem \[nonvex upper complexity\] can be found in Appendix \[App:Nonconvex-SVRG\]. \[nonvex upper complexity\] Under Assumption \[assumption 1\], let $x^*=\argmin_x F(x)$, $\overline{L}=(\sum_{i=1}^n\frac{L_i^2}{n^2p_i})^{\frac{1}{2}}$, $\kappa=\frac{L}{\mu}$, and $\eta=\frac{1}{2}\min\{\frac{1}{L}, (\frac{1}{\overline{L}^2m})^{\frac{1}{2}}\}$. Then the Prox-SVRG in Algorithm \[alg\_1\] satisfies: $$\begin{aligned} \label{nonconvex linear convergence} \mathbb{E}[F({x}^{k})-F(x^*)]&\leq \cO(\rho^{k})[F(x^0)-F(x^*)],\\ \text{ for }\rho&=\frac{1}{1+\frac{1}{2}m\eta\mu}\label{nonconvex linear rate}.\end{aligned}$$ Hence for $m=\min\{n,2\}$, in order to obtain an $\epsilon$-optimal solution in terms of function value, the SVRG in Algorithm \[alg\_1\] needs at most $$\begin{aligned} \label{nonconvex K} K=\cO\big((\frac{n}{\ln{(1+\frac{n}{4\kappa})}}+\frac{n}{\ln(1+(\frac{n\mu^2}{4\overline{L}^2})^{1/2})}+\kappa+\sqrt{n}\frac{\Bar{L}}{\mu})\ln\frac{1}{\epsilon}\big)+2n\end{aligned}$$ gradient evaluations in expectation. The complexity of nonconvex SVRG using the original analysis of [@Allen-Zhu2017_katyusha] would have been $$\begin{aligned} K=\cO\big((n+\kappa+\sqrt{n}\frac{\Bar{L}}{\mu})\ln\frac{1}{\epsilon}\big) \end{aligned}$$ Hence we have obtained a similar logarithmic speedup as we obtained in Corollary \[Cor:OptimalUpperBound\]. In Theorem \[nonvex upper complexity\], the optimal choice of the probability distribution $P=\{p_1,p_2,...,p_n\}$ on $\{1,2,...,n\}$ is $p_i=\frac{L_i^2}{\sum_{i=1}^n L_j^2}$ for $i=1,2,...,n$, and $\overline{L}=(\frac{\sum_{i=1}^n L_i^2}{n})^{\frac{1}{2}}$. Upper Complexity Bound for Convex SVRG {#App:UpperComplexity} ====================================== and follows directly from the analysis of [@XiaoZhang2014_proximal Thm 3.1] with slight modification. For the linear rate $\rho$ in , we have $$\begin{aligned} \rho&\overset{(\mathrm{a})}{\leq} 2(\frac{1}{\mu\eta m}+4L_Q\eta+\frac{1}{m})\\ &\overset{(\mathrm{b})}{=} 2(\frac{1}{\mu\eta m}+2\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}})+\frac{2}{m}\\ &\overset{(\mathrm{c})}{=} 2\Big(\frac{1}{\mu m}2L_Q\kappa_Q^{-\frac{1}{2}}m^{\frac{1}{2}}+2\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}\Big)+\frac{2}{m}\\ &=8\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}+\frac{2}{m}\\ &\overset{(\mathrm{d})}{\leq}8\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}+2\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}\\ &=10\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}},\end{aligned}$$ where (a) is by $\eta=\frac{\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}}{2L_Q}\leq\frac{1}{22L_Q}\leq \frac{1}{8L_Q}$, (b) is by $\eta = \frac{\kappa_Q^{\frac{1}{2}}m^{-\frac{1}{2}}}{2L_Q}$, (c) is by $\frac{1}{\eta}= 2L_Qm^{\frac{1}{2}}\kappa_Q^{-\frac{1}{2}}$, and (d) follows from $\kappa_Q^{\frac{1}{2}}m^{\frac{1}{2}}\geq 1$. Therefore, the epoch complexity (i.e. the number of epochs required to reduce the suboptimality to below $\epsilon$) is $$\begin{aligned} K_0&=\lceil\frac{1}{\ln(\frac{1}{10}m^{\frac{1}{2}}\kappa_Q^{-\frac{1}{2}})}\ln\frac{F(x^0)-F(x^*)}{\epsilon}\rceil\\ &\leq \frac{1}{\ln(\frac{1}{10}m^{\frac{1}{2}}\kappa_Q^{-\frac{1}{2}})}\ln\frac{F(x^0)-F(x^*)}{\epsilon}+1\\ &=\frac{2}{\ln(1.21+\frac{1}{100}\frac{n}{\kappa_Q})}\ln\frac{F(x^0)-F(x^*)}{\epsilon}+1\\ &=\cO\big(\frac{1}{\ln(1.21+ \frac{n}{100\kappa_Q})}\ln\frac{1}{\epsilon}\big)+1\end{aligned}$$ where $\lceil \cdot \rceil$ is the ceiling function, and the second equality is due to $m=n+121\kappa_Q$. Hence, the gradient complexity is $$\begin{aligned} K&=(n+m)K_0\\ &\leq \cO\big(\frac{n+\kappa_Q}{\ln (1.21+\frac{n}{100\kappa_Q})}\ln\frac{1}{\epsilon}\big)+n+121\kappa_Q,\end{aligned}$$ which is equivalent to . Lower Complexity Bound for Convex SVRG {#App:LowerBound} ====================================== [@ArjevaniShamir2016_dimensionfreea Def. 2] An optimization algorithm is called a Canonical Linear Iterative (CLI) optimization algorithm, if given a function $F$ and initialization points $\{w^{0}_i\}_{i\in J}$, where $J$ is some index set, it operates by iteratively generating points such that for any $i\in J$, $$w^{k+1}_i = \sum_{j\in J} O_F(w^k_j; \theta^k_{ij}),\quad k=0,1,...$$ holds, where $\theta^k_{ij} $ are parameters chosen, stochastically or deterministically, by the algorithm, possibly depending on the side-information. $O_F$ is an oracle parameterized by $\theta^k_{ij}$. If the parameters do not depend on previously acquired oracle answers, we say that the given algorithm is oblivious. Lastly, algorithms with $|J|\leq p$, for some $p\in \mathbb{N}$, are denoted by p-CLI. In [@ArjevaniShamir2016_dimensionfreea], two types of oblivious oracles are considered. The generalized first order oracle for $F(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)$ $$O(w; A, B, C, j)=A\nabla f_j(w)+Bw+C, \quad A,B\in\mathbb{R}^{d\times d}, C\in \mathbb{R}^d, j\in [n].$$ The steepest coordinate descent oracle for $F(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)$ is given by $$O(w;i,j)=w+t^*e_i, \quad t^*\in\argmin_{t\in \mathbb{R}}f_j(w_1,...,w_{i-1}, w+t, w_{i+1},...,w_d), j\in [n],$$ where $e_i$ is the $i$th unit vector. SDCA, SAG, SAGA, SVRG, SARAH, etc. without proximal terms are all $p-$CLI oblivious algorithms. We now state the full version of Theorem \[lower complexity of SVRG, SARAH\]. \[lower complexity OPCLI\] For any oblivious p-CLI algorithm $A$, for all $\mu,L,k$, there exist $L$-smooth, and $\mu$-strongly convex functions $f_i$ such that at least[^12]: $$\begin{aligned} K\p{{\epsilon}} &=\tilde{\Omega}\p{\p{\frac{n}{1+(\ln(\frac{n}{\kappa}))_+}+\sqrt{n\kappa}}\ln{\frac{1}{\epsilon}}+n} \label{Lower Bound Oblivious PCLI append}\end{aligned}$$ iterations are needed for $A$ to obtain expected suboptimality $\EE\sp{f\p{K\p{{\epsilon}}}-f\p{X^*}}<{\epsilon}$. In this proof, we use lower bound given in [@ArjevaniShamir2016_dimensionfreea Thm 2], and refine its proof for the case $n\geq \frac{1}{3}\kappa$. [@ArjevaniShamir2016_dimensionfreea Thm 2] gives the following lower bound, $$\begin{aligned} K(\epsilon)\geq \Omega(n+\sqrt{n(\kappa-1)}\ln{\frac{1}{\epsilon}}).\label{case 1}\end{aligned}$$ Some smaller low-accuracy terms are absorbed are ignored, as is done in [@ArjevaniShamir2016_dimensionfreea]. For the case $n\geq \frac{1}{3}\kappa$, the proof of [@ArjevaniShamir2016_dimensionfreea Thm 2] tells us that, for any $k\geq 1$, there exist $L-$Lipschitz differentiable and $\mu-$strongly convex quadratic functions $f^k_1, f^k_2,...,f^k_n$ and $F^k=\frac{1}{n}\sum_{i=1}^n f^k_i$, such that for any $x^0$, the $x^K$ produced after $K$ gradient evaluations, we have[^13] $$\mathbb{E}[F^K(x^K)-F^K(x^*)]\geq \frac{\mu}{4}(\frac{nR\mu}{L-\mu})^2(\frac{\sqrt{1+\frac{\kappa-1}{n}}-1}{\sqrt{1+\frac{\kappa-1}{n}}+1})^{\frac{2K}{n}},$$ where $R$ is a constant and $\kappa=\frac{L}{\mu}$. Therefore, in order for $\epsilon\geq\mathbb{E}[F(x^k)-F(x^*)]$, we must have $$\epsilon \geq \frac{\mu}{4}(\frac{nR\mu}{L-\mu})^2(\frac{\sqrt{1+\frac{\kappa-1}{n}}-1}{\sqrt{1+\frac{\kappa-1}{n}}+1})^{\frac{2K}{n}}=\frac{\mu}{4}(\frac{nR\mu}{L-\mu})^2(1-\frac{2}{1+\sqrt{1+\frac{\kappa-1}{n}}})^{\frac{2k}{n}}.$$ Since $1+\frac{1}{3}x\leq \sqrt{1+x}$ when $0\leq x \leq 3$, and $0\leq \frac{\kappa-1}{n}\leq \frac{\kappa}{n}\leq 3$, we have $$\epsilon\geq \frac{\mu}{4}(\frac{nR\mu}{L-\mu})^2(1-\frac{2}{2+\frac{1}{3}\frac{\kappa-1}{n}})^{\frac{2K}{n}},$$ or equivalently, $$K\geq \frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln\big(\frac{\frac{\mu}{4}(\frac{nR}{\kappa-1})^2}{\epsilon}\big).$$ As a result, $$\begin{aligned} K&\geq\frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln\frac{1}{\epsilon}+\frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln\big(\frac{\mu}{4}(\frac{nR}{\kappa-1})^2\big)\\ &=\frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln\frac{1}{\epsilon}+\frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln(\frac{\mu R^2}{24})+\frac{n}{\ln(1+\frac{6n}{\kappa-1})}\ln\frac{6n}{\kappa-1}.\end{aligned}$$ Since $\frac{\ln\frac{6n}{\kappa-1}}{\ln(1+\frac{6n}{\kappa-1})}\geq \frac{\ln2}{\ln 3}$ when $\frac{n}{\kappa -1}\geq \frac{n}{\kappa}\geq \frac{1}{3}$, for small $\epsilon$ we have $$\begin{aligned} K&\geq \frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln\frac{1}{\epsilon}+\frac{n}{2\ln(1+\frac{6n}{\kappa-1})}\ln(\frac{\mu R^2}{24})+\frac{\ln 2}{\ln 3}n\nonumber\\ &=\Omega\big(\frac{n}{\ln(1+\frac{6n}{\kappa-1})}\ln\frac{1}{\epsilon}\big)+\frac{\ln 2}{\ln 3}n\\ &= \Omega(\frac{n}{1+(\ln(n/\kappa))_+}\ln(1/\epsilon)+n)\label{case 2}\end{aligned}$$ Now the expression in is valid for $n\geq\frac{1}{3}\kappa$. When $n<\frac{1}{3}\kappa$, the lower bound in is asymptotically equal to $\Omega(n\ln(1/\epsilon)+n)$, which is dominated by . Hence the lower bound in is valid for all $\kappa,n$. We may sum the lower bounds in and to obtain . This is because given an oblivious p-CLI algorithm, we may simply chose the adversarial example that has the corresponding greater lower bound. Lower Complexity Bound for SDCA {#App:lower complexity of SDCA} =============================== Let $\phi_i(t)=\frac{1}{2}t^2$, $\lambda=\mu$, and $y_i$ be the $i$th column of $Y$, where $Y=c(n^2I+J)$ and $J$ is the matrix with all elements being $1$, and $c=(n^4+2n^2+n)^{-1/2}(L-\mu)^{1/2}$. Then $$\begin{aligned} f_i(x)&=\frac{1}{2}(x^Ty_i)^2+\frac{1}{2}\mu\|x\|^2, \\ F(x)&=\frac{1}{2n}\|Y^Tx\|^2+\frac{1}{2}\mu\|x\|^2,\\ D(\alpha)&=\frac{1}{n\mu}(\frac{1}{2n}\|Y\alpha\|^2+\frac{1}{2}\mu\|\alpha\|^2).\end{aligned}$$ Since $$\|y_i\|^2=c^2\big((n^2+1)^2+n-1\big)=c^2(n^4+2n^2+n)=L-\mu,$$ $f_i$ is $L-$smooth and $\mu-$strongly convex, and that $x^*=\mathbf{0}$. We also have $$\nabla D(\alpha)=\frac{1}{n\mu}(\frac{1}{n}Y^2\alpha+\mu\alpha)=\frac{1}{n\mu}\big((c^2n^3I+2nc^2J+c^2J)\alpha +\mu\alpha\big),$$ So for every $k\geq 0$, minimizing with respect to $\alpha_{i_k}$ as in yields the optimality condition: $$\begin{aligned} 0&=e_{i_k}^T \nabla D(\alpha^{k+1})\\ &=\frac{1}{n\mu}\big(c^2n^3\alpha^{k+1}_{i_k}+2c^2n(\sum_{j\neq i_k} \alpha^{k}_j+\alpha^{k+1}_{i_k})+c^2(\sum_{j\neq i_k} \alpha^{k}_j+\alpha^{k+1}_{i_k})+\mu \alpha^{k+1}_{i_k}\big).\end{aligned}$$ Therefore, rearranging yields: $$\alpha^{k+1}_{i_k}=-\frac{(c^2+2c^2n)}{c^2n^3+2c^2n+c^2+\mu}\sum_{j\neq i_k}\alpha^k_j=-\frac{(c^2+2c^2n)}{c^2n^3+2c^2n+c^2+\mu}(e_{i_k}^T(J-I)\alpha^k).$$ As a result, $$\alpha^{k+1}=(I-e_{i_k}e_{i_k}^T)\alpha^k-\frac{(c^2+2c^2n)}{c^2n^3+2c^2n+c^2+\mu}(e_{i_k}e_{i_k}^T(J-I)\alpha^k).$$ Taking full expectation on both sides gives $$\mathbb{E}\alpha^{k+1}=\Big((1-\frac{1}{n})I-\frac{(c^2+2c^2n)}{c^2n^3+2c^2n+c^2+\mu}\frac{J-I}{n}\Big)\mathbb{E}\alpha^k\triangleq T\EE\alpha^k.$$ for linear operator $T$. Hence we have by Jensen’s inequality: $$\begin{aligned} \EE\n{x^{k}}^{2} & =n^{-2}\mu^{-2}\EE\n{Y\alpha^{k}}^{2}\\ & \geq n^{-2}\mu^{-2}\n{Y\EE\alpha^{k}}^{2}\\ & =n^{-2}\mu^{-2}\n{YT^{k}\alpha^{0}}^{2}\end{aligned}$$ We let $\alpha^{0}=\p{1,\ldots,1}$, which is an vector of $T$. Let us say the corresponding eigenvalue for $T$ is $\theta$: $$\begin{aligned} \EE\n{x^{k}}^{2} & \geq\theta^{2k}n^{-2}\mu^{-2}\n{Y\alpha^{0}}^{2}\\ & =\theta^{2k}\n{x^{0}}^{2}\label{eq:Theta-bound}\end{aligned}$$ We now analyze the value of $\theta$: $$\begin{aligned} \theta & =(1-\frac{1}{n})-\frac{(c^{2}+2c^{2}n)}{c^{2}n^{3}+2c^{2}n+c^{2}+\mu}\frac{n-1}{n}\\ & =1-\frac{1}{n}-\frac{1+2n}{n^{3}+2n+1+\mu c^{-2}}\frac{n-1}{n}\\ & \geq1-\frac{1}{n}-\frac{1+2n}{n^{3}+2n+1}\\ & \geq1-\frac{2}{n}\end{aligned}$$ for $n>2$. This in combination with yields . Nonconvex SVRG Analysis {#App:Nonconvex-SVRG} ======================= Without loss of generality, we can assume $x^*=\mathbf{0}$ and $F(x^*)=0.$ According to lemma 3.3 and Lemma 5.1 of [@Allen-Zhu2018_katyusha], for any $u\in\RR^d$, and $\eta\leq\frac{1}{2}\min\cp{\frac{1}{L},\frac{1}{\sqrt{m}\bar{L}}}$ we have $$\mathbb{E}[F(x^{j+1})-F(u))]\leq \mathbb{E}[-\frac{1}{4m\eta}\|x^{j+1}-x^j\|^2+\frac{\langle x^j-x^{j+1}, x^j-u\rangle}{m\eta}-\frac{\mu}{4}\|x^{j+1}-u\|^2],$$ or equivalently, $$\mathbb{E}[F(x^{j+1})-F(u))]\leq \mathbb{E}[\frac{1}{4m\eta}\|x^{j+1}-x^j\|^2+\frac{1}{2m\eta}\|x^j-u\|^2-\frac{1}{2m\eta}\|x^{j+1}-u\|^2-\frac{\mu}{4}\|x^{j+1}-u\|^2].$$ Setting $u=x^*=0$ and $u=x^j$ yields the following two inequalities: $$\begin{aligned} F(x^{j+1})&\leq \frac{1}{4m\eta}(\|x^{j+1}-x^j\|^2+2\|x^j\|^2-2(1+\frac{1}{2}m\eta\mu)\|x^{j+1}\|^2),\label{1}\\ F(x^{j+1})-F(x^j)&\leq-\frac{1}{4m\eta}(1+m\eta\mu)\|x^{j+1}-x^j\|^2.\label{2}\end{aligned}$$ Define $\tau=\frac{1}{2}m\eta\mu$, multiply $(1+2\tau)$ to , then add it to yields $$2(1+\tau)F(x^{j+1})-F(x^j)\leq \frac{1}{2m\eta}(1+2\tau)\big(\|x^j\|^2-(1+\tau)\|x^{j+1}\|\big).$$ Multiplying both sides by $(1+\tau)^j$ gives $$2(1+\tau)^{j+1}F(x^{j+1})-(1+\tau)^jF(x^j)\leq \frac{1}{2m\eta}(1+2\tau)\big((1+\tau)^j\|x^j\|^2-(1+\tau)^{j+1}\|x^{j+1}\|\big).$$ Summing over $j=0,1,...,k-1$, we have $$(1+\tau)^k F(x^k)+\sum_{j=0}^{k-1}(1+\tau)^j F(x^j)-F(x^0)\leq \frac{1}{2m\eta}(1+2\tau)(\|x^0\|^2-(1+\tau)^k\|x^k\|^2).$$ Since $F(x^j)\geq 0$, we have $$F({x}^k)(1+\tau)^k \leq F(x^0)+\frac{1}{2m\eta}(1+2\tau)\|x^0\|^2.$$ By the strong convex of $F$, we have $F(x^0)\geq \frac{\mu}{2}\|x^0\|^2$, therefore $$F({x}^k)(1+\tau)^k\leq F(x^0)(2+\frac{1}{2\tau}),$$ Finally, $\eta=\frac{1}{2}\min\{\frac{1}{L}, (\frac{1}{\overline{L}^2m})^{\frac{1}{2}}\}$ gives $$\frac{1}{\tau}=4\max\{\frac{\kappa}{m}, (\frac{\overline{L}^2}{m\mu^2})^{\frac{1}{2}}\}\leq 4(\frac{\kappa}{m}+(\frac{\overline{L}^2}{m\mu^2})^{-\frac{1}{2}}),$$ which yields $$F(x^k)\leq (1+\tau)^{-k}F(x^0)\big(2+2(\frac{\kappa}{m}+(\frac{\overline{L}^2}{m\mu^2})^{-\frac{1}{2}})\big). $$ To prove , we notice that $$\tau=\frac{1}{4}\min\{\frac{m}{\kappa}, (\frac{m\mu^2}{\overline{L}^2})^{\frac{1}{2}}\},$$ so we have $$\frac{1}{\ln(1+\tau)}\leq \frac{1}{\ln(1+\frac{m}{4\kappa})}+\frac{1}{\ln\big(1+(\frac{m\mu^2}{4\overline{L}})^{\frac{1}{2}}\big)}$$ Now for small $\epsilon$, the epoch complexity can be written as $$\begin{aligned} K_0&=\lceil\frac{1}{\ln(1+\tau)}\ln\frac{F(x^0)(2+2(\frac{\kappa}{m}+(\frac{\overline{L}^2}{m\mu^2})^{-\frac{1}{2}}))}{\epsilon}\rceil\\ &\leq\cO\Big((\frac{1}{\ln(1+\frac{m}{4\kappa})}+\frac{1}{\ln\big(1+(\frac{m\mu^2}{4\overline{L}})^{\frac{1}{2}}\big)})\ln\frac{1}{\epsilon}\Big)+1.\end{aligned}$$ Since $m=\min\{2,n\}$, we have a gradient complexity of $$K=(n+m)K_0\leq\cO\Big((\frac{n}{\ln(1+\frac{n}{4\kappa})}+\frac{n}{\ln\big(1+(\frac{n\mu^2}{4\overline{L}})^{\frac{1}{2}}\big)})\ln\frac{1}{\epsilon}\Big)+2n.$$ And this is equivalent to the expression in . [^1]: Corresponding author: <RobertHannah89@gmail.com> [^2]: <yanli@math.ucla.edu> [^3]: <daniel.v.oconnor@gmail.com> [^4]: <WotaoYin@math.ucla.edu> [^5]: A function $f$ is $L$-smooth if it has an $L$-Lipschitz gradient $\nabla f$ [^6]: SDCA must be modified however with a dummy regularizer. [^7]: We define $(a)_+$ as $\max\cp{a,0}$ for $a\in\RR$. [^8]: I.e. the complexity cannot be improved among a very broad class of finite-sum algorithms. [^9]: Specifically, we improve the analysis of Theorem 2 from this paper. [^10]: We absorb some smaller low-accuracy terms (high $\epsilon$) as is common practice. Exact lower bound expressions appear in the proof. [^11]: This is the Hilbert space of sequence $(x_{i})_{i=1}^\infty$ with $\sum_{i=1}^\infty x_i^2<\infty$ [^12]: We absorb some smaller low-accuracy terms (high $\epsilon$) as is common practice. Exact lower bound expressions appear in the proof. [^13]: note that for the SVRG in Algorithm \[alg\_1\] with $\psi=0$, each update in line $7$ is regarded as an iteration.

--- bibliography: - 'bibliografia.bib' ---  UNIVERSIDAD DE BUENOS AIRES Facultad de Ciencias Exactas y Naturales Departamento de Matemática **Metaestabilidad para una EDP con blow-up y la dinámica FFG en modelos diluidos** Tesis presentada para optar al título de Doctor de la Universidad de Buenos Aires en el área Ciencias Matemáticas **Santiago Saglietti** Director de tesis: Pablo Groisman Consejero de estudios: Pablo Groisman Buenos Aires, 2014 Fecha de defensa : 27 de Junio del 2014 {#section .unnumbered} [**Metaestabilidad para una EDP con blow-up y la dinámica FFG en modelos diluidos**]{} **Resumen** Esta tesis consiste de dos partes, en cada una estudiamos la estabilidad bajo pequeñas perturbaciones de ciertos modelos probabilísticos en diferentes contextos. En la primera parte, estudiamos pequeñas perturbaciones *aleatorias* de un sistema dinámico determinístico y mostramos que las mismas son inestables, en el sentido de que los sistemas perturbados tienen un comportamiento cualitativo diferente al del sistema original. Más precisamente, dado $p > 1$ estudiamos soluciones de la ecuación en derivadas parciales estocástica $${\partial}_t U = {\partial}^2_{xx} U + U|U|^{p-1} + \varepsilon \dot{W}$$ con condiciones de frontera de Dirichlet homogéneas y mostramos que para $\varepsilon > 0$ pequeños éstas presentan una forma particular de inestabilidad conocida como *metaestabilidad*. En la segunda parte nos situamos dentro del contexto de la mecánica estadística, donde estudiamos la estabilidad de medidas de equilibrio en volumen infinito bajo ciertas perturbaciones *determinísticas* en los parámetros del modelo. Más precisamente, mostramos que las medidas de Gibbs para una cierta clase general de sistemas son continuas con respecto a cambios en la interacción y/o en la densidad de partículas y, por lo tanto, estables bajo pequeñas perturbaciones de las mismas. También estudiamos bajo qué condiciones ciertas configuraciones típicas de estos sistemas permanecen estables en el límite de temperatura cero $T \to 0$. La herramienta principal que utilizamos para nuestro estudio es la realización de estas medidas de equilibrio como distribuciones invariantes de las dinámicas introducidas en [@FFG1]. Referimos al comienzo de cada una de las partes para una introducción de mayor profundidad sobre cada uno de los temas. [*Palabras claves:*]{} ecuaciones en derivadas parciales estocásticas, metaestabilidad, blow-up, medidas de Gibbs, procesos estocásticos, redes de pérdida, Pirogov-Sinai. {#section-1 .unnumbered} [**Metastability for a PDE with blow-up and the FFG dynamics in diluted models**]{} **Abstract** This thesis consists of two separate parts: in each we study the stability under small perturbations of certain probability models in different contexts. In the first, we study small *random* perturbations of a deterministic dynamical system and show that these are unstable, in the sense that the perturbed systems have a different qualitative behavior than that of the original system. More precisely, given $p > 1$ we study solutions to the stochastic partial differential equation $${\partial}_t U = {\partial}^2_{xx} U + U|U|^{p-1} + \varepsilon \dot{W}$$ with homogeneous Dirichlet boundary conditions and show that for small $\varepsilon > 0$ these present a rather particular form of unstability known as *metastability*. In the second part we situate ourselves in the context of statistical mechanics, where we study the stability of equilibrium infinite-volume measures under small *deterministic* perturbations in the parameters of the model. More precisely, we show that Gibbs measures for a general class of systems are continuous with respect to changes in the interaction and/or density of particles and, hence, stable under small perturbations of them. We also study under which conditions do certain typical configurations of these systems remain stable in the zero-temperature limit $T \to 0$. The main tool we use for our study is the realization of these equilibrium measures as invariant distributions of the dynamics introduced in [@FFG1]. to the beginning of each part for a deeper introduction on each of the subjects. [*Key words*]{}: stochastic partial differential equations, , stochastic blow-up, measures, loss networks, Pirogov-Sinai. Agradecimientos {#agradecimientos .unnumbered} =============== Un gran número de personas han contribuido, de alguna manera u otra, con la realización de este trabajo. Me gustaría agradecer: 1. A mi director, Pablo Groisman, por todo. Por su constante apoyo y durante la elaboración de esta Tesis. Por su infinita paciencia y gran Por estar siempre para atender mis inquietudes, y por recibirme todas y cada una de las veces con una sonrisa y la mejor onda. Por compartir conmigo su manera de concebir y hacer matemática, lo que ha tenido un gran impacto en mi formación como matemático y es, para mí, de un valor incalculable. Por su amistad. Por todo. 2. A Pablo Ferrari, por estar siempre dispuesto a darme una mano y a discutir sobre matemática conmigo. Considero realmente un privilegio haber tenido la oportunidad de pensar problemas juntos y entrar en contacto con su forma de ver la matemática. Son muchísimas las cosas que he aprendido de él en estos últimos cuatro años, y es por ello que le voy a estar siempre inmensamente agradecido. 3. A los jurados de esta Tesis: Pablo De Napoli, Mariela Sued y Aernout Van Enter. Por leerla y darme sus sugerencias, con todo el esfuerzo y tiempo que ello requiere. 4. A Nicolas Saintier, por su entusiasmo en mi trabajo y su colaboración en esta Tesis. 5. A Roberto Fernández y Siamak Taati, por la productiva estadía en Utrecht. 6. A Inés, Matt y Leo. Por creer siempre en mí y enseñarme algo nuevo todos los días. 7. A Maru, por las muchas tardes de clase, estudio, charlas, chismes y chocolate. 8. A mis hermanitos académicos: Anita, Nico, Nahuel, Sergio L., Sergio Y. y Julián. Por todos los momentos compartidos, tanto de estudio como de amistad. 9. A Marto S., Pablo V., Caro N. y los chicos de la 2105 (los de ahora y los de antes). 10. A Adlivun, por los buenos momentos y la buena música. 11. A la (auténtica) banda del Gol, por ser los amigos incondicionales que son. 12. A Ale, por haber estado siempre, en las buenas y (sobre todo) en las malas. 13. A mi familia, por ser mi eterno sostén y apoyo. $$\text{Gracias!}$$ Introducción a la Parte I {#introducción-a-la-parte-i .unnumbered} ========================= Las ecuaciones diferenciales han probado ser de gran utilidad para modelar un amplio rango de fenómenos físicos, químicos y biológicos. Por ejemplo, una vasta clase de ecuaciones de evolución, conocidas como ecuaciones en derivadas parciales parabólicas surgen naturalmente en el estudio de fenómenos tan diversos como la difusión de un fluido a través de un material poroso, el transporte en un semiconductor, las reacciones químicas acopladas con difusión espacial y la genética de poblaciones. En todos estos casos, la ecuación representa un modelo aproximado del fenómeno y por lo tanto es de interés entender cómo su descripción puede cambiar si es sujeta a pequeñas perturbaciones aleatorias. Nos interesa estudiar ecuaciones del tipo $$\label{intro0esp} {\partial}_t U = {\partial}^2_{xx} U + f(U)$$ con condiciones de frontera de Dirichlet en $[0,1]$, donde $f: {{\mathbb R}}\rightarrow {{\mathbb R}}$ es una fuente localmente Lipschitz. Dependiendo del dato inicial, es posible que las soluciones de esta ecuación no se encuentren definidas para todo tiempo. Decimos entonces que estamos ante la presencia del fenómeno de *blow-up* o *explosión*, i.e. existe $\tau > 0$ tal que la solución $U$ se encuentra definida para todo tiempo $t < \tau$ y además satisface $\lim_{t \rightarrow \tau^-} \| U(t,\cdot)\|_\infty = +\infty$. Agregando una pequeña perturbación aleatoria al sistema se obtiene la ecuación en derivadas parciales estocástica $$\label{intro1esp} {\partial}_t U = {\partial}^2_{xx} U + f(U) + \varepsilon \dot{W}$$ donde $\varepsilon > 0$ es un parámetro pequeño y $\dot{W}$ es ruido blanco espacio-temporal. Uno puede preguntarse entonces si existen diferencias cualitativas en comportamiento entre el sistema determinístico y su perturbación estocástica. Para tiempos cortos ambos sistemas deberían comportarse de manera similar, ya que en este caso el ruido será típicamente de un orden mucho menor que los términos restantes en el miembro derecho de . Sin embargo, debido a los incrementos independientes y normalmente distribuidos del ruido uno espera que, si es dado el tiempo suficiente, éste eventualmente alcanzará valores suficientemente grandes como para inducir un cambio de comportamiento significativo en . Estamos interesados en entender qué cambios pueden ocurrir en el fenómeno de blow-up debido a esta situación y, más precisamente, cuáles son las propiedades asintóticas cuando $\varepsilon \rightarrow 0$ del tiempo de explosión de para los diferentes datos iniciales. En particular, para sistemas como en con un único equilibrio estable $\phi$, uno espera el siguiente panorama: 1. Para datos iniciales en el dominio de atracción del equilibrio estable, el sistema estocástico es inmediatamente atraído hacia el equilibrio. Una vez cerca de éste, los términos en el miembro derecho de se vuelven despreciables de manera tal que el proceso puede ser luego empujado lejos del equilibrio por acción del ruido. Estando lejos de $\phi$, el ruido vuelve a ser superado por los términos restantes en el miembro derecho de y esto permite que el patrón anterior se repita: un gran número de intentos de escapar del equilibrio, seguidos de una fuerte atracción hacia el mismo. 2. Eventualmente, luego de muchos intentos frustrados, el proceso logra escaparse del dominio de atracción de $\phi$ y alcanza el dominio de explosión, aquel conjunto de datos iniciales para los cuales la solución de explota en tiempo finito. Como la probabilidad de un evento tal es muy baja, esperamos que este tiempo de escape sea exponencialmente grande. Más aún, debido al gran número de intentos que fueron necesarios, esperamos que este tiempo muestre escasa memoria del dato inicial. 3. Una vez dentro del dominio de explosión, el sistema estocástico es forzado a explotar por la fuente $f$, que se convierte en el término dominante. Este tipo de fenómeno se conoce como *metaestabilidad*: el sistema se comporta por un tiempo muy largo como si estuviera bajo equilibrio, para luego realizar una transición abrupta hacia el equilibrio real (en nuestro caso, hacia infinito). La descripción anterior fue probada rigurosamente en [@GOV] en el contexto finito-dimensional para sistemas del tipo $$\dot{U}=- \nabla S(U)$$ donde $U$ es un potencial de doble pozo. En este contexto, el comportamiento metaestable es observado en la manera en que el sistema estocástico viaja desde cualquiera de los pozos hacia el otro. Luego, en [@MOS] y [@B1], el problema análogo infinito-dimensional fue investigado, obteniendo resultados similares. El enfoque general sugerido en [@GOV] para establecer el comportamiento metaestable en este tipo de sistemas es estudiar el escape de un dominio acotado $G$ satisfaciendo: 1. $G$ contiene al equilibrio estable $\phi$ y a los equilibrios inestables de mínima energía. 2. Existe una región ${\partial}^*$ en la frontera de $G$ tal que: 1. El “costo” para el sistema de alcanzar ${\partial}^*$ comenzando desde $\phi$ es el mismo que el costo de alcanzar cualquiera de los equilibrios inestables de mínima energía. 2. Con probabilidad que tiende a uno cuando $\varepsilon \rightarrow 0^+$ el sistema estocástico comenzando en ${\partial}^*$ alcanza el verdadero equilibrio antes de un tiempo acotado $\tau^*$ independiente de $\varepsilon$. La construcción de este dominio para el potencial de doble pozo finito-dimensional fue llevada a cabo en [@GOV]. En el marco infinito-dimensional, sin embargo, este tipo de resultados fueron probados sin seguir estrictamente el enfoque de [@GOV]: la pérdida de memoria asintótica fue lograda en [@MOS] sin acudir a ningún dominio auxiliar, mientras que el restante panorama fue establecido en [@B1] considerando un dominio que tiene a los equilibrios inestables de mínima energía en su frontera y por lo tanto no cumple (ii). Un dominio de tales características no puede ser utilizado como sugiere [@GOV] para obtener la pérdida de memoria asintótica, pero es utilizado en [@B1] de todas maneras puesto a que dicha pérdida de memoria ya había sido probada en [@MOS] por otros métodos. Nosotros hemos decidido aferrarnos al enfoque general sugerido en [@GOV] para estudiar nuestro sistema ya que quizás éste sea el más sencillo de seguir y, además, ya que provee un único marco general sobre el cual se pueden probar todos los resultados que nos interesan. Más aún, para seguirlo deberemos introducir herramientas que son también útiles para estudiar otros tipos de problemas, como el escape de un dominio con un único equilibrio. En nuestro trabajo también consideraremos ecuaciones de tipo gradiente, pero la situación en nuestro contexto es más delicada que en la del modelo del potencial de doble pozo. En efecto, la construcción del dominio $G$ dependerá en gran medida de la geometría del potencial asociado a la ecuación, la cual en general será más complicada que la dada por el potencial de doble pozo. Además, (ii) en la descripción del dominio $G$ dada arriba es usualmente una consecuencia directa de las estimaciones de grandes desvíos disponibles para el sistema estocástico. No obstante, la validez de estas estimaciones en todos los casos depende de un control apropiado sobre el crecimiento de las soluciones de . Como estaremos enfocándonos específicamente en trayectorias que explotan en un tiempo finito, está claro que para esta última parte un nuevo enfoque será necesario en nuestro problema, uno que involucre un estudio cuidadoso del fenómeno de blow-up. Desafortunadamente, cuando se trata con perturbaciones de ecuaciones diferenciales con blow-up, entender cómo se modifica el comportamiento del tiempo de explosión o incluso mostrar la existencia del fenómeno de blow-up mismo no es para nada una tarea fácil en la mayoría de los casos. No existen resultados generales al respecto, ni siquiera para perturbaciones no aleatorias. Esta es la razón por la cual el enfoque usual a este tipo de problemas es considerar modelos particulares. En esta primera parte estudiamos el comportamiento metaestable de la ecuación con blow-up $$\label{intro2} {\partial}_t U = {\partial}^2_{xx} U + U|U|^{p-1}$$ con condiciones de frontera de Dirichlet homogéneas en $[0,1]$, para un cierto parámetro Hemos elegido esta ecuación particular ya que ha sido tomada como problema modelo por la comunidad de EDP, dado que exhibe las principales características de interés que aparecen en la presencia de blow-up (ver por ejemplo los libros [@QS; @SGKM] o las notas [@BB; @GV]). También, trabajamos con una variable espacial unidimensional dado que no existen soluciones de para dimensiones más altas en el sentido tradicional. La Parte I está organizada de la siguiente manera. En el Capítulo 1 damos las definiciones necesarias y los resultados preliminares para ayudarnos a tratar nuestro problema, como también así detallamos los resultados principales que hemos obtenido. El Capítulo 2 se enfoca en el tiempo de explosión para el sistema estocástico para datos iniciales en el dominio de explosión. La construcción del dominio auxiliar $G$ en nuestro contexto es llevada a cabo en el Capítulo 3, mientras que estudiamos el escape de $G$ en el capítulo siguiente. En el Capítulo 5 establecemos el comportamiento metaestable para soluciones con datos iniciales en el dominio de atracción del equilibrio estable. En el Capítulo 6 estudiamos una variante finito-dimensional de nuestro problema original e investigamos qué resultados pueden obtenerse en este marco simplificado. Finalmente, incluimos al final un apéndice con algunos resultados auxiliares a ser utilizados durante nuestro análisis. Introduction to Part I {#introduction-to-part-i .unnumbered} ====================== Differential equations have proven to be of great utility to model a wide range of , chemical and biological phenomena. For example, a broad class of evolution , known as semilinear parabolic partial differential equations, naturally arise in the study of as diverse as diffusion of a fluid through a porous material, transport in a semiconductor, coupled chemical reactions with spatial diffusion and population genetics. In all these cases, the equation represents an approximated model of the phenomenon and thus it is of interest to understand how its description might change if subject to small random perturbations. We are concerned with studying equations of the sort $$\label{intro0} {\partial}_t U = {\partial}^2_{xx} U + f(U)$$ with homogeneous Dirichlet boundary conditions on $[0,1]$, where $f: {{\mathbb R}}\rightarrow {{\mathbb R}}$ is a locally Lipschitz source. Depending on the initial datum, it is possible that solutions to this equation are not defined for all times. We then say we are in the presence of a *blow-up* phenomenon, i.e. there exists $\tau > 0$ such that the solution $U$ is defined for all times $t < \tau$ and verifies $\lim_{t \rightarrow \tau^-} \| U(t,\cdot)\|_\infty = +\infty$. Adding a small random perturbation to the system yields the stochastic partial differential equation $$\label{intro1} {\partial}_t U = {\partial}^2_{xx} U + f(U) + \varepsilon \dot{W}$$ where $\varepsilon > 0$ is a small parameter and $\dot{W}$ is space-time white noise. One can then wonder if there are any qualitative differences in behavior between the deterministic system and its stochastic perturbation. For short times both systems should behave similarly, since in this case the noise term will be typically of much smaller order than the remaining terms in the right hand side of . However, due to the independent and normally distributed increments of the perturbation, one expects that when given enough time the noise term will eventually reach sufficiently large values so as to induce a significant change of behavior in . We are interested in understanding what changes might occur in the blow-up phenomenon due to this situation and, more precisely, which are the asymptotic properties as $\varepsilon \rightarrow 0$ of the explosion time of for the different In particular, for systems as in with a unique stable equilibrium $\phi$, one expects 1. For initial data in the domain of attraction of the stable equilibrium, the stochastic system is immediately attracted towards this equilibrium. Once near it, the terms in the right hand side of become negligible and so the process is then pushed away from the equilibrium by noise. Being away from $\phi$, the noise becomes overpowered by the remaining terms in the right hand side of and this allows for the previous pattern to repeat itself: a large number of attempts to escape from the followed by a strong attraction towards it. 2. Eventually, after many frustrated attempts, the process succeeds in escaping the domain of attraction of $\phi$ and reaches the domain of explosion, i.e. the set of initial data for which blows up in finite time. Since the probability of such an event is very small, we expect this escape time to be exponentially large. Furthermore, due to the large number of attempts that are necessary, we expect this time to show little memory of the initial data. 3. Once inside the domain of explosion, the stochastic system is forced to explode by the dominating source term $f$. This type of phenomenon is known as *metastability*: the system behaves for a very long time as if it were under equilibrium, but then performs an abrupt transition towards the real equilibrium (in our case, towards infinity). The former description was proved rigorously in [@GOV] in the finite-dimensional setting for systems of the sort $$\dot{U}=- \nabla S(U)$$ where $U$ is a double-well potential. In their context, metastable behavior is observed in the way in which the stochastic system travels from one of the wells to the other. Later, in [@MOS] and [@B1], the analogous infinite-dimensional problem was investigated, obtaining similar results. The general approach suggested in [@GOV] to establish metastable behavior in these kind of systems is to study the escape from a bounded domain $G$ satisfying the following: 1. $G$ contains the stable equilibrium $\phi$ and all the unstable equilibria of minimal energy. 2. There exists a region ${\partial}^*$ in the boundary of $G$ such that: 1. The “cost” for the system to reach ${\partial}^*$ starting from $\phi$ is the same as the cost to reach any of the unstable equilibria of minimal energy. 2. With overwhelming probability as $\varepsilon \rightarrow 0^+$ the stochastic system arrives at the real equilibrium before a bounded time $\tau^*$ independent of $\varepsilon$. The construction of this domain for the finite-dimensional double-well potential was carried out in [@GOV]. In the infinite-dimensional setting, however, these type of results were proved without strictly following this approach: the asymptotic loss of memory was achieved in [@MOS] without resorting to any auxiliary domain, while the remaining parts of the picture were settled in [@B1] by considering a domain which has the unstable equilibria of minimal energy in its boundary and hence does not satisfy (ii). Such a domain cannot be used as suggested in [@GOV] to obtain the asymptotic loss of memory, but it is used nonetheless in [@B1] since this loss of memory had already been established in [@MOS] by different methods. We have decided to hold on to this general approach introduced in [@GOV] to study our system since it is perhaps the easiest one to follow and, also, since it provides with a unique general framework on which to prove all results of interest. Furthermore, in order to follow it we will need to introduce tools which are also useful for treating other type of problems, such as the escape from a domain with only one equilibrium. In our work we shall also consider gradient-type equations, but the situation in our context is more delicate than in the double-well potential model. Indeed, the construction of the domain $G$ will clearly rely on the geometry of the potential associated to the equation, which in general, will be more complicated than the one given by the double-well potential. Furthermore, (ii) in the description of the domain $G$ above is usually a direct consequence of the large deviations estimates available for the stochastic system. The validity of these estimates always relies, however, on a proper control of the growth of solutions to . Since we will be focusing specifically on trajectories which blow up in finite time, it is clear that for this last part a new approach is needed in our setting, one that involves a careful study of the blow-up phenomenon. Unfortunately, when dealing with perturbations of differential equations with blow-up, understanding how the behavior of the blow-up time is modified or even showing existence of the blow-up phenomenon itself is by no means an easy task in most cases. There are no general results addressing this matter, not even for nonrandom perturbations. This is why the usual approach to this kind of problems is to consider particular models. In this first part we study metastable behavior for the following equation with blow-up: $$\label{intro2} {\partial}_t U = {\partial}^2_{xx} U + U|U|^{p-1}$$ with homogeneous Dirichlet boundary conditions on $[0,1]$, for some fixed parameter $p > 1$. We chose this particular equation since it has been taken as a model problem for the PDE community as it exhibits some of the essential interesting features which appear in the presence of blow-up (see the books [@QS; @SGKM] or the surveys [@BB; @GV]). Also, we work with a one-dimensional space variable since there are no solutions to for higher dimensions in the traditional sense. Part I is organized as follows. In Chapter 1 we give the necessary definitions and preliminary results to help us address our problem, as well as detail the main results we have obtained. Chapter 2 focuses on the explosion time of the stochastic system for initial data in the domain of explosion. The construction of the auxiliary domain $G$ in our context is performed in Chapter 3, we study the escape from $G$ in the following chapter. In Chapter 5 we establish metastable behavior for solutions with initial data in the domain of attraction of the stable equilibrium. In Chapter 6 we study a finite-dimensional variant of our original problem and investigate which results can be obtained for this simplified setting. Finally, we include at the end an appendix with some auxiliary results to be used throughout our analysis. Preliminaries ============= The deterministic PDE --------------------- Consider the partial differential equation $$\label{MainPDE} \left\{\begin{array}{rll} {\partial}_t U &= {\partial}^2_{xx}U + g(U) & \quad t>0 \,,\, 0<x<1 \\ U(t,0)& =0 & \quad t>0 \\ U(t,1) & = 0 & \quad t>0 \\ U(0,x) &=u(x) & \quad 0<x<1 \end{array}\right.$$ where $g : {{\mathbb R}}\rightarrow {{\mathbb R}}$ is given by $g(u)=u|u|^{p-1}$ for a fixed $p > 1$ and $u$ belongs to the space of continuous functions defined on $[0,1]$ with homogeneous Dirichlet boundary conditions $$C_{D}([0,1])= \{ v \in C([0,1]) : v(0)=v(1)=0 \}.$$ Equation can be reformulated as $$\label{formalPDE} {\partial}_t U = - \frac{\partial S}{\partial \varphi} (U)$$ where the *potential* $S$ is the functional on $C_D([0,1])$ given by $$S(v) = \left\{ \begin{array}{ll} \displaystyle{\int_0^1 \left[\frac{1}{2} \left(\frac{dv}{dx}\right)^2 - \frac{|v|^{p+1}}{p+1}\right]} & \text{ if $v \in H^1_0((0,1))$} \\ \\ +\infty & \text{ otherwise.}\end{array}\right.$$ Here $H^1_0((0,1))$ denotes the Sobolev space of square-integrable functions defined on $[0,1]$ with square-integrable weak derivative which vanish at the boundary $\{0,1\}$. Recall that $H^1_0((0,1))$ can be embedded into $C_D([0,1])$ so that the potential is indeed well defined. We refer the reader to the Appendix for a review of some of the main properties of $S$ which shall be required throughout our work. The formulation on is interpreted as the validity of $$\int_0^1 {\partial}_t U(t,x) \varphi(x)dx = \lim_{h \rightarrow 0} \frac{S(U + h\varphi) - S(U)}{h}$$ for any $\varphi \in C^1([0,1])$ with $\varphi(0)=\varphi(1)=0$. It is known that for any $u \in C_D([0,1])$ there exists a unique solution $U^{u}$ to equation defined on some maximal time interval $[0,\tau^{u})$ where $0 < \tau^{u} \leq +\infty$ is called the *explosion time* of $U^u$ (see [@QS] for further details). In general this solution will belong to the space $$C_D([0,\tau^u) \times [0,1]) = \{ v \in C( [0,\tau^{u}) \times [0,1]) : v(\cdot,0)=v(\cdot,1) \equiv 0 \}.$$ However, whenever we wish to make its initial datum $u$ explicit we will do so by saying that the solution belongs to the space $$C_{D_{u}}([0,\tau^{u}) \times [0,1]) = \{ v \in C( [0,\tau^{u}) \times [0,1]) : v(0,\cdot)=u \text{ and }v(\cdot,0)=v(\cdot,1) \equiv 0 \}.$$ The origin $\mathbf{0} \in C_D([0,1])$ is the unique stable equilibrium of the system and is in fact asymptotically stable. It corresponds to the unique local minimum of the potential $S$. There is also a family of unstable equilibria of the system corresponding to the remaining critical points of the potential $S$, all of which are saddle points. Among these unstable equilibria there exists only one of them which is nonnegative, which we shall denote by $z$. It can be shown that this equilibrium $z$ is in fact strictly positive for $x \in (0,1)$, symmetric with respect to the axis $x=\frac{1}{2}$ (i.e. $z(x)=z(1-x)$ for every $x \in [0,1]$) and that is of both minimal potential and minimal norm among the unstable equilibria. More precisely, one has the following characterization of the unstable equilibria. \[equilibrios\] A function $w \in C_D([0,1]) $ is an equilibrium of the system there exists $n \in {{\mathbb{Z}}}$ such that $w = z^{(n)}$, where for each $n \in {{\mathbb N}}$ we define $z^{(n)} \in C_D([0,1])$ by the formula $$z^{(n)}(x)= \left\{ \begin{array}{rl} n^{\frac{2}{p-1}}z( nx - [nx] ) & \text{ if $[nx]$ is even} \\ \\ - n^{\frac{2}{p-1}}z( nx - [nx] )& \text{ if $[nx]$ is odd}\end{array}\right.$$ and also define $z^{(-n)}:= - z^{(n)}$ and $z^{(0)}:= \mathbf{0}$. Furthermore, for each $n \in {{\mathbb{Z}}}$ we have $$\| z^{(n)} \|_\infty = |n|^{\frac{2}{p-1}}\|z\|_\infty \hspace{2cm}\text{ and }\hspace{2cm} S(z^{(n)}) = |n|^{2 \left(\frac{p+1}{p-1}\right)} S(z).$$ It is simple to verify that for each $n \in {{\mathbb{Z}}}$ the function $z^{(n)}$ is an equilibrium of the system and that each $z^{(n)}$ satisfies both $\| z^{(n)} \|_\infty = |n|^{\frac{2}{p-1}}\|z\|_\infty$ and $S(z^{(n)}) = |n|^{2 \left(\frac{p+1}{p-1}\right)} S(z)$. Therefore, we must only check that for any equilibrium of the system $w \in C_D([0,1]) - \{ \mathbf{0}\}$ there exists $n \in {{\mathbb N}}$ such that $w$ coincides with either $z^{(n)}$ or $-z^{(n)}$. Thus, for a given equilibrium $w \in C_D([0,1]) - \{ \mathbf{0}\}$ let us define the sets $$G^+ = \{ x \in (0,1) : w(x) > 0 \} \hspace{2cm}\text{ and }\hspace{2cm}G^- = \{x \in (0,1) : w(x) < 0 \}.$$ Since $w \neq \mathbf{0}$ at least one of these sets must be nonempty. On the other hand, if only one of them is nonempty then, since $z$ is the unique nonnegative equilibrium different from $\mathbf{0}$, we must have either $w=z$ or $w=-z$. Therefore, we may assume that both $G^+$ and $G^-$ are nonempty. Notice that since $G^+$ and $G^-$ are open sets we may write them as $$G^+ = \bigcup_{k \in {{\mathbb N}}} I^{+}_k \hspace{2cm}\text{ and }\hspace{2cm}G^- = \bigcup_{k \in {{\mathbb N}}} I^-_k$$ where the unions are disjoint and each $I^{\pm}_k$ is a (possibly empty) open interval. Our first task now will be to show that each union is in fact finite. For this purpose, let us take $k \in {{\mathbb N}}$ and suppose that we can write $I^+_k = (a_k, b_k)$ for some $0\leq a_k < b_k \leq 1$. It is easy to check that $\tilde{w}_k : [0,1] \rightarrow {{\mathbb R}}$ given by $$\tilde{w}_k (x) = (b_k - a_k)^{\frac{2}{p-1}} w( a_k + (b_k-a_k) x)$$ is a nonnegative equilibrium of the system different from $\mathbf{0}$ and thus it must be $\tilde{w}_k = z$. This, in particular, implies that $\| w \|_\infty \geq (b_k - a_k)^{- \frac{2}{p-1}} \| \tilde{w}_k \|_\infty = (b_k - a_k)^{- \frac{2}{p-1}} \| z \|_\infty$ from where we see that an infinite number of nonempty $I^+_k$ would contradict the fact that $\| w \|_\infty < +\infty$. Therefore, we conclude that $G^+$ is a finite union of open intervals and that, by an analogous argument, the same holds for $G^-$. Now, by Hopf’s Lemma (see [@E p. 330]) we obtain that $\partial_x z(0^+) > 0$ and $\partial_x z(1^-) < 0$. In particular, this tells us that for each $I_k^+$ we must have $d( I^+_k , G^+ - I^+_k ) > 0$, i.e. no two plus intervals lie next to each other, since that would contradict the differentiability of $w$. Furthermore, we must also have $d(I^+_k, G^-) = 0$, i.e. any plus interval lies next to a minus interval, since otherwise we would have a plus interval lying next to an interval in which $w$ is constantly zero, a fact which again contradicts the differentiability of $w$. Therefore, from all this we conclude that plus and minus intervals must be presented in alternating order, and that their closures must cover all of the interval $[0,1]$. Finally, since $z$ is symmetric with respect to $x=\frac{1}{2}$ we obtain that $\partial_x z(0^+) = - \partial_x z(1^-)$. This implies that all intervals must have the same length, otherwise we would once again contradict the differentiability of $w$. Since the measures of the intervals must add up to one, we see that their length must be $l=\frac{1}{n}$ where $n$ denotes the total amount of intervals. This concludes the proof. Regarding the behavior of solutions to the equation we have the following result, whose proof was given in [@CE1]. \[descomp1\] Let $U^u$ be the solution to equation with initial datum $u \in C_D([0,1])$. Then one of these two possibilities must hold: 1. $\tau^{u} < +\infty$ and $U^u$ blows up as $t \rightarrow \tau^{u}$, i.e. $\lim_{t \rightarrow \tau^{u}} \|U^u(t,\cdot)\|_\infty = +\infty$ 2. $\tau^{u} = +\infty$ and $U^u$ converges (in the $\| \cdot \|_\infty$ norm) to a stationary solution as $t \rightarrow +\infty$, i.e. a critical point of the potential $S$. Theorem \[descomp1\] is used to decompose the space $C_D([0,1])$ of initial data into three parts: $$\label{decomp12} C_D([0,1]) = \mathcal{D}_{\mathbf{0}} \cup \mathcal{W} \cup \mathcal{D}_e$$ where $\mathcal{D}_{\mathbf{0}}$ denotes the stable manifold of the origin $\mathbf{0}$, $\mathcal{W}$ is the union of all stable manifolds of the unstable equilibria and $\mathcal{D}_e$ constitutes the domain of explosion of the system, i.e. the set of all initial data for which the system explodes in finite time. It can be seen that both $\mathcal{D}_{\mathbf{0}}$ and $\mathcal{D}_e$ are open sets and that $\mathcal{W}$ is the common boundary separating them. The following proposition gives a useful characterization of the domain of explosion $\mathcal{D}_e$. Its proof is can be found on [@QS Theorem 17.6]. \[caract\] Let $U^u$ denote the solution to with initial datum $u \in C_D([0,1])$. Then $$\mathcal{D}_e = \{ u \in C_D([0,1]) : S( U^u (t, \cdot) ) < 0 \text{ for some }0 \leq t < \tau^u \}.$$ Furthermore, we have $\lim_{t \rightarrow (\tau^u)^-} S( U^u(t,\cdot) ) = -\infty$. As a consequence of these results one can obtain a precise description of the domains $\mathcal{D}_{\mathbf{0}}$ and $\mathcal{D}_e$ in the region of nonnegative data. The following theorem can be found on [@CE2]. \[descomp2\] $\,$ 1. Assume $u \in C_D([0,1])$ is nonnegative and such that $U^u$ is globally defined and converges to $z$ as $t \rightarrow +\infty$. Then for $v \in C_D([0,1])$ we have that 1. $\mathbf{0} \lneq v \lneq u \Longrightarrow U^v$ is globally defined and converges to $\mathbf{0}$ as $t \rightarrow +\infty$. 2. $u \lneq v \Longrightarrow U^v$ explodes in finite time. 2. For every nonnegative $u \in C_D([0,1])$ there exists $\lambda_c^u > 0$ such that for every $\lambda > 0$ 1. $0 < \lambda < \lambda_c^u \Longrightarrow U^{\lambda u}$ is globally defined and converges to $\mathbf{0}$ as $t \rightarrow +\infty$. 2. $\lambda = \lambda_c^u \Longrightarrow U^{\lambda u}$ is globally defined and converges to $z$ as $t \rightarrow +\infty$. 3. $\lambda > \lambda_c^u \Longrightarrow U^{\lambda u}$ explodes in finite time. From this result we obtain the existence of an unstable manifold of the saddle point $z$ which is contained in the region of nonnegative initial data and shall be denoted by $\mathcal{W}^z_u$. It is $1$-dimensional, has nonempty intersection with both $\mathcal{D}_{\mathbf{0}}$ and $\mathcal{D}_e$ and joins $z$ with $\mathbf{0}$. By symmetry, a similar description also holds for the opposite unstable equilibrium $-z$. Figure \[fig1\] depicts the decomposition together with the unstable manifolds $\mathcal{W}^{\pm z}_u$. structure of the remaining unstable equilibria given by Proposition \[equilibrios\] one can verify for each of them the analogue of (ii) in Theorem \[descomp2\]. This is detailed in the following proposition. ![The phase diagram of equation .[]{data-label="fig1"}](dibujo1-santi.eps){width="8cm"} \[descomp3\] If $w \in C_D([0,1]) - \{\mathbf{0}\}$ is an equilibrium of the system then for every $\lambda > 0$ we have that 1. $0 < \lambda < 1 \Longrightarrow U^{\lambda w}$ is globally defined and converges to $\mathbf{0}$ as $t \rightarrow +\infty$. 2. $\lambda = 1 \Longrightarrow U^{\lambda w}$ is globally defined and satisfies $U^{\lambda w} \equiv w$. 3. $\lambda > 1 \Longrightarrow U^{\lambda w}$ explodes in finite time. Let us suppose that $w \equiv z^{(n)}$ for some $n \in {{\mathbb{Z}}}- \{0\}$. Then for any $\lambda > 0$ the solution to with initial datum $\lambda w$ is given by the formula $$U^{\lambda w}(t,x) = \left\{ \begin{array}{rl} |n|^{\frac{2}{p-1}}U^{\text{sg}(n) \lambda z} (|n|^2t,|n|x - [|n|x] ) & \text{ if $[|n|x]$ is even} \\ \\ - |n|^{\frac{2}{p-1}}U^{\text{sg}(n) \lambda z}(|n|^2t,|n|x - [|n|x])& \text{ if $[|n|x]$ is odd}\end{array}\right.$$ where $\text{sg}(n):=\frac{n}{|n|}$ and $U^{\pm \lambda z}$ is the solution to with initial datum $\pm \lambda z$, respectively. That is, $U^{\lambda w}$ is obtained from $U^{\lambda z}$ by performing the analogous procedure to the one explained in Proposition \[equilibrios\] to obtain $w$ from $z$. Indeed, this follows from an argument similar in spirit to the one given for Proposition \[equilibrios\] which exploits the facts that $U^{\lambda z}$ is symmetric, it verifies $U^{-\lambda z}=-U^{\lambda z}$ and also that it vanishes on the boundary of $[0,1]$. Having this formula for $U^{\lambda w}$, now the result follows at once from Theorem \[descomp2\]. Brownian sheet -------------- Throughout our work we consider perturbations of given by additive white noise. This noise term can be regarded as the formal time derivative of a Brownian sheet process. We say that a stochastic process $W=\{W(t,x) : (t,x) \in {{\mathbb R}}^+ \times [0,1]\}$ is a *Brownian sheet* if it satisfies the following properties: 1. $W$ has continuous paths, i.e. $(t,x) \mapsto W{(t,x)}(\omega)$ is continuous for every $\omega \in \Omega$. 2. $W$ is a centered Gaussian process with covariance structure given by $$\text{Cov}( W{(t,x)} , W{(s,y)} ) = (t \wedge s)(x \wedge y)$$ for every $(t,x),(s,y) \in {{\mathbb R}}^+ \times [0,1]$. We refer to [@P; @RY] for the construction of a such a process and a list of its basic properties, as well as the fundamentals of the theory of stochastic integration with respect to it. Definition of solution for the SPDE ----------------------------------- In this first part we study stochastic partial differential equations of the form $$\label{MainSPDE} \left\{\begin{array}{rll} {\partial}_t X &= {\partial}^2_{xx}X + f(X) + \varepsilon \dot{W}& \quad t>0 \,,\, 0<x<1 \\ X(t,0)&=X(t,1)=0 & \quad t>0 \\ X(0,x) &=u(x) \end{array}\right.$$where $\varepsilon > 0$ is some parameter, $u \in C_D([0,1])$ and $f: {{\mathbb R}}\rightarrow {{\mathbb R}}$ is a locally Lipschitz source. It is possible that such equations do not admit strong solutions in the usual sense as these may not be globally defined but instead defined *up to an explosion time*. In the following we review the usual definition of solution when the source is globally Lipschitz as well as formalize the idea of explosion and properly define the concept of solutions in the case of locally Lipschitz sources. ### Definition of strong solution for globally Lipschitz sources We begin by fixing a probability space $(\Omega,{{\mathcal F}},P)$ in which we have defined a $\{ W{(t,x)} : (t,x) \in {{\mathbb R}}^+ \times [0,1]\}$. For every $t \geq 0$ we define $${{\mathcal G}}_t = \sigma( W{(s,x)} : 0 \leq s \leq t , x \in [0,1])$$ and denote its augmentation by ${{\mathcal F}}_t$.[^1] The family $({{\mathcal F}}_t)_{t \geq 0}$ constitutes a filtration on $(\Omega,{{\mathcal F}})$. A *strong solution* of the equation on the probability space $(\Omega,{{\mathcal F}},P)$ with the Brownian sheet $W$ is a stochastic process $$X = \{ X{(t,x)} : (t,x) \in {{\mathbb R}}^+ \times [0,1]\}$$ satisfying the following properties: - $X$ has continuous paths taking values in ${{\mathbb R}}$. - $X$ is adapted to the filtration $({{\mathcal F}}_t)_{t \geq 0}$, i.e. for every $t \geq 0$ the mapping $$(\omega,x) \mapsto X{(t,x)}(\omega)$$ is ${{\mathcal F}}_t \otimes {{\mathcal B}}([0,1])$-measurable. - If $\Phi$ denotes the fundamental solution of the heat equation on the interval $[0,1]$ with homogeneous Dirichlet boundary conditions, which is given by the formula $$\Phi(t,x,y) = \frac{1}{\sqrt{4\pi t}} \sum_{n \in {{\mathbb{Z}}}} \left[ \exp\left( - \frac{(2n+y -x)^2}{4t} \right) - \exp\left( - \frac{(2n+y +x)^2}{4t} \right)\right],$$ then $P$-almost surely we have $$\int_0^1 \int_{0}^{t} |\Phi(t -s,x,y) f(X(s,y))|dsdy < +\infty \hspace{0,5cm} \,\forall\,\,\, 0\leq t < + \infty$$ and $$X(t,x) = I_H(t,x)+ I_N(t,x)\hspace{0,5cm}\,\forall\,\,\, (t,x) \in {{\mathbb R}}^+ \times [0,1],$$ where $I_H$ and $I_N$ are respectively defined by the formulas $$I_H(t,x) = \int_0^1 \Phi(t,x,y)u(y)dy$$ and $$I_N(t,x) = \int_0^{t} \int_0^1 \Phi(t-s,x,y) \left(f(X(s,y))dyds + \varepsilon dW(s,y)\right).$$ It is well known that if $f$ satisfies a global Lipschitz condition then for any initial datum $u \in C_D([0,1])$ there exists a unique strong solution to the equation on $(\Omega,{{\mathcal F}},P)$. Furthermore, this strong solution satisfies the strong Markov property and also behaves as a weak solution in the sense described in the following lemma. See [@W] for details. \[weaksol\] If $X$ is a strong solution to with initial datum $u \in C_D([0,1])$ then for every $\varphi \in C^2((0,1)) \cap C_D([0,1])$ we have $P$-almost surely $$\int_0^1 X(t,x)\varphi(x)dx = I_H^\varphi(t) + I_N^\varphi(t) \hspace{0,5cm} \,\forall\,\,\, 0\leq t < + \infty$$ where for each $t \geq 0$ $$I_H^\varphi(t) = \int_0^1 u(x)\varphi(x)dx$$ and $$I_N^\varphi(t) = \int_0^t\int_0^1 \left( \left(X(s,x)\varphi''(x) + f(X(s,x))\varphi(x)\right)dxds + \varepsilon \varphi(s,x)dW(s,x)\right).$$ ### Solutions up to an explosion time Just as in the previous section we begin by fixing a probability space $(\Omega,{{\mathcal F}},P)$ in which we have defined a $\{ W{(t,x)} : (t,x) \in {{\mathbb R}}^+ \times [0,1]\}$ and consider its augmented generated filtration $({{\mathcal F}}_t)_{t \geq 0}$. A *solution up to an explosion time* of the equation on $(\Omega,{{\mathcal F}},P)$ with respect to $W$ is a stochastic process $X = \{ X{(t,x)} : (t,x) \in {{\mathbb R}}^+ \times [0,1]\}$ satisfying the following properties: - $X$ has continuous paths taking values in $\overline{{{\mathbb R}}}:={{\mathbb R}}\cup \{\pm \infty\}$. - $X$ is adapted to the filtration $({{\mathcal F}}_t)_{t \geq 0}$. - If we define $\tau^{(n)} := \inf\{t>0 : \|X{(t,\cdot)}\|_{\infty}=n\}$ then for every $n \in {{\mathbb N}}$ we have $P$-a.s. $$\int_0^1\int_{0}^{t\wedge{\tau^{(n)}}} |\Phi(t\wedge \tau^{(n)} -s,x,y)f(X(s,y))|dsdy < +\infty \hspace{0,5cm} \,\forall\,\,\, 0\leq t < + \infty$$ and $$X(t \wedge \tau^{(n)},x) = I_H^{(n)}(t,x)+ {\varepsilon}I_N^{(n)}(t,x)\hspace{0,5cm}\,\forall\,\,\, (t,x) \in {{\mathbb R}}^+ \times [0,1],$$ where $$I_H^{(n)}(t,x) = \int_0^1 \Phi(t \wedge \tau^{(n)},x,y)u(y)dy$$ and $$I_N^{(n)}(t,x) = \int_0^{t} \int_0^1 \mathbbm{1}_{\{s \leq \tau^{(n)}\}} \Phi(t\wedge \tau^{(n)} -s,x,y) \left( f(X(s,y))dyds + \varepsilon dW(s,y)\right)$$ with $\Phi$ being the fundamental solution of the heat equation as before. We call $\tau:=\lim_{n \rightarrow +\infty} \tau^{(n)}$ the [*explosion time*]{} for $X$. Let us notice that the assumption of over $\overline{{{\mathbb R}}}$ implies that 1. $\tau = \inf \{ t > 0 : \|X{(t,\cdot)}\|_\infty =+\infty\}$ 2. $\|X{(\tau^-,\cdot)}\|_\infty = \|X{(\tau,\cdot)}\|_\infty =+\infty\,\,\mbox{ on }\,\,\{ \tau < +\infty\}.$ We stipulate that $X{(t,\cdot)}\equiv X(\tau,\cdot)$ for $t \geq \tau$ whenever $\tau < +\infty$ but we shall not assume that $\lim_{t \to +\infty} X{(t,\cdot)}$ exists if $\tau=+\infty$. Furthermore, observe that since any initial datum $u \in C_D([0,1])$ verifies $\|u\|_\infty < +\infty$ we always have $P( \tau > 0) = 1$ and also that if $P(\tau = +\infty)=1$ then we are left with the usual definition of strong solution. In can be shown that for $f \in C^1({{\mathbb R}})$ there exists a unique solution $X$ of up to an explosion time. Furthermore, if $f$ is globally Lipschitz then the solution is globally defined in the sense that $P(\tau = +\infty)=1$. Finally, it is possible to prove that this solution $X$ maintains the strong Markov property, i.e. if $\tilde \tau$ is a stopping time of $X$ then, conditional on $\tilde\tau<\tau$ and $X{(\tilde\tau,\cdot)}=w$, the future $\{ X{(t + \tilde \tau,\cdot)} \colon 0 < t<\tau-\tilde\tau\}$ is independent of the past $\{ X{(s,\cdot)} \colon 0 \leq s\le \tilde\tau \}$ and identical in law to the solution of with We refer to [@IM] for details. Freidlin-Wentzell estimates {#secLDP} --------------------------- One of the main tools we shall use to study the solutions to are the large deviations estimates we briefly describe next. We refer to [@FJL; @B1; @SOW] for further details. Let $X^{u,{\varepsilon}}$ be the solution to the SPDE $$\label{MainSPDE2} \left\{\begin{array}{rll} {\partial}_t X^{u,{\varepsilon}} &= {\partial}^2_{xx}X^{u,{\varepsilon}} + f(X^{u,{\varepsilon}}) + \varepsilon \dot{W}& \quad t>0 \,,\, 0<x<1 \\ X^{u,{\varepsilon}}(t,0)&=X^{u,{\varepsilon}}(t,1)=0 & \quad t>0 \\ X^{u,{\varepsilon}}(0,x) &=u(x) \end{array}\right.$$ where $u \in C_D([0,1])$ and $f: {{\mathbb R}}\to {{\mathbb R}}$ is bounded and satisfies a *global* Lipschitz condition. Let us also consider $X^{u}$ the unique solution to the deterministic equation $$\label{MainPDE2} \left\{\begin{array}{rll} {\partial}_t X^u &= {\partial}^2_{xx}X^u + f(X^u) & \quad t>0 \,,\, 0<x<1 \\ X^u(t,0)&=X^u(t,1)=0 & \quad t>0 \\ X^u(0,x) &=u(x). \end{array}\right.$$ Given $u \in C_D([0,1])$ and $T > 0$, we consider the metric space of continuous functions $$C_{D_u}([0,T] \times [0,1]) = \{ v \in C([0,T]\times[0,1]) : v(0,\cdot)=u \text{ and }v(\cdot,0)=v(\cdot,1)\equiv 0 \}$$ with the distance $d_T$ induced by the supremum norm, i.e. for $v,w \in C_{D_u}([0,T]\times[0,1])$ $$d_T(v,w) := \sup_{(t,x) \in [0,T]\times [0,1]} | v(t,x) - w(t,x) |,$$ and define the rate function $I^u_T : C_{D_u}([0,T]\times [0,1]) \rightarrow [0,+\infty]$ by the formula $$I^u_T (\varphi) = \left\{ \begin{array}{ll} \displaystyle{\frac{1}{2} \int_0^T \int_0^1 |{\partial}_t \varphi - {\partial}_{xx} \varphi - f(\varphi)|^2} & \text{ if }\varphi \in W^{1,2}_2([0,T]\times[0,1]) \,,\,\varphi(0,\cdot) = u \\ \\ +\infty & \text{otherwise.}\end{array}\right.$$ Here $W^{1,2}_2([0,T]\times[0,1])$ is the closure of $C^\infty([0,T] \times [0,1])$ with respect to the norm $$\| \varphi \|_{W^{1,2}_2} = \left( \int_0^T \int_0^1 \left[ |\varphi|^2 + |{\partial}_t \varphi|^2 + |{\partial}_x \varphi|^2 + |{\partial}_{xx} \varphi|^2\right]\right)^\frac{1}{2},$$ i.e. the Sobolev space of square-integrable functions defined on $[0,T]\times [0,1]$ with one square-integrable weak and two square-integrable weak space derivatives. The following estimates hold: 1. For any $\delta > 0$, $h > 0$ there exists $\varepsilon_0 > 0$ such that $$\label{LDP1} P\left( d_T ( X^{u,\varepsilon}, \varphi ) < \delta \right) \geq e^{- \frac{ I^u_T(\varphi) + h }{\varepsilon^2}}$$ for all $0 < \varepsilon < \varepsilon_0$, $u \in C_D([0,1])$ and $\varphi \in C_{D_u}([0,T]\times [0,1])$. 2. For any $\delta > 0$, $h > 0$, $s_0 > 0$ there exists $\varepsilon_0 > 0$ such that $$\label{LDP2} \sup_{u \in C_D([0,1])} P\left( d_T ( X^{u,\varepsilon}, J^u_T(s)) \geq \delta \right) \leq e^{-\frac{s-h}{\varepsilon^2}}$$ for all $0 < \varepsilon < \varepsilon_0$ and $0 < s \leq s_0$, where $$J^u_T(s) = \{ \varphi \in C_{D_u}([0,T]\times [0,1]) : I^u_T(\varphi) \leq s \}.$$ 3. For any $\delta > 0$ there exist $\varepsilon_0 > 0$ and $C > 0$ such that $$\label{grandes1} \sup_{u \in C_D([0,1])} P\left( d_T \left( X^{u,\varepsilon},X^{u}\right) > \delta \right) \leq e^{-\frac{C}{\varepsilon^2}}$$for all $0 < \varepsilon < \varepsilon_0$. The first and second estimates are equivalent to those obtained in [@FJL], except for the uniformity in the initial datum. This uniformity can be obtained as in [@B1] by exploiting the fact that $f$ is bounded and Lipschitz. On the other hand, the last estimate is in fact implied by the second one. Indeed, if $V^{\mathbf{0},\varepsilon}$ and $V^{\mathbf{0}}$ respectively denote the solutions to and with initial datum $\mathbf{0}$ and source term $f \equiv 0$, then (iii) is obtained from (ii) upon noticing that there exists $K > 0$ depending on $f$ such that for any $u \in C_D([0,1])$ $$\label{LDP3} d_T \left( X^{u,\varepsilon},X^{u}\right) \leq e^{KT} d_T ( V^{\mathbf{0},\varepsilon}, V^{\mathbf{0}} )$$ and that given $\delta > 0$ there exists $s_0 > 0$ such that $$\label{LDP4} \{ d_T ( V^{\mathbf{0},\varepsilon}, V^{\mathbf{0}} ) > \delta \} \subseteq \left\{ d_T ( V^{\mathbf{0},\varepsilon}, \tilde{J}_T(s_0)) > \frac{\delta}{2}\right\}$$ where for $s \geq 0$ we set $$\tilde{J}^{\mathbf{0}}_T(s) = \{ \varphi \in C_{D_{\mathbf{0}}}([0,T]\times [0,1]) : \tilde{I}^{\mathbf{0}}_T(\varphi) \leq s \}$$ and $\tilde{I}^{\mathbf{0}}_T$ is the rate function obtained by setting $u=\mathbf{0}$ and $f \equiv 0$ in the definition above. The estimate in is obtained as in [@B1] whereas the inclusion in follows from the fact that the level sets $\tilde{J}^{\mathbf{0}}_T(s)$ are compact for all $s \geq 0$ and also that the rate function $\tilde{I}^{\mathbf{0}}_T$ vanishes only at $V^{\mathbf{0}}$. Truncations of the potential and localization {#trunca} --------------------------------------------- The large deviations estimates given on Section \[secLDP\] demand a global Lipschitz condition on the source term $f$ which is unfortunately not satisfied for our model. Even though large deviations estimates have been obtained for systems with locally Lipschitz sources (see for example [@FJL; @A]), these always rely on some sort of a priori control on the growth of solutions. Hence, we cannot hope to obtain similar results for our system in the study of the explosion time. Nonetheless, the use of localization techniques will help us solve this problem and allow us to take advantage of the estimates on Section \[secLDP\]. we give details about the localization procedure to be employed in the study of our system. For every $n \in {{\mathbb N}}$ let $G^{(n)} : {{\mathbb R}}\longrightarrow {{\mathbb R}}$ be a smooth function such that $$\label{gtruncada} G^{(n)}(u) = \left\{\begin{array}{ll} \frac{|u|^{p+1}}{p+1} &\,\,\text{if}\,\,|u| \leq n\\ 0 &\,\,\text{if}\,\,|u| \geq 2n \end{array}\right.$$ and consider the potential $S^{(n)}$ given by the formula $$S^{(n)}(v)= \left\{\begin{array}{ll}\displaystyle{\int_0^1 \left[\frac{1}{2} \left(\frac{dv}{dx}\right)^2 - G^{(n)}(v)\right]} & \text{ if }v \in H^1_0 ((0,1)) \\ \\ +\infty & \text{ otherwise.}\end{array}\right.$$ For every $u \in C_D([0,1])$ there exists a unique solution $U^{(n),u}$ to the partial differential equation $${\partial}_t U = - \frac{\partial S^{(n)}}{\partial \varphi}(U)$$ with initial datum $u$. Since the source $g_n:=\left(G^{(n)}\right)'$ is globally Lipschitz, this solution $U^{(n),u}$ is globally defined and describes the same trajectory as the solution to starting at $u$ until $\tau^{(n),u}$, the escape time from the ball $$B_n :=\{ v \in C_D([0,1]) : \| v \|_\infty \leq n \}.$$ In the same way, for each $\varepsilon > 0$ there exists a unique solution $U^{(n),u,\varepsilon}$ to the stochastic partial differential equation $$\label{eqtruncada} {\partial}_t U = - \frac{\partial S^{(n)}}{\partial \varphi}(U) + \varepsilon \dot{W}$$ with initial datum $u$ and it is globally defined. Moreover, since for $n \leq m$ the functions $G_n$ and $G_m$ coincide on $B_n$ by uniqueness of the solution we have that $U^{(n),u,\varepsilon}$ and $U^{(m),u,\varepsilon}$ coincide until the escape from $B_n$. Therefore, if we write $$\label{tiempostau} \tau^{(n), u}_\varepsilon = \inf \{ t \geq 0 : \|U^{(n),u,\varepsilon}(t,\cdot)\|_\infty \geq n \}, \qquad \tau_\varepsilon^u:=\lim_{n \rightarrow +\infty} \tau^{(n),u}_\varepsilon,$$ then for $t < \tau_\varepsilon^u$ we have that $U^{u,\varepsilon}(t) := \lim_{n \rightarrow +\infty} U^{(n),\,u,\,\varepsilon}(t)$ is well defined and constitutes the solution to until the explosion time $\tau^u_\varepsilon$ with initial datum $u$. Let us observe that for each $n \in {{\mathbb N}}$ this solution $U^{u,\,\varepsilon}$ coincides with $U^{(n),\,u,\,\varepsilon}$ until the escape from $B_n$. Furthermore, each $U^{(n),u,\varepsilon}$ is a positive recurrent Markov process which almost surely hits any open set in $C_D([0,1])$ in a finite time. Finally, since each $g_n$ is bounded and Lipschitz we have that for every $n \in {{\mathbb N}}$ the family $\left(U^{(n),u,\varepsilon}\right)_{\varepsilon > 0}$ satisfies the large deviations estimates given in Section \[secLDP\]. Hereafter, whenever we refer to the solution of we shall mean the solution constructed in this particular manner. Main results ------------ Our purpose in this first part of the thesis is to study the asymptotic behavior as $\varepsilon \rightarrow 0$ of $U^{u,\varepsilon}$, the solution to the equation , for the different initial data $u \in C_{D}([0,1])$. the main results we have obtained in this regard. From now onwards we shall write $P_u$ to denote the law of the stochastic process $U^{u,\varepsilon}$. Whenever the initial datum is made clear in this way we shall often choose to drop the superscript $u$ from the remaining notation for simplicity purposes. Our first result is concerned with the continuity of the explosion time for initial data in the domain of explosion $\mathcal{D}_e$. In this case one expects the stochastic and deterministic systems to both exhibit a similar behavior for $\varepsilon > 0$ sufficiently small, since then the noise will not be able to grow fast enough so as to overpower the quickly exploding source term. We show this to be truly the case for $u \in \mathcal{D}_e$ such that **Theorem I**. Let $\mathcal{D}^*_e$ be the set of those $u \in \mathcal{D}_e$ such that $U^u$ explodes only through one side, i.e. $U^{u}$ remains bounded either from below or above until its explosion time $\tau^u$. Then given $\delta > 0$ and a bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}^*_e$ there exists a constant $C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( |\tau_\varepsilon - \tau| > \delta ) \leq e^{-\frac{C}{\varepsilon^2}}.$$ The main differences in behavior between the stochastic and deterministic systems appear for initial data in $\mathcal{D}_{{\mathbf 0}}$, where metastable behavior is observed. According to the characterization of metastability for stochastic processes given in the articles [@CGOV] and [@GOV], metastable behavior is given by two facts: the time averages of the process remain stable until an abrupt transition occurs and then a different value is attained; furthermore, the time of this transition is unpredictable in the sense that, when suitably rescaled, it should have an exponential distribution. We manage to establish this description rigorously for our system whenever $1 < p < 5$, where $p$ is the parameter in the source term of . This rigorous description is contained in the remaining results. We begin by defining for each $\varepsilon > 0$ the scaling coefficient $$\label{defibeta} \beta_{\varepsilon}= \inf \{ t \geq 0 : P_{\mathbf{0}} ( \tau_{\varepsilon}> t ) \leq e^{-1} \}$$ and show that the family $(\beta_\varepsilon)_{\varepsilon > 0}$ verifies $\lim_{\varepsilon \rightarrow 0} \varepsilon ^{2}\log\beta_{\varepsilon} = \Delta$, where $\Delta := 2(S(z) - S(\mathbf{0}))$. In fact, we shall prove the following stronger statement which details the asymptotic order of magnitude of $\tau^u_\varepsilon$ for initial data $u \in \mathcal{D}_{{\mathbf 0}}$. **Theorem II**. Given $\delta > 0$ and a bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive distance from ${\partial}\mathcal{D}_{\mathbf{0}}$ we have $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} \left| P_u \left( e^{\frac{\Delta - \delta}{\varepsilon^2}} < \tau_\varepsilon < e^{\frac{\Delta + \delta}{\varepsilon^2}}\right)-1\right|\right]=0.$$ Next we show the asymptotic loss of memory of $\tau^u_\varepsilon$ for initial data $u \in \mathcal{D}_{{\mathbf 0}}$. **Theorem III**. Given $\delta > 0$ and a bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive distance from ${\partial}\mathcal{D}_{\mathbf{0}}$ we have for any $t > 0$ $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} \left| P_{u} (\tau_{\varepsilon} > t\beta_{\varepsilon}) - e^{-t} \right| \right] = 0.$$ Finally, we show the stability of time averages of continuous functions evaluated along paths of the process starting in $\mathcal{D}_{\mathbf{0}}$, i.e. they remain close to the value of the These time averages are taken along intervals of length going to infinity and times may be taken as being almost (in a suitable scale) the explosion time. This tells us that, up until the explosion time, the system spends most of its time in a small **Theorem IV**. There exists a sequence $(R_\varepsilon)_{\varepsilon > 0}$ with $\lim_{\varepsilon \rightarrow 0} R_\varepsilon = +\infty$ and $\lim_{\varepsilon \rightarrow 0} \frac{R_\varepsilon}{\beta_\varepsilon} = 0$ such that given $\delta > 0$ for any bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive we have $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \frac{1}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(\mathbf{0})\right| > \delta \right) \right] = 0$$ for any bounded continuous function $f: C_D([0,1]) \rightarrow {{\mathbb R}}$. Theorem I is proved in Chapter 2, the remaining results are proved in Perhaps the proof of Theorem I is where one can find the most differences with other works in the literature dealing with similar problems. In these works, the analogue of can be obtained as a direct consequence of the large deviations estimates for the system. However, since in our case Theorem I particularly focuses on trajectories of the process as it escapes any bounded domain, the estimates on will not be of any use for the proof. Thus, a new approach is needed, one which is different from previous approaches in the literature and does not rely on large deviations estimates. The remaining results were established in [@B1; @MOS] for the tunneling time in an infinite-dimensional double-well potential model, i.e. the time the system takes to go from one well to the bottom of the other one. Our proofs are similar to the ones found in these references, although we have the additional difficulty of dealing with solutions which are not globally defined. Resumen del Capítulo 1 ---------------------- En este primer capítulo introducimos las nociones y conceptos preliminares para poder estudiar nuestro problema. La EDP con blow-up que vamos a considerar es $$\left\{\begin{array}{rll} {\partial}_t U &= {\partial}^2_{xx}U + g(U) & \quad t>0 \,,\, 0<x<1 \\ U(t,0)& =0 & \quad t>0 \\ U(t,1) & = 0 & \quad t>0 \\ U(0,x) &=u(x) & \quad 0<x<1 \end{array}\right.$$ donde $g : {{\mathbb R}}\rightarrow {{\mathbb R}}$ viene dada por $g(u)=u|u|^{p-1}$ para $p > 1$ y $u$ pertenece al espacio $$C_{D}([0,1])= \{ v \in C([0,1]) : v(0)=v(1)=0 \}.$$ Dicha ecuación puede reformularse como $${\partial}_t U = - \frac{\partial S}{\partial \varphi} (U)$$ donde el potencial $S$ es el funcional en $C_D([0,1])$ dado por $$S(v) = \left\{ \begin{array}{ll} \displaystyle{\int_0^1 \left[\frac{1}{2} \left(\frac{dv}{dx}\right)^2 - \frac{|v|^{p+1}}{p+1}\right]} & \text{ si $v \in H^1_0((0,1))$} \\ \\ +\infty & \text{ en caso contrario.}\end{array}\right.$$ El origen $\mathbf{0} \in C_D([0,1])$ es el único equilibrio estable del sistema y es, de hecho, asintóticamente estable. Corresponde al único mínimo local del potencial $S$. Existe también una familia de equilibrios inestables del potencial $S$, todos ellos puntos de ensilladura. Entre estos equilibrios inestables existe un único equilibrio que es no negativo, $z$. Puede mostrarse que $z$ es de hecho estrictamente positivo en $(0,1)$ y tanto de mínima energía como norma entre los equilibrios inestables. Además, $C_D([0,1])$ puede descomponerse en tres partes: $$C_D([0,1]) = \mathcal{D}_{\mathbf{0}} \cup \mathcal{W} \cup \mathcal{D}_e$$ donde $\mathcal{D}_{\mathbf{0}}$ denota la variedad estable del origen, $\mathcal{W}$ es la unión de todas las variedades estables de los equilibrios inestables y $\mathcal{D}_e$ constituye el dominio de explosión del sistema, i.e. el conjunto de todos aquellos datos iniciales $u$ para los cuales el sistema explota en un tiempo finito $\tau^u$. Puede verse que tanto $\mathcal{D}_{\mathbf{0}}$ como $\mathcal{D}_e$ son conjuntos abiertos y que $\mathcal{W}$ es la frontera común que los separa. Además, existe una variedad inestable $\mathcal{W}^z_u$ del punto de ensilladura $z$ contenida en la región de datos no negativos. La misma es $1$-dimensional, tiene intersección no vacía tanto con $\mathcal{D}_{\mathbf{0}}$ como con $\mathcal{D}_e$ y une a $z$ con $\mathbf{0}$. Por simetría, una descripción análoga también vale para el equilibrio inestable opuesto $-z$. La Figura \[fig1\] describe esta descomposición. Las perturbaciones estocásticas que consideramos son de la forma $$\label{formalSPDEresumen} {\partial}_t U^{{\varepsilon}} = - \nabla S + \varepsilon \dot{W}$$ donde $W$ es una sábana Browniana. Definimos formalmente el concepto de solución a una ecuación de este tipo, lo cual excede el marco tradicional ya que las mismas podrían no estar definidas globalmente sino hasta un tiempo de explosión $\tau_\varepsilon$ finito. Estudiamos además dos propiedades importantes de las soluciones a este tipo de ecuaciones: la propiedad fuerte de Markov y el principio de grandes desvíos para los sistemas truncados asociados. Por último, terminamos el capítulo presentando los resultados que habremos de probar en los capítulos siguientes. Incluimos una breve descripción de los mismos aquí. Nuestro primer resultado es con respecto a la continuidad del tiempo de explosión para datos iniciales en $\mathcal{D}_e$. En este caso uno espera que que los sistemas estocástico y determinístico exhiban ambos un comportamiento similar para $\varepsilon > 0$ suficientemente pequeño, ya que entonces el ruido no tendrá el tiempo suficiente como para crecer lo necesario para sobrepasar al término de la fuente que está explotando. Mostramos que esto es en efecto así para los casos en que $u \in \mathcal{D}_e$ es tal que la solución $U^u$ de con dato inicial $u$ permanece acotada por un lado. **Teorema I**. Sea $\mathcal{D}^*_e$ el conjunto de aquellos $u \in \mathcal{D}_e$ tales que $U^u$ explota sólo por un lado, i.e. $U^{u}$ permanece acotada ya sea inferior o superiormente hasta su tiempo de explosión $\tau^u$. Entonces dado $\delta > 0$ y un conjunto acotado $\mathcal{K} \subseteq \mathcal{D}_e^*$ a una distancia positiva de ${\partial}\mathcal{D}^*_e$ existe $C > 0$ tal que $$\sup_{u \in \mathcal{K}} P ( |\tau_\varepsilon^u - \tau_0^u| > \delta ) \leq e^{-\frac{C}{\varepsilon^2}}.$$ donde $\tau^u_\varepsilon$ denota el tiempo de explosión de $U^{u,\varepsilon}$, la solución de con dato inicial $u$. Las principales diferencias en comportamiento entre ambos sistemas surgen para datos iniciales en $\mathcal{D}_{{\mathbf 0}}$, donde se presenta el fenómeno de metaestabilidad. De acuerdo con [@GOV], el comportamiento metaestable viene dado por dos hechos: los promedios temporales del proceso permanecen estables hasta que ocurre una transición abrupta y luego un valor diferente se obtiene; más aún, el tiempo en que ocurre esta transición es impredecible en el sentido de que, bajo una normalización apropiada, debería tener una distribución exponencial. Logramos establecer esta descripción rigurosamente para nuestro sistema para los casos en que $1 < p < 5$, donde $p$ es el parámetro en el término no lineal de la fuente en . Esta descripción rigurosa abarca los restantes resultados. **Teorema II**. Dado $\delta > 0$ y un conjunto acotado $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ a una distancia positiva de ${\partial}\mathcal{D}_{\mathbf{0}}$ tenemos $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} \left| P \left( e^{\frac{\Delta - \delta}{\varepsilon^2}} < \tau^u_\varepsilon < e^{\frac{\Delta + \delta}{\varepsilon^2}}\right)-1\right|\right]=0.$$ **Teorema III**. Dado $\delta > 0$ y un conjunto acotado $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ a una distancia positiva de ${\partial}\mathcal{D}_{\mathbf{0}}$ tenemos para cualquier $t > 0$ $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} \left| P (\tau_{\varepsilon}^u > t\beta_{\varepsilon}) - e^{-t} \right| \right] = 0.$$ donde para cada $\varepsilon > 0$ definimos el coeficiente de normalización $\beta_\varepsilon$ como $$\beta_{\varepsilon}= \inf \{ t \geq 0 : P_{\mathbf{0}} ( \tau_{\varepsilon}> t ) \leq e^{-1} \}.$$ **Teorema IV**. Existe una sucesión $(R_\varepsilon)_{\varepsilon > 0}$ con $\lim_{\varepsilon \rightarrow 0} R_\varepsilon = +\infty$ y $\lim_{\varepsilon \rightarrow 0} \frac{R_\varepsilon}{\beta_\varepsilon} = 0$ tal que dado $\delta > 0$ para cualquier conjunto acotado $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ a una distancia positiva de $\mathcal{W}$ $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \frac{1}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(\mathbf{0})\right| > \delta \right) \right] = 0$$ para cualquier función continua $f: C_D([0,1]) \rightarrow {{\mathbb R}}$. Asymptotic behavior of $\tau_\varepsilon^u$ for $u \in \mathcal{D}_e$ ===================================================================== In this chapter we investigate the continuity properties of the explosion time $\tau_\varepsilon^u$ for initial data in the domain of explosion $\mathcal{D}_e$. Our purpose is to show that under suitable conditions on the initial datum $u \in \mathcal{D}_e$ the explosion time $\tau_\varepsilon^u$ of the stochastic system converges in probability to the deterministic explosion time $\tau^u$. To make these conditions more precise, let us consider the sets of initial data in $\mathcal{D}_e$ which explode only through $+\infty$ or $-\infty$, i.e. $$\mathcal{D}_e^+ = \left\{ u \in \mathcal{D}_e : \inf_{(t,x) \in [0,\tau^u) \times [0,1]} U^u(t,x) > -\infty \right\}$$ and $$\mathcal{D}_e^- = \left\{ u \in \mathcal{D}_e : \sup_{(t,x) \in [0,\tau^u) \times [0,1]} U^u(t,x) < +\infty \right\}.$$ Notice that $\mathcal{D}_e^+$ and $\mathcal{D}_e^-$ are disjoint and also that they satisfy the relation $\mathcal{D}_e^- = - \mathcal{D}_e^+$. Furthermore, we shall see below that $\mathcal{D}_e^+$ is an open set. Let us write $\mathcal{D}_e^*:= \mathcal{D}_e^+ \cup \mathcal{D}_e^-$. The result we are to prove is the following. \[contexp\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}_e^*$ and $\delta > 0$ there exists a constant $C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( |\tau_\varepsilon - \tau| > \delta ) \leq e^{- \frac{C}{\varepsilon^2}}.$$ We shall split the proof of Theorem \[contexp\] into two parts: proving first a lower bound and then an upper bound for $\tau_\varepsilon$. The first one is a consequence of the continuity of solutions to with respect to ${\varepsilon}$ on intervals where the deterministic solution The precise estimate is contained in the following proposition. \[convergenciainferior0\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_e$ and $\delta > 0$ there exists a constant $C > 0$ such that $$\label{convergenciainferior} \sup_{u \in \mathcal{K}} P_u ( \tau_\varepsilon < \tau - \delta ) \leq e^{- \frac{C}{\varepsilon^2}}.$$ Let us observe that by Proposition \[G.2\] we have that $\inf_{u \in \mathcal{K}} \tau^u > 0$ so that we may assume without loss of generality that $\tau^u > \delta$ for all $u \in \mathcal{K}$. Now, for each $u \in \mathcal{D}_e$ let us define the quantity $$M_u := \sup_{0 \leq t \leq \max\{0, \tau^u - \delta \}} \|U^{u}(t,\cdot)\|_\infty.$$ By resorting to Proposition \[G.2\] once again, we obtain that the application $u \mapsto M_u$ is both upper semicontinuous and finite on $\mathcal{D}_e$ and hence, with the aid of Propositions \[G.1\] and \[A.2\], we conclude that $M:= \sup_{u \in \mathcal{K}} M_u < +\infty$. Similarly, since the mapping $u \mapsto \tau^u$ is continuous and finite on $\mathcal{D}_e$ (see Corollary \[contdetexp\] below for proof of this fact) we also obtain that $\mathcal{T}:= \sup_{u \in \mathcal{K}} \tau^u < +\infty$. Hence, for $u \in \mathcal{K}$ we get $$P_u (\tau^u_\varepsilon < \tau^u - \delta ) \leq P_u \left( d_{\tau^u - \delta}\left(U^{M_u+1,\varepsilon},U^{M_u+1}\right) > \frac{1}{2}\right) \leq P_u \left( d_{\mathcal{T} - \delta}\left(U^{M+1,\varepsilon},U^{M+1}\right) > \frac{1}{2}\right).$$ By the estimate we conclude . To establish the upper bound we consider for each $u \in \mathcal{D}_e^+$ the process $$Z^{u,\varepsilon} := U^{u,\varepsilon} - V^{\mathbf{0},\varepsilon}$$ where $U^{u,\varepsilon}$ is the solution of with initial datum $u$ and $V^{\mathbf{0},\varepsilon}$ is the solution of with source term $f \equiv 0$ and initial datum $\mathbf{0}$ constructed from the same Brownian sheet as $U^{u,\varepsilon}$. Let us observe that $Z^{u,\varepsilon}$ satisfies the random partial differential equation $$\label{randomPDE} \left\{\begin{array}{rll} {\partial}_t Z^{u,\varepsilon} &= {\partial}^2_{xx}Z^{u,\varepsilon} + g(Z^{u,\varepsilon} - V^{\mathbf{0},\varepsilon}) & \quad t>0 \,,\, 0<x<1 \\ Z^{u,\varepsilon}(t,0)&=Z^{u,\varepsilon}(t,1)=0 & \quad t>0 \\ Z^{u,\varepsilon}(0,x) &=u(x). \end{array}\right.$$Furthermore, since $V^{\mathbf{0},\varepsilon}$ is globally defined and remains bounded on finite time intervals, we have that $Z^{u,\varepsilon}$ and $U^{u,\varepsilon}$ share the same explosion time. Hence, to obtain the desired upper bound on $\tau^u_\varepsilon$ we may study the behavior of $Z^{u,\varepsilon}$. The advantage of this approach is that, in general, the behavior of $Z^{u,\varepsilon}$ will be easier to understand than that of $U^{u,\varepsilon}$. Indeed, each realization of $Z^{u,\varepsilon}$ is the solution of a partial differential equation which one can handle by resorting to standard arguments in PDE theory. Now, a straightforward calculation using the mean value theorem shows that whenever $\| V^{\mathbf{0},\varepsilon} \|_\infty < 1$ the process $Z^{u,\varepsilon}$ satisfies the inequality $$\label{eqcomparacion} {\partial}_t Z^{u,\varepsilon} \geq {\partial}^2_{xx} Z^{u,\varepsilon} + g(Z^{u,\varepsilon}) - h|Z^{u,\varepsilon}|^{p-1} - h$$ where $h := p2^{p-1}\| V^{\mathbf{0},\varepsilon} \|_\infty > 0$. Therefore, in order to establish the upper bound on $\tau_\varepsilon^u$ one may consider for $h > 0$ the solution $\underline{Z}^{(h),u}$ to the equation $$\label{randomPDE2} \left\{\begin{array}{rll} {\partial}_t \underline{Z}^{(h),u} &= {\partial}^2_{xx}\underline{Z}^{(h),u} + g(\underline{Z}^{(h),u}) - h|\underline{Z}^{(h),u}|^{p-1} - h & \quad t>0 \,,\, 0<x<1 \\ \underline{Z}^{(h),u}(t,0)&=\underline{Z}^{(h),u}(t,1)=0 & \quad t>0 \\ \underline{Z}^{(h),u}(0,x) &=u(x). \end{array}\right.$$ and obtain a convenient upper bound for the explosion time of this new process valid for every $h$ sufficiently small. If we also manage to show that for $h$ suitably small the process $\underline{Z}^{(h),u}$ explodes through $+\infty$, then the fact that $Z^{u,\varepsilon}$ is a supersolution to will yield the desired upper bound on the explosion time of $Z^{u,\varepsilon}$, if $\| V^{\mathbf{0},\varepsilon} \|_\infty$ remains small enough. To show this, however, we will need to impose the additional condition that $u \in \mathcal{D}_e^+$. Lemma \[expestimate\] below contains the proper estimate on $\underline{\tau}^{(h),u}$, the explosion time of $\underline{Z}^{(h),u}$. For each $h \geq 0$ we define the potential $\underline{S}^{(h)}$ on $C_D([0,1])$ associated to the equation by the formula $$\underline{S}^{(h)}(v)\left\{ \begin{array}{ll} \displaystyle{\int_0^1 \left[\frac{1}{2} \left(\frac{dv}{dx}\right)^2 - \frac{|v|^{p+1}}{p+1} + hg(v) + hv\right]} & \text{ if $v \in C_D \cap H^1_0([0,1])$ }\\ \\ +\infty & \text{ otherwise.}\end{array}\right.$$ Notice that $\underline{S}^{(0)}$ coincides with our original potential $S$. Moreover, it is easy to check that for all $h \geq 0$ the potential $\underline{S}^{(h)}$ satisfies all properties established for $S$ in the Appendix. \[expestimate\] Given $\delta > 0$ there exists $M > 0$ such that: 1. For every $0 \leq h < 1$ any $u \in C_D([0,1])$ such that $\underline{S}^{(h)}(u) \leq - \frac{M}{2}$ verifies $\underline{\tau}^{(h),u} < \frac{\delta}{2}$. 2. Given $K > 0$ there exist constants $\rho_{M,K}, h_{M,K} > 0$ depending only on $M$ and $K$ such that any $u \in C_D([0,1])$ satisfying $S(u) \leq - M$ and $\| u\|_\infty \leq K$ verifies $$\sup_{v \in B_{\rho_{M,K}}(u)} \underline{\tau}^{(h),v} < \delta$$ for all $0 \leq h < h_{M,K}$. Given $\delta > 0$ let us begin by showing that (i) holds for an appropriate choice of $M$. Thus, for fixed $M > 0$ and $0 \leq h < 1$, let $u \in C_D([0,1])$ be such that $\underline{S}^{(h)}(u) \leq - \frac{M}{2}$ and consider the application $\phi^{(h),u}: [0, \tau^{(h),u}) \rightarrow {{\mathbb R}}^+$ given by the formula $$\phi^{(h),u}(t) = \int_0^1 \left(\underline{Z}^{(h),u}(t,x)\right)^2dx.$$ It is simple to verify that $\phi^{(h),u}$ is continuous and that for any $t_0 \in (0,\tau^{(h),u})$ it satisfies $$\label{eqpoten1} \frac{d\phi^{(h),u}}{dt}(t_0) \geq - 4\underline{S}^{(h)}(u^{(h)}_{t_0}) + 2 \int_0^1 \left[ \left(\frac{p-1}{p+1}\right) |u^{(h)}_{t_0}|^{p+1} - h \left(\frac{p+2}{p}\right)|u^{(h)}_{t_0}|^p - h|u^{(h)}_{t_0}|\right]$$ where we write $u^{(h)}_{t_0}:=\underline{Z}^{(h),u}(t_0,\cdot)$ for convenience. Hölder’s inequality reduces to $$\label{eqpoten2} \frac{d\phi^{(h),u}}{dt}(t_0) \geq - 4\underline{S}^{(h)}(u^{(h)}_{t_0}) + 2\left[ \left(\frac{p-1}{p+1}\right) \|u^{(h)}_{t_0}\|_{L^{p+1}}^{p+1} - h\left(\frac{p+2}{p}\right)\|u^{(h)}_{t_0}\|_{L^{p+1}}^{p} - h\|u^{(h)}_{t_0}\|_{L^{p+1}}\right].$$ Observe that, by definition of $\underline{S}^{(h)}$ and the fact that the map $t \mapsto \underline{S}^{(h)}(u^{(h)}_t)$ is decreasing, we obtain the inequalities $$\frac{M}{2} \leq -\underline{S}^{(h)}(u^{(h)}_{t_0}) \leq \frac{1}{p+1}\|u^{(h)}_{t_0}\|_{L^{p+1}}^{p+1} + h\|u^{(h)}_{t_0}\|_{L^{p+1}}^p + h\|u^{(h)}_{t_0}\|_{L^{p+1}}$$ from which we deduce that by taking $M$ sufficiently large one can force $\|u^{(h)}_{t_0}\|_{L^{p+1}}$ to be large enough so as to guarantee that $$\left(\frac{p-1}{p+1}\right) \|u^{(h)}_{t_0}\|_{L^{p+1}}^{p+1} - h \left(\frac{p+2}{p}\right)\|u^{(h)}_{t_0}\|_{L^{p+1}}^p - h\|u^{(h)}_{t_0}\|_{L^{p+1}} \geq \frac{1}{2}\left(\frac{p-1}{p+1}\right) \|u^{(h)}_{t_0}\|_{L^{p+1}}^{p+1}$$ is satisfied for any $0 \leq h < 1$. Therefore, we see that if $M$ sufficiently large then for all $0 \leq h < 1$ the application $\phi^{(h),u}$ satisfies $$\label{eqpoten3} \frac{d\phi^{(h),u}}{dt}(t_0) \geq 2M + \left(\frac{p-1}{p+1}\right) \left(\phi^{(h),u}(t_0)\right)^{\frac{p+1}{2}}$$ for every $t_0 \in (0,\tau^{(h),u})$, where to obtain we have once again used Hölder’s inequality and the fact that the map $t \mapsto \underline{S}^{(h)}(u^{(h)}_t)$ is decreasing. Now, it is not hard to show that the solution $y$ of the ordinary differential equation $$\left\{\begin{array}{l} \dot{y} = 2M + \left(\frac{p-1}{p+1}\right) y^{\frac{p+1}{2}} \\ y(0) \geq 0 \end{array}\right.$$ explodes before time $$T = \frac{\delta}{4} + \frac{2^{\frac{p+1}{2}}(p+1)}{(p-1)^2(M\delta)^{\frac{p-1}{2}}}.$$ Indeed, either $y$ explodes before time $\frac{\delta}{4}$ or $\tilde{y}:= y( \cdot + \frac{\delta}{4})$ satisfies $$\left\{\begin{array}{l} \dot{\tilde{y}} \geq \left(\frac{p-1}{p+1}\right) \tilde{y}^{\frac{p+1}{2}} \\ \tilde{y}(0) \geq \frac{M\delta}{2} \end{array}\right.$$ which can be seen to explode before time $$\tilde{T}=\frac{2^{\frac{p+1}{2}}(p+1)}{(p-1)^2(M\delta)^{\frac{p-1}{2}}}$$ by performing the standard integration method. If $M$ is taken sufficiently large then $T$ can be made strictly smaller than $\frac{\delta}{2}$ which, by , implies that $\tau^{(h),u} < \frac{\delta}{2}$ as desired. Now let us show statement (ii). Given $K > 0$ let us take $M > 0$ as above and consider $u \in C_D([0,1])$ satisfying $S(u) \leq -M$ and $\|u\|_\infty \leq K$. Using Propositions \[S.1\] and \[Lyapunov\] adapted to the system we may find $\rho_{M,K} > 0$ sufficiently small so as to guarantee that for some small $0 < t_{u} < \frac{\delta}{2}$ any $v \in B_{\rho_{M,K}}(u)$ satisfies $$\underline{S}^{(h)}(\underline{Z}^{(h),v}(t_{u},\cdot)) \leq \underline{S}^{(h)}(u) +\frac{M}{4}$$ for all $0 \leq h < 1$. Notice that this is possible since the constants appearing in Propositions \[S.1\] adapted to this context are independent from $h$ provided that $h$ remains bounded. These constants still depend on $\| u \|_\infty$ though, so that the choice of $\rho_{M,K}$ will inevitably depend on both $M$ and $K$. Next, let us take $0 < h_{M,K} < 1$ so as to guarantee that $\underline{S}^{(h)}(u) \leq - \frac{3M}{4}$ for every $0 \leq h < h_{M,K}$. Notice that, since $\underline{S}^{(h)}(u) \leq S(u) + h(K^p + K),$ it is possible to choose $h_{M,K}$ depending only on $M$ and $K$. Thus, for any $v \in B_{\rho_{M,K}}(u)$ we obtain $\underline{S}^{(h)}(\underline{Z}^{(h),v}(t_{u},\cdot)) \leq - \frac{M}{2}$ which, by the choice of $M$, implies that $\tau^{(h),v} < t_u + \frac{\delta}{2} < \delta$. This concludes the proof. Let us observe that the system $\overline{Z}^{(0),u}$ coincides with $U^u$ for every $u \in C_D([0,1])$. Thus, by the previous lemma we obtain the following corollary. \[contdetexp\] The application $u \mapsto \tau^u$ is continuous on $\mathcal{D}_e$. Given $u \in \mathcal{D}_e$ and $\delta > 0$ we show that there exists $\rho > 0$ such that for all $v \in B_\rho(u)$ we have $$-\delta + \tau^u < \tau^v < \tau^u + \delta.$$ To see this we first notice that by Proposition \[G.2\] there exists $\rho_1 > 0$ such that $-\delta + \tau^u < \tau^v$ for any $v \in B_{\rho_1}(u)$. On the other hand, by (i) in Lemma \[expestimate\] we may take $M, \tilde{\rho_2} > 0$ such that $\tau^{\tilde{v}} < \delta$ for any $\tilde{v} \in B_{\tilde{\rho_2}}(\tilde{u})$ with $\tilde{u} \in C_D([0,1])$ such that $S(\tilde{u}) \leq - M$. For this choice of $M$ by Proposition \[caract\] we may find some $0 <t_M < t^u$ such that $S( U^u(t_M,\cdot) ) \leq -M$ and using Proposition \[G.2\] we may take $\rho_2 > 0$ such that $U^v(t_M,\cdot) \in B_{\tilde{\rho_2}}(U^u(t_M,\cdot))$ for any $v \in B_{\rho_2}(u)$. This implies that $\tau^v < t_M + \delta < t^u + \delta$ for all $v \in B_{\rho_2}(u)$ and thus by taking $\rho = \min\{ \rho_1,\rho_2\}$ we obtain the result. The following two lemmas provide the necessary tools to obtain the uniformity in the upper bound claimed in Theorem \[contexp\]. \[lemacontsup\] Given $M > 0$ and $u \in \mathcal{D}_e$ let us define the quantities $$\mathcal{T}_M^u = \inf\{ t \in [0,\tau^u) : S( U^u(t,\cdot) ) < - M \} \hspace{1cm}\text{ and }\hspace{1cm}\mathcal{R}_M^u = \sup_{0 \leq t \leq \mathcal{T}_M^u} \| U^u(t,\cdot)\|_\infty.$$ Then the applications $u \mapsto \mathcal{T}_M^u$ and $u \mapsto \mathcal{R}_M^u$ are both upper semicontinuous on $\mathcal{D}_e$. We must see that the sets $\{ \mathcal{T}_M < \alpha\}$ and $\{ \mathcal{R}_M < \alpha \}$ are open in $\mathcal{D}_e$ for all $\alpha > 0$. But the fact that $\{ \mathcal{T}_M < \alpha\}$ is open follows at once from Proposition \[S.1\] and $\{ \mathcal{R}_M < \alpha\}$ is open by Proposition \[G.2\]. \[lemacontsup2\] For each $u \in \mathcal{D}_e^+$ let us define the quantity $$\mathcal{I}^u:= \inf_{(t,x) \in [0,\tau^u) \times [0,1]} U^u(t,x).$$ Then the application $u \mapsto I^u$ is lower semicontinuous on $\mathcal{D}_e^+$. Notice that $\mathcal{I}^u \geq 0$ for any $u \in \mathcal{D}_e^+$ since $U^u(t,0)=U^u(t,1)=0$ for all $t \in [0,\tau^u)$. Therefore, it will suffice to show that the sets $\{ \alpha < \mathcal{I} \}$ are open in $\mathcal{D}^+_e$ for every $\alpha < 0$. With this purpose in mind, given $\alpha < 0$ and $u \in \mathcal{D}_e^+$ such that $\alpha < \mathcal{I}^u$, take $\beta_1,\beta_2 < 0$ such that $\alpha < \beta_1 < \beta_2 < \mathcal{I}^u$ and let $y$ be the solution to the ordinary differential equation $$\label{contsup2eq} \left\{\begin{array}{l} \dot{y} = - |y|^p \\ y(0) = \beta_2.\end{array}\right.$$ Define $t_\beta := \inf \{ t \in [0,t_{max}^y) : y(t) < \beta_1 \}$, where $t_{max}^y$ denotes the explosion time of $y$. Notice that by the lower semicontinuity of $S$ for any $M > 0$ we have $S( U^u( \mathcal{T}^u_M, \cdot) ) \leq -M$ and thus, by Lemma \[expestimate\], we may choose $M$ such that $$\label{contsupeq4} \sup_{v \in B_\rho( U^u( \mathcal{T}^u_M,\cdot) )} \tau^v < t_\beta$$ for some small $\rho > 0$. Moreover, if $\rho < \mathcal{I}^u - \beta_2$ then every $v \in B_\rho( U^u( \mathcal{T}^u_M,\cdot) )$ satisfies $\inf_{x \in [0,1]} v(x) \geq \beta_2$ so that $U^v$ is in fact a supersolution to the equation . By this implies that $v \in \mathcal{D}_e^+$ and $\mathcal{I}^v \geq \beta_1 > \alpha$. On the other hand, by Proposition \[G.2\] we may take $\delta > 0$ sufficiently small so that for every $w \in B_\delta(u)$ we have $\mathcal{T}^u_M < \tau^w$ and $$\sup_{t \in [0,\mathcal{T}^u_M]} \| U^w(t,\cdot) - U^u(t,\cdot)\|_\infty < \rho.$$ Combined with the previous argument, this yields the inclusion $B_\delta(u) \subseteq \mathcal{D}_e^+ \cap \{ \alpha < \mathcal{I} \}$. In particular, this shows that $\{ \alpha < \mathcal{I} \}$ is open and thus concludes the proof. The preceding proof shows, in particular, that the set $\mathcal{D}_e^+$ is open. The conclusion of the proof of Theorem \[contexp\] is contained in the next proposition. \[convsup\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}^*_e$ and $\delta > 0$ there exists a constant $C > 0$ such that $$\label{convergenciasuperior} \sup_{u \in \mathcal{K}} P_u ( \tau_\varepsilon > \tau + \delta ) \leq e^{- \frac{C}{\varepsilon^2}}.$$ Since $\mathcal{D}_e^- = - \mathcal{D}_e^+$ and $U^{-u}= - U^u$ for $u \in C_D([0,1])$, without any loss of generality we may assume that $\mathcal{K}$ is contained in $\mathcal{D}_e^+$. Let us begin by noticing that for any $M > 0$ $$\mathcal{T}_M := \sup_{u \in \mathcal{K}} \mathcal{T}_M^u < +\infty \hspace{1cm}\text{ and }\hspace{1cm}\mathcal{R}_M:= \sup_{u \in \mathcal{K}} \mathcal{R}^u_M < +\infty.$$ Indeed, by Propositions \[G.1\] and \[A.2\] we may choose $t_0 > 0$ sufficiently small so that the orbits $\{ U^{u}(t,\cdot) : 0 \leq t \leq t_0 , u \in \mathcal{K} \}$ remain uniformly bounded and the family $\{ U^{u}(t_0,\cdot) : u \in \mathcal{K} \}$ is contained in a compact set $\mathcal{K}' \subseteq \mathcal{D}_e^+$ at a positive distance But then we have $$\mathcal{T}_M \leq t_0 + \sup_{u \in \mathcal{K}'} \mathcal{T}_M^u \hspace{1cm}\text{ and }\hspace{1cm}\mathcal{R}_M \leq \sup_{0 \leq t \leq t_0, u \in \mathcal{K}} \| U^u(t,\cdot) \|_\infty + \sup_{u \in \mathcal{K}'} \mathcal{R}^u_M$$ and both right hand sides are finite due to Lemma \[lemacontsup\] and the fact that $\mathcal{T}_M^u$ and $\mathcal{R}_M$ are both finite for each $u \in \mathcal{D}_e$ by Proposition \[caract\]. Similarly, by Lemma \[lemacontsup2\] we also have $$\mathcal{I}_{\mathcal{K}}:= \inf_{u \in \mathcal{K}} \mathcal{I}^u > - \infty.$$ Now, for each $u \in \mathcal{K}$ and $\varepsilon > 0$ by the Markov property we have for any $\rho > 0$ $$\label{descompexp} P_u ( \tau_\varepsilon > \tau + \delta ) \leq P( d_{\mathcal{T}_M}( U^{(\mathcal{R}_M+1),u,\varepsilon}, U^{(\mathcal{R}_M+1),u}) > \rho ) + \sup_{v \in B_{\rho}(U^u( \mathcal{T}^u_M, \cdot))} P_v ( \tau_\varepsilon > \delta).$$ The first term on the right hand side is taken care of by so that in order to show it only remains to deal with the second term by choosing $M$ and $\rho$ appropriately. The argument given to deal with this term is similar to that of the proof of Let $y$ be the solution to the ordinary differential equation $$\label{convsupeq2} \left\{\begin{array}{l} \dot{y} = - |y|^p - |y|^{p-1} - 1\\ y(0) = \mathcal{I}_{\mathcal{K}} - \frac{1}{2}.\end{array}\right.$$ Define $t_{\mathcal{I}} := \inf \{ t \in [0,t_{max}^y) : y(t) < \mathcal{I}_{\mathcal{K}} - 1 \}$, where $t_{max}^y$ denotes By Lemma \[expestimate\], we may choose $M$ such that $$\label{convsupeq4} \sup_{v \in B_{\rho_M}( U^u( \mathcal{T}^u_M,\cdot) )} \tau^{(h),v} < \min\{\delta, t_\mathcal{I}\}$$ for all $0 \leq h < h_M$, where $\rho_M > 0$ and $h_M > 0$ are suitable constants. The key observation here is that, since $\mathcal{R}_M < +\infty$, we may choose these constants so as not to depend on $u$ but rather on $M$ and $\mathcal{R}_M$ themselves. Moreover, if $\rho_M < \frac{1}{2}$ then every $v \in B_{\rho_M}( U^u( \mathcal{T}^u_M,\cdot) )$ satisfies $\inf_{x \in [0,1]} v(x) \geq \mathcal{I}_{\mathcal{K}} - \frac{1}{2}$ so that $\underline{Z}^{(h),v}$ is in fact a supersolution to the equation for all $0 \leq h < \min\{h_M,1\}$. By the former implies that $\underline{Z}^{(h),v}$ explodes through $+\infty$ and that it remains bounded from below by $\mathcal{I}_{\mathcal{K}} - 1$ until its explosion time which, by , is smaller than $\delta$. In particular, we see that if $\|V^{\mathbf{0},\varepsilon}\|_\infty < \min \{ 1, \frac{h_M}{p2^{p-1}}\}$ then $Z^{v,\varepsilon}$ explodes before $\underline{Z}^{(h),v}$ does, so that we have that $\tau_\varepsilon < \delta$ under such conditions. Hence, we conclude that $$\sup_{v \in B_{\rho_M}(U^u( \mathcal{T}^u_M, \cdot))} P_v ( \tau_\varepsilon > \delta) \leq P \left( \sup_{t \in [0,\delta]} \|V^{\mathbf{0},\varepsilon}(t,\cdot) \|_\infty \leq \min \left\{ 1, \frac{h_M}{p2^{p-1}} \right\} \right)$$ which, by recalling the estimate , gives the desired control on the second term in the right hand side of . Thus, by taking $\rho$ equal to $\rho_M$ in , we obtain the result. This last proposition in fact shows that for $\delta > 0$ and a given bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}^*_e$ there exist constants $M,C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( \tau_\varepsilon > \mathcal{T}_M^u + \delta ) \leq e^{- \frac{C}{\varepsilon^2}}.$$ By exploiting the fact $\mathcal{T}_M < +\infty$ for every $M > 0$ we obtain the following useful corollary. \[exploacot\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}_e^*$ there exist constants $\tau^*, C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( \tau_\varepsilon > \tau^*) \leq e^{- \frac{C}{\varepsilon^2}}.$$ Resumen del Capítulo 2 ---------------------- En este capítulo investigamos la continuidad del tiempo de explosión $\tau_\varepsilon^u$ para datos iniciales en el dominio de $\mathcal{D}_e$. Mostramos que el tiempo de explosión $\tau_\varepsilon^u$ del sistema estocástico converge en probabilidad al tiempo de explosión determinístico $\tau^u$ uniformemente sobre compactos de $\mathcal{D}_e^*$, el conjunto de aquellos datos iniciales $u \in \mathcal{D}_e$ para los cuales la solución $U^{u}$ explota por un único lado, i.e. permanece acotada en alguna dirección (inferior o superiormente) hasta el tiempo de explosión. Más precisamente, tenemos el siguiente resultado. **Teorema**. Para cualquier conjunto acotado $\mathcal{K} \subseteq \mathcal{D}_e^*$ a una distancia positiva de ${\partial}\mathcal{D}_e^*$ y $\delta > 0$ existe una constante $C > 0$ tal que $$\sup_{u \in \mathcal{K}} P_u ( |\tau_\varepsilon - \tau| > \delta ) \leq e^{- \frac{C}{\varepsilon^2}}.$$ Dividimos la demostración de este resultado en dos partes, las cotas y . La cota inferior es una consecuencia directa de la estimación de grandes desvíos . En efecto, como la solución $U^u$ permanece acotada en $[0,\tau^u - \delta)$, la condición $\tau^u_\varepsilon < \tau^u - \delta$ implica que los sistemas estocástico y determinístico deben necesariamente separarse antes de tiempo $\tau^u - \delta$ y, por lo tanto, que lo mismo debe suceder para los sistemas truncados para los que se tiene . La demostración de la cota superior consiste en estudiar el proceso $Z^{u,\varepsilon}$ definido como la solución del problema . Dicho proceso posee el mismo tiempo de explosión que $U^{u,\varepsilon}$ pero es más sencillo de tratar debido a que cada realización del mismo resuelve una ecuación diferencial en derivadas parciales que puede estudiarse mediante técnicas usuales. Además, dado $0 < h < 1$ es posible mostrar que para cada realización $\omega$ en un conjunto $\Omega_h^\varepsilon$ con probabilidad que tiende a uno cuando $\varepsilon \rightarrow 0$ la trayectoria $Z^{u,\varepsilon}(\omega)$ es supersolución de . A partir de esto, la estrategia que adoptamos para probar si $u \in D_e^+$ (ver ) es mostrar que para un conjunto de realizaciones $\Omega^{u,\varepsilon} \subseteq \Omega_h^\varepsilon$ con probabilidad que tiende a uno (exponencialmente rápido en $\frac{1}{\varepsilon^2}$) cuando $\varepsilon \rightarrow 0$ suceden dos cosas: 1. La solución de explota antes de tiempo $\tau^u + \delta$. 2. La solución de explota por $+\infty$, i.e. permanece acotada inferiormente hasta el tiempo de explosión. Se sigue de la descripción anterior que para toda realización en $\Omega^{u,\varepsilon}$ el tiempo de explosión de $Z^{u,\varepsilon}$ (y por lo tanto $\tau^u_\varepsilon$) es menor a $\tau^u + \delta$, lo cual implica a partir de . Para mostrar (i) utilizamos técnicas de ecuaciones similares a las que figuran en [@CE1] y (ii) se deduce del hecho de que $u \in \mathcal{D}_e^+$. Por simetría se obtienen los mismos resultados para $\mathcal{D}_e^-$. Por último, la uniformidad sobre compactos se obtiene a partir de los resultados en el Apéndice. Construction of an auxiliary domain {#dominioauxiliar} =================================== As suggested in [@GOV], to study the behavior of the explosion time for initial data in $\mathcal{D}_\mathbf{0}$ it is convenient to consider an auxiliary bounded domain $G$ satisfying the conditions stated in the Introduction. By doing so we can then reduce our problem to a characterizing the escape from this domain. This is simpler because for it we may assume that the source term $g$ in is Lipschitz, as the escape only depends on the behavior of the system while it remains inside a bounded region. It is then that the large deviations estimates of Section \[secLDP\] can be applied. To succeed in the construction of such a domain, we must first understand the behavior of exploding trajectories in the stochastic system. This is the purpose behind the following results. \[compacidad\] If $p < 5$ then for any $a > 0$ the sets $\{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq a \}$ are bounded. Let $a > 0$ and for $v \in \{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq a \}$ consider $\psi : {{\mathbb R}}_{\geq 0} \rightarrow {{\mathbb R}}_{\geq 0}$ given by the formula $$\psi(t):= \int_0^1 (U^v(t,\cdot))^2.$$ A direct computation shows that for every $t_0 > 0$ the function $\psi$ satisfies $$\frac{d\psi(t_0)}{dt} = - 4 S( U^v(t,\cdot) ) + 2\left(\frac{p-1}{p+1}\right) \int_0^1 |U^v(t,\cdot)|^{p+1}.$$ By Proposition \[Lyapunov\] and Hölder’s inequality we then obtain $$\frac{d\psi(t_0)}{dt} \geq - 4 a + 2\left(\frac{p-1}{p+1}\right) (\psi(t_0))^{\frac{p+1}{2}}$$ which implies that $\psi(0) \leq B:= \left[2a\left(\frac{p+1}{p-1}\right)\right]^{\frac{2}{p+1}}$ since otherwise $\psi$ (and therefore $U^v$) would explode in finite time. Now, by the Gagliardo-Niremberg interpolation inequality (recall that $v$ is absolutely continuous since $S(v) < +\infty$) $$\| v \|_\infty^2 \leq C_{G-N} \| v \|_{L^2} \| {\partial}_x v \|_{L^2},$$ we obtain $$\int_0^1 |v|^{p+1} \leq \| v \|_{L^2}^2 \|v \|_{\infty}^{p-1} \leq C_{G-N} B^{\frac{p+3}{4}} \| {\partial}_x v \|_{L^2}^{\frac{p-1}{2}} \leq C_{G-N} B^{\frac{p+3}{4}} (2a + \int_0^1 |v|^{p+1})^{\frac{p-1}{4}}$$ which for $p < 5$ implies the bound $$\label{bound1} \int_0^1 |v|^{p+1} \leq B':=\max \left\{ 2a , \left[C_{G-N} B^{\frac{p+3}{4}}2^{\frac{p-1}{4}}\right]^{\frac{4}{5-p}}\right\}.$$ Since $S(v) \leq a$ we see that implies the bound $\| {\partial}_x v \|_{L^2} \leq \sqrt{2B'}$ and therefore we conclude $$\| v \|_\infty \leq \sqrt{ C_{G-N} \sqrt{2BB'}}$$ which shows that $\{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq a \}$ is bounded. The proof of Lemma \[compacidad\] is the only instance throughout our entire work in which the assumption $p < 5$ is used. If $p \geq 5$ is such that the sets $\{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq a \}$ remain bounded for every $a > 0$, then all of our results remain valid for this choice of $p$. As a matter of fact, we shall only require the weaker condition that there exists $\alpha > 0$ such that the set $\{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq S(z) + \alpha \}$ is bounded. However, determining the validity of this condition for arbitrary $p > 1$ does not seem to be an easy problem. \[dominio1\] The potential $S$ satisfies $\displaystyle{\lim_{n \rightarrow +\infty} \left[ \inf_{u \in \partial B_n \cap \mathcal{D}_{\mathbf{0}}} S(u) \right] = +\infty.}$ Given $M > 0$, by Lemma \[compacidad\] we may take $N \in {{\mathbb N}}$ such that for every $n \geq N$ the set $\{ u \in \mathcal{D}_{\mathbf{0}} : 0 \leq S(u) \leq M \}$ is contained in $B_n^\circ$. Now, let us consider $u \in \partial B_n \cap \mathcal{D}_\mathbf{0}$. Since $u \in \mathcal{D}_\mathbf{0}$, we know that $S(u) \geq 0$ by Proposition \[Lyapunov\]. Therefore, if $u \in \partial B_n$ for $n \geq N$, in particular we have that $u \notin \{ u \in \mathcal{D}_{\mathbf{0}} : 0 \leq S(u) \leq M \}$ and thus it must be $S(u) > M$. This proves the claim. Given $T > 0$ and $\varphi \in C_{D}([0,T] \times [0,1])$ we define the *rate* $I(\varphi)$ of $\varphi$ by the formula $$I(\varphi) := I^{(n),\varphi(0,\cdot)}_T(\varphi)$$ for any $n \in {{\mathbb N}}$ larger than $\| \varphi \|_{\infty}$, where $I^{(n),\varphi(0,\cdot)}_T$ denotes the rate function associated to the system with $f=g_n$. Notice that $I(\varphi)$ does not depend on the choice of $n$. We say that a function $\varphi \in C_D([0,T]\times [0,1])$ is *regular* if both derivatives ${\partial}_t \varphi$ and ${\partial}^2_{xx} \varphi$ exist and belong to $C_D ([0,T]\times [0,1])$. \[costo\] Given $T > 0$ for any $\varphi \in C_{D} \cap W^{1,2}_2 ([0,T] \times [0,1])$ such that ${\partial}^2_{xx}\varphi(0,\cdot)$ exists and belongs to $C_D([0,1])$ we have that $$\label{cotainftasa} I(\varphi) \geq 2 \left[\sup_{0 \leq T' \leq T} \left( S(\varphi(T',\cdot))-S(\varphi(0,\cdot))\right)\right].$$ Let us begin by assuming that $\varphi$ is regular and take $N \in {{\mathbb N}}$ larger than $\|\varphi\|_\infty$. Using the identity $(x-y)^2 = (x+y)^2 -4xy$ for $x,y \in {{\mathbb R}}$ and $0 \leq T' \leq T$ we obtain that $$\begin{aligned} I(\varphi) = I^{(N),\varphi(0,\cdot)}_T(\varphi) & = \frac{1}{2} \int_0^T \int_0^1 |{\partial}_t \varphi - {\partial}_{xx}^2 \varphi - g_N(\varphi)|^2 \geq \frac{1}{2} \int_0^{T'} \int_0^1 |{\partial}_t \varphi - {\partial}_{xx}^2 \varphi - g_N(\varphi)|^2 \\ \\ & = \frac{1}{2} \int_0^{T'} \int_0^1 \left[|{\partial}_t \varphi + {\partial}_{xx}^2 \varphi + g_N(\varphi)|^2 - 2\left( {\partial}_{xx}^2 \varphi + g_N(\varphi)\right){\partial}_t \varphi\right]\\ \\ & = \frac{1}{2} \int_0^{T'} \left[ \left( \int_0^1 |{\partial}_t \varphi + {\partial}_{xx}^2 \varphi + g_N(\varphi)|^2\right) + 2 \frac{d S^{(N)}( \varphi(t,\cdot) )}{dt}\right]\\ \\ & \geq 2 \left(S^{(N)}(\varphi(T',\cdot)) - S^{(N)}(\varphi(0,\cdot))\right) =2\left( S(\varphi(T',\cdot))-S(\varphi(0,\cdot))\right)\end{aligned}$$ where the last equality follows from the fact that both $S^{(N)}$ and $S$ coincide inside $B_N$. Taking supremum on $T'$ yields the result in this particular case. Now, if $\varphi$ is not necessarily regular then by [@FJL Theorem 6.9] we may take a sequence $(\varphi_n)_{n \in {{\mathbb N}}}$ of regular functions converging to $\varphi$ on $C_{D_{\varphi(0,\cdot)}}([0,T]\times[0,1])$ and also such that $\lim_{n \rightarrow +\infty} I(\varphi_n) = I(\varphi)$. The result in the general case then follows from the validity of for regular functions and the lower semicontinuity of $S$. In order to properly interpret the content of Proposition \[costo\] we need to introduce the concept of *quasipotential* for our system. We do so in the following definitions. Given $u,v \in C_D([0,1])$ a *path from $u$ to $v$* is a continuous function $\varphi \in C_{D}([0,T] \times [0,1])$ for some $T > 0$ such that $\varphi(0,\cdot)=u$ and $\varphi(T,\cdot)=v$. Given $u,v \in C_D([0,1])$ we define the *quasipotential* $V(u,v)$ by the formula $$V(u,v)= \inf \{ I(\varphi) : \varphi \text{ path from $u$ to $v$}\}.$$ Furthermore, given a subset $B \subseteq C_D([0,1])$ we define the quasipotential from $u$ to $B$ as $$V(u,B):= \inf \{ V(u,v) : v \in B \}.$$ We refer the reader to the Appendix for a review of some of the main properties of $V$ which shall be required throughout our work. In a limiting sense, made rigorous through the large deviations principle established in Section \[secLDP\], the quasipotential $V(u,v)$ represents the energy cost for the stochastic system to travel from $u$ to (an arbitrarily small neighborhood of) $v$. In light of all these definitions we see that the energy cost for the stochastic system starting from $\mathbf{0}$ to explode in a finite time while remaining inside $\mathcal{D}_\mathbf{0}$ is infinite. Indeed, combining Propositions \[dominio1\], \[costo\] and \[A.4\] we see that $\lim_{n \rightarrow +\infty} V(\mathbf{0},\partial B_{n} \cap \mathcal{D}_{\mathbf{0}})=+\infty$ which implies that a path from $\mathbf{0}$ to infinity lying inside $\mathcal{D}_0$ should have, at least formally, an infinite rate. Thus, were the stochastic system starting from $\mathbf{0}$ to explode, it would have to do so by stepping outside $\mathcal{D}_{\mathbf{0}}$ and In view of Proposition \[costo\], the system will typically wish to cross $\mathcal{W}$ through $\pm z$ since the energy cost for performing such a feat is the lowest there. Hence, if we wish the problem of escaping the domain $G$ to capture the essential characteristics of the explosion phenomenon in the stochastic system (at least when starting from $\mathbf{0}$) then it is important to guarantee that the escape from this domain occurs by passing through (an arbitrarily small neighborhood of) $\pm z$. Not only this, but we also require that once the system escapes this domain $G$ then it explodes with overwhelming probability in a quick fashion, i.e. before a certain time $\tau^*$ which does not depend on $\varepsilon$. More precisely, we wish to consider a bounded domain $G \subseteq C_D([0,1])$ verifying the following properties: \[assumpg\]$\,$ 1. There exists $r_{\mathbf{0}}>0$ such that $B_{2r_\mathbf{0}} \subseteq \mathcal{D}_{\mathbf{0}} \cap G$. 2. There exists $c > 0$ such that $B_c \subseteq B_{r_\mathbf{0}}$ and for all $v \in B_c$ the solution $U^{v}$ to with initial datum $v$ is globally defined and converges to $\mathbf{0}$ without escaping $B_{r_\mathbf{0}}$. 3. There exists a closed subset ${\partial}^{\pm z}$ of the boundary $\partial G$ which satisfies 1. $V(\mathbf{0},\partial G - \partial^{\pm z} ) > V(\mathbf{0},\partial^{\pm z}) = V( \mathbf{0}, \pm z)$. 2. ${\partial}^{\pm z}$ is contained in $\mathcal{D}_e^*$ and at a positive distance from its boundary. In principle, we have seen that such a domain is useful to study the behavior of the explosion time whenever the initial datum of the stochastic system is (close to) the origin. Nevertheless, as we shall later see, when starting inside $\mathcal{D}_\mathbf{0}$ the system will typically visit a small neighborhood of the origin before crossing $\mathcal{W}$ and thus such a choice of $G$ will also be suitable to study the explosion time for arbitrary initial data in $\mathcal{D}_\mathbf{0}$. The construction of the domain $G$ is done as follows. Since $\mathcal{D}_{\mathbf{0}}$ is open we may choose $r_{\mathbf{0}} > 0$ such that $B_{3r_{\mathbf{0}}}$ is contained in $\mathcal{D}_{\mathbf{0}}$. Moreover, by the asymptotic stability of $\mathbf{0}$ we may choose $c > 0$ verifying (ii) in Conditions \[assumpg\]. Now, given $\zeta_1 > 0$ by Lemma \[compacidad\] we may take $n_0 \in {{\mathbb N}}$ such that $n_0 > 3r_{\mathbf{0}}$ and the set $\{ u \in \overline{\mathcal{D}_{\mathbf{0}}} : 0 \leq S(u) \leq S(z) + \zeta_1 \}$ is contained in the interior of the ball $B_{n_0-1}$. We then define the pre-domain $\tilde{G}$ as $$\label{predomain} \tilde{G}:= B_{n_0} \cap \overline{\mathcal{D}_{\mathbf{0}}}.$$ Notice that since both $B_{n_0}$ and $\overline{\mathcal{D}_\mathbf{0}}$ are closed sets we have that $$\partial \tilde{G} = \left( \mathcal{W} \cap B_{n_0}\right) \cup \left( \partial B_{n_0} \cap \mathcal{D}_\mathbf{0}\right)$$ which, by the particular choice of $n_0$ and Proposition \[Lyapunov\], implies $\min_{u \in {\partial}\tilde{G}} S(u) =S(z)$. By Propositions \[costo\] and \[A.4\] we thus obtain $V(\mathbf{0},{\partial}\tilde{G}) \geq \Delta$. Next, if for $u \in C_D([0,1])$ we let $u^-$ denote the negative part of $u$, i.e. $u^- = \max \{ - u, 0 \}$, then since $z^- = \mathbf{0}$ we may find $\tilde{r}_z > 0$ such that $u^- \in \mathcal{D}_{\mathbf{0}}$ for any $u \in B_{\tilde{r}_z}(z)$. Finally, if for $r > 0$ we write $B_{r}(\pm z) := B_{r}(z) \cup B_{r}(-z)$ and take $r_{z} > 0$ such that $r_z < \frac{\tilde{r}_z}{2}$, $B_{2r_z}(\pm z)$ is contained in the interior of $B_{n_0}$ and $z$ is the unique equilibrium point of the system lying inside $B_{r_z}(z)$, then we define our final domain $G$ as $$G= \tilde{G} \cup B_{r_z}(\pm z).$$ Let us now check that this domain satisfies all the required conditions. We begin by noticing that (i) and (ii) in Conditions \[assumpg\] are immediately satisfied by the Now, let us also observe that for any $r > 0$ $$\label{lejosdelminimo} \inf\{ S(u) : u \in {\partial}\tilde{G} - B_{r}(\pm z)\} > S(z).$$ Indeed, if this were not the case then there would exist $(u_k)_{k \in {{\mathbb N}}} \subseteq \left[\mathcal{W}\cap B_{n_0} - B_{r}(\pm z)\right]$ such that $\lim_{k \rightarrow +\infty} S(u_k) = S(z)$. By Proposition \[G.1\] we have that there exists $t_0 > 0$ sufficiently small satisfying $$\sup_{k \in {{\mathbb N}}} \left[\sup_{t \in [0,t_0]} \| U^{u_k}(t,\cdot) \|_\infty\right] < +\infty \hspace{1cm}\text{ and }\hspace{1cm} \inf_{k \in {{\mathbb N}}} \| U^{u_k}(t_0,\cdot) - (\pm z) \|_\infty > \frac{r}{2}$$ and therefore by Proposition \[A.2\] we may conclude that there exists a subsequence $(u_{k_j})_{j \in {{\mathbb N}}}$ such that $U^{u_{k_j}}(t_0,\cdot)$ converges to a limit $u_{\infty} \in C_D([0,1])$ as $j \rightarrow +\infty$. Since the potential $S$ is lower semicontinuous and $\mathcal{W}$ is both closed and invariant under the deterministic flow, by Proposition \[Lyapunov\] we conclude that $u_\infty = \pm z$ which contradicts the fact that the sequence $(U^{u_{k_j}}(t_0,\cdot))_{j \in {{\mathbb N}}}$ is at a positive distance from these equilibriums. Hence, we obtain the validity of . In particular, this implies that $V(\mathbf{0}, {\partial}\tilde{G} - B_{r}(\pm z)) > \Delta$ for any $r > 0$. Let us then take $\zeta_2 > 0$ such that $\Delta + \zeta_2 < V(\mathbf{0},{\partial}\tilde{G} - B_{\frac{r_z}{2}}(\pm z))$ and define $$\tilde{{\partial}}^z:= \{ u \in {\partial}B_{r_z}(z) \cap \overline{\mathcal{D}_e} : V(\mathbf{0},u) \leq \Delta + \zeta_2 \}.$$ Notice that $d(\tilde{{\partial}}^z, \mathcal{W}) > 0$. Indeed, if this were not the case we would have sequences $(u_k)_{k \in {{\mathbb N}}} \subseteq \mathcal{W}$ and $(v_k)_{k \in {{\mathbb N}}} \subseteq \tilde{{\partial}}^z$ such that $\lim_{k \rightarrow +\infty} d(u_k,v_k)=0$. The growth estimates on the Appendix section then imply that there exists $t_1 > 0$ sufficiently small such that $$\lim_{k \rightarrow +\infty} d(U^{u_k}(t_1,\cdot),U^{v_k}(t_1,\cdot))=0\hspace{0.2cm}\text{and}\hspace{0.2cm} \frac{r_z}{2} < \inf_{k \in {{\mathbb N}}} d(U^{v_k}(t_1,\cdot),z) \leq \sup_{k \in {{\mathbb N}}} d(U^{v_k}(t_1,\cdot),z) < 2r_z.$$ By Proposition \[A.2\] we obtain that for some appropriate subsequence we have $$\lim_{j \rightarrow +\infty} U^{u_{k_j}}(t_1,\cdot) =\lim_{j \rightarrow +\infty} U^{v_{k_j}}(t_1,\cdot) = v_\infty.$$ Observe that $v_\infty \in \mathcal{W} \cap B_{n_0} - B_{\frac{r_z}{2}}(\pm z)$ and thus that $v_\infty \in {\partial}\tilde{G}- B_{\frac{r_z}{2}}(\pm z)$. Furthermore, by the lower semicontinuity of $V(\mathbf{0},\cdot)$ and the fact that the mapping $t \mapsto V(\mathbf{0},U^u(t,\cdot))$ is monotone decreasing for any $u \in C_D([0,1])$ (see the Appendix section for details), we obtain that $V(\mathbf{0},v_\infty) \leq \Delta + \zeta_2$ which, together with the previous observation, implies the contradiction $\Delta + \zeta_2 \geq V(\mathbf{0},{\partial}\tilde{G} - B_{\frac{r_z}{2}}(\pm z))$. Hence, we see that $d(\tilde{{\partial}}^z, \mathcal{W}) > 0$ and thus we may define $${\partial}^z = \left\{ u \in {\partial}B_{r_z}(z) \cap \overline{\mathcal{D}_e} : d(u,\mathcal{W}) \geq \frac{d(\tilde{{\partial}}^z,\mathcal{W})}{2} \right\}$$ and set ${\partial}^{\pm z}:= {\partial}^z \cup (-{\partial}^z)$. Since one can easily check that $${\partial}G = [ {\partial}\tilde{G} - B_{r_z}(\pm z) ] \cup [{\partial}B_{r_z}(\pm z) \cap \overline{D_e} ]$$ we conclude that $V(\mathbf{0},{\partial}G - {\partial}^{\pm z}) \geq \Delta + \zeta_2$. On the other hand, by proceeding similarly to the proof of Lemma \[cotasuplema0\] below, one can show that $V(\mathbf{0},{\partial}^z)=V(\mathbf{0},\tilde{{\partial}}^z)= V(\mathbf{0},\pm z)=\Delta$, from which one obtains that $$V(\mathbf{0},{\partial}G - {\partial}^{\pm z}) > V(\mathbf{0},{\partial}^z) = V(\mathbf{0},\pm z).$$ Furthermore, by the comparison principle and the choice of $\tilde{r}_z$ we have Therefore, since we clearly have $d({\partial}^{\pm z}, \mathcal{W}) > 0$ by definition of ${\partial}^{\pm z}$, upon recalling that ${\partial}^{\pm z} \subseteq \mathcal{D}_e$ and $r_z < \frac{\tilde{r}_z}{2}$ we see that ${\partial}^{\pm z} \subseteq \mathcal{D}_e^+$ and $d({\partial}^{\pm z}, {\partial}\mathcal{D}_e^*) \geq \min\{d({\partial}^{\pm z}, \mathcal{W}), \frac{\tilde{r}_z}{2}\} > 0$, so that condition (iii) also holds. See Figure \[fig2\]. ![The auxiliary domain $G$[]{data-label="fig2"}](dibujo2-santi.eps){width="8cm"} \[obsequivG\] Let us notice that, by Corollary \[exploacot\], ($\bullet \bullet$) in Conditions \[assumpg\] implies that there exist constants $\tau^*,C > 0$ such that $$\sup_{u \in {\partial}^{\pm z}} P_u (\tau_\varepsilon > \tau^* ) \leq e^{-\frac{C}{\varepsilon^2}}$$ for all $\varepsilon > 0$ sufficiently small. Since ($\bullet$) guarantees that the escape from $G$ will typically take place through ${\partial}^{\pm z}$, this tells us that both $\tau_\varepsilon$ and $\tau_\varepsilon({\partial}G)$ are asymptotically equivalent, so that it will suffice to study the escape from $G$ in order to establish each of our results. Resumen del Capítulo 3 ---------------------- En este capítulo damos la construcción del dominio auxiliar acotado con las características discutidas en la Introducción. La razón por la cual llevamos a cabo tal construcción es porque nos permite reducir nuestro problema original al de estudiar cómo el sistema estocástico se escapa de dicho dominio. La ventaja de esto reside en que para estudiar este nuevo problema podemos asumir que la fuente es globalmente Lipschitz, dado que el escape depende únicamente del comportamiento del sistema mientras se encuentra en una región acotada. Es entonces que se pueden aplicar las estimaciones de grandes desvíos de la Sección \[secLDP\]. Para poder precisar qué condiciones debe cumplir nuestro dominio, definimos primero el quasipotencial $V$ siguiendo la Definición \[defipotentialV\]. Dados $u,v \in C_D([0,1])$, el quasipotencial $$V(u,v)= \inf\{ I^u_T(\varphi) : \varphi \in C_D([0,T] \times [0,1]), \varphi(0,\cdot)= u, \varphi(T,\cdot) = v \}$$ representa el costo para el sistema estocástico (en términos de las estimaciones de la Sección \[secLDP\]) de ir desde $u$ hasta (un entorno arbitrariamente pequeño de) $v$. Asimismo, dado un conjunto $B \subseteq C_D([0,1])$, el quasipotencial $$V(u,B)= \inf \{ V(u,v) : v \in B \}$$ representa el costo para el sistema de ir desde $u$ hasta $B$. Luego, el dominio $G$ de interés debe cumplir con las siguientes características: 1. Existe $r_{\mathbf{0}}>0$ tal que $B_{2r_\mathbf{0}} \subseteq \mathcal{D}_{\mathbf{0}} \cap G$. 2. Existe $c > 0$ tal que $B_c \subseteq B_{r_\mathbf{0}}$ y para todo $v \in B_c$ la solución $U^{v}$ de con dato inicial $v$ está globalmente definida y converge a $\mathbf{0}$ sin escapar de $B_{r_\mathbf{0}}$. 3. Existe un subconjunto cerrado ${\partial}^{\pm z}$ de la frontera $\partial G$ que satisface 1. $V(\mathbf{0},\partial G - \partial^{\pm z} ) > V(\mathbf{0},\partial^{\pm z}) = V( \mathbf{0}, \pm z)$. 2. ${\partial}^{\pm z}$ está contenido en $\mathcal{D}_e^*$ y a una distancia positiva de su frontera. La estabilidad asintótica del origen $\mathbf{0}$ garantiza que cualquier dominio que contenga a un entorno del origen va a satisfacer (i) y (ii). Por otro lado, mostramos que si el parámetro $p$ en la fuente satisface $1 < p < 5$ entonces existe un dominio acotado que además satisface (iii). Dicho dominio será la porción de $\mathcal{D}_0$ contenida en una bola de centro en $\mathbf{0}$ y radio apropiadamente grande, unida a entornos pequeños de los equilibrios inestables de mínima energía, $\pm z$ (ver Figura \[fig2\]). La principal dificultad a la hora de demostrar que el dominio así construido cumple con las características buscadas yace en la falta de compacidad del espacio $C_D([0,1])$. Sin embargo, nos fue posible lidiar con este problema apelando a algunos de los resultados contenidos en el Apéndice. Por último, la restricción $p < 5$ surge de limitaciones geométricas impuestas por el potencial $S$. Esperamos que una construcción similar sea posible aún en el caso $p \geq 5$, aunque no tenemos una prueba. The escape from $G$ {#secescapedeg} =================== The behavior of the explosion time for initial data $u\in \mathcal{D}_{\mathbf{0}}$ is proved by showing that, with overwhelming probability as $\varepsilon \to 0$, the stochastic system describes the following path: 1. The system enters the neighborhood of the origin $B_c$ before some finite time $T$ which does not depend on ${\varepsilon}$. 2. Once inside $B_c$ the system remains in $G$ for a time of order $e^{\frac{\Delta}{{\varepsilon}^2}}$ and then escapes from $G$ through $\partial^{\pm z}$ since the barrier imposed by the potential is the lowest there. 3. After escaping $G$ through $\partial^{\pm z}$ the system explodes before some finite time $\tau^*$ which does not depend on ${\varepsilon}$. The fact that the domain $G$ is bounded allows us to assume that the source term $g$ is globally Lipschitz if we wish to study the behavior of our system while it Indeed, we may consider $n_0 \in {{\mathbb N}}$ from the definition of $G$ and study the behavior of the solution to for $n=n_0+1$ since it coincides with our process until the For this reason, in the following we shall often drop the superscript $(n_0+1)$ in the notation $U^{(n_0+1),u,\varepsilon}$ unless it is completely necessary. Our aim in this section is to obtain a complete and precise understanding of (ii) in the description above, since by (iii) the explosion time will inherit all the of the escape time from $G$. In particular, we are interested in the asymptotic magnitude and distribution of this escape time, as well as in a good understanding of which are the typical paths that lead the stochastic system outside of our bounded domain $G$. a bounded domain with these characteristics was first studied in [@GOV] for a finite-dimensional double-well potential, and later investigated in [@B1] for its infinite-dimensional analogue. The results we present in this chapter are an adaptation to our setting of the results featured in these references. Other references dealing with similar problems include [@MOS; @B2; @FW]. Hereafter, $B_c$ will denote the neighborhood of the origin highlighted in Conditions \[assumpg\]. Also, for a given closed set $\Gamma \subseteq C_D([0,1])$ we write $$\tau_\varepsilon^u (\Gamma) := \inf\{ t \geq 0 : U^{u,\varepsilon}(t,\cdot) \in \Gamma \}.$$ Asymptotic magnitude of $\tau_\varepsilon({\partial}G)$ ------------------------------------------------------- We begin our study of the escape from $G$ by studying its asymptotic magnitude as $\varepsilon \rightarrow 0$. The precise result we are to show is the following. \[ecotsuplema1\] Given $\delta > 0$ we have $$\lim_{\varepsilon \rightarrow 0 } \left[\sup_{u \in B_c} \left| P_{u} \left( e^{\frac{\Delta - \delta}{\varepsilon^{2}}} < \tau_{\varepsilon}(\partial G) < e^{\frac{\Delta + \delta}{\varepsilon^{2}}}\right)-1 \right|\right] = 0.$$ We shall split the proof of this result into two parts over Sections \[secupperbound\] and \[seclowerbound\], the first of them dealing with the upper bound and the second one with the lower bound. ### Upper bound on $\tau_\varepsilon(\partial G)$ {#secupperbound} Our first goal is to establish the upper bound for the escape time from $G$ contained in the following theorem. \[cotsuplema1\] For any $\delta > 0$ we have $$\label{cotsuplema1eq1} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in G} P_{u} \left( \tau_{\varepsilon}(\partial G) > e^{\frac{\Delta + \delta}{\varepsilon^{2}}}\right) \right] = 0.$$ The main idea behind the proof is to show that there exist paths escaping the domain $G$ with rates arbitrarily close to $\Delta$, so that the typical time one must wait for any of these paths to be described by the stochastic system is of lesser asymptotic order than $e^{\frac{\Delta + \delta}{\varepsilon^{2}}}$. The precise estimate is contained in the following lemma. \[cotasuplema0\] Given $\delta > 0$ there exists $T^{(\delta)} > 0$ such that for each $u \in G$ there exists a set of paths $\mathcal{E}_{u,T^{(\delta)}} \subseteq C_{D_u}([0,T^{(\delta)}] \times [0,1])$ satisfying 1. Every path in $\mathcal{E}_{u,T^{(\delta)}}$ escapes $G$ before time $T^{(\delta)}$. 2. For any $\varepsilon > 0$ sufficiently small we have $\inf_{u \in G} P_u ( U^\varepsilon \in \mathcal{E}_{u,T^{(\delta)}} ) \geq T^{(\delta)}e^{-\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}}.$ We will prove the lemma with the aid of the large deviations principle established for our system. The idea of the proof is to show that for each $u \in G$ there exists a path $\varphi^u \in C_{D_u}([0,T^{(\delta)}]\times [0,1])$ starting at $u$ with rate less than $\Delta + \frac{\delta}{3}$ and such that not only does $\varphi^u$ itself escape from $G$ before time $T^{(\delta)}$, but also any path sufficiently close to $\varphi^u$ does so as well. We construct $\varphi^{u}$ explicitly for each $u \in G$. Each path $\varphi^{u}$ will consist of several pieces, each of which must either follow the trajectory described by the deterministic system (sometimes in the opposite direction) or be a linear interpolation between nearby elements of $C_D \cap W^2_2([0,1])$. In view of this last possibility, we first need to establish some control on the contribution of these linear interpolations to Thus, let us consider a velocity one linear interpolation $s$ between two points $u,w \in B_{n_0+1}$, i.e. $s: [0,\|w-v\|_\infty] \times [0,1] \to {{\mathbb R}}$ given by $$s(t,x)= u(x) + t \cdot \left(\frac{w(x)-u(x)}{\|w-u\|_\infty}\right)\text{ for $(t,x) \in [0,\|w-u\|_\infty] \times [0,1]$},$$ and suppose that both $u$ and $w$ have $W^2_2([0,1])$-norm bounded by some constant $M > 0$. Then we have $$I(s) = \frac{1}{2}\int_0^{\|w-u\|_\infty} \int_0^1 | {\partial}_t s - {\partial}^2_{xx} s - g_{n_0}(s) |^2 \leq \int_0^{\|w-u\|_\infty} \int_0^1 \left[ | {\partial}_t s |^2 + |{\partial}^2_{xx} s + g_{n_0}(s) |^2 \right].$$ Since $\| {\partial}_t s \|_\infty = 1$ by construction and ${\partial}^2_{xx} s (t,\cdot) = \partial^2_{xx}u + t \cdot \left(\frac{{\partial}^2_{xx}w-{\partial}^2_{xx}u}{\|w-u\|_\infty}\right)$ we obtain that $$I(s) \leq C_{n_0,M} \|w-v\|_\infty$$ where $C_{n_0,M} > 0$ is a constant depending only on $\| g_{n_0+1}\|_\infty$ and $M$. Taking this into consideration, by using Propositions \[G.1\], \[G.2\] and \[A.1\] in the Appendix we may take a time $T_0 > 0$ sufficiently small such that $$\sup_{t \in [0,T_0]} \| U^u(t,\cdot) - u \|_\infty < \frac{1}{2} \hspace{1.2cm}\text{ and }\hspace{1cm} \sup_{t \in [0,T_0]} \| U^u(T_0,\cdot) - U^v(T_0,\cdot) \|_\infty \leq 2 \| u - v\|_\infty$$ for any $u,v \in B_{n_0+1}$, a constant $H > 0$ such that $\| U^{u}(T_0,\cdot)\|_{W^2_2([0,1])} \leq H$ for any $u \in B_{n_0+1}$ and a distance $r > 0$ such that any linear interpolation between elements in $B_{n_0+1}$ with $W^2_2([0,1])$-norm bounded by $2H$ and at a distance smaller than $2r$ has rate less than $\frac{\delta}{9}$. We may assume that both $T_0$ and $r$ are sufficiently small, e.g. $T_0 < 1$ and $r < \min\{ \frac{1}{4}, c\}$. We then define the set $$\mathcal{W}_{(r^-)} = \{ u \in C_D([0,1]) : d(u, \mathcal{W}) < r\}.$$ The construction of $\varphi^{u}$ is done as follows. The first step is to follow the deterministic flow for a time period of length $T_0$. The remainder of the construction will vary according to where $v_0:=U^{u}(T_0,\cdot)$ is located. We describe the different scenarios below. - If $v_0 \in B_{n_0+r}^c$ then $\varphi^u$ has already escaped $G$ and reached a distance from $G$ which is greater than $r$. The construction in this scenario ends here. - If $v_0 \in B_{n_0+r} \cap \mathcal{W}_{(r^-)}$ then: 1. We choose $v \in \mathcal{W}$ such that $d(v_0,v) < r$ and first let $\varphi^u$ follow once again the deterministic flow for a time period of length $T_0$ and afterwards describe the linear interpolation between the points $U^{v_0}(T_0, \cdot)$ and $v_1:= U^{v}(T_0,\cdot)$ in time $T_1^u = d(U^{v_0}(T_0, \cdot),v_1)$. Notice that $U^{v_0}(T_0, \cdot)$ and $v_1$ lie inside $B_{n_0 +1}$, are at a distance smaller than $2r$ from each other and both have $W^2_2([0,1])$-norm less than $H$ by the choice of $T_0$ and the fact that both $v_0$ and $v$ lie inside $B_{n_0 + \frac{1}{2}}$. 2. From there we let the path $\varphi^u$ follow the deterministic flow $U^{v_1}$ until the time $T_2^u = \inf \{ t \geq T_0 : d( U^{v_1}(t,\cdot), z^{(n)}) \leq r \}$ for some $n \in {{\mathbb{Z}}}- \{ 0 \}$. 3. If for some $t \in [0,T_2^u]$ we have $U^{v_1}(t,\cdot) \notin B_{n_0+r}$ then once again we have that $\varphi^u$ has already escaped $G$ and reached a distance from $G$ greater than $r$, in which case we end the construction here. 4. If this is not the case then we continue $\varphi^u$ by describing the linear interpolation between $v_2:=U^{v_1}(T_2^u,\cdot)$ and $v_3:=(1+r)z^{(n)}$ in time $T_3^u= d(v_2,v_3) \leq 2r$. Notice that $v_2$ and $v_3$ lie inside $B_{n_0 +1}$ and both have $W^2_2([0,1])$-norm less than $2H$ since we have that $v_2 = U^{U^{v_1}(T_2^u - T_0,\cdot)}(T_0,\cdot)$ and $U^{z^{(n)}}(T_0,\cdot)=z^{(n)}$. 5. Finally, we follow once again the deterministic flow $U^{v_3}$ until we reach a distance from $G$ greater than $r$ in a finite time $T^u_4$ which depends only on $r$ and $z^{(n)}$. Notice that this is possible due to the fact that $v_3 \in \mathcal{D}_e$ by Proposition \[descomp3\]. - If $v_0 \in B_{n_0 + r} \cap \mathcal{D}_{\mathbf{0}} \cap \mathcal{W}_{(r^-)}^c$ then: 1. From there we let the path $\varphi^u$ follow the deterministic flow $U^{v_0}$ until the time $T_5^u = \inf \{ t \geq T_0 : U^{u}(t,\cdot) \in B_r\}$. 2. Next we fix $u^* \in W^z_u \cap {\partial}B_r$ and consider $T^*= \inf \{ t \geq T_0 : U^{u^*}(t,\cdot) \in B_r \}$, a time which only depends on the choice of $u^*$. We then continue $\varphi^u$ by describing the linear interpolation between $v_4:= U^{u}(T_5^u,\cdot)$ and $v_5:= U^{u^*}(T^*,\cdot)$ in time $T_6^u = d(v_4,v_5) \leq 2r$. Notice that $v_4$ and $v_5$ lie inside $G$ since $r < c$ and both have $W^2_2([0,1])$-norm less than $H$ by a similar argument to the one given above. 3. Once on $\mathcal{W}^z_u$ we let $\varphi^u$ follow the reverse deterministic flow until the time $T_7^u = \inf \{ t \geq 0 : U^{v_5}(-t,\cdot) \in G \cap \mathcal{W}_{(r^-)}\}$ which does not depend on $u$, but rather on the choice of $u^*$ instead. 4. We can then continue as in the second scenario, by noticing that $U^{v_5}(-T_7^u,\cdot)$ belongs to $B_{n_0}$ and has $W^2_2([0,1])$-norm less than $H$ since $\mathcal{W}^z_u \cap \mathcal{D}_{\mathbf{0}} \subseteq G$ by the mere construction of $G$. - If $v_0 \in B_{n_0 + r} \cap \mathcal{D}_{e} \cap \mathcal{W}_{(r^-)}^c$ then we let the path $\varphi^u$ follow the deterministic flow $U^{v_0}$ until we reach a distance from $G$ greater than $r$ in a finite time $T^u_8$. If built in this way, the path $\varphi^u$ verifies all the required properties. Indeed, we have that: - Each $\varphi^u$ belongs to $C_{D_u} \cap W^{1,2}_2 ([0,T^u]\times [0,1])$ for some $T_u > 0$ (the sum of the corresponding $T^u_i$) since by construction $\varphi^u$ is piecewise differentiable. - The total time length $T^u$ of the path $\varphi^u$ is uniformly bounded in $u \in G$. Indeed, the total time which $\varphi^u$ spends following the deterministic flow is uniformly bounded by Proposition \[A.3\] since there are only finitely many equilibrium On the other hand, there are at most three linear interpolations in $\varphi^u$, each of which lasts a time of length less than one. Finally, the time spent by $\varphi^u$ following the reverse deterministic flow is finite and does not depend on $u$. - Each $\varphi^u$ has total rate less than $\Delta + \frac{\delta}{3}$. Indeed, its total rate can be computed as the sum of the rate of each of its pieces. We have already seen that each linear interpolation has rate less than $\frac{\delta}{9}$ and, since in any case there at most three of them, their total contribution is less than $\frac{\delta}{3}$. On the other hand, the pieces in which $\varphi^u$ follows the deterministic flow have zero rate. Finally, if $\varphi^u$ follows the reverse deterministic flow (i.e. ${\partial}_t \varphi^u = - ({\partial}^2_{xx} \varphi^u + g(\varphi^u)) )$ during the time interval $[t_1,t_2]$ then, similarly to Proposition \[Lyapunov\], we have $$\frac{1}{2}\int_{t_1}^{t_2} \int_0^1 |{\partial}_t \varphi^u - ({\partial}^2_{xx} \varphi^u + g(\varphi^u))|^{2} = 2 \int_{t_1}^{t_2} \frac{d S(\varphi^u(t,\cdot))}{dt} = 2 (S(\varphi^u(t_2)) - S(\varphi^u(t_1)))$$ from where, upon recalling that $\varphi^u(t_2),\varphi^u(t_1) \in \mathcal{W}^z_u$, we obtain that the rate of this last piece is less than $\Delta$. - Any path at a distance strictly less than $r$ from $\varphi^u$ in the supremum norm must also escape from $G$ before $T^u$, since by this time $\varphi^u$ reaches a distance $r$ from $G$. Let us notice that for each $u \in G$ we have built a path $\varphi^u$ on the time interval $[0,T^u]$, but we wish all constructed paths to be defined on a same time interval. For this reason, we consider $T^{(\delta)}:= \sup_{u \in G} T^u < +\infty$ and extend all $\varphi^u$ to the time interval $[0,T^{(\delta)}]$ by following the deterministic flow. It is easy to check that these extended paths maintain the aforementioned properties. We then define the set $\mathcal{E}_{u,T^{(\delta)}}$ for each $u \in G$ as $$\mathcal{E}_{u,T^{(\delta)}}:=\{ \psi \in C_{D_u}([0,T^{(\delta)}] \times [0,1]) : d_{T^{(\delta)}}(\psi , \varphi^u ) < r\}.$$ It is clear that each $\mathcal{E}_{u,T^{(\delta)}}$ verifies condition (i) by construction, whereas (ii) follows from the large deviations estimate (i) in Section \[secLDP\]. Now, if we write $T^+_\varepsilon = e^{\frac{\Delta + \delta}{\varepsilon^2}}$ and split the interval $[0,T^+_\varepsilon]$ into subintervals of length $T^{(\delta)}$ given by Lemma \[cotasuplema0\], then by the Markov property for the solution of we obtain $$\label{boundonguniform} P_u ( \tau_{\varepsilon}(\partial G) > T^+_\varepsilon) \leq P_u \left( \bigcap_{k=1}^{m_\varepsilon} \{ U^\varepsilon (t,\cdot) \in G \text{ for all } t \in [(k-1)T^{(\delta)} , kT^{(\delta)}] \}\right) \leq (1-\alpha_\varepsilon)^{m_\varepsilon}$$ where $\alpha_\varepsilon:=T^{(\delta)}e^{-\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}}$, $m_\varepsilon:=\left\lfloor\frac{T^+_\varepsilon}{T^{(\delta)}}\right\rfloor$ and for the second inequality we used that for $u \in G$ $$\{ U^{u,\varepsilon}(t,\cdot) \in G \text{ for all }t \in [0,T^{(\delta)}]\} \subseteq \{ U^{u,\varepsilon} \notin \mathcal{E}_{u,T^{(\delta)}}\}.$$ Since the bound is uniform on $G$, by taking $\varepsilon \rightarrow 0$ a direct ### Lower bound on $\tau_\varepsilon(\partial G)$ {#seclowerbound} Our next purpose is to establish the lower bound on $\tau_\varepsilon({\partial}G)$ contained in the There exists $r > 0$ sufficiently small such that $$\label{cotainferioreq0} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in {\partial}B_{r}} P_u \left( \tau_\varepsilon ({\partial}G) < e^{\frac{\Delta - \delta}{\varepsilon^2}}\right) \right] = 0.$$ The key to establishing this lower bound is to observe that for initial data in a small neighborhood of the origin the path described by the stochastic system while it remains inside $G$ will typically consist of several failed attempts to reach ${\partial}G$ followed by one last attempt which is successful and thus leads to the escape from $G$. Each of these failed attempts is an excursion drifting away from the origin which, having failed to reach ${\partial}G$, later returns to (a small neighborhood of) the origin. Hence, the desired lower bound on the time the process needs in order to escape from $G$ can be obtained upon giving suitable bounds on the number and length of these excursions. To accomplish this we consider, given constants $r_1,r_2 > 0$ such for each $u \in {\partial}B_{r_1}$ and $\varepsilon > 0$ the increasing sequence of stopping times $$\left\{\begin{array}{l} \eta_0 = 0\\ \\ \sigma_0 = \inf \{ t \geq 0 : U^{u,\varepsilon}_t \in {\partial}B_{r_2}\} \end{array}\right.$$ and for $n \in {{\mathbb N}}_0$ $$\left\{\begin{array}{l} \eta_{n+1} = \inf \{ t > \sigma_n : U^{u,\varepsilon}_t \in {\partial}B_{r_1} \cup (\partial \tilde{G})_{(d)}\}\\ \\ \sigma_{n+1} = \inf \{ t > \eta_{n+1} : U^{u,\varepsilon}_t \in {\partial}B_{r_2}\} \end{array}\right.$$ where ${\partial}\tilde{G}$ is the pre-domain defined in and $d > 0$ is taken such that $B_{r_2}$ and $({\partial}\tilde{G})_{(d)}$ are at positive distance from each other, where $$(\partial \tilde{G})_{(d)}:=\{ u \in C_D([0,1]) : d(u,\partial \tilde{G}) \leq d \}.$$ These positive constants $r_1,r_2$ and $d$ will be later taken conveniently small Also, whenever any of the sets involved is empty we take the corresponding time as $+\infty$. We then define the Markov chain $(Z^{u,\varepsilon}_n)_{n \in {{\mathbb N}}}$ by the formula $$Z^{u,\varepsilon}_n=U^{u,\varepsilon}_{\eta_n}$$ for each $n \in {{\mathbb N}}_0$, and set $\vartheta^u_\varepsilon := \min\{ n \in {{\mathbb N}}: Z^{u,\varepsilon}_n \in (\partial \tilde{G})_{(d)}\}$. Since the process $U^{u,\varepsilon}$ escapes $\tilde{G}$ in a finite time almost surely and we are only interested in events occurring before $\tau_\varepsilon({\partial}\tilde{G})$, we need not worry about the possibility of $Z_n^\varepsilon$ not being well defined. Let us notice then that, since for any $u \in {\partial}B_{r_1}$ we have $\tau^u_\varepsilon ({\partial}\tilde{G}) \leq \tau^u_\varepsilon ({\partial}G)$ by the continuity of the trajectories of $U^u$, for each $\delta > 0$ we obtain the bound $$\label{cotainferioreq1} \sup_{u \in {\partial}B_{r_1}} P_u \left( \tau_\varepsilon ({\partial}G) < e^{\frac{\Delta - \delta}{\varepsilon^2}}\right) \leq \sup_{u \in {\partial}B_{r_1}} P_u( \nu_\varepsilon \leq k_\varepsilon ) + \sup_{u \in {\partial}B_{r_1}} P_u \left( \eta_{k_\varepsilon} < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right)$$ where $k_\varepsilon \in {{\mathbb N}}$ is to be determined next for each $\varepsilon > 0$. Thus, we see that to obtain it suffices to show that, for a suitable choice of $(k_\varepsilon)_{\varepsilon > 0}$, both terms in the right hand side of vanish as $\varepsilon \rightarrow 0$. We will do this with the aid of the following two lemmas, which are slight modifications of two results originally appearing in [@B1]. \[lemab1\] Let $F,B \subseteq C_D([0,1])$ be bounded sets and suppose $\psi \in C_D([0,1])$ is such that $d(\psi, F) > 3r$ for some $r > 0$. Then for any $h,T > 0$ there exists $r^* > 0$ such that $$\sup_{u \in B_{\rho}(\psi)} P_u ( \tau_\varepsilon( F_{(r)}) \leq \min \{ T , \tau_\varepsilon( B^c )\} ) \leq e^{-\frac{V(\psi,F_{(2r)}) - h}{\varepsilon^2}}$$ for any $0 < \rho < r^*$. \[lemab2\] Let $B \subseteq C_D([0,1])$ be bounded and closed. If for $e > 0$ we consider $$C_e := \bigcup_{n \in {{\mathbb{Z}}}} B_e(z^{(n)}).$$ then given $K > 0$ there exists $T > 0$ such that $$\sup_{u \in B} P_u \left( \min\{ \tau(C_e), \tau( B^c )\} > T \right) \leq e^{- \frac{K}{\varepsilon^2}}.$$ Now, in order to deal with the first term in the right hand side of it will suffice to establish the bound $$\label{cotainferioreq2} \sup_{u \in {\partial}B_{r_1}} P_u ( \vartheta_\varepsilon = 1 ) \leq e^{- \frac{\Delta - \frac{\delta}{4}}{\varepsilon^2}}$$ for $\varepsilon > 0$ sufficiently small. Indeed, if we prove then by the Markov property of $Z^\varepsilon$ we obtain $$\inf_{u \in {\partial}B_{r_1}} P_u( \vartheta_\varepsilon > n ) \geq \left(1 - e^{- \frac{\Delta - \frac{\delta}{4}}{\varepsilon^2}}\right)^n$$ for every $n \in {{\mathbb N}}$ and $\varepsilon > 0$ sufficiently small, which implies the inequality $$\label{cotainferioreq5} \sup_{u \in {\partial}B_{r_1}} P_u( \vartheta_\varepsilon \leq k_\varepsilon ) \leq 1 - \left(1 - e^{- \frac{\Delta - \frac{\delta}{4}}{\varepsilon^2}}\right)^{k_\varepsilon}$$ whose right hand side vanishes as $\varepsilon \rightarrow 0$ if we set for example $k_\varepsilon := \left[\exp\left({\frac{\Delta - \frac{\delta}{2}}{\varepsilon^2}}\right)\right]+1$. Thus, let us check that holds. Notice that the strong Markov property for $\sigma_0$ yields $$\sup_{u \in {\partial}B_{r_1}} P_u( \vartheta_\varepsilon = 1 ) \leq \sup_{u \in {\partial}B_{r_2}} P_u \left( \tau_\varepsilon ((\partial \tilde{G})_{(d)}) = \tau_\varepsilon ({\partial}B_{r_1} \cup (\partial \tilde{G})_{(d)})\right)$$ so that to check it will suffice to find $T > 0$ such that $$\label{cotainferioreq3} \sup_{u \in {\partial}B_{r_2}} P_u \left( \tau_\varepsilon ((\partial \tilde{G})_{(d)}) \leq T\right) \leq e^{- \frac{ \Delta - \frac{\delta}{8}}{\varepsilon^2}}$$ and $$\label{cotainferioreq4} \sup_{u \in {\partial}B_{r_2}} P_u \left( \tau_\varepsilon ({\partial}B_{r_1} \cup (\partial \tilde{G})_{(d)}) > T\right) \leq e^{- \frac{ \Delta - \frac{\delta}{8}}{\varepsilon^2}}$$ for every $\varepsilon > 0$ sufficiently small. Now, let us observe that, since the potential $S$ attains in $\pm z$ its minimum on $\partial \tilde{G}$, by proceeding as in the proof of Lemma \[cotasuplema0\] one can show that $V(\mathbf{0},{\partial}\tilde{G})=\Delta$. Therefore, by conducting a similar argument to the one given in the construction of $G$, it follows that the set $$\mathcal{V} := \left\{ u \in C_D([0,1]) : V(\mathbf{0},u) \leq \Delta - \frac{\delta}{16} \right\}$$ is contained in $\tilde{G}$ and satisfies $d( \mathcal{V},\mathcal{W} ) > 0$. Furthermore, since $\mathcal{V}$ is contained in $\overline{D_\mathbf{0}}$ and the set $\{ u \in \overline{D_\mathbf{0}} : 0 \leq S(u) \leq S(z) \}$ is contained in $B_{n_0 -1}$, then Proposition \[costo\] implies that $d( \mathcal{V}, {\partial}B_{n_0}) > 0$ and thus we obtain that $d(\mathcal{V}, {\partial}\tilde{G}) \geq \min\{ d( \mathcal{V},\mathcal{W} ) , d( \mathcal{V}, {\partial}B_{n_0})\} > 0$. In particular, if we take $d < \frac{d(\mathcal{V}, {\partial}\tilde{G})}{3}$ then we have $V(\mathbf{0}, (\partial \tilde{G})_{(2d)}) \geq \Delta - \frac{\delta}{16}$ and therefore by Lemma \[lemab1\] we conclude that for a fixed $T > 0$ and $r_2 > 0$ sufficiently small $$\sup_{u \in {\partial}B_{r_2}} P_u ( \tau_\varepsilon ( ({\partial}\tilde{G})_{(d)}) \leq T ) \leq e^{- \frac{ \Delta - \frac{\delta}{8}}{\varepsilon^2}}$$ provided that $\varepsilon > 0$ is sufficiently small, which yields . On the other hand, if we take $0 < e < \min\{ r_1 , d\}$ then we have that $B_e( z^{(n)} ) \subseteq ({\partial}\tilde{G})_{(d)}$ for any $n \in {{\mathbb{Z}}}- \{ 0 \}$ such that $z^{(n)} \in \tilde{G}$. In particular, we see that $$\label{cotainferioreq6} \sup_{u \in {\partial}B_{r_2}} P_u \left( \tau_\varepsilon ({\partial}B_{r_1} \cup (\partial \tilde{G})_{(d)}) > T\right) \leq \sup_{u \in {\partial}B_{r_2}} P_u \left( \min\{ \tau(C_e), \tau( (\partial \tilde{G})_{(d)} )\} > T \right).$$ It then follows from Lemma \[lemab2\] that $T > 0$ can be taken sufficiently large so that holds. Together with , this yields and establishes the convergence to zero of the first term in the right hand side of . Thus, it only remains to establish the same convergence for the second term. Notice that by Proposition \[G.1\] there exists some $T_{r_2} > 0$ such that for any $u \in {\partial}B_{r_2}$ the system $U^u$ spends a time of length at least $T_{r_2}$ before reaching $B_{\frac{r_2}{2}}$. Furthermore, we may assume that $r_2$ is sufficiently small so that the path described by $U^u$ until time $T_{r_2}$ is at a distance greater than $\frac{r_2}{2}$ from $({\partial}\tilde{G})_{(d)}$. By the strong Markov property and we conclude that for $\varepsilon > 0$ sufficiently small $$\inf_{u \in {\partial}B_{r_1}} P_u ( \eta_1 \geq T_{r_2} ) \geq \inf_{u \in {\partial}B_{r_2}} P_u \left( d_{T_{r_2}}\left( U^{(n_0),\varepsilon},U^{(n_0)} \right) < \frac{r_2}{2} - r_1 \right) \geq \frac{2}{3}.$$ Let us observe that since for any $k \in {{\mathbb N}}$ we have the inequality $\eta_k \geq \sum_{i=1}^k T_{r_2} \mathbbm{1}_{\{\eta_i - \eta_{i-1} \geq T_{r_2}\}}$, by definition of $k_\varepsilon$ we obtain that for $\varepsilon > 0$ sufficiently small $$\sup_{u \in {\partial}B_{r_1}} P_u \left( \eta_{k_\varepsilon} < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right) \leq \sup_{u \in {\partial}B_{r_1}} P_u \left( \frac{\eta_{k_\varepsilon}}{k_\varepsilon} < e^{- \frac{\delta}{2\varepsilon^2}} \right) \leq P \left( \frac{1}{k_\varepsilon}\sum_{i=1}^{k_\varepsilon} X_i < \frac{e^{- \frac{\delta}{2\varepsilon^2}}}{T_{r_2}}\right)$$ where $(X_i)_{i \in {{\mathbb N}}}$ is a sequence of i.i.d. Bernoulli random variables with parameter $p=\frac{2}{3}$. The result now follows at once from the law of large numbers. The escape route ---------------- We are now interested in characterizing the typical route that the stochastic system describes to escape from $G$. By the considerations made at the beginning of this Section we expect this typical path to escape $G$ by going through the region of the boundary with the lowest quasipotential, namely ${\partial}^{\pm z}$. More precisely, we wish to prove the \[teoescape0\] If $c > 0$ is given by (ii) in Conditions \[assumpg\] then $$\label{escape0} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_c} P_u \left( U^{\varepsilon}(\tau_\varepsilon(\partial G),\cdot) \notin \partial^{\pm z}\right)\right] = 0.$$ To prove this result we shall need to establish the following two crucial facts: 1. For each $r > 0$ strictly smaller than $c$ $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_c} P_u\left(\tau_\varepsilon(\partial G) < \tau\left(B_r\right)\right)\right]=0.$$ 2. For any $r > 0$ sufficiently small $$\label{escape1} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in \partial B_r} P_u \left( U^{\varepsilon}(\tau_\varepsilon(\partial G),\cdot) \notin \partial^{\pm z} \right) \right] = 0.$$ Indeed, follows immediately from (i) and (ii) by applying the strong Hence, it will suffice to establish (i) and (ii). Assertion (i) is shown in the following lemma. \[escapelemai\] For each $r > 0$ strictly smaller than $c$ we have $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_c} P_u\left(\tau_\varepsilon(\partial G) < \tau\left(B_r\right)\right)\right]=0.$$ Notice that by choice of $c$ we have that any $u \in B_c$ reaches the neighborhood $B_{\frac{r}{2}}$ in a finite time $T^u_r$ while remaining at a distance greater than $r_{\mathbf{0}}$ in $\partial G$ and $\mathcal{W}$. By Proposition \[A.3\] we therefore conclude that $T_r = \sup_{u \in B_c} T^u_r$ must be finite. Hence, for $u \in B_c$ we obtain $$P_u \left( \tau_\varepsilon(\partial G) < \tau_\varepsilon\left(B_r\right) \right) \leq \sup_{v \in B_c} P_v \left( d_{T_r}(U^{(n_0),\varepsilon},U^{(n_0)}) \geq r_{\mathbf{0}}\wedge \frac{r}{2}\right)$$ which, by and the uniformity of the bound in $u \in B_c$, implies the result at once. In order to prove assertion (ii) we first show that it suffices to study the path described by the stochastic system since its last visit to (a small neighborhood of) the origin. this by resorting to a Markov chain similar to the one introduced in the preceding section to establish the lower bound. More precisely, given constants $r_1,r_2 > 0$ such that $r_1 < \frac{r_2}{2}$ and $r_2 < c$, for each $u \in {\partial}B_{r_1}$ and $\varepsilon > 0$ we consider the increasing sequence of stopping times $$\left\{\begin{array}{l} \eta_0 = 0\\ \\ \sigma_0 = \inf \{ t \geq 0 : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{r_2}\} \end{array}\right.$$ and for $n \in {{\mathbb N}}_0$ $$\left\{\begin{array}{l} \eta_{n+1} = \inf \{ t > \sigma_n : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{r_1} \cup \partial G\}\\ \\ \sigma_{n+1} = \inf \{ t > \eta_{n+1} : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{r_2}\} \end{array}\right.$$ with the convention that $\inf \emptyset = +\infty$. We then define the Markov chain $(Z^{u,\varepsilon}_n)_{n \in {{\mathbb N}}}$ as $$Z^{u,\varepsilon}_n:=U^{u,\varepsilon}(\eta_n,\cdot)$$ for each $n \in {{\mathbb N}}_0$ and set $\vartheta^u_\varepsilon := \min \{ n \in {{\mathbb N}}: Z^{u,\varepsilon}_n \in {\partial}G\}$. Just as in the previous section, for our purposes we will not need to worry about the possibility of $Z_n^\varepsilon$ not being well defined. Also, the constants $r_1$ and $r_2$ will be later taken conveniently small It is not hard to see that for any $u \in {\partial}B_{r_1}$ the strong Markov property yields $$\label{cociente} P_u \left( U^{\varepsilon}(\tau_\varepsilon(\partial G),\cdot) \notin \partial^{\pm z} \right) \leq \sup_{v \in {\partial}B_{r_1}} \frac{ P_v ( Z_1^\varepsilon \in {\partial}G - {\partial}^{\pm z} )}{P_v ( Z_1^\varepsilon \in {\partial}G)}$$ from which we conclude that in order to show (ii) it will suffice to give a lower bound for the denominator and an upper bound for the numerator such that the quotient of these bounds goes to zero with $\varepsilon$. The following lemma, whose proof can be found in [@B1], provides the desired lower bound. \[lemab0\] Let us suppose that $r_1$ is sufficiently small so as to guarantee that for any $u \in {\partial}B_{r_1}$ the deterministic orbit $\{ U^u(t,\cdot) : t \geq 0 \}$ does not intersect ${\partial}B_{\frac{r_2}{2}}$. Then for all $\varepsilon > 0$ sufficiently small $$\inf_{u \in {\partial}B_{r_1}} P_u( Z_1^\varepsilon \in {\partial}G) \geq e^{- \frac{\Delta + k r_2}{\varepsilon^2} },$$ where $k > 0$ is a constant which does not depend on the choice of $r_1$ and $r_2$. The upper bound on the numerator in is more involved and requieres several steps. We start by considering, for fixed constants $d,e > 0$ such that $e < r_z$ and $d < \frac{r_z - e}{2}$ (which will be later made conveniently small), the stopping times $$\tau_\varepsilon^0 = \tau_\varepsilon \left( ({\partial}G - {\partial}^{\pm z})_{(d)} \right) \hspace{2cm}\text{ and }\hspace{2cm} \tau_\varepsilon^2 = \tau_\varepsilon \left( B_e( \pm z) \right)$$ where $$({\partial}G - {\partial}^{\pm z})_{(d)} := \{ u \in C_D([0,1]) : d(u, {\partial}G - {\partial}^{\pm z}) \leq d \}.$$ where the initial datum is implicit. Notice that $d$ is such that $({\partial}G - {\partial}^{\pm z})_{(d)}$ and $B_e( \pm z)$ are disjoint. Then we can decompose the event in the numerator into three disjoint parts: $$\label{decomp} A:=\{ Z_1^{u,\varepsilon} \in {\partial}G - {\partial}^{\pm z}\} = \bigcup_{i=0}^2 [A \cap \{ \tau_\varepsilon^i = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}]$$ where, for notational convenience, we have set $\tau_\varepsilon^1$ as some fixed time $T > 0$ which is to be conveniently determined later and $\tau_\varepsilon^3$ as the escape time $\tau_\varepsilon({\partial}G)$ from our domain $G$. Observe that the set $A \cap \{\tau_\varepsilon^3 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}$ is empty and is therefore left out of the decomposition. Thus, in order to provide an upper bound for the numerator we see that it will suffice to estimate the probabilities of each of the sets in the decomposition. ### Upper bound on $P(A \cap \{ \tau_\varepsilon^0 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})$ To bound the probability of the first set we observe that if $d > 0$ is taken sufficiently small then $$\label{cotacuasipotencial} V(\mathbf{0},({\partial}G - {\partial}^{\pm z})_{(2d)}) > \Delta.$$ Indeed, since $$({\partial}G - {\partial}^{\pm z})_{(2d)} = ( {\partial}\tilde{G} - B_{r_z}(\pm z))_{(2d)} \cup ( [{\partial}B_{r_z}(\pm z) \cap \overline{\mathcal{D}_e}] - {\partial}^{\pm z})_{(2d)}$$ to establish it will suffice to show that $$\label{cotacuasipotencial2} V(\mathbf{0},( {\partial}\tilde{G} - B_{r_z}(\pm z))_{(2d)}) > \Delta\hspace{0.7cm}\text{ and }\hspace{0.7cm}V(\mathbf{0},( [{\partial}B_{r_z}(\pm z) \cap \overline{\mathcal{D}_e}] - {\partial}^{\pm z})_{(2d)}) > \Delta.$$ From the definition of ${\partial}^{\pm z}$ it easily follows that if we take $2d < \frac{d( \tilde{{\partial}}^z, \mathcal{W})}{2}$ then the sets $\tilde{{\partial}}^z$ and $( [{\partial}B_{r_z}(z) \cap \overline{\mathcal{D}_e}] - {\partial}^{z})_{(2d)}$ are disjoint so that the second inequality in is settled. To obtain the first inequality we begin by noticing that by an argument analogous to the one employed in the construction of $G$ there exists $\alpha > 0$ such that $$\inf\{ S(u) : u\in [\mathcal{W} \cap B_{n_0+1}] - B_{\frac{r_z}{2}}(\pm z) \} > \alpha > S(z).$$ Hence, to establish the first inequality in it will suffice to consider the set $$\mathcal{V}'= \{ u \in C_D([0,1]) : V(\mathbf{0},u) \leq 2 \min\{\alpha, S(z) + \zeta_1\} \}$$ and show that $d( \mathcal{V}', {\partial}\tilde{G} - B_{r_z}(\pm z) ) > 0$. But, if this were not so, then there would exist sequences $(u_k)_{k \in {{\mathbb N}}} \subseteq \mathcal{V}'$ and $(v_k)_{k \in {{\mathbb N}}} \subseteq {\partial}\tilde{G} - B_{r_z}(\pm z)$ such that $d(u_k,v_k) \rightarrow 0$ as $k \rightarrow +\infty$. By Propositions \[G.2\] and \[A.2\] we may find subsequences $(u_{k_j})_{j \in {{\mathbb N}}},(v_{k_j})_{j \in {{\mathbb N}}}$ and a time $t > 0$ such that $$\lim_{j \rightarrow +\infty} U^{u_{k_j}}(t,\cdot) =\lim_{j \rightarrow +\infty} U^{v_{k_j}}(t,\cdot) = v_\infty$$ for some limit $v_\infty \in C_D([0,1])$. Let us observe that, since we have $v_\infty=\lim_{j \rightarrow +\infty} U^{u_{k_j}}(t,\cdot)$, the lower semicontinuity of $V(\mathbf{0},\cdot)$ and the fact that the mapping $t \mapsto V(\mathbf{0},U^u(t,\cdot))$ is monotone decreasing for any $u \in C_D([0,1])$ together imply that $$\label{cotacuasipotencial3} V(\mathbf{0},v_\infty) \leq 2 \min\{\alpha, S(z) + \zeta_1\}.$$ At least one of the following possibilities must then occur: 1. If $v_{k} \in {\partial}B_{n_0} \cap \mathcal{D}_{\mathbf{0}}$ for infinitely many $k \in {{\mathbb N}}$, then $(k_j)_{j \in {{\mathbb N}}}$ and $t > 0$ can be taken so as to guarantee that the condition $U^{v_{k_j}}(t,\cdot) \notin B_{n_0 -1}$ is satisfied for every $j \in {{\mathbb N}}$. Since $\overline{\mathcal{D}_{\mathbf{0}}}$ is closed and invariant under the deterministic flow we therefore conclude that $v_\infty \in \overline{\mathcal{D}_{\mathbf{0}}} - B_{n_0 -1}^\circ$ and thus that $S(v_\infty) > S(z)+\zeta_1$. In particular, we obtain that $V(\mathbf{0},v_\infty) > 2(S(z)+\zeta_1)$, a fact which contradicts . 2. If $v_{k} \in \mathcal{W} \cap B_{n_0}$ for infinitely many $k \in {{\mathbb N}}$, then $(k_j)_{j \in {{\mathbb N}}}$ and $t > 0$ can be taken so as to guarantee that $U^{v_{k_j}}(t,\cdot) \in B_{n_0+1} - B_{\frac{2}{3}r_z}(\pm z)$ is satisfied for every $j \in {{\mathbb N}}$. Since $\mathcal{W}$ is closed and invariant under the deterministic flow we then conclude that $v_\infty \in [\mathcal{W} \cap B_{n_0+1}] - B_{\frac{r_z}{2}}(\pm z)$ and thus that $S(v_\infty) > \alpha$. In particular, we obtain that $V(\mathbf{0},v_\infty) > 2\alpha$, which again contradicts . Now that we have shown , Lemma \[lemab1\] guarantees that if $r_1$ is sufficiently small then there exists $h > 0$ such that $$\sup_{u \in {\partial}B_{r_1}} P_u (A \cap \{ \tau_\varepsilon^0 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}) \leq \sup_{u \in {\partial}B_{r_1}} P_u ( \tau_\varepsilon^0 \leq \min\{ T, \tau_\varepsilon({\partial}G) \} ) \leq e^{- \frac{\Delta + h }{\varepsilon^2}}$$ which provides the desired upper bound. Indeed, notice that if $T > 0$ is fixed and $r_2 > 0$ is taken sufficiently small then the upper bound obtained and Lemma \[lemab0\] together yield $$\label{decomp1} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in {\partial}B_{r_1}} \frac{P_u (A \cap \{ \tau_\varepsilon^0 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})}{P_u ( Z_1^\varepsilon \in {\partial}G ) }\right] = 0.$$ ### Upper bound on $P(A \cap \{ \tau_\varepsilon^1 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})$ To bound the probability of the second set we observe that any unstable equilibrium of the system lying inside $G$ and different from $\pm z$ must necessarily belong to ${\partial}G - {\partial}^{\pm z}$. This implies that if we take $e < \min\{r_1,d\}$ then $$\sup_{u \in {\partial}B_{r_1}} P_u(A \cap \{ \tau_\varepsilon^1 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}) \leq \sup_{u \in {\partial}B_{r_1}} P_u ( T \leq \min\{ \tau_\varepsilon(C_e), \tau_\varepsilon({\partial}G)\} )$$ where $C_e$ is defined as in Lemma \[lemab2\]. Hence, by the same lemma we obtain for $T > 0$ sufficiently large the upper bound $$\sup_{u \in {\partial}B_{r_1}} P_u(A \cap \{ \tau_\varepsilon^1 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}) \leq e^{-\frac{ \Delta + 1}{\varepsilon^2}}.$$ Together with Lemma \[lemab0\] this upper bound yields $$\label{decomp2} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in {\partial}B_{r_1}} \frac{P_u (A \cap \{ \tau_\varepsilon^1 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})}{P_u ( Z_1^\varepsilon \in {\partial}G ) }\right] = 0.$$ ### Upper bound on $P(A \cap \{ \tau_\varepsilon^2 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})$ To conclude the proof it only remains to give an upper bound on the probability of the third set in the right hand side of . If we write $D^2 = \{ \tau_\varepsilon^2 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \}$ then notice that by the strong Markov property one has $$P_u (A \cap D^2 ) \leq {{\mathbb E}}( \mathbbm{1}_{D^2} P_{U^{u,\varepsilon}(\tau_\varepsilon^2,\cdot)} ( U^\varepsilon(\tau_\varepsilon({\partial}G),\cdot) \in {\partial}G - {\partial}^{\pm z} , \tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G) )).$$ The next lemma provides a suitable upper bound on the probability inside the expectation. There exists a constant $C > 0$ such that for every $e > 0$ sufficiently small one has $$\label{eqlemab3} \sup_{v \in B_e(\pm z)} \frac{ P_v ( U^\varepsilon(\tau_\varepsilon({\partial}G),\cdot) \in {\partial}G - {\partial}^{\pm z} , \tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G) )}{P_v (\tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G))} \leq e^{-\frac{C}{\varepsilon^2}}$$ for every $\varepsilon > 0$ sufficiently small. The idea is to consider once again a suitable embedded Markov chain. Fix $f > 0$ such that $B_f(\pm z)$ is at a positive distance from $({\partial}G - {\partial}^{\pm z})_{(d)}$ and assume that $e < \frac{f}{2}$. Consider then the stopping times $$\left\{\begin{array}{l} \tilde{\eta}_0 = 0\\ \\ \tilde{\sigma}_0 = \inf \{ t \geq 0 : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{f}(\pm z) \} \end{array}\right.$$ and for $n \in {{\mathbb N}}_0$ $$\left\{\begin{array}{l} \tilde{\eta}_{n+1} = \inf \{ t > \tilde{\sigma}_n : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{e}(\pm z) \cup \partial G\}\\ \\ \tilde{\sigma}_{n+1} = \inf \{ t > \tilde{\eta}_{n+1} : U^{u,\varepsilon}(t,\cdot) \in {\partial}B_{f}(\pm z)\} \end{array}\right.$$ with the convention that $\inf \emptyset = +\infty$. We then define the Markov chain $(W^{u,\varepsilon}_n)_{n \in {{\mathbb N}}}$ as $$W^{u,\varepsilon}_n:=U^{u,\varepsilon}({\tilde{\eta}_n},\cdot)$$ for $n \in {{\mathbb N}}_0$ and set $\tilde{\vartheta}^u_\varepsilon := \min \{ n \in {{\mathbb N}}: W^{u,\varepsilon}_n \in {\partial}G\}$. To show it $$\sup_{v \in B_e(\pm z)} \frac{ P_v ( W_1^\varepsilon \in {\partial}G - {\partial}^{\pm z} , \tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G), \tilde{\vartheta}_\varepsilon=1 ) }{ P_v( \tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G), \tilde{\vartheta}_\varepsilon = 1)} \leq e^{- \frac{C}{\varepsilon^2}}$$ holds for all $\varepsilon > 0$ sufficiently small provided $e > 0$ is chosen adequately. To see this, that $V( \pm z , {\partial}G)= 0$ and thus, by Lemma \[lemab0\], we obtain the lower bound $e^{- \frac{K f}{\varepsilon^2}}$ for the denominator, where $K > 0$ does not depend on the choice of both $e$ and $f$. On the other hand, since $$V(\mathbf{0}, ({\partial}G - {\partial}^{\pm z})_{(2d)} ) \leq V(\mathbf{0}, \pm z) + V( \pm z , ({\partial}G - {\partial}^{\pm z})_{(2d)})$$ we see that $V(\pm,({\partial}G - {\partial}^{\pm z})_{(2d)}) > 0$ and thus, with the aid of Lemmas \[lemab1\] and \[lemab2\], bound $e^{- \frac{h}{\varepsilon^2}}$ for the numerator, for some small constant $h > 0$. The lemma follows at once by taking $f$ sufficiently small. We are now ready to finish the proof of Theorem \[teoescape0\]. Indeed, by we obtain $$P_u (A \cap D^2 ) \leq e^{-\frac{C}{\varepsilon^2}}{{\mathbb E}}( \mathbbm{1}_{D^2} P_{U^{u,\varepsilon}(\tau_\varepsilon^2,\cdot)} (\tau_\varepsilon( B_{r_1} ) > \tau_\varepsilon({\partial}G) )) \leq e^{-\frac{C}{\varepsilon^2}}P_u ( Z_1^\varepsilon \in {\partial}G)$$ for every $u \in {\partial}B_{r_1}$ provided that $e > 0$ is taken sufficiently small. This implies that $$\label{decomp3} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in {\partial}B_{r_1}} \frac{P_u (A \cap \{ \tau_\varepsilon^2 = \min\{ \tau_\varepsilon^0, \tau_\varepsilon^1, \tau_\varepsilon^2, \tau_\varepsilon^3 \} \})}{P_u ( Z_1^\varepsilon \in {\partial}G ) }\right] = 0.$$ and thus concludes the proof. By using , the same argument given here to prove Theorem \[teoescape0\] can be used to show that for any $\delta > 0$ $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_c} P_u \left( U^\varepsilon \left( \tau_\varepsilon({\partial}\tilde{G}),\cdot \right) \notin B_\delta(\pm z) \right) \right] = 0.$$ This result tells us, perhaps in a more explicit manner than Theorem \[teoescape0\] does, that for sufficiently small $\varepsilon > 0$ the escape from $\tilde{G}$ of the stochastic system and thus its route towards explosion typically involves passing through the unstable equilibria with minimal potential, namely $\pm z$, at least whenever the initial datum is close enough to the origin. By the argument to be used in the proof of Theorem \[taumagnitude\] below, this implies that the same fate holds for arbitrary initial data in $\mathcal{D}_{\mathbf{0}}$. Asymptotic loss of memory of $\tau_\varepsilon({\partial}G)$ {#sec7asymp} ------------------------------------------------------------ Our next goal is to show the asymptotic loss of memory as $\varepsilon \rightarrow 0$ of $\tau_\varepsilon^u ({\partial}G)$ for $u \in G$. To this end for each $\varepsilon > 0$ we define the normalization coefficient $\gamma_\varepsilon > 0$ by the relation $$P_{\mathbf{0}}( \tau_\varepsilon(\partial G) > \gamma_\varepsilon ) = e^{-1}.$$ Notice that $\gamma_\varepsilon$ is well defined since $\tau_\varepsilon^{\mathbf{0}}$ is a continuous almost surely finite random variable, with a strictly increasing distribution function. The result we aim to prove reads as follows. \[escapeteo1\] For every $\varepsilon > 0$ consider the function $\nu_\varepsilon : {{\mathbb R}}_{\geq 0} \rightarrow [0,1]$ given by $$\nu_\varepsilon (t) = P_{\mathbf{0}} (\tau_\varepsilon (\partial G) > t\gamma_\varepsilon).$$ Then 1. There exists $(\delta_\varepsilon)_{\varepsilon > 0} \subseteq {{\mathbb R}}_{> 0}$ satisfying $\lim_{\varepsilon \rightarrow 0} \delta_\varepsilon = 0$ and such that for any $s,t > 0$ $$\label{1.49}\nu_\varepsilon(s + \delta_\varepsilon)\nu_\varepsilon(t) - \psi_\varepsilon(s,t) \leq \nu_\varepsilon (s+t) \leq \nu_\varepsilon (s) \nu_\varepsilon (t - \delta_\varepsilon) + \psi_\varepsilon(s,t)$$ where $\psi_\varepsilon(s,t)$ is a function which for any fixed $t_0 > 0$ verifies $$\label{1.50}\lim_{\varepsilon \rightarrow 0}\left[\sup_{s \geq0 \,,\,t\geq t_0}\psi_\varepsilon(s,t)\right]= 0.$$ 2. There exists $\rho > 0$ such that for every $t \geq 0$ $$\label{convunie} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_\rho} |P_u( \tau_\varepsilon(\partial G) > t\gamma_\varepsilon) - e^{-t}|\right] = 0.$$ ### Coupling of solutions with small initial data The key element in the proof of Theorem \[escapeteo1\] is the fact that, uniformly over any pair $u,v$ of initial data in a small neighborhood of the origin the corresponding escape times $\tau_\varepsilon^u({\partial}G)$ and $\tau_\varepsilon^v({\partial}G)$ possess the same asymptotic distribution. We shall establish this fact rigorously on Lemma \[escapelema1\] below with the aid of a suitable coupling between More precisely, for $n_0 \in {{\mathbb N}}$ as in the definition of $G$ the result we require is the following. \[coupling0\] There exists $\rho > 0$ such that for any pair of initial data $u,v \in B_\rho$ and $\varepsilon > 0$ sufficiently small there exists a coupling of $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),v,\varepsilon}$ satisfying $$\label{coupling2} U^{(n_0+1),u,\varepsilon}(t,\cdot) \equiv U^{(n_0+1),v,\varepsilon}(t,\cdot) \hspace{0.5cm}\text{ for all }t \geq \eta^{\varepsilon}_{u,v},$$ where $$\eta^\varepsilon_{u,v} := \inf\{ t \geq 0 : U^{(n_0+1),u,\varepsilon}(t,\cdot) \equiv U^{(n_0+1),v,\varepsilon}(t,\cdot)\}.$$ Furthermore, $\eta^\varepsilon_{u,v}$ satisfies $$\label{coupling1} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u,v \in B_\rho} P \left( \eta^\varepsilon_{u,v} \geq \frac{1}{\varepsilon^3} \right)\right]=0.$$ The existence of a coupling fulfilling these characteristics was first established in [@M] for a class of stochastic differential equations with periodic boundary $$\label{MainSPDEMueller} \left\{\begin{array}{rll} {\partial}_t U^u &= {\partial}^2_{xx}U^u - \alpha U^u + a(U^u) + b(U^u)\dot{W} & \quad t>0 \,,\, x \in S^1 \\ U^u(0,x) &= u(x) &\quad x \in S^1 \end{array}\right.$$ where $\alpha > 0$ is fixed parameter, $a: {{\mathbb R}}\rightarrow {{\mathbb R}}$ is a Lipschitz nonincreasing function and $b: {{\mathbb R}}\rightarrow {{\mathbb R}}$ is a positive, Lipschitz function bounded away from both zero and infinity. In this work, Mueller considers an arbitrary pair of continuous functions $u,v \in C(S^1)$ as initial data and shows that the coupling time $\eta^\varepsilon_{u,v}$ is almost surely finite. Later this very same result was adapted in [@B2] to a particular system verifying the assumptions in [@M] but including Dirichlet boundary conditions instead of periodic ones. Furthermore, in this second work Brassesco shows the asymptotic estimate for the coupling time $\eta^\varepsilon_{u,v}$. Unfortunately, for our system the coefficient $a$ fails to be increasing so that the results on [@B2] cannot be directly applied. Nevertheless, it is still possible to obtain Theorem \[coupling0\] by performing some minor adjustments to the proof given there, although this comes at the expense of losing the almost sure finiteness of the coupling time and also a certain freedom in the choice of initial data. In what follows we present a brief summary of the , highlighting the main differences with [@B2] and explaining how to deal with each of them. We refer the reader to [@M] and [@B2] for the remaining details. Let us begin by observing that it will suffice to show the coupling for initial data $u,v$ such that $u \geq v$. Indeed, it follows from the proof in this case that if $w:= \max\{u,v\}$ then we can construct the three solutions $U^{(n_0+1),u,\varepsilon}$, $U^{(n_0+1),v,\varepsilon}$ and $U^{(n_0+1),w,\varepsilon}$ in the same probability space so that $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),w,\varepsilon}$ are identical after the time $\eta^\varepsilon_{u,w}$ and also $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),w,\varepsilon}$ are identical after the time $\eta^\varepsilon_{v,w}$. It then follows that $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),v,\varepsilon}$ must become identical after the $\eta^\varepsilon_{u,v} \leq \max\{ \eta^\varepsilon_{u,w}, \eta^\varepsilon_{v,w}\}$, so that it suffices to estimate the desired probability in for the simpler case $u \geq v$. Thus, we assume that $u \geq v$ and given two independent Brownian sheets $W_1$ and $W_2$ we consider the pair $(U^{(n_0+1),u,\varepsilon},U^{(n_0+1),v,\varepsilon})$ which satisfies $${\partial}_t U^{(n_0+1),u,\varepsilon} = - \frac{\partial S^{(n_0+1)}}{\partial \varphi}(U^{(n_0+1),u,\varepsilon}) + {\varepsilon}\dot{W_1}$$ and $${\partial}_t U^{(n_0+1),v,\varepsilon} = - \frac{\partial S^{(n_0+1)}}{\partial \varphi}(U^{(n_0+1),v,\varepsilon}) + {\varepsilon}\left( \left( \sqrt{1- \min\{|E|, 1\}} \dot{W_1} + \sqrt{\min\{|E|,1\}} \dot{W_2} \right)\right)$$ with initial data $u$ and $v$ respectively and where $$E:=U^{(n_0+1),u,\varepsilon} - U^{(n_0+1),v,\varepsilon}.$$ Thus, both $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),v,\varepsilon}$ are solutions with the appropriate initial data, constructed in the same probability space but with respect to different white noises. Following Lemma 3.1 of [@M] it is possible to construct the pair $(U^{(n_0+1),u,\varepsilon},U^{(n_0+1),v,\varepsilon})$ in such a way that the process $E$ is nonnegative almost surely for $\varepsilon > 0$ sufficiently small. Then, using the weak formulation of solutions to available on Lemma \[weaksol\] one can immediately see as in [@B2] that if we write $$U(t) = \int_0^1 E(t,x) \sin (\pi x) dx$$ then $U$ satisfies $$\label{EcuaU} U(t)= U(t) + \int_0^t C(s)ds + M_t$$ where $$\label{drift} C(s) = \int_0^1 \left( g_{n_0+1}\left(U^{(n_0+1),u,\varepsilon}(s,x)\right)-g_{n_0+1}\left(U^{(n_0+1),v,\varepsilon}(s,x)\right) - \pi^2 E(s,x)\right) \sin(\pi x) dx.$$ and $M$ is a continuous martingale with respect to the filtration generated by $W_1$ and $W_2$ satisfying $$\label{compensator} \langle M \rangle(t) = 2{\varepsilon}^2 \int_0^t\int_0^1 \frac{\min\{ E(s,x), 1\}}{1 + \sqrt{1-\min\{E(s,x),1\}}}dxds.$$ Notice that in order for to hold, it was necessary to introduce a $C^\infty$ function in $C_D([0,1])$ in the definition of $U$. We selected $\sin(\pi x)$ as in [@B2] but the same reasoning also holds for other nonnegative $C^\infty$ functions. Now, from we easily obtain $$\frac{\langle M \rangle (t)}{dt} \geq \varepsilon^2 \int_0^1 \sin^2 (\pi x) \min\{ E(t,x), 1\}dx.$$ Using Hölder’s inequality we obtain that $$\int_0^1 \sin(\pi x) \min\{E(t,x),1\}dx \leq \left(\int_0^1 \sin^2(\pi x) \min\{E(t,x),1\}dx\right)^{\frac{5}{8}} \left(\int_0^1 \sin^{-\frac{2}{3}}(\pi x)dx\right)^{\frac{3}{8}}$$ which implies the estimate $$\begin{aligned} \frac{\langle M \rangle (t)}{dt}& \geq K\varepsilon \left[\int_0^1 \sin(\pi x) \min\{E(t,x),1\}dx\right]^{\frac{8}{5}}\\ \\ & \geq K\varepsilon \left[\int_0^1 \sin(\pi x) \frac{E(t,x)}{\max\{E(t,x),1\}}dx\right]^{\frac{8}{5}}\\ \\ & \geq \frac{K\varepsilon^2}{\left[\sup_{x \in [0,1]} \max\{E(t,x),1\}\right]^\frac{8}{5}}\left(U(t)\right)^{\frac{8}{5}}\end{aligned}$$ where $$K := \left(\int_0^1 \sin^{-\frac{2}{3}}(\pi x)dx\right)^{-\frac{3}{5}}.$$ Thus, we conclude that there exists an adapted process $D$ such that for all $t \geq 0$ $$\frac{\langle M \rangle (t)}{dt} = \left( U(t) \right)^{\frac{8}{5}}D(t)$$ and $$\label{boundtimechange} D(t) \geq \frac{K\varepsilon^2}{\left[\sup_{x \in [0,1]} \max\{E(t,x),1\}\right]^\frac{8}{5}}.$$ Next, we introduce the time change $$\varphi(t) = \int_0^t D(s)ds$$ and consider the time-changed process $$X(t) := U( \varphi^{-1}(t)).$$ In [@M] it is shown that whenever the condition $$\label{conditiontimechange} \max\left\{ \sup_{t \geq 0} {{\mathbb E}}\left( \sup_{x \in [0,1]} U^{(n_0+1),u,\varepsilon}(t,x) \right) , \sup_{t \geq 0} {{\mathbb E}}\left( \sup_{x \in [0,1]} U^{(n_0+1),v,\varepsilon}(t,x) \right) \right\} < +\infty$$is met then $\lim_{t \rightarrow +\infty} \varphi(t) = +\infty$ so that the process $X$ is globally defined. Unfortunately, in this article condition is seen to hold only under the presence of the linear term in , i.e. $\alpha > 0$, which is missing in our system. However, since we are interested in achieving the coupling between the solutions before they escape the domain $G$, modify the source term $g_{n_0 +1}$ outside $B_{n_0 +1}$ in such a way that is satisfied without it affecting our plans. Hence, by and the definition of $\varphi$, it can be seen that $X$ satisfies $$X(t) = U(0) + \int_0^t \tilde{C}(s) ds + \int_0^t \left(X(t)\right)^{\frac{8}{10}} dB_t$$ for a certain Brownian motion $B$ and $\tilde{C}$ given by the formula $$\tilde C(t):=C(\varphi^{-1}(t)) \frac{1}{\varphi'(\varphi^{-1}(t))}.$$ Now, Itô’s formula yields that, up until its arrival time at zero, the process $Y: = 5 X^{\frac{1}{5}}$ satisfies $$Y(t) = 5\left(U(0)\right)^{\frac{1}{5}} +\int_0^t \left(\frac{\tilde C(s)}{\left(Y(s)\right)^4}-\frac{2}{5Y(s)}\right)ds + B_t$$ In both [@M] and [@B2] the corresponding term $\tilde{C}$ is nonpositive, so that $Y$ is guaranteed to hit zero before the time the Brownian motion $B$ takes to reach $-5\left(U(0)\right)^{\frac{1}{5}}$. However, in our case the term $\tilde{C}$ may eventually take positive values so that some additional work is needed in order to arrive at the same conclusion. Notice that a straightforward calculation using the definition of $\varphi$ and the bound shows that if for some $h> 0$ the term $C(t)$ is nonpositive for all $t \leq e^{\frac{h}{\varepsilon^2}}$ then $\tilde{C}(t)$ is also nonpositive but only for all $t \leq e^{\frac{h}{2\varepsilon^2}}$. Hence, if we set $$\gamma:= \inf \{ t \geq 0 : B_t = -5\left(U(0)\right)^{\frac{1}{5}}\}$$ then $Y$ is guaranteed to hit zero before $\gamma$ provided that there exists $h > 0$ such that $\gamma \leq e^{\frac{h}{2\varepsilon^2}}$ and the term $C(t)$ is nonpositive for all $t \leq e^{\frac{h}{\varepsilon^2}}$. Now, $Y(t)=0$ implies that $U(\varphi^{-1}(t))=0$ and ultimately that $E( \varphi^{-1}(t), x ) \equiv \mathbf{0}$ which means that both solutions $U^{(n_0+1),u,\varepsilon}$ and $U^{(n_0+1),v,\varepsilon}$ coincide at time $t$. But, since the process $E$ is governed by the differential equation $${\partial}_t E = {\partial}^{2}_{xx} E + \left(g_{n_0+1}(U^{(n_0+1),u,\varepsilon}) - g_{n_0+1}(U^{(n_0+1),v,\varepsilon})\right) + 2\varepsilon^2 \frac{\min\{ E(s,x), 1\}}{1 + \sqrt{1-\min\{E(s,x),1\}}}\dot{W},$$ we see that once the solutions meet each other, they remain identical forever afterwards and thus the coupling is achieved. Hence, we conclude that the pair $(U^{(n_0+1),u,\varepsilon},U^{(n_0+1),v,\varepsilon})$ satisfies and furthermore that, if there exists $h > 0$ such that $\gamma \leq e^{\frac{h}{2\varepsilon^2}}$ and the term $C(t)$ is nonpositive for all $t \leq e^{\frac{h}{\varepsilon^2}}$, then $\eta^\varepsilon_{u,v}$ is bounded from above by $\varphi^{-1}(\gamma)$. Our goal now is then to find $h > 0$ such that both these conditions are satisfied with overwhelming probability as $\varepsilon > 0$ tends to zero for all $u,v$ in a small neighborhood of the origin. Notice that, since $g'_{n_0 + 1}$ is continuous and $g'_{n_0+1}(0)= g_{n_0+1}(0) = 0$, there exists $\delta > 0$ sufficiently small such that $\sup_{|y|\leq \delta} g_{n_0+1}(y) \leq \pi^2$. Therefore, if for some $t \geq 0$ we have that $U^{(n_0+1),u,\varepsilon}(t,\cdot)$ and $U^{(n_0+1),v,\varepsilon}(t,\cdot)$ both belong to $B_\delta$ then by we obtain $$C(t) \leq \int_0^1 \left[\sup_{|y| \leq \delta} g'_{n_0+1}(y) - \pi^2\right] E(t,x) \sin(\pi x) dx \leq 0.$$ Since one can show as in the proof of that for every $r > 0$ sufficiently small one has $$\inf_{u \in {\partial}B_r} S(u) > S(\mathbf{0})=0$$ then by the methods applied in Section \[seclowerbound\] we conclude that there exists $0 < \rho < \delta$ sufficiently small and $h > 0$ such that $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_\rho} P_u \left( \tau_\varepsilon ({\partial}B_\delta) \leq e^{\frac{h}{\varepsilon^2}} \right) \right] = 0.$$ Thus, by all these considerations we see that if $\delta < \frac{1}{2}$ then for $\varepsilon > 0$ sufficiently small $$\label{cotacoupling0} P\left( \eta^\varepsilon_{u,v} \geq \frac{1}{\varepsilon^3} \right) \leq 2 \left( P\left( \gamma > \frac{K}{\varepsilon} \right) + \sup_{u \in B_\rho} P_u \left( \tau_\varepsilon ({\partial}B_\delta) \leq e^{\frac{h}{\varepsilon^2}} \right) \right)$$ where we have used the fact that if $U^{(n_0+1),u,\varepsilon}(t,\cdot)$ and $U^{(n_0+1),v,\varepsilon}(t,\cdot)$ both belong for all $t \in [0,\frac{1}{\varepsilon^3}]$ and $\delta < \frac{1}{2}$ then $\sup_{x \in [0,1]} |E(t,x)| < 1$ for all $t \in [0,\frac{1}{\varepsilon^3}]$ Since the bound obtained in holds for all pairs $u,v \in B_{\rho}$ and $\gamma$ is we conclude and so Theorem \[coupling0\] is proved. As a direct consequence of Theorem \[coupling0\] we now obtain the following lemma which establishes the claim in the beginning of the section regarding the asymptotic distribution of the escape time $\tau_{\varepsilon}({\partial}G)$ for initial data in a small neighborhood of the origin. \[escapelema1\] There exists $\rho > 0$ such that for every $t_0 > 0$ $$\lim_{\varepsilon \rightarrow 0}\left[ \sup_{u,v \in B_{\rho}} \left[ \sup_{t > t_0} |P_u(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon}) - P_{v}(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon})|\right]\right] = 0.$$ Let $\rho > 0$ be as in Theorem \[coupling0\] and given a pair of initial data $u,v \in B_\rho$ let us consider the coupling $\left( U^{(n_0+1),u,\varepsilon}, U^{(n_0+1),v,\varepsilon}\right)$ constructed in the aforementioned theorem. Since by the results established in Section \[seclowerbound\] for any given $t_0 > 0$ there exists $\varepsilon_0 > 0$ such that $t_0 \gamma_\varepsilon > \frac{1}{\varepsilon^3}$ for all $0 < \varepsilon < \varepsilon_0$, then for all $t \geq t_0$ we have $$|P_u(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon}) - P_{v}(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon})| \leq P ( \eta^\varepsilon_{u,v} \geq t\gamma_\varepsilon ) \leq P\left( \eta^\varepsilon_{u,v} \geq \frac{1}{\varepsilon^2}\right)$$ for all $0 < \varepsilon < \varepsilon_0$, so that the result follows at once by . In [@MOS] the authors study the asymptotic distribution of the tunneling in a double-well potential model, i.e. the time needed for the stochastic system to go from one well to a small neighborhood of the bottom of the other one. They show that, under proper normalization, the tunneling time converges in distribution to an exponential random variable, as it also happens with $\tau^\varepsilon({\partial}G)$ in our case. To do this they show an analogue of Lemma \[escapelema1\] but using a different technique, which is based on an exponential loss of memory of the initial datum. The joining of trajectories due to the attractive drift in the final part of the motion plays an essential role in their argument, and so the reasoning no longer works, for example, when studying the exit time from a bounded region containing only the attractor. However, we point out that the approach we introduce here relying on the coupling of solutions does not have the same limitation, and so it can also be used to study these other type of problems. This is shown in detail in [@B2]. ### Proof of Theorem \[escapeteo1\] We shall need the results contained in following lemma for the proof of Theorem \[escapeteo1\]. \[escapelema2\] Let us consider $ 0 < \alpha < \Delta$ and define $\eta_\varepsilon := e^{\frac{\alpha}{\varepsilon^2}}$. Then 1. $\lim_{\varepsilon \rightarrow 0}\frac{\eta_\varepsilon}{\gamma_\varepsilon} = 0$ 2. $\lim_{\varepsilon \rightarrow 0} \left[ \displaystyle{\sup_{u \in G} P_u ( \tau_{\varepsilon}(\partial G) > \eta_\varepsilon \,,\, \tau_{\varepsilon}(B_{\rho}) > \eta_\varepsilon)}\right] = 0$ for any $\rho > 0$. Let us notice that by the bounds established for $\tau_\varepsilon({\partial}G)$ in Sections \[secupperbound\] and \[seclowerbound\] we have that $$\lim_{\varepsilon \rightarrow 0} \varepsilon^2 \log \gamma_\varepsilon = \Delta$$ from where (i) immediately follows. Next, we establish (ii) with the aid from the large deviations principle as in Lemma \[cotasuplema0\]. We must show that there exists a time $T > 0$ such that for each $u \in G$ there exists a set of paths $\mathcal{E}_{u,T} \subseteq C_{D_u}([0,T]\times [0,1])$ satisfying 1. Every path in $\mathcal{E}_{u, T}$ reaches $\partial G \cap B_\rho$ before times $T$. 2. $\inf_{x \in G} P_u (U^{\varepsilon} \in \mathcal{E}_{u, T}) \geq \tilde{\alpha}_{\varepsilon}$, where $\tilde{\alpha}_{\varepsilon}:= Te^{-\frac{\alpha}{2\varepsilon^{2}}}.$ Once again, it suffices to show that for each $u \in G$ there exists $\varphi^u \in C_{D_u}([0,T]\times [0,1])$ starting at $u$ with rate less than $\frac{\alpha}{3}$ and such that not only does $\varphi^u$ reach $\partial G \cup B_\rho$ before time $T$, but also any path sufficiently close to $\varphi^u$ does so as well. The construction of such a $\varphi^u$ is similar to the one given in the proof of Lemma \[cotasuplema0\]. The remainder of the proof follows once again from the large deviations principle valid for our system. With Lemmas \[escapelema1\] and \[escapelema2\] at our disposal, we are now ready to prove Theorem \[escapeteo1\]. Given $s > 0$ let us define $$R^{u,s}_{\varepsilon} = \inf \{ r > s\gamma_\varepsilon : U^{u,\varepsilon}(r,\cdot) \in B_\rho\}$$ where $\rho > 0$ is given by Lemma \[escapelema1\]. We may then decompose $\nu_\varepsilon (t+s)$ as $$\nu_\varepsilon (t+s) = P_{\mathbf{0}} ( \tau_\varepsilon(\partial G) > (s+t)\gamma_\varepsilon \,,\, R^{s}_{\varepsilon} > s\gamma_\varepsilon + \eta_\varepsilon) + P_{\mathbf{0}} ( \tau_\varepsilon(\partial G) > (s+t)\gamma_\varepsilon \,,\, R^{s}_{\varepsilon} \leq s \gamma_\varepsilon + \eta_\varepsilon).$$ Let us observe that for $u \in G$ the Markov property yields $$P_u ( \tau_\varepsilon(\partial G) > s\gamma_\varepsilon + \eta_\varepsilon \,,\, R^{s}_{\varepsilon} > s\gamma_\varepsilon + \eta_\varepsilon) \leq \sup_{u \in G} P_u \left(\tau_\varepsilon(\partial G) > \eta_\varepsilon \,,\, \tau_\varepsilon(B_\rho) > \eta_\varepsilon\right).$$ Thus, by Lemma \[escapelema2\] we conclude that for any fixed $t_0 > 0$ $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{s\geq 0\,,\,t\geq t_0} \left[\sup_{u\in G} P_u (\tau_\varepsilon(\partial G) > (s+t)\gamma_\varepsilon \,,\, R^{s}_{\varepsilon} > s\gamma_\varepsilon + \eta_\varepsilon)\right]\right] = 0.$$ To establish (i) it suffices then to give proper upper and lower bounds on the second term of the decomposition. But by applying the strong Markov property with respect to the stopping time $R^{\mathbf{0},s}_\varepsilon$ we obtain $$P_\mathbf{0} (\tau_\varepsilon(\partial G) > (s+t)\gamma_\varepsilon \,,\, R^{s}_{\varepsilon} \leq s\gamma_\varepsilon + \eta_\varepsilon) \leq P_\mathbf{0}(\tau_\varepsilon(\partial G)> s\gamma_\varepsilon) \left[\sup_{u \in B_\rho} P_u (\tau_\varepsilon(\partial G) > t\gamma_\varepsilon - \eta_\varepsilon)\right]$$ and $$P_\mathbf{0} (\tau_\varepsilon(\partial G) > (s+t)\gamma_\varepsilon \,,\, R^{s}_{\varepsilon} \leq s\gamma_\varepsilon + \eta_\varepsilon) \geq P_\mathbf{0} ( R^{s}_{\varepsilon} \leq s \gamma_\varepsilon + \eta_\varepsilon)\left[\inf_{u \in B_{\rho}}P_u(\tau_\varepsilon (\partial G) > t\gamma_\varepsilon)\right].$$ From this we immediately obtain (i) by using Lemmas \[escapelema1\] and \[escapelema2\]. Now, assertion (ii) will follow immediately from Lemma \[escapelema1\] once we manage to show that for every $t > 0$ we have $$\label{casop} \lim_{\varepsilon \rightarrow 0} \nu_\varepsilon(t)=e^{-t}.$$ To see this, let us first observe that by applying (i) successively we obtain $$\left\{ \begin{array}{l}\nu_\varepsilon (2k) \leq [\nu_\varepsilon (2 - \delta_\varepsilon)]^k + \sum_{i=1}^{k-1} \psi_{\varepsilon} (2i,2)\\ \\ \nu_\varepsilon(1)= e^{-1}\leq [\nu_\varepsilon (\frac{1}{k} - \delta_\varepsilon)]^k + \sum_{i=1}^{k-1} \psi_{\varepsilon} (\frac{i}{k},\frac{1}{k}). \end{array}\right.$$ Thus, given $0 < \delta < 1$ and $k \in {{\mathbb N}}$ such that $e^{-k} < \frac{\delta}{2}$ and $(1- \delta)^k < \frac{e^{-1}}{2}$, in light of we may take $\varepsilon_0 > 0$ such that for every $\varepsilon \leq \varepsilon_0$ the following conditions hold: - $\sum_{i=1}^{k-1} \psi_{\varepsilon} (2i,2) < \frac{\delta}{2},$ - $\sum_{i=1}^{k-1} \psi_{\varepsilon} (\frac{i}{k},\frac{1}{k}) < \frac{e^{-1}}{2},$ - $2 - \delta_\varepsilon > 1,$ - $\frac{1}{k} - \delta_\varepsilon > \frac{1}{2k}.$ Under these conditions it can be seen that $\nu_\varepsilon(2k) < \delta \:,\: \nu_\varepsilon (\frac{1}{2k}) > 1 - \delta$ for every $\varepsilon \leq \varepsilon_0$. In particular, this implies that any sequence $(\varepsilon_j)_{j \in {{\mathbb N}}} \subseteq {{\mathbb R}}_{ > 0}$ with $\lim_{j \rightarrow +\infty} \varepsilon_j = 0$ satisfies that the family $(\nu_{\varepsilon_j})_{j \rightarrow +\infty}$ is tight, i.e. $$\lim_{k \rightarrow +\infty} \left[ \inf_{j \in {{\mathbb N}}} \left[ \nu_{\varepsilon_j}(k) - \nu_{\varepsilon_j}(k^{-1}) \right] \right] = 1.$$ Therefore, by Prohorov’s theorem we see that in order to establish we must only check that any sequence $(\nu_{\varepsilon_j})_{j \in {{\mathbb N}}}$ which is weakly convergent has the mean one exponential distribution as its limit. But if we denote this limit by $\nu$, then (i) implies that $\nu$ must satisfy the memory loss property, i.e. for every $s,t > 0$ $$\nu(s+t)=\nu(s)\nu(t)$$ and thus it must be $\nu(t)=e^{-\lambda t}$ for some $\lambda \geq 0$. By recalling that $\nu_\varepsilon(1)=e^{-1}$ for every $\varepsilon > 0$ we see that $\lambda = 1$. This concludes the proof. Resumen del Capítulo 4 ---------------------- Este capítulo se encuentra dedicado a estudiar, para datos iniciales $u$ en un entorno pequeño de $\mathbf{0}$, el fenómeno del escape del dominio $G$ construido en el Capítulo 3 por parte del sistema estocástico $U^{u,\varepsilon}$. El problema del escape de un dominio acotado con estas características fue originalmente estudiado en [@GOV] para el caso de un potencial de doble pozo finito-dimensional, y luego investigado en [@B1] en su variante infinito-dimensional. Los resultados que presentamos en este capítulo son una adaptación a nuestro contexto de los resultados que aparecen en dichas referencias. El primer resultado caracteriza el orden de magnitud asintótico de $\tau^u_\varepsilon({\partial}G)$, el tiempo de salida del dominio $G$. **Teorema**. Dado $\delta > 0$ se tiene $$\lim_{\varepsilon \rightarrow 0 } \left[\sup_{u \in B_c} \left| P_{u} \left( e^{\frac{\Delta - \delta}{\varepsilon^{2}}} < \tau_{\varepsilon}(\partial G) < e^{\frac{\Delta + \delta}{\varepsilon^{2}}}\right)-1 \right|\right] = 0,$$ donde $B_c$ es el entorno del origen resaltado en la construcción de $G$. Para probar este resultado mostraremos por separado la cota superior y la inferior . La cota superior se sigue del hecho de que dado $\delta > 0$ existe $T > 0$ tal que para todo $u \in G$ existe una trayectoria $\varphi^u \in C_D([0,T] \times [0,1])$ con $\varphi^u(0)=u$ y de tasa $I^u_T(\varphi^u) < \Delta + \frac{\delta}{3}$ tal que toda trayectoria suficientemente cercana a $\phi^u$ se escapa de $G$ antes de tiempo $T$. Usando la estimación podemos concluir entonces que $$\inf_{u \in G} P( \tau^u_\varepsilon ({\partial}G) \leq T ) \geq e^{- \frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}}$$ de modo tal que, por la propiedad de Markov, el tiempo que $U^{u,\varepsilon}$ tarde en escapar de $G$ será típicamente menor a $T e^{\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}}$. Observando que $T e^{\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}} \ll e^{\frac{\Delta + \delta}{\varepsilon^2}}$ cuando $\varepsilon \rightarrow 0$ se concluye el resultado. Para la cota inferior se divide el intervalo $[0,\tau^u_\varepsilon({\partial}G)]$ en subintervalos disjuntos que corresponden a las excursiones que realiza el sistema $U^{u,\varepsilon}$ alejándose de $\mathbf{0}$ en busca de ${\partial}G$. En todas estas excursiones el sistema $U^{u,\varepsilon}$ fracasa en llegar a ${\partial}G$ exceptuando la última de ellas, donde finalmente consigue el éxito y alcanza ${\partial}G$. Es posible mostrar que cada una de estas excursiones tiene típicamente una longitud mayor a cierto $T' > 0$ y que la probabilidad de que sea exitosa es inferior a $e^{-\frac{\Delta - \frac{\delta}{2}}{\varepsilon^2}}$, de modo tal que el tiempo que $U^{u,\varepsilon}$ tarde en escapar de $G$ será típicamente mayor a $T e^{\frac{\Delta - \frac{\delta}{2}}{\varepsilon^2}}$. Observando que $T e^{\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}} \gg e^{\frac{\Delta - \delta}{\varepsilon^2}}$ cuando $\varepsilon \rightarrow 0$ se concluye el resultado. El siguiente resultado obtenido en este capítulo muestra que el sistema $U^{u,\varepsilon}$ típicamente se escapa de ${\partial}G$ por ${\partial}^{\pm z}$. **Teorema**. Si $B_c$ es el entorno de $\mathbf{0}$ resaltado en la construcción de $G$ entonces $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_c} P_u \left( U^{\varepsilon}(\tau_\varepsilon(\partial G),\cdot) \notin \partial^{\pm z}\right)\right] = 0.$$ Para probar este resultado es necesario nuevamente dividir el intervalo $[0,\tau^u_\varepsilon({\partial}G)]$ en las distintas excursiones que realiza el sistema $U^{u,\varepsilon}$ alejándose de $\mathbf{0}$ en busca de ${\partial}G$, y estimar la probabilidad de que en la última de ellas el sistema haya alcanzado ${\partial}^{\pm z}$. Por la propiedad de Markov esto coincide con estimar la probabilidad de que la excursión inicial haya alcanzado ${\partial}^{\pm z}$ condicionada a ser exitosa. Como $V(\mathbf{0}, {\partial}G - {\partial}^{\pm z}) > V(\mathbf{0}, {\partial}^{\pm z})$, i.e. el costo para el sistema $U^{\mathbf{0},\varepsilon}$ de escapar de $G$ es menor si lo hace por ${\partial}^{\pm z}$, con ayuda de las estimaciones de grandes desvíos es posible probar que dicha probabilidad condicional tiende a cero cuando $\varepsilon \rightarrow 0$ y se obtiene así el resultado. El último resultado de este capítulo concierne la distribución asintótica del tiempo de escape $\tau^u_\varepsilon ({\partial}G)$. Concretamente, mostramos que bajo una normalización adecuada, $\tau^u_\varepsilon ({\partial}G)$ converge en distribución a una variable aleatoria exponencial. **Teorema**. Si para cada $\varepsilon > 0$ definimos el coeficiente $\gamma_\varepsilon > 0$ mediante la relación $$P_{\mathbf{0}}(\tau_\varepsilon ({\partial}G) > \gamma_\varepsilon ) = e^{-1}$$ entonces existe $\rho > 0$ tal que para todo $t \geq 0$ se tiene $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_\rho} |P_u( \tau_\varepsilon(\partial G) > t\gamma_\varepsilon) - e^{-t}|\right] = 0.$$ Para demostrar este resultado, primero tratamos el caso $u= \mathbf{0}$ separadamente. Para probar el resultado en este caso, si definimos $\nu_\varepsilon(t) = P_{\mathbf{0}}(\tau_\varepsilon ({\partial}G) > t\gamma_\varepsilon )$, bastará con verificar que 1. La familia de distribuciones $(\nu_\varepsilon)_{\varepsilon > 0}$ es asintóticamente acotada en probabilidad, i.e. uniformemente sobre $\varepsilon > 0$ suficientemente chico. 2. Cualquier límite por subsucesiones de $\nu_\varepsilon$ cuando $\varepsilon \rightarrow 0$ es exponencial de Tanto (i) como (ii) se obtienen fácilmente una vez que se demuestran las desigualdades en . La dificultad más importante a la hora de mostrar yace en verificar que la distribución asintótica de $\tau^u_\varepsilon({\partial}G)$ es la misma para datos iniciales $u$ en un entorno suficientemente pequeño de $\mathbf{0}$, i.e. existe $\rho > 0$ tal que para todo $t_0 > 0$ $$\lim_{\varepsilon \rightarrow 0}\left[ \sup_{u,v \in B_{\rho}} \left[ \sup_{t > t_0} |P_u(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon}) - P_{v}(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon})|\right]\right] = 0.$$ Para probar esto recurrimos a un acoplamiento entre soluciones del sistema estocástico para datos en un entorno del origen similar al estudiado en [@B2]. Con este resultado, a partir del caso $u=\mathbf{0}$ se deduce inmediatamente el caso $u \in B_\rho$. Asymptotic behavior of $\tau^u_\varepsilon$ for $u \in \mathcal{D}_\mathbf{0}$ ============================================================================== Asymptotic properties of $\tau_\varepsilon$ for initial data in $\mathcal{D}_\mathbf{0}$ {#sec8} ---------------------------------------------------------------------------------------- In this section we devote ourselves to establishing the asymptotic properties as $\varepsilon \rightarrow 0$ of the explosion time $\tau_\varepsilon^u$ for arbitrary $u \in \mathcal{D}_{\mathbf{0}}$. Our first result in this direction, detailed on the following theorem, is concerned with its asymptotic magnitude. \[taumagnitude\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive distance from $\mathcal{W}$ and $\delta > 0$ $$\label{taumagnitud1} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} P_u \left( e^{\frac{\Delta - \delta}{\varepsilon^2}} < \tau_\varepsilon < e^{\frac{\Delta + \delta}{\varepsilon^2}}\right)\right]=1.$$ First let us suppose that $\mathcal{K}= B_{c}$. In this case the continuity of trajectories and the strong Markov property imply that for $u \in B_{c}$ we have $$P_u \left( \tau_\varepsilon < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right) \leq P_u \left( \tau_\varepsilon (\partial G) < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right)$$ and $$P_u \left( \tau_\varepsilon > e^{\frac{\Delta + \delta}{\varepsilon^2}} \right) \leq P_u \left( \tau_\varepsilon(\partial G) > e^{\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}} \right) + P_u \left( U(\tau_\varepsilon(\partial G),\cdot) \notin {\partial}^{\pm z} \right) + \sup_{v \in {\partial}^{\pm z}} P_v ( \tau_\varepsilon > \tau^* )$$ for every $\varepsilon > 0$ sufficiently small, from which we can conclude the result in this case by the results in Section \[secescapedeg\]. Now, let us observe that for any $u \in \mathcal{D}_{\mathbf{0}}$ the system $U^u$ reaches the set $B_{\frac{c}{2}}$ in a finite time $\tau^u(B_{\frac{c}{2}})$ while remaining at all times inside the ball $B_{r^u}$ where $r^u:= \sup_{t \geq 0} \| U^u(t,\cdot) \|_\infty$. Therefore, if $\mathcal{K}$ is now any bounded set contained in $\mathcal{D}_{\mathbf{0}}$ at a positive distance from $\mathcal{W}$ then we have that $\tau_{\mathcal{K},\frac{c}{2}} := \sup_{u \in \mathcal{K}} \tau^u(B_{\frac{c}{2}})$ and $r_{\mathcal{K}}:=\sup_{u \in \mathcal{K}} r^u$ are both finite. Indeed, of $\tau_{\mathcal{K},\frac{c}{2}}$ follows at once from Proposition \[A.3\] whereas $r_{\mathcal{K}}$ is finite since by Proposition \[G.1\] one may find $t_0 > 0$ sufficiently small such that $\sup_{u \in \mathcal{K}} \left[ \sup_{t \in [0,t_0]} \| U^u(t,\cdot) \|_\infty \right]$ is finite. That $\sup_{u \in \mathcal{K}} \left[ \sup_{t \geq t_0} \| U^u(t,\cdot) \|_\infty \right]$ is finite then follows as in the proof of Proposition \[A.3\] due to the fact that the mapping $u \mapsto r^u$ is both upper semicontinuous and finite on $\mathcal{D}_{\mathbf{0}}$. Using the strong Markov property we can then obtain the bounds $$P_u \left( \tau_\varepsilon < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right) \leq P_u \left( \tau_\varepsilon(B_c) > \tau_{\mathcal{K},\frac{c}{2}} \right) + P_u \left( \tau_\varepsilon \leq \tau_{\mathcal{K},\frac{c}{2}}\right) + \sup_{v \in B_c} P_v \left( \tau_\varepsilon < e^{\frac{\Delta - \delta}{\varepsilon^2}} \right)$$ and $$P_u \left( \tau_\varepsilon > e^{\frac{\Delta + \delta}{\varepsilon^2}} \right) \leq P_u \left( \tau_\varepsilon(B_c) > \tau_{\mathcal{K},\frac{c}{2}} \right) + \sup_{v \in B_c} P_v \left( \tau_\varepsilon > e^{\frac{\Delta + \frac{\delta}{2}}{\varepsilon^2}} \right)$$ for any $u \in \mathcal{K}$ and $\varepsilon > 0$ sufficiently small. But let us observe that for $u \in \mathcal{K}$ we have $$\label{convunibola0} P_u \left( \tau_\varepsilon(B_c) > \tau_{\mathcal{K},\frac{c}{2}} \right) \leq P_u\left( d_{\tau_{\mathcal{K},\frac{c}{2}}}( U^{(r_{\mathcal{K}}+1),\varepsilon}, U^{(r_{\mathcal{K}}+1)}) > \min\left\{\frac{c}{2}, \frac{1}{2}\right\} \right)$$ and $$P_u \left( \tau_\varepsilon \leq \tau_{\mathcal{K},\frac{c}{2}}\right) \leq P_u \left( d_{\tau_{\mathcal{K},\frac{c}{2}}}( U^{(r_{\mathcal{K}}+1),\varepsilon}, U^{(r_{\mathcal{K}}+1)}) > 1\right).$$ Now the uniform bounds given by allow us to conclude the result. The next proposition shows that, for initial data in a small neighborhood of the origin, both the explosion time and the escape time from $G$ are asymptotically of the same order of magnitude. We will use this fact to conclude that the explosion time $\tau_\varepsilon$ shares the same asymptotic distribution with the escape time from $G$ and thus obtain the asymptotic loss of memory for $\tau_\varepsilon$. \[nescapelema3\] If $\tau^* > 0$ is taken as in Remark \[obsequivG\] then $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_c} P_u ( \tau_\varepsilon > \tau_\varepsilon (\partial G) + \tau^* )\right] = 0.$$ For any $u \in B_c$ we have by the strong Markov property that $$P_u ( \tau_\varepsilon > \tau_\varepsilon (\partial G) + \tau^* ) \leq \sup_{v \in B_c} P_v \left( U^\varepsilon (\tau_\varepsilon(\partial G), \cdot) \notin \partial^{\pm z} \right) + \sup_{v \in {\partial}^{\pm z}} P_v ( \tau_\varepsilon > \tau^*).$$ We may now conclude the result by Theorem \[teoescape0\] and Remark \[obsequivG\]. \[nescapecor0\] Let $\rho > 0$ be as in Lemma \[escapelema1\] and for each $\varepsilon > 0$ define $\beta_\varepsilon$ as in . Then 1. $\lim_{\varepsilon \rightarrow 0} \frac {\beta_\varepsilon}{\gamma_\varepsilon} = 1$. 2. $\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_\rho} |P_u (\tau_\varepsilon (\partial G) > t\beta_\varepsilon ) - e^{-t}| \right]= 0.$ Let us first notice that by the upper bound for the explosion time we have that $\beta_\varepsilon$ is finite for every $\varepsilon > 0$ sufficiently small. Now, the continuity of trajectories implies that $$P_{\mathbf{0}}(\tau_\varepsilon (\partial G) > \beta_\varepsilon ) \leq P_\mathbf{0}( \tau_\varepsilon > \beta_\varepsilon ) \leq e^{-1},$$ from where we conclude $\gamma_\varepsilon \leq \beta_\varepsilon$ and, thus, that $\liminf_{\varepsilon \rightarrow 0} \frac {\beta_\varepsilon}{\gamma_\varepsilon} \geq 1$. Let us now suppose that $\limsup_{\varepsilon \rightarrow 0} \frac {\beta_\varepsilon}{\gamma_\varepsilon} > 1$. Then there would exist $\lambda > 0$ and a sequence $(\varepsilon_j)_{j \in {{\mathbb N}}} \subseteq {{\mathbb R}}_{> 0}$ with $\lim_{j \rightarrow +\infty} \varepsilon_j = 0$ such that for all $j \in {{\mathbb N}}$ sufficiently large $$e^{-1} < P_\mathbf{0}( \tau_{\varepsilon_j} > \beta_{\varepsilon_j} - 1 ) \leq P_{\mathbf{0}} \left(\tau_{\varepsilon_j}(\partial G) > \left(1+ \lambda_0\right)\gamma_{\varepsilon_j}\right) + P_{\mathbf{0}} \left( \tau_{\varepsilon_j} > \tau_{\varepsilon_j} (\partial G) + \tau^* \right).$$ Taking the limit on the right hand side of this inequality with $j \rightarrow +\infty$, by we arrive at the contradiction $e^{-1} \leq e^{-(1+\lambda)}$. We thus conclude that $\limsup_{\varepsilon \rightarrow 0} \frac {\beta_\varepsilon}{\gamma_\varepsilon} \leq 1$ which implies (i). Notice that (i) itself implies by Theorem \[escapeteo1\] that for every $t > 0$ $$\lim_{\varepsilon \rightarrow 0} P_{\mathbf{0}} ( \tau_\varepsilon(\partial G) > t \beta_\varepsilon ) = e^{-t}.$$ We can now establish (ii) by following the proof of Theorem \[escapeteo1\] since one can show as in Lemma \[escapelema1\] that for every $t_0 > 0$ $$\lim_{\varepsilon \rightarrow 0}\left[ \sup_{u,v \in B_{\rho}} \left[ \sup_{t > t_0} |P_u(\tau_{\varepsilon}(\partial G) > t\beta_{\varepsilon}) - P_{v}(\tau_{\varepsilon}(\partial G) > t\beta_{\varepsilon})|\right]\right] = 0.$$ We are now ready to show the asymptotic exponential distribution of the for arbitrary initial data in $\mathcal{D}_{\mathbf{0}}$. This is contained in the following theorem. \[nteoasint\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive distance from $\mathcal{W}$ we have $$\label{convunicompact} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} |P_u (\tau_{\varepsilon} > t\beta_{\varepsilon}) - e^{-t}| \right] = 0.$$ for every $t > 0$. Let us consider the radius $\rho > 0$ given by Lemma \[escapelema1\] and suppose $\rho \leq c$ where $c$ is taken as in Conditions \[assumpg\]. Then from the inequalities $$P_u( \tau_\varepsilon(\partial G) > t\beta_\varepsilon ) \leq P_u (\tau_{\varepsilon} > t\beta_{\varepsilon}) \leq P_u( \tau_\varepsilon(\partial G)> t\beta_\varepsilon - \tau^*) + P_u ( \tau_\varepsilon > \tau_\varepsilon({\partial}G) + \tau^*)$$ for $u \in B_\rho$ one can easily verify, using (ii) in Corollary \[nescapecor0\] and Proposition \[nescapelema3\], that $$\label{convunibola} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_\rho} | P_u (\tau_\varepsilon > t \beta_\varepsilon ) - e^{-t} | \right] = 0.$$ Now, given a bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive distance from $\mathcal{W}$, take $\tau_{\mathcal{K},\frac{c}{2}} > 0$ as in the proof of Theorem \[taumagnitude\]. The strong Markov property implies for $$\inf_{v \in \mathcal{K}} P_v ( \tau_\varepsilon (B_c) \leq \tau_{\mathcal{K},\frac{c}{2}}) \inf_{v \in B_c} P( \tau_\varepsilon > t\beta_\varepsilon) \leq P_u (\tau_\varepsilon > t\beta_\varepsilon)$$ and $$P_u (\tau_\varepsilon > t\beta_\varepsilon) \leq \sup_{v \in \mathcal{K}} P_v ( \tau_\varepsilon (B_c) > \tau_{\mathcal{K},\frac{c}{2}}) + \sup_{v \in B_c} P( \tau_\varepsilon > t\beta_\varepsilon - \tau_{\mathcal{K},\frac{c}{2}}).$$ From these we may conclude by recalling and . Stability of time averages -------------------------- Our purpose in this final section is to show the stability of time averages along typical paths of the stochastic system up until (almost) the explosion time. The precise statement we wish to show is that of the following theorem. There exists a sequence $(R_\varepsilon)_{\varepsilon > 0}$ with $\lim_{\varepsilon \rightarrow 0} R_\varepsilon = +\infty$ and $\lim_{\varepsilon \rightarrow 0} \frac{R_\varepsilon}{\beta_\varepsilon} = 0$ such that given $\delta > 0$ for any bounded set $\mathcal{K} \subseteq \mathcal{D}_{\mathbf{0}}$ at a positive we have $$\label{average} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \frac{1}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(\mathbf{0})\right| > \delta \right) \right] = 0$$ for any bounded continuous function $f: C_D([0,1]) \rightarrow {{\mathbb R}}$. This result was originally established in [@GOV] for the double-well potential model in the finite-dimensional setting. Later the analogous result in the infinite-dimensional setting was obtained in [@B1]. We present here an adaptation of those proofs to our model. Let us observe that it suffices to show the result for the particular case $\mathcal{K}=B_c(\mathbf{0})$. Indeed, if we take $\tau_{\mathcal{K},\frac{c}{2}} > 0$ as in the proof of Theorem \[taumagnitude\] then, since $R_\varepsilon > k$ holds for $\varepsilon > 0$ sufficiently small, the strong Markov property then implies that for every $u \in \mathcal{K}$ and bounded continuous function $f: C_D([0,1]) \rightarrow {{\mathbb R}}$ we have $$P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \vartheta^\varepsilon_t(f) \right| > \delta \right) \leq \sup_{v \in \mathcal{K}} P( \tau_\varepsilon(B_c) > \tau_{\mathcal{K},\frac{c}{2}}) + \sup_{v \in B_c} P_v \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \vartheta^\varepsilon_t(f) \right| > \frac{\delta}{2} \right)$$ where for $0 \leq t < \tau_\varepsilon - R_\varepsilon$ we write $$\vartheta^{u,\varepsilon}_t(f) = \frac{1}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{u,\varepsilon}(s,\cdot))ds - f(\mathbf{0}).$$ Furthermore, by Proposition \[nescapelema3\] we see that in fact it will suffice to show that $$\label{averagelocal1} \lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in B_c} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon({\partial}G) - 2R_\varepsilon}\left| \vartheta^\varepsilon_t(f) \right| > \delta \right)\right] = 0.$$This provides the advantage of only having to consider paths inside a bounded domain. Finally, let us notice that in order to obtain for any bounded continuous function $f :C_D([0,1]) \rightarrow {{\mathbb R}}$, it will be enough to show only for the class of functions $\mathbbm{1}_\theta$ with $\theta > 0$ sufficiently small, where $\mathbbm{1}_\theta$ denotes the indicator function of the ball $B_\theta$. Indeed, this follows from that for and $\theta > 0$ one has $$|\vartheta^{u,\varepsilon}_t(f)| \leq \sup_{u \in B_\theta}|f(u)-f(\mathbf{0})|- 2 \|f\|_\infty \vartheta^{u,\varepsilon}_t(\mathbbm{1}_\theta)$$ for every $0 \leq t < \tau_\varepsilon - R_\varepsilon$. Thus, let us fix $\delta, \theta > 0$ and for each $u \in B_c$ and $l \in {{\mathbb N}}_0$ let us define the set $$A^{u,\varepsilon}_l:= \{ |\vartheta^{u,\varepsilon}_{lR_\varepsilon}(\mathbbm{1}_\theta)| \leq \delta \}$$ with the convention that $|\vartheta^{u,\varepsilon}_{lR_\varepsilon}(\mathbbm{1}_\theta)|=+\infty$ whenever $l \geq l^u_\varepsilon$, where $$l^u_\varepsilon := \inf \{ l \in {{\mathbb N}}_0 : \tau^u_\varepsilon({\partial}G) \leq (l+1)R_\varepsilon \}.$$ Let us observe that the validity of for $f=\mathbbm{1}_\theta$ will follow if we manage to show that, for $(R_\varepsilon)_{\varepsilon}$ as in the statement of the theorem, one has $$\label{averagelocal2} \lim_{\varepsilon \rightarrow 0} \left[\inf_{u \in B_c} P_u\left( \left[\bigcap_{0 \leq l < l_\varepsilon} A^\varepsilon_l \right] \cap \{ l_\varepsilon > 1 \} \right)\right]=1.$$ Now, for each $u \in B_c$ and $K_\varepsilon \geq 2$ we have $$\begin{aligned} P_u\left( \left[\bigcap_{0 \leq l < l_\varepsilon} A^\varepsilon_l \right] \cap \{ l_\varepsilon > 1 \} \right)& = \sum_{L=2}^\infty P_u\left( \left[\bigcap_{0 \leq l < l_\varepsilon} A^\varepsilon_l \right] \cap \{ l_\varepsilon = L \} \right)\\ \\ & = P_u ( l_\varepsilon > 1 ) - \sum_{L=2}^\infty P_u\left( \left[\bigcup_{0 \leq l < l_\varepsilon} (A^\varepsilon_l)^c \right] \cap \{ l_\varepsilon = L \} \right)\\ \\ & \geq P_u ( K_\varepsilon \geq l_\varepsilon > 1 ) - \sum_{L=2}^{[K_\varepsilon]} P_u\left( \left[\bigcup_{0 \leq l < l_\varepsilon} (A^\varepsilon_l)^c \right] \cap \{ l_\varepsilon = L \} \right)\end{aligned}$$ so that we may obtain provided that we can choose the sequences $(R_\varepsilon)_{\varepsilon > 0}$ and $(K_\varepsilon)_{\varepsilon}$ in such a way that: 1. $\lim_{\varepsilon \rightarrow 0} \left[\inf_{u \in B_c} P_u ( K_\varepsilon \geq l_\varepsilon > 1 )\right] = 1$ 2. $\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_c} \sum_{L=2}^{[K_\varepsilon]} P_u \left( \left[\bigcup_{0 \leq l < l_\varepsilon} (A^\varepsilon_l)^c \right] \cap \{ l_\varepsilon = L \} \right) \right] =0$. Since by definition of $l^u_\varepsilon$ for every $u \in B_c$ we have $$P_u ( K_\varepsilon \geq l_\varepsilon > 1 ) = P_u ( 2R_\varepsilon < \tau_\varepsilon({\partial}G) \leq (K_\varepsilon + 1)R_\varepsilon),$$ by Theorem \[ecotsuplema1\] we see that (i) follows if for each $\varepsilon > 0$ we choose $R_\varepsilon = e^{\frac{\alpha}{\varepsilon^2}}$ with $0 < \alpha < \Delta$ and $K_\varepsilon = e^{\frac{\gamma}{\varepsilon^2}}$ with $\gamma > \Delta - \alpha$. Therefore, it only remains to check that (ii) holds for this choice of the sequences $(R_\varepsilon)_{\varepsilon > 0}$ and $(K_\varepsilon)_{\varepsilon}$. But notice that for each $u \in B_c$ we have $$\begin{aligned} \sum_{L=2}^{[K_\varepsilon]} P_u \left( \left[\bigcup_{0 \leq l < l_\varepsilon} (A^\varepsilon_l)^c \right] \cap \{ l_\varepsilon = L \} \right)& \leq \sum_{L=2}^{[K_\varepsilon]} P_u \left( \left[\bigcup_{0 \leq l < l_\varepsilon} (A^\varepsilon_l)^c \right] \cap \{ l_\varepsilon > l \} \right)\\ \\ & \leq \sum_{L=2}^{[K_\varepsilon]} \sum_{l=0}^L P_u \left( (A^\varepsilon_l)^c \cap \{ l_\varepsilon > l \} \right)\\ \\ & \leq K_\varepsilon^2 \sup_{0 \leq l < K_\varepsilon} P_u\left( (A^\varepsilon_l)^c \cap \{ l_\varepsilon > l \} \right)\\ \\ & \leq K_\varepsilon^2 \sup_{u \in G} P_u \left( (A^\varepsilon_0)^c \cap \{ l_\varepsilon > 0 \} \right)\end{aligned}$$ where in the last inequality we have used the Markov property. The fact that (ii) holds now follows from the following proposition. This concludes the proof. If $0 < \theta < c$ where $c$ is given by Conditions \[assumpg\] then there exists $a > 0$ such that given $\delta > 0$ for $\varepsilon > 0$ sufficiently small we have $$\label{averagelocal3} \sup_{u \in G} P_u \left( (A^\varepsilon_0)^c \cap \{ l_\varepsilon > 0 \} \right) \leq e^{-\frac{\delta}{16} e^{\frac{a}{\varepsilon^2}}}.$$ Notice that, since $\theta < c$, we have that $B_\theta \subseteq \mathcal{D}_{\mathbf{0}}$ and thus that $V(\mathbf{0}, {\partial}B_\theta) > 0$. Thus, by the methods in Section \[seclowerbound\] one can show that there exist sufficiently small $0 < r < \theta$ and $0 < b < \alpha$ such that $$\label{aleq4} \lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in B_r} P_u \left( \tau_\varepsilon({\partial}B_\theta) \leq e^{\frac{b}{\varepsilon^2}} \right)\right] = 0.$$ Now, for each $\varepsilon > 0$ let us set $t_\varepsilon:=e^{\frac{b}{\varepsilon^2}}$, $N_\varepsilon:= \left[ \frac{R_\varepsilon}{t_\varepsilon}\right]$ and for $1 \leq i \leq N_\varepsilon$ and $u \in G$ define the random variable $$Y^{u,\varepsilon}_i = \left\{ \begin{array}{ll}0 &\text{ if $U^{u,\varepsilon}$ visits $B_d$ in $[(i-1)t_\varepsilon, (i-1)t_\varepsilon + \sqrt{t_\varepsilon})$ and then}\\ & \text{ spends the rest of the time interval $[(i-1)t_\varepsilon,it_\varepsilon)$ in $B_\theta$}\\ \\ 1 & \text{ otherwise.}\end{array}\right.$$ Since $\lim_{\varepsilon \rightarrow 0} \frac{R_\varepsilon}{t_\varepsilon} = +\infty$ and $\lim_{\varepsilon \rightarrow 0} t_\varepsilon = +\infty$, for $\varepsilon > 0$ sufficiently small and any $u \in G$ we have that $$\begin{aligned} P_u \left( (A^\varepsilon_0)^c \cap \{ l_\varepsilon > 0 \} \right) &= P_u \left( \left|\frac{1}{R_\varepsilon}\int_0^{R_\varepsilon} \mathbbm{1}_\theta( U^\varepsilon(t,\cdot))dt - \right|> \delta , \tau_\varepsilon({\partial}G) > R_\varepsilon \right) \\ \\ & \leq P_u \left( \left|\frac{1}{N_\varepsilon t_\varepsilon}\int_0^{N_\varepsilon t_\varepsilon} \mathbbm{1}_\theta( U^\varepsilon(t,\cdot))dt - \right|> \frac{\delta}{2} , \tau_\varepsilon({\partial}G) > N_\varepsilon t_\varepsilon \right) \\ \\ & \leq P_u \left( \frac{1}{N_\varepsilon} \sum_{i=1}^{N_\varepsilon} Y^\varepsilon_i > \frac{\delta}{2} - \frac{1}{\sqrt{t_\varepsilon}} , \tau_\varepsilon({\partial}G) > N_\varepsilon t_\varepsilon \right)\\ \\ & \leq P_u \left( \sum_{i=1}^{N_\varepsilon} Y^\varepsilon_i > \frac{\delta}{4}N_\varepsilon , \tau_\varepsilon({\partial}G) > N_\varepsilon t_\varepsilon \right)\\ \\ & \leq e^{-\frac{\delta}{4}N_\varepsilon} {{\mathbb E}}_u\left( \mathbbm{1}_{\{\tau_\varepsilon({\partial}G) > N_\varepsilon t_\varepsilon\}}e^{ \sum_{i=1}^{N_\varepsilon} Y^\varepsilon_i} \right)\\ \\ & \leq e^{-\frac{\delta}{4}N_\varepsilon} \left[\sup_{v \in G} {{\mathbb E}}_v \left( \mathbbm{1}_{\{\tau_\varepsilon({\partial}G) > t_\varepsilon\}}e^{Y^\varepsilon_1}\right)\right]^{N_\varepsilon}\end{aligned}$$ where the last inequality is a consequence of the Markov property. Now $$\begin{aligned} \sup_{v \in G} {{\mathbb E}}_v \left( \mathbbm{1}_{\{\tau_\varepsilon({\partial}G) > t_\varepsilon\}}e^{Y^\varepsilon_1}\right)& \leq \sup_{v \in G} \left[ P_v( Y^\varepsilon_1 = 0 , \tau_\varepsilon({\partial}G) > t_\varepsilon) + e P_v ( Y^\varepsilon_1 = 1 , \tau_\varepsilon({\partial}G) > t_\varepsilon) \right] \\ \\ & = \sup_{v \in G} \left[ P_v(\tau_\varepsilon({\partial}G) > t_\varepsilon) + (e-1) P_v ( Y^\varepsilon_1 = 1 , \tau_\varepsilon({\partial}G) > t_\varepsilon) \right] \\ \\ & \leq e^{(e-1) \sup_{v \in G}P_v ( Y^\varepsilon_1 = 1 , \tau_\varepsilon({\partial}G) > t_\varepsilon)}\end{aligned}$$ where in the last inequality we have used the fact that $1+x \leq e^x$ is valid for all $x \geq 0$. But let us observe that $$\sup_{v \in G}P_v ( Y^\varepsilon_1 = 1 , \tau_\varepsilon({\partial}G) > t_\varepsilon) \leq \sup_{v \in G} P( \tau_\varepsilon(B_r) > \sqrt{t_\varepsilon} , \tau_\varepsilon({\partial}G) > t_\varepsilon ) + \sup_{v \in B_r} P_v ( \tau_\varepsilon( {\partial}B_\theta) \leq t_\varepsilon )$$ where each term in the right hand side tends to zero as $\varepsilon \rightarrow 0$ by Lemma \[escapelema2\] and respectively. Thus, for $\varepsilon > 0$ sufficiently small we obtain that $$\sup_{u \in G} P_u \left( (A^\varepsilon_0)^c \cap \{ l_\varepsilon > 0 \} \right) \leq e^{-\frac{\delta}{8} N_\varepsilon} \leq e^{-\frac{\delta}{16} \frac{R_\varepsilon}{t_\varepsilon}} = e^{-\frac{\delta}{16} e^{\frac{a}{\varepsilon^2}}}$$ where $a = \alpha - b > 0$. Resumen del Capítulo 5 ---------------------- En este capítulo damos la demostración de los Teoremas II, III y IV en la sección de resultados del Capítulo 1. Los Teoremas II y III se deducen de los resultados del Capítulo 4 para el tiempo de escape del dominio $G$ dado que $\tau^u_\varepsilon$ y $\tau^u_\varepsilon({\partial}G)$ son asintóticamente equivalentes. Más precisamente, existe $\tau^* > 0$ tal que $$\lim_{\varepsilon \rightarrow 0} \left[\sup_{u \in G} | P_u ( \tau_\varepsilon ({\partial}G) \leq \tau_\varepsilon \leq \tau_\varepsilon ({\partial}G) + \tau^* ) - 1 |\right] = 0.$$ La desigualdad $\tau_\varepsilon ({\partial}G) \leq \tau_\varepsilon$ vale siempre como consecuencia de la continuidad de las trayectorias de $U^{u,\varepsilon}$. Por otro lado, la segunda desigualdad $\tau_\varepsilon \leq \tau_\varepsilon ({\partial}G) + \tau^*$ se deduce de los resultados del Capítulo 2, puesto que $U^{u,\varepsilon}$ se escapa de $G$ típicamente por ${\partial}G$ y, además, ${\partial}G$ es un subconjunto cerrado de $\mathcal{D}_e^*$ a una distancia positiva de la frontera. Por último, como $\tau^u_\varepsilon$ y $\tau^u_\varepsilon({\partial}G)$ son asintóticamente equivalentes, podemos suponer que en los promedios ergódicos en el enunciado del Teorema IV figura $\tau^u_\varepsilon({\partial}G)$ en lugar de $\tau^u_\varepsilon$. La demostración del Teorema IV en este caso sigue los pasos de [@GOV] y [@B1]. A finite-dimensional problem ============================ In this chapter we study the asymptotic properties of the explosion time for small random perturbations of a particular ordinary differential equation with blow-up. This can be seen as a finite-dimensional version of our original problem. However, the equation we consider in this chapter is not the finite-dimensional analogue of the original equation , and thus one cannot perform the exact same analysis of the previous chapters. We decided to include this variant here for a number of reasons. First, because it serves as an example of the lack of an unified approach to treat perturbations of differential equations with blow-up: in general, different systems require different techniques to study them. We also do it to show that the finite-dimensional structure can simplify matters to some extent, allowing us to achieve more general results than for the infinite-dimensional alternative. Finally, we do it to show that the ideas developed in this first part are not restricted to equations with Dirichlet boundary conditions. The analysis of this chapter can be found in more detail in [@GS]. Preliminaries ------------- ### The deterministic system We consider small random perturbations of the following ODE $$\label{1.1} \left\{\begin{array}{lcll} U'_1 &= &\frac{2}{h^2} ( -U_1 + U_2 ),\\ U'_i &= &\frac{1}{h^2} ( U_{i+1} - 2U_i + U_{i-1} ) &\,\,\, 2 \leq i \leq d-1,\\ U'_d &= &\frac{2}{h^2} ( -U_d + U_{d-1} +hg(U_d) )\\ U(0) &= &u. \end{array}\right.$$ Here $g\colon {{\mathbb R}}\to {{\mathbb R}}$ is a reaction term given by $g(x) = (x^+)^p -x$ for $p>1$ These kind of systems arise as spatial discretizations of diffusion equations with nonlinear boundary conditions of Neumann type. In fact, it is well known that as $h\to 0$ solutions to this system converge to solutions of the PDE $$\left \{\begin{array}{rcll} {\partial}_t U(t,x) & = & {\partial}^2_{xx}U(t,x) & 0<x<1, 0\le t<T,\\ {\partial}_x U(t,0) & = & 0 & 0\le t <T,\\ {\partial}_x U(t,1) &= & g(U(t,1)) & 0\le t <T,\\ U(0,x) & = & U_0(x) & 0\le x \le 1. \end{array} \right.$$ For details on this convergence see [@DER]. Equation can be written in matrix form as $$\label{A1} dU = \Big(-AU + \frac2h g(U_d)e_d\Big)dt$$ for some positive definite $A \in {{\mathbb R}}^{d \times d}$ and where $e_d$ denotes the $d$-th canonical vector on ${{\mathbb R}}^d$. The field $b(u):= -AU + \frac2hg(U_d)e_d$ is of gradient type, i.e. $b=-\nabla S$, with given by $$S(u) = \frac{1}{2} \langle Au , u \rangle - \frac{2}{h}\Big(\frac{\;\;\;|u_d^+|^{p+1}}{p+1} - \frac{\,\,{u_d}^2}{2}\Big).$$ This potential satisfies all properties shown for its analogue in . It has exactly two critical points, $\1:=(1,\dots,1)$ and the origin, both of them hyperbolic. The origin $0$ is the unique local minimum of the potential $S$ while $\1$ is a saddle point. Furthermore, we have a decomposition of ${{\mathbb R}}^d$ similar to (see [@AFBR; @GS] for details). Indeed, we have $${{\mathbb R}}^d = \mathcal{D}_0 \cup \mathcal{W}^s_1 \cup \mathcal{D}_e$$ where $\mathcal{D}_0$ denotes the stable manifold of the origin, $\mathcal{W}^s_{1}$ is the stable manifold of $\1$ and $\mathcal{D}_e$ is the domain of explosion. Once again the sets $\mathcal{D}_0$ and $\mathcal{D}_e$ are open in ${{\mathbb R}}^d$ and the origin is an asymptotically stable equilibrium of the system. $\mathcal{W}^s_{1}$ is a manifold of The saddle point $\1$ also admits an unstable manifold, $\mathcal W^u_\1$. This unstable manifold is contained in ${{\mathbb R}}^d_+$ and has dimension one. Furthermore, it has nonempty intersection with both $\mathcal{D}_0$ and $\mathcal{D}_e$ and joins $\1$ with the origin. An illustration of this decomposition is given in Figure \[fig:inclination\] for the $2$-dimensional case. Finally, we have the finite-dimensional analogue of Theorem \[descomp2\], originally proved in [@AFBR]. ![The phase diagram of equation .[]{data-label="fig:inclination"}](df.eps){width="8cm"} ### The stochastic system We study random perturbations of given by additive white-noise. More precisely, we consider stochastic differential equations of the form $$\label{Aestoc} dU^{{\varepsilon}} = \Big(-AU^{{\varepsilon}} + \frac2h g(U_d^{{\varepsilon}})e_d\Big)dt + {\varepsilon}dW$$ for ${\varepsilon}>0$ small and where $W=(W_1, \dots,W_d)$ a $d-$dimensional standard Brownian motion. Given a probability space $(\Omega,{{\mathcal F}}, P)$ and a standard $d$-dimensional Brownian motion $W$, we say that a stochastic process $U^\varepsilon = (U^\varepsilon(t))_{t \geq 0}$ is a solution up to an explosion time of on $(\Omega, \mathcal{F}, P)$ and with respect to $W$ if it satisfies the following: - $U^\varepsilon$ has continuous paths taking values in ${{\mathbb R}}^d \cup \{ \infty\}$ - $U^\varepsilon$ is adapted to the augmented filtration generated by $W$. - For every $n \in {{\mathbb N}}$ we have $P$-almost surely $$\int_{0}^{t\wedge{\tau^{(n)}}} |b(U^\varepsilon(s))|\,ds < +\infty \hspace{0,5cm} \,\forall\,\,\, 0\leq t < + \infty$$ and $$U^\varepsilon({t\wedge \tau^{(n)}}) = U^\varepsilon(0) + \int_{0}^{t} b(U^\varepsilon(s))\mathbbm{1}_{\{s \leq \tau^{(n)}\}}\,ds + \varepsilon W({t \wedge \tau{(n)}}); \,\,\,\forall\,\,\, 0\leq t < + \infty.$$ where $\tau^{(n),\varepsilon}:=\inf \{ t \geq 0 : \|U^\varepsilon(t)\|_\infty \geq n\}$. - $U^\varepsilon$ has the strong Markov property. We call $\tau^\varepsilon:= \lim_{n \rightarrow +\infty} \tau^{(n),\varepsilon}$ the *explosion time* of $U^{\varepsilon}$. As in the previous chapters, we shall write $U^{u,{\varepsilon}}$ to denote the unique solution to with initial datum $u \in {{\mathbb R}}^d$ and also write $U^u$ to denote the corresponding solution to . Furthermore, for each $n \in {{\mathbb N}}$ we consider the truncation $S^{(n)}$ of the potential $S$ given by $$S^{(n)}(u) = \frac{1}{2}\langle Au, u \rangle - \frac{2}{h} G_n(u_d)$$ where $G_n : {{\mathbb R}}\longrightarrow {{\mathbb R}}$ is a function of class $C^2$ satisfying that $$\label{gtruncada2} G_n(u) = \left\{\begin{array}{ll} \frac{|u^+|^{p+1}}{p+1} - \frac{u^2}{2} &\,\,\text{if}\,\,u \leq n\\ 0 &\,\,\text{if}\,\,u \geq 2n. \end{array}\right.$$ The unique solution $U^{(n),u}$ to the equation $\dot{U}^{(n),u} = -\nabla S^{(n)}(U^{(n),u})$ with is globally defined and coincides with $U^u$ until the escape from the *unbounded* set $$\Pi^n := \{ u \in {{\mathbb R}}^d : u_d < n \}.$$ Similarly, for $\varepsilon > 0$ the unique solution $U^{(n),u,\varepsilon}$ to the equation $$d{U}^{(n),u,\varepsilon} = -\nabla S^{(n)}(U^{(n),u,\varepsilon}) dt + \varepsilon dW$$ with initial datum $u$ is globally defined and coincides with $U^{u,\varepsilon}$ until the escape from $\Pi^n$. Moreover, since the field $-\nabla S^{(n)}$ is globally Lipschitz, the family of solutions $\left(U^{(n),u,\varepsilon}\right)_{\varepsilon > 0}$ satisfies the analogous large deviations estimates of Section \[secLDP\] with rate function $$I^u_T(\varphi)=\left\{\begin{array}{ll}\displaystyle{\frac12\int_0^T|\dot \varphi(s) + \nabla S^{(n)}(\varphi(s))|^2 ds} & \mbox{if $\varphi$ is absolutely continuous and $\varphi(0)=u$}\\ \\ +\infty & \mbox{otherwise} \end{array} \right.$$ Finally, for each $\varepsilon > 0$ and $u \in {{\mathbb R}}^d$ the process $U^{(n),u,\varepsilon}$ is positive recurrent. ### Main results We now state the main results obtained in this finite-dimensional setting. Some of these results are more refined than their infinite-dimensional counterparts. This is due to the friendlier finite-dimensional setting and also to the convenient choice of reaction term. We maintain the notation of Chapter 1. Our first result is concerned with the almost sure existence of blow-up for arbitrary initial data and noise parameter. **Theorem I**. For any $u \in {{\mathbb R}}^d$ and $\varepsilon > 0$ we have $P_u(\tau_{\varepsilon}< +\infty) = 1$. Let us notice that for the infinite-dimensional system we were only able to show that $$\lim_{\varepsilon \rightarrow 0} P_u ( \tau_\varepsilon < +\infty ) = 1.$$ This is because, by the particular choice of reaction term $g$, solutions in this setting only explode in one direction, so that comparison arguments can be successfully applied. Next, we study the asymptotic behavior of the explosion time for initial data in $\mathcal{D}_e$. **Theorem II**. Given $\delta > 0$ and a compact set $\mathcal{K} \subseteq \mathcal{D}_e$ there exists $C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( |\tau_\varepsilon - \tau_0| > \delta ) \leq e^{-\frac{C}{\varepsilon^2}}$$ for every $\varepsilon > 0$ sufficiently small. Finally, we show metastable behavior for solutions of with initial data $u \in \mathcal{D}_0$. We have the following results. **Theorem III**. Given $\delta > 0$ and a compact set $\mathcal{K} \subseteq \mathcal{D}_0$ we have $$\lim_{\varepsilon \rightarrow +\infty} \left[ \sup_{u \in \mathcal{K}} \left|P_u \left( e^{\frac{\Delta - \delta}{\varepsilon^2}} < \tau_\varepsilon < e^{\frac{\Delta + \delta}{\varepsilon^2}}\right)- 1\right|\right]=0$$ where $\Delta:=2(S(\1)-S(0))$. **Theorem IV**. If for each $\varepsilon > 0$ we define the scaling coefficient $$\beta_{\varepsilon}= \inf \{ t \geq 0 : P_0 ( \tau_{\varepsilon}> t ) \leq e^{-1} \}$$ then $\lim_{\varepsilon \rightarrow 0} \varepsilon ^{2}\log\beta_{\varepsilon} = \Delta$ and for each compact set $\mathcal{K} \subseteq \mathcal{D}_0$ and $t > 0$ we have $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} \left|P_{u} (\tau_{\varepsilon} > t\beta_{\varepsilon}) - e^{-t}\right| \right]=0.$$ **Theorem V**. There exists a sequence $(R_\varepsilon)_{\varepsilon > 0}$ with $\lim_{\varepsilon \rightarrow 0} R_\varepsilon = +\infty$ and $\lim_{\varepsilon \rightarrow 0} \frac{R_\varepsilon}{\beta_\varepsilon} = 0$ such that given $\delta > 0$ for any compact set $\mathcal{K} \subseteq \mathcal{D}_{0}$ we have $$\lim_{\varepsilon \rightarrow 0} \left[ \sup_{u \in \mathcal{K}} P_u \left( \sup_{0 \leq t \leq \tau_\varepsilon - 3R_\varepsilon}\left| \frac{1}{R_\varepsilon}\int_t^{t+R_\varepsilon} f(U^{\varepsilon}(s,\cdot))ds - f(0)\right| > \delta \right) \right] = 0$$ for any bounded continuous function $f: {{\mathbb R}}\rightarrow {{\mathbb R}}$. With the exception of Theorem I, the proof of the remaining results follow very closely the ideas featured in the previous chapters for the infinite-dimensional problem (perhaps with even fewer technical difficulties). Thus, we include here only the parts of the analysis which differ from the ones given in the previous setting. The main differences appear on the proof of Theorem II and the construction of the auxiliary domain $G$. For the latter, we shall exploit the fact that we are in a finite-dimensional setting to obtain a different construction of $G$, one which does not rely so heavily on the structure of the potential $S$. In particular, this will allow us to obtain our results for every $p > 1$ instead of $p \in (1,5)$. Finally, the results presented here can be extended to more general systems than . We refer the reader to [@GS] for details on possible extensions. Almost sure blow-up in the stochastic model {#section4} ------------------------------------------- \[estoc.exp\] In this section we devote ourselves to the proof of Theorem I. The idea is to show that, conditioned on non-explosion, the system is guaranteed to enter a specific region of space in which we can prove that explosion occurs with total probability. From this we can conclude that non-explosion must happen with zero probability. We do this by comparison with an adequate Ornstein-Ühlenbeck process. Indeed, let $Y^{y,\,\varepsilon}$ be the solution to $$\label{OU} dY^{y,\,\varepsilon}=-\Big(AY^{y,\,\varepsilon} + \frac{2}{h} Y^{y,\,\varepsilon}_d e_d\Big)\,dt + \varepsilon dW$$ with initial datum $y \in {{\mathbb R}}^d$. Notice that the drift term in is linear and given by a negative definite matrix. Hence, $Y^{y,{\varepsilon}}$ is in fact a $d$-dimensional Ornstein-Ühlenbeck process which admits an invariant distribution supported in ${{\mathbb R}}^d$. Since we also have convergence to this equilibrium measure for any initial distribution, the hitting time of $Y^{y,{\varepsilon}}$ of any open set is finite almost surely. On the other hand, the drift term of is smaller or equal than $b$ so that by the comparison principle we conclude that if $u \geq y$ then $U^{u,{\varepsilon}}(t) \ge Y^{y,{\varepsilon}}$ holds almost surely for as long as $U^{u,{\varepsilon}}$ is finite. From here, Theorem I follows at once from the next lemma and the strong Markov property. If we consider the set $$\Theta^M:=\{ y \in {{\mathbb R}}^d : y_k \geq 0 \,\,\text{for all}\,\,0\leq k \leq d-1\,,\, y_d \geq M \},$$ then we have $$\lim_{M\rightarrow \infty } \left[\inf_{y\in \Theta^M} P_y(\tau_{\varepsilon}<\infty)\right] =1.$$ Consider the auxiliary process $Z^{y,\,\varepsilon}:= U^{y,\,\varepsilon} - {\varepsilon}W$. Notice that this process verifies the random differential equation $$dZ^{y,{\varepsilon}}=b(Z^{y,{\varepsilon}} +{\varepsilon}W)dt, \quad Z^{y,{\varepsilon}}(0)=y.$$ Also observe that $Z^{y,{\varepsilon}}$ has the same explosion time as $U^{y,{\varepsilon}}$. For each $k \in {{\mathbb N}}$ let us define the set $A_k:=\{\sup_{0\leq t \leq 1} |W_d(t)|\le k\}$. On $A_k$ we have that $Z^{y,{\varepsilon}}$ verifies the inequality $$\label{ineq.z} \frac{dZ^{y,\,\varepsilon}}{dt} \ge -AZ^{y,\,\varepsilon} - \frac{4}{h^2}{\varepsilon}k \sum e_i+ \frac{2}{h} ((Z^{y,\,\varepsilon}_d - \varepsilon k)_+^p - Z^{y,\,\varepsilon}_d - \varepsilon k) e_d.$$ Observe that can be written as $$\frac{dZ^{y,\,\varepsilon}}{dt} \ge QZ^{y,\,\varepsilon} + q + (Z^{y,\,\varepsilon}_d - \varepsilon k)_+^p e_d \ge QZ^{y,\,\varepsilon} + q ,$$ where $Q\in{{\mathbb R}}^{d\times d}$ verifies a comparison principle and $q\in{{\mathbb R}}^d$ both depend on ${\varepsilon}, h$ and $k$, but not on $M$. This allows us to conclude the inequality $Z_{d-1}^{y,\,\varepsilon} \ge -(M + |q|){\mathrm exp}(|Q|)$ for all $0\le t \le \min\{1,\tau_\varepsilon^y\}$. In particular, for all $0\le t \le \min\{1,\tau_\varepsilon^y\}$ the last coordinate verifies the inequality $$\left\{\begin{array}{ll} \frac{dZ_d^{y,\,\varepsilon}}{dt} \ge -\alpha_1 M + \alpha_2 Z_d^{y,\,\varepsilon} + \alpha_3(Z^{y,\,\varepsilon}_d)^p & \\ \\ Z_d^{y,\,\varepsilon}(0)\ge M \end{array} \right.$$ for positive constants $\alpha_1, \alpha_2, \alpha_3$ which do not depend on $M$. It is a straightforward calculation to check that solutions to this one-dimensional inequality blow up in a finite time that converges to zero as Therefore, for each $k \in {{\mathbb N}}$ there exists $M_k$ such that $P(A_k)\leq \inf_{y \in \Theta^{M}} P_y(\tau_{\varepsilon}<\infty)$ for all $M \geq M_k$. Since $\lim_{k \rightarrow +\infty} P(A_k)=1$, this concludes the proof. Convergence of $\tau^u_\varepsilon$ for initial data in $\mathcal{D}_e$ {#estoc.conv} ----------------------------------------------------------------------- Our purpose in this section is to prove Theorem II. We shall only give the upper bound for the explosion time $\tau^u_\varepsilon$ since the lower bound can be obtained exactly as in We need the following lemma, whose proof can be found on [@GS]. \[prinmax\] If $U$ is a solution to then for every $t \geq 0$ $$\label{pmaximo} \max_{k=1,\dots,d} |U_k (t)| \leq \max \{ \max_{k=1,\dots,d} |U_k(0)| , \max_{0\leq s \leq t} U_d(s)\}.$$ The upper bound for the explosion time is given in the following proposition. For any $\delta > 0$ and compact set $\mathcal{K} \in \mathcal{D}_e$ there exists $C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( \tau_\varepsilon > \tau_0 + \delta ) \leq e^{- \frac{C}{\varepsilon^2}}$$for every $\varepsilon > 0$ sufficiently small. Given $u \in \mathcal{K}$, let $Y^{u}$ be the solution to the ordinary differential equation $$\dot{Y}^{u}=-\Big(AY^{u} + \frac{2}{h} Y^{u,\,\varepsilon}_d e_d\Big)$$ with initial datum $u$. Let us notice that we have $U^{u} \geq Y^{u}$ for as long as $U^{u}$ is defined by Now, since $Y^{u}$ is the solution to a linear system of ordinary differential equations whose associated matrix is symmetric and negative definite, we get that there exists $\rho_{\mathcal{K}} \in {{\mathbb R}}$ such that for all $u \in \mathcal{K}$ every coordinate of $U^{u}$ remains bounded from below by $\rho_{\mathcal{K}} + 1$ up until $\tau^u_0$. Thus, if for $\rho \in {{\mathbb R}}$ and $M > 0$ we write $$\Theta_{\rho}^M:=\{ y \in {{\mathbb R}}^d : y_k \geq \rho \,\,\text{for all}\,\,0\leq k \leq d-1\,,\, y_d \geq M \}$$ then by Lemma \[prinmax\] we conclude that $T_u:= \inf \{ t \geq 0 : U^{u}_t \in \Theta_{\rho_{\mathcal{K}}+1}^{M+1}\}$ is finite. Furthermore, since $U^{M+2,\,u}$ agrees with $U^{u}$ until the escape from $\Pi^{M+2}$, we obtain the expression $$T_u= \inf \{ t \geq 0 : U^{M+2,\,u}_t \in \Theta_{\rho_{\mathcal{K}+1}}^{M+1}\}.$$ Taking $T_{\mathcal{K}}:=\sup_{u \in \mathcal{K}} T_u <+\infty$ we may compute $$\begin{aligned} P_u\big(\tau_{\varepsilon}(\Theta_{\rho_{\mathcal{K}}}^M) > T_u\big) &\leq P_u\big( \pi^{M+2}_\varepsilon \wedge \tau_{\varepsilon}(\Theta_{\rho_{\mathcal{K}}}^M) > T_u \big) + P_u\big(\pi^{M+2}_\varepsilon \leq T_u\,,\,\tau_{\varepsilon}(\Theta_{\rho_{\mathcal{K}}}^M) > T_u \big) \\ \\ & \leq 2 \sup_{v \in {{\mathbb R}}^d} P_v \Big(\sup_{0\le t \le T_{\mathcal{K}}} |U^{M+2,\,\varepsilon}(t) - U^{M+2}(t)| > 1 \Big).\end{aligned}$$ On the other hand, by the strong Markov property for $U^{u,\varepsilon}$ we get $$\label{cotaexpfinito} P_u \big( \tau_\varepsilon > \tau_0 + \delta \big) \leq P_u \big( \tau_\varepsilon > T_u + \delta \big) \leq \sup_{y \in \Theta_{\rho_{\mathcal{K}}}^M} P_y ( \tau_\varepsilon > \delta) + \sup_{u \in \mathcal{K}} P_u\big(\tau_{\varepsilon}(\Theta_{\rho_{\mathcal{K}}}^M) > T_u\big).$$ Thus, by and the previous computation, in order to finish the proof it will suffice to show that the first term on the right hand side of tends to zero exponentially fast in $\frac{1}{\varepsilon^2}$ as $\varepsilon \rightarrow 0$ for an adequate choice of $M$. To see this we consider for $\varepsilon > 0$ and $y \in \Theta_{\rho_{\mathcal{K}}}^M$ the processes $Y^{y,\,\varepsilon}$ and $Z^{y,\,\varepsilon}$ defined by $$dY^{y,\,\varepsilon}=-\Big(AY^{y,\,\varepsilon} + \frac{2}{h} Y^{y,\,\varepsilon}_d e_d\Big)\,dt + \varepsilon dW$$ and $Z^{y,\,\varepsilon}:= U^{y,\,\varepsilon} - Y^{y,\,\varepsilon}$, respectively. Notice that $Y^{y,\,\varepsilon}$ is globally defined and thus that both $U^{y,\,\varepsilon}$ and $Z^{y,\,\varepsilon}$ have the same explosion time. Furthermore, $Z^{y,\,\varepsilon}$ is the solution of the random differential equation $$dZ^{y,\,\varepsilon}=-\Big(AZ^{y,\,\varepsilon} + \frac{2}{h}\Big( \Big[\big(U_d^{y,{\varepsilon}}\big)^{+}\Big]^p - Z^{y,\,\varepsilon}_d\Big)e_d\Big)\,dt.$$ The continuity of trajectories allows us to use the Fundamental Theorem of Calculus to show that almost surely $Z^{y,\,\varepsilon}(\omega)$ is a solution to the ordinary differential equation $$\label{rde1} \dot{Z}^{y,\,\varepsilon}(t)(\omega) = -AZ^{y,\,\varepsilon}(\omega) + \frac{2}{h}\Big( \Big[\big(U_d^{y,{\varepsilon}}\big)^{+}\Big]^p(\omega) - Z^{y,\,\varepsilon}_d(\omega)\Big)e_d.$$ Then, for each $y \in \Theta_{\rho_{\mathcal{K}}}^M$ and $\varepsilon > 0$ let $\Omega^y_\varepsilon$ be a set of probability one in which holds. Notice that for every $\omega \in \Omega^y_\varepsilon$ we have the inequality $$\label{rde2} \dot{Z}^{y,\,\varepsilon}(\omega) \geq -AZ^{y,\,\varepsilon}(\omega) - \frac{2}{h}Z^{y,\,\varepsilon}_d(\omega)e_d.$$ By the comparison principle we conclude that $Z^{y,\,\varepsilon}(\omega)\geq 0 $ for every $\omega \in \Omega^y_\varepsilon$ and, therefore, that the inequality $U^{y,\,\varepsilon}(\omega) \geq Y^{y,\,\varepsilon}(\omega)$ holds for as long as $U^{y,\,\varepsilon}(\omega)$ is defined. For each $y \in \Theta_{\rho_{\mathcal{K}}}^M$ and $\varepsilon > 0$ let us also consider the set $$\tilde{\Omega}^y_\varepsilon = \Big\{ \omega \in \Omega : \sup_{0\le t\le \delta} | Y^{y,\,\varepsilon}(\omega,t)- Y^{y}(\omega,t)| \leq 1 \,,\, \sup_{0 \leq t \leq \delta} |\varepsilon W(\omega,t)| \leq 1 \Big\}.$$ Our goal is to show that if $M$ is appropriate then for each $y \in \Theta_{\rho_{\mathcal{K}}}^M$ explodes before time $\delta$ for all $\omega \in \Omega^y_\varepsilon \cap \tilde{\Omega}^y_\varepsilon$. From this we get that $$\inf_{y \in \Theta_{\rho_{\mathcal{K}}}^M} P(\tilde{\Omega}^y_\varepsilon) = \inf_{y \in \Theta_{\rho_{\mathcal{K}}}^M} P( \Omega^y_\varepsilon \cap \tilde{\Omega}^y_\varepsilon) \leq \inf_{y \in \Theta_{\rho_{\mathcal{K}}}^M} P_y (\tau_\varepsilon \leq \delta ).$$ and so by we may conclude the result. Hence, let us take $y \in \Theta_{\rho_{\mathcal{K}}}^M$, $\omega \in \Omega^y_\varepsilon \cap \tilde{\Omega}_\varepsilon$ and suppose that $U^{y,\,\varepsilon}(\omega)$ is defined in $[0,\delta]$. Notice that since $\omega \in \Omega^y_\varepsilon \cap \tilde{\Omega}_\varepsilon$ then the $(d-1)$-th coordinate of $Y^{y,\,\varepsilon}(\omega,t)$ is bounded from below by $\rho_{\mathcal{K}} - 1$ for all $t \in [0,\delta]$. By comparison we know that the $(d-1)$-th coordinate of $U^{y,\,\varepsilon}_t(\omega,t)$ is bounded from below by $\rho_{\mathcal{K}} - 1$ as well. From here we deduce that the last coordinate of $U^{y,\varepsilon}(\omega)$ verifies the integral equation $${U}^{y,\,\varepsilon}_d(\omega,t) \geq {U}^{y,\,\varepsilon}_d(\omega,s) + \int_s^t \frac{2}{h²}\,\Big(- U^{y,\,\varepsilon}_d(\omega,r) +\rho_{\mathcal{K}} -1 + hg\big( U^{y,\,\varepsilon}_d(\omega,r)\big) \Big)\,dr - 1$$for $s < t$ in the interval $[0,\delta]$. We can take $M \in {{\mathbb N}}$ sufficiently large so as to guarantee that there exists a constant $\alpha > 0$ such that for all $m\geq M$ we have $$\label{lcotafea} \frac{2}{h²}\,\big(-m +\rho_{\mathcal{K}} -1 + hg(m) \big) \geq \alpha m^p.$$ If we recall that $U^{y,\,\varepsilon}_d(\omega,0) \geq M$ then our selection of $M$ implies that $${U}^{y,\,\varepsilon}_d(\omega,t) \geq M-1 + \alpha\int_0^t \big( U^{y,\,\varepsilon}_d(\omega,u)\big)^p\,du$$ for every $t \in [0,\delta]$. But if this inequality holds and $M$ is sufficiently large, one can check that $U^{y,\,\varepsilon}(\omega)$ explodes before time $\delta$, a fact which contradicts our assumptions. Therefore, if $y \in \Theta_{\rho_{\mathcal{K}}}^M$ and $\omega \in \Omega^y_\varepsilon \cap \tilde{\Omega}_\varepsilon$ then $U^{y,\,\varepsilon}(\omega)$ explodes before time $\delta$, Construction of an auxiliary domain {#construction-of-an-auxiliary-domain} ----------------------------------- In this final section we present the alternative construction of the auxiliary domain $G$. The reader will notice that the finite-dimensional environment plays an essential role in the construction. Despite this fact, we point out that the only other ingredient which is relevant in this alternative construction is the validity of an analogue of Theorem \[descomp2\], so that one may carry out the same construction in other systems with a similar description. We wish to construct a bounded domain $G$ satisfying the following properties: 1. $G$ contains $\1$ and the origin. 2. There exists $c > 0$ such that $B_c \subseteq G$ and for all $u \in B_c$ the system $U^{u}$ is globally defined and tends to $0$ without escaping $G$. 3. There exists a closed subset ${\partial}^{\1}$ of the boundary $\partial G$ which satisfies: 1. $V(0,\partial G - \partial^{\1} ) > V(0,\partial^{\1}) = V(0, \1)$. 2. ${\partial}^{\1}$ is contained in $\mathcal{D}_e$ and at a positive distance from its boundary. The domain $G$ is constructed in the following manner: Since $S: {{\mathbb R}}^d \rightarrow {{\mathbb R}}$ is continuous and $S(\1) > S(0)$, we may take $c>0$ such that $S(u)< S(\1)$ for $u \in B_c$. Then, for each $u\in \partial B_c$ consider the ray $R_u:=\{ \lambda u : \lambda > 0\}$. Since the vector $\1$ is not tangent to $\W_\1^s$ at $\1$, we may take a sufficiently small neighborhood $V$ of $c\cdot \1$ such that for every $u \in V\cap \partial B_c$ the ray $R_u$ intersects $\W_\1^s \cap ({{\mathbb R}}_{> 0})^d$. For such $V$ we may then define $\bar{\lambda}_u=\inf\{ \lambda > 0 : \lambda u \in \W_\1^s\}$ for $u \in V\cap\partial B_c$. If we consider $$\eta:= \inf_{u \in \partial[V\cap\partial B_c]} \phi(\bar{\lambda}_u u) > \phi(\1)$$ where by $\partial[V\cap \partial B_c]$ we understand the boundary of $V\cap \partial B_c(0)$ as a $(d-1)$-dimensional manifold, then the fact that $S(U^u(t))$ is strictly decreasing allows us to shrink $V$ into a smaller neighborhood $V^*$ of $c\cdot \1$ such that $S(v)=\eta$ is satisfied for all $v \in \partial[V^*\cap \partial B_c]$. Let us also observe that since $\1$ is the only saddle point we can take $V$ sufficiently small so as to guarantee that $\max\{ S(\lambda u) : \lambda > 0\} \geq \eta$ for all $u \in \partial B_c\setminus V^*$. Then if we take the level curve $C_\eta = \{ x \in {{\mathbb R}}^d : S(x) = \eta \}$ every ray $R_u$ with $u \in \partial B_c\setminus V^*$ intersects $C_\eta$. With this we may define for each $u \in \partial B_c$ $$\lambda_u^*= \left\{\begin{array}{ll} \bar{\lambda}_u & \mbox{ if }\,\,u \in V^*\\ \\ \inf\{ \lambda > 0 : \lambda u \in C_\eta\} & \mbox{ if }\,\,u \in B_c(0)\setminus V^* \end{array} \right..$$ Notice that the mapping $u \mapsto \lambda^*_u$ is continuous. Thus, if $\tilde{G}:=\{ \lambda u : 0 \leq \lambda < \lambda^*_u\,,\, u \in \partial B_c\}$ then $\partial\tilde{G} =\{ \lambda^*_u u : u \in \partial B_c(0)\}$. To finish the construction of our domain we must make a slight radial expansion of $\tilde{G}$, i.e. for $\alpha > 0$ consider $G$ defined by the formula $$G:=\{\lambda u : 0 \leq \lambda < (1+\alpha)\lambda^*_u\,,\, u \in \partial B_c\}.$$ Observe that the finite-dimensional analogue of Theorem \[descomp2\] ensures that $G$ verifies (i). Since $\lambda^*_u > 1$ for all $u \in \partial B_c(0)$ then it must also verify (ii). Furthermore, if we define $\partial^\1:=\{(1+\alpha)\lambda^*(u) u : u \in \overline{V^*}\}$ then $\partial^\1$ is closed and contained in $\mathcal{D}_e$. By taking $\alpha > 0$ sufficiently small, the continuity of $S$ implies that (iii) holds as well. See Figure 6.2. {width="7cm"} \[fig:gaux\] Resumen del Capítulo 6 ---------------------- Estudiamos las propiedades asintóticas del tiempo de explosión para perturbaciones aleatorias por ruido blanco aditivo de la ecuación diferencial ordinaria $$\left\{\begin{array}{lcll} U'_1 &= &\frac{2}{h^2} ( -U_1 + U_2 ),\\ U'_i &= &\frac{1}{h^2} ( U_{i+1} - 2U_i + U_{i-1} ) &\,\,\, 2 \leq i \leq d-1,\\ U'_d &= &\frac{2}{h^2} ( -U_d + U_{d-1} +hg(U_d) )\\ U(0) &= &u \end{array}\right.$$ donde $h > 0$ es un parámetro fijo y $g : {{\mathbb R}}\rightarrow {{\mathbb R}}$ es un término de reacción dado por $g(x) = (x^+)^p -x$ para $p>1$. Este tipo de sistemas surge como discretizaciones espaciales de ecuaciones de difusión con condiciones de frontera no lineales de Neumann. De hecho, puede probarse que cuando $h\to 0$ las soluciones de este sistema convergen a las solución de la EDP $$\left \{\begin{array}{rcll} {\partial}_t U(t,x) & = & {\partial}^2_{xx}U(t,x) & 0<x<1, 0\le t<T,\\ {\partial}_x U(t,0) & = & 0 & 0\le t <T,\\ {\partial}_x U(t,1) &= & g(U(t,1)) & 0\le t <T,\\ U(0,x) & = & U_0(x) & 0\le x \le 1. \end{array} \right.$$ La ecuación diferencial ordinaria puede escribirse como en forma matricial como $$dU = - \nabla S,$$ donde $S$ viene dado por $$S(u) = \frac{1}{2} \langle Au , u \rangle - \frac{2}{h}\Big(\frac{\;\;\;|u_d^+|^{p+1}}{p+1} - \frac{\,\,{u_d}^2}{2}\Big).$$ para cierta matriz $A \in {{\mathbb R}}^{d \times d}$ definida positiva. Este potencial $S$ satisface las mismas propiedades que su análogo infinito-dimensional en . Tiene exactamente dos puntos críticos, $\1:=(1,\dots,1)$ y el origen, ambos ellos hiperbólicos. El origen $\mathbf{0}$ es el único mínimo local de $S$ mientras que $\1$ es un punto de ensilladura. Más aún, se tiene una descomposición de ${{\mathbb R}}^d$ análoga a la de . Las perturbaciones estocásticas que vamos a considerar son de la forma $$dU^{{\varepsilon}} = - \nabla S dt + {\varepsilon}dW$$ par ${\varepsilon}>0$ pequeño y donde $W=(W_1, \dots,W_d)$ es un movimiento Browniano $d$-dimensional estándar. La solución $U^\varepsilon$ de esta EDOE conserva las propiedades de la solución de . Los resultados que podemos probar en este contexto son esencialmente los mismos que para la EDP , con la excepción de que la restricción $p < 5$ desaparece en este contexto puesto que la geometría del potencial $S$ puede manejarse con mayor facilidad al estar definido sobre un espacio finito-dimensional como lo es ${{\mathbb R}}^d$ (el potencial de estaba definido $C_D([0,1])$, un espacio infinito-dimensional). Por otro lado, la elección particular del término $g$ (con término no lineal acotado inferiormente) nos permite probar además que el fenómeno de blow-up se hace presente casi seguramente. **Teorema**. Para cualquier $u \in {{\mathbb R}}^d$ y $\varepsilon > 0$ tenemos $P(\tau^u_{\varepsilon}< +\infty) = 1$. La idea de la demostración de este resultado consiste en mostrar que, condicionado a no explotar, el sistema estocástico alcanza inexorablemente una región particular del espacio en donde uno puede probar que el fenómeno de blow-up ocurre con probabilidad total. A partir de esto se deduce inmediatamente que la ausencia de blow-up debe darse con probabilidad nula. Mostramos que el proceso alcanza esta región particular mediante comparación con un proceso de Ornstein-Ühlenbeck adecuado, mientras que la explosión casi segura ocurre en esa región se obtiene mediante técnicas de ecuaciones similares a las empleadas durante el Capítulo 2. Con respecto a los resultados restantes, la demostración de los mismos sigue muy de cerca las ideas presentadas en los capítulos anteriores para el problema infinito-dimensional (con quizás menos dificultades técnicas). Incluimos en este capítulo únicamente las partes del análisis que difieren de aquellas dadas en el marco anterior. Estas aparecen en la demostración de lo que sería el Teorema I en este contexto y en la construcción del dominio auxiliar $G$. Con respecto al Teorema I, el potencial en este contexto tiene una estructura ligeramente diferente al infinito-dimensional, lo cual no nos permite adaptar por completo las ideas desarrolladas en el Capítulo 2 y nos obliga por lo tanto a introducir algunas variantes de las técnicas de ecuaciones empleadas durante éste. Finalmente, para la construcción de $G$ explotamos el marco finito-dimensional de este nuevo problema para proponer una nueva construcción que no impone la restricción $p < 5$. A partir de aquí, pueden obtenerse los resultados restantes para todo $p > 1$. Introducción a la Parte II {#introducción-a-la-parte-ii .unnumbered} ========================== La mecánica estadística del equilibrio intenta explicar el comportamiento macroscópico de sistemas en equilibrio térmico en términos de la interacción microscópica entre su gran número de constituyentes. Como un ejemplo típico, uno podría tomar un material ferromagnético como el hierro: sus constituyentes son entonces los spins de los imanes elementales en los sitios de un cierto reticulado de cristal. O también podemos pensar en una aproximación discreta de un gas real, en cuyo caso los constituyentes son los números de partículas en las celdas elementales de cualquier partición del espacio. El objeto central en cualquiera de estos sistemas es el Hamiltoniano que describe la interacción entre los constituyentes. Éste determina la energía relativa entre configuraciones que difieren únicamente microscópicamente. Los estados de equilibrio con respecto a la interacción dada son descritos por las medidas de Gibbs asociadas. Éstas son medidas de probabilidad en el espacio de configuraciones con probabilidades condicionales dadas respecto a configuraciones fijas fuera de regiones acotadas. Dichas probabilidades condicionales son determinadas por el factor de Boltzmann: la exponencial de la temperatura inversa multiplicada por la energía relativa. Esto le permite a uno calcular, al menos en principio, esperanzas en equilibrio y funciones de correlación espacial siguiendo el formalismo de Gibbs estándar. El conjunto de medidas de Gibbs para una interacción dada es un simplex cuyos vértices llamamos medidas de Gibbs extremales. Estas medidas extremales son de mayor importancia puesto que describen los posibles macroestados (o fases de equilibrio) de tro sistema físico. En un estado tal, las observables macroscópicas no fluctúan mientras que la correlación entre observaciones locales hechas a larga distancia entre ellas decaen a cero. Un aspecto muy importante en el estudio de cualquier sistema de la mecánica estadística del equilibrio es determinar cuando existe más de un posible macroestado para el sistema, un fenómeno conocido como transición de fase. Por la estructura de simplex del conjunto de medidas de Gibbs, la ocurrencia de transición de fase para un sistema dado es equivalente a la existencia de medidas de Gibbs múltiples (no necesariamente extremales). Uno de los modelos más famosos y mejor entendidos de la mecánica estadística es el modelo de Ising ferromagnético estándar en el reticulado ${{\mathbb{Z}}}^2$. En este modelo, sobre cada sitio del reticulado se tiene una variable de spin que toma puede tomar solamente los valores $+$ y $-$. La interacción es entre vecinos más cercanos y tiende a alinear spins vecinos en la misma dirección. Mediante los argumentos ingeniosos formulados en primera instancia por Peierls en 1936, la transición de fase en este modelo puede entenderse a través de la inspección de configuraciones típicas de contornos, i.e. líneas quebradas que separan los dominios con spins $+$ y $-$, respectivamente. Para temperaturas bajas, existe una fase $+$ que es realizada por un mar infinito de spins $+$ con islas finitas de spins $-$ (que a su vez pueden contener lagos de spins $+$, y así sucesivamente). En términos de contornos, este panorama equivale a que haya únicamente finitos contornos rodeando cada sitio del reticulado. También se tiene una fase $-$ que verifica la descripción simétrica. Por otro lado, por encima de cierta temperatura crítica no existe ningún camino infinito que una vecinos más cercanos con el mismo spin. Por lo tanto, para este modelo la estructura geométrica de las configuraciones típicas está bien entendida (ver [@PS; @G] por ejemplo). En general, sin embargo, se sabe mucho menos, y mucho menos es cierto. Aún así, ciertos aspectos de este análisis geométrico tienen amplias aplicaciones, al menos en ciertos regímenes del diagrama de fases. Estos “ciertos regímenes” son, por un lado, el régimen de alta temperatura (o, en el contexto de gases, baja densidad) y, al otro extremo, el comportamiento a baja temperatura (o altas densidades). A temperaturas altas o baja densidad, todas las consideraciones termodinámicas están basadas en el hecho de que la entropía domina sobre la energía. Esto es, la interacción entre los tituyentes no es lo suficientemente efectiva para forzar un ordenamiento macroscópico del sistema. Como resultado, los constituyentes se comportan aproximadamente al azar, no muy influenciados por otros constituyentes que se encuentran lejos. Así, el comportamiento del sistema es casi el de un sistema libre con componentes independientes. Esto significa, en particular, que en el centro de una caja grande típicamente vamos a encontrar aproximadamente las mismas configuraciones sin importar qué condiciones de frontera sean impuestas fuera de dicha caja. A bajas temperaturas o densidades grandes (cuando la interacción es suficientemente fuerte), el panorama de arriba ya no es válido. En realidad, las características específicas de la interacción entrarán en juego y determinarán las cualidades específicas de los macroestados. En muchos casos, el comportamiento a baja temperatura puede ser descrito como una perturbación aleatoria de un estado fundamental, i.e. una configuración fija de energía mínima. Luego, a bajas temperaturas podemos esperar que las fases de equilibrio se realicen como una de estado fundamental determinística, perturbada por finitas islas aleatorias en donde la configuración difiere con dicho estado fundamental. Esto significa que el patrón del estado fundamental puede percolar a través del espacio hasta el finito. Una manera prominente de confirmar este panorama general es provista por la llamada teoría de Pirogov-Sinai, descrita en detalle en [@Z2]. Esencialmente, esta teoría introduce en primer lugar una noción generalizada de contornos que puede ser utilizada por una amplia gama de sistemas y luego da condiciones que garantizan cuando existen solamente finitos de estos contornos alrededor de cada sitio del reticulado. Cuando esto suceda, un panorama similar al del modelo de Ising puede obtenerse. Más allá de lo poderosa que sea como herramienta, una de las desventajas de la teoría de Pirogov-Sinai es que la mayoría de sus aplicaciones se apoyan fuertemente en la convergencia absoluta de ciertas expansiones, llamadas expansiones en aglomerados, y esta convergencia muchas veces depende de resultados combinatorios profundos. Así, la teoría de Pirogov-Sinai constituye en realidad (al menos hasta cierto grado) un enfoque más combinatorio que probabilístico para entender las fases de equilibrio de un sistema físico dado. Como el problema matemático mismo se encuentra formulado dentro de un marco probabilístico, uno puede ver que este enfoque quizás no sea el más natural posible. Más aún, debido a la absoluta convergencia de las expansiones involucradas, uno obtiene gratuitamente la analiticidad de las funciones de correlación con respecto a los parámetros del modelo (como lo son la temperatura inversa o la densidad de partículas). Aunque la analiticidad es una buena propiedad para tener, es también un síntoma de que este enfoque es quizás demasiado fuerte y no óptimo desde el punto de vista probabilístico. Como una alternativa, en [@FFG1] los autores proveen un enfoque nuevo al estudio de este tipo de sistemas, uno que es puramente probabilístico. En lugar de depender de las expansiones en aglomerados para probar que existe una medida de equilibrio que satisface el panorama de mar con islas descrito arriba, ellos realizan esta medida como la distribución estacionaria de una red de pérdida que puede ser estudiada utilizando herramientas estándar y nociones de modelos probabilísticos y procesos. En este contexto, la existencia de la medida de equilibrio está relacionada con la ausencia de percolación en un proceso de percolación orientada. Más aún, muestran que la dinámica converge exponencialmente rápido a la medida buscada, de manera que este enfoque es también valioso para propósitos de simulación. Para ser precisos, en su trabajo los autores consideran únicamente el modelo de contornos de Peierls para la interacción de Ising a baja temperatura, mientras que en [@FFG2] discuten como algunas de estas ideas pueden ser extendidas a otros modelos. En esta segunda parte de la tesis introducimos una familia general de sistemas, la clase de los *modelos diluidos*, y mostramos que los resultados principales en [@FFG1] pueden extenderse a esta familia más amplia. El marco de modelos diluidos encaja perfectamente con el rango de aplicabilidad de este nuevo enfoque: los modelos diluidos son, quizás, la familia más amplia de modelos a la cual la dinámica presentada en [@FFG1] pueda aplicarse. Este marco incluye tanto modelos discretos como continuos de manera unificada, pero es lo suficientemente concreto como para que aún sea posible obtener un criterio general para la existencia de la medida de equilibrio. Concretamente, en esta segunda parte vamos a desarrollar el siguiente plan: 1. Introducir la familia de modelos diluidos y mostrar que para cualquier elemento en esta familia podemos definir una dinámica con las características mostradas en [@FFG1]. 2. Utilizar la construcción de la dinámica para obtener un criterio general para la unicidad de medidas de Gibbs en modelos diluidos y estudiar propiedades de este único equilibrio, como la propiedad de mixing exponencial. 3. Mostrar que la ausencia de percolación en el proceso de percolación orientada implica la continuidad de las funciones de correlación con respecto a los parámetros del modelo. 4. Explotar el marco general de los modelos diluidos y las características de sus dinámicas asociadas para mostrar que, bajo condiciones adecuadas, las medidas de equilibrio de sistemas discretos convergen, cuando son apropiadamente escaladas, a la medida de equilibrio de sistemas continuos en el régimen de alta temperatura o baja densidad. 5. Combinar las ideas y resultados previos con el marco de la teoría de Pirogov-Sinai para obtener algunos resultados fuera del rango de convergencia de las expansiones en 6. Combinar las ideas y resultados previos con el marco de la teoría de Pirogov-Sinai para obtener un algoritmo de simulación perfecta para una clase medianamente grande de medidas de equilibrio en el régimen de baja temperatura o alta densidad. La Parte II está organizada de la siguiente manera. En el Capítulo 7 proveemos del marco teórico en donde se definen los modelos diluidos. El Capítulo 8 se enfoca en la definición de modelos diluidos y alguna de sus propiedades básicas. En este capítulo también adaptamos algunas nociones elementales de la mecánica estadística a este marco de trabajo. El capítulo siguiente esta destinado a mostrar lo restante de (i) y (ii) en el plan de arriba. Los items (iii) y (iv) son establecidos en los Capítulos 10 y 11, respectivamente. Finalmente, los items (v) y (vi) se establecen en el Capítulo 12. Introduction to Part II {#introduction-to-part-ii .unnumbered} ======================= The purpose of equilibrium statistical mechanics is to describe macroscopic behavior of systems in thermal equilibrium in terms of the microscopic interactions among the great number of elements which constitute them. The most common example is the one of some ferromagnetic material like iron: its elements are then the spins of elementary magnets situated at the various sites of a given crystal lattice. Or we may also consider a lattice approximation to a real gas, in which case the elements are the particle numbers inside each of the cells of a given partition of space. In any of these systems, the central object is the Hamiltonian which describes the microscopic interaction between its elements by determining energy between configurations which differ only microscopically. The equilibrium states with respect to the given interaction are specified by the so called Gibbs measures associated to the model. These are probability measures on the space of configurations with given conditional probabilities relative to fixed configurations outside of bounded regions. These conditional probabilities are given by the Boltzmann factor, i.e. the exponential of the inverse temperature times the relative energy of the configuration. The set of Gibbs measures for a given interaction is a simplex whose vertices we call extremal Gibbs measures. These are most important since they describe the possible macrostates (or equilibrium phases) of our physical system. In such a state, we have that macroscopic observables do not fluctuate and also that the correlation between local observations made far apart from each other decays to zero. A very important aspect in the study of any system in equilibrium statistical mechanics is determining whether there exists more than one possible macrostate for the system, a phenomenon known as phase transition. By the simplex structure of the set of Gibbs measures, the occurrence of phase transition for a given system is equivalent to the existence of multiple (not necessarily extremal) Gibbs measures. One of the most famous and better understood models in statistical mechanics is the standard ferromagnetic Ising model on the square lattice. In this model, at each site of the lattice we have a spin variable which can take only two values, $+$ or $-$. The interaction is of nearest-neighbor and tends to align neighboring spins in the same direction. In 1936, Peierls showed that the phase transition in this model can be understood by looking at the typical configurations of contours: finite circuits separating the domains with plus and minus spins, respectively. For low temperatures, there exists a plus phase which is realized by an infinite sea of plus spins with finite islands of minus spins (which may further contain lakes of plus spins, and so on). In terms of contours, this picture corresponds to there being only finitely many contours surrounding any site in the lattice. One also has a minus phase verifying the symmetric description. On the other hand, above a certain critical temperature there is no infinite path joining nearest neighbors with the same spin value. Thus, for this model the geometric structure of typical configurations is well understood (see [@PS; @G] for example). In general, however, much less is known and a similar description may not always hold. Still, certain aspects of the geometric analysis performed by Peierls have wide applications, at least in certain regimes of the phase diagram. These regimes are, on the one hand, the high-temperature (or low-density of particles in a lattice gas setting) regime and, on the other hand, the low-temperature behavior (or high-density of particles). At high temperatures, the behavior is dictated by the fact that entropy dominates over energy. That is, the interaction between the elements of our system is not strong enough to enforce a macroscopic ordering of it. As a consequence, the elements of our system behave more or less at random, without being much influenced by other elements which are far apart. Thus, the behavior of the system is almost like that of a free system with independent components. In particular, we have that deep inside a large region we will typically encounter more or less the same configurations no matter which boundary conditions are imposed outside this region. However, at low temperatures (i.e. when the interaction is strong enough), the scenario described above no longer holds. Instead, the particular characteristics of the interaction will come into play and determine the specific features of the low temperature macrostates. In many cases, the low temperature behavior can be seen as a random perturbation of a ground state: a fixed configuration having minimal energy. Therefore, one expects, for sufficiently low temperatures, the equilibrium phases to be obtained as a deterministic ground state configuration perturbed by finite random islands on which the configuration disagrees with the corresponding ground state. In particular, the configuration pattern imposed by the ground state percolates in space to infinity. One way in which to show this general picture is provided by Pirogov-Sinai theory, described in detail in [@Z2]. Essentially, this theory first introduces a generalized notion of contours which can be used for a wide range of systems and then gives conditions which guarantee when are there only finitely many of such contours surrounding each site in the lattice. Whenever this is the case, a similar picture to the one for the Ising model can be obtained. As powerful a tool as it may be, one of the disadvantages of Pirogov-Sinai theory is that most of its applications rely heavily on the absolute convergence of certain expansions, known as the cluster expansions, and this convergence often depends on deep combinatorial results. Thus, in fact Pirogov-Sinai theory constitutes (at least to some degree) more of a combinatorial approach to understanding the equilibrium phases of a given physical system rather than a probabilistic one. Since the mathematical problem itself is posed within the probabilistic framework, one can see that this approach is perhaps not the most natural one to have. Furthermore, due to the absolute convergence of the expansions involved, one obtains for free the analyticity of correlation functions with respect to the parameters in the model (e.g. inverse temperature or density of particles). Though analyticity is a very nice property to have, it is also a symptom that this approach is perhaps too strong and not optimal from the probabilistic point of view. As an alternative, in [@FFG1] the authors provide a fresh new approach to study this type of systems, one which is purely probabilistic. Instead of relying on cluster expansions to prove that there exists an equilibrium measure satisfying the sea with islands picture described above, they realize this measure as the stationary distribution of a loss network dynamics that can be studied using standard tools and notions from probabilistic models and processes. In this context, the existence of the equilibrium measure is related to the absence of percolation in an oriented percolation process. Furthermore, they show that the dynamics converges exponentially fast to the desired measure, so that this approach is also valuable for simulation purposes. Strictly speaking, in their work the authors only consider the model of Peierls contours for the Ising interaction at low temperature (i.e. low density of contours), while in [@FFG2] they discuss how some of these ideas can be extended to other models in the low density regime. In this second part of the thesis we introduce a general family of systems called and show that the main results obtained in [@FFG1] can be extended to this broader family. The framework of diluted models perfectly fits the range of applicability of this new approach: diluted models are, perhaps, the broadest family of models for which the dynamics presented in [@FFG1] may be applied. This framework covers both discrete and continuum systems in an unified way, while remaining concise enough so that a general criterion for the existence of the equilibrium measure can still be obtained. Concretely, in this second part of the thesis we carry out the following plan: 1. Introduce the family of diluted models and show that for any element in this family we can define a dynamics with the characteristics shown in [@FFG1]. 2. Use the construction of the dynamics to obtain a general criterion for uniqueness of Gibbs measures in diluted models and study properties of this unique equilibrium, such as exponential mixing. 3. Show that the absence of percolation in the oriented percolation process implies the continuity of correlation functions with respect to the parameters of the model. 4. Exploit the general framework of diluted models and the characteristics of their associated dynamics to show that, under suitable conditions, equilibrium measures of discrete systems converge, when properly rescaled, to equilibrium measures of continuum systems in the high temperature (or low density) regime. 5. Combine the previous ideas and results with the framework of Pirogov-Sinai theory to obtain results outside the range of convergence of cluster expansions. 6. Combine the previous ideas and results with the framework of Pirogov-Sinai theory to obtain a perfect simulation algorithm for a large class of equilibrium measures in the low temperature (or high density) regime. This extends the previous results in [@FFG1] obtained for the high temperature (or low density) regime. Part II is organized as follows. In Chapter 7 we provide the theoretical setting in which diluted models are defined. Chapter 8 focuses on the definition of diluted models and some of their basic properties. In this chapter we also adapt some elementary notions from statistical mechanics to this framework. The following chapter is devoted to showing the remainder of (i) and (ii) in the plan above. Items (iii) and (iv) are established in Chapters 10 and 11, respectively. Finally, items (v) and (vi) are settled in Chapter 12. Preliminaries ============= Since we intend the class of diluted models to include discrete and continuum models, we are interested in adopting a general framework which a priori makes no distinction between both types of systems. The correct framework is that of particle configurations, which we introduce now. We refer to [@K; @DVJ2; @DVJ1] for further details. Particle configurations ----------------------- Throughout this second part we fix two locally compact complete separable metric spaces: the *allocation space* or *lattice* $(S,d_S)$ and the *animal set* or *spin set* $(G,d_G)$. Typical examples of allocation spaces include $S={{\mathbb{Z}}}^d$ or $S={{\mathbb R}}^d$ for some $d \in {{\mathbb N}}$, whereas the spin set can range from finite sets such as $G=\{+,-\}$, to uncountable sets such as $G=S^{d-1}$. The product space $S \times G$ is also a locally compact complete separable metric space if endowed with the product metric $d_S + d_G$. For convenience, we shall denote an element $(x,\gamma) \in S \times G$ simply by $\gamma_x$, which we interpret as an animal $\gamma$ positioned at location $x$. Given a metric space $(X,d)$ we write: - ${{\mathcal B}}_X$ for the class of all Borel subsets of $X$. - ${{\mathcal B}}^0_X$ for the class of all Borel subsets of $X$ with compact closure. Let $\xi$ be a measure on $(S\times G, {{\mathcal B}}_{S \times G})$. - $\xi$ is said to be a *Radon measure* if $\xi(B) < +\infty$ for every $B \in {{\mathcal B}}^0_{S \times G}$. - $\xi$ is said to be a *particle configuration* if $\xi(B) \in {{\mathbb N}}_0$ for every $B \in {{\mathcal B}}^0_{S \times G}$. A Radon measure $\xi$ on $(S\times G, {{\mathcal B}}_{S \times G})$ is a particle configuration if and only if there exist a countable set $Q_\xi \subseteq S \times G$ and $m_\xi: Q_\xi \to {{\mathbb N}}$ such that $$\label{standardrep} \xi = \sum_{\gamma_x \in Q_\xi} m_\xi(\gamma_x) \delta_{\gamma_x}$$ with $\delta_{\gamma_x}$ being the Dirac measure centered at $\gamma_x$. We call the *standard representation* of the particle configuration $\xi$. Thus any particle configuration on $S \times G$ may also be regarded as a locally finite point configuration on $S \times G$ where the points are allowed to have varied multiplicities. we shall often view particle configurations in this manner if convenient. With this in mind, we define the *support* of $\xi$ as $$[\xi]:=\{ (\gamma_x,i) \in (S\times G)\times {{\mathbb N}}: \gamma_x \in Q_\xi \text{ and }i \leq m_\xi(\gamma_x)\}$$ which is merely the set of points that constitute $\xi$ as a point configuration counted with their respective multiplicities. If we only wish to consider the set of points in $\xi$ without regard for their multiplicities then we shall write $\langle \xi \rangle$, i.e. the projection onto $S \times G$ of $[\xi]$. A measure $\xi$ on $(S\times G, {{\mathcal B}}_{S \times G})$ is said to be of *locally finite allocation* if it satisfies $\xi(\Lambda \times G) < +\infty$ for every $\Lambda \in {{\mathcal B}}^0_S$. $\,$ - We shall write $\mathcal{N}(S\times G)$ to denote the space of all particle configurations on $S \times G$ which are of locally finite allocation. - Given $\Lambda \in {{\mathcal B}}^0_S$ we write $\mathcal{N}(\Lambda \times G)$ to denote the space of all particle configurations of locally finite allocation which are supported on $\Lambda \times G$. The space $\mathcal{N}(S\times G)$ of particle configurations ------------------------------------------------------------- ### Restriction and superposition of particle configurations Given $\xi \in \mathcal{N}(S \times G)$ and $A \in {{\mathcal B}}_{S \times G}$ we define the *restriction* of $\xi$ to $A$ as the particle configuration $\xi_{A}$ given for every $B \in {{\mathcal B}}_{S \times G}$ by the formula $$\xi_A (B) = \xi (A \cap B).$$ Equivalently, if $\xi = \sum_{\gamma_x \in Q_\xi} m(\gamma_x) \delta_{\gamma_x}$ we define $\xi_A$ through the standard representation $$\xi_A = \sum_{\gamma_x \in Q_\xi \cap A} m(\gamma_x) \delta_{\gamma_x}.$$ Given $\sigma,\eta \in \mathcal{N}(S \times G)$ we define their *superposition* as the particle configuration $\sigma \cdot \eta$ given for every $B \in {{\mathcal B}}_{S \times G}$ by the formula $$\sigma \cdot \eta (B) = \sigma(B) + \eta (B).$$ \[obsiden\] Given $\Lambda \in {{\mathcal B}}^0_S$ there is a natural identification between $\mathcal{N}(S \times G)$ and $\mathcal{N}(\Lambda \times G) \times \mathcal{N}(\Lambda^c \times G)$ given by the restriction and superposition operations, i.e. we have that the applications $$\begin{array}{ccc} \mathcal{N}(S \times G) \overset{r}{\longrightarrow} \mathcal{N}(\Lambda \times G)\times \mathcal{N}(\Lambda^c \times G) & &\mathcal{N}(\Lambda \times G)\times \mathcal{N}(\Lambda^c \times G) \overset{s}{\longrightarrow} \mathcal{N}(S \times G)\\ \hspace{-0.35cm}\xi \longmapsto (\xi_{\Lambda \times G},\xi_{\Lambda^c \times G}) & \text{ and }& \hspace{2.5cm} (\sigma,\eta) \longmapsto \sigma\cdot \eta. \end{array}$$ are bijections and have each other as their respective inverse. ### Measurable structure The space $\mathcal{N}(S \times G)$ can be endowed with a measurable space structure by considering the $\sigma$-algebra ${{\mathcal F}}$ generated by the counting events $$\label{salgebra} {{\mathcal F}}= \sigma\left( \{ \xi \in \mathcal{N}(S \times G) : \xi(B) = k \} : k \in {{\mathbb N}}_0 \text{ and } B \in {{\mathcal B}}^0_{S\times G} \right).$$ Furthermore, for any $A \in {{\mathcal B}}_{S\times G}$ we define ${{\mathcal F}}_A$, the $\sigma$-*algebra of events occurring in* $A$, by considering only the counting events inside $A$, i.e. $${{\mathcal F}}_A = \sigma\left( \{ \xi \in \mathcal{N}(S \times G) : \xi(B) = k \} : k \in {{\mathbb N}}_0 \text{ and } B \in {{\mathcal B}}^0_A \right).$$ Alternatively, if for every $B \in {{\mathcal B}}_{S\times G}$ we define the respective counting random variable $N_B : \mathcal{N}(S \times G) \to {{\mathbb N}}_0$ by the formula $N_B(\eta) = \eta(B)$ then for each $A \in {{\mathcal B}}_{S\times G}$ the $\sigma$-algebra ${{\mathcal F}}_A$ can also be defined as the one generated by the counting random variables inside $A$, i.e. $${{\mathcal F}}_A = \sigma\left( N_B : B \in {{\mathcal B}}^0_A\right).$$ For any $\Lambda \in {{\mathcal B}}^0_s$ the identification $\mathcal{N}(S \times G)=\mathcal{N}(\Lambda \times G) \times \mathcal{N}(\Lambda^c \times G)$ on Remark \[obsiden\] is in fact a measurable isomorphism when endowing each space with the $\sigma$-algebras ${{\mathcal F}}$ and ${{\mathcal F}}_{\Lambda \times G} \otimes {{\mathcal F}}_{\Lambda^c \times G}$, respectively. $\,$ 1. A function $f: \mathcal{N}(S\times G) \to {{\mathbb R}}$ is called a *local function* if there exists $\Lambda \in {{\mathcal B}}^0_S$ such that $f$ is ${{\mathcal F}}_{\Lambda \times G}$-measurable. 2. An event $A \in {{\mathcal F}}$ is called a *local event* if $\mathbbm{1}_A$ is a local function. 3. Given a function $f: \mathcal{N}(S\times G) \to {{\mathbb R}}$ we define its *measurability support* as $$\Lambda_f = \bigcap_{\Lambda \in \mathcal{D}_f} \overline{\Lambda}$$ where $\mathcal{D}_f = \{ \Lambda \in {{\mathcal B}}_S : f \text{ is }{{\mathcal F}}_{\Lambda \times G}\text{-measurable}\}$. That is, $\Lambda_f$ is the $\Lambda \in {{\mathcal B}}_S$ such that $f$ is ${{\mathcal F}}_{\Lambda \times G}$-measurable. Notice that if $f$ is local then $\Lambda_f \in {{\mathcal B}}^0_S$. Notice that a function $f: \mathcal{N}(S\times G) \to {{\mathbb R}}$ is ${{\mathcal F}}_{\Lambda\times G}$-measurable if and only if $f(\sigma)=f(\eta)$ whenever $\sigma,\eta \in \mathcal{N}(S \times G)$ are such that $\sigma_{\Lambda \times G}=\eta_{\Lambda \times G}$. ### Topological structure The space $\mathcal{N}(S \times G)$ can also be endowed with a topological structure. We think of any two particle configurations $\xi,\eta$ as close to each other whenever the particles in $\xi$ lying inside some sufficiently large compact set $K$ can be matched with nearby particles of $\eta$ and viceversa. The precise definitions are given below. Given $\delta > 0$ and $\xi,\eta \in \mathcal{N}(S)$ we say that $\xi$ is $\delta$-*embedded* in $\eta$ if there exists an injective application $p:[\xi] \to [\eta]$ such that $d\left( \pi_{S\times G}(x,i) , \pi_{S\times G}(p(x,i))\right) < \delta$ for each $(x,i) \in [\xi]$, where $\pi_{S\times G} : (S\times G) \times {{\mathbb N}}\to S$ is the projection onto $S \times G$ and $d=d_S + d_G$ is the metric on $S\times G$. We denote it by $\xi \preceq_{\delta} \eta$. Given a particle configuration $\xi \in \mathcal{N}(S \times G)$, a compact set $K \subseteq S \times G$ and $\delta > 0$ we define the $(K,\delta)$-neighborhood of $\xi$ by the formula $$(\xi)_{K,\delta} = \{ \eta \in \mathcal{N}(S \times G) : \xi_K \preceq_\delta \eta \text{ and }\eta_K \preceq_\delta \xi \}.$$ We define the *vague topology* on $\mathcal{N}(S \times G)$ as the one generated by the basis $$\mathfrak{B} = \{ (\xi)_{K,\delta} : \xi \in \mathcal{N}(S \times G) , K \subseteq S \times G\text{ compact and }\delta > 0\}.$$ It can be shown that $\mathcal{N}(S \times G)$ admits a metric which is consistent with the vague topology under which it is complete and separable. The $\sigma$-algebra ${{\mathcal F}}$ defined in is actually the Borel $\sigma$-algebra given by the vague topology on $\mathcal{N}(S \times G)$. Poisson processes on $S \times G$ --------------------------------- We shall call any random element of $\mathcal{N}(S \times G)$ a *point process* on $S \times G$. Throughout this part we shall work with many different point processes on $S \times G$, all of them related in one way or another to the Poisson point process, which we define below. Let $\nu$ be a measure on $(S \times G,{{\mathcal B}}_{S \times G})$ with locally finite allocation. $\nu$ is defined as the unique measure $\pi^\nu$ on $\mathcal{N}(S \times G)$ which satisfies $$\pi^\nu ( \{ \xi \in \mathcal{N}(S \times G) : \xi(B_i) = k_i \text{ for all }i=1,\dots,n \} ) = \prod_{i=1}^n \frac{e^{-\nu(B_i)} \left(\nu(B_i)\right)^{k_i}}{k_i!}$$ for all $k_1,\dots,k_n \in {{\mathbb N}}_0$, disjoint $B_1,\dots,B_n \in {{\mathcal B}}^0_{S \times G}$ and $n \in {{\mathbb N}}$. \[defiPoissonasd\] Let $\nu$ be a measure on $(S \times G,{{\mathcal B}}_{S \times G})$ with locally finite allocation. A point process $X$ on $S \times G$ is called a *Poisson process* with intensity measure $\nu$ if it is distributed according to $\pi^\nu$, i.e. for every $n \in {{\mathbb N}}$ and all choices of disjoint sets $B_1,\dots,B_n \in {{\mathcal B}}^0_{S \times G}$ the random variables $X(B_1),\dots,X(B_n)$ are independent and have a Poisson distribution with respective means $\nu(B_1),\dots,\nu(B_n)$. It follows from Definition \[defiPoissonasd\] that if $X$ is a and we consider $\Lambda \in {{\mathcal B}}^0_S$ then, conditioned on the event $\{X(\Lambda \times G) = n\}$, the locations of these $n$ particles inside $\Lambda \times G$ are independent and distributed according to $\frac{\nu}{\nu(\Lambda \times G)}$. The next proposition found in [@M2 Proposition 3.1] generalizes this idea to obtain a convenient formula for the integral of functions with respect to the restricted Poisson measures. For any $\Lambda \in {{\mathcal B}}^0_S$ and any bounded nonnegative $f: \mathcal{N}(\Lambda \times G) \to {{\mathbb R}}$ we have the formula $$\label{poisson} \int f(\sigma) d\pi^\nu_{\Lambda}(\sigma) = \sum_{n=0}^\infty \frac{e^{-\nu(\Lambda \times G)}}{n!} \int_{(\Lambda \times G)^n} f\left(\sum_{i=1}^n \delta_{\gamma_x^i}\right) d\nu^n(\gamma_x^1,\dots,\gamma_x^n)$$ where $\nu^n$ denotes the $n$-fold product measure of $\nu$. In our work we shall, among other things, establish limit theorems for point processes. Therefore, we shall require a notion of convergence which is appropriate for our purposes. The most familiar notion available is that of convergence in distribution. We say that a sequence $(X_n)_{n \in {{\mathbb N}}}$ of point processes on $S \times G$ converges *in distribution* to a point process $X$ on $S \times G$ if $$\lim_{n \rightarrow +\infty}{{\mathbb E}}(f(X_n)) = {{\mathbb E}}(f(X))$$ for every bounded continuous function $f:\mathcal{N}(S \times G) \to {{\mathbb R}}$. We denote it by $X_n \overset{d}{\longrightarrow} X_n$. At some points throughout our work the use of local functions shall be much more natural than that of continuous ones. Under such circumstances we shall adopt instead the notion of local convergence for point processes, which we introduce next. \[localconvergence\] We say that a sequence $(X_n)_{n \in {{\mathbb N}}}$ of point processes on $S \times G$ converges *locally* to a point process $X$ on $S \times G$ if $$\lim_{n \rightarrow +\infty}{{\mathbb E}}(f(X_n)) = {{\mathbb E}}(f(X))$$ for every bounded local function $f:\mathcal{N}(S \times G) \to {{\mathbb R}}$. We denote it by $X_n \overset{loc}{\longrightarrow} X_n$. It is important to notice that in most cases local functions need not be continuous. Therefore, in general the notions of local convergence and convergence in distribution do not coincide. Nevertheless, since local functions are always dense in the space of uniformly continuous functions with the supremum norm, we get the following result. Local convergence implies convergence in distribution. Resumen del Capítulo 7 ---------------------- En este primer capítulo de la segunda parte introducimos el marco teórico sobre el cuál definiremos todos los modelos que nos interesará estudiar. Todos estos modelos serán casos particulares de configuraciones de partículas aleatorias. Dado un espacio de posiciones $S$ y otro de spines $G$, una configuración de partículas $\xi$ en $S \times G$ es una medida sobre $S \times G$ que admite la representación $$\xi = \sum_{(x,\gamma) \in Q_\xi} m_\xi (x,\gamma) \delta_{(x,\gamma)}$$ para cierto conjunto numerable $Q_\xi \subseteq S \times G$ sin puntos de acumulación y una función $m_\xi : Q_\xi \rightarrow {{\mathbb N}}$. Denotamos por $\mathcal{N}(S \times G)$ al espacio de aquellas configuraciones de partículas $\xi$ en $S \times G$ que satisfacen que $Q_\xi \cap (\Lambda \times G)$ es finito para todo subconjunto acotado $\Lambda$ de $S$. El espacio $\mathcal{N}(S \times G)$ tiene una estructura de espacio medible bajo la $\sigma$-álgebra generada por los eventos de conteo, i.e. $${{\mathcal F}}= \sigma\left( \{ \xi \in \mathcal{N}(S \times G) : \xi(B) = k \} : k \in {{\mathbb N}}_0 \text{ y } B \subseteq S \text{ boreliano acotado} \right).$$ También existe una topología natural en este espacio, la topología vaga, que es la generada por la base $$\mathfrak{B} = \{ (\xi)_{K,\delta} : \xi \in \mathcal{N}(S \times G) , K \subseteq S \times G\text{ compacto y }\delta > 0\},$$ donde el entorno $(\xi)_{K,\delta}$ viene dado por $$(\xi)_{K,\delta} = \{ \eta \in \mathcal{N}(S \times G) : \xi_K \preceq_\delta \eta \text{ and }\eta_K \preceq_\delta \xi \}.$$ y cuando, dadas dos configuraciones de partículas $\xi$ y $\eta$, por $\xi_K \preceq_\delta \eta$ entendemos que existe una correspondencia inyectiva (teniendo en cuenta la multiplicidad) entre las partículas de $\xi$ dentro de $K$ y las de $\eta$ tal que las partículas correspondidas están a distancia menor que $\delta$ entre sí. Puede verse que esta topología es metrizable y que así $\mathcal{N}(S \times G)$ resulta completo y separable. Provistos de este marco teórico, podemos definir el más básico de los modelos de interés que es el Proceso de Poisson. Dada una medida $\nu$ en $S \times G$ con $\nu(\Lambda \times G) < +\infty$ para todo $\Lambda \subseteq S$ acotado, decimos que una configuración de partículas aleatoria $X$ es un proceso de Poisson en $S \times G$ si 1. Para cada $B \subseteq S \times G$ boreliano la variable aleatoria $X(B)$ tiene distribución Poisson de parámetro $\nu(B)$ 2. Si $B_1,\dots,B_n$ son borelianos disjuntos de $S \times G$ entonces las variables aleatorias $X(B_1),\dots,X(B_n)$ son independientes. Todos los demás modelos que estudiemos en esta segunda parte se podrán obtener, de una manera u otra, a partir de un proceso de Poisson adecuado. Por último, introducimos las nociones de convergencia en distribución y local para configuraciones de partículas aleatorias en $S \times G$. Decimos que una sucesión $(X_n)_{n \in {{\mathbb N}}}$ de configuraciones de partículas aleatorias converge a otra $X$ en distribución si $$\lim_{n \rightarrow +\infty}{{\mathbb E}}(f(X_n)) = {{\mathbb E}}(f(X))$$ para toda función $f:\mathcal{N}(S \times G) \to {{\mathbb R}}$ continua y acotada, mientras que decimos que lo hace localmente si vale lo anterior para toda función acotada local (i.e., que depende del estado de la configuración sólo dentro de una región acotada) en lugar de continua. Puede verificarse que la convergencia local implica la convergencia en distribución. Diluted models {#him} ============== Definition of a diluted model ----------------------------- In this section we formally define the models which we shall study throughout this work. Diluted models on $\mathcal{N}(S \times G)$ are always defined by specifying two characteristic elements: a measure $\nu$ on $S\times G$ called the *intensity measure* and a family $H$ of measurable functions $$H_{\Lambda} : \mathcal{N}(\Lambda \times G) \times \mathcal{N}(S\times G) \to [0,+\infty]$$ called the *local Hamiltonians*, both satisfying the conditions on Assumptions \[assump\] below. Essentially, the measure $\nu$ will be responsible for the way in which particles in $G$ are distributed throughout the location space $S$ while the Hamiltonians $H_{\Lambda}$ will determine how these particles interact among themselves. Also, the family $H$ shall be referred to as the *Hamiltonian*. We establish the following applications: 1. For $\Lambda \in {{\mathcal B}}^0_S$ and $\eta \in \mathcal{N}(S\times G)$ we define $H_{\Lambda|\eta}: \mathcal{N}(\Lambda \times G) \to [0,+\infty]$ by the formula $$H_{\Lambda|\eta} = H_{\Lambda}( \cdot , \eta).$$ $H_{\Lambda|\eta}$ shall be called the *local Hamiltonian on $\Lambda$ with boundary condition $\eta$*. 2. For $\eta \in \mathcal{N}(S\times G)$ we define $\Delta E_{\eta} : S \times G \to [-\infty,+\infty]$ by the formula $$\Delta E_{\eta} (\gamma_x) = H_{\{x\}|\eta}( \eta_{\{x\}} + \delta_{\gamma_x} ) - H_{\{x\}|\eta}(\eta_{\{x\}}).\footnote{Here we adopt the convention $\infty - \infty = \infty$.}$$ $\Delta E_{\eta}$ shall be called the *energy leap function with base configuration $\eta$*. It represents the energy cost for the model to add the particle $\gamma_x$ to its current configuration whenever the latter is given by $\eta$. \[assump\] We assume that the pair $(\nu,H)$ satisfies the following: 1. *Locally finite allocation*. For every $\Lambda \in \mathcal{B}^0_S$ the measure $\nu$ satisfies $\nu(\Lambda \times G) < +\infty$. 2. *Diluteness condition*. $H_{\Lambda|\eta}( \emptyset ) = 0$ for every $\Lambda \in {{\mathcal B}}^0_S$ and $\eta \in \mathcal{N}(S \times G)$. 3. *Bounded energy loss*. $$-\infty < \Delta E := \inf_{\substack{ \eta \in \mathcal{N}(S \times G) \\ \gamma_x \in S \times G }} \Delta E_{\eta} (\gamma_x) < +\infty.$$ 4. *Integrable local interaction range*. If we define a relation $\rightharpoonup$ on $S\times G$ by setting $$\tilde{\gamma}_y \rightharpoonup \gamma_x \Longleftrightarrow \exists\,\, \eta \in \mathcal{N}(S\times G) \text{ with }\Delta E_{\eta}(\gamma_x) \neq \Delta E_{\eta + \delta_{\tilde{\gamma}_y}} (\gamma_x)$$ then for every $B \in {{\mathcal B}}_{S \times G}$ the *interaction range* of $B$ defined as the set $$I(B)= \{ \tilde{\gamma}_y \in S \times G : \exists\,\, \gamma_x \in B \text{ such that } \tilde{\gamma}_y \rightharpoonup \gamma_x \}$$ is measurable and each $\Lambda \in \mathcal{B}^0_S$ satisfies $\nu(I(\Lambda \times G)) < +\infty$. 5. *Consistent Hamiltonian*. 1. Given $\Delta, \Lambda \in {{\mathcal B}}^0_S$ such that $\Delta \subseteq \Lambda$ and any $\eta \in \mathcal{N}(S\times G)$ $$H_{\Lambda|\eta}(\sigma) = H_{\Delta|\sigma_{(\Lambda - \Delta) \times G} \cdot \eta_{\Lambda^c \times G}}(\sigma_{\Delta \times G}) + H_{\Lambda - \Delta| \emptyset_{\Lambda \times G}\cdot \eta_{\Lambda^c \times G}}(\sigma_{(\Lambda - \Delta) \times G})$$ for every $\sigma \in \mathcal{N}(\Lambda \times G)$. 2. For every $\Lambda \in {{\mathcal B}}^0_S$, $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$ $$H_{\Lambda|\eta}(\sigma + \delta_{\gamma_x}) = H_{\Lambda|\eta}(\sigma) + \Delta E_{\sigma_{\Lambda \times G} \cdot \eta_{\Lambda^c \times G}} (\gamma_x).$$ 6. *Interaction measurability of the Hamiltonian*. 1. For every $\Lambda \in {{\mathcal B}}^0_S$ the application $H_\Lambda$ is $({{\mathcal F}}_{\Lambda \times G} \otimes {{\mathcal F}}_{(\Lambda^c \times G) \cap I(\Lambda \times G)})$-measurable. 2. For every $\gamma_x \in S \times G$ the application $\eta \mapsto \Delta E_{\eta} (\gamma_x)$ is ${{\mathcal F}}_{I(\{\gamma_x\})}$-measurable. In what follows we shall refer to the different diluted models by the pair $(\nu,H)$ which defines them. We shall say that a given diluted model is of *bounded local interaction range* whenever $I(\Lambda \times G)$ is bounded for every $\Lambda \in \mathcal{B}^0_S$. Also, whenever $\tilde{\gamma}_y \rightharpoonup \gamma_x$ we shall say that $\tilde{\gamma}_y$ has an . Notice that this impact relation $\rightharpoonup$ need not be symmetric. \[defibgd\] Given $\Lambda \in \mathcal{B}^0_S$ and a particle configuration $\eta \in \mathcal{N}( S \times G)$ we define the corresponding *Boltzmann-Gibbs distribution* $\mu_{\Lambda|\eta}$ by the formula $$\label{Gibbs1} \mu_{\Lambda|\eta} = \omega_{\Lambda|\eta} \times \delta_{\eta_{\Lambda^c}}$$ where we identify $\mathcal{N}(S \times G)$ with $\mathcal{N}( \Lambda \times G) \times \mathcal{N}(\Lambda^c \times G)$ and $\omega^\eta_\Lambda$ is the probability measure on $\mathcal{N}(\Lambda \times G)$ defined through the relation $$d\omega_{\Lambda|\eta} = \frac{e^{-H_{\Lambda|\eta}}}{Z_{\Lambda|\eta}} d\pi^\nu_\Lambda$$ with $\pi^\nu_\Lambda$ denoting the Poisson measure on $\mathcal{N}(\Lambda \times G)$ of intensity measure $\nu_{\Lambda \times G}$ and $$Z_{\Lambda|\eta}=\displaystyle{\int_{\mathcal{N}(\Lambda \times G)} e^{-H_{\Lambda|\eta}(\sigma)} d\pi^\nu_\Lambda(\sigma)}$$ serving as a normalizing constant. Notice that due to Assumptions \[assump\] we have $$Z_{\Lambda|\eta} \geq \pi^\nu ( N_{\Lambda \times G} = 0 ) = e^{- \nu(\Lambda \times G) } > 0$$ and thus $\omega_{\Lambda|\eta}$ is well defined. \[tradition\] We would like to point out that for the discrete setting, i.e. when $S={{\mathbb{Z}}}^d$, this is not the standard way in which most lattice systems are defined. Traditionally, discrete systems are defined on the configuration space $G^{{{\mathbb{Z}}}^d}$ for a given spin set $G$, so that in each configuration all sites in the lattice are assigned exactly one spin. In this context, a model is regarded as diluted whenever the element $0$ belongs to the spin set $G$ and a site with $0$-spin is understood as an empty site, i.e. devoid of any particles. Furthermore, for any given $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in G^{{{\mathbb{Z}}}^d}$, the corresponding Boltzmann-Gibbs distribution $\mu_\Lambda^\eta$ (notice the difference in notation) in this context is usually defined by the formula $$\mu_{\Lambda}^\eta(\sigma) = \frac{\mathbbm{1}_{\{\sigma_{\Lambda^c} \equiv \eta_{\Lambda^c}\}}}{Z_{\Lambda}^\eta} e^{- \beta \sum_{B \subseteq {{\mathbb{Z}}}^d : B \cap \Lambda \neq \emptyset} \Phi_B(\sigma)}$$ where $\beta > 0$ is a parameter known as the *inverse temperature*, $\sigma_{\Lambda^c}$ indicates the restriction of $\sigma$ to the region $\Lambda^c$ and also for every $B \subseteq {{\mathbb{Z}}}^d$ the function $\Phi_B : G^{{{\mathbb{Z}}}^d} \rightarrow \overline{{{\mathbb R}}}$ depends only on the values of spins inside $B$. The family $\Phi = (\Phi_B)_{B \subseteq {{\mathbb{Z}}}^d}$ is known as the *potential*. Nonetheless, we prefer to adopt the definition given in Definition \[defibgd\] as it will allow us to treat discrete and continuum systems in the same manner. This will be necessary for Chapter 11, where we study the convergence of discrete systems towards continuum ones. The Boltzmann-Gibbs distribution $\mu_{\Lambda|\eta}$ is meant to describe the local behavior of the model inside the bounded set $\Lambda$ once the configuration outside $\Lambda$ has been fixed as $\eta$. It then seems natural to expect Boltzmann-Gibbs distributions to exhibit some sort of consistency among themselves. This is indeed true, as our next proposition shows. For $\Delta \subseteq \Lambda \in \mathcal{B}^0_S$ and $\eta \in \mathcal{N}(S \times G)$ we have the following consistency property: $$\label{consistencia} \mu_{\Lambda|\eta}(A) = \int_{\mathcal{N}(S\times G)} \mu_{\Delta|\xi} (A) d\mu_{\Lambda|\eta} (\xi)$$for every $A \in {{\mathcal F}}$. Notice that if we identify $\mathcal{N}(S \times G)$ with $\mathcal{N}(\Delta \times G) \times \mathcal{N}((\Lambda - \Delta)\times G) \times \mathcal{N}(\Lambda^c \times G)$ and for every $\xi \in \mathcal{N}(S \times G)$ we write $\xi = (\xi^{(1)},\xi^{(2)},\xi^{(3)})$ accordingly then, since we have $\pi^\nu = \pi^\nu_{\Delta} \times \pi^\nu_{\Lambda -\Delta} \times \pi^\nu_{\Lambda^c}$, using the Fubini-Tonelli theorem and (i) in the the consistent Hamiltonian property we obtain $$\begin{aligned} \mu_{\Lambda|\eta}(A) &= \frac{1}{Z_{\Lambda|\eta}} \int_{\mathcal{N}\left( (\Lambda - \Delta) \times G \right)} \int_{\mathcal{N}(\Delta \times G)} e^{-H_{\Lambda|\eta}(\xi^{(1)}\cdot\xi^{(2)})} \mathbbm{1}_{\{(\xi^{(1)},\xi^{(2)},\eta^{(3)}) \in A\}}d\pi^\nu_{\Delta}(\xi^{(1)})d\pi^\nu_{\Lambda - \Delta}(\xi^{(2)})\\ \\ & = \frac{1}{Z_{\Lambda|\eta}} \int_{\mathcal{N}\left( (\Lambda - \Delta) \times G \right)} Z_{\Delta|\xi^{(2)} \cdot \eta^{(3)}} \mu_{\Delta|{\xi^{(2)} \cdot \eta^{(3)}}} (A) e^{-H_{\Lambda - \Delta| \eta^{(3)}}(\xi^{(2)})}d\pi^\nu_{\Lambda - \Delta}(\xi^{(2)})\\ \\ & = \frac{1}{Z_{\Lambda|\eta}} \int_{\mathcal{N}\left( (\Lambda - \Delta) \times G \right)} \int_{\mathcal{N}(\Delta \times G)} e^{-H_{\Lambda|\eta}(\xi^{(1)}\cdot\xi^{(2)})} \mu_{\Delta|\xi^{(1)}\cdot \xi^{(2)} \cdot \eta^{(3)}} (A) d\pi^\nu_{\Delta}(\xi^{(1)}) d\pi^\nu_{\Lambda - \Delta}(\xi^{(2)})\\ \\ & = \int_{\mathcal{N}(S\times G)} \mu_{\Delta|\xi} (A) d\mu_{\Lambda|\eta} (\xi)\end{aligned}$$for every $A \in {{\mathcal F}}$ which concludes the proof. \[Gibbs2\] A probability measure $\mu$ on $\mathcal{N}( S \times G)$ is called a *Gibbs measure* for the diluted model with intensity measure $\nu$ and Hamiltonian family $H$ if $$\mu(A) = \int_{\mathcal{N}( S \times G)} \mu_{\Lambda|\eta}(A) \,d\mu(\eta)$$for every $\Lambda \in \mathcal{B}^0_S$ and $A \in {{\mathcal F}}$. Notice that by Definition \[Gibbs2\] a probability measure $\mu$ is a Gibbs measure if and only if it is consistent, in the sense of equation , with every Boltzmann-Gibbs distribution, each of which describes the local equilibrium state of the model inside some bounded region. Hence, we may think of Gibbs measures as those representing the global equilibrium states of our model. For this reason they are sometimes referred to as *infinite-volume* . The next proposition validates this choice of terminology. \[limitegibbs\] Let $(\nu,H)$ be a diluted model of bounded local interaction range and $\mu$ be a probability measure on $\mathcal{N}(S \times G)$ such that $$\mu_{\Lambda|\eta} \overset{loc}{\longrightarrow} \mu$$ as $\Lambda \nearrow S$ for some $\eta \in \mathcal{N}(S \times G)$. Then $\mu$ is a Gibbs measure for the model $(\nu,H)$. Let us first notice that, by the interaction measurability of $H$ in for any $\Delta \in \mathcal{B}^0_S$ and $\xi \in \mathcal{N}(S\times G)$ we have $$H_{\Delta|\xi}=H_{\Delta|\xi_{I(\Delta \times G)}}.$$ From this we obtain that for any local event $A \in {{\mathcal F}}$ the mapping $$(\sigma,\xi) \mapsto \mathbbm{1}_{\{\sigma \cdot \xi_{\Delta^c \times G} \in A\}}e^{-H_{\Delta|\xi}(\sigma)}$$ is $\left({{\mathcal F}}_{\Delta \times G} \otimes {{\mathcal F}}_{(\Lambda_A \times G) \cup I(\Delta \times G)}\right)$-measurable which implies that the mapping $\xi \mapsto \mu_{\Delta|\xi}(A)$ is ${{\mathcal F}}_{(\Lambda_A \times G) \cup I(\Delta \times G)}$-measurable. Since the model is of bounded local interaction range we get that for any local event $A \in {{\mathcal F}}$ the mapping $\xi \mapsto \mu_{\Delta|\xi}(A)$ is local as well. Therefore, $\mu_{\Lambda_n|\eta} \overset{loc}{\longrightarrow} \mu$ implies that for any local event $A$ we have $$\int \mu_{\Delta|\xi} (A)\,d\mu(\xi)= \lim_{\Lambda \nearrow S} \int \mu_{\Delta|\xi} (A)\,d\mu_{\Lambda|\eta} (\xi)= \lim_{\Lambda \nearrow S} \mu_{\Lambda|\eta} (A)= \mu(A)$$ where the second equality follows from the consistency property of the Boltzmann-Gibbs distributions and the fact that $\Lambda \nearrow S$. Since the class of local events is closed under intersections and it generates ${{\mathcal F}}$, we see that $$\mu(A) = \int_\Omega \mu_{\Delta|\xi}(A) \,d\mu(\xi)$$ for any $A \in {{\mathcal F}}$ and $\Delta \in \mathcal{B}^0_S$ which allows us to conclude $\mu$ is a Gibbs measure. Standard compactness arguments imply that Gibbs measures for traditional lattice systems (see Remark \[tradition\]) always exist. In the continuum setting the situation is much more delicate, although one can show that, under some reasonable additional assumptions on the pair $(\nu,H)$ (known as the almost sure Feller property), every diluted model of bounded interaction range admits at least one Gibbs measure. Whenever a diluted model admits more than one Gibbs measure we say that the model exhibits a *phase transition*. In this second part of the thesis we shall be specifically interested in studying properties of Gibbs measures for diluted models in general, as well as developing tools to establish the occurrence (or absence) of phase transitions. For the latter, the property detailed on the following proposition will play an important role. Let $H$ be a Hamiltonian inducing a measurable local interaction range, i.e. $I(B)$ is measurable for every $B \in \mathcal{B}_{S \times G}$. A particle configuration $\sigma \in \mathcal{N}(S\times G)$ is of *finite local interaction range* with respect to $H$ if $\sigma( I(\Lambda \times G) )< +\infty$ for every $\Lambda \in \mathcal{B}^0_{S}$. We write $\mathcal{N}_H(S \times G)$ to denote the space of all particle configurations which are of finite local with respect to $H$. \[lfir\] If the pair $(\nu,H)$ satisfies Assumptions \[assump\] then 1. $\mathcal{N}_H(S \times G)$ is a measurable subset of $\mathcal{N}(S \times G )$. 2. Every Gibbs measure of the diluted model $(\nu,H)$ is supported on $\mathcal{N}_H(S \times G)$. If $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq \mathcal{B}^0_{S}$ is such that $\Lambda_n \nearrow S$ then we can write $$\mathcal{N}_H(S \times G) = \bigcap_{n \in {{\mathbb N}}} \{ \sigma \in \mathcal{N}(S\times G) : \sigma ( I(\Lambda_n \times G) ) < + \infty \}.$$ Since for each $n \in {{\mathbb N}}$ the set $I(\Lambda_n \times G)$ is measurable by Assumptions \[assump\] then we obtain that the sets $\{ \sigma \in \mathcal{N}(S\times G) : \sigma ( I(\Lambda_n \times G) ) < + \infty \}$ are also measurable for every $n \in {{\mathbb N}}$ and (i) follows at once from this. To establish (ii) it suffices to show that for each $n \in {{\mathbb N}}$ $$\label{lfirsup} \int_{\mathcal{N}(S \times G)} \sigma(I(\Lambda_n \times G)) d\mu(\sigma) = \sup_{k \in {{\mathbb N}}} \int_{\mathcal{N}(S \times G)} \sigma(I(\Lambda_n \times G) \cap \Lambda_k) d\mu(\sigma) < +\infty.$$ But since $\mu$ is a Gibbs measure for each $n,k \in {{\mathbb N}}$ we have $$\begin{aligned} \int \sigma(I(\Lambda_n \times G) \cap \Lambda_k) d\mu(\sigma)&= \int \left[\int \sigma(I(\Lambda_n \times G) \cap \Lambda_k) d\mu_{I(\Lambda_n \times G) \cap \Lambda_k | \xi}(\sigma) \right] d\mu(\xi)\\ \\ &\leq \int \left[ \int \frac{\sigma(I(\Lambda_n \times G) \cap \Lambda_k)}{Z_{I(\Lambda_n \times G) \cap \Lambda_k|\xi}} d \pi^\nu_{I(\Lambda_n \times G) \cap \Lambda_k} (\sigma) \right] d\mu(\xi)\\ \\ &\leq \frac{\displaystyle{\int \sigma(I(\Lambda_n \times G)) d\pi^\nu(\sigma)}}{ \pi^\nu ( N_{I(\Lambda_n \times G)}=0)} = e^{\nu(I(\Lambda_n \times G))}\nu( I(\Lambda_n \times G))\end{aligned}$$ which, by Assumptions \[assump\], establishes and thus concludes the proof. Some examples of diluted models {#examples} ------------------------------- ### The Widom-Rowlinson model The Widom-Rowlinson model is a classical hardcore interaction model in which particles located on ${{\mathbb R}}^d$ for $d \geq 2$ may be of two different types, $(+)$-particles and $(-)$-particles, and any two particles of different type are forbidden to become within a certain distance $r > 0$ of each other. In the present context of diluted models, the is defined as the diluted model on $\mathcal{N}({{\mathbb R}}^d \times \{+,-\})$ specified by 1. The intensity measure $\nu^{\lambda_+,\lambda_-}$ defined as $$\nu^{\lambda_+,\lambda_-} = \left( \lambda_+ \mathcal{L}^d \times \delta_+ \right) + \left(\lambda_- \mathcal{L}^d \times \delta_-\right)$$ where $\lambda_+,\lambda_- > 0$ are two fixed parameters known as the *fugacities* of $(+)$-particles and $(-)$-particles respectively and $\mathcal{L}^d$ denotes the Lebesgue measure on ${{\mathbb R}}^d$. 2. The Hamiltonian $H$ given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb R}}^d}$ and $\eta \in \mathcal{N}({{\mathbb R}}^d \times \{+,-\})$ by the formula $$H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y )$$ where $$\label{wru} U(\gamma_x,\tilde{\gamma}_y) := \left\{ \begin{array}{ll} +\infty &\text{if }\gamma \neq \tilde{\gamma}\text{ and }\|x-y\|_\infty \leq r\\ 0 &\text{otherwise}\end{array}\right.$$ and $$e_{\Lambda}(\sigma|\eta) := \{ (\gamma_x ,\tilde{\gamma}_y) \in \langle \sigma \cdot \eta_{\Lambda^c \times G} \rangle^2 : x \in \Lambda \}.$$ Thus, in this model the measure $\omega_{\Lambda|\eta}$ in becomes the distribution of a superposition of two independent homogeneous Poisson processes of respective intensities $\lambda_+$ and $\lambda_-$ conditioned on the event that no particle inside $\Lambda$ has a particle of the opposite type (including also particles in $\eta_{\Lambda^c \times \{+,-\}}$) at a distance smaller than $r$ from them. There also exists a discrete version of the Widom-Rowlinson model, first introduced by Lebowitz and Gallavoti in [@L]. In the traditional setting, this model is defined on the configuration space $\{+,0,-\}^{{{\mathbb{Z}}}^d}$ through the Boltzmann-Gibbs distributions given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in \{+,0,-\}^{{{\mathbb{Z}}}^d}$ by the formula $$\label{wrdbgd} \mu_{\Lambda}^\eta(\sigma) = \frac{\mathbbm{1}_{\{\sigma_{\Lambda^c} \equiv \eta_{\Lambda^c}\}}}{Z_{\Lambda}^\eta} e^{-\sum_{B : B \cap \Lambda \neq \emptyset} \Phi_B(\sigma)}$$ where for each $B \subseteq {{\mathbb{Z}}}^d$ the interaction $\Phi_B$ is given by $$\label{wrdht} \Phi_B(\sigma) = \left\{ \begin{array}{ll} (+\infty) \mathbbm{1}_{\{ \sigma(x) \times \sigma(y)= - \}} & \text{ if $B=\{x,y\}$ with $\|x-y\|_\infty \leq r$} \\ \\ - (\mathbbm{1}_{\{\sigma(x)= +\}} \log \lambda_+ + \mathbbm{1}_{\{\sigma(x)= -\}} \log \lambda_- ) & \text{ if $B=\{x\}$} \\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ Notice that the inverse temperature $\beta$ is missing in : since pair interactions are either $0$ or $+\infty$, it is customary to set $\beta = 1$ and vary only the fugacity parameters in the model. In the present setting of diluted models, this discrete version of the Widom-Rowlinson model is defined as the diluted model on $\mathcal{N}({{\mathbb{Z}}}^d \times \{+,-\})$ specified by 1. The intensity measure $\nu^{\lambda_+,\lambda_-}$ defined as $$\nu^{\lambda_+,\lambda_-} = \left( \lambda_+ c^d \times \delta_+ \right) + \left(\lambda_- c^d \times \delta_-\right)$$ where $c^d$ denotes the counting measure on ${{\mathbb{Z}}}^d$. 2. The Hamiltonian $H$ given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in \mathcal{N}({{\mathbb{Z}}}^d \times \{+,-\})$ by the formula $$\label{wrhd} H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y ) + \sum_{x \in \Lambda} V_x(\sigma)$$ where the pair interaction $U$ is the same as in and for each $x \in \Lambda$ we set $$V_x (\sigma):= \left\{ \begin{array}{ll} +\infty &\text{if } \sigma( \{x\} \times \{+,-\} ) > 1 \\ 0 &\text{otherwise.}\end{array}\right.$$ The term $V$ is introduced to allow at most one as in the traditional setting. in which to define this discrete version within the setting of diluted models is to leave the term $V$ out of and then avoid the possibility of multiple particles of the same type per site by considering the projected Boltzmann-Gibbs distributions $\langle \mu_{\Lambda|\eta} \rangle$ instead, i.e. by considering particle configurations without any regard for their respective particle multiplicities. Both possibilities are indeed equivalent, but the latter introduces a change in fugacities: for a choice of fugacities $\lambda_{\pm}$ in the second alternative, one recovers the traditional discrete Widom-Rowlinson model with fugacities $e^{\lambda_\pm} - 1$. For this reason, unless explicitly stated otherwise, we shall always work with the first of these alternatives. We refer the reader to [@GHM] for a review of the general results known for these models. ### The Widom-Rowlinson model with generalized interactions There are several interesting generalizations of the Widom-Rowlinson model. One possible generalization which has been well studied is to consider a model in which nearby pairs of particles of the opposite type are not necessarily forbidden, but merely More precisely, given a decreasing function $h: {{\mathbb R}}^+ \rightarrow [0,+\infty]$ with bounded support we define the Widom-Rowlinson model with *interspecies repulsion function* $h$ by replacing the previous pair interaction $U$ in by $$U_h(\gamma_x,\tilde{\gamma}_y) := h(|x-y|)\mathbbm{1}_{\{\gamma \neq \tilde{\gamma}\}} = \left\{ \begin{array}{ll} h(\|x-y\|_\infty) &\text{if }\gamma \neq \tilde{\gamma}\\ 0 &\text{otherwise.}\end{array}\right.$$ We may also add a type-independent repulsion by taking a second decreasing function $j: {{\mathbb R}}^+ \rightarrow [0,+\infty]$ with bounded support and redefining the Hamiltonian as $$H_{\Lambda|\eta}(\sigma)= \sum_{\{\gamma_x ,\tilde{\gamma}_y\} \in e^u_{\Lambda}(\sigma|\eta)} h(\|x-y\|_\infty)\mathbbm{1}_{\{\gamma \neq \tilde{\gamma}\}} + j(\|x-y\|_\infty)$$ where $$e^u_{\Lambda}(\sigma|\eta) = \{ \{\gamma_x, \tilde{\gamma}_y\} \subseteq \langle \sigma \cdot \eta_{\Lambda^c \times G} \rangle : \gamma_x \neq \tilde{\gamma}_y, \{x,y\} \cap \Lambda \neq \emptyset \}.$$ Notice that for $h:= (+\infty) \mathbbm{1}_{[0,r]}$ and $j \equiv 0$ we obtain the original Widom-Rowlinson model. We refer the reader to [@GH], where these type of generalizations were investigated. ### The thin rods model Another possible generalization of the Widom-Rowlinson model is to consider $q \geq 3$ types of particles instead of just two. Furthermore, one could have different *exclusion radii* $r_{ij}$ for the different pairs of types of particles $1 \leq i < j \leq q$. This asymmetric generalization of the original model was studied in [@BKL]. We introduce here a different asymmetric variant which is also featured on the former reference. Given $q \geq 3$ and $l> 0$ we consider in ${{\mathbb R}}^2$ a system consisting of rods of length $2l$ and zero width positioned anywhere throughout the plane. We assume that these rods may possess $q$ different orientations specified by some fixed angles $0 \leq \theta_1 < \dots < \theta_q < \pi$ measured with respect to the $x$-axis. Finally, the interaction between these rods is that no two rods are allowed to intersect each other. More precisely, if we set $$L_i = \{ t \cdot (\cos\theta_i, \sin \theta_i) : t \in [-l,l] \}$$ then the thin rods model is defined as the diluted model on $\mathcal{N}({{\mathbb R}}^2 \times \{1,\dots,q\})$ 1. The intensity measure $$\nu := \sum_{i=1}^q \lambda_i \mathcal{L}^d \times \delta_{\theta_{i}}$$ where for $1 \leq i \leq q$ the parameter $\lambda_i > 0$ is the fugacity of rods of orientation $\theta_i$. 2. The Hamiltonian $$H_{\Lambda|\eta}(\sigma) := \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y )$$ where $$\label{tru} U(\gamma_x,\tilde{\gamma}_y) := \left\{ \begin{array}{ll} +\infty &\text{if }(L_\gamma + x) \cap (L_{\tilde{\gamma}} + y) \neq \emptyset\\ 0 &\text{otherwise.}\end{array}\right.$$ In general, given a probability measure $\rho$ on $S^1_*:=[0,\pi)$ we define the thin rods model with fugacity $\lambda$, rod length $2l$ and orientation measure $\rho$ as the diluted model on $\mathcal{N}({{\mathbb R}}^2 \times S^1_*)$ specified by the intensity measure $\nu^\lambda := \lambda \mathcal{L}^2 \times \rho$ and the Hamiltonian $H$ given by . Notice that this broader definition allows for an infinite number of possible orientations depending on the measure $\rho$. Also, notice that the original model with $q$ orientations is recovered by setting $\lambda = \lambda_1 + \dots + \lambda_q$ and $\rho = \frac{1}{\lambda} \sum_{i=1}^q \lambda_i \delta_{\theta_{i}}$. ### The tolerant Widom-Rowlinson model Yet another variant to consider is the tolerant Widom-Rowlinson model, in which particles can tolerate up to $k \in {{\mathbb N}}$ particles of the opposite type within a distance $r > 0$ from them. In this case, the intensity measure remains unchanged while the Hamiltonian $H$ becomes $$H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma^1_{x_1} , \dots, \gamma^{k+1}_{x_n}) \in e_{\Lambda}^k(\sigma|\eta)} U(\gamma^1_{x_1} , \dots, \gamma^{k+1}_{x_n})$$ where $$U(\gamma^1_{x_1},\dots, \gamma^{k+1}_{x_n}) := \left\{ \begin{array}{ll} +\infty &\text{if }\gamma^1 \neq \gamma^2 = \dots = \gamma^{k+1}\text{ and } \max_{j=2,\dots,k+1} \|x_1-x_j\|_\infty \leq r \\ 0 & \text{otherwise}\end{array}\right.$$ and $$e_{\Lambda}^k(\sigma|\eta) := \{ (\gamma^1_{x_1} , \dots, \gamma^{k+1}_{x_n}) \in \langle \sigma \cdot \eta_{\Lambda^c \times G} \rangle^{k+1} : x_1 \in \Lambda \}.$$ Observe that, unlike all previous models, the interactions featured here are not pairwise. This fact is of interest since most continuum models featured in the literature present only pairwise interactions. A discrete analogue of this model is also available just as it was in the original setting. can be deduced from its continuum counterpart in the same way as before, so we omit it here. ### The symbiotic model This is an example of a model in which the interactions involved are of attractive type instead of repulsive. It features particles of two types, *hosts* and *parasites*, which interact in the following way: the hosts spread freely throughout ${{\mathbb R}}^d$ without any care for the location of parasites, whereas the parasites prefer to locate themselves near the hosts. More precisely, the symbiotic model is defined as the diluted model on $\mathcal{N}({{\mathbb R}}^d \times \{h,p\})$ with intensity measure $$\nu^{\lambda_h,\lambda_p} = (\lambda_h \mathcal{L}^d \times \delta_{h}) + (\lambda_p \mathcal{L}^d \times \delta_{p})$$ and Hamiltonian given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb R}}^d}$ and $\eta \in \mathcal{N}({{\mathbb R}}^d \times \{h,p\})$ by the formula $$H_{\Lambda|\eta}(\sigma) = \sum_{p_x \in \sigma \,:\, x \in \Lambda} J\mathbbm{1}_{\{ \sigma( B^*(x,r) \times \{h\}) = 0 \}}$$ where $\lambda_{h},$ $\lambda_p$, $J$ and $r$ are all positive constants, and for $x \in {{\mathbb R}}^d$ we set $$B^*(x,r):=\{ y \in {{\mathbb R}}^d : 0 < \|x - y \|_2 < r\}.$$Unlike all the previous examples, notice that for this model we have $\Delta E < 0$. Furthermore, this is the example of a model in which the impact relation is not symmetric: parasites do not have any impact on hosts, whereas hosts always have an impact on nearby parasites. ### An inconsistent example: the Ising contours model {#exampleisingc} A very important tool in the study of phase transitions is the use of contour models. Perhaps one of the most popular examples in statistical mechanics of a contour model is the Ising contours model (also known as Peierls contours), which arises as a geometrical representation of the Ising model on ${{\mathbb{Z}}}^d$ and was used originally by Peierls to establish the occurrence of phase transition at low temperatures. The Ising model is not a diluted model itself in the sense that particles are forced to occupy *every* site of the lattice.[^2] Nonetheless, it fits within the framework of Boltzmann-Gibbs distributions The Ising model (with zero external field) is defined on the configuration space $\{+,-\}^{{{\mathbb{Z}}}^d}$ through the Boltzmann-Gibbs distributions given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in \{+,-\}^{{{\mathbb{Z}}}^d}$ by the formula $$\mu_{\Lambda}^\eta(\sigma) = \frac{\mathbbm{1}_{\{\sigma_{\Lambda^c} \equiv \eta_{\Lambda^c}\}}}{Z_{\Lambda}^\eta} e^{-\beta \sum_{B : B \cap \Lambda \neq \emptyset} \Phi_B(\sigma)}$$ with $Z_{\Lambda}^\eta$ being the normalizing constant, and for each $B \subseteq {{\mathbb{Z}}}^d$ $$\label{hising} \Phi_B(\sigma) = \left\{ \begin{array}{ll} \mathbbm{1}_{\{ \sigma(x)\sigma(y)=-1\}} & \text{ if $B=\{x,y\}$ with $\|x-y\|_1 = 1$} \\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ Now, the contour representation arises in the study of Boltzmann-Gibbs distributions with a constant boundary condition, i.e. either $\eta(x)=+$ or $\eta(x)=-$ for all $x \in {{\mathbb{Z}}}^d$. To fix ideas let us consider the $(+)$-boundary condition and denote the corresponding Boltzmann-Gibbs distribution on the volume $\Lambda$ by $\mu_{\Lambda}^+.$ From we immediately see that the weight assigned by $\mu_{\Lambda}^+$ to each configuration which is positively aligned outside $\Lambda$ depends only on the amount of misaligned nearest neighbors spins in the configuration. With this in mind, one may introduce the following alternative representation of any such configuration which keeps track of misaligned spins: 1. Consider the edge set $e({{\mathbb{Z}}}^d)$ consisting of all bonds joining nearest neighbors in ${{\mathbb{Z}}}^d$. 2. For each bond $e \in e({{\mathbb{Z}}}^d)$ consider the plaquette $p(e)$: the unique $(d-1)$-dimensional unit cube with vertices in the dual lattice intersecting $e$ in a perpendicular manner. Recall that the dual lattice $({{\mathbb{Z}}}^d)^*$ is defined as $$({{\mathbb{Z}}}^d)^*:=\left\{ \left(x_1+\frac{1}{2}, \dots, x_d + \frac{1}{2}\right) : x \in {{\mathbb{Z}}}^d\right\}.$$ 3. We call any collection of plaquettes a *surface*. We shall say that a surface $P$ is *closed* if every $(d-2)$-dimensional face of $P$ is shared by an even number of 4. We say that two plaquettes are adjacent if they share a $(d-2)$-dimensional face. is said to be *connected* if for every pair of plaquettes in $P$ there exists a sequence of pairwise adjacent plaquettes joining them. 5. A *contour* is then defined as a connected and closed surface. Two contours $\gamma$ and $\gamma'$ are said to be *incompatible* if they share a $(d-2)$-dimensional face, in which case we denote this fact by $\gamma \not \sim \gamma'$. 6. Given a configuration $\sigma$ which is positively aligned outside $\Lambda$ we may assign to it a family $\Gamma_\sigma$ of pairwise compatible contours lying inside $\Lambda^*$, the smallest subset of the dual lattice which contains $\Lambda$. Indeed, given any such configuration $\sigma$ we may consider the surface $P_\sigma$ consisting of those plaquettes $p(e)$ such that the bond $e$ joins misaligned spins in $\sigma$. This surface $P_\sigma$ is split into maximal connected components, each of which is a contour. If $\Gamma_\sigma$ denotes the collection of these maximal components, we immediately see that $\Gamma_\sigma$ satisfies all desired requirements. 7. If $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ is simply connected[^3] then for any given family $\Gamma$ of compatible contours lying inside $\Lambda^*$ there exists a unique $\sigma^+_\Gamma$ such that $\Gamma_{\sigma^+_\Gamma} = \Gamma$. Indeed, the value of $\sigma^+_\Gamma(x)$ for $x \in \Lambda$ can be computed as $(-1)^{n_\Gamma(x)}$, where $n_\Gamma(x)$ denotes the total number of contours in $\Gamma$ around $x$. We call $\sigma^+_\Gamma$ the $(+)$-*alignment* of $\Gamma$. From the considerations made above it is clear that for any configuration $\sigma \in \{+,-\}^{{{\mathbb{Z}}}^d}$ in the support of $\mu^+_\Lambda$ one has that $$\mu_{\Lambda}^+(\sigma) = \frac{1}{Z_{\Lambda}^+}e^{-\beta \sum_{\gamma \in \Gamma_\sigma} |\gamma|}$$ where $|\gamma|$ denotes the total number of plaquettes in $\gamma$. Thus, if we are only interested in understanding the behavior of the system with a positively aligned boundary condition, ourselves to the study of the interactions between the different contours, which give rise to a contour model. Before we can introduce it, we make some conventions. We define the spin set $G$ as the space of possible contour shapes, i.e. without any regard for their position on $({{\mathbb{Z}}}^d)^*$, and identify each contour $\gamma$ with an element $\gamma_x \in ({{\mathbb{Z}}}^d)^* \times G$: whereas $x$ will be the location of the minimal vertex in $\gamma$ when ordered according to the dictionary order in $({{\mathbb{Z}}}^d)^*$. See Figure \[fig3\] for a possible example. Finally, we define the Ising contours model as the model on $\mathcal{N}(({{\mathbb{Z}}}^d)^* \times G)$ with intensity measure ![Ising contours in the dual lattice $({{\mathbb{Z}}}^2)^*$ for the $(+)$-boundary condition.[]{data-label="fig3"}](contornosSanti.eps){width="8cm"} $$\label{imic} \nu^\beta(\gamma_x):= e^{-\beta |\gamma_x|}$$ and Hamiltonian $$\label{hic} H_{\Lambda|\Gamma'} (\Gamma) = \left\{ \begin{array}{ll} +\infty & \text{ if either $\Gamma$ is incompatible, $\Gamma \not \sim \Gamma'_{\Lambda^c \times G}$ or $\Gamma \not \subseteq \Lambda$}\\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ where $\Gamma \subseteq \Lambda$ indicates that all contours in $\Gamma$ lie entirely inside $\Lambda$. Notice that it is because of this last restriction in the Hamiltonian that the model fails to satisfy Assumptions \[assump\]. Indeed, one has that: 1. $\Delta E_{\eta} \equiv +\infty$ for all $\eta \in \mathcal{N}(({{\mathbb{Z}}}^d)^* \times G)$ since contours contain more than one vertex. Therefore, for each $\gamma_x \in S \times G$ the quantity $\Delta_\eta(\gamma_x)$ fails to represent what it should: the energy cost for the infinite-volume system to add the particle $\gamma_x$ when the current configuration of the system is given by $\eta$. To fix this problem one considers instead the *localized energy leap functions* $\Delta E_{\Lambda|\eta} : S \times G \rightarrow [-\infty,+\infty]$ defined as $$\Delta E_{\Lambda | \eta} (\gamma_x ) = \left\{ \begin{array}{ll} +\infty & \text{ if either $\gamma_x \not \sim \eta $ or $\gamma_x \not \subseteq \Lambda$}\\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ Notice that for every $\eta \in \mathcal{N}(({{\mathbb{Z}}}^d)^* \times G)$ and $\gamma_x \in S \times G$ one has that $$\label{elising} \Delta E^*_{\eta}(\gamma_x) := \lim_{\Lambda \nearrow ({{\mathbb{Z}}}^d)^*} \Delta E_{\Lambda|\eta} (\gamma_x) = (+\infty)\mathbbm{1}_{\{ \gamma_x \not \sim \eta \}}.$$ Thus, for the localized energy leap functions one recovers in the limit as $\Lambda \nearrow ({{\mathbb{Z}}}^d)^*$ the correct notion of energy cost. Taking this into consideration, one also redefines the impact relation by incompatibility, i.e. $\tilde{\gamma}_y \rightharpoonup \gamma_x$ if and only if $\tilde{\gamma}_y \not \sim \gamma_x$. 2. The consistent Hamiltonian property is not satisfied. As a consequence, one can easily check that the resulting Boltzmann-Gibbs distributions are also inconsistent in the sense of . In this context, Gibbs measures as defined in Definition \[Gibbs2\] could fail to exist. However, it is still possible for these contour model to admit “infinite-volume Boltzmann-Gibbs distributions” in the sense described by Proposition \[limitegibbs\]. These limiting measures will be of particular interest to us since, as we will see on Chapter \[chapterffg\], their existence implies a phase transition in the Ising model. Despite the fact that not all conditions on Assumptions \[assump\] are satisfied, under the minor adjustments suggested above most of the analysis carried out in the next chapter for diluted models will also hold in this context, which is why we decided to include this model among the examples even if it is not a diluted model as we understand them. Another fact worth mentioning is that this is the only given example which is of unbounded local interaction range. Finally, we point out the following crucial relation between the Boltzmann-Gibbs distributions in the original Ising and Ising contours models: for any simply connected $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and family $\Gamma$ of compatible contours lying inside $\Lambda^*$ we have $$\label{dualidadising} \mu^+_\Lambda ( \sigma^+_{\Gamma}) = \mu_{\Lambda^*|\emptyset}(\Gamma).$$ An analogous contour representation is also available for Boltzmann-Gibbs distributions with negatively aligned boundary condition. We refer to [@PS] for a of the Ising model and the standard proof of phase transition using its contour representation. Resumen del Capítulo 8 ---------------------- Definimos en este capítulo la clase de modelos diluidos, que habremos de estudiar en lo que resta de la segunda parte. Esencialmente, un modelo diluido se define a partir de sus distribuciones de Boltzmann-Gibbs, medidas sobre el espacio de configuraciones $\mathcal{N}(S \times G)$ que describen el comportamiento local del modelo en volúmenes finitos sujeto sujeto a distintas condiciones de frontera. Concretamente, fijada una medida $\nu$ en $S \times G$, dado $\Lambda \subseteq S$ acotado y $\eta \in \mathcal{N}(S \times G)$ se define la distribución de Boltzmann-Gibbs $\mu_{\Lambda|\eta}$ en $\Lambda$ con respecto a la condición de frontera $\eta$ mediante la fórmula $$\mu_{\Lambda|\eta} = \omega_{\Lambda|\eta} \times \delta_{\eta_{\Lambda^c}}$$ donde identificamos $\mathcal{N}(S \times G) = \mathcal{N}( \Lambda \times G) \times \mathcal{N}(\Lambda^c \times G)$ y $\omega^\eta_\Lambda$ es la medida de probabilidad en $\mathcal{N}(\Lambda \times G)$ definida a través de la relación $$d\omega_{\Lambda|\eta} = \frac{e^{-H_{\Lambda|\eta}}}{Z_{\Lambda|\eta}} d\pi^\nu_\Lambda.$$ Aquí $\pi^\nu_\Lambda$ denota la distribución de un proceso de Poisson en $\mathcal{N}(\Lambda \times G)$ con intensidad $\nu_{\Lambda \times G}$, cuyo rol es el de distribuir las partículas dentro de $\Lambda$, mientras que $H_{\Lambda|\eta}$ es lo que se denomina el Hamiltoniano relativo a $\Lambda$ con condición de frontera $\eta$, encargado de asignar un peso a las distintas configuraciones de acuerdo a la interacción que haya entre las partículas que la conforman. De esta manera, si $H$ denota a la familia de Hamiltonianos locales, el par $(\nu, H)$ determina por completo al modelo. Asumimos que $(\nu,H)$ cumple las condiciones dadas en \[assump\]. De particular interés en cada modelo son las medidas de Gibbs asociadas al mismo, es decir, las medidas $\mu$ sobre $\mathcal{N}(S \times G)$ que son compatibles con las distribuciones de Boltzmann-Gibbs para cualquier volumen $\Lambda$, i.e. $$\mu(\cdot) = \int_{\mathcal{N}(S \times G)} \mu_{\Lambda|\eta}(\cdot) d\mu(\eta).$$ Las medidas de Gibbs representan los distintos posibles estados de equilibrio globales del modelo. En efecto, mostramos que bajo ciertas condiciones adicionales sobre el par $(\nu,H)$, cualquier límite local de las distribuciones de Boltzmann-Gibbs $\mu_{\Lambda|\eta}$ cuando $\Lambda \nearrow S$ es una medida de Gibbs. En lo que resta, nos interesará estudiar qué condiciones garantizan existencia y unicidad o multiplicidad de medidas de Gibbs. Por último, culminamos el capítulo mostrando que algunos modelos clásicos, como lo son el modelo de Widom-Rowlinson (tanto en su versión continua como discreta) y el de contornos de Ising, caen dentro de la familia de modelos diluidos. Explicamos en detalle la dualidad entre el modelo de Ising original y su modelo de contornos asociado, que será de vital importancia para los desarrollos del Capítulo 12. Agregamos además algunos otros ejemplos, como los modelos de Widom-Rowlinson tolerante y el simbiótico, para mostrar la flexibilidad de nuestras definiciones y la amplitud de nuestro marco teórico. The Fernández-Ferrari-Garcia dynamics {#chapterffg} ===================================== In this chapter we study the Fernández-Ferrari-Garcia dynamics first . In their work the authors focus on the Ising contours model and show that, for a sufficiently large value of the inverse temperature $\beta$, the infinite-volume Boltzmann-Gibbs distribution of this contour model can be realized as the unique invariant measure of these dynamics. Later on [@FFG2], the authors investigate the possibility of using this new approach as a perfect simulation scheme for Gibbs measures of a number of systems with exclusion in the low density or extreme temperature regime. Our purpose now is to the same endeavor in general for the broader family of diluted models. The main ideas in this chapter are those originally featured in [@FFG1]: the majority of the results presented here are direct generalizations of those found there. Nonetheless, some of the proofs we give in this chapter are different from the ones in the original article, since not all of their proofs can be adapted to continuum models. Given an intensity measure $\nu$ and a Hamiltonian $H$ satisfying Asssumptions \[assump\], the essentials of the associated Fernández-Ferrari-García dynamics as follows: 1. At rate $e^{-\Delta E}$ the birth of new animals is proposed with intensity given by $\nu$. 2. Each $\gamma_x$ proposed for birth will be effectively born with probability $e^{-(\Delta E_{\eta}(\gamma_x)-\Delta E)}$, where $\eta$ is the state of the system at the time in which the birth of $\gamma_x$ is proposed. 3. Every effectively born animal has an independent random exponential lifetime of parameter 1. 4. After its lifetime has expired, each animal dies and vanishes from the configuration. Our aim in this section is to make this description rigorous as well as to study some of the basic properties enjoyed by these dynamics. We shall begin by introducing the dynamics restricted to a finite volume and then treat the more delicate scenario of infinite volume. All processes defined below shall be subsets of the product space $\mathcal{C}= (S \times G) \times {{\mathbb R}}\times {{\mathbb R}}^+$. The elements of $\mathcal{C}$ shall be called *cylinders* since any $(\gamma_x, t, s) \in \mathcal{C}$ can be seen as a cylinder on $S \times {{\mathbb R}}$ of axis $\{x\}\times [t,t+s]$ and diameter $\gamma$. However, we shall prefer to describe each cylinder $C=(\gamma_x,t,s)\in \mathcal{C}$ in terms of its *basis* $\gamma_x$, its *time of birth* $t$ and its *lifespan* $s$. We shall denote these three features of $C$ by $basis(C)$, $b_C$ and $l_C$, respectively. Following this line of thought, we can identify any random element $\mathcal{V} \in {{\mathcal C}}$ with a birth and death process on $S \times G$ by means of its time sections, i.e. if for each $t \in {{\mathbb R}}$ we define the random particle configuration $\mathcal{V}_t \in \mathcal{N}(S \times G)$ by the formula $$\mathcal{V}_t ( \{ \gamma_x\}) = \#\{ C \in \mathcal{V} : basis(C)=\gamma_x \text{ and }b_C \leq t < b_C + l_C \}$$ for every $\gamma_x \in S \times G$, then $(\mathcal{V}_t)_{t \in {{\mathbb R}}}$ constitutes a birth and death process on $S \times G$. we thus interpret any cylinder $(\gamma_x,t,s)$ as a particle $\gamma$ being born at time $t$ on location $x$ which lives on for a period of length $s$. Local dynamics {#localdynamics} -------------- We begin our formal introduction of the Fernández-Ferrari-García dynamics (referred to as FFG dynamics from now on) by fixing a bounded set $\Lambda \in \mathcal{B}^0_{S}$ and a particle configuration $\eta \in \mathcal{N}(S\times G)$, and defining the dynamics on the finite volume $\Lambda \times G$ with $\eta$ acting as a boundary condition. Though it may seem misleading at first, we choose to build these local dynamics from infinite-volume processes since it will provide a clear and direct way in which to couple all local dynamics together. Consider a Poisson process $\Pi$ on $\mathcal{C}$ with intensity measure $\phi_\nu = \nu \times e^{-\Delta E} \mathcal{L} \times \mathcal{E}^1$, where $\mathcal{L}$ is the Lebesgue measure on ${{\mathbb R}}$ and $\mathcal{E}^1$ is the exponential distribution of parameter 1. We shall refer to $\Pi$ as the *free process*, whose time evolution can be described as follows: 1. At rate $e^{-\Delta E}$ animals are born with intensity given by $\nu$, regardless of the impact preexisting animals may have upon them. 2. Each animal has an independent random exponential lifetime of parameter 1. 3. After its lifetime has expired, each animal dies and vanishes from the configuration. This free process constitutes, as its name suggests, a stationary non-interacting birth and death process whose invariant measure is $\pi^\nu$. To be able to define the dynamics we need to add an additional component to $\Pi$: to each cylinder in $\Pi$ we will attach an independent random variable uniformly distributed on $[0,1]$, which will be called its *flag*. Each flag will be used to determine the success of the associated cylinder’s attempted birth in the dynamics. One way in which to attach these flags would be to replace $\Pi$ with the marked Poisson process $\overline{\Pi}$ on $\mathcal{C} \times [0,1]$ with intensity measure $\overline{\phi}_\nu :=\phi_\nu \times \mathcal{L}_{[0,1]}$. Thus, elements of $\overline{\Pi}$ can be seen as cylinders in $\Pi$ together with their respective independent flags in $[0,1]$. For any given $(\gamma_x,t,s) \in \Pi$ we shall denote its corresponding flag by $F(\gamma_x,t,s)$.[^4] Finally, recalling the identification $\mathcal{N}(S \times G) = \mathcal{N}(\Lambda \times G) \times \mathcal{N}(\Lambda^c \times G)$ we define the local FFG process $\mathcal{K}^{\Lambda|\eta}$ on $\Lambda \times G$ with boundary condition $\eta$ by the formula $$\label{keptformula1} \mathcal{K}^{\Lambda|\eta} = \{ (\gamma_x,t,s) \in \Pi_{\Lambda \times G} : F(\gamma_x,t,s) < M(\gamma_x | \mathcal{K}^{\Lambda|\eta}_{t^-}) \} \times \{ (\gamma_x,t,s) \in \mathcal{C}: \gamma_x \in \eta_{\Lambda^c \times G}\}$$ where for $\gamma_x \in S \times G$ and $\xi \in \mathcal{N}(S\times G)$ we use the notation $M(\gamma_x|\xi):=e^{-(\Delta E_{\xi}(\gamma_x)-\Delta E)}$ and $\Pi_{\Lambda \times G}$ denotes the restriction of $\Pi$ to $(\Lambda \times G) \times {{\mathbb R}}\times {{\mathbb R}}^+$. In other words, $\mathcal{K}^{\Lambda|\eta}$ is the process obtained as a thinning of the free process inside $\Lambda$ given by the rule on with the addition of a boundary condition $\eta$ outside $\Lambda$ which must be kept fixed for all times, i.e. $(\mathcal{K}^{\Lambda|\eta}_t)_{\Lambda^c \times G} = \eta_{\Lambda^c \times G}$ for every $t \in {{\mathbb R}}$. Notice that the self-referential nature of the thinning rule in could lead to $\mathcal{K}^{\Lambda|\eta}$ not being well defined. Indeed, let us introduce some definitions that will help us give further details on this matter. \[defiances\]$\,$ 1. Given $C, \tilde{C} \in {{\mathcal C}}$ we say that $\tilde{C}$ is a *first generation ancestor* of $C$ and write $\tilde{C} \rightharpoonup C$ whenever $$basis(\tilde{C}) \rightharpoonup basis(C)\hspace{1.5cm} \text{ and }\hspace{1.5cm} b_{\tilde{C}} < b_C < b_{\tilde{C}} + l_{\tilde{C}}.$$ We shall denote the set of all first generation ancestors of a given $C \in {{\mathcal C}}$ by $\mathcal{P}(C)$. 2. For $C \in {{\mathcal C}}$ we define $\A_1(C):=\Pi_{\mathcal{P}(C)}$ and for $n \in {{\mathbb N}}$ we set $$\A_{n+1}(C)= \bigcup_{\tilde{C} \in \mathcal{A}_n(C)} \A_1(\tilde{C}).$$ We define the *clan of ancestors* of $C$ in $\Pi$ as $$\A(C):= \bigcup_{n \in {{\mathbb N}}} \A_n(C).$$ Furthermore, for $\Lambda \in {{\mathcal B}}^0_S$ and $n \in {{\mathbb N}}$ we define the $n$-th generation of ancestors of $C$ restricted to $\Lambda$ as $$\A_{n}^\Lambda(C) = \{ \tilde{C} \in \A_n(C) : basis(\tilde{C}) \in \Lambda \times G\}.$$ We define $\A^\Lambda(C)$, the clan of ancestors of $C$ restricted to $\Lambda$, in an analogous fashion. 3. For $t \in {{\mathbb R}}$ and $\Lambda \in {{\mathcal B}}^0_S$ let us define the *clan of ancestors of $\Lambda \times G$ at time $t$* as $$\mathcal{A}^t(\Lambda \times G) := \bigcup_{n \in {{\mathbb N}}_0} \mathcal{A}^t_{n}(\Lambda \times G)$$ where $\mathcal{A}^t_0(\Lambda \times G) := \{ C \in \Pi : basis(C) \in \Lambda \times G \text{ , } b_C \leq t < b_C + l_C \}$ and for $n \in {{\mathbb N}}$ $$\mathcal{A}^t_n(\Lambda \times G) := \bigcup_{C \in \mathcal{A}^t_0 (\Lambda \times G)} \mathcal{A}_n(C).$$ For any $\Delta \in {{\mathcal B}}^0_S$ such that $\Lambda \subseteq \Delta$ we define $\A^{t,\Delta}(\Lambda \times G)$, the clan of ancestors of $\Lambda \times G$ at time $t$ restricted to $\Delta \times G$, in the same manner as above. Having defined the notion of ancestors in the dynamics, we now return to discuss the good definition of $\mathcal{K}^{\Lambda|\eta}$. Notice that if we wish to determine whether a given cylinder $C=(\gamma_x,t,s) \in \Pi_{\Lambda \times G}$ belongs to $\mathcal{K}^{\Lambda|\eta}$ or not then first we need to specify the configuration $\mathcal{K}^{\Lambda|\eta}_{t^-}$ in order to evaluate whether the condition on is satisfied. To be more accurate, due to Assumptions \[assump\] we will only need to specify $\mathcal{K}^{\Lambda|\eta}_{t^-}$ inside the set $I( \{\gamma_x\})$. However, since $\mathcal{K}^{\Lambda|\eta}$ is known outside $\Lambda$ as it coincides with $\eta$ for all times, it remains to specify $\mathcal{K}^{\Lambda|\eta}_{t^-}$ inside $I( \{\gamma_x\}) \cap (\Lambda \times G)$. Therefore, recalling Definition \[defiances\], we see that to determine the fate of $C$ we must first determine the fate of all its first generation ancestors with bases in $\Lambda \times G$, i. e. cylinders in $\A^\Lambda_1(C)$. But this task itself involves determining the fate of a second generation of ancestors of $C$, those cylinders with bases in $\Lambda \times G$ being ancestors to cylinders in $\A^\Lambda_1(C)$. In general, to determine if $C$ belongs to $\mathcal{K}^{\Lambda|\eta}$ we must study the fate of every cylinder in $\A^{\Lambda}(C)$, the clan of ancestors of $C$ restricted to $\Lambda$. If $\A^{\Lambda}(C)$ were to span over an infinite number of generations then it may be impossible to decide whether to keep $C$ or not and, therefore, $\mathcal{K}^{\Lambda|\eta}$ may not be well defined in this situation. On the other hand, if we were able to guarantee that for every cylinder $C \in \Pi_{\Lambda \times G}$ the restricted clan $\A^\Lambda(C)$ spans only over a finite number of generations then $\mathcal{K}^{\Lambda|\eta}$ would be well defined. Indeed, since $M(\gamma_x | \mathcal{K}^\sigma_{t^-}) = M(\gamma_x | \eta_{\Lambda^c \times G})$ for any cylinder $(\gamma_x, t,s) \in \ Pi_{\Lambda \times G}$ with no ancestors preceding it, we have that the fate of every cylinder in the last generation of ancestors restricted to $\Lambda$ of a given cylinder $C$ can be decided upon inspecting their respective flags (and nothing else) and thus it will also be possible to determine the fate of all their descendants, including $C$. More precisely, take $C \in \Pi_{\Lambda \times G}$ and let $N$ be a nonnegative integer such that $\mathcal{A}^\Lambda_n(C) = \emptyset$ for every $n > N$. If we set $$K_N^\Lambda(C):= \{ (\tilde{\gamma}_y,r,l) \in \mathcal{A}^\Lambda_N(C) : F(\tilde{\gamma}_y,r)< M(\tilde{\gamma}_y|\eta_{\Lambda^c \times G})\}$$ and for $1 \leq i \leq N-1$ inductively define $$K_i^\Lambda(C) = K_{i+1}^\Lambda(C) \cup \{ (\tilde{\gamma}_y,r,l) \in \mathcal{A}^\Lambda_i(C) : F(\tilde{\gamma}_y,r,l)< M(\tilde{\gamma}_y|K^\Lambda_{i+1}(C))\}$$ then the cylinder $C \in \Pi_{\Lambda \times G}$ will be kept if and only if $$F(C) < M(\tilde{\gamma}_y|K^\Lambda_1(C)).$$ In other words, to decide if a cylinder $C \in \Pi_{\Lambda \times G}$ is kept one could conduct the following procedure: 1. If $C$ has no first generation ancestors, i.e. $\mathcal{A}^\Lambda_1(C)=\emptyset$, then the value of its flag $u$ alone will determine whether $C$ is kept or not. Otherwise, the value of $u$ will decide if $C$ is kept once we determine the fate of all the first generation ancestors of $C$. 2. To decide whether any given first generation ancestor $\tilde{C} \in \mathcal{A}^\Lambda_1(C)$ is kept, one must repeat step (i) for $\tilde{C}$ instead of $C$. 3. Since the clan of ancestors of $C$ restricted to $\Lambda$ possesses only a finite number of generations, one must go backwards in time and examine a previous generation of ancestors only a finite amount of times (at most $N$ times) so that it is ultimately possible to determine whether $C$ is kept. Therefore, we are left to answer the question of under which conditions do the clans of ancestors restricted to $\Lambda$ possess only a finite amount of generations. Fortunately, under Assumptions \[assump\] this is always the case. This is the content of the following proposition. \[localfinit\] For every $\Lambda \in {{\mathcal B}}^0_S$ we have that $\A^{t,\Lambda}(\Lambda \times G)$ is finite for all $t \in {{\mathbb R}}$. Since $\Pi$ is a stationary process whose invariant measure $\pi^\nu$ satisfies $$\pi^\nu( \{ \xi \in \mathcal{N}(S \times G) : \xi( \Lambda \times G) = 0 \} ) = e^{-\nu(\Lambda \times G)} > 0$$ we have that the entrance times $(t_i(\Lambda))_{i \in {{\mathbb{Z}}}}$ to the set $\{ \xi \in \mathcal{N}(S \times G) : \xi( \Lambda \times G) = 0 \}$ are well defined and satisfy $t_i(\Lambda) \rightarrow \pm \infty$ as $i \rightarrow \pm \infty$. In particular, for every $t \in {{\mathbb R}}$ there exists $i_0 \in {{\mathbb{Z}}}$ such that $t_{i_0-1}(\Lambda) \leq t < t_{i_0}(\Lambda)$. Since $\Pi_{t_i(\Lambda)}(\Lambda \times G) = 0$ for each $i \in {{\mathbb{Z}}}$ by definition, this implies that there exist (random) $k < r \in {{\mathbb{Z}}}$ such that $$\label{contan} \A^{t,\Lambda}(\Lambda \times G) \subseteq \Pi_{(\Lambda \times G) \times [t,t_{i_0}] \times {{\mathbb R}}^+} \subseteq \Pi_{(\Lambda \times G) \times [k,r] \times {{\mathbb R}}^+}.$$ Since for every $k < r \in {{\mathbb{Z}}}$ we have $\phi((\Lambda \times G) \times [k,r] \times {{\mathbb R}}^+) = (r-k)\nu(\Lambda \times G) < +\infty$ by Assumptions \[assump\], with probability one we have that for every $k < r \in {{\mathbb{Z}}}$ the random particle configurations $\Pi_{(\Lambda \times G) \times [k,r] \times {{\mathbb R}}^+}$ are all finite. By this concludes the proof. By the discussion above, Proposition \[localfinit\] yields that for $\Lambda \in {{\mathcal B}}^0_S$ and $\eta \in \mathcal{N}(S\times G)$ the process $\mathcal{K}^{\Lambda|\eta}$ is well defined and constitutes an interacting birth and death process. Moreover, $\mathcal{K}^{\Lambda|\eta}$ is stationary due to the time translational invariance of its construction and that of $\Pi$. Infinite-volume dynamics ------------------------ ### Stationary dynamics As stated before, some complications arise when lifting the restriction of finite volume in the dynamics. The procedure to define the unrestricted FFG process in the entire space $S \times G$ is completely analogous to that of the finite volume case: it suffices to take $\Lambda=S$ everywhere in the previous section. Thus, the FFG process $\mathcal{K}$ on the whole space $S \times G$ is defined by the formula $$\label{keptformula2} \mathcal{K} = \{ (\gamma_x,t,s) \in \Pi : F(\gamma_x,t,s) \leq M(\gamma_x | \mathcal{K}_{t^-}) \}.$$ Following the analysis of the previous section in this context, we see that in order for $\mathcal{K}$ to be well defined we must guarantee that for every cylinder $C \in \Pi$ its clan of ancestors $\A(C)$ spans only over a finite number of generations. Notice that the argument used in Proposition \[localfinit\] will not go through this time as in general we have $\nu(S \times G)=+\infty$. Therefore, we will need to impose additional conditions on both $\nu$ and $H$ besides those on Assumptions \[assump\] to guarantee that $\mathcal{K}$ is well defined in this case. This is the content of the next proposition. \[finit1\] If there exists a measurable function $q: S \times G \to {{\mathbb R}}$ satisfying $\inf_{\gamma_x \in S \times G} q(\gamma_x) \geq 1$ and such that $$\label{hdiluted} \alpha_q:=\sup_{\gamma_x \in S\times G} \left[\frac{e^{-\Delta E}}{q(\gamma_x)} \int_{I(\{\gamma_x\})} q(\tilde{\gamma}_y)d\nu(\tilde{\gamma}_y)\right] < 1$$ then $\mathcal{A}^t(\Lambda \times G)$ is finite for every $t \in {{\mathbb R}}$ and $\Lambda \in {{\mathcal B}}^0_S$ almost surely. Whenever holds we say that $\nu$ and $H$ satisfy the (F1)-diluteness condition with size function $q$ and that the associated model is *heavily diluted*. Thus, whenever dealing with a heavily diluted model we have that $\mathcal{K}$ is well defined and constitutes a stationary interacting birth and death process on the entire space $S \times G$. We postpone the proof of Proposition \[finit1\] until Section \[ancestors\]. ### Dynamics on a bounded time window {#forwarddynamics} One can wonder whether it is possible that, upon relaxing the conditions on Proposition \[finit1\], the FFG process remains well defined on the infinite volume but perhaps only for a bounded time window, i.e. if given $t_1 < t_2 \in {{\mathbb R}}$ we replace $\phi$ in the construction above by $$\phi_{[t_1,t_2]}= \nu \times e^{-\Delta E} \mathcal{L}_{[t_1,t_2]} \times \mathcal{E}^1.$$ This will occur if for every $\Lambda \in {{\mathcal B}}^0_{S}$ one has that $$\label{clanbounded} \A^{[t_1,t_2]}(\Lambda \times G) := \{ C \in \A^{t_2}(\Lambda \times G) : birth(C) \geq t_1 \}$$ spans only over a finite number of generations. The next proposition shows that this is the case whenever the coefficient $\alpha_q$ defined on is finite. \[finit2\] If there exists a measurable function $q: S \times G \to {{\mathbb R}}$ satisfying $\inf_{\gamma_x \in S \times G} q(\gamma_x) \geq 1$ and such that $$\label{wdiluted} \alpha_q:=\sup_{\gamma_x \in S\times G} \left[\frac{e^{-\Delta E}}{q(\gamma_x)} \int_{I(\{\gamma_x\})} q(\tilde{\gamma}_y)d\nu(\tilde{\gamma}_y)\right] < +\infty$$ then $\mathcal{A}^{[t_1,t_2]}(\Lambda \times G)$ is finite for every $t_1 < t_2 \in {{\mathbb R}}$ and $\Lambda \in {{\mathcal B}}^0_{S}$ almost surely. Whenever holds we say that $\nu$ and $H$ satisfy the (F2)-diluteness condition with size function $q$ and that the associated model is *well diluted*. Let us observe that due to Proposition \[finit2\] we have that whenever a model is well diluted it is possible to define the FFG dynamics as a forward time evolution on ${{\mathbb R}}^+$ for any initial condition $\sigma \in \mathcal{N}(S \times G)$. Indeed, given any particle configuration $\sigma \in \mathcal{N}(S \times G)$ and a family $(L_{(\gamma_x,i)})_{(\gamma_x,i)\in [\sigma]}$ of i.i.d. exponential random variables of parameter 1 independent of $\Pi$ we may set $$\overline{\Pi}^\sigma = \overline{\Pi} \cup \{(\gamma_x,0, L_{(\gamma_x,i)},0) : (\gamma_x,i) \in [\sigma] \}$$ and define $(\mathcal{K}^\sigma_t)_{t \geq 0}$ by the formula $$\label{keptformula3} \mathcal{K}^\sigma = \{ (\gamma_x,t,s) \in \Pi^\sigma : F(\gamma_x,t,s) \leq M(\gamma_x | \mathcal{K}_{t^-}) \}$$ where $\Pi^\sigma$ denotes the projection of $\overline{\Pi}^\sigma$ onto $\mathcal{C}^+:= (S\times G) \times {{\mathbb R}}^+ \times {{\mathbb R}}^+$ and by convention we set $\mathcal{K}_{0^-}\equiv \emptyset$. Notice that, even though by Proposition \[finit2\] we have that $\A^{[0,t]}(\Lambda \times G)$ is finite for every $t \geq 0$ and $\Lambda \in {{\mathcal B}}^0_S$ almost surely, the clan of ancestors associated to these forward dynamics contains also cylinders corresponding to the initial configuration $\sigma$ and therefore it may not be finite (unless $\sigma$ has a finite local interaction range). Nonetheless, under the (F2)-diluteness condition it will always span over a finite number of generations and so $\mathcal{K}^\sigma$ is ultimately well defined. Furthermore, since we have assigned a $0$ flag value to every particle in the initial condition $\sigma$, we get that the initial condition is always kept in $\mathcal{K}^\sigma$ even if $\sigma$ is a particle configuration forbidden by $H$. We prove Proposition \[finit2\] in the next section. Finiteness criteria for the clan of ancestors {#ancestors} --------------------------------------------- The aim of this section is to give the proofs of Propositions \[finit1\] and \[finit2\], and to investigate some of their consequences as well. In both proofs we shall make use of the crucial fact that each clan of ancestors can be contained in the offspring of some branching process. This is the content of the following lemma. \[domilema\] Given $\Lambda \in {{\mathcal B}}^0_S$ and $t \in {{\mathbb R}}$ there exists a family of random sets $({{\mathcal B}}_n)_{n \in {{\mathbb N}}_0} \subseteq \mathcal{C}$ such that 1. $\A^t_0(\Lambda \times G) = {{\mathcal B}}_0$ 2. $\displaystyle{\bigcup_{i=0}^{n} \A^t_{i}(\Lambda \times G) \subseteq \bigcup_{i=0}^{n} {{\mathcal B}}_i}$ for every $n \in {{\mathbb N}}$ 3. Conditional on $({{\mathcal B}}_i)_{0\leq i \leq n}$, ${{\mathcal B}}_{n+1}$ is a Poisson process with intensity measure $\displaystyle{\sum_{C \in {{\mathcal B}}_n} \phi_{\mathcal{P}(C)}.}$ Consider the space $\M_t$ of particle configurations $\zeta$ on $\mathcal{C}_t:=(S\times G)\times (-\infty,t] \times {{\mathbb R}}^+$ such that 1. $\zeta$ is finite 2. No two cylinders in $\zeta$ have the same time of birth. For $\zeta \in \M_t$ we shall set $\A_0(\zeta):=\zeta$ and for $n \in {{\mathbb N}}$ write $$\displaystyle{\A_n(\zeta) := \bigcup_{C \in \zeta} \A_n(C)}.$$ Furthermore, suppose that we have ordered the elements of $\zeta$ in some particular way. Then, if $C_1 \preceq \dots \preceq C_k$ denote the ordered elements of $\zeta$, for each $i=1,\dots,k$ we define $$\mathcal{P}_\zeta(C_i) = \mathcal{P}(C_i) - \bigcup_{j=1}^{i-1} [C_j \cup\mathcal{P}(C_j)].$$To define the family $({{\mathcal B}}_n)_{n \in {{\mathbb N}}_0}$ first we shall fix $\zeta \in \M_t$ and construct a collection of sets $({{\mathcal B}}_n(\zeta))_{n \in {{\mathbb N}}_0}$ such that for every $n \in {{\mathbb N}}_0$ the following properties are satisfied: 1. ${{\mathcal B}}_n(\zeta)$ belongs to $\M_t$ almost surely. 2. Conditional on ${{\mathcal B}}_0(\zeta),\dots,{{\mathcal B}}_n(\zeta)$, the random set ${{\mathcal B}}_{n+1}(\zeta)$ is a Poisson process on ${{\mathcal C}}_t$ with intensity measure $\sum_{C \in {{\mathcal B}}_n(\zeta)} \phi_{\mathcal{P}(C)}.$ 3. $\displaystyle{\bigcup_{i=0}^{n} \A_i(\zeta) \subseteq \bigcup_{i=0}^{n} {{\mathcal B}}_i(\zeta).}$ We start by setting ${{\mathcal B}}_0(\zeta):=\zeta$ and now proceed with the construction of the set ${{\mathcal B}}_1(\zeta)$. First we order the elements of $\zeta$ according to their times of birth, i.e. $\zeta=\{C_1,\dots,C_k\}$ where $0 \leq b_{C_1} < \dots < b_{C_k} \leq t$. Then continue by considering a collection $\Pi^{(1,1)},\dots,\Pi^{(1,k)}$ of independent Poisson processes on ${{\mathcal C}}$ of intensity measure $\phi$ such that $\Pi^{(1,1)}=\Pi$ and defining for each $i=1,\dots,k$ $${{\mathcal B}}_\zeta(C_i) := \Pi^{(1,i)}_{\mathcal{P}(C_i) - \mathcal{P}_\zeta(C_i)} \cup \Pi_{\mathcal{P}_\zeta(C_i)}.$$ If we set $\displaystyle{{{\mathcal B}}_1(\zeta):=\bigcup_{i=1}^k {{\mathcal B}}_\zeta(C_i)}$ then ${{\mathcal B}}_1(\zeta)$ satisfies the properties stated above. Indeed: 1. Each ${{\mathcal B}}_\zeta(C_i)$ is a Poisson process with intensity measure $\phi_{\mathcal{P}(C_i)}$ by virtue of the $\Pi^{(1,1)},\dots,\Pi^{(1,k)}$ and the disjointness of $\mathcal{P}(C_i) - \mathcal{P}_\zeta(C_i)$ and $\mathcal{P}_\zeta(C_i)$. 2. The independence of ${{\mathcal B}}_\zeta(C_1),\dots,{{\mathcal B}}_\zeta(C_k)$ follows from the independence of the $\Pi^{(1,i)}$ and the fact that $\mathcal{P}_\zeta(C_i) \cap \mathcal{P}_\zeta(C_j)=\emptyset$ for $i \neq j$. Together with (1) this gives (ii). 3. Property (iii) follows upon noticing that for $i=1,\dots,k$ $$\A_1(C_i) - \bigcup_{j=1}^{i-1} \A_1(C_j) \subseteq \Pi_{\mathcal{P}_\zeta(C_i)}.$$ 4. Property (i) is also a consequence of (1) and (2) since for each $i=1,\dots,k$ $$\phi_\nu(\mathcal{P}(C_i)) = e^{-\Delta E} \int_{I(basis(C_i))} \int_{-\infty}^{b_{C_i}} \int_{b_{C_i} - t}^{+\infty} e^{-s}ds dt d\nu < +\infty.$$ Having constructed ${{\mathcal B}}_1(\zeta)$, we now define the next generations in an inductive manner. For this we shall need to consider an array of ${{\mathbb N}}\times {{\mathbb N}}$ Poisson processes on ${{\mathcal C}}$ such that: 1. $\Pi^{(n,k)}$ is a Poisson process with intensity measure $\phi$ for every $n,k \in {{\mathbb N}}$ 2. $\Pi^{(n,1)}=\Pi$ for every $n \in {{\mathbb N}}$ 3. The processes $\{ \Pi^{(n,k)} : n \in {{\mathbb N}}, k \geq 2\}$ are independent of each other and also of $\Pi$. Suppose now that we have constructed the first $n$ generation of sets ${{\mathcal B}}_1(\zeta),\dots,{{\mathcal B}}_n(\zeta)$ and let us construct the next generation, ${{\mathcal B}}_{n+1}(\zeta)$. Order each of the constructed generations separately by their times of birth and write for each $j=1,\dots,n$ $${{\mathcal B}}_j(\zeta) = \{ C^{(j,1)}, \dots, C^{(j,k_j)} \}.$$ Now let us consider the joint configuration $\zeta^{(n)}=\{ C^{(j,i)} : 1 \leq j \leq n \text{ and }1 \leq i \leq k_j \}$ ordered by the dictionary order, i.e. $C^{(j,i)} \preceq C^{(j',i')}$ if either $j < j'$ or $j=j'$ and $i \leq i'$. We then define $${{\mathcal B}}_{n+1}(\zeta): = \bigcup_{i=1}^{k_n} {{\mathcal B}}_{\zeta^ {(n)}}(C^{(n,i)})$$ where $${{\mathcal B}}_{\zeta^{(n)}}(C^{(n,i)}) = \Pi^{(n+1,i)}_{\mathcal{P}(C^{(n,i)}) - \mathcal{P}_\zeta(C^{(n,i)})} \cup \Pi_{\mathcal{P}_\zeta(C^{(n,i)})}.$$ Following a similar argument to the one given above it is possible show by inductive hypothesis that ${{\mathcal B}}_{n+1}(\zeta)$ satisfies properties (i), (ii) and (iii). Finally, having defined the collection $({{\mathcal B}}_n(\zeta))_{n \in {{\mathbb N}}_0}$ for each $\zeta \in \M_t$, for each $n \in {{\mathbb N}}_0$ we set $${{\mathcal B}}_n := {{\mathcal B}}_n\left( \mathcal{A}^t_0(\Lambda \times G)\right).$$ One can check that, by construction of $({{\mathcal B}}_n(\zeta))_{n \in {{\mathbb N}}_0}$, the family $({{\mathcal B}}_n)_{n \in {{\mathbb N}}}$ satisfies all the desired properties. This concludes the proof. Let us first fix $\Lambda \in {{\mathcal B}}^0_S$ and $t \in {{\mathbb R}}$ and consider the family of random sets $({{\mathcal B}}_n)_{n \in {{\mathbb N}}_0}$ satisfying the conditions in the Domination lemma. By condition (ii) of this lemma we see that if we wish to show that $\A^t(\Lambda \times G)$ is almost surely finite it will suffice to prove that $\sum_{n \in {{\mathbb N}}_0} |{{\mathcal B}}_n| <+ \infty$ almost surely. But this will follow immediately once we show that for every $n \in {{\mathbb N}}_0$ $$\label{cotabranching} {{\mathbb E}}\left( \sum_{C \in {{\mathcal B}}_n} q(basis(C)) \bigg| {{\mathcal B}}_0 \right) \leq \left(\sum_{C \in {{\mathcal B}}_0} q(basis(C))\right)\alpha^n_q.$$Indeed, if holds then since $\inf_{\gamma_x \in S \times G} q(\gamma_x) \geq 1$ we have $$\begin{aligned} P\left( \sum_{n \in {{\mathbb N}}_0} |{{\mathcal B}}_n| = +\infty \bigg| {{\mathcal B}}_0 \right) & \leq \lim_{k \rightarrow +\infty} P\left( \sum_{n \in {{\mathbb N}}_0} \sum_{C \in {{\mathcal B}}_n} q(basis(C)) > k \bigg| {{\mathcal B}}_0 \right) \\ \\ & \leq \lim_{k \rightarrow +\infty} \frac{\sum_{n \in {{\mathbb N}}_0} {{\mathbb E}}( \sum_{C \in {{\mathcal B}}_n} q(basis(C)) | {{\mathcal B}}_0 )}{k} \\ \\ & \leq \lim_{k \rightarrow +\infty} \frac{\sum_{C \in {{\mathcal B}}_0} q(basis(C))}{k(1-\alpha_q)} = 0\end{aligned}$$ since $P( |{{\mathcal B}}_0| < +\infty ) = 1$. From this we get that the unconditional probability is also null. Thus, in order to obtain we first notice that by (iii) and a simple calculation yields for every $n \in {{\mathbb N}}_0$ $$\label{tower} {{\mathbb E}}\left( \sum_{C \in {{\mathcal B}}_{n+1}} q(basis(C)) \Bigg| {{\mathcal B}}_n,\dots,{{\mathcal B}}_0\right) \leq \left(\sum_{C \in {{\mathcal B}}_{n}} q(basis(C))\right) \alpha_q.$$ Now, since clearly holds for $n=0$, the validity for every $n \in {{\mathbb N}}_0$ follows upon induction by applying and the tower property of conditional expectation. Finally, to show that with probability one this holds for every $\Lambda \in {{\mathcal B}}^0_S$ and $t \in {{\mathbb R}}$ simultaneously, we take $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_S$ such that $\Lambda_n \nearrow S$ and observe that (with probability one) given $\Lambda \in {{\mathcal B}}^0_S$ and $t \in {{\mathbb R}}$ there exists $n_0 \in {{\mathbb N}}$ and $r \in \mathbb{Q}$ such that $$\A^t(\Lambda \times G) \subseteq \A^r(\Lambda_{n_0} \times G).$$ Since there are only countable possibilities for $n_0$ and $r$ and we have shown that for every fixed pair $n,r$ the set $\A^r(\Lambda_{n} \times G)$ is finite almost surely, this yields the result. By the same reasoning as in the proof of Proposition \[finit1\] it will suffice to show that for each $r < l \in \mathbb{Q}$ and $n \in {{\mathbb N}}$ the random set $\A^{[r,l]}(\Lambda_n \times G)$ is almost surely finite, where $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_S$ is such that $\Lambda_n \nearrow S$. But we can show this by performing an inductive procedure once we manage to prove the following two facts: 1. There exists $\delta > 0$ such that if $0 < t-s < \delta$ then $\A^{[s,t]}(\Lambda \times G)$ is 2. If $|h| < \delta$ and $\A^{[s,t]}(\Lambda \times G)$ is finite almost surely then $\A^{[s-h,t]}(\Lambda \times G)$ is also finite almost surely. To establish (1) we fix $\Lambda \in {{\mathcal B}}^0_S$, $s < t$ and construct similarly to the Domination lemma a family of random sets $({{\mathcal B}}_n)_{n \in {{\mathbb N}}}$ satisfying 1. $\A^{[s,t]}_0(\Lambda \times G) = {{\mathcal B}}_0$ 2. $\displaystyle{\bigcup_{i=0}^{n} \A^{[s,t]}_{i}(\Lambda \times G) \subseteq \bigcup_{i=0}^{n} {{\mathcal B}}_i}$ for every $n \in {{\mathbb N}}$ 3. Conditional on $({{\mathcal B}}_i)_{0 \leq i \leq n}$, ${{\mathcal B}}_{n+1}$ is a Poisson process with intensity $\sum_{C \in {{\mathcal B}}_n} (\phi_{[s,t]})_{\mathcal{P}(C)}$ where for $n \in {{\mathbb N}}$ we set $\A^{[s,t]}_{n}(\Lambda \times G) = \A^{t}_{n}(\Lambda \times G) \cap \A^{[s,t]}(\Lambda \times G)$. By performing a similar computation to the one yielding we obtain $$\label{tower2} {{\mathbb E}}\left( \sum_{C \in {{\mathcal B}}_{n+1}} q(basis(C)) \Bigg| {{\mathcal B}}_n,\dots,{{\mathcal B}}_0\right) \leq \left(\sum_{C \in {{\mathcal B}}_{n}} q(basis(C))\right) \tilde{\alpha}_q.$$where $\tilde{\alpha}_q := \alpha_q (1- e^{-(t-s)})$. Since $\alpha_q < +\infty$, we may take $t-s$ small enough so as to guarantee that $\tilde{\alpha}_q < 1$. From this one obtains (1) by proceeding as in Proposition \[finit1\]. To see (2) we first notice that if $\A^{[s,t]}(\Lambda \times G)$ is finite then there exists a (random) set $\Lambda' \in {{\mathcal B}}^0_S$ such that the basis of every cylinder in $\A^{[s,t]}(\Lambda \times G)$ belongs to $\Lambda' \times G$. Furthermore, since $\nu(I(\Lambda'\times G)) < +\infty$ we have that there exists another (random) set $\Lambda'' \in {{\mathcal B}}^0_S$ such that the basis of every cylinder in $\Pi_{I(\Lambda'\times G) \times [s-h,s)\times {{\mathbb R}}^+}$ belongs to $\Lambda'' \times G$. Then, it is not hard to see that $$\label{ancesinduc} \A^{[s-h,t]}(\Lambda \times G) \subseteq A^{[s,t]}(\Lambda \times G) \cup A^{[s-h,s]}(\Lambda'' \times G)$$ Together with (1) (for $\Lambda''$ instead of $\Lambda$) , implies (2), which concludes the proof. When dealing with a heavily diluted model, the finiteness of every clan of ancestors forces the FFG dynamics to exhibit a loss of memory property. In particular, we obtain the convergence of the forward dynamics to the invariant measure of the stationary dynamics. More precisely, we have the following proposition. \[uni\] Let us suppose that $\nu$ and $H$ satisfy the (F1)-diluteness condition. Then for any initial particle configuration $\sigma \in \mathcal{N}_H(S\times G)$ as $t \rightarrow +\infty$ we have $$\mathcal{K}^\sigma_t \overset{loc}{\longrightarrow} \mathcal{K}_0.$$ Given a particle configuration $\sigma \in \mathcal{N}_H(S\times G)$ the idea is to construct a coupling of $\mathcal{K}^\sigma_t$ for each $t \geq 0$ together with $\mathcal{K}_0$ where the local convergence can be easily verified. To achieve this, recall that the forward FFG dynamics are built from a set $(L_{\gamma_x,i})_{(\gamma_x,i) \in [\sigma]}$ of exponential random variables of parameter 1 and a Poisson process $\overline{\Pi}$ on $\mathcal{C} \times [0,1]$ with intensity measure $\nu \times \mathcal{L}\times \mathcal{E}_1 \times \mathcal{L}_{[0,1]}$. More precisely, for each $\Lambda \in {{\mathcal B}}^0_S$ there exists a measurable function $\psi_\Lambda$ such that for each $t > 0$ $$(\mathcal{K}^\sigma_t)_{\Lambda \times G} = \psi_\Lambda \left( \mathcal{A}^{[0,t]}_\sigma(\Lambda \times G), F(\mathcal{A}^{[0,t]}_\sigma(\Lambda \times G))\right)$$ where $\mathcal{A}^{[0,t]}_\sigma(\Lambda \times G)$ denotes the clan of ancestors of $\Lambda \times G$ defined as in but using $\Pi^{\sigma}$ instead of $\Pi$ and $F(\mathcal{A}^{[0,t]}_\sigma(\Lambda \times G))$ denotes its corresponding set of flags. Furthermore, both the clan of ancestors and its flags are determined by the evolution of the process $\overline{\Pi}^\sigma$ in the time interval $[0,t]$, i.e., there exists a second measurable function $\theta$ such that $$(\mathcal{K}^\sigma_t)_{\Lambda \times G}= \psi_\Lambda \circ \theta \left(\left(\overline{\Pi}^{\sigma}_s\right)_{s \in [0,t]}\right)$$ Similarly, the stationary FFG process is defined for each $t \in {{\mathbb R}}$ and $\Lambda \in {{\mathcal B}}^0_S$ by the formula $$\label{statproc} (\mathcal{K}_t)_{\Lambda \times G} = \psi_\Lambda \circ \theta \left( \left(\overline{\Pi}_s\right)_{s \in (-\infty,t]}\right).$$ We shall construct our coupling by taking the Poisson process $\overline{\Pi}$ together with the family $(L_{\gamma_x,i})_{(\gamma_x,i) \in [\sigma]}$ of independent exponential random variables of parameter 1 and for each $t > 0$ defining the $t$-shifted free process $\overline{\Pi}^{\sigma,(t)}$ with initial condition $\sigma$ by the formula $\overline{\Pi}^{\sigma,\,(t)} = \overline{\Pi}^{\sigma}_0 \cup \overline{\Pi}^{\,(t)}$ where $$\overline{\Pi}^{\,(t)} = \{ (\gamma_x, r + t , s, u) \in \mathcal{C} \times [0,1] : (\gamma_x, r , s, u) \in \overline{\Pi} \,,\, r > - t \}.$$ We then define for each $t > 0$ the random particle configuration $X_t$ by the formula $$(X_t)_{\Lambda \times G} := \psi_\Lambda \circ \theta \left( \left(\overline{\Pi}^{\,\sigma,\,(t)}_s\right)_{s \in [0,t]} \right)$$ for every $\Lambda \in {{\mathcal B}}^0_S$ and set $X_\infty := \mathcal{K}_0$, where $\mathcal{K}$ is defined exactly as in , In other words, $X_t$ is the current state of the FFG process started at time $-t$ with initial condition $\sigma$ and underlying free process $\overline{\Pi}$, after having evolved for a time period of length $t$. Let us observe that each $X_t$ has the same distribution as $\mathcal{K}^\sigma_t$ by the time translational invariance of $\overline{\Pi}$. Moreover, this construction possesses a crucial property: the free processes $\overline{\Pi}^{\,\sigma,\,(t)}$ are “coupled backwards” with $\overline{\Pi}$, i.e. for $t > 0$ we have $$\label{coupleback} \overline{\Pi}^{\,\sigma,\,(t)}_{s} = \overline{\Pi}_{s-t}.$$ Using this property we shall prove that for any given $\Lambda \in \mathcal{B}^0_{S}$ one has $(X_t)_{\Lambda \times G} = (X_\infty)_{\Lambda \times G}$ for every $t$ sufficiently large, a fact from which we immediately obtain the validity of (i). Indeed, since the model is heavily diluted we know that $\mathcal{A}^0(\Lambda \times G)$ is finite almost surely. In particular, there exist (random) $t^*$ and $\Lambda' \in \mathcal{B}^0_{S}$ such that $$\mathcal{A}^0(\Lambda \times G) \subseteq (\Lambda' \times G) \times (-t^*,0] \times {{\mathbb R}}^+.$$Moreover, since the initial condition $\sigma$ has a locally finite interaction range with respect to $H$ we have that $\sigma(I(\Lambda'\times G)) < +\infty$ and so $t^\sigma_\Lambda := \sup_{(\gamma_x,i) \in [\sigma|_{I(B_\Lambda)}]} L_{\gamma_x,i} < +\infty$ as well. Therefore, if $t > t^* + t^\sigma_\Lambda$ then by we have that $$\mathcal{A}^{[0,t]}_\sigma(\Lambda \times G) = \{ (\gamma_x,r+t,s) : (\gamma_x,r,s) \in \mathcal{A}^{0}(\Lambda \times G)\}.$$ i.e., all the cylinders in the initial configuration $\sigma$ which could possibly interact with the clan of ancestors of the cylinders in $(\Pi^{\,\sigma,\,(t)}_t)_{\Lambda \times G}$ in the forward dynamics die out before reaching it and, as a consequence, this clan coincides with the ancestors of cylinders in $(\Pi_0)_{\Lambda \times G}$ in the stationary dynamics modulo some appropriate time shift. In particular, we get that $(X_t)_{\Lambda \times G} = (X_\infty)_{\Lambda \times G}$ if $t > t^*+t^\sigma_\Lambda$ as we wanted to show. Reversible measures for the FFG dynamics ---------------------------------------- The purpose of this section is to study the relationship between invariant measures for the FFG dynamics and Gibbs measures of the associated diluted model. More precisely, we will show that Gibbs measures are reversible for the corresponding FFG dynamics. Together with Proposition \[uni\], this will imply the existence of a unique Gibbs measure in all heavily diluted models. We begin our task by introducing the global and local evolution semigroups for the dynamics. However, in order to define the global semigroup we need the infinite-volume forward dynamics to be well defined. Hence, for the rest of this section we assume that the model under consideration is well diluted. Given $t > 0$, a bounded subset $\Lambda \in \mathcal{B}^0_{S}$ and a particle configuration $\eta \in \mathcal{N}(S \times G)$ we define the operators $S_t$ and $S^{\Lambda|\eta}_t$ on the class of bounded functions $f:\mathcal{N}(S \times G) \to {{\mathbb R}}$ by the formulas $$S_t(f) (\sigma) = {{\mathbb E}}\left(f(\mathcal{K}_t^\sigma)\right) \hspace{2cm}\text{ and }\hspace{2cm}S_t^{\Lambda|\eta} (f) (\sigma) = {{\mathbb E}}( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_t) )$$ where $\mathcal{K}^\sigma$ denotes the infinite-volume forward process with initial condition $\sigma$ and $\mathcal{K}^{\Lambda,\,\sigma,\,\eta}$ is the forward process on $\Lambda$ with boundary condition $\eta$ and initial condition $\sigma_{\Lambda \times G} \cdot \eta_{\Lambda^c \times G}$.[^5] The families of operators $(S_t)_{t \geq 0}$ and $(S_t^{\Lambda|\eta})_{t \geq 0}$ are called the *global evolution semigroup* and *local evolution semigroup on $\Lambda$ with boundary condition $\eta$*, respectively. Both families $(S_t)_{t \geq 0}$ and $(S_t^{\Lambda|\eta})_{t \geq 0}$ satisfy the semigroup property, i.e. $$S_{t} \circ S_s = S_{t+s} \hspace{2cm}\text{ and }\hspace{2cm}S_t^{\Lambda|\eta} \circ S_s^{\Lambda|\eta} = S_{t+s}^{\Lambda|\eta}$$ for every $t,s \geq 0$, and hence their name. Let $\mu$ be a measure on $\mathcal{N}(S \times G)$ and $\Lambda \in \mathcal{B}^0_{S}$. 1. We say that $\mu$ is *invariant* for the global FFG dynamics if for every $t \geq 0$ we have $$\int_{\mathcal{N}(S \times G)} S_t(f)(\sigma) d\mu(\sigma) = \int_{\mathcal{N}(S \times G)} f(\sigma) d\mu(\sigma)$$ for every bounded local function $f: \mathcal{N}(S \times G) \to {{\mathbb R}}$. 2. We say that $\mu$ is *invariant* for the local FFG dynamics on $\Lambda$ with boundary condition $\eta \in \mathcal{N}(S \times G)$ if 1. $\mu( \{ \xi \in \mathcal{N}({{\mathbb R}}^d \times G) : \xi_{\Lambda^c \times G}= \eta_{\Lambda^c \times G}\} ) = 1$ 2. For every $t \geq 0$ we have $$\int_{\mathcal{N}(S \times G)} S_t^{\Lambda|\eta}(f)(\sigma) d\mu(\sigma) = \int_{\mathcal{N}(S \times G)} f(\sigma) d\mu(\sigma)$$ for every bounded $\mathcal{F}_{\Lambda \times G}$-measurable function $f: \mathcal{N}(S \times G) \to {{\mathbb R}}$. 3. We say that $\mu$ is *reversible* for the global FFG dynamics if it also satisfies $$\label{rever} \int_{\mathcal{N}(S \times G)} g(\sigma) S_t(f)(\sigma) d\mu(\sigma) = \int_{\mathcal{N}(S \times G)} S_t(g)(\sigma) f(\sigma) d\mu(\sigma)$$ for every $t \geq 0$ and bounded local functions $f,g: \mathcal{N}(S \times G) \to {{\mathbb R}}$. 4. We say that $\mu$ is *reversible* for the local FFG dynamics on $\Lambda$ with boundary condition $\eta \in \mathcal{N}(S \times G)$ if it is invariant and also satisfies for all $t \geq 0$ $$\int_{\mathcal{N}(S \times G)} g(\sigma) S_t^{\Lambda|\eta}(f)(\sigma) d\mu(\sigma) = \int_{\mathcal{N}(S \times G)} S_t^{\Lambda|\eta}(g)(\sigma)f(\sigma) d\mu(\sigma)$$ for every $t \geq 0$ and bounded $\mathcal{F}_{\Lambda \times G}$-measurable functions $f,g: \mathcal{N}(S \times G) \to {{\mathbb R}}$. Our first step in studying the invariant measures for the FFG dynamics will be to derive an explicit formula for the generator of the local evolution semigroup valid for a sufficiently wide class of functions. Recall that, in general, the generator of a semigroup $(T_t)_{t \geq 0}$ is defined as $$L(f)(\sigma)= \frac{ dT_t(f)(\sigma) }{dt}\bigg|_{t=0} = \lim_{h \rightarrow 0^+} \frac{ T_{h}(f)(\sigma) - f(\sigma) }{h}$$ whenever $f$ is such that the limit exists for every particle configuration $\sigma \in \mathcal{N}(S \times G)$. \[generador\] For every $\Lambda \subseteq \mathcal{B}^0_{S}$ and $\eta \in \mathcal{N}(S \times G)$ the local evolution semigroup $(S^{\Lambda|\eta}_t)_{t \geq 0}$ has generator $L_{\Lambda|\eta}$ defined for any bounded ${{\mathcal F}}_{\Lambda \times G}$-measurable $f: \mathcal{N}(S \times G) \to {{\mathbb R}}$ by the formula $$L_{\Lambda|\eta}(f)(\sigma) = D_{\Lambda|\eta}(f)(\sigma) + B_{\Lambda|\eta}(f)(\sigma)$$ where $$D_{\Lambda|\eta}(f)(\sigma)=\sum_{ \gamma_x \in \langle \sigma_{\Lambda \times G} \rangle } \sigma(\gamma_x) \left(f( \sigma - \delta_{\gamma_x}) - f(\sigma)\right)$$ and $$B_{\Lambda|\eta}(f)(\sigma)=\int_{\Lambda \times G} e^{-\Delta E_{\sigma_{\Lambda \times G} \,\cdot\, \eta_{\Lambda^c \times G}}(\gamma_x)}\left(f( \sigma + \delta_{\gamma_x}) - f(\sigma)\right) d\nu(\gamma_x).$$ Given a bounded ${{\mathcal F}}_{\Lambda \times G}$-measurable function $f: \mathcal{N}(S \times G) \to {{\mathbb R}}$ we must show that for every particle configuration $\sigma \in \mathcal{N}(S \times G)$ we have $$\lim_{h \rightarrow 0^+} \frac{ {{\mathbb E}}( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) - f(\sigma)) }{h} = L(f)(\sigma).$$ If we write $B = \Lambda \times G$ then notice that we have the decomposition $$\frac{ {{\mathbb E}}( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))}{h}= \sum_{k=0}^\infty \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{B_k})}{h}$$ where $B_k = \{ \Pi( B \times (0,h] \times {{\mathbb R}}_+) = k \}$ for each $k \in {{\mathbb N}}$. We shall deal with each of these terms separately. If for every $j=1,\dots,\sigma(B)$ we write $L^{(j)}_B$ for the $j$-th order statistic of the family $(L_{(\gamma_x,i)})_{(\gamma_x,i)\in [\sigma_B]}$ then the term with $k=0$ can be decomposed into two parts $$\frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{B_0})}{h} = \frac{ {{\mathbb E}}( (f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))(\mathbbm{1}_{\{ L^{(1)}_B \leq h < L^{(2)}_B\} \cap B_0}+ \mathbbm{1}_{\{ L^{(2)}_B \leq h \}\cap B_0}))}{h}$$ since $f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h)=f(\sigma)$ on $\{ L^{(1)}_B > h\}\cap B_0$. Let us observe that due to the independence between $(L_{(\gamma_x,i)})_{(\gamma_x,i)\in [\sigma_B]}$ and $\Pi$, the first term in the right hand side can be rewritten as $$\sum_{(\gamma_x,i) \in [\sigma_B]} (f(\sigma - \delta_{\gamma_x}) - f(\sigma)) \frac{(1-e^{-h})e^{-(\sigma(B)-1 + \nu(B))h}}{h}$$ where $\nu(B) < +\infty$ by hypothesis. Thus, we obtain that $$\lim_{h \rightarrow 0^+} \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{\{ L^{(1)}_B \leq h < L^{(2)}_B\} \cap B_0})}{h} = D_{\Lambda|\eta}(f)(\sigma).$$ On the other hand, for the second term in the right hand side we have that $$\left| \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{\{ L^{(2)}_B \leq h \}\cap B_0})}{h}\right| \leq \frac{2\|f\|_\infty \binom{\sigma(B)}{2} (1-e^{-h})^2}{h} \longrightarrow 0$$ which establishes the case $k=0$. Now, to deal with the case $k=1$ notice that $$\frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{B_1})}{h} = \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))(\mathbbm{1}_{\{ L^{(1)}_B \leq h \} \cap B_1}+ \mathbbm{1}_{\{ L^{(1)}_B > h \}\cap B_1}))}{h}$$ where the first term in the right hand side satisfies $$\Bigg|\frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{\{ L^{(1)}_B \leq h \} \cap B_1})}{h}\Bigg| \leq 2 \|f\|_\infty (1-e^{-\sigma(B)h})e^{-\nu(B) h}\nu(B) \longrightarrow 0$$ and the second one equals $$\label{generaequ} \frac{ {{\mathbb E}}(( f( \sigma + (\Pi_h)_{B}) -f(\sigma))\mathbbm{1}_{\{ L^{(1)}_B > h , \Pi( C(B,h,\,\sigma|_{\Lambda \times G} \,\cdot\, \eta|_{\Lambda^c \times G}) ) = 1 \} \cap B_1})}{h}$$ where for $\xi \in \mathcal{N}(S \times G)$ we set $$C(B,h,\xi) = \{ (\gamma_x,t,s) \in \mathcal{C} : \gamma_x \in B , F(\gamma_x,t,s) \leq M(\gamma_x | \xi) , 0 < t \leq h , t+s > h \}.$$ By the expression on can be rewritten as $$\frac{e^{-(\sigma(B)+\nu(B))h}}{h} \int_0^h \int_{h-t}^\infty \left(\int_B e^{-\Delta E_{\sigma_{\Lambda \times G} \,\cdot\, \eta_{\Lambda^c \times G}}(\gamma_x)}\left(f( \sigma + \delta_{\gamma_x}) - f(\sigma)\right) d\nu(\gamma_x)\right) e^{-s} ds dt$$ from where a simple calculation yields $$\lim_{h \rightarrow 0^+} \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{\{ L^{(1)}_B > h \} \cap B_1})}{h} = B_{\Lambda|\eta}(f)(\sigma).$$ Finally, to deal with the cases when $k > 1$ let us observe that $$\left|\sum_{k=2}^\infty \frac{ {{\mathbb E}}(( f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_h) -f(\sigma))\mathbbm{1}_{B_k})}{h}\right| \leq \frac{2 \|f\|_\infty P( \Pi( B \times (0,h] \times {{\mathbb R}}_+) \geq 2 )}{h} \longrightarrow 0.$$ Together with the previous cases, this allows us to conclude the proof. Having established a proper formula for the generator of the local evolution semigroups we now show that the Boltzmann-Gibbs distributions are reversible for the local dynamics. We will do so with the aid of the following two lemmas. \[inv0\] Let $\Lambda \in {{\mathcal B}}^0_S$ and $\eta \in \mathcal{N}(S \times G)$. Then for for every pair $f,g$ of bounded ${{\mathcal F}}_{\Lambda \times G}$-measurable functions the Boltzmann-Gibbs distribution $\mu_{\Lambda|\eta}$ satisfies $$\label{genera} \int g L_{\Lambda|\eta}(f) d\mu_{\Lambda|\eta} = \int f L_{\Lambda|\eta}(g) d\mu_{\Lambda|\eta}.$$ By Proposition \[generador\] and symmetry it suffices to show that the two integrals $$\label{inta} \int_{\mathcal{N}(S \times G)}\left[\sum_{ \gamma_x \in \langle \sigma_{\Lambda \times G} \rangle} \sigma(\gamma_x)g(\sigma) f( \sigma - \delta_{\gamma_x})\right]d\mu_{\Lambda|\eta}(\sigma)$$ and $$\label{intb} \int_{\mathcal{N}(S\times G)} \left[\int_{\Lambda \times G} e^{-\Delta E_{\sigma_{\Lambda \times G} \,\cdot\, \eta_{\Lambda^c \times G}}(\gamma_x)}g(\sigma + \delta_{\gamma_x})f(\sigma)d\nu(\gamma_x)\right]d\mu_{\Lambda|\eta}(\sigma)$$ coincide. A simple calculation using yields that equals $$\frac{1}{Z_{\Lambda|\eta}}\sum_{n=1}^\infty \frac{e^{-\nu(B)}}{(n-1)!} \int_{B^n} g(\sigma^{(n)})f\left(\sigma^{(n)} - \delta_{\gamma^1_x}\right) e^{-H_{\Lambda|\eta}(\sigma^{(n)})} d\nu^n\left(\gamma_x^{(n)}\right)$$ and equals $$\frac{1}{Z_{\Lambda|\eta}}\sum_{n=0}^\infty \frac{e^{-\nu(B)}}{n!} \int_{B^n}e^{-H_{\Lambda|\eta}(\sigma^{(n)})} f(\sigma^{(n)})\left(\int_{B} e^{-\Delta E_{\sigma_{B} \,\cdot\, \eta_{B^c}}(\tilde{\gamma}_y)}g(\sigma^{(n)} + \delta_{\tilde{\gamma}_y})d\nu\left(\tilde{\gamma}_y\right)\right) d\nu^{n}\left(\gamma_x^{(n)}\right)$$ where we write $B=\Lambda \times G$, $\gamma_x^{(n)}=(\gamma_x^1,\dots,\gamma_x^n)$ and $\sigma^{(n)}= \sum_{i=1}^n \delta_{\gamma^i_x}$. The equality between and now follows upon a change of index in as a consequence of the Fubini-Tonelli theorem and (ii) in the consistent Hamiltonian property. \[inv1\] Let $f:\mathcal{N}(S \times G) \to {{\mathbb R}}$ be a bounded local function. Then for each particle configuration $\eta \in \mathcal{N}_H(S \times G)$ and $t \geq 0$ we have $$S^{\Lambda|\eta}_t(f) \longrightarrow S_t(f)$$ pointwise on $\mathcal{N}_H(S \times G)$ as $\Lambda \nearrow S$ for each $t \geq 0$. For $\Lambda \in \mathcal{B}^0_{S}$ consider the coupling of $\mathcal{K}^\sigma_t$ and $\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_t$ obtained by constructing these random particle configurations using the same Poisson process $\overline{\Pi}$ and exponential lifetimes $(L_{(\gamma_x,i)})_{(\gamma_x,i) \in [\sigma_{\Lambda \times G}]}$ for particles in $\sigma_{\Lambda \times G}$. Recall that by Proposition \[finit2\] we know that if $\sigma$ has a finite local interaction range then $\mathcal{A}^{[0,t]}_\sigma(\Lambda_f \times G)$ is almost surely finite and thus there exists $\Lambda' \in {{\mathcal B}}^0_S$ such that the basis of every cylinder in $\mathcal{A}^{[0,t]}_\sigma (\Lambda_f \times G)$ is contained in $\Lambda' \times G$. Furthermore, since $\eta$ is also of finite local interaction range, by taking any $\Lambda \in \mathcal{B}^0_{S}$ sufficiently large so that $\Lambda' \subseteq \Lambda$ and $\eta( I(\Lambda' \times G) \cap (\Lambda^c \times G) ) = 0$ we have that $(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_t)_ {\Lambda_f \times G} = (\mathcal{K}^{\sigma}_t)_{\Lambda_f \times G}$ and thus $f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_t) = f(\mathcal{K}^{\sigma}_t)$. In particular, we conclude that $f(\mathcal{K}^{\Lambda,\,\sigma,\,\eta}_t)$ converges almost surely to $f(\mathcal{K}^{\sigma}_t)$ as $\Lambda \nearrow S$. The assertion now follows at once from the dominated convergence theorem since $f$ is bounded. We are now ready to show the reversibility of Gibbs measures for the FFG dynamics. \[teoreversibilidad\] Any Gibbs measure $\mu$ for the diluted model $(\nu,H)$ is reversible for the global dynamics. Let $f$ and $g$ be bounded local functions and consider $\Lambda_0 \in \mathcal{B}^0_{S}$ such that $f$ and $g$ are both ${{\mathcal F}}_{\Lambda_0 \times G}$-measurable. Notice that by (i) in Lemma \[inv1\], Proposition \[lfir\] and the dominated convergence theorem we have that will hold if we show that $$\label{rever1} \int_{\mathcal{N}(S \times G)} g(\sigma) S^{\Lambda,\,\sigma}_t(f)(\sigma) d\mu(\sigma) = \int_{\mathcal{N}(S \times G)} f(\sigma) S^{\Lambda,\,\sigma}_t(g)(\sigma) d\mu(\sigma)$$ is satisfied for every $\Lambda \in \mathcal{B}^0_{S}$ with $\Lambda_0 \subseteq \Lambda$. Moreover, since $\mu$ is a Gibbs measure then we can rewrite as $$\int \int g(\sigma) S_t^{\Lambda|\eta}(f)(\sigma) d\mu_{\Lambda|\eta}(\sigma) d\mu(\eta)= \int \int f(\sigma) S_t^{\Lambda|\eta}(g)(\sigma) d\mu_{\Lambda|\eta}(\sigma) d\mu(\eta)$$ so that will follow if we prove that for each $\eta \in \mathcal{N}(S \times G)$ $$\label{rever2} \int_{\mathcal{N}(S \times G)} g(\sigma) S_t^{\Lambda|\eta}(f)(\sigma) d\mu_{\Lambda|\eta}(\sigma) = \int_{\mathcal{N}(S \times G)} f(\sigma) S_t^{\Lambda|\eta}(g)(\sigma) d\mu_{\Lambda|\eta}(\sigma)$$ holds for every $t \geq 0$ and $\Lambda \in \mathcal{B}^0_{S}$ containing $\Lambda_0$. In order to simplify the notation ahead, we shall fix $\Lambda \in \mathcal{B}^0_{S}$ containing $\Lambda_0$, $\eta \in \mathcal{N}(S\times G)$ and through the rest of the proof write $S_t := S^{\Lambda,\eta}_t$ for each $t \geq 0$ and $L:=L_{\Lambda | \eta}$. Now, to establish , given $0 < h \leq t$ let us begin by rewriting the left-hand side of as $$\label{rever3} h \int g L(S_{t-h}(f))d\mu_{\Lambda|\eta} + \int g\left[R_h(S_{t-h}(f)) + S_{t-h}(f)\right] d\mu_{\Lambda|\eta}$$ where for any bounded ${{\mathcal F}}_{\Lambda \times G}$-measurable function $u:\mathcal{N}(S \times G) \to {{\mathbb R}}$ the *error term* $R_h(u): \mathcal{N}(S \times G) \to {{\mathbb R}}$ is defined by the formula $$R_h(u)(\sigma) = S_h(u)(\sigma) - u(\sigma) - L(u)(\sigma)h.$$ By Lemma \[inv0\] we obtain that equals $$\label{rever4} h \int L(g) S_{t-h}(f)d\mu_{\Lambda|\eta} + \int g\left[R_h(S_{t-h}(f)) + S_{t-h}(f)\right] d\mu_{\Lambda|\eta}$$which, upon performing computations analogous to those made to obtain but in reverse order, turns into $$\label{rever5} \int S_h(g) S_{t-h}(f)d\mu_{\Lambda|\eta} + \int \left[ gR_h(S_{t-h}(f)) - R_h(g) S_{t-h}(f)\right]d\mu_{\Lambda|\eta}.$$ By iterating this procedure we ultimately obtain $$\label{reverfinal} \int g S_t(f) d\mu_{\Lambda|\eta} = \int S_t(g) f d\mu_{\Lambda|\eta} + \int R_{t,h}(g,f)d\mu_{\Lambda|\eta}$$ where $$R_{t,h}(g,f) = \sum_{k=1}^{\lceil\frac{t}{h}\rceil} \left( S_{(k-1)h} (g) R_h(S_{t-kh}(f)) - R_h(S_{(k-1)h} (g)) S_{t-kh}(f)\right).\footnote{There is a slight abuse of notation in the last term of the sum. The term corresponding to $k=\lceil\frac{t}{h}\rceil$ is actually $\left[S_{[\frac{t}{h}]h} (g) R_{t-[\frac{t}{h}]h}(f) - R_{t-[\frac{t}{h}]h}(S_{[\frac{t}{h}]h} (g)) f\right]$.}$$ Now, let us observe that from the proof of Proposition \[generador\] we get that for each bounded ${{\mathcal F}}_{\Lambda \times G}$-measurable function $u:\mathcal{N}(S \times G) \to {{\mathbb R}}$ and $\sigma \in \mathcal{N}(S \times G)$ there exists a positive constant $C$ depending only on $\nu(\Lambda \times G)$ such that for any $\eta \in \mathcal{N}(S \times G)$ and $0 < h < 1$ $$|R_h(u)(\sigma)| \leq C\|u\|_\infty (1+\sigma^2(\Lambda \times G)) h^2.$$ But since $$\begin{aligned} \int \sigma^2(\Lambda \times G) d\mu_{\Lambda|\eta}(\sigma) &\leq \frac{1}{Z_{\Lambda|\eta}}\int [\sigma(\Lambda \times G)]^2 d\pi^\nu_{\Lambda \times G}(\sigma) \\ \\ & \leq \frac{ \nu(\Lambda \times G) + \nu^2(\Lambda \times G)}{\pi^\nu ( N_{\Lambda \times G} = 0)} = e^{\nu(\Lambda \times G)}(\nu(\Lambda \times G) + \left(\nu(\Lambda \times G)\right)^2) < +\infty\end{aligned}$$ we obtain that there exists another positive constant $\hat{C}_t$, this time depending only on $\nu(\Lambda \times G)$ and $t$, such that for $0 < h < \min\{t,1\}$ we have $$\left|\int R_{t,h}(g,f)d\mu_{\Lambda|\eta}\right| \leq \hat{C}_t \|g\|_\infty \|f\|_\infty h \underset{h \to \,0}{\longrightarrow} 0.$$ Since the left-hand side of does not depend on $h$, by letting $h \rightarrow 0$ we conclude the proof. As a consequence of Theorem \[teoreversibilidad\] we obtain the following important result. \[teounigibbs\] $\\$ Let $(\nu,H)$ be a heavily diluted model on $\mathcal{N}(S \times G)$ and let $\mu$ be the invariant measure of the associated stationary global dynamics. Then the following holds: 1. For each $\Lambda \in \mathcal{B}^0_{S}$ and $\eta \in \mathcal{N}(S \times G)$ the Boltzmann-Gibbs distribution $\mu_{\Lambda|\eta}$ is the unique invariant measure of the local dynamics on $\Lambda$ with boundary condition $\eta$. 2. For any $\eta \in \mathcal{N}_H(S \times G)$ we have $\mu_{\Lambda|\eta} \overset{loc}{\longrightarrow} \mu$ as $\Lambda \nearrow S$. 3. $\mu$ is the unique Gibbs measure for the model. Statements (i) and (ii) in Theorem \[teounigibbs\] were obtained in [@FFG1] in the Ising contours model, while (iii) is a new result which we present here. Notice that, in general, (iii) is not a direct consequence of (ii) since the diluted model might not be of bounded local interaction range (see Proposition \[limitegibbs\]). Notice that, since the clans of ancestors are always finite for the local dynamics over any bounded region $\Lambda \in \mathcal{B}^0_{S}$, we can mimic the proof of Proposition \[uni\] to show that, for any $\eta \in \mathcal{N}(S \times G)$ and any bounded local function $f$, the local evolution $S^{\Lambda,\, \eta}_t (f)$ converges pointwise as $t \rightarrow +\infty$ to ${{\mathbb E}}( f( \mathcal{K}^{\Lambda|\eta}_0) )$, where $\mathcal{K}^{\Lambda|\eta}$ is the stationary local process on $\Lambda$ with boundary condition $\eta$. In particular, this implies that there is a unique invariant measure for the local dynamics on $\Lambda$ with boundary condition $\eta$ which coincides with the distribution of $\mathcal{K}^{\Lambda|\eta}$. Since by in the proof of Theorem \[teoreversibilidad\] we know that $\mu_{\Lambda|\eta}$ is invariant for these dynamics, we conclude (i). To see (ii), let us fix $\eta \in \mathcal{N}_H(S \times G)$ and given $\Lambda \in \mathcal{B}^0_{S}$ consider the coupling of $\mu$ and $\mu_{\Lambda|\eta}$ obtained by constructing the stationary local process $\mathcal{K}^{\Lambda|\eta}$ and the stationary global process $\mathcal{K}$ (which is well defined due to the heavy diluteness condition) using the same underlying process $\overline{\Pi}$. Now, given some bounded local function $f$, Proposition \[finit1\] gives that $\mathcal{A}^0(\Lambda_f \times G)$ is almost surely finite and thus there exists $\Lambda' \in {{\mathcal B}}^0_S$ such that the basis of every cylinder in $\mathcal{A}^0(\Lambda_f \times G)$ is contained in $\Lambda' \times G$. Furthermore, since $\eta$ is also of finite local interaction range, by taking any $\Lambda \in \mathcal{B}^0_{S}$ sufficiently large so that $\Lambda' \subseteq \Lambda$ and $\eta( I(\Lambda' \times G) \cap (\Lambda^c \times G) ) = 0$ we obtain that $(\mathcal{K}^{\Lambda|\eta}_0)_{\Lambda_f \times G} = (\mathcal{K}_0)_ {\Lambda_f \times G}$. Thus, we get that $f(\mathcal{K}^{\Lambda|\eta}_0)$ converges almost surely to $f(\mathcal{K}_0)$ as $\Lambda \nearrow S$ and, since $f$ is bounded, by the dominated convergence theorem and (i) we obtain (ii). , let us first show that $\mu$ is a Gibbs measure. Given $\Delta \in {{\mathcal B}}^0_S$ and a $A \in {{\mathcal F}}$, consider the mapping $g_{\Delta,A}:\mathcal{N}(S\times G) \rightarrow [0,1]$ given by the formula $$g_{\Delta,A}(\xi)=\mu_{\Delta|\xi}(A).$$ By the proof of Proposition \[limitegibbs\] we know that $g_{\Delta,A}$ is ${{\mathcal F}}_{(\Lambda_A \times G) \cup I(\Delta \times G)}$-measurable. Thus, to see that $\mu$ is a Gibbs measure it suffices to show that for any $\Lambda \in {{\mathcal B}}^0_S$ sufficiently large we have that $(\mathcal{K}^{\Lambda|\eta}_0)_{(\Lambda_A \times G) \cup I(\Delta \times G)} = (\mathcal{K}_0)_{(\Lambda_A \times G) \cup I(\Delta \times G)}$ holds for some $\eta \in \mathcal{N}_H(S \times G)$. Indeed, if this is the case then $$\lim_{\Lambda \nearrow S} g_{\Delta,A} (\mathcal{K}^{\Lambda|\eta}_0) = g_{\Delta,A}(\mathcal{K}_0)$$ and so by (ii), the consistency of the Boltzmann-Gibbs distributions and the dominated convergence theorem we have that $$\mu(A) = \lim_{\Lambda \nearrow S} \mu_{\Lambda|\eta}(A) = \lim_{\Lambda \nearrow S} {{\mathbb E}}(g_{\Delta,A} (\mathcal{K}^{\Lambda|\eta}_0)) = {{\mathbb E}}(g_{\Delta,A}(\mathcal{K}_0)) = \int \mu_{\Delta|\xi}(A) d\mu(\xi)$$ which proves that $\mu$ is a Gibbs measure. Now, since $\Pi_0( I(\Delta \times G) )< +\infty$ by the integrable local interaction range assumption, there exists $\Lambda^* \in {{\mathcal B}}^0_S$ such that $(\Pi_0)_{I(\Delta \times G)}$ is contained in $\Lambda^* \times G$. Hence, since $\mathcal{A}^0((\Lambda_A \cup \Lambda^*) \times G)$ is almost surely finite, by taking any $\Lambda \in \mathcal{B}^0_{S}$ sufficiently large so that the basis of every cylinder in $\mathcal{A}^0((\Lambda_A \cup \Lambda^*) \times G)$ is contained in $\Lambda \times G$ we conclude that $(\mathcal{K}^{\Lambda,\,\emptyset}_0)_{(\Lambda_A \times G) \cup I(\Delta \times G)} = (\mathcal{K}_0)_{(\Lambda_A \times G) \cup I(\Delta \times G)}$ as we wished to see. Having shown that $\mu$ is a Gibbs a measure, it only remains to show that it is unique. But if $\tilde{\mu}$ is a Gibbs measure for the model then by Proposition \[uni\] and assertion (ii) of Proposition \[lfir\] for any bounded local function $f$ we have that $$\lim_{t \rightarrow +\infty} \int_{\mathcal{N}(S \times G)} S_t (f) (\sigma) d\tilde{\mu} (\sigma) = \int_{\mathcal{N}(S \times G)} f(\sigma) d\mu(\sigma).$$ Since $\tilde{\mu}$ must also be an invariant measure for the global dynamics by Theorem \[teoreversibilidad\], from this we obtain $\tilde{\mu}=\mu$ and thus we conclude (iii). Notice that in Theorem \[teounigibbs\] we have actually showed that $\mu$ is the only invariant measure for the global dynamics which is supported on $\mathcal{N}_H(S \times G)$. Exponential mixing of Gibbs measures ------------------------------------ Our next goal is to study mixing properties for Gibbs measures of translation invariant heavily diluted models. Let us begin by settling what we understand by mixing properties and translation invariant models in this context. Throughout this section we assume that the allocation space is endowed with an operation $ + : S \times S \rightarrow S$ such that $(S,+)$ is a commutative group. We say that a diluted model $(\nu,H)$ is *translation invariant* if it satisfies the following properties: 1. $\nu$ is translation invariant: for every $a \in S$ we have $$\label{tidm} \nu = \nu \circ \tau_a^{-1}$$ where $\tau_a : S \times G \rightarrow S \times G$ is defined as $\tau_a(\gamma_x) = \gamma_{x+a}$. 2. $H$ is translation invariant: for every $\Lambda \in {{\mathcal B}}^0_{S}$, $\eta \in \mathcal{N}(S \times G)$ and $a \in S$ $$H_{\Lambda|\eta}( \tau_{-a} (\sigma )) = H_{\Lambda + a| \tau_a(\eta)}(\sigma)$$ for all $\sigma \in \mathcal{N}\left((\Lambda + a) \times G\right)$ with $\Lambda+a:=\{ x + a : x \in \Lambda\}$, where for any given particle configuration $\xi$ we define $\tau_a ( \xi)$ through the standard representation $$\tau_a(\xi) = \sum_{\gamma_x \in Q_\xi} m_\xi(\gamma_x) \delta_{\tau_a(\gamma_x)}.$$ Notice that as a straightforward consequence of Theorem \[teounigibbs\] we conclude that if $(\nu,H)$ is a translation invariant heavily diluted model then its unique Gibbs measure is translation invariant in the sense of . Also, observe that all models introduced in Section \[examples\] are translation invariant. A measure $\mu$ is said to satisfy the *exponential mixing property* when there exist $c_1,c_2 > 0$ such that for any pair of bounded local functions $f,g : \mathcal{N}(S \times G) \rightarrow {{\mathbb R}}$ with $d_S(\Lambda_f,\Lambda_g)$ sufficiently large (depending only on $c_1$) one has that $$\label{eqmixing} \left|\int f(\eta)g(\eta) d\mu(\eta) - \int f(\eta) d\mu(\eta)\int g(\eta) d\mu(\eta)\right| \leq \| f\|_\infty \|g\|_\infty e^{-c_1 d_S(\Lambda_f, \Lambda_g ) + c_2(\nu(\Lambda_f) + \nu(\Lambda_g))}.$$ Our purpose in this section is to show that Gibbs measures of translation invariant heavily diluted models satisfy the exponential mixing property. To achieve this, however, the size function $q$ will need to satisfy some further conditions. The necessary requirements are contained in Definition \[defigoodsize\] below. Consider the space $\M$ of particle configurations $\zeta$ on $\mathcal{C}$ such that 1. $\zeta$ is finite 2. No two cylinders in $\zeta$ have the same time of birth. Let us order the cylinders of $\zeta$ by time of birth, $\zeta = \{ C_1 , \dots, C_k \}$ with $b_{C_k} < \dots < b_{C_1}$. We say that $\zeta$ is an *ancestor family* if for every $j=2,\dots,k$ there exists $i < j$ such that $C_j$ is a first generation ancestor of $C_i$. The cylinder $C_1$ shall be referred to all cylinders of $\zeta$ are ancestors of $C_1$. \[defigoodsize\] Given a diluted model on $\mathcal{N}(S \times G)$ we say that is a *good size function* for the model if it satisfies the following properties: 1. $\inf_{\gamma_x \in S \times G} q(\gamma_x) \geq 1$. 2. Given two ancestor families $\zeta, \zeta' \in \M$ if $$\sum_{C \in \zeta} q(basis(C)) + \sum_{C' \in \zeta'} q(basis(C')) < d_S\left( \pi_S\left(basis(C_1)\right),\pi_S\left(basis(C'_1)\right)\right)$$ then none of the bases of $\zeta$ have an impact on any of the bases of $\zeta'$ and viceversa. denotes the metric in $S$ and $\pi_S : S \times G \rightarrow S$ is the projection onto $S$. 3. There exist $b_1,b_2 > 0$ such that for any $\Lambda \in {{\mathcal B}}_S^0$ $${{\mathbb E}}\left( e^{b_1 \sum_{C \in \mathcal{A}^0(\Lambda \times G)} q(C)} \right) \leq e^{b_2 \nu(\Lambda \times G)}.$$ The idea behind property (ii) is that, if $q$ is a good size function for the model, given an ancestor family $\zeta \in \M$ the quantity $\sum_{C \in \zeta} q(basis(C))$ should represent in some way the “reach” (or size) of the family. Hence, it is natural to ask that whenever the combined sizes of two ancestor families cannot overcome the distance between their roots then neither of the families has an impact on the other. On the other hand, notice that properties (i) and (iii) imply that the model under consideration admits exactly one Gibbs measure (see proof of Proposition \[finit1\]). \[teomixing\] If $(\nu,H)$ is a translation invariant heavily diluted model with a good size function $q$ then its unique Gibbs measure is exponentially mixing. Recall that for any $\Lambda \in {{\mathcal B}}^0_S$ there exists a measurable function $\psi_\Lambda$ such that $$\left(\mathcal{K}_0\right)_{\Lambda \times G} = \psi_\Lambda\left( \mathcal{A}^0_F(\Lambda) \right)$$ where we use the notation $\mathcal{A}^0_F(\Lambda):= \left( \mathcal{A}^0(\Lambda \times G), F( \mathcal{A}^0(\Lambda \times G) ) \right).$ Keeping this in mind, the left hand side of can be rewritten as $$\label{eqmixing2} \left| {{\mathbb E}}( f( \psi_{\Lambda_f}( \mathcal{A}^0_F(\Lambda_f))) g( \psi_{\Lambda_g}(\mathcal{A}^0_F(\Lambda_g))) - f( \psi_{\Lambda_f}( \mathcal{A}^0_F(\Lambda_f)))g( \psi_{\Lambda_g}(\tilde{\mathcal{A}}^0_F(\Lambda_g))) ) \right|$$ for $\tilde{\mathcal{A}}^0_F(\Lambda_g)$ carrying the same distribution as $\mathcal{A}^0_F(\Lambda_g)$ while being independent of $\mathcal{A}^0_F(\Lambda_f)$. Furthermore, if we construct the triple so as to also verify that $$\mathcal{A}^0_F(\Lambda_f) \sim \mathcal{A}^0_F(\Lambda_g) \Longrightarrow \mathcal{A}^0_F(\Lambda_g) = \tilde{\mathcal{A}}^0_F(\Lambda_g),$$ where $\mathcal{A}^0_F(\Lambda_f) \sim \mathcal{A}^0_F(\Lambda_g)$ means that none of the bases of $\mathcal{A}^0_F(\Lambda_f)$ have the bases of $\mathcal{A}^0_F(\Lambda_g)$ and viceversa, then we obtain the bound $$\label{eqmixing3} \left|\int f(\eta)g(\eta) d\mu(\eta) -\int f(\eta) d\mu(\eta)\int g(\eta) d\mu(\eta)\right| \leq 2 \| f \|_\infty \| g \|_\infty P( \mathcal{A}^0_F(\Lambda_f) \not \sim \mathcal{A}^0_F(\Lambda_g)).$$ The construction of this triple is similar in spirit to the one in the . The idea is to construct the families $\mathcal{A}^0_F(\Lambda_f)$ and $\mathcal{A}^0_F(\Lambda_g)$ using the same free process $\Pi$ and then to obtain $\tilde{\mathcal{A}}^0_F(\Lambda_g)$ by replacing those cylinders in $\mathcal{A}^0_F(\Lambda_g)$ which have an impact on $\mathcal{A}^0_F(\Lambda_f)$ (or receive an impact from $\mathcal{A}^0_F(\Lambda_f)$) and their ancestors with cylinders belonging to an independent We refer to [@FFG1]. Therefore, it suffices to produce a suitable bound for the right hand side of . Now, since $q$ is a good size function we have $$\begin{aligned} P( \mathcal{A}^0_F(\Lambda_f) \not \sim \mathcal{A}^0_F(\Lambda_g) ) & \leq P \left( \sum_{C \in \mathcal{A}^0(\Lambda_f \times G)} q(C) + \sum_{C \in \mathcal{A}^0(\Lambda_g \times G)} q(C) \geq d_S( \Lambda_f, \Lambda_g ) \right)\\ \\ & \leq P\left( \sum_{C \in \mathcal{A}^0(\Lambda_f \times G)} q(C) \geq \frac{d_{f,g}}{2} \right) + P\left( \sum_{C \in \mathcal{A}^0(\Lambda_g \times G)} q(C)\geq \frac{d_{f,g}}{2} \right)\end{aligned}$$ where we use the notation $d_{f,g}:= d_S(\Lambda_f,\Lambda_g)$. Now, since $q$ is a good size function, by the exponential Tchebychev inequality we have that there exist $b_1,b_2 > 0$ such that for any bounded local function $h:\mathcal{N}(S \times G)$ one has the estimate $$\label{eqmixing4} P\left(\sum_{C \in \mathcal{A}^0(\Lambda_h \times G)} q(C) \geq r \right) \leq e^{- b_1 r + b_2 \nu(\Lambda_h)}$$ for every $r > 0$, which yields the bound $$P( \mathcal{A}^0_F(\Lambda_f)\not \sim \mathcal{A}^0_F(\Lambda_g) ) \leq e^{- \frac{b_1}{2} d_S(\Lambda_f,\Lambda_g)}( e^{b_2 \nu(\Lambda_f)} + e^{b_2 \nu(\Lambda_g)}) \leq 2 e^{- \frac{b_1}{2} d_S(\Lambda_f,\Lambda_g) + b_2(\nu(\Lambda_f)+\nu(\Lambda_g))}$$ Thus, by taking $c_1=\frac{b_1}{3}$, $c_2=b_2$ and $d_S(\Lambda_f,\Lambda_g)$ sufficiently large we conclude . \[obsmixing\]Even though Theorem \[teounigibbs\] ensures that finding any size function $q$ which satisfies the (F1)-diluteness condition will be enough to conclude that the corresponding model is heavily diluted, if one wishes to obtain further properties of the unique Gibbs measure such as , then it is important for the size function $q$ to be of geometrical relevance within the context of the model, for example as (ii) in Definition \[defigoodsize\] suggests. We would like to point out that whenever checking if a certain size function is indeed a good size function for a , condition (iii) will be in general the The following result will be of much aid to us in this matter. A *Galton-Watson* process is a family $Z=(Z_n)_{n \in {{\mathbb N}}_0}$ of random variables taking values in ${{\mathbb N}}_0$ which satisfy for every $n \in {{\mathbb N}}_0$ the recurrence formula $$Z_{n+1} = \sum_{i=1}^{Z_n} X^{(n+1)}_i$$ where $(X^{(n+1)}_i)_{(i,n) \in {{\mathbb N}}\times {{\mathbb N}}_0}$ is a sequence of i.i.d. random variables taking values in ${{\mathbb N}}_0$. $Z_0$ is called the *initial distribution* of the Galton-Watson process $Z$, while the distribution of the random variables $X^{(n)}_i$ is called its *offspring distribution*. \[expbranching\] Let $Z$ be a Galton-Watson process with some offspring distribution $X$. If $Z_0 \equiv 1$ and ${{\mathbb E}}(X) < 1$ then there exists $b > 0$ which satisfies ${{\mathbb E}}( e^{b \sum_{n \in {{\mathbb N}}_0} Z_n} ) < +\infty$ such that ${{\mathbb E}}( e^{s X} ) < +\infty$. The following corollary illustrates the use of Theorem \[expbranching\] in this context. \[corbranching\] Let $(\nu,H)$ be a diluted model satisfying the following properties: 1. $\nu(I(\{ \gamma_x \})) = \nu(I(\{ \tilde{\gamma}_y \}))$ for every pair $\gamma_x,\tilde{\gamma}_y \in S \times G$. 2. $\nu(I(\{ \gamma_x\}) < e^{\Delta E}$ for every $\gamma_x \in S \times G$. Then there exist $b_1,b_2 > 0$ such that for any $\Lambda \in {{\mathcal B}}^0_S$ $$\label{corbranching1} {{\mathbb E}}\left( e^{b_1 \#(\mathcal{A}^0(\Lambda \times G))} \right) \leq e^{b_2 \nu(\Lambda \times G)}.$$In particular, any bounded size function for $(\nu,H)$ satisfies (iii) in Definition \[defigoodsize\]. Given $\Lambda \in {{\mathcal B}}^0_S$, let us consider the family ${{\mathcal B}}$ constructed in the Domination lemma. By definition of ${{\mathcal B}}=({{\mathcal B}}_n)_{n \in {{\mathbb N}}_0}$, for any $b_1 > 0$ we have that $${{\mathbb E}}\left( e^{b_1 \#(\mathcal{A}^0(\Lambda \times G))} \right) \leq {{\mathbb E}}\left( e^{b_1 \# {{\mathcal B}}} \right) = {{\mathbb E}}\left( \prod_{C \in {{\mathcal B}}_0} {{\mathbb E}}( e^{b_1 \#{{\mathcal B}}(C)} ) \right).$$ Now, by the assumptions on the pair $(\nu,H)$, we have that for every $C \in \mathcal{C}$ the family $Z^C=(Z_n^C)_{n \in {{\mathbb N}}_0}$ defined by the formula $Z_n^C = \# {{\mathcal B}}_n(C)$ is a Galton-Watson process whose distribution does not depend on $C$. Indeed, $Z^C$ has initial value 1 for all $C \in \mathcal{C}$ and Poisson offspring distribution with mean $e^{- \Delta E}\nu( I(\{basis(C)\}) )$ which does not depend on $C$. Furthermore, since $\nu(I(\{ basis(C)\}) < e^{\Delta E}$ by assumptions and the Poisson distribution has well defined exponential moments, for $b_1 > 0$ sufficiently small Theorem \[expbranching\] yields the existence of $\tilde{b}_2 > 1$ such that for all $C \in {{\mathcal C}}$ we have $$\tilde{b}_2:={{\mathbb E}}( e^{b_1 \# {{\mathcal B}}(C)} ) < +\infty.$$ Hence, we obtain that $${{\mathbb E}}\left( e^{b_1 \#(\mathcal{A}^0(\Lambda \times G))} \right) \leq {{\mathbb E}}( \tilde{b}_2^{\# {{\mathcal B}}_0} ) = e^{(\tilde{b}_2 - 1) e^{-\Delta E}\nu(\Lambda \times G)}$$ since $\# {{\mathcal B}}_0$ has Poisson distribution with mean $e^{-\Delta E}\nu(\Lambda \times G)$. By taking $b_2 = (\tilde{b}_2 - 1)e^{-\Delta E}$ we conclude the proof. The hypotheses in Corollary \[corbranching\] may seem restrictive, but in fact they can be relaxed: conditions ($\bullet$) and ($\bullet \bullet$) can be replaced by the weaker condition 1. $\sup_{\gamma_x \in S \times G} \nu(I(\{\gamma_x\})) < e^{\Delta E}.$ Indeed, if $(*)$ holds then one can obtain by enlarging ${{\mathcal B}}$ so that each individual in the enlarged process has Poisson offspring distribution with mean $e^{-\Delta E}\sup_{\gamma_x \in S \times G} \nu( I(\{\gamma_x\}))$. We leave the details to the reader. Applications {#teounigibssexamples} ------------ In the following we investigate which conditions are implied by Theorems \[teounigibbs\] and \[teomixing\] for the existence of a unique Gibbs measure and its exponential mixing property in some of the models introduced in Section \[examples\]. Bearing Remark \[obsmixing\] in mind, in each of the examples we proceed as follows: first we propose a function $q$ satisfying (i) and (ii) in Definition \[defigoodsize\] and then we investigate under which choice of parameters in the model are the (F1)-diluteness condition and (iii) in Definition \[defigoodsize\] also satisfied. ### Widom-Rowlinson model with generalized interactions Let us suppose that we have supp($h$)$=[0,m_h]$ and supp($j$)$=[0,m_j]$ for some $m_h,m_j > 0$. Then a proper choice of size function for this model would be $q(\gamma_x) = \max \{ 1 , m_h, m_j\}$. Now, by and the fact that $h$ and $j$ are both nonnegative we obtain that $\Delta E = 0$, which implies that $$\alpha_{WR}(\lambda^{\pm}, h,j) = 2^d \max \{ \lambda_+ m_j^d + \lambda_- \max\{ m_h^d , m_j^d \} , \lambda_- m_j^d + \lambda_+ \max\{ m_h^d , m_j^d \} \}.$$ Furthermore, since $q$ is constant we see that if $\alpha_{WR}(\lambda^{\pm}, h,j) < 1$ then ($*$) holds so that $q$ is a good size function for the model. Thus, we arrive at the following result. \[teounigibbswrm\]For $\alpha_{WR}(\lambda^{\pm}, h,j) < 1$ the Widom-Rowlinson model with and generalized interactions given by the pair $(h,j)$ admits a unique Gibbs measure. Furthermore, this Gibbs measure satisfies the exponential mixing property. Let us notice that for the original Widom-Rowlinson model with equal fugacities a simpler expression for $\alpha_{WR}$: $$\label{wrdc1} \alpha_{WR}(\lambda,r)= \lambda (2r)^d.$$ For the discrete Widom-Rowlinson model we obtain an analogue of Theorem \[teounigibbswrm\], but the (F1)-diluteness condition in this context is given by the coefficient $$\label{wrdc2} \alpha_{WR}^{discrete}(\lambda,r) = \lambda (2r)^d + \lambda,$$ where the extra term is due to the exclusion of equal-type particles with the same position. ### Thin rods model The size function in this model is given by the length of the rods, $q(\gamma_{x}):=\max\{1,2l\}$. Once again, the structure of the Hamiltonian implies that $\Delta E = 0$ and, since for every $\gamma_x \in {{\mathbb R}}^2 \times S^1_*$ we have $$I(\gamma_x) \subseteq \{ \tilde{\gamma}_y \in {{\mathbb R}}^2 \times S^1_* : \|x-y\|_2 \leq 2l \},$$ we obtain the bound $$\alpha_{TR}(\lambda,l) \leq 4 \lambda l^2 \sigma_2$$ where $\sigma_2$ denotes the Lebesgue measure of the $2$-dimensional unit ball in the $\| \cdot \|_2$ norm. Let us notice that, although this bound is valid for any choice of orientation measure $\rho$, it can be improved considerably provided that one has some knowledge on $\rho$. In any case, regardless of the particular choice of $\rho$ one may have, since $q$ is constant we have that ($*$) holds whenever $4\lambda l^2 < \frac{1}{\sigma_2}$. Thus, we obtain the following result. \[teounigibbstrm\] If $4 \lambda l^2 < \frac{1}{\sigma_2}$ then the thin rods model with fugacity $\lambda$ (and arbitrary orientation measure $\rho$) admits a unique Gibbs measure. Furthermore, this Gibbs measure satisfies the exponential mixing property. ### Ising contours model {#ffgisingc1} Since the Ising contours model does not satisfy Assumptions \[assump\], one needs to be careful when defining its FFG dynamics. We proceed as follows: 1. We replace $\Delta E$ in the construction of the local and global dynamics $$\Delta E^* := \inf_{\substack{ \eta \in \mathcal{N}(S \times G) \\ \gamma_x \in S \times G }} \Delta E^*_{\eta} (\gamma_x) \hspace{1cm}\text{ and }\hspace{1cm} \Delta E_{\Lambda} := \inf_{\substack{ \eta \in \mathcal{N}(S \times G) \\ \gamma_x \in S \times G }} \Delta E_{\Lambda|\eta} (\gamma_x).$$ 2. We define the local dynamics on $\Lambda$ by replacing $M(\gamma_x|\xi)$ in the construction with $$M_\Lambda(\gamma_x|\xi):=e^{- (\Delta E_{\Lambda|\xi}(\gamma_x)-\Delta E_{\Lambda})}.$$ 3. We define the global FFG dynamics by replacing $M(\gamma_x|\xi)$ in the construction with $$M^*(\gamma|\xi):=e^{-(\Delta E^*_{\xi}(\gamma_x)-\Delta E^*)}.$$ One can easily see that, by replacing the original energy leap functions with their localized versions, one recovers (ii) in the consistent Hamiltonian property of Assumptions \[assump\]. Furthermore, from the proof of Lemma \[inv0\] it is clear that this property alone is enough to show (i) of Theorem \[teounigibbs\]. Finally, from this and it is straightforward to also obtain (ii) of Theorem \[teounigibbs\] in this particular context. Thus, the (F1)-diluteness condition for the Ising contours model implies the existence of an infinite-volume limit measure. Since the natural choice of size function in this model is the size of , then yields $$\alpha_{IC}(\beta)= \sup_{\gamma_x \in ({{\mathbb{Z}}}^d)^* \times G} \left[ \frac{1}{|\gamma_x|} \sum_{\tilde{\gamma}_y : \tilde{\gamma}_y \not \sim \gamma_x} |\tilde{\gamma}_y| e^{-\beta|\tilde{\gamma}_y|}\right].$$ In particular, for $\alpha_{IC}< 1$ there exists a probability measure $\mu$ on $\mathcal{N}(({{\mathbb{Z}}}^d)^* \times G)$ which is the infinite-volume limit of Boltzmann-Gibbs distributions with empty boundary condition, i.e. $$\mu = \lim_{\Lambda \nearrow ({{\mathbb{Z}}}^d)^*} \mu_{\Lambda|\emptyset}.$$ In fact, Theorem \[teounigibbs\] produces a coupling $( (\mathcal{K}^{\Lambda|\emptyset}_0)_{\Lambda \subseteq ({{\mathbb{Z}}}^d)^*}, \mathcal{K}_0)$ of these measures satisfying the following property: for any $\Lambda_0 \in {{\mathcal B}}^0_{({{\mathbb{Z}}}^d)^*}$ there exists (a random) $\Delta \in {{\mathcal B}}^0_{({{\mathbb{Z}}}^d)^*}$ such that for all $\Lambda \subseteq ({{\mathbb{Z}}}^d)^*$ with $\Delta \subseteq \Lambda$ one has $$\label{isconvc} (\mathcal{K}^{\Lambda|\emptyset}_0)_{\Lambda_0 \times G} = \left(\mathcal{K}_0\right)_{\Lambda_0 \times G}.$$ Now, let us observe that the condition $\alpha_{IC} < 1$ implies that almost surely there exist only finitely many contours in $\Pi_0$ surrounding each point in the lattice ${{\mathbb{Z}}}^d$. Indeed, if $\gamma_x$ is a contour surrounding a given point in the lattice, for example the origin $0$, then $\gamma_x$ contains a plaquette which intersects the $x_1$-axis on some negative value $l(\gamma_x)$ which is at a distance not from the origin. With this, a straightforward argument using the translational invariance of the model gives the bound $$\sum_{\gamma_x : 0 \in \text{Int}(\gamma_x)} P( \gamma_x \in \Pi_0 ) \leq \sum_{\gamma_x : 0 \in \text{Int}(\gamma_x)} e^{-\beta |\gamma_x|} \leq \sum_{\gamma_x : \,p_0 \not \sim \,\gamma_x} |\gamma_x|e^{-\beta|\gamma_x|}=: \alpha^0_{IC}(\beta)$$ where the expression $0 \in \text{Int}(\gamma_x)$ means that $\gamma_x$ is a contour surrounding $0$, $p_0$ denotes a fixed plaquette in the dual lattice and the expression $p_0 \not \sim \gamma_x$ means that $p_0$ is adjacent to some plaquette in $\gamma_x$. Notice that the value of $\alpha^0_{IC}$ does not depend on the It is not hard to check that for any $\beta > 0$ one has the inequalities $$\alpha_{IC}(\beta) \leq \alpha^0_{IC}(\beta) \leq 2d \alpha_{IC}(\beta).$$ Thus, whenever $\alpha_{IC}(\beta) < 1$ we see that $\alpha^0_{IC}(\beta)$ is finite so that the Borel-Cantelli Lemma implies that almost surely $\Pi_0$ has only finitely many contours surrounding the origin. we conclude that the same conclusion must hold almost surely for *all* sites in the lattice ${{\mathbb{Z}}}^d$ simultaneously. Observe that whenever this holds it is possible to conduct an infinite-volume $(+)$-alignment $\sigma^+_{\mathcal{K}_0}$ of $\mathcal{K}_0$ as explained in Section \[exampleisingc\]. This spin configuration $\sigma^+_{\mathcal{K}_0}$ satisfies what is known as the $(+)$-*sea with islands picture*: there are always finitely many contours around each point in ${{\mathbb{Z}}}^d$ and thus there is no percolation of the minority spin $(-)$. In accordance to the discussion in the Introduction of Part II, we see that $\sigma^+_{\mathcal{K}_0}$ can thus be regarded as a small random perturbation of the constant $(+)$-configuration, where this small perturbation consists of finite islands on which $\sigma^+_{\mathcal{K}_0}$ disagrees with the $(+)$-configuration. Together with , the $(+)$-sea with islands picture implies that for any sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\Lambda_n \nearrow {{\mathbb{Z}}}^d$ one has the $(+)$-alignment of the finite-volume dynamics $\mathcal{K}^{\Lambda_n^*|\emptyset}_0$ converging to the infinite-volume $(+)$-alignment, i.e. $$\sigma^+_{\mathcal{K}^{\Lambda_n^*|\emptyset}_0} \overset{loc}{\rightarrow} \sigma^+_{\mathcal{K}_0}.\footnote{Here local convergence is defined in analogy with Definition \ref{localconvergence}, i.e. convergence of the expectation of bounded local functions, where a function $f: \{+,-\}^{{{\mathbb{Z}}}^d} \rightarrow {{\mathbb R}}$ is said to be \textit{local} if it depends only on the spin values inside some bounded set $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$.}$$ Thus, by there exists a probability measure $\mu^+$ on $\{+,-\}^{{{\mathbb{Z}}}^d}$ such that $$\mu^+_{\Lambda_n} \overset{loc}{\rightarrow} \mu^+.$$ Using Proposition \[limitegibbs\] we conclude that $\mu^+$ is a Gibbs measure for the Ising model on ${{\mathbb{Z}}}^d$ (the diluteness condition is not required throughout the proof). By symmetry we obtain that there exists another Gibbs measure $\mu^-$, which is realized as the , i.e. $\sigma^-_{\mathcal{K}_0} := - \sigma^+_{\mathcal{K}_0}$. However, since both “sea with islands” pictures cannot be satisfied simultaneously, we must have $\mu^+ \neq \mu^-$. Thus, we see that whenever $\beta$ is sufficiently large so as to guarantee that $\alpha_{IC}(\beta) < 1$ then the Ising model exhibits a phase transition. Now, with respect to establishing mixing properties for these measures, the exponential mixing property for the Ising contours model was studied in [@FFG1]. However, the true objects of interest here are the aligned measures $\mu^+$ and $\mu^-$, whose mixing properties cannot be immediately deduced from those of their underlying contour model. Indeed, local information on the original Ising model such as the spin at a given site in the lattice depends on the total amount of contours surrounding this site, which is highly non-local information in terms of contours. In general, given a bounded set $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$, the spin values of the configuration $\sigma^+_{\mathcal{K}_0}$ inside $\Lambda$ depend on the contour configuration $\mathcal{K}_0$ inside the set $$r^*(\Lambda) = \{ \gamma_x \in ({{\mathbb{Z}}}^d)^* \times G : p(\Lambda^*) \cap \text{V}(\gamma_x) \neq \emptyset\}$$ where $p(\Lambda^*)$ denotes the set of plaquettes with vertices in $\Lambda^*$ and $\text{V}(\gamma_x)$ denotes the *volume* of the contour $\gamma_x$, i.e. the set of points in $({{\mathbb{Z}}}^d)^*$ lying outside the only infinite component of $({{\mathbb{Z}}}^d)^* - \text{supp}(\gamma_x)$. In more precise terms, there exists a ${{\mathcal F}}_{r^*(\Lambda)}$-measurable function $\phi_{\Lambda}^+ : \mathcal{N}(({{\mathbb{Z}}}^d)^*\times G) \rightarrow \{+,-\}^{{{\mathbb{Z}}}^d}$ such that $$(\sigma^+_{\mathcal{K}_0})_{\Lambda} = \phi_{\Lambda}^+ (\mathcal{K}_0).$$ Thus, for every pair of bounded local functions $f,g : \{+,-\}^{{{\mathbb{Z}}}^d} \rightarrow {{\mathbb R}}$ we have that $$\label{eqmixingicm} \left|\mu^+(fg) - \mu^+(f)\mu^+(g)\right| = \left| \mu( (f\circ \phi_{\Lambda_f}^+)(g \circ \phi_{\Lambda_g}^+) ) -\mu( f\circ \phi_{\Lambda_f}^+)\mu(g \circ \phi_{\Lambda_g}^+)\right|$$ where we have used the standard integral notation $$\vartheta(f):= \int f(\eta)d\vartheta(\eta).$$ The problem with is that the functions on its right hand side are not local as functions on $\mathcal{N}(({{\mathbb{Z}}}^d)^*\times G)$. Nevertheless, if for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we had $\nu(r^*(\Lambda))< +\infty$ then we could construct a coupling between $\mathcal{A}^0_F(r^*(\Lambda_f))$ and $\mathcal{A}^0_F(r^*(\Lambda_g))$ as explained in the proof of Theorem \[teomixing\] and use it to obtain the bound $$\label{eqmixingicm2} \left|\mu^+(fg) - \mu^+(f)\mu^+(g)\right| \leq 2\|f\|_\infty\|g\|_\infty P(\mathcal{A}^0_F(r^*(\Lambda_f)) \not \sim \mathcal{A}^0_F(r^*(\Lambda_g))).$$ Fortunately, when $\alpha_{IC}(\beta) < 1$ this happens to be the case. Indeed, one has the estimate $$\nu(r^*(\Lambda)) \leq (\#\Lambda + 2d\#{\partial}\Lambda) \alpha_{IC}^0$$ where ${\partial}\Lambda := \{ x \in \Lambda : d(x,\Lambda^c) = 1 \}$, which follows upon noticing that if $\gamma_x \in r^*(\Lambda)$ then $\gamma_x$ is either surrounding a point in $\Lambda$ or containing a plaquette whose associated bond connects $\Lambda$ with $\Lambda^c$. Now, to estimate the probability in the right hand side of , by definition of $r^*(\Lambda)$ one can check that $$P( \mathcal{A}^0_F(r^*(\Lambda_f)) \not \sim \mathcal{A}^0_F(r^*(\Lambda_g)) ) \leq P \left( \sum_{C \in \mathcal{A}^0(r^*(\Lambda_f))} q(C) + \sum_{C \in \mathcal{A}^0(r^*(\Lambda_g))} q(C) \geq d_S( \Lambda_f, \Lambda_g ) \right).$$ However, the right hand side cannot be bounded as in the proof of Theorem \[teomixing\] by using Corollary \[corbranching\] directly: neither the size function $q$ is bounded nor is condition ($*$) satisfied. To fix this problem, for each $\Lambda \in {{\mathcal B}}^0_S$ we consider the branching process ${{\mathcal B}}$ dominating $\mathcal{A}^0(r^*(\Lambda))$ which can be constructed as in the proof of the In this process ${{\mathcal B}}$, the individuals are the different contours, each having an independent number of offspring which has Poisson distribution with mean proportional to their size. We wish to enlarge this branching process ${{\mathcal B}}$, so that the enlarged process $\overline{{{\mathcal B}}}$ satisfies: 1. $\sum_{C \in \mathcal{A}^0(r^*(\Lambda))} q(C) \leq \# \overline{{{\mathcal B}}}$, 2. All individuals in $\overline{{{\mathcal B}}}$ have the same offspring distribution. If we manage to do this, then we can proceed as in the proof of Corollary \[corbranching\] to bound the right hand side of , provided that the offspring distribution in $\overline{{{\mathcal B}}}$ has mean less than one. The way in which to achieve this is to enlarge ${{\mathcal B}}$ by considering plaquettes as individuals instead of whole contours. More precisely, the initial individuals in $\overline{{{\mathcal B}}}$ will be those plaquettes conforming the initial contours in ${{\mathcal B}}$ and for a given plaquette $p$ we define its offspring as follows: we first draw an independent number of contours containing $p$ with Poisson distribution of mean $\alpha_{IC}^0(\beta)$ and then regard all the plaquettes which constitute these contours as the offspring of $p$. The detailed construction of this enlarged process is similar to the one in the Domination Lemma, so we omit the details. Thus, a straightforward computation using Theorem \[expbranching\] yields that, whenever $\alpha^0_{IC}(\beta) < 1$, $b_1 > 0$ sufficiently small (depending on $\beta$) such that $${{\mathbb E}}( e^{b_1 \sum_{C \in \mathcal{A}^0(r^*(\Lambda))} q(C)} ) \leq e^{ b_1 \# \overline{B} } = e^{\nu^{\tilde{\beta}}(r^*(\Lambda)) - \nu^\beta(r^*(\Lambda))}$$ where we write the dependence of $\nu$ on the inverse temperature explicitly, and furthermore set $\tilde{\beta}:= \beta - \log \tilde{b}_2$ for a certain constant $$\tilde{b}_2:= {{\mathbb E}}( e^{b_1 \# \overline{{{\mathcal B}}}(p)} ) < +\infty$$ which is strictly larger than one and depends only on $b_1$ and $\beta$. Let us notice that $\tilde{\beta} < \beta$ and also that: 1. $\tilde{\beta}$ is a increasing function of $\beta$ satisfying $\tilde{\beta} \rightarrow +\infty$ as $\beta \rightarrow +\infty$, 2. $\tilde{\beta}$ is a increasing function of $b_1$ satisfying $\beta - \tilde{\beta} \rightarrow 0$ as $b_1 \rightarrow 0$. Thus, if we set $$\label{betas} \beta^*= \inf\{ \beta > 0 : \alpha^0_{IC}(\beta) < 1 \}\hspace{1cm}\text{ and }\hspace{1cm}\beta^{**}=\inf\{ \beta > 0 : \alpha^0_{IC}(\beta) < +\infty \}$$ then if $\beta > \beta^*$ we have that for each $\tilde{\beta} \in (\beta^{**},\beta)$ there exists a constant $c > 0$ such that $$\label{eqmixingicm3} \left|\mu^+(fg) - \mu^+(f)\mu^+(g)\right| \leq \|f\|_\infty\|g\|_\infty e^{- c d_S(\Lambda_f,\Lambda_g) + \nu^{\tilde{\beta}}(r^*(\Lambda_f)) + \nu^{\tilde{\beta}}(r^*(\Lambda_g))}.$$for $d_S(\Lambda_f,\Lambda_g)$ sufficiently large (depending only on $c$). Furthermore, if we take $\tilde{\beta} \in (\beta^*,\beta)$ then we obtain the simpler (yet weaker) formula $$\label{eqmixingicm4} \left|\mu^+(fg) - \mu^+(f)\mu^+(g)\right| \leq \|f\|_\infty\|g\|_\infty e^{- c d_S(\Lambda_f,\Lambda_g) + (2d+1)(\# \Lambda_f + \#\Lambda_g)}.$$ Clearly, the analogous conclusion also remains valid for the other Gibbs measure, $\mu^-$. analysis in the following theorem. \[teounigibbsising\] If $\beta > 0$ is sufficiently large so as to satisfy $\alpha_{IC}(\beta) < 1$ then: 1. The Ising model on ${{\mathbb{Z}}}^d$ admits two distinct Gibbs measures, $\mu^+$ and $\mu^-$. 2. The measures $\mu^{+}$ and $\mu^{-}$ can be obtained as the local limits $$\mu^{+} := \lim_{n \rightarrow +\infty} \mu^{+}_{\Lambda_n} \hspace{2cm}\text{ and }\hspace{2cm}\mu^{-} := \lim_{n \rightarrow +\infty} \mu^{-}_{\Lambda_n}$$ for any sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\Lambda_n \nearrow {{\mathbb{Z}}}^d$. 3. $\mu^+$ and $\mu^-$ satisfy the sea with islands picture for the $(+)$ and $(-)$ spins, respectively. 4. If also $\beta > \beta^*$ where $\beta^*$ is defined in , then both $\mu^+$ and $\mu^-$ also satisfy the exponential mixing property in the sense of and . Much sharper conditions than the one obtained here are known for the occurrence of phase transition in the Ising model. However, the argument presented here is of considerable relevance, since it shall be repeated when studying the applications of the FFG dynamics to the Pirogov-Sinai theory, where in general sharper conditions than the one given by Theorem \[teounigibbs\] are not known. As for the exponential mixing property, the range of validity provided by standard cluster expansion methods is strictly smaller than the one obtained here: these methods show that the exponential mixing property is satisfied as soon as $\beta > \beta'$, where $$\beta' = \inf\left\{ \beta > 0 : \sum_{\gamma_x : \,p_0 \not \sim \,\gamma_x} e^{q(\gamma_x)} e^{- \beta|\gamma_x|} < 1\right\}.$$ The coefficient $\beta'$ can be improved (see [@LM] for example), although these methods are not capable of getting rid of the exponential dependence in $q$. A remark on perfect simulation of Gibbs measures {#perfectsimulation} ------------------------------------------------ One of the advantages of the FFG dynamics is that not only do they yield a criterion for the existence and uniqueness of the equilibrium measure, but they also provide a way Indeed, if given a certain diluted model one wishes to obtain a perfect sample of its unique Gibbs measure on a finite volume $\Lambda \in {{\mathcal B}}^0_S$, then all one has to do is to obtain a perfect sample of the clan of ancestors $\mathcal{A}^0(\Lambda \times G)$ and afterwards perform the deleting procedure discussed in Section \[localdynamics\]. But by the definition of $\mathcal{A}^0(\Lambda \times G)$, one has that: 1. $\mathcal{A}^0_0(\Lambda \times G)$ is a Poisson process on $\mathcal{C}$ with intensity measure $\phi_{\mathcal{P}(\Lambda \times G)}$, where $$\mathcal{P}(\Lambda \times G):=\{ C \in \mathcal{C} : basic(C) \in \Lambda \times G, b_C \leq 0 < b_C + l_C \},$$ 2. Conditional on $\mathcal{A}^0_0(\Lambda \times G), \dots, \mathcal{A}^0_{n}(\Lambda \times G)$, the set $\mathcal{A}^0_{n+1}(\Lambda \times G) - \bigcup_{i=0}^n \mathcal{A}^0_{i}(\Lambda \times G)$ is a Poisson process on $\mathcal{C}$ with intensity measure $\phi$ restricted to the set $$\bigcup_{C \in \mathcal{A}^0_{n}(\Lambda \times G)} \mathcal{P}(C) - \left( \mathcal{P}(\Lambda \times G) \cup \bigcup_{i=1}^{n-1} \bigcup_{C \in \mathcal{A}^0_{i}(\Lambda \times G)} \mathcal{P}(C)\right).$$ Thus, to obtain a perfect sample of $\mathcal{A}^0(\Lambda \times G)$ one may proceed as follows: 1. Sample $\mathcal{A}^0_0(\Lambda \times G)$ from a Poisson process on $\mathcal{P}(\Lambda \times G)$ with intensity measure $\phi$. 2. Having sampled $\mathcal{A}^0_0(\Lambda \times G), \dots, \mathcal{A}^0_{n}(\Lambda \times G)$, obtain $\mathcal{A}^0_{n+1}(\Lambda \times G)$ by sampling from a Poisson process on $\bigcup_{C \in \mathcal{A}^0_{n}(\Lambda \times G)} \mathcal{P}(C)$ and discarding all those cylinders which are ancestors of cylinders in generations lesser than $n$. Since we are dealing with a diluted model, by the results on Section \[ancestors\] we have that eventually we will reach a step in which no new ancestors are added. Once that happens, the algorithm stops and the ancestor family constructed until that moment constitutes the perfect sample of $\mathcal{A}^0(\Lambda \times G)$. Upon conducting the deleting procedure on it as explained in Section \[localdynamics\], all the kept cylinders with bases in $\Lambda \times G$ which are alive at time $0$ constitute a perfect sample of the unique Gibbs measure on the finite volume $\Lambda$. We refer to [@FFG1; @FFG2] for alternative sampling algorithms, results on speed of convergence and comments on the user-impatience bias in the simulation scheme. Resumen del Capítulo 9 ---------------------- Aquí introducimos la dinámica de Fernández-Ferrari-Garcia desarrollada originalmente en [@FFG1] para el modelo de contornos de Ising y mostramos que se encuentra bien definida para la clase más general de modelos diluidos. Dado un par $(\nu,H)$ satisfaciendo las condiciones en \[assump\], podríamos resumir la dinámica de Fernández-Ferrari-Garcia asociada como sigue: 1. A tasa $e^{-\Delta E}$ se propone el nacimiento de nuevas partículas con intensidad $\nu$. 2. Cada partícula $\gamma_x$ propuesta para nacer lo hará efectivamente con probabilidad $e^{-(\Delta E_{\eta}(\gamma_x)-\Delta E)}$, donde $\eta$ es el estado del sistema al momento en que el nacimiento de la partícula $\gamma_x$ es propuesto. 3. Cada partícula que ha nacido efectivamente tiene un tiempo de vida aleatorio con distribución exponencial de parámetro 1. 4. Luego de que su tiempo de vida haya expirado, cada partícula muere y desaparece de la configuración. Estudiamos primero la dinámica sobre volúmenes finitos para luego dar condiciones que garanticen la existencia de la dinámica en el volumen infinito. Bajo estas condiciones, mostramos que toda medida de Gibbs para el modelo dado por $(\nu,H)$ es una medida invariante para la dinámica. Más aún, verificamos que bajo una condición adicional (la generalización de la originalmente propuesta en [@FFG1] para el modelo de contornos de Ising) la dinámica en volumen infinito posee una única medida invariante, de donde se deduce que bajo dicha condición existe una única medida de Gibbs y que ésta coincide con la medida invariante de la dinámica. Para probar esto, primero mostramos que la unicidad de medida invariante está garantizada por la ausencia de percolación en un proceso particular de percolación orientada dependiente. Luego, mostramos que dicho proceso puede ser dominado por otro de percolación independiente y que, bajo la condición propuesta en [@FFG1], éste resulta ser subcrítico. A partir de esto se concluye que hay ausencia de percolación en el proceso original, lo cual implica la unicidad buscada. Por último, probamos que, bajo la condición de unicidad, la medida invariante es límite local de distribuciones de Boltzmann-Gibbs y, por lo tanto, resulta ser también una medida de Gibbs para el modelo. Por el razonamiento anterior es, además, la única que existe. Estudiamos también cómo se traduce dicha condición de unicidad a algunos de los modelos introducidos en el Capítulo 8. Además, mostramos cómo este tipo de resultados pueden utilizarse en el modelo de contornos de Ising para probar la existencia de múltiples medidas de Gibbs en el modelo de Ising original a baja temperatura. También investigamos bajo qué condiciones adicionales la única medida invariante posee la propiedad de mixing exponencial; en el caso del modelo de contornos de Ising, adaptamos este análisis para obtener la propiedad de mixing exponencial para cada una de las medidas de Gibbs extremales en el régimen de baja temperatura. Para terminar, damos un algoritmo de simulación perfecta para la medida invariante (bajo la condición de unicidad) basado en la construcción hacia el pasado de la dinámica. Continuity of Gibbs measures {#chapterconvabs} ============================ In this chapter we show the continuity of the Gibbs measures for heavily diluted models with respect to their intensity measure and Hamiltonian in the This scenario typically includes continuity with respect to the parameters of the model such as fugacity of particles, inverse temperature and interaction range among others. The main result is contained in Theorem \[convabs\] below. One important aspect to point out is that we not only obtain the local convergence of the corresponding Gibbs measures, but in fact in the proof of we construct a coupling between these measures in which a rather strong form of almost sure convergence takes place: given a finite volume $\Lambda \in {{\mathcal B}}^0_S$, all realizations of these measures are *identical* on $\Lambda \times G$ for parameter values which are sufficiently close to the limit values. This is a distinctive feature of our approach since, in general, other methods used to establish these type of results (i.e. cluster expansion or disagreement percolation methods) are unable to obtain such a strong form of convergence, at least in the continuum setting. A general continuity result --------------------------- \[convabs\] Let $(\nu^\varepsilon,H^\varepsilon)_{\varepsilon \geq 0}$ be a family of diluted models such that 1. There exists a heavily diluted model $(\nu,H)$ satisfying 1. For every $\varepsilon \geq 0$ the intensity measure $\nu^\varepsilon$ is absolutely continuous with respect to $\nu$ with density $\frac{d \nu^\varepsilon}{d \nu}$ such that $$\label{controldensidad} 0 \leq \frac{d \nu^\varepsilon}{d \nu} \leq 1.$$ 2. For every $\varepsilon \geq 0$ and $\gamma_x \in S \times G$ we have $I^{H^\varepsilon}(\{\gamma_x\}) \subseteq I^H (\{\gamma_x\})$. 3. $$\Delta E^H \leq \inf_{\varepsilon \geq 0} \left[ \inf_{\substack{ \eta \in \mathcal{N}(S \times G) \\ \gamma_x \in S \times G }} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) \right]$$ \[controldensidad2\] where for each $\eta \in \mathcal{N}(S \times G)$ we define $$\Delta \tilde{E}^{H^\varepsilon}_\eta := \Delta E^{H^\varepsilon}_\eta - \log \left( \frac{d\nu^\varepsilon}{d \nu}\right).$$ 2. $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$. Then for each diluted model $(\nu^\varepsilon,H^\varepsilon)$ admits exactly one Gibbs measure $\mu^\varepsilon$ and as $\varepsilon \rightarrow 0^+$ $$\mu^{\varepsilon} \overset{loc}{\longrightarrow} \mu^0.$$ The model $(\nu,H)$ is called a *majorant model* for $(\nu^\varepsilon,H^\varepsilon)_{\varepsilon \geq 0}$. Let us start by noticing that, since $\nu^\varepsilon \ll \nu$, for every $\Lambda \in {{\mathcal B}}^0_S$ we have that $\pi^{\nu^\varepsilon}_\Lambda \ll \pi^{\nu}_\Lambda$ with density given by $$\label{density} \frac{d\pi^{\nu^\varepsilon}_\Lambda}{d\pi^{\nu}_\Lambda} (\sigma) = e^{-( \nu^\varepsilon(\Lambda \times G) - \nu(\Lambda \times G))} \prod_{\gamma_x \in [\sigma]} \frac{d\nu^\varepsilon}{d \nu}(\gamma_x),$$ a fact which can be deduced from . In particular, the Boltzmann-Gibbs distributions $\mu^\varepsilon_{\Lambda |\eta}$ specified by the pair $(\nu^\varepsilon,H^\varepsilon)$ are also be specified by $(\nu,\tilde{H}^\varepsilon)$, where $\tilde{H}^\varepsilon$ is given by the formula $$\tilde{H}^\varepsilon_{\Lambda|\eta} (\sigma) = H^\varepsilon_{\Lambda|\eta}(\sigma) - \sum_{\gamma_x \in [\sigma]} \log\left( \frac{d \nu^\varepsilon}{d \nu}(\gamma_x)\right).$$ It is not difficult to check that for each $\varepsilon \geq 0$ the pair $(\nu,\tilde{H}^\varepsilon)$ satisfies Assumptions \[assump\]. Moreover, since for every $\gamma_x \in S \times G$ it is possible to verify that $I^{\tilde{H}^\varepsilon}(\{\gamma_x\}) = I^{H^\varepsilon}(\{\gamma_x\})$, we have that the pair $(\nu,\tilde{H}^\varepsilon)$ also satisfies the (F1)-diluteness condition for every $\varepsilon \geq 0$, which guarantees that each diluted model $(\nu^\varepsilon, H^\varepsilon)$ admits exactly one Gibbs measure. To establish the local convergence we shall couple all the measures $\mu^\varepsilon$ simultaneously. For this purpose we consider the infinite-volume stationary FFG processes $\mathcal{K}^\varepsilon$ constructed by taking a Poisson process $\overline{\Pi}$ with intensity measure $\nu \times e^{-\Delta E^H} \mathcal{L} \times \mathcal{L}_{{{\mathbb R}}^+} \times \mathcal{U}[0,1]$ and setting $$\label{keptscaled} \mathcal{K}^\varepsilon = \{ (\gamma_x,t,s) \in \Pi : F(\gamma_x,t,s) < \tilde{M}^\varepsilon(\gamma_x | \mathcal{K}^\varepsilon_{t^-}) \}$$ where for each $\gamma_x \in S \times G$ and $\xi \in \mathcal{N}(S \times G)$ we define $$\tilde{M}^\varepsilon (\gamma_x |\xi) := e^{- (\Delta \tilde{E}^{H^\varepsilon}_\xi(\gamma_x) - \Delta E^H)}.$$ By the arguments given in the previous sections we see that for each $\varepsilon \geq 0$ the process $\mathcal{K}^\varepsilon$ is stationary with invariant measure $\mu^\varepsilon$. Thus it will suffice to show that as $\varepsilon \rightarrow 0^+$ $$\mathcal{K}^\varepsilon_0 \overset{loc}{\longrightarrow} \mathcal{K}^0_0.$$ Let us take then $\Lambda \in {{\mathcal B}}^0_S$ and consider the clan of ancestors of $\Lambda \times G$ ancestors at time 0 $\A^{0,H} ( \Lambda \times G)$ with respect to $H$. Notice that, following this notation, for every $\varepsilon \geq 0$ we have the inclusion $$\label{ancesinc0} A^{0,H^\varepsilon}(\Lambda \times G) \subseteq \A^{0,H} ( \Lambda \times G).$$ Furthermore, recall that $\A^{0,H}( \Lambda \times G)$ is finite almost surely since $(\nu,H)$ is a heavily diluted model. Now, since $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$, it follows that there exists (random) $\varepsilon_0 > 0$ such that if $0 \leq \varepsilon < \varepsilon_0$ then $$\label{ancesinc1} \mathcal{K}^\varepsilon_{\A^{0,H} ( \Lambda \times G)} = \mathcal{K}^0_{\A^{0,H} ( \Lambda \times G)}.$$ Indeed, if (random) $N \in {{\mathbb N}}$ is such that $\A^{0,H}_n(\Lambda \times G) = \emptyset$ for every $n > N$ almost surely then for every cylinder $C \in \A^{0,H}_N(\Lambda \times G)$ and $\varepsilon \geq 0$ we have that $$C \in \mathcal{K}^\varepsilon \Longleftrightarrow F(C) < \tilde{M}^\varepsilon( basis(C) | \emptyset )$$ from which we immediately obtain that for $\varepsilon$ (randomly) small enough $$\mathcal{K}^\varepsilon_{\A^{0,H}_N ( \Lambda \times G)} = \mathcal{K}^0_{\A^{0,H}_N( \Lambda \times G)}$$ and one may proceed with the succeeding generations by induction. But and together imply that for $0 \leq \varepsilon < \varepsilon_0$ we have $$(\mathcal{K}^\varepsilon_0)_{\Lambda \times G} = (\mathcal{K}^0_0)_{\Lambda \times G}$$ which establishes the local convergence and concludes the proof. We would like to point out that although the condition of the existence of a majorant model may seem restrictive at first, in practice all heavily diluted models admit such a majorant. Indeed, as we will see on Section \[examplescont\] below, most majorant models can be obtained by slightly increasing the fugacity (or decreasing the inverse temperature) and/or the interaction range of the limit model $(\nu^0,H^0)$. Since this limit model is heavily diluted by assumption, performing such an operation will yield once again a heavily diluted model. Finally, let us notice that the hypothesis in Theorem \[convabs\] can be relaxed a little bit. Indeed, needs to hold $\nu$-almost surely since under this condition we can always choose a version of $\frac{d\nu^\varepsilon}{d\nu}$ satisfying for every $\gamma_x \in S \times G$. Similarly, the convergence of the energy leap functions $\Delta \tilde{E}^{H^\varepsilon}_\eta$ for every $\eta \in \mathcal{N}(S \times G)$ can also be somewhat relaxed. The next definition explains how to do so. \[diset\] A measurable set $N \subseteq \mathcal{N}(S \times G)$ is said to be *dynamically impossible* for an intensity measure $\nu$ on $S \times G$ if it satisfies the following properties: 1. $\pi^\nu(N)=0$ 2. $\eta \in N^c \Longrightarrow \xi \in N^c$ for every $\xi \preceq \eta$, i.e. for every $\xi \in \mathcal{N}(S \times G)$ such that its standard representation satisfies $Q_\xi \subseteq Q_\eta$ and $m_\xi(\gamma_x) \leq m_\eta(\gamma_x)$ for every $\gamma_x \in Q_\xi$. 3. If $X \in \mathcal{N}( \mathcal{C} )$ satisfies $X_t \in N$ for some $t \in {{\mathbb R}}$ then there exists $h > 0$ such that $X_s \in N$ for every $s \in [t,t+h)$. Let us notice that if $N$ is a dynamically impossible set for the intensity measure $\nu$ then the corresponding Poisson process $\Pi^{\phi_\nu}$ on $\mathcal{C}$ satisfies $$P( \Pi_t^{\phi_\nu} \in N \text{ for some }t \in {{\mathbb R}})=0.$$ Indeed, we have that $$\begin{aligned} P( \Pi_t^{\phi_\nu} \in N \text{ for some }t \in {{\mathbb R}}) & = P( \Pi_r^{\phi_\nu} \in N \text{ for some }r \in \mathbb{Q} )\\ \\ & \leq \sum_{r \in \mathbb{Q}} P( \Pi_r^{\phi_\nu} \in N ) = \sum_{r \in \mathbb{Q}} \pi^{\nu}(N) = 0.\end{aligned}$$ If follows from (ii) in Definition \[diset\] and the proof of Theorem \[convabs\] that condition ($\bullet \bullet$) in the statement of Theorem \[convabs\] may be replaced with the following weaker condition: 1. There exists a dynamically impossible set $N$ for the intensity measure $\nu$ such that $$\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$$ for all $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$ satisfying $\eta, \eta + \delta_{\gamma_x} \in N^c$. Applications {#examplescont} ------------ In the following we show how Theorem \[convabs\] may be applied to the models in Section \[examples\]. ### Continuity in the inverse temperature for the Ising model Consider $\beta_0 > 0$ such that $\alpha_{IC}(\beta_0) < 1$ and a sequence $(\beta_\varepsilon)_{\varepsilon > 0} \subseteq {{\mathbb R}}^+$ For each $\varepsilon \geq 0$ we may consider the Ising contours model with inverse temperature $\beta_\varepsilon$, which is specified by the intensity measure $\nu^{\beta_\varepsilon}$ as in and Hamiltonian $H^\varepsilon$ as in , the latter being independent of $\varepsilon$. By Theorem \[convabs\] we have that for $\varepsilon \geq 0$ sufficiently small the corresponding Ising contours model admits an infinite-volume Boltzmann-Gibbs distribution $\mu^\varepsilon$ and, furthermore, that $$\mu^\varepsilon \overset{loc}{\rightarrow} \mu^0.$$ Indeed, it suffices to check that these models satisfy the hypothesis of Theorem \[convabs\] functions $\Delta E_\eta$ replaced by their modified versions $\Delta E^*_\eta$ defined in and the interaction ranges $I(\{\gamma_x\})$ given by incompatibility, see Section \[ffgisingc1\]). But notice that if for $0 < \beta^* < \beta_0$ such that $\alpha_{IC}(\beta^*) < 1$ we consider the Ising contours model with inverse temperature $\beta^*$ given by the pair $(\nu^{\beta^*},H)$ then we see that: 1. For $\varepsilon \geq 0$ such that $\beta^* < \beta_\varepsilon$ we have that $\nu^\varepsilon \ll \nu^{\beta^*}$ with density given by $$\label{controldensidad3} \frac{d \nu^\varepsilon}{d \nu^{\beta^*}} (\gamma_x) = e^{- (\beta_\varepsilon - \beta^*)|\gamma_x|}$$ which satisfies $0 \leq \frac{d \nu^\varepsilon}{d \nu^{\beta^*}} \leq 1$. 2. The validity of (ii) and (iii) in the hypothesis of the theorem follows at once from the fact that the Hamiltonian is the same in all the contour models under consideration. Furthermore, since $\beta_\varepsilon \rightarrow \beta_0$ it also follows that $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E^*}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E^*}^{H^0}_\eta (\gamma_x)$ $\eta \in \mathcal{N}({{\mathbb{Z}}}^d \times G)$ and $\gamma_x \in {{\mathbb{Z}}}^d \times G$, so that $(\nu^{\beta^*},H)$ Finally, let us notice that since we are always under the heavily diluted regime, all the FFG processes under consideration have well defined $(+)$-alignments and $(-)$-alignments. As a direct consequence we obtain the following result. \[convabsic\] For any $\beta_0 > 0$ such that $\alpha_{IC}(\beta_0) < 1$ we have that $$\lim_{\beta \rightarrow \beta_0} \mu^{+,\beta} = \mu^{+,\beta_0} \hspace{2cm}\text{ and }\hspace{2cm}\lim_{\beta \rightarrow \beta_0} \mu^{-,\beta} = \mu^{-,\beta_0}$$ where for $\beta > 0$ the measures $\mu^{+,\beta}$ and $\mu^{-,\beta}$ are respectively defined as the weak limits $$\mu^{+,\beta} := \lim_{n \rightarrow +\infty} \mu^{+,\beta}_{\Lambda_n} \hspace{2cm}\text{ and }\hspace{2cm}\mu^{-,\beta} := \lim_{n \rightarrow +\infty} \mu^{-,\beta}_{\Lambda_n}$$ for any increasing sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\bigcup_{n \in {{\mathbb N}}} \Lambda_n = {{\mathbb{Z}}}^d$. Let us observe that this continuity result is well known (even for arbitrary values of $\beta_0$) and is a direct consequence of the monotonicity properties of the Ising model. The advantage of the approach presented here is that it can be applied in the same manner to other contour models lacking these properties. We will do this in Chapter \[chapterpirogovsinai\]. ### Widom-Rowlinson model with generalized interactions For the Widom-Rowlinson model with generalized interactions we obtain the next result. \[convabswr\] Let us consider for $\varepsilon \geq 0$ the Widom-Rowlinson model with fugacities $\lambda^+_\varepsilon$ and $\lambda^-_\varepsilon$, interspecies repulsion function $h^\varepsilon$ and type-independent repulsion function $j^\varepsilon$. Let us assume that the following conditions hold: 1. $\lim_{\varepsilon \rightarrow 0^+} \lambda^+_\varepsilon = \lambda^+_0$ and $\lim_{\varepsilon \rightarrow 0^+} \lambda^-_\varepsilon = \lambda^-_0$. 2. $\lim_{\varepsilon \rightarrow 0^+} h^\varepsilon(r) = h^0(r)$ and $\lim_{\varepsilon \rightarrow 0^+} j^\varepsilon(r) = j^0(r)$ for every $r \geq 0$. 3. $\lim_{\varepsilon \rightarrow 0^+} m_{h^\varepsilon} = m_{h^0}$ and $\lim_{\varepsilon \rightarrow 0^+} m_{j^\varepsilon} = m_{j^0}$ where $m_{h^\varepsilon}$ and $m_{j^\varepsilon}$ are defined for each $\varepsilon \geq 0$ through the relation supp($h^\varepsilon$)$=[0,m_{h^\varepsilon}]$ and supp($j^\varepsilon$)$=[0,m_{j^\varepsilon}]$. 4. $\alpha_{WR}(\lambda^+_0,\lambda^-_0,h^0,j^0) < 1$. Then for $\varepsilon \geq 0$ sufficiently small there exists a unique Gibbs measure $\mu^\varepsilon$ of the associated Widom-Rowlinson model. Furthermore, we have the convergence $$\mu^\varepsilon \overset{loc}{\rightarrow} \mu^0.$$ It suffices to see that this family of models is under the hypothesis of Theorem \[convabs\]. For this purpose, consider $L > m_{h^0}$ and define the interspecies repulsion function $$h:= \sup_{0 \leq \varepsilon < \varepsilon_L} h^\varepsilon$$ where $\varepsilon_L > 0$ is such that $m_{h^\varepsilon} < L$ for all $0 \leq \varepsilon < \varepsilon_L$. Next, take $K > m_{j^0}$ and define the type independent repulsion function $j$ in the analogous manner. Let us observe that $h$ and $j$ are both monotone decreasing, have bounded support and also satisfy $h^\varepsilon \leq h$ and $j^\varepsilon \leq j$ for every $\varepsilon \geq 0$ sufficiently small. Furthermore, we may take $L$ and $K$ sufficiently close to $m_{h^0}$ and $m_{j^0}$ respectively so as to guarantee that there exist $\lambda^+ > \lambda^+_0$, $\lambda^- > \lambda^-_0$ such that $$\alpha_{WR}(\lambda^+,\lambda^-,h,j) < 1.$$ Finally, if we consider the Widom-Rowlinson model with fugacities $\lambda^+$ and $\lambda^-$, interspecies repulsion function $h$ and type-independent repulsion function $j$ then for $\varepsilon \geq 0$ sufficiently small this model acts as a majorant. Indeed, in the notation of Theorem \[convabs\] we have: 1. $\nu^\varepsilon \ll \nu$ with density given by $$\frac{d\nu^\varepsilon}{d\nu} (\gamma_x) = \frac{\lambda^+_\varepsilon}{\lambda^+}\mathbbm{1}_{\{\gamma = +\}} + \frac{\lambda^-_\varepsilon}{\lambda^-}\mathbbm{1}_{\{\gamma = -\}}$$ which satisfies $0 \leq \frac{d\nu^\varepsilon}{d\nu} \leq 1$ for $\varepsilon \geq 0$ sufficiently small. 2. \(ii) is a direct consequence from the fact that $h^\varepsilon \leq h$ and $j^\varepsilon \leq j$ for every $\varepsilon \geq 0$ sufficiently small. 3. \(iii) follows from the fact that $\Delta E^H = 0$ and $\Delta E^{H^\varepsilon}=0$ for all $\varepsilon \geq 0$ since all the interactions are repulsive. 4. $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$ for all $\eta \in \mathcal{N}( {{\mathbb R}}^d \times \{+,-\})$ and $\gamma_x \in {{\mathbb R}}^d \times \{+,-\}$ such that both $\eta$ and $\eta + \delta_{\gamma_x}$ are outside the dynamically impossible set $$N=\{ \xi \in \mathcal{N}({{\mathbb R}}^d \times \{+,-\}) : \exists\,\, \gamma_x \neq \tilde{\gamma}_y \in \langle \xi \rangle \text{ such that } \|x-y\|_\infty \in \{ m_h^0 , m_j^0 \} \}$$ by assumptions (i), (ii) and (iii) in the statement of the theorem. By Theorem \[convabs\] this concludes the proof. Observe that Theorem \[convabswr\] shows, in the particular example of the heavily diluted Widom-Rowlinson model, that Gibbs measures of softcore converge as the force tends to infinity towards the Gibbs measure of its corresponding hardcore analogue. It is clear from the proof of Theorem \[convabs\] that this behavior also holds for other systems in a similar situation. ### Thin rods model For the thin rods model Theorem \[convabs\] yields the following result. \[convabstr\] Let us consider for each $\varepsilon \geq 0$ the thin rods model with fugacity $\lambda^\varepsilon$, rod length $2l^\varepsilon$ and orientation measure $\rho^\varepsilon$. Assume that the following conditions hold: 1. $\lim_{\varepsilon \rightarrow 0^+} \lambda^\varepsilon = \lambda^0$. 2. $\lim_{\varepsilon \rightarrow 0^+} l^\varepsilon = l^0$. 3. There exists a probability measure $\rho$ on the circle such that $\rho^\varepsilon \ll \rho$ for every $\varepsilon \geq 0$ with density $\frac{d \rho^\varepsilon}{d \rho}$ satisfying $0 \leq \frac{d \rho^\varepsilon}{d \rho} \leq 1$. 4. $4\lambda^0 (l^0)^2 \sigma_2 < 1$. Then for $\varepsilon \geq 0$ sufficiently small there exists a unique Gibbs measure $\mu^\varepsilon$ of the associated thin rods model. Furthermore, we have the convergence $$\mu^\varepsilon \overset{loc}{\rightarrow} \mu^0.$$ The proof of this result is similar to that of Theorems \[convabsic\] and \[convabswr\] so we omit it here. Nonetheless, we would like to point out that, just as in Section \[teounigibssexamples\], condition (iv) in the statement of Theorem \[convabstr\] may be relaxed provided that we have further knowledge on the measures $(\rho^\varepsilon)_{\varepsilon \geq 0}$. Resumen del Capítulo 10 ----------------------- En este capítulo mostramos la continuidad de las medidas de Gibbs de modelos altamente diluidos, i.e. bajo la condición de unicidad dada en el Capítulo 9, con respecto a su medida de intensidad $\nu$ y Hamiltoniano $H$ en el caso absolutamente continuo. Este escenario típicamente incluye continuidad con respecto a los parámetros del modelo como pueden ser la densidad de partículas, la temperatura inversa y el rango de interacción entre otros. El resultado principal está contenido en el Teorema \[convabs\] arriba. Esencialmente, éste garantiza continuidad con respecto a pequeños cambios en $(\nu,H)$ que sean absolutamente continuos en $\nu$ bajo la existencia de un modelo mayorante (ver Teorema \[convabs\]) que sea altamente diluido. La demostración consiste en acoplar de manera conveniente las medidas de Gibbs del modelo original y el modificado mediante la construcción hacia el pasado de la dinámica de Fernández-Ferrari-Garcia. Es para poder construir este acoplamiento de manera exitosa que se requiere la existencia de un modelo mayorante. Un aspecto importante a destacar de este resultado es que prueba la continuidad con respecto a la convergencia local de medidas de probabilidad, mientras que hasta ahora sólo era conocida la continuidad con respecto a la convergencia en distribución que es, al menos en el contexto continuo, más débil que la local. Por último, discutimos algunas aplicaciones de este resultado. Mostramos que en la práctica los modelos mayorantes siempre existen bajo la condición de unicidad, y que típicamente se pueden obtener mediante un ligero incremento en la densidad de partículas (o disminución de la temperatura inversa) y/o del rango de interacción del modelo $(\nu,H)$. Obtenemos así, entre otros resultados, la continuidad de la medida de Gibbs con respecto a la densidad de partículas y rango de exclusión en el modelo de Widom-Rowlinson (tanto continuo como discreto), continuidad con respecto a la temperatura inversa para cada medida de Gibbs extremal en el modelo de Ising a baja temperatura y con respecto a la medida de orientación en el modelo de las varas finas. Discretization of Gibbs measures {#secdis} ================================ In this chapter we establish the continuity of Gibbs measures in heavily diluted models with respect to discretization procedures. We begin by introducing a formal definition of discretization and then move on to establish a general continuity result in this scenario. Finally, we conclude with some examples and applications. A general discretization result ------------------------------- A metric space $X$ is called *absolutely locally compact* if for any $\delta > 0$ the closed balls $\overline{B}(x,\delta)$ are compact for all $x \in X$. $\,$ - If $X$ is an absolutely locally compact metric space then for any $\delta > 0$ and compact set $K \subseteq G$ the $\delta$-neighborhood of $K$ denoted by $K^{(\delta)}$ has compact closure. - If $X$ and $Y$ are absolutely locally compact metric spaces then so is $X \times Y$. A metric space $X$ is called a *discretizable* if it is complete, separable and absolutely locally compact. If $X$ and $Y$ are discretizable metric spaces then so is the product $X \times Y$. Throughout the rest of the section we shall with diluted models where both $S$ and $G$ are discretizable metric spaces so that $S \times G$ remains a discretizable metric space under the product metric $d_{S\times G}:= d_S + d_G$. \[discdefi\] A family $(D_\varepsilon)_{\varepsilon > 0}$ of measurable applications $D_\varepsilon : S \times G \to S\times G$ is called a *discretization family* if for every $\varepsilon > 0$ and $\gamma_x \in S \times G$ one has $$\label{rho} d_{S \times G}( D_\varepsilon(\gamma_x), \gamma_x ) \leq \varepsilon.$$ The application $D_\varepsilon$ shall be called the $\varepsilon$-discretization operator. $\,$ 1. *Spatial discretization*. Let us consider $S = {{\mathbb R}}^d$ and $G=\{1,\dots,q\}$ for some $q \in {{\mathbb N}}$. For each $\varepsilon > 0$ we define $\varepsilon$-discretization operator $D_\varepsilon$ by the formula $$D_\varepsilon ( x, \gamma ) = ( x_\varepsilon , \gamma )$$ where if $x=(x_1,\dots,x_d) \in {{\mathbb R}}^d$ we set $$\label{spatialdiscret} x_\varepsilon := \left( \varepsilon \left[\frac{x_1}{\varepsilon}\right],\dots,\varepsilon \left[\frac{x_d}{\varepsilon}\right]\right).$$ 2. *Spin discretization*. Let us consider $S = {{\mathbb R}}^2$ and $G = S^1_* := [0,\pi)$. For each $\varepsilon > 0$ we define $\varepsilon$-discretization operator $D_\varepsilon$ by the formula $$\label{spindiscret} D_\varepsilon ( x, \gamma ) = \left( x , \varepsilon \left[\frac{\theta}{\varepsilon}\right] \right)$$ Notice that, in order to remain faithful to the idea of discretization, in Definition \[discdefi\] it would be natural to also require the image of $D_\varepsilon$ to be countable for every $\varepsilon > 0$. However, this extra assumption is not needed for our results and so we leave From now onwards, to simplify the notation we shall write $\gamma_x^\varepsilon$ instead of $D_\varepsilon (\gamma_x)$. Now, let us consider some fixed discretization family $(D_\varepsilon)_{\varepsilon > 0}$ on the space $S \times G$. Given a Poisson process $\overline{\Pi}$ on $\mathcal{C} \times [0,1]$ with intensity measure $\overline{\phi}_\nu$ we may define for each $\varepsilon > 0$ the $\varepsilon$-discretized process $\overline{\Pi}^\varepsilon$ (or simply $\varepsilon$-process) by the formula $$\label{piepsilon} \overline{\Pi}^\varepsilon := \{ ( \gamma_x^\varepsilon, t, s, u ) \in \mathcal{C} \times [0,1] : (\gamma_x, t, s, u ) \in \overline{\Pi}\}.$$Let us observe that $\overline{\Pi}^\varepsilon$ is a Poisson process on $\mathcal{C} \times [0,1]$ with intensity measure $\overline{\phi}_{\nu_\varepsilon}$ where $\nu_\varepsilon$ denotes the $\varepsilon$-*discretized intensity measure* defined by the formula $$\nu_\varepsilon := \nu \circ D_\varepsilon^{-1}.$$ Furthermore, establishes a one-to-one correspondence between cylinders of With this in mind, we shall write $C_\varepsilon$ to denote the $\varepsilon$-cylinder in $\Pi^\varepsilon$ which corresponds to the cylinder $C \in \Pi$, i.e., if $C=(\gamma_x,t,s)$ then we shall set $C_\varepsilon = (\gamma_x^\varepsilon,t,s )$. \[convaga\] For each $t \in {{\mathbb R}}$ we have $\Pi^\varepsilon_t \overset{as}{\longrightarrow} \Pi_t$ as $\varepsilon \rightarrow 0^+$ with the vague topology. Straightforward consequence of the following lemma. \[lemaconvaga\] Let $\xi \in \mathcal{N}(S \times G)$ and for each $\varepsilon > 0$ consider the configuration $\xi^\varepsilon$ defined by the standard representation $$\xi^\varepsilon = \sum_{\gamma_x \in Q_\xi} m(\gamma_x)\delta_{\gamma_x^\varepsilon}.\footnote{The fact that for every $\varepsilon > 0$ the configuration $\xi^\varepsilon$ is indeed \textit{locally finite} follows from the absolute local compactness of $S \times G$.}$$ Then with respect to the vague topology in $\mathcal{N}(S \times G)$ we have $\lim_{\varepsilon \rightarrow 0^+} \xi^\varepsilon = \xi.$ It suffices to show that for each compact set $K \subseteq S \times G$ and $\delta > 0$ there exists $\varepsilon_0 > 0$ small enough such that $\xi^\varepsilon \in (\xi)_{K,\delta}$ for all $0 <\varepsilon < \varepsilon_0$. Notice that if we take $$\varepsilon_0 := \left(\frac{1}{2}\min\{ d_{S \times G}\left(\gamma_x,\tilde{\gamma}_y\right) : \gamma_x\neq \tilde{\gamma}_y \in [\xi_{K_\delta}]\}\right)\wedge \delta > 0$$ where $K_\delta = \{ \gamma_x \in S \times G : d_{S \times G} (\gamma_x, K) < \delta \}$ is the $\delta$-neighborhood of $K$ then for every $0 < \varepsilon < \varepsilon_0$ we have that $\xi^\varepsilon \in (\xi)_{K,\delta}$ since: 1. By the mere definition of $D_\varepsilon$ to each point of $\xi^\varepsilon_K$ we can assign at least one point of $\xi$ at a distance smaller than $\varepsilon$ (without any regard for their respective multiplicities). Moreover, since $\varepsilon < \varepsilon_0$ we have that there is at most one point of $\xi$ in these conditions so that the multiplicity must be preserved. Thus we may define $p: [\xi^\varepsilon_K] \to [\xi]$ by the formula $$p(\gamma_x^\varepsilon, i)=\left(D_\varepsilon^{-1}(\gamma_x^\varepsilon),i\right)$$ which is clearly injective. This shows that $\xi^\varepsilon_K \preceq_\delta \xi$. 2. Once again, by definition of $D_\varepsilon$ to each point of $\xi_K$ we can assign a point of $\xi^\varepsilon$ at a distance smaller than $\varepsilon$ (without any regard for their respective multiplicities). Moreover, since $0 < \varepsilon < \varepsilon_0$ this assignation is injective. Hence, if $p:[\xi_K] \to [\xi^\varepsilon]$ is defined by the formula $$p(\gamma_x, i)=\left(\gamma_x^\varepsilon,i\right).$$ then $p$ is injective for $0 < \varepsilon < \varepsilon_0$, which shows that $\xi_K \preceq_\delta \xi^\varepsilon$. \[convdis\] Let $(D_\varepsilon)_{\varepsilon > 0}$ be a discretization family and consider a family of diluted models $(\nu^\varepsilon,H^\varepsilon)_{\varepsilon \geq 0}$ such that 1. There exists a heavily diluted model $(\nu,H)$ satisfying 1. For every $\varepsilon \geq 0$ the intensity measure $\nu^\varepsilon$ satisfies $$\nu^\varepsilon = \nu \circ D_\varepsilon^{-1}$$ where $D_0$ is set as the identity operator, i.e. $\nu^0 = \nu$. 2. For every $\varepsilon \geq 0$ and $\gamma_x \in S \times G$ we have that $D_\varepsilon^{-1}\left(I^{H^\varepsilon}(\{\gamma^\varepsilon_x\})\right) \subseteq I^H (\{\gamma_x\})$, i.e. if $\tilde{\gamma}^\varepsilon_y \rightharpoonup_{H^\varepsilon} \gamma_x^\varepsilon$ for some $\varepsilon \geq 0$ then $\tilde{\gamma}_y \rightharpoonup_{H} \gamma_x$. 3. $\Delta E^H \leq \inf_{\varepsilon \geq 0} \Delta E^{H^\varepsilon}.$ 2. $\lim_{\varepsilon \rightarrow 0^+} \Delta E^{H^\varepsilon}_{\eta^\varepsilon} (\gamma^\varepsilon_x) = \Delta E^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$. Then each diluted model $(\nu^\varepsilon,H^\varepsilon)$ admits exactly one Gibbs measure $\mu^\varepsilon$ and as $\varepsilon \rightarrow 0^+$ $$\mu^{\varepsilon} \overset{d}{\longrightarrow} \mu^0.$$ The model $(\nu,H)$ is called a *majorant model* for $(\nu^\varepsilon,H^\varepsilon)_{\varepsilon \geq 0}$. Let us start by showing that each model $(\nu^\varepsilon,H^\varepsilon)$ admits exactly one Gibbs measure for every $\varepsilon \geq 0$. To do this let us consider a Poisson process $\overline{\Pi}$ on $\mathcal{C} \times [0,1]$ with intensity measure $\nu \times e^{-\Delta E^H} \mathcal{L} \times \mathcal{L}_{{{\mathbb R}}^+} \times \mathcal{U}[0,1]$ and and its corresponding discretizations $(\overline{\Pi}^\varepsilon)_{\varepsilon > 0}$. By the the proof of Theorem \[teounigibbs\] we see that it suffices to show that for each $\varepsilon \geq 0$ and $\Lambda \in {{\mathcal B}}^0_S$ the clan of ancestors at time 0 with respect to the Hamiltonian $H^\varepsilon$ and underlying free process $\Pi^\varepsilon$ is finite almost surely. But this follows from the heavy diluteness of $(\nu,H)$ since $$\label{ancesincdis} \mathcal{A}^{0,H^\varepsilon}(\Lambda \times G) \subseteq D_\varepsilon \left( \A^{0,H}(\overline{\Lambda_{\varepsilon}} \times G)\right)$$ where $\overline{\Lambda_{\varepsilon}}$ denotes the closed $\varepsilon$-neighborhood of $\Lambda$ and for $\Gamma \subseteq \mathcal{C}$ we set $$D_\varepsilon (\Gamma) = \{ (\gamma^\varepsilon_x , t,s ) \in \mathcal{C} : (\gamma_x,t,s) \in \Gamma \}.$$ This settles the first statement. To establish the local convergence, we shall proceed as in the proof of Theorem \[convabs\]. We couple all measures $\mu^\varepsilon$ simultaneously by considering the infinite-volume stationary FFG processes $\mathcal{K}^\varepsilon$ defined as $$\label{keptscaled2} \mathcal{K}^\varepsilon = \{ (\gamma_x^\varepsilon,t,s) \in \Pi : F(\gamma_x^\varepsilon,t,s) < M^\varepsilon(\gamma_x^\varepsilon | \mathcal{K}^\varepsilon_{t^-}) \}$$ where for each $\gamma_x \in S \times G$ and $\xi \in \mathcal{N}(S \times G)$ we define $$M^\varepsilon (\gamma_x |\xi) := e^{- (\Delta E^{H^\varepsilon}_\xi(\gamma_x) - \Delta E^H)}.$$ Just as in the proof of Theorem \[convaga\], for each $\varepsilon \geq 0$ the process $\mathcal{K}^\varepsilon$ is stationary with invariant measure $\mu^\varepsilon$ and thus it will suffice to show that as $\varepsilon \rightarrow 0^+$ $$\mathcal{K}^\varepsilon_0 \overset{as}{\longrightarrow} \mathcal{K}^0_0.$$ Let us take then a compact set $K \in {{\mathcal B}}^0_{S\times G}$ and $\Lambda \in {{\mathcal B}}^0_S$ such that $K \subseteq \Lambda \times G$. Now, since $\lim_{\varepsilon \rightarrow 0^+} \Delta E^{H^\varepsilon}_{\eta^\varepsilon} (\gamma^\varepsilon_x) = \Delta E^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$, it follows that there exists (random) $\varepsilon_0 > 0$ such that if $0 \leq \varepsilon < \varepsilon_0$ then $$\label{ancesincdisc1} \mathcal{K}^\varepsilon_{D_\varepsilon(\A^{0,H} ( \Lambda_1 \times G))} = D_\varepsilon \left(\mathcal{K}^0_{\A^{0,H} ( \Lambda_1 \times G)}\right),$$ where $\Lambda_1$ denotes the $1$-neighborhood of $\Lambda$. Indeed, if (random) $N \in {{\mathbb N}}$ is such that $\A^{0,H}_n(\Lambda_1 \times G) = \emptyset$ for every $n > N$ almost surely then for every cylinder $C \in \A^{0,H}_N(\Lambda_1 \times G)$ and $\varepsilon \geq 0$ we have that $$C^\varepsilon \in \mathcal{K}^\varepsilon \Longleftrightarrow F(C) < M^\varepsilon( basis(C^\varepsilon) | \emptyset )$$ from which we immediately obtain that for $\varepsilon$ (randomly) small enough $$\mathcal{K}^\varepsilon_{D_\varepsilon(\A^{0,H}_N ( \Lambda_1 \times G))} = D_\varepsilon \left(\mathcal{K}^0_{\A^{0,H}_N( \Lambda_1 \times G)}\right)$$ and one may proceed with the succeeding generations by induction using inclusion . From and the inclusion $$D^{-1}_\varepsilon (K) \subseteq \Lambda_1 \times G$$ valid for every $0 <\varepsilon < 1$ one can show as in the proof of Lemma \[lemaconvaga\] that given $\delta > 0$ for $\varepsilon$ (randomly) small enough we have $\mathcal{K}^\varepsilon_0 \in (\mathcal{K}^0_0)_{K,\delta}$. This establishes the almost sure convergence and thus concludes the proof. Just as it was the case for Theorem \[convabs\], majorant models in this context are fairly easy to obtain, and one can generally do so by slightly “inflating” the interaction of the corresponding limit model in some appropriate sense. Also, let us notice that the proof of does not only yield convergence in distribution but in fact provides a coupling between the corresponding Gibbs measures in which the convergence takes place in the stronger almost sure sense. Once again, we stress this fact since other methods used to obtain these type of results (i.e. cluster expansion or disagreement percolation methods) in general cannot produce such a coupling. Finally, condition ($\bullet \bullet$) in the statement of the theorem may be replaced by the following weaker condition: 1. There exists a dynamically impossible set $N$ for the intensity measure $\nu$ such that $$\lim_{\varepsilon \rightarrow 0^+} \Delta E^{H^\varepsilon}_{\eta^\varepsilon} (\gamma^\varepsilon_x) = \Delta E^{H^0}_\eta (\gamma_x)$$ for all $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$ satisfying $\eta, \eta + \delta_{\gamma_x} \in N^c$. Applications {#applications} ------------ We now discuss two applications of Theorem \[convdis\] related to models in Section \[examples\]. ### Widom-Rowlinson model As a direct consequence of Theorem \[convdis\] one obtains that Gibbs measures in discrete heavily diluted models converge, when properly rescaled, towards the Gibbs measure of the analogous continuum model. As an example we study the particular case of the Widom-Rowlinson model. Other models may be handled in the same fashion. \[diswr\] For $\lambda^0,r^0 > 0$ such that $\lambda^0 (2r^0)^d < 1$ we have the following: 1. The continuum Widom-Rowlinson model on $\mathcal{N}({{\mathbb R}}^d \times \{+,-\})$ with fugacity $\lambda^0$ and exclusion radius $r^0$ admits exactly one Gibbs measure, which we shall denote by $\mu^0$. 2. For $ 0 < \varepsilon < \sqrt[d]{\frac{1}{\lambda^0} - (2r^0)^d}$ the discrete Widom-Rowlinson model on $\mathcal{N}({{\mathbb{Z}}}^d \times \{+,-\})$ with fugacity $\varepsilon^d\lambda^0$ and exclusion radius $\frac{r^0}{\varepsilon}$ admits exactly one Gibbs measure $\tilde{\mu}^\varepsilon$. 3. Provided $0 < \varepsilon < \sqrt[d]{\frac{1}{\lambda^0} - (2r^0)^d}$ as $\varepsilon \rightarrow 0^+$ we have $$\tilde{\mu}^\varepsilon \circ i_\varepsilon^{-1} \overset{d}{\longrightarrow} \mu^0.$$ where for each $\varepsilon > 0$ we define the *shrinking map* $i_\varepsilon : {{\mathbb{Z}}}^d \times \{+,-\} \to {{\mathbb R}}^d \times \{+,-\}$ by the formula $$i_\varepsilon ( x , \gamma ) = (\varepsilon\cdot x, \gamma).$$ The first two statements are a direct consequence of To show (iii), first we consider the spatial discretization family $(D_\varepsilon)_{\varepsilon \geq 0}$ given by and for each $\varepsilon \geq 0$ set the intensity measure $\nu^\varepsilon$ as $$\nu^\varepsilon := \nu \circ D_\varepsilon^{-1}$$ where $$\nu := \left( \lambda \mathcal{L}^d \times \delta_+ \right) + \left(\lambda \mathcal{L}^d \times \delta_-\right).$$ Then we set the Hamiltonian $H^0$ as in Section \[examples\], i.e. $$H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y )$$ where $$\label{wrud} U(\gamma_x,\tilde{\gamma}_y) := \left\{ \begin{array}{ll} +\infty &\text{if }\gamma \neq \tilde{\gamma}\text{ and }\|x-y\|_\infty \leq r^0\\ 0 &\text{otherwise.}\end{array}\right.$$ Finally, for every $\varepsilon > 0$ we consider the Hamiltonian $H^\varepsilon$ defined for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb R}}^d}$ and by the formula $$H^\varepsilon_{\Lambda|\eta}(\sigma) := \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e^\varepsilon_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y ) + \sum_{x_\varepsilon \in \Lambda} V_{x_\varepsilon}(\sigma)$$ where $$e^\varepsilon_{\Lambda}(\sigma|\eta) := \{ (\gamma_x ,\tilde{\gamma}_y) \in \langle \sigma^\varepsilon_{\Lambda \times G} \cdot \eta_{\Lambda^c \times G} \rangle^2 : x \in \Lambda \},$$ the pair interaction $U$ is the same as in , $x_\varepsilon$ is defined as in and $$V_{x_\varepsilon}(\sigma) := \left\{ \begin{array}{ll} +\infty &\text{if } \sigma( \{x_\varepsilon\} \times \{+,-\} ) > 1 \\ 0 &\text{otherwise.}\end{array}\right.$$ Now, the crucial observation is that for every $\varepsilon > 0$ the diluted model specified by the pair $(\nu^\varepsilon, H^\varepsilon)$ is essentially the shrunken version of the discrete Widom-Rowlinson model of fugacity $\varepsilon^d \lambda^0$ and exclusion radius $\frac{r^0}{\varepsilon}$. More precisely, for every $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\varepsilon > 0$ we have $$\label{igualdaddis} \mu^\varepsilon_{i_\varepsilon(\Lambda)|\emptyset} = \tilde{\mu}^\varepsilon_{\Lambda|\emptyset} \circ i_\varepsilon^{-1}$$ where $\tilde{\mu}^\varepsilon_{\Lambda|\emptyset}$ is the Boltzmann-Gibbs distribution with empty boundary condition associated to the discrete Widom-Rowlinson model whereas $\mu^\varepsilon_{\Lambda|\emptyset}$ is the one associated to $(\nu^\varepsilon,H^\varepsilon)$. Thus, by taking the limit as $\Lambda \nearrow {{\mathbb{Z}}}^d$, Theorem \[teounigibbs\] yields for $0 < \varepsilon < \sqrt[d]{\frac{1}{\lambda^0} - (2r^0)^d}$ $$\mu^\varepsilon = \tilde{\mu}^\varepsilon \circ i_{\varepsilon}^{-1}$$ where $\mu^\varepsilon$ is the unique Gibbs measure of the diluted model given by the pair $(\nu^\varepsilon, H^\varepsilon)$. Hence, it suffices to show that the family $(\nu^\varepsilon,H^\varepsilon)$ is under the hypothesis of Theorem \[convdis\]. But notice that if for $\delta > 0$ we define the Hamiltonian $H$ by the formula $$H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U^\delta( \gamma_x , \tilde{\gamma}_y ) + \sum_{\gamma_x \in \Lambda} V_{x}^\delta( \sigma )$$ where $$U^\delta(\gamma_x,\tilde{\gamma}_y) := \left\{ \begin{array}{ll} +\infty &\text{if }\gamma \neq \tilde{\gamma}\text{ and }\|x-y\|_\infty \leq r^0 + \delta\\ 0 &\text{otherwise.}\end{array}\right.$$ $$V_{x}^\delta( \sigma ) := \left\{ \begin{array}{ll} +\infty &\text{if } \sigma( \{y \in {{\mathbb R}}^d : \|x - y \|_\infty \leq \delta\} \times \{+,-\} ) > 1 \\ 0 &\text{otherwise}\end{array}\right.$$ then for $\delta > 0$ small the diluted model $(\nu,H)$ acts a majorant for $\varepsilon \geq 0$ sufficiently small. Indeed, we have that 1. \(i) holds trivially by the choice of measures $\nu^\varepsilon$. 2. \(ii) holds for all $\varepsilon < \frac{\delta}{2}$ by definition of $D_\varepsilon$. 3. \(iii) holds since $\Delta E^H = 0 = \inf_{\varepsilon \geq 0} \Delta E^{H^\varepsilon}$ due to the fact that all interactions considered are repulsive. 4. $\lim_{\varepsilon \rightarrow 0^+} \Delta E^{H^\varepsilon}_{\eta^\varepsilon} (\gamma_x^\varepsilon) = \Delta E^{H^0}_\eta (\gamma_x)$ for all $\eta \in \mathcal{N}( {{\mathbb R}}^d \times \{+,-\})$ and $\gamma_x \in {{\mathbb R}}^d \times \{+,-\}$ such that both $\eta$ and $\eta + \delta_{\gamma_x}$ are outside the dynamically impossible set $N_1 \cup N_2$ where $$N_1=\{ \xi \in \mathcal{N}({{\mathbb R}}^d \times \{+,-\}) : \exists\,\, \gamma_x \neq \tilde{\gamma}_y \in \langle \xi \rangle \text{ such that } \|x-y\|_\infty = r^0 \}$$ and $$N_2 = \{ \xi \in \mathcal{N}({{\mathbb R}}^d \times \{+,-\}) : \sigma( \{x\} \times \{+,-\} ) > 1 \text{ for some }x \in {{\mathbb R}}^d \}.$$ If we take $\delta > 0$ such that $\lambda^0(2(r^0+\delta))^d < 1$ then the model $(\nu,H)$ is heavily diluted. Thus, by Theorem \[convdis\] we obtain the result. ### Thin rods model Another application of Theorem \[convdis\] is to study the limit of the thin rods model when the number of possible orientations tends to infinity. As expected, under the heavily diluted regime we have the following result. Given $\lambda, l > 0$ and a probability measure $\rho$ on $S^1_*$, for each $\varepsilon > 0$ consider the thin rods model on $\mathcal{N}({{\mathbb R}}^2 \times S^1_*)$ with fugacity $\lambda$, rod length $2l$ and orientation measure $$\rho^\varepsilon = \sum_{i=0}^{n^\varepsilon} w^\varepsilon(i) \delta_{i \varepsilon}$$ where $n^\varepsilon:= \left[\frac{\pi}{\varepsilon}\right]$ and $w^\varepsilon(i):= \rho( \{\theta \in S^1_* : \left[\frac{\pi}{\varepsilon}\right]=i \} )$. If $4\lambda l^2 \sigma_2 < 1$ then for every $\varepsilon > 0$ there exists a unique Gibbs measure $\mu^\varepsilon$ of the corresponding thin rods model. Furthermore, as $\varepsilon \rightarrow 0^+$ we have $$\mu^\varepsilon \overset{d}{\rightarrow} \mu^0$$ where $\mu^0$ is the unique Gibbs measure of the thin rods model with fugacity $\lambda$, and orientation measure $\rho$. We omit the proof of this result since it goes very much along the lines of but using the spin discretization family introduced in instead of the spatial one. ### Some important remarks on discretization procedures Suppose that we have some continuum heavily diluted model and let $\mu$ denote its unique Gibbs measure. One could then ask what can be said about the discretized measures $\mu^\varepsilon := \mu \circ D_\varepsilon^{-1}$ for $\varepsilon > 0$. For example, 1. Is it true that $\mu^\varepsilon$ is a Gibbs measure for the corresponding discrete model? 2. If not, is it close to the actual Gibbs measure of the discrete system? The examples discussed above show that we cannot expect (i) to be true. Indeed, for example in the Widom-Rowlinson model the discretized Gibbs measures $\mu^\varepsilon$ can assign positive weight to particle configurations in which particles of opposite type are within the exclusion radius; this is because certain allowed configurations in the may violate the exclusion radius restriction when discretized. Therefore, in general it is not enough to discretize the continuum Gibbs measure to obtain the Gibbs measure of the discrete system. What Theorem \[convdis\] in fact shows is that obtaining the actual discrete Gibbs measure demands a more complicated procedure: one has to discretize the free process and then do the deleting procedure all over again. Nevertheless, implies that $\mu^\varepsilon$ is indeed close to the Gibbs measure of the discrete system. On a similar note, observe that when trying to simulate Gibbs measures of continuum systems using the FFG dynamics, practical limitations prevent the inclusion of all possible configurations in the simulation, and so one inevitably has to replace the original model by a discretized version of it. What Theorem \[convdis\] also shows is that no problems arise by this replacement, since by the simulated discrete measure will be close to the original continuum one. Resumen del Capítulo 11 ----------------------- Mostramos aquí la continuidad de medidas de Gibbs en modelos altamente diluidos con respecto a procesos de discretización, i.e. la convergencia de modelos discretos a modelos continuos a nivel de las medidas de Gibbs correspondientes. En general, los modelos discretos a los que hacemos referencia pueden ser de dos tipos: con espacio de ubicaciones discreto (el modelo de Widom-Rowlinson discreto, por ejemplo) o con espacio de spines discreto (el modelo de varas finas con finitas orientaciones posibles). El resultado principal está contenido en el Teorema \[convdis\] arriba. Nuevamente, se obtiene la convergencia (en distribución) de modelos discretos a sus análogos continuos bajo la existencia un modelo mayorante altamente diluido. La demostración consiste una vez más en acoplar las medidas de Gibbs de los modelos discretos junto a la del modelo continuo límite mediante la construcción hacia el pasado de la dinámica FFG. Aquí es donde se vuelve evidente la necesidad de un marco teórico que nos permita encarar por igual la dinámica tanto en modelos discretos como continuos. Cabe destacar que el problema de la continuidad con respecto a discretizaciones no ha sido muy estudiado hasta ahora, y que muchas de las técnicas de mayor influencia dentro de la mecánica estadística (como por ejemplo la teoría de Pirogov-Sinai) no se encuentran, en principio, preparadas para lidiar con este tipo de problemas (especialmente para discretizaciones en el espacio de spins). No obstante, en nuestro contexto este tipo de problemas pueden plantearse y resolverse de manera natural. Luego, a manera de aplicación mostramos que bajo el régimen de unicidad el modelo de Widom-Rowlinson discreto apropiadamente escalado converge, cuando la densidad de partículas tiende a cero y el radio de exclusión tiende a infinito, al correspondiente modelo de Widom-Rowlinson continuo. También mostramos que el modelo de varas finas con $n$ orientaciones converge cuando $n \rightarrow +\infty$ al modelo con un continuo de orientaciones, nuevamente bajo el régimen de unicidad. Por último, discutimos algunas conclusiones que pueden sacarse a partir del resultado probado. En primer lugar, la demostración del Teorema \[convdis\] muestra que al discretizar una medida de Gibbs en un modelo continuo no se obtiene, en general, una medida de Gibbs del correspondiente modelo discreto pero que, sin embargo, el resultado se encuentra razonablemente próximo de esta última. Por otro lado, el Teorema \[convdis\] también garantiza que para simular numéricamente medidas de Gibbs de modelos continuos bajo el régimen de unicidad es razonable simular medidas de Gibbs para modelos que sean aproximaciones discretas de los mismos, ya que éstas serán una buena aproximación de las verdaderas medidas de interés. Applications to Pirogov-Sinai theory {#chapterpirogovsinai} ==================================== In this final chapter we combine the ideas of previous chapters with the framework of Pirogov-Sinai theory to show that some of the typical results in this theory can be obtained without the use of cluster expansions. Moreover, we show that this allows us to enlarge the traditional range of validity of the theory in some cases. This constitutes a step towards completing the approach first proposed in [@FFG1]. As a byproduct, we obtain a perfect simulation algorithm for systems at the low temperature or high density regime. For simplicity, we shall discuss the framework of Pirogov-Sinai theory and its applications only in some particular cases, but the experienced reader will understand how to extend these ideas to the general setting. We follow the presentation of this theory given in [@Z1]. Discrete $q$-Potts model of interaction range $r$ ------------------------------------------------- Consider the discrete model on $\{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ defined in the traditional manner through the Boltzmann-Gibbs distributions given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ by $$\mu_{\Lambda}^\eta(\sigma) = \frac{\mathbbm{1}_{\{\sigma_{\Lambda^c} \equiv \eta_{\Lambda^c}\}}}{Z_{\Lambda}^\eta} e^{- \beta \sum_{B : B \cap \Lambda \neq \emptyset} \Phi_B(\sigma_{B})}$$ with $\beta > 0$ denoting the inverse temperature, $Z_{\Lambda}^\eta$ being the normalizing constant and $$\label{hpotts} \Phi_B (\sigma_B):= \left\{\begin{array}{ll} \mathbbm{1}_{\{ \sigma(x) \neq \sigma(y)\,,\,\|x-y\|_1 \leq r\}} & \text{ if $B=\{x,y\}$} \\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ This model, known as the $q$-Potts model of interaction range $r$, can be interpreted as a direct generalization of the Ising model presented in Section \[exampleisingc\]. We define the set $\mathcal{R}:=\{ \eta_1,\dots,\eta_q\}$ of reference configurations, where for $i=1,\dots,q$ the configuration $\eta_i$ corresponds to the $i$-aligned configuration, i.e. For convenience purposes, for every $i=1,\dots,q$ we shall denote the corresponding Boltzmann-Gibbs distribution on $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ simply by $\mu^i_{\Lambda}$ instead of $\mu^{\eta_i}_\Lambda$ shall be of particular interest to us since, as we will see, each of them will represent a different equilibrium state of the system at low temperature. To show this fact we shall need to introduce as in the original Ising model the notion of contour with respect to each of these reference configurations. We proceed as follows. For $i=1,\dots,q$ we shall say that the site $x \in {{\mathbb{Z}}}^d$ is $i$-correct for a given configuration $\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ whenever $\sigma(y)=i$ for every $y \in {{\mathbb{Z}}}^d$ such that $\| x - y\|_1 \leq r$. We label a site as *incorrect* with respect to $\sigma$ if it fails to be $i$-correct for all $i=1,\dots,q$. We define the *defect set* $D_\sigma$ of the configuration $\sigma$ as the set of all incorrect sites with respect to $\sigma$. The restriction of $\sigma$ to any one of the finite connected components of $D_\sigma$ will be called a *contour* of $\sigma$ and the corresponding component will be called the *support* of this contour. Given a contour $\gamma$, the space ${{\mathbb{Z}}}^d - \text{supp}(\gamma)$ is divided into a finite number of connected components, only one of which is infinite. We call one of this components a $i$-component if its neighboring spins in $\gamma$ all have the value $i$. Notice that every component of ${{\mathbb{Z}}}^d - \text{supp}(\gamma)$ is an $i$-component for some $i \in \{1,\dots,q\}$. If the infinite component in ${{\mathbb{Z}}}^d - \text{supp}(\gamma)$ is a $i$-component we say that $\gamma$ is a $i$-contour and denote this fact by $\gamma^i$. For any $j \in \{1,\dots,q\}$ we denote by $\text{Int}_{j}(\gamma)$ the union of all finite $j$-components of ${{\mathbb{Z}}}^d - \text{supp}\gamma$. Then we set $$\text{Int}(\gamma) := \bigcup_{j=1}^q \text{Int}_j(\gamma)\hspace{1cm}\text{V}(\gamma):= \text{supp}(\gamma) \cup \text{Int}(\gamma) \hspace{1cm}\text{Ext}(\gamma):= {{\mathbb{Z}}}^d - \text{V}(\gamma).$$ See Figure \[fig4\] for a possible configuration of Pirogov-Sinai contours in the Ising model. Finally, if $\gamma$ is a contour of a configuration $\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ then define the energy of $\gamma$ as ![Pirogov-Sinai contours for the Ising model on ${{\mathbb{Z}}}^2$.[]{data-label="fig4"}](contornosSanti2.eps){width="8cm"} $$\Phi(\gamma) := \sum_{B \subseteq {{\mathbb{Z}}}^d} \frac{|B \cap \text{supp}(\gamma)|}{|B|} \Phi_B(\sigma_B).$$ Notice that this value does not depend on the choice of $\sigma$, only on $\gamma$. Now, notice that each configuration $\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ with a finite defect set $D_\sigma$ defines a unique family of contours $\Gamma_\sigma$ from which it can be completely recovered. Furthermore, we have the following result. \[proprepresentacionps\] For $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ let us consider a configuration $\sigma$ such that $D_\sigma \subseteq \Lambda$ and $\sigma(x)=i$ for every $x \in {{\mathbb{Z}}}^d$ lying in the unique infinite connected component of ${{\mathbb{Z}}}^d - D_\sigma$. If $\Gamma_{\sigma}$ denotes the family of contours associated to $\sigma$, then we have $$\label{ps1} \mu^{i}_\Lambda (\sigma) = \frac{1}{Z^{i}_\Lambda} e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)}.$$ For each finite $B \subseteq {{\mathbb{Z}}}^d$ let us write $$\Phi_B (\sigma_B) = \sum_{x \in B} \frac{1}{|B|} \Phi_B(\sigma_B) = \sum_{\gamma \in \Gamma_\sigma} \frac{|B \cap \text{supp}(\gamma)|}{|B|} \Phi_B(\sigma_B) + \frac{|B \cap ({{\mathbb{Z}}}^d - D_\sigma)|}{|B|} \Phi_B(\sigma_B).$$ Notice that $\sigma$ is necessarily constant on any $B=\{x,y\}$ such that $B \cap ({{\mathbb{Z}}}^d - D_\sigma) \neq \emptyset$, which implies that $\Phi_B(\sigma_B) = 0$ for such $B$. Thus, summing over all $B$ such that $B \cap \Lambda \neq \emptyset$ we immediately obtain . Looking at , one might be tempted to proceed as for the Ising contours model on Section \[exampleisingc\]. However, the situation now is more complicated than it was before: it is no longer true that each family of contours with disjoint supports corresponds to some configuration in $\{1,\dots,q\}^{{{\mathbb{Z}}}^d}$. Indeed, besides having disjoint supports, nested contours must have matching internal and external labels for the whole family to correspond to some configuration. In particular, if one defines the $q$-Potts contour model by analogy with what was done on Section \[exampleisingc\], the resulting model will violate the bounded energy loss condition, i.e. $\Delta E^* = -\infty$, so that one cannot associate an FFG dynamics to it. Indeed, there exist contour configurations which are forbidden because they carry nested contours with mismatched labels on them, but that can be turned into admissible configurations by adding a suitable contour in between. The energy leap function associated to an addition of this sort is thus $-\infty$. Nonetheless, the following procedure by Minlos and Sinai [@MS1; @MS2] will help solve this problem. Let us fix $i=1,\dots,q$ and consider the contour model on $\mathcal{N}({{\mathbb{Z}}}^d \times G^i)$, where $G^i$ denotes the space of all $i$-contour shapes, given by the intensity measure $$\label{impottsc} \nu^i(\gamma^i_x):= e^{-\beta \Phi(\gamma^i_x)}$$ and the Hamiltonian $$\label{hpottsc} H^i_{\Lambda|\Gamma'} (\Gamma) = \left\{ \begin{array}{ll} +\infty & \text{ if either $\Gamma$ is incompatible, $\Gamma \not \sim \Gamma'_{\Lambda^c \times G}$ or $\Gamma \not \subset \Lambda$}\\ \\ 0 & \text{ otherwise.}\end{array}\right.$$where we say that two contours $\gamma,\gamma'$ are incompatible whenever $d_1( \text{supp}(\gamma), \text{supp}(\gamma') ) \leq 1$ and the expression $\Gamma \subset \Lambda$ indicates that $d_1(V(\gamma^i), \Lambda^c) > 1 \text{ for every contour } \gamma^i \in \Gamma$. Notice that, as it happened in the Ising contours model, this contour model will also fail to satisfy Assumptions \[assump\]. Thus, when working with this model we will have to take the necessary precautions already described for the Ising contours model, we omit them here. Also, observe that in this model we still have that compatible families of contours will not, in general, correspond to actual configurations in $\{1,\dots,q\}^{{{\mathbb{Z}}}^d}$. Nonetheless, this artificial contour model no longer violates the bounded energy loss condition since its interactions are given only by intersections and Given a family $\Gamma$ of contours with disjoint support we say that $\gamma \in \Gamma$ is an *exterior contour* of $\Gamma$ if $\gamma$ is not contained in the interior of any other contour of $\Gamma$. The set of all exterior contours of $\Gamma$ shall be denoted by $\text{Ext}(\Gamma)$. Furthermore, given a configuration $\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ with a finite defect set $D_\sigma$, we denote the set of all exterior contours of $\Gamma_\sigma$ by $\text{Ext}(\sigma)$. \[propps\] Let $\Gamma$ be a finite family of $i$-contours which are pairwise compatible. Then for any $\Lambda \in B^0_{{{\mathbb{Z}}}^d}$ such that $d_1(\text{V}(\gamma), \Lambda^c) > 1$ for all $\gamma \in \Gamma$ we have that $$\frac{ \mu^{i}_{\Lambda}( \{\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d} : \text{Ext}(\sigma)= \Gamma \} ) }{ \mu^{i}_\Lambda ( \{ \sigma : d_1(\text{V}(\gamma), \Lambda^c) > 1 \text{ for all }\gamma \in \Gamma_\sigma\} )}= \mu_{\Lambda|\emptyset}(\{ \Gamma' \in \mathcal{N}(\Lambda \times G^i) : \text{Ext}(\Gamma') = \Gamma \})$$ where $\mu_{\Lambda|\emptyset}$ is the Boltzmann-Gibbs distribution of the model $(\nu^i,H^i)$. We can assume that all contours in $\Gamma$ are exterior. Thus, we have to check that $$\label{ps2} \frac{\sum_{ \sigma : \text{Ext}(\sigma) = \Gamma } e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)}}{ \sum_{\sigma \in C^i(\Lambda)} e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)}} = \frac{\sum_{ \Gamma' : \text{Ext}(\Gamma') = \Gamma } e^{-\beta \sum_{\gamma^i \in \Gamma'} \Phi(\gamma^i) }}{ \sum_{\Gamma' \in D^i(\Lambda)} e^{-\beta \sum_{\gamma^i \in \Gamma'} \Phi(\gamma^i) }}$$ where the sums in the left hand side are only over configurations $\sigma$ in the support of $\mu^{i}_{\Lambda}$, those in the right hand side are only over families $\Gamma'$ of compatible $i$-contours and, finally, where we have set $$C^i(\Lambda):=\{ \sigma : d_1(\text{V}(\gamma), \Lambda^c) > 1 \text{ for all }\gamma \in \Gamma_\sigma\}$$ and $$D^i(\Lambda):=\{ \Gamma' : d_1(V(\gamma^i), \Lambda^c) > 1 \text{ for all } \gamma^i \in \Gamma' \}.$$ We show that both numerators and both denominators in are respectively identical. We shall proceed by induction. Given a contour $\gamma$ we define its *level* as the maximum $n \in {{\mathbb N}}_0$ such that there exists a sequence of contours $\gamma_0, \dots, \gamma_n$ such that $\gamma = \gamma_0$ and $\text{supp}(\gamma_{i}) \subseteq \text{Int}(\gamma_{i-1})$ for all $i=1,\dots,n$. Furthermore, define the level of a family $\Gamma$ of contours as the maximum level of any contour $\gamma \in \Gamma$ and also define the level of $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ as the maximum level of any contour $\gamma$ such that $V(\gamma) \subseteq \Lambda$. Thus, we shall proceed by induction on the level of both $\Gamma$ and $\Lambda$, respectively. If $\Gamma$ has level zero then we have that $$\text{Ext}(\sigma)=\Gamma \Longleftrightarrow \Gamma_\sigma = \Gamma \hspace{2cm}\text{ and }\hspace{2cm} \text{Ext}(\Gamma')= \Gamma \Longleftrightarrow \Gamma' = \Gamma.$$ which immediately implies that both numerators in are identical in this case. We would like to point out the importance in the previous argument of the fact that $d_1(\text{V}(\gamma), \Lambda^c) > 1$ for all $\gamma \in \Gamma$. Indeed, notice that the sum in the numerator of the left hand side of is over all $\sigma$ in the support of $\mu^{i}_\Lambda$ and that, for arbitrary $\Lambda$, it could very well happen that there are no configurations $\sigma$ in the support of $\mu^{i}_\Lambda$ such that $\Gamma_\sigma=\Gamma$. there exists $\gamma \in \Gamma$ such that , that is, when $\Lambda$ is not simply connected and $\Gamma$ has a contour whose interior labels come into conflict with the boundary configuration $i$. If this were to be the case, then both numerators would not coincide. However, due to the assumption that $d_1(\text{V}(\gamma), \Lambda^c) > 1$ for all $\gamma \in \Gamma$, we can rule out this possibility and thus conclude as we have. if $\Lambda$ has level zero we have that $$\sigma \in C^i(\Lambda) \Longrightarrow \Gamma_\sigma \in D^i(\Lambda) \hspace{0.5cm}\text{ and }\hspace{0.5cm}\Gamma' \in D^i(\Lambda) \Longrightarrow \exists\,\, \sigma \in C^i(\Lambda) \text{ such that }\Gamma'=\Gamma_\sigma$$ since for $\sigma$ in the support of $\mu^{i}_\Lambda$ we have that all contours in $\Gamma_\sigma$ must be exterior $i$-contours. Thus we conclude that both denominators in are equal as well. Next, if $\Gamma$ has level $n+1$ then, by reordering the sum in the left hand side by summing independently over configurations in the interior of each $\gamma \in \Gamma$, we obtain that $$\sum_{ \sigma : \text{Ext}(\sigma) = \Gamma } e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)} = \prod_{\gamma^i \in \Gamma}\left[e^{-\beta \Phi(\gamma^i) } \prod_{j=1}^q \left( \sum_{\sigma \in C^j(\text{Int}_j(\gamma^i))} e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)}\right)\right]$$ where we have used the fact that all contours involved are compatible and, furthermore, that $\text{Int}_j(\gamma)$ has simply connected components for every contour $\gamma \in \Gamma$ and $j=1,\dots,q$. Now, since $\text{Int}_j(\gamma)$ is of level no greater than $n$ for every contour $\gamma \in \Gamma$ and $j=1,\dots,q$, by inductive hypothesis we conclude that $$\label{ps3} \sum_{ \sigma : \text{Ext}(\sigma) = \Gamma } e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)} = \prod_{\gamma^i \in \Gamma}\left[e^{-\beta \Phi(\gamma^i) } \prod_{j=1}^q \left( \sum_{\Gamma^j \in D^j(\text{Int}_j(\gamma^i))} e^{-\beta \sum_{\gamma^j \in \Gamma^j} \Phi(\gamma^j)}\right)\right]$$Furthermore, by the symmetry between spins in the $q$-Potts model, for each $j=1,\dots,q$ we have that $$\label{ps4} \sum_{\Gamma^j \in D^j(\text{Int}_j(\gamma^i))} e^{-\beta \sum_{\gamma^j \in \Gamma^j} \Phi(\gamma^j)} = \sum_{\Gamma^i \in D^i(\text{Int}_j(\gamma^i))} e^{-\beta \sum_{\gamma^i \in \Gamma^i} \Phi(\gamma^i)}$$ so that the sums in the right hand of become only over $i$-contours. Then, by reversing the summation order, we obtain the numerator in the right hand side of . Finally, if $\Lambda$ has level $n+1$ then we decompose the sum in the denominator of the left hand side of over all compatible families of exterior $i$-contours and use the fact that all these families have level no greater than $n+1$, so that for each of them the numerators in coincide by the argument given above. As a consequence of Proposition \[propps\] we have that for every the measure $\mu^{i}_{\Lambda}$ can be obtained by first sampling the external contours and then sampling the spin configuration in the interior of each contour finite-volume Boltzmann-Gibbs distributions. The precise statement is given in Proposition \[corps\] below. Given $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we define its *interior boundary* ${\partial}\Lambda$ as $${\partial}\Lambda := \{ x \in \Lambda : d_1(x,\Lambda^c) = 1\}$$ and its $r$-*interior* $\Lambda^\circ$ as $$\Lambda^\circ := \{x \in \Lambda : d_1(x,{\partial}\Lambda) > r \}.$$ \[corps\] For simply connected $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\sigma \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ with $\mu^i_{\Lambda^\circ}(\sigma) > 0$ we have $$\label{corpsequation} \mu^i_{\Lambda^\circ} (\sigma) = \mu_{\Lambda|\emptyset}( \{\Gamma' : \text{Ext}(\Gamma') = \text{Ext}(\Gamma_\sigma)\}) \prod_{\gamma^i \in \text{Ext}(\Gamma_\sigma)} \prod_{j=1}^q \mu^j_{\text{Int}^\circ_j(\gamma)}(\sigma^j_{\text{Int}^\circ_j(\gamma)})$$ where, for each $j=1,\dots,q$ and $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$, the configuration $\sigma^j_{\Delta} \in \{1,\dots,q\}^{{{\mathbb{Z}}}^d}$ is defined by the formula $$\sigma^j_{\Delta}(x) := \left\{\begin{array}{ll} \sigma(x) & \text{ if $x \in \Delta$ }\\ \\ j & \text{ if $x \notin \Delta$.}\end{array}\right.$$ Let us notice the following equivalences: $$\mu^i_{\Lambda^\circ}(\sigma) > 0 \Longleftrightarrow \sigma(x)=i \text{ for all $x \in {{\mathbb{Z}}}^d - \Lambda^\circ$} \Longleftrightarrow \text{$d(\text{supp}(\gamma^i),\Lambda^c) > 1$ for all $\gamma^i \in \Gamma_\sigma.$}$$ Since $\Lambda$ is simply connected, this implies in fact that $d(\text{V}(\gamma^i),\Lambda^c) > 1$ for all $\gamma^i \in \Gamma_\sigma$. Thus, by the consistency of Boltzmann-Gibbs distributions and Proposition \[propps\] we have $$\mu^i_{\Lambda^\circ} (\sigma) = \frac{\mu^i_{\Lambda}(\sigma)}{\mu^i_{\Lambda}(C^i(\Lambda))}= \mu_{\Lambda|\emptyset}(\{ \Gamma' : \text{Ext}(\Gamma') = \text{Ext}(\sigma) \})\mu^i_{\Lambda}(\sigma | \{ \sigma' : \text{Ext}(\sigma') = \text{Ext}(\sigma)\}).$$ Now, notice that if a configuration $\sigma'$ in the support of $\mu^i_{\Lambda}$ is such that $\text{Ext}(\sigma') = \text{Ext}(\sigma)$ then the spin values of $\sigma'$ outside $\text{Int}^\circ(\text{Ext}(\sigma)):=\bigcup_{\gamma^i \in \text{Ext}(\sigma)} \text{Int}^\circ(\gamma^i)$ are fully determined. More precisely, we have the following identity of events $$\{ \sigma' : \text{Ext}(\sigma') = \text{Ext}(\sigma)\} = \{ \sigma' : \sigma'_{{{\mathbb{Z}}}^d - \text{Int}^\circ(\text{Ext}(\sigma))} = \sigma_{{{\mathbb{Z}}}^d - \text{Int}^\circ(\text{Ext}(\sigma))}\}.$$ This, combined with the fact that spins in different interiors never interact (either because they are too far apart or they have the same value), gives . As a consequence of Proposition \[corps\], we obtain the following \[corposta\] Given a simply connected set $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$, let $Y$ be a random $i$-contour collection distributed according to $\mu_{\Lambda|\emptyset}$ and $X=\{ X^j_{\Delta} : \Delta \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}, j=1,\dots,q\}$ be a family of random spin configurations satisfying: 1. $X$ is independent of $Y$. 2. The random elements $X^j_\Delta$ are all independent and with distribution $\mu^j_\Delta$, respectively. If we take the random spin configuration $Z$ with external contours matching those of $Y$ and with internal spin configuration given by the corresponding random i.e. the random spin configuration $Z$ defined by the formula $$Z(x)=\left\{\begin{array}{ll} i & \text{ if }x \in \bigcap_{\gamma^i \in \text{Ext}(Y)} \text{Ext}(\gamma^i)\\ \\ Y_\Lambda(x) & \text{ if }x \in \bigcup_{\gamma^i \in \text{Ext}(Y)} \text{supp}(\gamma^i) \\ \\ j & \text{ if }x \in \text{Int}_j(\gamma^i)-\text{Int}^\circ_j(\gamma^i) \text{ for }\gamma^i \in \text{Ext}(Y) \text{ and }j=1,\dots,q \\ \\ X^j_{\text{Int}^\circ_j(\gamma^i)}(x) & \text{ if }x \in \text{Int}^\circ_j(\gamma^i) \text{ for }\gamma^i \in \text{Ext}(Y),\end{array}\right.$$ then $Z$ is distributed according to $\mu^i_{\Lambda^\circ}$. We call $Z$ the The key advantage of this simulation scheme is that it can also be carried out in an infinite volume with the help of the FFG dynamics. Indeed, consider the coefficient $$\alpha_{q\text{-Potts}}(\beta) := \sup_{\gamma^i_x \in {{\mathbb{Z}}}^d \times G^i} \left[ \frac{1}{|\gamma^i_x|} \sum_{\tilde{\gamma}^i_y \not \sim \gamma^i_x} |\tilde{\gamma}^i_y| e^{-\beta \Phi(\tilde{\gamma}^i_y)} \right].$$ where, for a given contour $\gamma$, we denote the cardinal of its support by $|\gamma|$. Notice that, by the symmetry of the $q$-Potts model, the coefficient $\alpha$ does not depend on the choice of $i$. Now, if $\alpha_{q\text{-Potts}}< 1$ then $(\nu^i, H^i)$ admits an infinite-volume Gibbs distribution $\mu$ obtained as the stationary measure of the corresponding FFG dynamics $\mathcal{K}^i$. Furthermore, it follows as in the Ising contours model that the free process at time $0$, $\Pi^i_0$, has only finitely many contours surrounding any point in ${{\mathbb{Z}}}^d$ and, thus, external contours in $\mathcal{K}_0^i$ are well defined. Hence, if we consider a family $X$ independent of $\mathcal{K}^i$ as in Corollary \[corposta\], then it is possible to conduct the $i$-alignment of $\mathcal{K}_0^i$ with respect to $X$. By repeating a similar analysis to the one carried out for the Ising contours model, one can verify that the distribution of this $i$-alignment is a Gibbs measure for the $q$-Potts model. Furthermore, by following the ideas discussed in the proof of Theorem \[teomixing\], it is possible to construct for any pair $f,g$ of bounded local functions a triple $$\left\{ \left(\mathcal{A}^0_F(r(\Lambda_f)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_f^c}\right),\left(\mathcal{A}^0_F(r(\Lambda_g)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right), \left(\tilde{\mathcal{A}}^0_F(r(\Lambda_g)), (\tilde{X}^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right) \right\}$$ where for $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we define $r(\Lambda):=\{ \gamma_x^i \in {{\mathbb{Z}}}^d \times G^i : \text{V}(\gamma_x^i) \cap \Lambda \neq \emptyset\}$, such that 1. $\left(\mathcal{A}^0_F(r(\Lambda_f)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_f^c}\right)$ and $\left(\tilde{\mathcal{A}}^0_F(r(\Lambda_g)), (\tilde{X}^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right)$ are independent, 2. $\left(\mathcal{A}^0_F(r(\Lambda_g)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right)$ and $\left(\tilde{\mathcal{A}}^0_F(r(\Lambda_g)), (\tilde{X}^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right)$ have the same distribution, 3. $\mathcal{A}^0(r(\Lambda_f) \sim \mathcal{A}^0(r(\Lambda_g)) \Longrightarrow \left(\mathcal{A}^0_F(r(\Lambda_g)), (X^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right) =\left(\tilde{\mathcal{A}}^0_F(r(\Lambda_g)), (\tilde{X}^j_\Delta)_{\Delta \not \subseteq \Lambda_g^c}\right).$ By combining these elements as in the proof of Theorem \[teounigibbsising\], we get the following result. \[teopotts\]If $\beta > 0$ is sufficiently large so as to satisfy $\alpha_{q\text{-Potts}}(\beta) < 1$ then: 1. The $q$-Potts model on ${{\mathbb{Z}}}^d$ of interaction range $r$ admits $q$ distinct Gibbs measures, which we denote by $\mu^i$ for $i=1,\dots,q$. 2. For each $i=1,\dots,q$ the measure $\mu^{i}$ can be obtained as the local limit $$\mu^{i} := \lim_{n \rightarrow +\infty} \mu^{i}_{\Lambda_n^\circ}$$ for any sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\Lambda_n \nearrow {{\mathbb{Z}}}^d$. 3. For each $i=1,\dots,q$ the measure $\mu^i$ satisfies the $i$-sea with islands picture. 4. If also $\beta > \beta^*$ where $$\beta^* := \inf \left \{ \beta > 0 \,:\, \sum_{\gamma^i_x : d_1(0, \text{supp}(\gamma^i_x)) \leq 1} |\gamma^i_x| e^{-\beta \Phi(\gamma^i_x)} < 1 \right\}$$ then each $\mu^i$ is exponentially mixing in the sense of and . In principle it is not clear why for $\beta > 0$ sufficiently large one should have $\alpha_{q\text{-Potts}}(\beta) < 1$ or even $\alpha_{q\text{-Potts}}(\beta) < +\infty$ for that matter. Indeed, this will depend on the behavior of contour energies $\Phi(\gamma)$ in the limit as $|\gamma| \rightarrow +\infty$. Fortunately, it is not hard to see that all contours $\gamma$ in the $q$-Potts model satisfy the *Peierls bound* $$\Phi(\gamma) \geq \frac{|\gamma|}{2}.$$ This bound guarantees that for $\beta > 0$ sufficiently large one effectively has $\alpha_{q\text{-Potts}}(\beta) < 1$, so that the results of Theorem \[teopotts\] have true meaning. One can also adapt the arguments featured in Chapter \[chapterconvabs\] to this context and obtain the following continuity result. \[teopotts2\] For any $\beta_0 > 0$ such that $\alpha_{q\text{-Potts}}(\beta_0) < 1$ and any $i=1,\dots,q$ we have the local convergence $$\lim_{\beta \rightarrow \beta_0} \mu^{i,\beta} = \mu^{i,\beta_0}.$$ Let us consider a sequence $(X^\beta)_{\beta > 0}$ of families as in Corollary \[corposta\] (where we write the dependence in the inverse temperature explicitly) such that for each $j=1,\dots,q$ and $\Delta \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we have the almost sure convergence $$\lim_{\beta \rightarrow \beta_0} X^{j,\beta}_\Delta = X^{j,\beta_0}_{\Delta}.$$ Such a sequence can be constructed using the standard coupling from the past methods (see e.g. [@HA]). Notice then that, by Corollary \[corposta\] and the discussion following it, obtain the result it will suffice to couple the FFG dynamics $\mathcal{K}^{i,\beta}$ independently of $(X^\beta)_{\beta > 0}$ so that as $\beta \rightarrow \beta_0$ we have $$\mathcal{K}_0^{i,\beta} \overset{loc}{\longrightarrow} \mathcal{K}_0^{i,\beta_0}.$$ This, however, can be done as in Chapter \[chapterconvabs\]. Once again, we would like to point out that these results can also be obtained through standard Pirogov-Sinai theory. Nevertheless, the range of $\beta > 0$ for which these results hold under the standard theory is strictly smaller than the one obtained standard methods give these results for $\beta > \beta'$ where $$\beta' = \inf\left\{ \beta > 0 : \sum_{\gamma^i_x : d(0, \text{supp}(\gamma^i_x)) \leq 1} e^{|\gamma^i_x|} e^{- \beta \Phi(\gamma^i_x)} < 1\right\}.$$ Due to the fact that traditional Pirogov-Sinai theory relies on the convergence of certain once again we obtain in the threshold value an exponential dependence in the size of contours which is only linear for our approach. Furthermore, another strength of our approach is that it provides us with a perfect simulation scheme for We believe that such a scheme has not been developed before. It is essentially contained in Corollary \[corposta\] and the discussion following it (see also the discussion on Section \[perfectsimulation\]). Discrete Widom-Rowlinson model ------------------------------ In this section we show how the ideas discussed for the $q$-Potts model on ${{\mathbb{Z}}}^d$ can be adapted to establish a phase transition in the discrete Widom-Rowlinson model on ${{\mathbb{Z}}}^d$ with fugacity $\lambda > 0$ and interaction range $r \in {{\mathbb N}}$. Recall that this model is traditionally defined on the configuration space $\{+,0,-\}^{{{\mathbb{Z}}}^d}$ through the Boltzmann-Gibbs distributions given for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $\eta \in \{+,0,-\}^{{{\mathbb{Z}}}^d}$ by the formula $$\label{wrdbgdps} \mu_{\Lambda}^\eta(\sigma) = \frac{\mathbbm{1}_{\{\sigma_{\Lambda^c} \equiv \eta_{\Lambda^c}\}}}{Z_{\Lambda}^\eta} e^{-\sum_{B : B \cap \Lambda \neq \emptyset} \Phi_B(\sigma)}$$ where for each $B \subseteq {{\mathbb{Z}}}^d$ the interaction $\Phi_B$ is given by $$\label{wrdht2} \Phi_B(\sigma) = \left\{ \begin{array}{ll} (+\infty) \mathbbm{1}_{\{ \sigma(x) \times \sigma(y)= - \}} & \text{ if $B=\{x,y\}$ with $\|x-y\|_\infty \leq r$} \\ \\ - \mathbbm{1}_{\{\sigma(x)= \pm\}} \log \lambda & \text{ if $B=\{x\}$} \\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ We wish to show that for sufficiently large values of $\lambda > 0$ the model admits at least two different Gibbs measures. The idea is to follow the procedure for the $q$-Potts model. Once again we fix a set of reference configurations $\mathcal{R}:=\{ \eta^+,\eta^0,\eta^- \}$, where for each $i \in \{+,0,-\}$ we let $\eta^i$ denote the configuration on $\{+,0,-\}^{{{\mathbb{Z}}}^d}$ with constant Also, for each $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we let $\mu^i_\Lambda$ denote the Boltzmann-Gibbs distribution on $\Lambda$ with boundary configuration $\eta^i$. The notion of contour for this model is defined in exactly the same way as for the $q$-Potts model, except for the fact that in the definition of correct points here we use the supremum distance $d_\infty$ instead of $d_1$. However, since in this model no longer true that $\Phi_B(\eta^i)=0$ for every $\eta^i \in \mathcal{R}$, the expression for the energy of becomes slightly different: for an $i$-contour $\gamma^i$ belonging to some we define its energy $\Phi(\gamma^i)$ by the formula $$\Phi(\gamma^i):= \sum_{B \subseteq {{\mathbb{Z}}}^d} \frac{|B \cap \text{supp}(\gamma^i)|}{|B|} \Phi_B(\sigma_B) - e_i|\gamma^i|$$ where $e_i$ is the *mean energy* of the configuration $\eta^i$ defined as $$e_i := \sum_{B \subseteq {{\mathbb{Z}}}^d : 0 \in B} \frac{1}{|B|} \Phi_B(\eta^i_B)= \left\{ \begin{array}{ll} - \log \lambda & \text{ if $i=\pm$} \\ \\ 0 & \text{ if $i=0$.}\end{array}\right.$$ By proceeding as in the proof of Proposition \[proprepresentacionps\] one can show the following. For any $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ let us consider a configuration $\sigma$ such that $D_\sigma \subseteq \Lambda$ and $\sigma(x)=i$ for every $x \in {{\mathbb{Z}}}^d$ lying in the unique infinite connected component of ${{\mathbb{Z}}}^d - D_\sigma$. If $\Gamma_{\sigma}$ denotes the family of contours associated to $\sigma$, then we have $$\label{wrps1} \mu^{i}_\Lambda (\sigma) = \frac{1}{Z^{i}_\Lambda} e^{- \left(\sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma) + \sum_{u \in \{+,0,-\}} e_u |\Lambda_u|\right)}$$ where for each $u \in \{+,0,-\}$ we let $\Lambda_u$ denote the set of all points in $\Lambda$ which are either $u$-correct or belong to an $u$-contour of $\sigma$. The next step is to fix a spin $i \in \{+,0,-\}$ and define a contour model on $\mathcal{N}({{\mathbb{Z}}}^d \times G^i)$, where $G^i$ is the space of all $i$-contour shapes, satisfying the bounded energy loss condition and preserving the distribution of exterior contours. With this purpose in mind, we first define for $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $j \in \{+,0,-\}$ the *diluted partition function* $$Z^j(\Delta) := \sum_{\sigma \in C^j(\Delta)} e^{-\sum_{B : B \cap \Lambda} \Phi_B(\sigma)}$$ where $C^j(\Delta):=\{ \sigma \in \text{supp}(\mu^j_\Delta) : d_1(\text{V}(\gamma), \Lambda^c) > 1 \text{ for all }\gamma \in \Gamma_\sigma\}$, and then consider the contour model with intensity measure $\nu^i$ given by the formula $$\label{wrimpottsc} \nu^i(\gamma^i_x):= \left[\prod_{j \neq i} \frac{Z^j(\text{Int}_j(\gamma^i_x))}{Z^i(\text{Int}_j(\gamma^i_x))}\right]e^{-\Phi(\gamma^i_x)}$$ and Hamiltonian $H^i$ defined as $$\label{hpottsc2} H^i_{\Lambda|\Gamma'} (\Gamma) = \left\{ \begin{array}{ll} +\infty & \text{ if either $\Gamma$ is incompatible, $\Gamma \not \sim \Gamma'_{\Lambda^c \times G}$ or $\Gamma \not \subset \Lambda$}\\ \\ 0 & \text{ otherwise.}\end{array}\right.$$ Notice that the intensity measure carries an additional product of partition functions which was missing in the $q$-Potts model. This is again attributed to the fact that not all reference configurations in this model have zero mean energy. By proceeding as in the previous section we obtain for this context the analogues of Propositions \[propps\], \[corps\] and Corollary \[corposta\]. Thus, if for $i \in \{+,0,-\}$ we define the coefficient $$\label{dcwrps007} \alpha_i (\lambda) = \sup_{\gamma^i_x \in {{\mathbb{Z}}}^d \times G^i} \left[ \frac{1}{|\gamma^i_x|} \sum_{\tilde{\gamma}^i_y \not \sim \gamma^i_x} |\tilde{\gamma}^i_y| \left[\prod_{j \neq i} \frac{Z^j(\text{Int}_j(\gamma^i_x))}{Z^i(\text{Int}_j(\gamma^i_x))}\right] e^{-\Phi_\lambda(\tilde{\gamma}^i_y)} \right]$$ then the condition $\alpha_i(\lambda) < 1$ implies the existence of a Gibbs measure $\mu^i$ associated which fulfills the description in Theorems \[teopotts\] and \[teopotts2\]. In particular, since $\alpha_+ = \alpha_-$ , the condition $\alpha_+(\lambda) < 1$ already implies a phase transition for the model. However, it is not clear why this condition should be fulfilled by any value of $\lambda$. Indeed, this will not only depend on the behavior of the energies $\Phi(\gamma^+_x)$ as $|\gamma^+_x| \rightarrow +\infty$ on but also on the growth of the additional factor $$\label{pwrps007} \prod_{j \neq +} \frac{Z^j(\text{Int}_j(\gamma^+_x))}{Z^+(\text{Int}_j(\gamma^+_x))} = \frac{Z^0(\text{Int}_0(\gamma^+_x))}{Z^+(\text{Int}_0(\gamma^+_x))}.$$ Fortunately, it can be seen that for $\lambda > 0$ sufficiently large the condition $\alpha_+(\lambda) < 1$ is satisfied. Indeed, first let us notice that a straightforward combinatorial argument yields, for any $+$-contour $\gamma^+_x$, the Peierls bound $$\label{wrpeierls} \Phi(\gamma^+_x) = (+\infty)\mathbbm{1}_{\{\Phi(\gamma^+_x) = +\infty\}} + \#\{ y \in \text{supp}(\gamma^+_x) : \sigma(y) = 0\} \log \lambda \mathbbm{1}_{\{\Phi(\gamma^+_x)< +\infty\}} \geq \frac{|\gamma_x^+|}{(2r)^d}\log \lambda.$$ On the other hand, the following lemma shows that the additional factor in remains bounded for large values of $\lambda$. \[lemacontrol\] There exist constants $c_1,c_2 > 0$, which depend only on $d$ and $r$, for any $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ $$\label{lemacontroleq} \frac{Z^0(\Delta)}{Z^+(\Delta)} \leq \left( \frac{2^{c_1}}{\lambda^{c_2}}\right)^{\# {\partial}_e \Delta},$$ where ${\partial}_e \Delta:= \{ y \in \Delta^c : d_1(y,\Delta) =1 \}$. For $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ we define its boundary and $r$-interior by the respective formulas $${\partial}\Delta = \{ x \in \Delta : d_1(x, \Delta^c) = 1\} \hspace{1cm}\text{ and }\hspace{1cm}\Delta^\circ = \{ x \in \Delta : d_\infty(x, {\partial}\Delta) > r \}.$$ Now, notice that we have the identities $$Z^0(\Delta) = Z^0_{\Delta^\circ} \hspace{2cm}\text{ and }\hspace{2cm}Z^+(\Delta)= \lambda^{|\Delta| - |\Delta^\circ|} Z^+_{\Delta^\circ}$$ where $Z^0_{\Delta^\circ}$ and $Z^+_{\Delta^\circ}$ are the normalizing constants given in . Furthermore, we have $$Z^0_{\Delta^\circ} = Z^+_{\Delta^\circ} + \sum_{\sigma \in A^0_+(\Delta^\circ)} e^{- \sum_{B: B \cap \Delta^\circ \neq \emptyset} \Phi_B(\sigma_B)}$$ where $A^0_+(\Delta^\circ)$ is the set of all configurations $\sigma$ in the support of $\mu^0_{\Delta^\circ}$ such that $\sigma_{\Delta^\circ} \cdot \eta^+_{{{\mathbb{Z}}}^d - \Delta^\circ}$ does not belong to the support of $\mu^+_{\Delta^\circ}$. Observe that a configuration $\sigma$ in the support of $\mu^0_{\Delta^\circ}$ belongs to $A^0_+(\Delta^\circ)$ if and only if $\sigma(x)=-$ for some $x \in \Delta^\circ$ with $d_\infty(x,{\partial}_e (\Delta^\circ)) \leq r$. Now, for a fixed configuration $\sigma$ in the support of $\mu^0_{\Delta^\circ}$, we say that two given sites $x,y \in \text{supp}(\sigma):=\{ z \in \Delta^\circ : \sigma(z) \neq 0 \}$ are $\sigma$-*connected* if $\|x-y\|_\infty \leq r$. Notice that if are $\sigma$-connected then $\sigma$ must assign both sites $x$ and $y$ the same spin value. components of $\text{supp}(\sigma)$ with respect to this notion of connection will be called $\sigma$-*components*. Next, assign to $\sigma$ the configuration $\sigma^+$ defined by the formula $$\sigma^+(x):=\left\{ \begin{array}{ll} + & \text{ if either $x \in {{\mathbb{Z}}}^d - \Delta^\circ$ or $x$ belongs a $\sigma$-component $C \in C_\sigma({\partial}_e (\Delta^\circ))$ } \\ \\ \sigma(x) & \text{ otherwise,}\end{array}\right.$$ where $C_\sigma({\partial}_e (\Delta^\circ))$ is the set $\sigma$-components $C$ of $\text{supp}(\sigma)$ satisfying $d_\infty(C,{\partial}_e(\Delta^\circ))\leq r$. $\sigma^+$ is obtained from $\sigma$ by flipping to $+$ the spin of all $\sigma$-components which interact with the boundary ${\partial}_e(\Delta^\circ)$ and also the spin of sites outside $\Delta^\circ$. Notice that $\sigma^+$ belongs to the support of $\mu^+_{\Delta^\circ}$ and that both $\sigma$ and $\sigma^+$ have the same energy in $\Delta^\circ$, i.e. $$\sum_{B: B \cap \Delta^\circ \neq \emptyset} \Phi_B(\sigma_B) = \sum_{B: B \cap \Delta^\circ \neq \emptyset} \Phi_B(\sigma^+_B).$$ Hence, since there are at most $2^{\# C_{\sigma^+}({\partial}_e (\Delta^\circ))}$ different configurations $\sigma'$ in which can be assigned the same configuration $\sigma^+$ and there exists $\tilde{c}_1 > 0$ such that for any configuration $\sigma$ there can be at most $\tilde{c}_1 (\# {\partial}_e(\Delta^\circ))$ $\sigma$-components in $C_\sigma({\partial}_e (\Delta^\circ))$, $$\sum_{\sigma \in A^0_+(\Delta^\circ)} e^{- \sum_{B: B \cap \Delta^\circ \neq \emptyset} \Phi_B(\sigma_B)} \leq 2^{\tilde{c}_1 (\# {\partial}_e (\Delta^\circ))} Z^+_{\Delta^\circ}$$ which implies that $$\frac{Z^0(\Delta)}{Z^+(\Delta)} \leq \frac{1+2^{\tilde{c}_1 (\# {\partial}_e (\Delta^\circ))}}{ \lambda^{|\Delta| - |\Delta^\circ|}}.$$ From here a straightforward calculation allows us to conclude the result. It follows from the Peierls bound on the energy of contours and the previous lemma that for $\lambda > 0$ sufficiently large one has $\alpha_+(\lambda) < 1$. Thus, we obtain the following result. \[teowrpt\]If $\lambda > 0$ is sufficiently large so as to satisfy $\alpha_{+}(\lambda) < 1$ then: 1. The discrete Widom-Rowlinson model on ${{\mathbb{Z}}}^d$ with exclusion radius $r$ admits two distinct Gibbs measures, $\mu^+$ and $\mu^-$. 2. The measures $\mu^{+}$ and $\mu^-$ can be obtained as the local limits $$\mu^{+}= \lim_{n \rightarrow +\infty} \mu^{+}_{\Lambda_n^\circ}\hspace{1cm}\text{ and }\hspace{1cm}\mu^{-}= \lim_{n \rightarrow +\infty} \mu^{-}_{\Lambda_n^\circ}$$ for any sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\Lambda_n \nearrow {{\mathbb{Z}}}^d$. 3. The measures $\mu^+$ and $\mu^-$ satisfy the sea with islands picture for the $+$ and $-$ spins, respectively. 4. The measures $\mu^+$ and $\mu^-$ are continuous in $\lambda$, i.e. if $\lambda_0$ is such that $\alpha_{+}(\lambda_0) < 1$ then we have the local convergence $$\lim_{\lambda \rightarrow \lambda_0} \mu^{\pm,\lambda} = \mu^{\pm,\lambda_0}.$$ 5. If also $\lambda > \lambda^*$, where $$\lambda^* := \inf \left \{ \lambda > 0 \,:\, \sum_{\gamma^+_x : d_1(0, \text{supp}(\gamma^+_x)) \leq 1} |\gamma^+_x| \left[\frac{Z^0(\text{Int}_0(\gamma^+_x))}{Z^+(\text{Int}_0(\gamma^+_x))}\right]e^{-\Phi_\lambda(\gamma^+_x)} < 1 \right\},$$ then both $\mu^+$ and $\mu^-$ are exponentially mixing in the sense of and . Even though we have established the occurrence of phase transition for large values of the fugacity $\lambda$, one could still wonder what can be said about the remaining $0$-spin. Similar combinatorial arguments to the ones given above yield, for any $0$-contour $\gamma^0_x$, the inverse Peierls bound $$\Phi(\gamma_x^0) \leq - \frac{|\gamma_x^0|}{(2r)^d} \log \lambda$$ and the lower bound $$\frac{Z^\pm(\Delta)}{Z^0(\Delta)} \geq \left( \frac{\lambda^{c_2}}{2^{c_1}}\right)^{\# {\partial}_e \Delta}$$ for any $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$, which implies that for every $\lambda > 0$ sufficiently large one actually has $\alpha_0(\lambda) = +\infty$ and thus our entire argument breaks down for the $0$-spin. This suggests that the reference configuration $\eta^0$ is *unstable* for large fugacities, in the sense that there is no Gibbs measure of the model satisfying the $0$-sea with islands picture. The other reference configurations $\eta^+$ and $\eta^-$ are thus regarded as *stable*. As a final remark, we would like to point out that the occurrence of a phase transition in the discrete Widom-Rowlinson model can be established by other methods beside cluster expansion, such as random cluster representations (see e.g. [@GHM]). However, the use of such representations usually leads to a less complete picture than that of Theorem \[teowrpt\] for very large fugacities: these cannot be used to show neither the $\pm$-sea with islands picture nor the continuity in $\lambda$, and also these representations alone are not enough to establish exponential mixing properties. Furthermore, random cluster representations in general are not robust, in the sense that they may work very well for certain models but then fail to work whenever these models are subject to slight perturbations. As an example we have the $k$-tolerant Widom-Rowlinson model, for which one expects a similar behavior to the original Widom-Rowlinson model for small values of $k$, but whose random cluster representations become extremely more difficult to handle: each cluster of particles may not have a unique $+$ or $-$ spin value inside them, and the number of spin changes allowed inside each cluster will vary depending on its own geometry. However, these considerations do not interfere with the general argument given in this section, so that the Peierls bound in and the upper bound in Lemma \[lemacontrol\] still remain valid for the $k$-tolerant Widom-Rowlinson model provided that $k$ is small enough to guarantee that for any incorrect point of an admissible configuration there exists at least one empty site at a distance no greater than $r$ from it. We can therefore conclude the following result without any additional difficulties. \[tolerantpt\] For $\lambda > 0$ sufficiently large and $k,r \in {{\mathbb N}}$ such that $k < (r+1)^d - 1$, Widom-Rowlinson model with fugacity $\lambda$ and exclusion radius $r$ admits two distinct Gibbs measures, $\mu^+$ and $\mu^-$, each satisfying the description of Theorem \[teowrpt\]. Continuum Widom-Rowlinson model ------------------------------- The ideas discussed in the previous section for the discrete Widom-Rowlinson model can be adapted to the continuum setting to obtain analogous results. In this section we comment briefly on how to perform such adaptation. Given $\lambda, r > 0$, recall that the Widom-Rowlinson model on ${{\mathbb R}}^d$ with fugacity $\lambda$ and exclusion radius $r > 0$ is defined as the diluted model on $\mathcal{N}({{\mathbb R}}^d \times \{+,-\})$ given by $(\nu,H)$, where $$\nu = \lambda \mathcal{L}^d \times ( \delta_+ + \delta_- )$$ and $$H_{\Lambda|\eta}(\sigma)= \sum_{(\gamma_x ,\tilde{\gamma}_y) \in e_{\Lambda}(\sigma|\eta)} U( \gamma_x , \tilde{\gamma}_y )$$ with $$\label{wrups} U(\gamma_x,\tilde{\gamma}_y) := \left\{ \begin{array}{ll} +\infty &\text{if }\gamma \neq \tilde{\gamma}\text{ and }\|x-y\|_\infty \leq r\\ 0 &\text{otherwise.}\end{array}\right.$$Now, consider the tiling $\mathcal{T}$ of ${{\mathbb R}}^d$ given by the cubic cells $(\mathcal{Q}_x)_{x \in {{\mathbb{Z}}}^d}$, where $$\mathcal{Q}_x := l \cdot (x + [0,1)^d ),$$ where $l > 0$ is sufficiently small so that particles in any two adjacent cells are by the Hamiltonian $H$ to be of opposite type. Here we understand adjacent cells of the graph ${{\mathbb{Z}}}^d$ with the supremum distance, i.e. we say that two cells $\mathcal{Q}_x, \mathcal{Q}_y$ are adjacent if $d_\infty(\mathcal{Q}_x,\mathcal{Q}_y)=0$ where $d_\infty$ is the supremum distance on ${{\mathbb R}}^d$. Given a configuration $\sigma \in \mathcal{N}({{\mathbb R}}^d \times G)$ we assign it the configuration $s_\sigma \in \{+,0,-,*\}^{{{\mathbb{Z}}}^d}$ defined by the formula $$s_\sigma (x) = \left\{\begin{array}{ll}+ & \text{ if there are only particles of type $(+)$ in $\mathcal{Q}_x$}\\ \\ - & \text{ if there are only particles of type $(-)$ in $\mathcal{Q}_x$}\\ \\ 0 & \text{ if there are no particles in $\mathcal{Q}_x$}\\ \\ * & \text{ if there are particles of both types in $\mathcal{Q}_x$.}\end{array}\right.$$ We call $s_\sigma$ the *skeleton* of $\sigma$. Notice that for configurations $\sigma$ which are not forbidden we have in fact that $s_\sigma(x) \neq *$ for all $x \in {{\mathbb{Z}}}^d$. Next, fix a set of reference configurations $$\mathcal{R}=\{\eta^+, \eta^0,\eta^-\} \subseteq \mathcal{N}({{\mathbb R}}^d \times \{+,-\})$$ such that for each $i \in \{+,0,-\}$ the configuration $\eta^i$ satisfies $s_{\eta^i}(x)=i$ for all $x \in {{\mathbb{Z}}}^d$. Then, for $i \in \{+,0,-\}$ say that a site $x \in {{\mathbb{Z}}}^d$ is $i$-correct with respect to the configuration $s_\sigma \in \{+,0,-,i\}^{{{\mathbb{Z}}}^d}$ if $s_\sigma(y)=i$ for all $y \in {{\mathbb{Z}}}^d$ with $\|x-y\|_\infty \leq \frac{r+1}{l}$ and label a site as incorrect if it is not $i$-correct for any $i \in \{+,0,-\}$. Finally, we proceed to define the contours of the configuration $s_\sigma$ by analogy with the previous sections. Observe that contours are not defined for particle configurations on $\mathcal{N}({{\mathbb R}}^d \times \{+,-\})$ but rather for their skeletons. Now, given an $i$-contour $\gamma^i$ belonging to some skeleton configuration, energy $\Phi(\gamma^i)$ $$\Phi(\gamma^i) := \left(- \log \int_{\mathcal{N}(\text{supp}(\gamma^i) \times \{+,-\})} \mathbbm{1}_{\{s_\sigma = \gamma^i\}} e^{-H_{\text{supp}(\gamma^i)|\emptyset}(\sigma)} d\pi^\nu_{\text{supp}(\gamma^i)}(\sigma)\right) - e_i |\text{supp}(\gamma^i)|$$ where $$e_i = - \log \int_{\mathcal{N}(\mathcal{Q}_0 \times \{+,-\})} \mathbbm{1}_{\{s_\sigma = i \}} d\pi^\nu_{\text{supp}(\gamma^i)}(\sigma) = \left\{\begin{array}{ll} - \log (1-e^{-\lambda l^d}) + \lambda l^d & \text{ if $i= \pm$} \\ \\ 2\lambda l^d & \text{ if $i=0$.}\end{array}\right.$$ We can write Boltzmann-Gibbs distributions in terms of the energy of contours as follows. For any $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ let us consider a skeleton configuration $s'$ such that $D_{s'} \subseteq \Lambda$ and $s'(x)=i$ for every $x \in {{\mathbb{Z}}}^d$ lying in the infinite component of ${{\mathbb{Z}}}^d - D_{s'}$. If $\Gamma_{s'}$ denotes the family of contours associated to $s'$, then we have $$\label{wrcps1} \mu^{\eta^i}_{\Lambda_l} (\{ \sigma \in \mathcal{N}({{\mathbb R}}^d \times \{+,-\}) : s_\sigma = s' \}) = \frac{1}{Z_{\Lambda|\eta^i}} e^{- \left(\sum_{\gamma \in \Gamma_{s'}} \Phi(\gamma) + \sum_{u \in \{+,0,-\}} e_u |\Lambda(u)|\right)}$$ where $$\Lambda_l :=\bigcup_{x \in \Lambda} \mathcal{Q}_x$$ and for each $u \in \{+,0,-\}$ we let $\Lambda(u)$ denote the set of all points in $\Lambda$ which are either $u$-correct or belong to an $u$-contour of $s^*$. This follows from the consistent Hamiltonian property on Assumptions \[assump\] by noticing that particles lying inside correct cells do not interact with particles and also that particles inside incorrect cells do not interact with particles in correct cells. We omit the details. We then fix a spin $i \in \{+,0,-\}$ and consider the contour model on $\mathcal{N}({{\mathbb{Z}}}^d \times G^i)$, where $G^i$ is the space of $i$-contour shapes, with Hamiltonian as in and intensity measure $$\nu^i(\gamma^i_x):= \left[\prod_{j \neq i} \frac{Z^j(\text{Int}_j(\gamma^i_x))}{Z^i(\text{Int}_j(\gamma^i_x))}\right]e^{-\Phi(\gamma^i_x)}$$ where for each $\Delta \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $j \in \{+,0,-\}$ we set $$Z^j(\Delta) := \sum_{s_\sigma \in C^j(\Delta)} e^{- \left(\sum_{\gamma \in \Gamma_{s_\sigma}} \Phi(\gamma) + \sum_{u \in \{+,0,-\}} e_u |\Delta (u)|\right)}$$ for $C^j(\Delta):=\{ s_\sigma : s_\sigma(x) = j \text{ for all $x \in {{\mathbb{Z}}}^d - \Delta$ and } d_1(\text{V}(\gamma), \Delta^c) > 1 \text{ for all }\gamma \in \Gamma_{s_\sigma}\}$. It is easy to verify that this contour model satisfies the bounded energy loss condition and that it also preserves the distribution of exterior contours in skeleton configurations. Thus, in order to repeat the analysis of the previous section, it only remains to show an analogue of Corollary \[corposta\], i.e. we need to show how to conduct an $(i)$-alignment of contour configurations such that, whenever these are distributed according to $(\nu^i,H^i)$, the resulting particle configurations carry the correct Boltzmann-Gibbs distributions. set $\Lambda \in {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ and $i \in \{+,-\}$, we proceed as follows: 1. Consider a contour configuration $Y$ with distribution $\mu_{\Lambda|\emptyset}$ specified by $(\nu^i,H^i)$. 2. Place an $i$-particle independently on each cell belonging to the set $\Lambda^\circ \cap \bigcap_{\gamma^i \in \text{Ext}(Y)}$, so that each of these cells carries at least one $(i)$-particle. Here we let $\Lambda^\circ$ denote the $\frac{r+1}{l}$-interior of $\Lambda$, as in the previous section. Then, on top of these particles place a Poisson process of $(i)$-particles on $(\Lambda^\circ \cap \bigcap_{\gamma^i \in \text{Ext}(Y)})_l$ with fugacity $\lambda$, which will account for the remaining $(i)$-particles in each of these cells. 3. Proceed as in (ii) for cells in $\text{Int}_j(\gamma^i)-\text{Int}^\circ_j(\gamma^i) \text{ for }\gamma^i \in \text{Ext}(Y) \text{ and } j \in \{+,0,-\}$. 4. For every contour $\gamma^i \in \text{Ext}(Y)$ place a particle independently on each cell belonging to $\text{supp}(\gamma^i)$ of the type $\gamma^i$ indicates by the acceptance-rejection method, so that the resulting configuration of particles inside the support of $\gamma^i$ is not forbidden by $H$. $\gamma^i$ assigns a $0$-spin to some cell then place no particle in that cell. Then, place the remaining particles of the cells belonging to $\text{supp}(\gamma^i)$ according to the FFG dynamics on the finite volume $\left(\text{supp}(\gamma^i)\right)_l$ with empty boundary condition, but keeping the particles of the acceptance-rejection method alive for all times $t \in {{\mathbb R}}$. 5. For each contour $\gamma^i \in \text{Ext}(Y)$ place the particles inside the cells belonging to $\text{Int}(\gamma^i)$ with the FFG dynamics on the finite volume $\left(\text{Int}(\gamma^i)\right)_l$ with boundary condition given by the portion of particle configuration constructed in (iii). 6. Finally, place particles in the remaining cells according to $\eta^i$. It can be seen that this configuration carries the distribution $\mu_{(\Lambda^\circ)_l|\eta^i}$ specified by $(\nu,H)$. We may then proceed as in the previous section to show a phase transition for large $\lambda$. The Peierls bound for the energy of contours and Lemma \[lemacontrol\] still hold in this setting; the latter being a consequence of the argument given in Lemma \[lemacontrol\] combined with and the Fubini theorem for the Thus, we obtain the following result. \[wrptps\] Given $\lambda, r > 0$ let us suppose that $\lambda > 0$ is sufficiently large so that $$\alpha_+ (\lambda,r) = \sup_{\gamma^+_x \in {{\mathbb{Z}}}^d \times G^+} \left[ \frac{1}{|\gamma^+_x|} \sum_{\tilde{\gamma}^+_y \not \sim \gamma^+_x} |\tilde{\gamma}^+_y| \left[\frac{Z^0(\text{Int}_0(\gamma^+_x))}{Z^+(\text{Int}_0(\gamma^+_x))}\right] e^{-\Phi_{\lambda,r}(\tilde{\gamma}^+_y)} \right] < 1.$$Then: 1. The continuum Widom-Rowlinson model on ${{\mathbb R}}^d$ with exclusion radius $r$ admits two distinct Gibbs measures, $\mu^+$ and $\mu^-$. 2. The measures $\mu^{+}$ and $\mu^-$ can be obtained as the local limits $$\mu^{+}= \lim_{n \rightarrow +\infty} \mu_{(\Lambda_n^\circ)_l| \eta^+}\hspace{1cm}\text{ and }\hspace{1cm}\mu^{-}= \lim_{n \rightarrow +\infty} \mu_{(\Lambda_n^\circ)_l|\eta^-}$$ for any sequence $(\Lambda_n)_{n \in {{\mathbb N}}} \subseteq {{\mathcal B}}^0_{{{\mathbb{Z}}}^d}$ of simply connected sets with $\Lambda_n \nearrow {{\mathbb{Z}}}^d$. 3. The measures $\mu^+$ and $\mu^-$ satisfy the sea with islands picture for the $+$ and $-$ spins, respectively. 4. The measures $\mu^+$ and $\mu^-$ are continuous in $\lambda$ and $r$, i.e. if $\lambda_0,r_0 > 0$ are such that $\alpha_{+}(\lambda_0,r_0) < 1$ then we have the local convergence $$\lim_{(\lambda,r) \rightarrow (\lambda_0,r_0)} \mu^{\pm,\lambda,r} = \mu^{\pm,\lambda_0,r_0}.$$ 5. If also $\lambda > \lambda^*$, where $$\lambda^* := \inf \left \{ \lambda > 0 \,:\, \sum_{\gamma^+_x : d_1(0, \text{supp}(\gamma^i_x)) \leq 1} |\gamma^+_x| \left[\frac{Z^0(\text{Int}_0(\gamma^+_x))}{Z^+(\text{Int}_0(\gamma^+_x))}\right]e^{-\Phi_{\lambda,r}(\gamma^+_x)} < 1 \right\},$$ then both $\mu^+$ and $\mu^-$ are exponentially mixing in the sense of and . Notice that Theorem \[wrptps\] gives also continuity with respect to the exclusion radius. This is due to the fact that given $r_0 > 0$ one can choose the size $l$ of the tiling so that for any $r > 0$ sufficiently close to $r_0$ the definition of correct point in the skeleton configuration remains unaltered. From this observation, a straightforward argument analogous to the one in Chapter \[chapterconvabs\] gives the result. Moreover, since the Peierls bound and the estimation in Lemma \[lemacontrol\] still hold in this continuum setting, one can also obtain an analogue of Theorem \[tolerantpt\] showing a phase transition in the $k$-tolerant Widom-Rowlinson model for sufficiently small values of $k$. Unfortunately, for the Widom-Rowlinson model with generalized interactions one can see that a more careful argument is needed than the one given here, we shall not pursue it here. Finally, we observe that the exact same analysis given here carries over to show a phase transition in the thin rods model for evenly spaced angles. The condition of evenly spaced is important, since it gives the symmetry between orientations which was used for the Widom-Rowlinson model. The result is the following. Given $n \geq 2$ consider the thin rods model of fugacity $\lambda > 0$, rod length $2r > 0$ and orientation measure $\rho$ given by $$\rho = \frac{1}{n} \sum_{i=0}^{n-1} \delta_{\frac{i}{n} \pi}.$$ Then, for $\lambda > 0$ sufficiently large (depending on $n$ and $r$) the model admits $n$ distinct Gibbs measures, $(\mu^i)_{i=1,\dots,n}$, each of which satisfies the description on Theorem \[wrptps\]. As a final remark we point out that this same analysis can be carried out in the same manner for more general models. However, for this analysis to be fruitful, one needs to guarantee a Peierls bound on the energy contours and a suitable control on the ratios of diluted partition functions like the one given in Lemma \[lemacontrol\]. While in general the Peierls bound will not bring much problems, an estimate as in is a priori not easy to obtain. Cluster expansion methods offer further tools to control these terms although at the cost of narrowing the range of validity of the results. Nevertheless, even if we cannot control the ratios of diluted partition functions directly and we must rely on cluster expansion methods to obtain any results, the approach discussed here still guarantees that in that smaller range one can still perform perfect simulation of the corresponding equilibrium measures. Resumen del Capítulo 12 ----------------------- En este último capítulo combinamos las ideas en los capítulos anteriores con el marco de la teoría de Pirogov-Sinai para obtener algunos resultados bajo el régimen de no unicidad. Concretamente, la teoría de Pirogov-Sinai nos provee de un procedimiento sistemático para trasladar el estudio de las propiedades macroscópicas (existencia y propiedades de medidas de Gibbs) en sistemas bajo el régimen de no unicidad, i.e. alta densidad de partículas o baja temperatura, al estudio de un modelo de contornos adecuado a baja densidad que puede ser tratado con las técnicas desarrolladas en los capítulos previos. De hecho, aprovechando estas técnicas nos es posible extender el rango de validez de algunos resultados con respecto al ofrecido por la teoría de Pirogov-Sinai clásica. Por simplicidad, discutimos estas ideas primero para el modelo de Potts con $q$ spines ($q \geq 2$), en donde la simetría entre spines dada por el modelo facilita el análisis y nos evita algunas cuestiones técnicas. Para este modelo, mostramos que a temperaturas bajas existe para cada spin $i=1,\dots,q$ una medida de Gibbs $\mu^i$ que verifica un escenario de mar con islas para el spin $i$, análogo al descrito en la Introducción para el modelo de Ising. En particular, las medidas $\mu^i$ son singulares entre sí, lo cual implica una transición de fase para el modelo a bajas temperaturas. Además, probamos que cada medida $\mu^i$ es exponencialmente mixing y continua como función de la temperatura (con la noción de convergencia local). Luego, tratamos el caso del modelo de Widom-Rowlinson discreto, en donde la falta de simetría entre los spins introduce algunas complicaciones en el análisis. No obstante, conseguimos resultados análogos a los obtenidos para el modelo de Potts: si la densidad de partículas es suficientemente alta entonces existen dos medidas de Gibbs $\mu^+$ y $\mu^-$ con las propiedades descritas arriba (para los spins $+$ y $-$, respectivamente). Por otro lado, por la falta de simetría entre los spins $(\pm)$ y el $0$, puede verse que no existe una medida con estas características asociada al spin $0$ restante. A continuación, mencionamos cómo el análisis realizado en los dos casos anteriores puede generalizarse a modelos discretos generales. También mostramos cómo pueden aplicarse estas ideas en modelos continuos (con espacio de spines finito y con simetría entre spins), obteniendo así los mismos resultados para el modelo de Widom-Rowlinson continuo y el modelo de varas finas con $n$ orientaciones equidistantes. Por último, en todos estos casos mostramos cómo puede ser aprovechada la dinámica FFG para simular perfectamente las distintas medidas bajo consideración. Cabe destacar que hasta ahora sólo se conocían métodos de simulación perfecta a baja temperatura para el modelo de Ising original. Nuestro análisis extiende estos métodos a una gama mucho más amplia de modelos, tanto discretos como continuos. Comparison principle -------------------- Let $f$ and $g$ be Lipschitz functions on ${{\mathbb R}}$ and, given $u,v \in C([0,1])$, consider $U^u$ and $U^v$ the solutions of the equation $${\partial}_t U = {\partial}^2_{xx} U + f(U) + g(U)\dot{W}$$ with initial data $u$ and $v$, respectively, and boundary conditions satisfying $$P( U(t,\cdot)|_{{\partial}[0,1]} \geq V(t,\cdot)|_{{\partial}[0,1]} \text{ for all $t \geq 0$}) = 1.$$ Then, if $u \geq v$ we have that $$P\left( U^u(t,x) \geq U^v(t,x) \text{ for all }t \geq 0, x \in [0,1] \right)=1.$$ A proof of this theorem can be found on [@DNKXM p. 130]. Notice that if $g \equiv 0$ one obtains a comparison principle for deterministic partial differential equations. Growth and regularity estimates ------------------------------- \[G.1\]Given a bounded set $B \subseteq C_D([0,1])$ there exists $t_B > 0$ such that - $\tau^u > t_B$ for any $u \in B$ - There exists $b: [0,t_B] \rightarrow {{\mathbb R}}^+$ such that $\lim_{t \rightarrow 0^+} b(t) = 0$ and for any $t \in [0,t_B]$ $$\sup_{u \in B} \| U^u(t,\cdot) - u \|_\infty \leq b(t).$$ Let $n \in {{\mathbb N}}$ be such that $B \subseteq B_{n -1}$. Notice that for $u \in B$ the truncated system $U^{(n),u}$ verifies $$\partial^2_{xx} U^{(n),u} - \| g_n \|_\infty \leq {\partial}_t U^{(n),\varepsilon} \leq \partial^2_{xx} U^{(n),u} + \| g_n \|_\infty$$ and so, by the comparison principle there exists $b : {{\mathbb R}}^+ \rightarrow {{\mathbb R}}^+$ such that $\lim_{t \rightarrow 0^+} b(t) = 0$ and for any $t > 0$ $$\sup_{u \in B} \| U^{(n),u}(t,\cdot) - u \|_\infty \leq b(t).$$ Now, if we take $t_B > 0$ such that $b(t) < 1$ for all $t \in [0,t_B]$ then for any such $t$ we have that $U^{(n),u}(t,\cdot) \in B_n^\circ$ and so $U^{(n),u}(t,\cdot)$ coincides with $U^{u}(t,\cdot)$. In particular, we see that $\tau^u > t$ and also that $$\sup_{u \in B} \| U^u(t,\cdot) - u \|_\infty \leq b(t)$$ which concludes the proof. \[G.2\]The following local and pointwise growth estimates hold: 1. Given a bounded set $B \subseteq C_D([0,1])$ there exist $C_B, t_B > 0$ such that - $\tau^u > t_B$ for any $u \in B$ - For any pair $u,v \in B$ and $t \in [0,t_B]$ $$\|U^u(t,\cdot) - U^v(t,\cdot) \|_\infty \leq e^{C_B t} \| u - v\|_\infty.$$ 2. Given $u \in C_D([0,1])$ and $t \in [0,\tau^u)$ there exist $C_{u,t}, \delta_{u,t} > 0$ such that - $\tau^v > t$ for any $v \in B_{\delta_{u,t}}(u)$ - For any $v \in B_{\delta_{u,t}}(u)$ and $s \in [0,t]$ $$\|U^u(s,\cdot) - U^v(s,\cdot) \|_\infty \leq e^{C_{u,t} s} \|u - v\|_\infty.$$ Given a bounded set $B \subseteq C_D([0,1])$ let us take $n \in {{\mathbb N}}$ and $t_B > 0$ as in the proof of Proposition \[G.1\]. Since $g_n$ globally Lipschitz, there exists a constant $C_n > 0$ such that for any pair $u,v \in C_D([0,1])$ and $t \geq 0$ $$\label{G.2eq1} \|U^{(n),u}(t,\cdot) - U^{(n),v}(t,\cdot) \|_\infty \leq e^{C_n t} \| u - v\|_\infty.$$ In particular, since for every $u \in B$ both $U^{(n),u}(t,\cdot)$ and $U^{u}(t,\cdot)$ coincide for all $t \in [0,t_B]$, we see that if we take $C_B := C_n$ then from and Proposition \[G.1\] we obtain (i). On the other hand, given $u \in C_D([0,1])$ and $t < t^u$ there exists $n \in {{\mathbb N}}$ such that $\sup_{s \in [0,t]} \|U^u(s,\cdot)\|_\infty \leq n-1$ and a constant $C_n > 0$ such that for any pair $u,v \in C_D([0,1])$ and $s \geq 0$ $$\label{G.2eq2} \|U^{(n),u}(s,\cdot) - U^{(n),v}(s,\cdot) \|_\infty \leq e^{C_n s} \| u - v\|_\infty.$$ Then by taking $C_{u,t}:=C_n$ and $\delta_{u,t} < e^{-C_n t}$ from we see that for any $v \in B_{\delta_{u,t}}$ both $U^{(n),v}(s,\cdot)$ and $U^{v}(s,\cdot)$ coincide for all $s \in [0,t]$ and so (ii) immediately follows. \[A.1\] If $u \in C_D([0,1])$ then $\partial^2_{xx} U^{u}$ exists for any $t \in (0,\tau^u)$. Furthermore, for any bounded set $B \subseteq C_D([0,1])$ there exists a time $t_B > 0$ such that 1. $\tau^u > t_B$ for any $u \in B$ 2. For any $t \in (0,t_B)$ we have $\sup_{u \in B} \| \partial^2_{xx} U^{u} (t,\cdot) \|_{\infty} < + \infty.$ Given $u \in C_D([0,1])$ and $t \in (0,\tau^u)$ let us take $n \in {{\mathbb N}}$ such that $$\sup_{s \in [0,t]} \|U^u(s,\cdot)\|_\infty \leq n.$$ It follows from this choice of $n$ that $U^u$ and $U^{(n),u}$ coincide on the interval $[0,t]$, so that it suffices to show that $\partial^2_{xx} U^{(n),u}(t,\cdot)$ exists. For this purpose, recall that $U^{(n),u}$ satisfies the integral equation $$\label{A.1eq1} U^{(n),u}(t,x) = \int_0^1 \Phi(t,x,y)u(y)dy + \int_0^{t} \int_0^1 \Phi(t-s,x,y)g_n(U^{(n),u})(s,y))dyds$$ where $\Phi$ denotes the fundamental solution of the heat equation with Dirichlet boundary conditions given by the formula $$\Phi(t,x,y) = \frac{1}{\sqrt{4\pi t}} \sum_{n \in {{\mathbb{Z}}}} \left[ \exp\left( - \frac{(2n+y -x)^2}{4t} \right) - \exp\left( - \frac{(2n+y +x)^2}{4t} \right)\right].$$ The equation can be rewritten as $$U^{(n),u}(t,x) = \int_{{\mathbb R}}\frac{e^{-\frac{(y-x)^2}{4t}}}{\sqrt{4\pi t}} \overline{u}(y)dy + \int_0^{t} \int_{{\mathbb R}}\frac{e^{-\frac{(y-x)^2}{4(t-s)}}}{\sqrt{4\pi (t-s)}} \overline{g_n(U^{(n),u})}(s,y)dyds$$ where, for $v \in C_D([0,1])$, $\overline{v}$ denotes its odd $2$-periodic extension to the whole real line and, for $\varphi \in C_D([0,T] \times [0,1])$, we set $\overline{\varphi}$ as $\overline{\varphi}(t,x) := \overline{\varphi(t,\cdot)}(x)$ for all $(t,x) \in [0,T]\times [0,1]$. Since $g_n$ and $u$ are both bounded it follows that the spatial derivative ${\partial}_x U^{(n),u}(t,\cdot)$ exists and satisfies $$\label{A.1eq2} {\partial}_x U^{(n),u}(t,x) = \int_{{\mathbb R}}\frac{\xi e^{-\xi^2}}{\sqrt{\pi t}} \overline{u}(x + \sqrt{4t} \xi)d\xi + \int_0^{t} \int_{{\mathbb R}}\frac{\eta e^{-\eta^2}}{\sqrt{\pi (t-s)}}\overline{g_n(U^{(n),u})}(s,x+\sqrt{4(t-s)}\eta)d\eta ds$$ where we have performed the changes of variables $$\xi = \frac{y-x}{\sqrt{4t}}\hspace{2cm}\text{ and }\hspace{2cm}\eta = \frac{y-x}{\sqrt{4(t-s)}}.$$ Consequently, we obtain the bound $$\| {\partial}_x U^{(n),u}(t,\cdot)\|_\infty \leq \left( \frac{1}{\sqrt{\pi t}} + \frac{\sqrt{t}}{\sqrt{\pi}}\right)(\|u\|_\infty + \|g_n\|_\infty).$$ Let us observe that if we set $\psi := U^{(n),u}(\frac{t}{2},\cdot)$ then $U^{(n),u}(t,\cdot) = U^{(n),\psi}(\frac{t}{2},\cdot)$ and by applied to the time $\frac{t}{2}$ we obtain that $\frac{d\psi}{dx}$ exists and satisfies $$\left\|\frac{d\psi}{dx}\right\|_\infty \leq \left( \frac{\sqrt{2}}{\sqrt{\pi t}} + \frac{\sqrt{t}}{\sqrt{2\pi}}\right)(\|u\|_\infty + \|g_n\|_\infty).$$ Furthermore, since $g_n'$ is bounded and for any $s \in (0,t)$ we have $$\| {\partial}_x U^{(n),u}(s,\cdot)\|_\infty \leq \left( \frac{1}{\sqrt{\pi s}} + \frac{\sqrt{s}}{\sqrt{\pi}}\right)(\|u\|_\infty + \|g_n\|_\infty)$$ then it follows that ${\partial}^2_{xx} U^{(n),u}(t,\cdot)$ exists and satisfies $$\begin{aligned} {\partial}^2_{xx} U^{(n),u}(t,x) &= {\partial}^2_{xx} U^{(n),\psi}\left(\frac{t}{2},x\right) = \int_{{\mathbb R}}\frac{\sqrt{2}\xi e^{-\xi^2}}{\sqrt{\pi t}} \frac{d \overline{\psi}}{dx}(x + \sqrt{2t} \xi)d\xi\\ \\ &+ \int_0^{\frac{t}{2}} \int_{{\mathbb R}}\frac{\sqrt{2}\eta e^{-\eta^2}}{\sqrt{\pi (t-2s)}}\left(\overline{g_n'(U^{(n),u})}\cdot\overline{{\partial}_x U^{(n),\psi}}\right)(s,x+\sqrt{2(t-2s)}\eta)d\eta ds.\end{aligned}$$Consequently, we obtain the bound $$\|{\partial}^2_{xx} U^{(n),u}(t,\cdot)\|_\infty \leq \left( \left( \frac{2}{\pi t} + \frac{1}{\pi} \right) + \|g'_n\|_\infty \int_0^{\frac{t}{2}} \frac{\frac{1}{\sqrt{\pi s}} + \frac{\sqrt{s}}{\sqrt{\pi}}}{\sqrt{\pi(\frac{t}{2}-s)}} ds \right)(\|u\|_\infty + \|g_n\|_\infty).$$ By observing that $$\int_0^{\frac{t}{2}} \frac{\frac{1}{\sqrt{s}} + \sqrt{s}}{\sqrt{\frac{t}{2}-s}} ds \leq 4 + t$$ we conclude that $$\label{A.1eq3} \|{\partial}^2_{xx} U^{(n),u}(t,\cdot)\|_\infty \leq \frac{1}{\pi} \left( 1 + \frac{2}{t} + \|g'_n\|_\infty(4+t)\right)(\|u\|_\infty + \|g_n\|_\infty).$$ Finally, given a bounded set $B \subseteq C_D([0,1])$ then the uniformity of the bound in $B$ follows from since by Proposition \[G.1\] we may take $n \in {{\mathbb N}}$ such that $$\sup_{u \in B, t \in [0,t_B]} \| U^{u}(t,\cdot)\|_\infty \leq n$$ for some $t_B > 0$ such that $\tau^u > t_B$ for all $u \in B$. \[A.2\] For any bounded set $B \subseteq C_D([0,1])$ there exists $t_B > 0$ such that 1. $\tau^u > t_B$ for any $u \in B$ 2. For any $t \in (0,t_B)$ there exist $R_t , N_t > 0$ such that for every $u \in B$ the function $U^{u}(t,\cdot)$ belongs to the compact set $$\gamma_{R_t,N_t} = \{ v \in C_D([0,1]) : \|v\|_\infty \leq R_t \,,\, |v(x)-v(y)| \leq N_t |x-y| \text{ for all }x,y \in [0,1]\}.$$ By Proposition \[G.1\] there exists $t_B > 0$ such that $\tau^u > t_B$ for every $u \in B$ and for each $t \in (0,t_B)$ there exists $R_t > 0$ such that $$\sup_{u \in B, s \in [0,t]} \| U^u(s,\cdot) \|_\infty \leq R_t.$$ It then follows from the proof of Proposition \[A.1\] that $$\sup_{u \in B} \| {\partial}_x U^{u}(t,\cdot)\|_\infty \leq \left( \frac{1}{\sqrt{\pi t}} + \frac{\sqrt{t}}{\sqrt{\pi}}\right)( R_t + \|g_{R_t}\|_\infty):=N_t$$ which by the mean value theorem implies that $U^{u}(t,\cdot) \in \gamma_{R_t,N_t}$ for all $u \in B$. \[A.7\] The following local and pointwise growth estimates hold: 1. Given a bounded set $B \subseteq C_D([0,1])$ there exists $t_B > 0$ such that - $\tau^u > t_B$ for any $u \in B$ - For any $t \in (0,t_B)$ there exists $C_{t,B} > 0$ such that for all $u,v \in B$ $$\| {\partial}_x U^{u}(t,\cdot) - {\partial}_x U^v (t,\cdot) \|_\infty \leq C_{t,B} \| u - v\|_\infty.$$ 2. Given $u \in C_D([0,1])$ and $t \in (0,\tau^u)$ there exist $C_{u,t}, \delta_{u,t} > 0$ such that - $\tau^v > t$ for any $v \in B_{\delta_{u,t}}(u)$ - For any $v \in B_{\delta_{u,t}}(u)$ $$\| {\partial}_x U^{u}(t,\cdot) - {\partial}_x U^v (t,\cdot) \|_\infty \leq C_{u,t} \| u - v\|_\infty.$$ Notice that if $t_B > 0$ is such that $\tau^u > t_B$ for every $u \in B$ and for each $t \in (0,t_B)$ there exists $R_{B,t} > 0$ such that $$\sup_{u \in B, s \in [0,t]} \| U^u(s,\cdot) \|_\infty \leq R_{B,t},$$ then it follows from that $$\| {\partial}_x U^{u}(t,\cdot) - {\partial}_x U^v (t,\cdot) \|_\infty \leq \left[\left( \frac{1}{\sqrt{\pi t}} + \frac{\sqrt{t}}{\sqrt{\pi}}\right)(1 + \|g'_{R_t}\|_\infty)\right] \| u - v\|_\infty.$$ which shows (i). Now, (ii) follows in the same way upon noticing that by Proposition \[G.2\] given $t \in (0,\tau^u)$ there exist $R_{u,t}, \delta_{u,t} > 0$ such that $\tau^v > t$ for any $v \in B_{\delta_{u,t}}(u)$ and $$\sup_{v \in B_{\delta_{u,t}}(u), s \in [0,t]} \| U^v(s,\cdot) \|_\infty \leq R_{u,t}.$$ \[A.3\] For any equilibrium point $w$ of the deterministic system let us consider its stable manifold $\mathcal{W}^w$ defined as $$\mathcal{W}^{w}:=\{ u \in C_D([0,1]) : U^{u} \text{ is globally defined and }U^{u}(t,\cdot) \underset{t \rightarrow +\infty}{\longrightarrow} w\}.$$ Notice that $\mathcal{W}^{\mathbf{0}}=\mathcal{D}_{\mathbf{0}}$. Then for any bounded set $B \subseteq \mathcal{W}^w$ there exists $t_B > 0$ such that for any $t_0 \in [0,t_B]$ and $r>0$ we have $$\sup_{u \in B} \left[ \inf \{ t \geq t_0 : d( U^u(t,\cdot), w) \leq r \} \right] < +\infty$$ whenever one of the following conditions hold: 1. $w \neq \mathbf{0}$ 2. $w = \mathbf{0}$ and $B$ is at a positive distance from $\mathcal{W} := \bigcup_{n \in {{\mathbb{Z}}}- \{0\}} \mathcal{W}^{z^{(n)}}$. Furthermore, if $B \subseteq \mathcal{D}_e$ is a bounded set at a positive distance from $\mathcal{W}$ then for any $n \in {{\mathbb N}}$ we have $$\sup_{u \in B} \tau^{(n),u} < +\infty.$$ Let us suppose first that $w \neq \mathbf{0}$. Then, since $\mathcal{W}^w$ is a closed set, by Proposition \[A.2\] we have that the family $\{ U^u(t_B ,\cdot) : u \in B \}$ is contained in a compact set $B' \subseteq \mathcal{W}^{w}$ for some suitably small $t_B > 0$. Hence, we obtain that $$\sup_{u \in B} \left[ \inf \{ t \geq t_0 : d( U^u(t,\cdot), w) \leq r \} \right] \leq t_B + \sup_{v \in B'} \left[ \inf \{ t \geq 0 : d( U^v(t,\cdot), w) < r \} \right]$$ Since the application $v \mapsto \inf \{ t \geq 0 : d( U^v(t,\cdot), w) < r \}$ is upper semicontinuous and finite on $\mathcal{W}^w$, we conclude that the right hand side is finite and thus the result follows in this case. Now, if $w = \mathbf{0}$ then once again by Proposition \[A.2\] we have that the family $\{ U^u(t_B ,\cdot) : u \in B \}$ is contained in a compact set $B' \subseteq \mathcal{D}_{\mathbf{0}}$ but this time by Proposition \[G.1\] we may choose $t_B > 0$ sufficiently small so as to guarantee that $B'$ is at a positive distance from $\mathcal{W}$. From here we conclude the proof as in the previous case. Finally, the last statement of the proposition is proved in a completely analogous fashion. Properties of the potential $S$ ------------------------------- \[Lyapunov\] The mapping $t \mapsto S( U^u(t,\cdot) )$ is monotone decreasing and continuous for any $u \in H^1_0((0,1))$. First, observe that a direct calculation shows that for any $u \in C_D([0,1])$ and $t_0 > 0$ $$\begin{aligned} \frac{d S( U^u(t,\cdot) )}{dt}(t_0) &= \int_0^1 \left(\partial_x U^u(t_0,\cdot) \partial^2_{xt} U^u(t_0,\cdot) - g(U^u(t_0,\cdot)) \partial_t U^u(t_0,\cdot)\right) \\ \\ & = - \int_0^1 \left( \partial^2_{xx} U^u(t_0,\cdot) + g(U^u(t_0,\cdot)) \right) \partial_t U^u(t_0,\cdot) \\ \\ &= - \int_0^1 \left(\partial_t U^u(t_0,\cdot)\right)^2 \leq 0.\end{aligned}$$ On the other hand, it is well known (see [@QS] p.75) that the mapping $t \mapsto U^{u}{(t,\cdot)}$ is continuous on $H^1_0((0,1))$ whenever $u \in H^1_0((0,1))$ and on $L^{p+1}([0,1])$ when $u \in C_D([0,1])$. In particular, we see that $t \mapsto S( U^u(t,\cdot) )$ is continuous at $t_0=0$ and so, by the previous calculation, we conclude the result. \[A.4\] The potential $S$ is lower semicontinuous. Let $(v_k)_{k \in {{\mathbb N}}} \subseteq C_D([0,1])$ be a sequence converging to some limit $v_\infty \in C_D([0,1])$. We must check that $$\label{continferiors} S(v_\infty) \leq \liminf_{k \rightarrow +\infty} S(v_k).$$ Notice that since $(v_k)_{k \in {{\mathbb N}}}$ is convergent in the supremum norm we have, in particular, that $$\label{cotalp} \sup_{k \in {{\mathbb N}}} \| v_k \|_{L^{p+1}} < +\infty$$ and therefore that $\liminf_{k \rightarrow +\infty} S(v_k) > -\infty$. Hence, by passing to an subsequence if necessary, we may assume that the limit in exists and is finite so that, in particular, we have that the sequence $(S(v_k))_{k \in {{\mathbb N}}}$ remains bounded. This implies that $v_k$ is absolutely continuous for every $k \in {{\mathbb N}}$ and, furthermore, by we conclude that the sequence $(v_k)_{k \in {{\mathbb N}}}$ is bounded in $H^1_0((0,1))$. Hence, there exists a subsequence $(v_{k_j})_{j \in {{\mathbb N}}}$ which is weakly convergent in $H^1_0((0,1))$ and strongly convergent in $L^2([0,1])$ to some limit $v_\infty^*$. Notice that since $(v_k)_{k \in {{\mathbb N}}}$ converges in the supremum norm to $v_\infty$, it also converges in $L^q$ for every $q \geq 1$. In particular, we have that $v^*_\infty = v_\infty$ and thus, by the lower semicontinuity of the $H^1_0$-norm with respect in the weak topology we conclude that $$\| {\partial}_x v_\infty \|_{L^2} \leq \liminf_{j \rightarrow +\infty} \| {\partial}_x v_{k_j}\|_{L^2}.$$ Finally, since $(v_k)_{k \in {{\mathbb N}}}$ converges to $v_\infty$ in $L^{p+1}$ and we have $S(u) = \frac{1}{2} \| {\partial}_x u \|_{L^2}^2 - \frac{1}{p+1} \| u \|_{L^{p+1}}^{p+1}$ for all $u \in H^1_0$, we obtain . \[S.1\] Given $u \in C_D([0,1])$ and $t \in (0,\tau^u)$ there exist $C_{u,t}, \delta_{u,t} > 0$ such that - $\tau^v > t$ for any $v \in B_{\delta_{u,t}}(u)$ - For any $v \in B_{\delta_{u,t}}(u)$ one has $$\| S(U^{u}(t,\cdot)) - S(U^v (t,\cdot) ) \|_\infty \leq C_{u,t} \| u - v\|_\infty.$$ This is a direct consequence of Propositions \[A.7\] and \[G.2\]. Properties of the quasipotential $V$ ------------------------------------ \[A.5\] The mapping $u \mapsto V(\mathbf{0},u)$ is lower semicontinuous on $C_D([0,1])$. Let $(u_k)_{k \in {{\mathbb N}}} \subseteq C_D([0,1])$ be a sequence converging to some limit $u_\infty \in C_D([0,1])$. We must check that $$\label{continferiorv} V(\mathbf{0},u_\infty) \leq \liminf_{k \rightarrow +\infty} V(\mathbf{0},v_k).$$ If $S(u_\infty)=+\infty$ then by Proposition \[costo\] we see that $V(\mathbf{0},u_\infty)=+\infty$ and thus by the lower semicontinuity of $S$ we conclude that $\lim_{v \rightarrow u} V(\mathbf{0},v)=+\infty$ which establishes in this particular case. Now, if $S(u_\infty)< +\infty$ then, by the lower semicontinuity of $S$ and the continuity in time of the solutions to , given $\delta > 0$ there exists $t_0 > 0$ sufficiently small such that $S(U^{u_\infty}(t_0,\cdot)) > S(u_\infty) - \frac{\delta}{2}$. Moreover, by Proposition \[G.2\] we may even assume that $t_0$ is such that $$\| U^{u_k}(t_0,\cdot) - U^{u_\infty}(t_0,\cdot) \|_\infty \leq 2 \| u_k - u_\infty\|_\infty$$ for any $k \in {{\mathbb N}}$ sufficiently large. Thus, given $k$ sufficiently large and a path $\varphi_k$ we construct a path $\varphi_{k,\infty}$ from $\mathbf{0}$ to $u_\infty$ by the following steps: 1. We start from $\mathbf{0}$ and follow $\varphi_k$ until we reach $u_k$. 2. From $u_k$ we follow the deterministic flow $U^{u_k}$ until time $t_0$. 3. We then join $U^{u_k}(t_0,\cdot)$ and $U^{u_\infty}(t_0,\cdot)$ by a linear interpolation of speed one. 4. From $U^{u_\infty}(t_0,\cdot)$ we follow the reverse deterministic flow until we reach $u_\infty$. By the considerations made in the proof of Lemma \[cotasuplema0\] it is not hard to see that there exists $C > 0$ such that for any $k \in {{\mathbb N}}$ sufficiently large we have $$I(\varphi_{k,\infty}) \leq I(\varphi_k) + C \| u_k - u_\infty\|_\infty + \delta$$ so that we ultimately obtain $$V(\mathbf{0},u_\infty) \leq \liminf_{k \rightarrow +\infty} V(\mathbf{0},u_k) + \delta.$$ Since $\delta > 0$ can be taken arbitrarily small we conclude . \[A.6\] For any $u,v \in C_D([0,1])$ the map $t \mapsto V\left(u,U^v(t,\cdot)\right)$ is decreasing. Given $0 \leq s < t$ and a path $\varphi$ from $u$ to $U^{u}(s,\cdot)$ we may extend $\phi$ to a path $\tilde{\varphi}$ from $u$ to $U^{u}(t,\cdot)$ simply by following the deterministic flow afterwards. It follows that $$V\left(u,U^v(t,\cdot)\right) \leq I(\tilde{\varphi}) = I(\varphi)$$ which, by taking infimum over all paths from $u$ to $U^{u}(s,\cdot)$, yields the [^1]: This means that ${{\mathcal F}}_t = \sigma( {{\mathcal G}}_t \cup \mathcal{N})$ where $\mathcal{N}$ denotes the class of all $P$-null sets of ${{\mathcal G}}_\infty = \sigma( {{\mathcal G}}_t : t \in {{\mathbb R}}^+)$. [^2]: Although there exists an equivalent representation of the Ising model as a lattice gas which falls into the category of diluted models as presented in this chapter. See [@OV p. 154]. [^3]: We say that $\Lambda \subseteq {{\mathbb{Z}}}^d$ is *simply connected* if the set $\bigcup_{x \in \Lambda} (x + [-\frac{1}{2},\frac{1}{2}]^d)$ is simply connected in ${{\mathbb R}}^d$. [^4]: Notice that for each $(\gamma_x,t,s) \in \Pi$ its flag is the unique $u \ in [0,1]$ such that $(\gamma_x,t,s,u) \in \overline{\Pi}$. Thus, there is no ambiguity in this choice of notation. [^5]: The forward dynamics on a finite volume are defined in the natural manner, following the approach of Section \[forwarddynamics\].

--- abstract: 'We comment on the paper of S. Postnikov et al. in Phys. Rev. D 82, 024016 (2010) and give a modified formula that needs to be taken into account when calculating the tidal Love number of neutron stars in case a first order phase-transition occurs at non-zero pressure. We show that the error made when using the original formula tends to zero as $p \rightarrow 0$ and we estimate the maximum relative error to be $\sim 5\%$ if the density discontinuity is at larger densities.' author: - 'János Takátsy$^{1,2}$' - 'Péter Kovács$^{1,2}$' title: 'Comment on “Tidal Love numbers of neutron and self-bound quark stars”' --- In Ref. [@postnikov2010] the authors investigated the qualitative differences between the tidal Love numbers of self-bound quark stars and neutron stars. In Eq. (14) they derived an expression for the extra term that should be subtracted from the logarithmic derivative $y(r)$ of the metric perturbation $H(r)$ in case there is a first-order phase transition in the equation of state (EoS). The authors applied this formula to quark stars where there is a core-crust phase transition at or below neutron-drip pressure. Since then multiple papers have included or applied this formula explicitly using EoSs with first-order phase transitions at non-negligible pressures ([*e.g.*]{} [@zhao2018; @han2019]). However, when the pressure $p_d$ corresponding to the density discontinuity is non-negligible compared to the central energy density of the neutron star, Eq. (14) of Ref. [@postnikov2010] should be modified as shown below. In this comment we derive the correct formula and estimate the error made when using the other formula instead. It needs to be added, that although Ref. [@han2019] contains the uncorrected formula, the results presented in the paper were calculated using the correct relation, as it was reported by the authors and also verified by the authors of Ref. [@postnikov2010]. This also applies to more recent publications including the same authors [@han2019b; @chatziioannou2020]. Moreover, despite using the erroneous formula, the results of Ref. [@zhao2018] are also mainly unaffected by this error, since they only provide approximate analytic fits for the ratios of tidal deformabilities of the two components in binary neutron stars. Thus, uncertainties of a few percent are inherently contained in these fits, which encompass the errors of individual tidal deformabilities. The corrected fits – as it was claimed by the authors of Ref. [@postnikov2010] – are negligibly different from the reported fits in Ref. [@zhao2018]. The tidal $l=2$ tidal Love number can be expressed the following way: $$\begin{aligned} k_2 &= \frac{8}{5} (1-2 \beta)^2 \beta^5 [2 \beta (y_R-1)-y_R+2]\nonumber\\ &\times \{2 \beta [4 (y_R+1) \beta^4+(6 y_R-4) \beta^3+(26-22 y_R) \beta^2\nonumber\\ &+3 (5 y_R-8) \beta-3 y_R+6]+3 (1-2 \beta)^2\nonumber\\ &\times[2 \beta (y_R-1)-y_R+2]\ln \left(1-2\beta\right)\}^{-1} , \label{eq:k2}\end{aligned}$$ where $\beta=M/R$ is the compactness parameter of the neutron star and $y_R=y(R)=[rH'(r)/H(r)]_{r=R}$ with $H(r)$ being a function related to the quadrupole metric perturbation (see [*e.g.*]{} [@damour2009]). $y_R$ is obtained by solving the following first-order differential equation: $$\begin{aligned} ry'(r)&+y(r)^2+r^2 Q(r) \nonumber\\ &+ y(r)e^{\lambda(r)}\left[1+4\pi r^2(p(r)-\varepsilon(r))\right] = 0 , \label{eq:y}\end{aligned}$$ where $\varepsilon$ and $p$ are the energy density and pressure, respectively, and $$\begin{aligned} Q(r)=4\pi e^{\lambda(r)}\left(5\varepsilon(r)+9p(r)+\frac{\varepsilon(r)+p(r)}{c_s^2(r)}\right) \nonumber\\ -6\frac{e^{\lambda(r)}}{r^2}-(\nu'(r))^2 . \label{eq:Q}\end{aligned}$$ Here $c_s^2=\mathrm{d}p/\mathrm{d}\varepsilon$ is the sound speed squared, while $e^{\lambda(r)}$, $\nu(r)$ metric functions are given by $$\begin{aligned} e^{\lambda(r)} &= \left[1-\frac{2m(r)}{r}\right]^{-1} \label{eq:tov_e} , \\ \nu'(r) &= \dfrac{2[m(r)+4\pi r^3 p(r)]}{r^2 - 2 m(r) r} \label{eq:tov_nu} ,\end{aligned}$$ with the line element for the unperturbed star defined as $$\mathrm{d}s^2 = e^{\nu(r)}\mathrm{d}t^2 - e^{\lambda(r)}\mathrm{d}r^2 - r^2(\mathrm{d}\vartheta^2 + \sin^2\vartheta \, \mathrm{d}\varphi^2),$$ and where $m(r)$ and $p(r)$ are calculated through the Tolman-Oppenheimer-Volkoff equations [@tolman1939; @oppenheimer1939]: $$\begin{aligned} m'(r) &= 4\pi r^2 \varepsilon(r) , \label{eq:tov_m} \\ p'(r) &= - [\varepsilon(r)+p(r)]\dfrac{m(r)+4\pi r^3 p(r)}{r^2 - 2 m(r) r} .\label{eq:tov_p}\end{aligned}$$ In case there is a first-order phase transition in the EoS, there is a jump of $\Delta\varepsilon$ in the energy density at constant pressure, hence $c_s^2=0$ in that region and the term in Eq. (\[eq:Q\]) containing $1/c_s^2$ diverges. Expressing $1/c_s^2$ in the vicinity of the density discontinuity: $$\frac{1}{c_s^2} = \frac{\mathrm{d}\varepsilon}{\mathrm{d}p}\bigg|_{p\neq p_d} + \delta(p-p_d) \Delta \varepsilon . \label{eq:cs2}$$ Changing the delta-function to a function in the radial position $r$, inserting Eq. (\[eq:cs2\]) into Eq. (\[eq:y\]) and integrating over an infinitesimal distance around $r_d$ one obtains: $$y(r_d^+) - y(r_d^-) = -4\pi r_d e^{\lambda(r_d)} [\varepsilon(r_d)+p(r_d)] \frac{\Delta \varepsilon}{|p'(r_d)|} .$$ Using Eq. (\[eq:tov\_p\]) we get: $$\begin{aligned} y(r_d^+) - y(r_d^-) &= -\frac{4\pi r_d^3 \Delta \varepsilon}{m(r_d)+4\pi r_d^3 p(r_d)}\nonumber\\ &= -\frac{\Delta \varepsilon}{\tilde{\varepsilon}/3+p(r_d)} , \label{eq:ydisc}\end{aligned}$$ where $\tilde{\varepsilon}=m(r_d)/(4\pi r_d^3/3)$ is the average energy density of the inner ($r<r_d$) region. Eq. (\[eq:ydisc\]) shows that there is an extra $p(r_d)$ term in the denominator as compared to Eq. (14) of Ref. [@postnikov2010]. We see that if the phase transition is at very low densities compared to the central energy density then $p(r_d)/\tilde{\varepsilon}\rightarrow0$ [^1] and we get back the formula in Ref. [@postnikov2010]. ![\[fig:css\]Illustration of the EoS in the constant-sound-speed construction [@alford2013; @han2019]. At $p_\mathrm{trans}$ a quark matter part with a constant sound speed of $c_\mathrm{QM}$ is attached to the nuclear matter EoS after an energy density jump of $\Delta\varepsilon$.](CSS_EoS){width="48.00000%"} We investigated the difference caused by applying the two different formulas using a constant-sound-speed construction (see Fig. \[fig:css\]) [@alford2013; @han2019]: $$\varepsilon(p)= \bigg\{\begin{array}{lr} \varepsilon_\mathrm{NM}(p) &p<p_\mathrm{trans}\\ \varepsilon_\mathrm{NM}(p_\mathrm{trans}) + \Delta \varepsilon + c_\mathrm{QM}^{-2}(p-p_\mathrm{trans}) \quad &p>p_\mathrm{trans} \end{array},$$ where we fixed $c_\mathrm{QM}^2 = 1$ as in Ref. [@han2019], while varying the values of $p_\mathrm{trans}$ (through $n_\mathrm{trans}\equiv n_\mathrm{NM}(p_\mathrm{trans})$) and $\Delta \varepsilon$. For the nuclear matter (NM) part we chose the SFHo [@steiner2013] and DD2 [@typel2010] as two representative EoSs. We varied the baryon number density at the phase transition $n_\mathrm{trans}$ between $n_0$ and $3.5 n_0$ with $n_0=0.16$ fm$^{-3}$ being the nuclear saturation density. The strength of the phase transition $\Delta\varepsilon/\varepsilon_\mathrm{trans}$ was varied between $0$ and $3$, where $\varepsilon_\mathrm{trans}\equiv\varepsilon_\mathrm{NM}(p_\mathrm{trans})$. ![\[fig:k2ex\]Tidal Love number – neutron star mass relations for the SFHo (orange line) and DD2 (blue line) EoSs, as well as for EoSs obtained from the constant-sound-speed construction. The different number pairs denote different values of $n_\mathrm{trans}/n_0$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}$, respectively. The tidal Love numbers calculated using Eq. (\[eq:ydisc\]) (solid lines) are reduced by a few percent compared to the ones calculated using Eq. (14) of Ref. [@postnikov2010] (dashed lines).](k2_examp){width="48.00000%"} {width="48.00000%"} {width="48.00000%"} In Fig. \[fig:k2ex\] we show some examples of tidal Love number – neutron star mass relations. For EoSs with first-order phase transitions, the Love numbers are reduced when using Eq. (\[eq:ydisc\]) (red and green solid lines) compared to using the formula in Ref. [@postnikov2010] (red and green dashed lines). The maximum relative difference in the tidal Love number as a function of the two parameters defining our constant-sound-speed EoSs is shown in Fig. \[fig:k2err\]. We see that the maximum relative difference reaches its maximum at $n_\mathrm{trans}/n_0\approx2.5$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}\approx1.5$ for the SFHo EoS, and at $n_\mathrm{trans}/n_0\approx2.0$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}\approx1.5$ for the DD2 EoS, however, it does not exceed $5\%$ for the whole parameter range. The relative difference also diminishes as we go to lower densities, as it is expected. Acknowledgement {#acknowledgement .unnumbered} =============== J. Takátsy and P. Kovács acknowledge support by the NRDI fund of Hungary, financed under the FK19 funding scheme, project no. FK 131982. P. Kovács also acknowledges support by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. [99]{} S. Postnikov, M. Prakash, and J. M. Lattimer, Tidal Love Numbers of Neutron and Self-Bound Quark Stars, Phys. Rev. D **82**, 024016 (2010), arXiv:1004.5098 \[astro-ph.SR\]. T. Zhao and J. M. Lattimer, Tidal Deformabilities and Neutron Star Mergers, Phys. Rev. D **98**, 063020 (2018), arXiv:1808.02858 \[astro-ph.HE\]. S. Han and A. W. Steiner, Tidal deformability with sharp phase transitions in (binary) neutron stars, Phys. Rev. D **99**, 083014 (2019), arXiv:1810.10967 \[nucl-th\]. T. Damour and A. Nagar, Relativistic tidal properties of neutron stars, Phys. Rev. D **80**, 084035 (2009), arXiv:0906.0096 \[gr-qc\]. R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. **55**, 364 (1939). J. Oppenheimer and G. Volkoff, On Massive neutron cores, Phys. Rev. **55**, 374 (1939). M. G. Alford, S. Han, and M. Prakash, Generic conditions for stable hybrid stars, Phys. Rev. D **88**, 083013 (2013), arXiv:1302.4732 \[astro-ph.SR\]. A. W. Steiner, M. Hempel, and T. Fischer, Core-collapse supernova equations of state based on neutron star observations, Astrophys. J. **774**, 17 (2013), arXiv:1207.2184 S. Typel, G. Ropke, T. Klahn, D. Blaschke, and H. Wolter, Composition and thermodynamics of nuclear matter with light clusters, Phys. Rev. C **81**, 015803 (2010), arXiv:0908.2344 \[nucl-th\]. S. Han, M. A. A. Mamun, S. Lalit, C. Constantinou, M. Prakash, Treating quarks within neutron stars, Phys. Rev. D **100**, 103022 (2019), arXiv:1906.04095 \[astro-ph.HE\]. K. Chatziioannou, S. Han, Studying strong phase transitions in neutron stars with gravitational waves, Phys. Rev. D **101**, 044019 (2020), arXiv:1911.07091 \[gr-qc\] [^1]: It is worth to note here that although $\tilde{\varepsilon}$ – the average energy density of the inner core – is not equal to the central energy density $\varepsilon_c$, it falls to the same order of magnitude ($\tilde{\varepsilon}/\varepsilon_c \gtrsim 0.25 - 0.5$ for $M>0.5$ $M_\odot$).

--- abstract: | The experimental results relevant for the understanding of the microscopic dynamics in liquid metals are reviewed, with special regards to the ones achieved in the last two decades. Inelastic Neutron Scattering played a major role since the development of neutron facilities in the sixties. The last ten years, however, saw the development of third generation radiation sources, which opened the possibility of performing Inelastic Scattering with X rays, thus disclosing previously unaccessible energy-momentum regions. The purely coherent response of X rays, moreover, combined with the mixed coherent/incoherent response typical of neutron scattering, provides enormous potentialities to disentangle aspects related to the collectivity of motion from the single particle dynamics. If the last twenty years saw major experimental developments, on the theoretical side fresh ideas came up to the side of the most traditional and established theories. Beside the raw experimental results, therefore, we review models and theoretical approaches for the description of microscopic dynamics over different length-scales, from the hydrodynamic region down to the single particle regime, walking the perilous and sometimes uncharted path of the generalized hydrodynamics extension. Approaches peculiar of conductive systems, based on the ionic plasma theory, are also considered, as well as kinetic and mode coupling theory applied to hard sphere systems, which turn out to mimic with remarkable detail the atomic dynamics of liquid metals. Finally, cutting edges issues and open problems, such as the ultimate origin of the anomalous acoustic dispersion or the relevance of transport properties of a conductive systems in ruling the ionic dynamic structure factor are discussed. author: - 'Tullio Scopigno$^{1}$' - 'Giancarlo Ruocco$^{1}$' - 'Francesco Sette$^{2}$' title: ' Microscopic dynamics in liquid metals: the experimental point of view. ' --- Introduction ============ Liquid metals are an outstanding example of systems combining great relevance in both industrial applications and basic science. On the one hand they find broad technological application ranging from the production of industrial coatings (walls of refinery coker, drill pipe for oil search) to medical equipments (reconstructive devices, surgical blades) or high performance sporting goods. Most metallic materials, indeed, need to be refined in the molten state before being manufactured. On the other hand liquid metals, in particular the monoatomic ones, have been recognized since long to be the prototype of simple liquids, in the sense that they encompass most of the physical properties of real fluids without the complications which may be present in a particular system [@BALUCANI]. In addition to that, metallic fluids such as molten sodium, having similar density and viscosity as water, find application as coolant in nuclear reactors. The thermodynamic description of liquid metals can be simplified by assuming a few parameters. Usually, if compound formation is weak physical theory alone can be used while, if there is strong compound formation, chemical theory alone is used. The lowest-melting liquid metals are those that contain heavier elements, and this may be due to an increase in ease of creating a free-electron solution. Alkali metals are characterized by low melting points, and they tend to follow trends. Binary associating liquids show a sharp melting point, with the most noticeable example being mercury ($T_m=234$ K). Melting points can be lowered by introducing impurities into the metal. Often, to this purpose, another metal with a low melting point is used. Mixing different metals may often result in a solution that is eutectic. In other words, from Henry’s law it is understood that a melting point depression occurs, and the system becomes more disordered as a result of the perturbation to the lattice. This is the case, for instance, of the well known eutectic Pb-Tin alloy, widely used in soldering applications ($T_m=453$ K). Until the sixties the understanding of the physical properties of metals proceeded rather slowly. It was John Ziman, indeed, who made the theory of liquid metals respectable for the first time [@ZIMAN], and the Faber-Ziman theory, developed in 1961-63 and dealing with electronic and transport properties, is attractively introduced in Faber’s book, which is an excellent treatise of the physical properties of liquid metals [@FABER]. The other text which can be considered a classic is March’s book [@MARCHLM], along with the more recent [@MARCHLMCT], which provides a comprehensive overview over liquid metals. It is from these texts that a first clear definition of liquid metal can be outlined. At first glance, indeed, the words “liquid metal” are self-explanatory: by definition any metal heated to its melting point can be cast in this category. Liquid metals, however, are implicitly understood to be less general than the above definition, and no literature clearly states an exact definition. Although no precise agreement has been made, there are certain characteristics shared by liquid metals, descending from a close interplay between ionic structure, electronic states and transport properties. The book of Shimoji [@SHIMOJI] deals with the fundamentals of liquid metals in an elementary way, covering the developments achieved after the first book by March. It does not address, however, the dynamical properties in great detail. Addison’s book [@ADDISON] is much like March’s general book, but is more focused on applications of alkali metals, especially on their use in organic chemistry. In addition, Addison discusses many methods for purifying and working with liquid alkali metals. March is more theoretical whereas Addison is practical, but both authors focus on a thermodynamic explanation of liquid metals. For an appealing general introduction to the physics and chemistry of the liquid-vapor phase transition (beyond the scope of this review) the reader should certainly make reference to [@HENSEL], which also provides a bird eye view of the practical applications of fluid metals, such as high-temperature working fluid or key ingredients for semiconductor manufacturing. There are, then, a number of books which are more general and more specific at the same time, in the sense that they deal with with the wider class of simple liquids (including noble fluids, hard sphere fluids etc.), but they are mainly concerned with structural and dynamical properties only [@BALUCANI; @HANSEN; @BY; @MARCH; @EGELSTAFF]. They are practically ineludible for those aiming at a rigorous approach to the statistical mechanics description of the liquid state. It can be difficult to find an exhaustive updated database of the physical properties of liquid metals, especially as far as dynamics is concerned. But the handbooks of [@ida] and [@OSE] are remarkable exceptions, with the second one specifically addressing liquid alkali metals. Historical background --------------------- Early phenomenological approaches to the study of relaxation dynamic in fluids can be dated back to the end of the nineteen century [@max_visco; @kel_visco]. Only in the mid twentieth century, however, it was realized that a deeper understanding of the physical properties of liquids could have been reached only through a microscopic description of the atomic dynamics. This became possible through the achievements of statistical mechanics which provided the necessary tools, such as correlation functions, integral equations etc. The mathematical difficulties related to the treatment of real liquids brought to the general attention the importance of simple liquids, as systems endowed with the rich basic phenomenology of liquids but without the complications arising, for instance, by orientational and vibrational degrees of freedom. As a consequence, the end of the fifties saw major experimental efforts related to the development of Inelastic Neutron Scattering (INS) facilities which, as we shall see, constitutes a privileged probe to access the microscopic dynamics in condensed matter and, in particular, in the liquid state [@egel_pio]. A sizable library of experimental data on liquid metals has been constituted since then, realizing the prototypical structural and dynamical properties of these systems, representative of the whole class of liquids. In the sixties, the advent and the broad diffusion of computational facilities brought a new era for two main reasons: on the one side, realistic computer simulation experiments become possible [@sch_sim], on the other side the new computation capabilities greatly facilitated the interpretation of INS experiments. For instance, new protocols for accurate estimates of the multiple scattering contribution affecting neutron scattering were proposed [@cop_multiplo]. The theoretical framework of the Inelastic Neutron Scattering, and the guidelines to interpret the results, have been reviewed in the classical textbooks [@LOVESEY; @MARSHALL]. The dynamics of liquid metals has been extensively investigated by INS and computer simulations with the main purpose of ascertaining the role of the mechanisms underlying both collective and single-particle motions at the microscopic level. In the special case of collective density fluctuations, after the seminal inelastic neutron scattering study by Copley and Rowe [@cop_rb] and the famous molecular dynamics simulation of Rahman [@rahman_sim] in liquid rubidium, the interest in performing more and more accurate experiments is continuously renewed: it was soon realized, indeed, that well-defined oscillatory modes could be supported even outside the strict hydrodynamic region. In molten alkali metals, moreover, this feature is found to persist down to wavelengths of one or two interparticle distances, making these systems excellent candidates to test the various theoretical approaches developed so far for the microdynamics of the liquid state. Up to ten years ago the only experimental probe appropriate to access the atomic dynamics over the interparticle distance region were thermal neutrons, and using this probe fundamental results have been gained. There are, however, certain limitations of this technique which can restrict its applicability: First, the presence of an incoherent contribution to the total neutron scattering cross section. If on one hand this allow to gather a richer information, being simultaneously sensitive to collective and single particle dynamics, on the other hand poses the problem of decoupling the two contributions, when aiming at the determination of collective properties only (i.e. of the coherent dynamic structure factor $S(Q,\omega)$). In liquid sodium, for instance, the incoherent cross section dominates; even in more favorable cases (Li, K) at small $Q$ the intensity of the collective contribution is low, and its extraction requires a detailed knowledge of the single particle dynamics. The second reason is dictated by the need of satisfying both the energy and momentum conservation laws which define the $(Q-E)$ region accessible to the probe [@BALUCANI]. Roughly speaking, when the sound speed of the system exceeds the velocity of the probing neutrons ($\sim 1500$ m/s for thermal neutrons) collective excitations can hardly be detected for $Q$ values below, let’s say, $Q_m,$ the position of the main diffraction peak of the sample, which is the region where collective properties show the richer phenomenology. As we shall see in section \[sec\_klim\], given a certain kinematic region accessible to neutrons (basically ruled by their thermal energy), by virtue of the $m^{-1/2}$ dependence of the sound velocity of an atomic system, the higher its atomic number, the wider is the accessible energy-momentum region of the excitations which can be studied. Taking as an example alkali metals, indeed, accurate INS data are available for heavier elements such as rubidium ($v \sim 1260$ m/s) [@cop_rb; @chi_rb; @pas_rb] and cesium ($v\sim 970$ m/s) [@bod_cs], while more difficulties are met in the case of lighter atoms. In particular, lithium represents the most critical case due to its high sound speed ($v \sim 4500$ m/s) and to the weak scattering cross section which, moreover, results from comparable values of the coherent and incoherent contributions: for this reason INS aiming to the study of collective properties of Li represented a very hard challenge [@dej_li; @dej_phd]. From a general point of view the main outcome of most of these early INS experiments, as far as collective properties are concerned, is the evidence of inelastic excitations in $S(Q,\omega)$ which have been necessarily analyzed within simple models such as the damped harmonic oscillator (DHO) [@fak_dho], suitable to extract reliable and resolution-corrected information on the peak positions but not about the detail of the whole lineshape. Some additional information have been achieved, for instance, in the case of cesium [@bod_cs], where information about an average relaxation time have been extracted utilizing Lovesey’s viscoelastic model [@lov_visco] or, more recently, in molten potassium, where a generalized hydrodynamic treatment as the one described in sec. \[sec\_collnonhydro\] is undertaken [@cab_k] and electronic screening effects have been explicitly taken into account [@bov_k]. Paralleling the development of INS facilities, new ideas arose on both theoretical and numerical fields from 1975 and in the intervening decades, driven by kinetic theory applied to Enskog’s fluid [@desh_hyd; @desh_hyd0; @kag_hs; @all_hs; @all_hs1], allowing to describe the hydrodynamic region in terms of three pole approximation, or to reproduce the dynamic structure factor at wavelengths comparable to the inverse mean particle distance in terms of extended heat mode [@coh_hs]. Kinetic approaches where eventually complemented by memory function formalism and by Mode Coupling theory [@Wah_kin; @sjog_kin; @sjog_kin1; @sjog_sw; @desh_mc; @got_na; @sjog_coll; @beng_kinhs]. Turning the attention to numerical advances in the liquid metals field, the major achievement are probably related to the introduction of the pseudo-potential concept [@HARRISON; @aus_psp; @heine_psp; @stroud_psp] which, beside offering a deeper comprehension of physical properties such as electrical resistivity, provided a clue for realistic numerical simulations. In Molecular Dynamics, indeed, the choice of a realistic interatomic potential - i.e. a potential model able to reproduce structural properties - is crucial to allow the determination of the dynamics of the system via the integration of the classical Newton equations. Exploiting the pseudo-potential theory, it has been possible to express the atomic interaction as a sum of pairwise interactions, ruled by an effective density dependent interaction. In this respect, one of the most successful expressions is the effective potential proposed for alkali metals [@pri_psp]. The numerical simulation framework is particularly useful since the single particle and the collective dynamics can easily be investigated within technical restrictions due to the finite box size (defining the minimum accessible wavevector) and computation time (related to the statistical quality and to the energy resolution of the calculated spectra). Broadly speaking the features of the atomic collective motion i.e. the details of $S(Q,\omega)$ lineshape, as outcome of MD run, turns out to be less noisy and more straightforward than the correspondent INS results: no absolute normalization is required, no mixing between coherent/incoherent dynamics occurs and, above all, basically no resolution corrections are needed. The major experimental breakthrough, however, happened in the last ten years when x-rays came up by the side of neutrons to study the collective dynamics in a similar frequency and wavelength region. The intriguing theoretical possibility of performing Inelastic X ray scattering [@BURKEL] became real thanks to the advent of the third generation sources [@mas_strum], disclosing previously unaccessible tasks in the physics of disordered systems [@ruo_nat; @set_sci; @sin_al2o3; @sco_sci]. In this case, the cross section is mainly coherent, and the combination of the two techniques can in principle serve to disentangle the two contributions. Unfortunately, such complementarity has not yet been exploited at full. Dynamical aspects ----------------- There are mainly two routes to approach the dynamics of a viscous melt. The first one stems from a quasi-crystalline picture, and relies on the observation that, often, the diffusion coefficient $D$ linearly depends on $1/T$. The same dependence, in fact, is induced in crystalline solids by vacancies and defects. This analogy suggests that diffusion in liquids is an activated process, and many attempts have been done to relate the activation energy in the liquid to the thermodynamics of its solid. The other point of view is the kinetic theory, a gas-like picture where the correlation functions are different in view of the density which is typically much higher than in the gas state. Within this framework an expression for the behavior of diffusion coefficient and viscosity can be gained in terms of the friction coefficient $\xi$. When a particle of the melt is moving with constant velocity $v$ a net retarding force results from the different rate between front and back collisions of the form $F=-\xi v$. For an hard sphere gas it turns out from Fick’s law of diffusion that $\xi=\frac{8ng(\sigma)\sigma ^2 }{3}\sqrt{\pi mk_B T}$, where $\rho$ is the atomic density, $m$ the atom mass, and $4\pi \sigma^2 g(\sigma)$ the density probability of finding two units at distance $\sigma$. The diffusion coefficient is therefore $D=\frac{k_B T}{\xi}$, while the viscosity $\eta=\frac{\xi}{3\pi \sigma}$ [@long_diff]. These simple expressions turn out to describe remarkably well the dynamics of liquid metals as long as the one of other simple fluids. However, better quantitative agreement can be obtained introducing the velocity autocorrelation function $\psi(\Delta t)=\frac{\langle \mathbf{v}(t+\Delta t) \cdot \mathbf{v}(t)\rangle}{\langle v^2 \rangle}$, whose time integral determines the diffusion coefficient. The $\psi(t)$ plays a central role in liquid dynamics, not only for providing a rigorous way to calculate $D$, but also because through its Fourier transform one can grasp an insight into the detail of the interatomic interactions. It can be accessed either by molecular simulations (the calculation trivially follows from its definition) or experimentally, mainly by Inelastic Neutron Scattering (INS) with the methods detailed in section \[sec\_exp\]. Broadly speaking, $\psi(\Delta t)$ is related to the knowledge of $G_s(r,t)$ i.e. the probability that a given particle travels a distance $r$ in the time interval $t$. INS is always sensitive to a combination of $G_s(r,t)$ and $G_d(r,t)$, being this latter quantity the probability of finding two distinct particles at a space distance $r$ and time distance $t$. The way to separate these two contributions mixed in the instrumental response is one of the major conundrum of the neutron scattering technique. The direct knowledge of the coherent response, gained by means of IXS, opens the possibility to the understanding of the high frequency modes that have been seen to survive since the famous INS experiments on the highly coherent scatterer liquid Rubidium [@cop_rb]. In particular, from a solid point of view, the main issue is to ascertain the relation between these modes and the phonon excitations in the corresponding polycrystal just below the melting. In a perfect crystal (i.e. a periodic assembly of atoms or ions), indeed, atomic dynamics is mainly vibrational, characterized by normal modes which are plane waves, due to the harmonic nature of the interatomic forces. Consequently, a well defined ratio (the sound velocity) exists between the frequency and the wavevector of the density fluctuations, named in this case *phonon*. In real crystals, therefore, the energy spectrum of a scattered probe results in sharp peaks, whose linewidth is related to the presence of anharmonicities or lattice defects. In a liquid things are more involved: other effects arise besides anharmonicity, both structural (the average atomic positions are randomly distributed) and dynamical (mass diffusion and activated processes join the purely vibrational motion). But even in this complex scenario distinct peaks survive and one can extract dispersion relations. Interestingly, the relation between crystals and liquids involves more than the mere similarity between the sound velocity of the crystalline acoustic branches and of the mode observed in the liquid (i.e. the low $Q$ limit of the dispersion relation). Despite the lack of periodicity intrinsic to the inherent liquid structure, indeed, the presence of some residual correlation (testified by the oscillation in static structure factor) seems to warrant a support for the existence of umklapp modes similar to the one existing in crystals. Such processes, i.e. the presence of inelastic modes characterized by wavevectors which differ by multiples of the reciprocal lattice spacing, have been early reported by means of INS in liquid lead [@rand_umk; @coc_umk; @dor_umk] and more recently in liquid lithium by means of IXS [@scop_prbumk]. As correctly pointed out by Faber, the presence of these excitations does not imply the existence of genuine high $Q$ modes, it rather indicates that umklapp processes may occur in liquid as much as in solid [@FABER]. From the liquid point of view the interest in this phenomenology lies in the challenging extension of the simple hydrodynamics, describing the density fluctuations in the long wavelenght limit, down to the lenghtscale of the mean interparticle distances. As it will be shown in the following, such an extension relies on serious and sometimes not fully justified assumptions necessary to walk in the uncharted and perilous territory between hydrodynamics and single particle regimes. Peculiarities of liquid metals ------------------------------ Apparently at odds with the previously mentioned classification of liquid metals as prototype of simple liquids, even in the simplest monoatomic case, metallic fluids are actually two component systems. The interplay between electron and ions, indeed, is an intrinsic aspect of liquid metals, and a rigorous approach should therefore mimic the formalism utilized for binary mixtures. For many aspect, however, one might be interested in ionic properties only, and as far as atomic dynamics is concerned, this seems to be the case. In such circumstance, one can look at a liquid metal as an ionic assembly whose interaction is mediated by the conduction electron gas. The treatment is in such way reduced to a one component system, like for noble fluids one can introduce a pairwise interaction, but this latter will be ruled by a density dependent pseudopotential. Although within the pseudopotential approach many results for liquid metals are qualitatively similar to those for ordinary non conductive fluids, some remarkable differences exist. One of the most relevant of these differences concerns peculiar structural properties involving short range order: in several liquid metals the static structure factor exhibits an asymmetry or even a shoulder just above the main peak. The origin of this anomaly has been highly debated, and ascribed to the peculiar shape of the interaction potential in those metals in which the hard sphere description fails [@tsay_ga]. More specifically, it can be interpreted in terms of a repulsive interaction composed by an hard sphere part plus an adjacent ledge induced by electronic effects, by a curvature change occurring at the nearest neighbor distance, or by the interplay with Friedl oscillations. The reported peculiarities extend also to the dynamics: while in the long wavelength limit they are expected to behave similarly to non conductive fluids, at finite wavevectors their departure from ordinary hydrodynamics can be in principle influenced by the high values of the thermal conductivity [@FABER; @sing_pre; @scop_comm; @sing_rep]. One of the most striking quantitative differences with ordinary fluids concerns the “visibility” of the inelastic features, i.e. the inelastic to elastic ratio, which seems to be related to the softness of the interaction potential [@cana_ljlm; @baluc1_sim]. This latter is responsible, for example, for the very favorable inelastic to elastic ratio which makes alkali metals ideal systems to study collective properties. Though the whole dynamics of liquid metals seems to be conveniently rationalized treating them as ordinary fluids interacting via an effective, density dependent, pairwise interaction potential, there is an alternative route which explicitly takes into account electronic screening effect on the ionic dynamics, relying on the introduction of a suitable model for the wavelength dependent dielectric function. In this way one his able to test different approximations comparing the predictions for the mechanical compressibility (or, equivalently, for the sound velocity) with the experimental values [@burns_prb; @bov_k]. Why this review --------------- As previously pointed out, some books offer a broad coverage of the physics and chemistry of liquid metals, but none of them is focused on the dynamical aspects. On the other side, books dealing with liquid dynamics do not spot on liquid metals in great detail. Egelstaff’s review [@egel_rev] offers an exhaustive coverage of transport phenomena in liquids but dynamical properties of liquid metals are marginally addressed. The Copley and Lovesey review [@coplov_rev] provides an invaluable sight on the dynamics of simple liquids, and it presents many results obtained for liquid metals, but again does not emphasize the peculiarity of these systems. Moreover, it also addresses in detail numerical results and last, but not least, it is necessarily not up to date given the large amount of experiments recently performed. The last decade, indeed, has seen important advances in the dynamics of liquid metals, especially on the experimental side, driven by the advent of new X-ray facilities and by the upgrades of the neutron ones. On the theoretical side, in the eighthes, impressive results have been achieved working on hard sphere models and/or mode coupling approaches which, again, remain uncovered in the Copley and Lovesey review. To conclude, it is certainly worth mentioning the more recent review article by P. Verkerk [@verk_rev], which offers a clear overview of the theoretical models developed sofar for liquid dynamics, presenting a selection of experimental results for liquified rare gases, molten metals and binary mixtures. Given this background, it seemed to us helpful to focus on the experiments on liquid metals, and to discuss and summarize the results and their interpretations in terms of the existing theories, trying to emphasizes advantages and weakness of each approach. This turned out to be a difficult task, given the broadness of the matter, and we necessarily had to make some choices. We left out, for instance, mixtures and alloys, and we tried to focus on the most recent experimental achievements, say of the last ten years: in most cases we quickly reference to older results, unless they are particularly relevant in view of the most recent ones. The review is organized as follows. In section \[sec\_bg\] we present different theoretical approaches to the dynamics of liquid metals. We develop in parallel subsections the treatment of the self and collective properties which are, in turn, organized according to the different wavelength domains: hydrodynamic, non-hydrodynamic, single particle. We also include a subsection dealing with hard sphere treatment and one presenting the ionic plasma approach, which is peculiar of conductive systems. In section \[sec\_exp\] we describe the experimental approach to the investigation of microscopic dynamics in liquids, outlining the basics of the inelastic scattering problem. Since the case of X rays is relatively newer, we decided to treat it in detail, but continuous reference is made to neutron scattering in an effort to emphasize merit, drawbacks and complementarities of the two methods. Section \[sec\_res\] is the bulk part of this paper, in which the experimental results are reviewed and ordered element by element. Here, we make constant reference to section \[sec\_bg\] to recall the different approaches utilized by different authors to describe the experimental result. In section \[sec\_sum\], finally, we try to summarize the arising scenario, pointing out the issues which, in our opinion, deserves further investigations and trying to draw, when possible, some conclusive pictures. Theoretical background \[sec\_bg\] ================================== General overview ---------------- ### Some basic definitions The investigation of microscopic dynamics of an ensemble of $N$ identical atomic or molecular units usually proceeds through the study of correlation functions of dynamical variables, i.e. of functions of the phase space variables, defined as the $6N$ positions ${\bf r}_i(t)$ and momenta ${\bf p}_i(t)=m{\bf v}_i(t)$ of the particles. Relevant dynamical variables are those stemming from the microscopic density $\rho({\bf r},t)$ (whose average is related to the number density $\rho=\langle \rho({\bf r},t) \rangle$), momentum density ${\bf J}({\bf r},t)$ and kinetic energy density E([**r**]{},t): $$\begin{aligned} && \rho({\bf r},t)\doteq \frac{1}{\sqrt N} \sum_{i} \delta({\bf r}-{\bf r}_i(t)) \nonumber \\ && {\bf J}({\bf r},t) \doteq \frac{1}{\sqrt N} \sum_{i} {\bf v}_i(t) \delta({\bf r}-{\bf r}_i(t)) \nonumber \\ && E({\bf r},t) \doteq \frac{1}{\sqrt N} \sum_{i} \frac{1}{2}m v^2_i(t) \delta({\bf r}-{\bf r}_i(t)) \label{3dv}\end{aligned}$$ In many cases, the study of the dynamics of a tagged particle $i$ can be of interest, and it relies on similar definitions for the single particle dynamical variables: $$\begin{aligned} && \rho_s({\bf r},t)\doteq \delta({\bf r}-{\bf r}_i(t)) \nonumber \\ && {\bf J}_s({\bf r},t)\doteq {\bf v}_i(t) \delta({\bf r}-{\bf r}_i(t)) \nonumber \\ && E_s({\bf r},t)\doteq \frac{1}{2}m v^2_i(t) \delta({\bf r}-{\bf r}_i(t)) \nonumber\end{aligned}$$ The well known van Hove distribution functions $G_d(r,t)$ and $G_s(r,t)$ are related in the classic (not quantum) case to the microscopic self and collective densities through: $$\begin{aligned} G({\bf r},t)=G_s({\bf r},t)+G_d({\bf r},t) \nonumber\end{aligned}$$ with $$\begin{aligned} G_s({\bf r},t)&=&\frac{1}{N}\left \langle \sum_{i} \delta({\bf r}+{\bf r}_i(0)-{\bf r}_i(t))\right \rangle \nonumber \\ G_d({\bf r},t)&=&\frac{1}{N}\left \langle \sum_{i}\sum_{j \neq i} \delta({\bf r}+{\bf r}_j(0)-{\bf r}_i(t))\right \rangle \nonumber\end{aligned}$$ As we shall see, experiments usually give information on the correlation functions in the reciprocal, $Q$, space. Therefore, it can be useful to define the space Fourier transform of the microscopic quantities previously introduced: $$\begin{aligned} &&\rho({\bf Q},t)=\frac{1}{\sqrt N} \sum_{i} e^{-i{\bf Q} \cdot {\bf r}_i(t)} \nonumber \\ && {\bf J}({\bf Q},t)=\frac{1}{\sqrt N} \sum_{i} {\bf v}_i(t) e^{-i{\bf Q} \cdot {\bf r}_i(t)} \nonumber \\ && E({\bf Q},t)=\frac{1}{\sqrt N} \sum_{i} \frac{1}{2}m v^2_i(t) e^{-i{\bf Q} \cdot {\bf r}_i(t)} \nonumber\end{aligned}$$ and similarly for the single particle variables. The time evolution of these microscopic variables is cross-linked and can be studied by replacing them with their average over small but statistically significant volumes. A closed set of equations, basically conservation laws and constitutive relations, can easily be written for these averages [^1]. Once the expression for $a(\mathbf{Q},t)$ is known ($a=\rho,\mathbf J, E$) one can calculate $$\Phi_{\alpha \beta}(Q,t)=\left \langle a_\alpha^*({\bf Q}) e^{\mathcal{L}t} a_\beta({\bf Q}) \right \rangle_N \label{cf}$$ In which $a_\alpha({\bf Q})$ are the appropriate dynamical variables, and $\mathcal{L}$ is the Liouville operator ruling the time evolution in the configurational space. The $\langle ... \rangle_N$ indicates thermal averages evaluated over the $N$-particle ensemble (from now on we will omit the subscript $N$). In particular, the autocorrelation functions of the microscopic density (both self and collective) play a privileged role, and we define, therefore: $$\begin{aligned} && F(Q,t)=\langle \rho (Q,t)\rho(-Q,0) \rangle \label{fqt}\\ && S(Q)=\langle |\rho(Q,0)|^2 \rangle \nonumber \\ && \Phi(Q,t)=\Phi_{11}(Q,t)=\frac{F(Q,t)}{S(Q)} \nonumber \\ && F_s(Q,t)=\langle \rho_s (Q,t)\rho_s(-Q,0) \rangle \nonumber \\ &&\Phi_s(Q,t)=F_s(Q,t) \nonumber\end{aligned}$$ Consequently $$\begin{aligned} \Phi(Q,t)&=&\frac{1}{N S(Q)} \sum_{i,j} \left \langle e^{i{\bf Q} \cdot {\bf r}_i(0)}e^{-i{\bf Q} \cdot {\bf r}_j(t)} \right \rangle \nonumber \\ &=&\frac{1}{N S(Q)} \sum_{i,j} \left \langle e^{i{\bf Q} \cdot {\bf r}_i(0)}e^{\mathcal{L}t}e^{-i{\bf Q} \cdot {\bf r}_j(0)} \right \rangle \label{daf} \\ \Phi_s(Q,t)&=&\frac{1}{N} \sum_{i} \left \langle e^{i{\bf Q} \cdot {\bf r}_i(0)}e^{-i{\bf Q} \cdot {\bf r}_i(t)} \right \rangle \label{daf_s}\end{aligned}$$ The above autocorrelation functions $F(Q,t)$ and $F_s(Q,t)$ are, in fact, connected through their time-Fourier transform to the self and collective dynamic structure factor $S_s(Q,\omega)$ and $S(Q,\omega)$, respectively. These latter, in turn, are the experimentally accessible quantities in neutrons and X-rays inelastic scattering experiments and therefore, in the following, we will mainly refer to the density-density correlation functions. ### Spectral moments Before illustrating some models for the evolution of the density autocorrelation function, it is worth to recall here some basic relations involving the frequency moments of the dynamic structure factor. These can be very useful to the experimentalist, as a way to normalize the data (an example is given in section \[sec\_fetdq\], as well as to any theoretical approach, as a direct test of sum rules - for example, we shall see how the hydrodynamic expression for the density-density time correlation function is valid up to the second frequency moment, because in hydrodynamics the liquid is treated as continuum without atomic structure, and such an information on structure and interatomic potentials appears only within the fourth and higher frequency moments of $S(Q,\omega)$. By expanding the density autocorrelation function in a Taylor series, one can easily find a connection with the frequency moments of the dynamic structure factor as: $$\frac{d^n F(Q,t)}{dt^n} \Bigg |_{t=0} = (-i)^n \int_{-\infty}^{+\infty} \omega^n S(Q,\omega) d\omega=(-i)^n\langle \omega^n \rangle$$ This relation holds for the frequency moments of both the collective ($\langle \omega^n \rangle_S$) and the self ($\langle \omega^n \rangle_{S_s}$) dynamic structure factor. From the previous definitions, it easily follows that $\langle \omega^0 \rangle_S=S(Q)$ and $\langle \omega^0 \rangle_{S_s}=1$. The first frequency momentum are, on the other side, $\langle \omega^1 \rangle_S=\langle \omega^1 \rangle_{S_s}=\frac{\hbar Q^2}{2m}$ and therefore are zero for any classical theory, characterized by symmetric spectral functions. The second frequency moments are $\langle \omega^2 \rangle_S=\langle \omega^2 \rangle_{S_s}=\frac{k_B T Q^2}{m}+O(\hbar ^2)$. Higher order spectral moments depend on the details of the microscopic interactions, and can be analytically derived for additive pairwise interatomic potential [@dej_ema]. Some examples will be given in sections \[sec\_mfs\] and \[sec\_mf\], while pratical usage of sum rules for normalization purposes will be outlined in section \[sec\_fetdq\]. ### Quantum aspects The models that we will illustrate in the next sections have been developed for a classical system. The main effect of quantum-mechanical corrections stems from the well-known inequality of the positive and negative-frequency parts of the spectra, connected by the detailed balance factor $e^{\hbar \omega / k_B T}$. Additional sources of non-classical behavior, such as those associated with a finite value of the de Broglie wavelength $\Lambda =\left( 2\pi \hbar ^2/mk_BT\right) ^{1/2}$, are small (for lithium at melting $\Lambda $ is only $0.11$ times the average interparticle distance, and this ratio decreases for heavier metals) and can safely be neglected. Since the effects of the detailed balance are clearly visible in the experimentally measured dynamic structure factors, we briefly discuss a possible procedure to account for this quantum feature in a consistent way, while preserving the inherent advantages of the classical description. In doing this, for the sake of clarity we shall denote all the previous classical quantities with the subscript [*cl*]{}, while the notation [*q*]{} will refer to the quantum case. The natural theoretical counterpart of the classical density correlation function is the so called Kubo canonical relaxation function [@kubo_quantum] $$K_q(Q,t)=\frac 1{\beta N}\sum_{i,j}\int_0^\beta d\lambda \left\langle e^{-i{\bf Q\cdot }\widehat{{\bf r}}_i(0)}e^{-\lambda \widehat{H}}e^{i{\bf Q\cdot }\widehat{{\bf r}}_j(t)}e^{\lambda \widehat{H}}\right\rangle \label{kubo}$$ where $\beta =1/k_BT$ and the angular brackets denote now a quantum statistical average. In the classical limit ($\beta \rightarrow 0,$ $\hbar \rightarrow 0$) the quantum operators become classical commuting dynamical variables and $K_q(Q,t)\rightarrow F_{cl}(Q,t).$ It can be shown [@lov_visco] that $K_q(Q,t)$ is a real even function of time, so that its spectrum $K_q(Q,\omega )$ is an even function of frequency. On the other hand, the experimental scattering cross section involves the Fourier transform $S_q(Q,\omega )$ of the quantum density correlator $F_q(Q,t)=(1/N)\sum_{i,j}\left\langle e^{-i{\bf Q\cdot }\widehat{{\bf r}}_i(0)}e^{i{\bf Q\cdot }\widehat{{\bf r}}_j(t)}\right\rangle .$ The relation between $S_q(Q,\omega )$ and $K_q(Q,\omega )$ reads [@LOVESEY] $$S_{q}(Q,\omega )=\frac{\beta \hbar \omega }{1-e^{-\beta \hbar \omega }}K_{q}(Q,\omega )$$ and, as can be easily checked, satisfies the detailed balance condition. Moreover, it can be seen that the relation $$\langle \omega^{2n}\rangle _K=\frac 2{\beta \hbar }\langle \omega^{2n-1}\rangle _S \label{dispari}$$ connects the even frequency moments of $K_q$ with the odd ones of $S_q.$ In addition to that, the same memory function framework which will be outlined in sections \[sec\_mfs\] and \[sec\_le\] can be phrased for the Kubo relaxation function and for its Laplace transform $\widetilde{K}_q(Q,s)$ By virtue of all these properties, in a situation where the quantum aspects not associated with detailed balance are marginal, it is reasonable (although not strictly rigorous) to identify the spectrum $K_q(Q,\omega )$ with the classical quantity $S_{cl}(Q,\omega )$ so that $$S_{q}(Q,\omega )\simeq \frac{\beta \hbar \omega }{1-e^{-\beta \hbar \omega }}S_{cl}(Q,\omega ) \label{squantclass}$$ Having assumed such a correspondence, from now on we will drop out the subscript [*cl*]{} and refer to the classical quantities as in fact done at the beginning of this section. The transformation (\[squantclass\]) allows one to test classical models against experimental data. It is worth to point out, however, that it alters the frequency moments: as shown in the previous section, for instance, it introduces a $\hbar ^2$ correction to the second frequency moment, though this effect has been shown to be hardly noticeable in liquid metals [@scop_jpc]. Single particle dynamics in the hydrodynamic regime {#sec_selfhydro} --------------------------------------------------- The time evolution of the single particle density can be easily obtained through the continuity equation and the constitutive relation (Fick’s law) relating density and current variables: $$\begin{aligned} &&\dot{\rho}_s({\bf r},t)+ \nabla \cdot {\bf J}_s({\bf r},t)=0 \nonumber \\ &&{\bf J}_s({\bf r},t)=-D\nabla \rho_s({\bf r},t) \nonumber\end{aligned}$$ It is worth to stress that while the first equation is exact, the second is a phenomenological “closure”. Combining the two equations, one gets the diffusion equation straightforward: $$\dot{\rho}_s({\bf r},t)=D\nabla ^2 \rho_s({\bf r},t) \nonumber$$ which, in the reciprocal space, has the solution $$\rho_s(Q,t)=\rho_s(Q)e^{-DQ^2t} \nonumber$$ The normalized autocorrelation function of the single particle density is now obtained as $$\Phi_s(Q,t)=\langle \rho_s(Q,t)\rho_s(-Q) \rangle=e^{-DQ^2t} \nonumber$$ while its fourier transform, the self dynamic structure factor, will read $$S_s(Q,\omega)=\frac{1}{\pi} \frac{DQ^2}{\omega^2+(DQ^2)^2} \label{fick}$$ i.e. a Lorenzian function centered at $\omega=0$ with FWHM equal to $2DQ^2$. It is worth to point out how, in the hydrodynamic limit, the diffusion coefficient is related to the dynamic structure factor as $D=\lim_{Q\to 0}\frac{\omega^2}{Q^2}\pi S_s(Q,\omega)$. Finally, for completeness, the corresponding Van-Hove self correlation function is: $$G_s(r,t)=\frac{1}{(4\pi Dt)^{3/2}} e^{-r^2/4Dt} \label{Gs_hydro}$$ Collective dynamics in the hydrodynamic regime {#sec_collhydro} ---------------------------------------------- In a similar manner as in the previous section, one can build again a set of closed equations but, in this case, Fick’s law does not apply, and the situation is more involved. The constitutive relations, indeed, couple together the three conservation laws for the microscopic variables density, momentum and energy, which, in terms of the correspondent fluxes reads: $$\begin{aligned} &&\dot{\rho}({\bf r},t)+ \nabla \cdot {\bf J}({\bf r},t)=0 \label{3idro} \\ &&\dot{\bf J}({\bf r},t)+ \nabla \cdot {\bf \sigma}({\bf r},t)=0 \nonumber \\ &&\dot{E}({\bf r},t)+ \nabla \cdot {\bf H}({\bf r},t)=0 \nonumber\end{aligned}$$ were we have defined the momentum flux $\mathbf \sigma (\mathbf r ,t)$ and the energy flux $\mathbf H(\mathbf r ,t)$ $$\begin{aligned} \sigma_{\alpha, \beta}(\mathbf r ,t)&=&\delta_{\alpha, \beta}P(\mathbf r ,t)-\eta \left ( \frac{\partial u_\alpha (\mathbf r ,t)}{\partial r_\beta} + \frac{\partial u_\beta (\mathbf r ,t)}{\partial r_\alpha} \right ) \nonumber \\ &+& \delta_{\alpha, \beta}(\frac{2}{3}\eta-\xi) \mathbf \nabla \cdot \mathbf u(\mathbf r ,t) \nonumber \\ \mathbf H(\mathbf r ,t)&=& h \mathbf u(\mathbf r ,t)-\kappa \mathbf \nabla T(\mathbf r ,t) \nonumber\end{aligned}$$ here $\eta$ and $\xi$ are the shear and bulk viscosities, $P$ and $T$ are the local pressure and temperature fields, $h$ is the enthalpy density and $\kappa$ is the thermal conductivity. Two additional constitutive relations (the Navier Stokes equation and the Fourier law) call into play the two additional thermodynamic variables pressure $P$ and temperature $T$. Invoking the thermal equilibrium and the equations of state one obtains a closed set of equations which can be solved to get the density density correlation function. The detailed derivation of $\rho(Q,t)$ is beyond the purpose of the present review, and can be easily retrieved in classical textbooks [@BERNE; @HANSEN]. Here we will only recall the final *approximate* results [^2] which are: $$\begin{aligned} &&\frac{\rho(Q,t)}{\rho(Q)}=\left[ \left( \frac{\gamma-1}{\gamma} \right ) e^{-D_T Q^2t} + \frac{1}{\gamma}e^{-\Gamma Q^2 t} \cos c_sQt \right] \nonumber \\ &&\frac{S(Q,\omega)}{S(Q)}=\frac{1}{2\pi}\left[ \left( \frac{\gamma-1}{\gamma} \right ) \frac{2D_TQ^2}{\omega^2+(D_TQ^2)^2}\right] \nonumber \\ && + \frac{1}{\gamma}\left[ \frac{\Gamma Q^2}{(\omega+c_sQ)^2+(\Gamma Q^2)^2} + \frac{\Gamma Q^2}{(\omega-c_sQ)^2+(\Gamma Q^2)^2} \right] \nonumber \\ \label{S_hidro}\end{aligned}$$ having defined $$\begin{aligned} &&\gamma=\frac{c_P}{c_V} \nonumber \\ &&D_{T}=\frac{\kappa}{\rho m C_{P}} \nonumber \\ &&\Gamma=\frac{1}{2\rho m}\left[ \frac{4}{3}\eta_s + \eta_B +\frac{(\gamma-1)\kappa}{c_P} \right] \label{gidro}\end{aligned}$$ In the above expressions $c_P$ and $c_V$ are the specific heat ratios at constant pressure and volume, $\eta_s$ and $\eta_B$ are the shear and bulk viscosities, $\kappa$ is the thermal conductivity and $c_s=\sqrt{\frac{\gamma}{m}\left (\frac{\partial P}{\partial \rho}\right )}_T$ is the adiabatic sound velocity. The classical hydrodynamics, therefore, predicts in the long wavelength limit ($Q \rightarrow 0$) a frequency spectrum for the density fluctuations constituted by two main features. The central part of the spectrum is dominated by a quasi-elastic, non propagating mode related to entropy fluctuations (Rayleigh component) of linewidth $\Gamma_{qe}=2D_T Q^2$, which reflects the fact that thermal fluctuations decay over a finite lifetime $\tau=2/\Gamma_{qe}$. Beside, two symmetrically shifted inelastic components peaked at frequency $\omega_s=\pm c_s Q$ are the signature of propagating pressure waves (Brillouin doublet), which are damped by a combination of viscous and thermal effects. The ratio between the Rayleigh and the Brillouin component is given by the Landau-Placzeck ratio $$\frac{I_R}{2 I_B}=\gamma -1 \label{lp}$$ Usually, the hydrodynamic regime is investigated by visible light scattering spectroscopy (BLS, Brillouin Light Scattering). In the case of liquid metals, the light scattering study of density fluctuations, is prevented by the non-transparent nature of these systems [^3]. Inelastic scattering experiment can, in fact, only be performed by means of higher energy photons (X-rays) or by neutrons, but in both cases the probed wavevectors are fairly outside the strict hydrodynamic region. The way the Brillouin triplet evolves at finite $Q$ is far from being fully understood, though some simplified phenomenological models have been proposed in the past [@mcg_hyd]. Primarily, one should account for the frequency dependence of the transport coefficients which, as we shall see, corresponds to abandon the hypothesis of a Markovian dynamics. Second, once the wavevector approaches the inverse interparticle distances, structural effects are expected in the form of a $Q$ dependence of all the thermodynamic quantities. Last, but not least, it is highly questionable whether the role of the thermal and viscous processes remains well separated at high $Q$. In particular, specifically in the case of liquid metals, due to the high thermal conductivity one expects $D_TQ^2$ to become soon of the order of the brillouin frequency, so that entropy and density fluctuations become closely interwove. Strangely enough, this aspect, with a few exceptions [@FABER] did not receive much attention in the past, although lately it has been the matter of some debate [@sing_pre; @scop_comm; @sing_rep]. Before coming in the discussion of the evolution of $S(Q,\omega)$ at increasing $Q$ values, above the hydrodynamic limit, it is worth to discuss another analytically solvable case: the high $Q$ limit. The short wavelength limit \[sec\_swl\] --------------------------------------- In the previous sections we examined the self and collective motion at very small wavevectors and frequencies. In the opposite regime, i.e. at short distances and timescales, the particles of a fluid are expected to move as they were free, as it happens in an ideal gas. Since in this case the behaviors of different particles are uncorrelated ($G_d(r,t)=0$), the self and collective dynamic structure factors coincides in this limit. The correlation function of Eq. (\[fqt\]) can be easily calculated for a free particle, i.e. for ${\bf r}_i(t)={\bf r}_i(0)+{\bf v}_i t$. The classical free particle correlation function is: $$F(Q,t)=\frac{1}{N} \sum_i \left\langle e^{-i{\bf Q} \cdot \frac{{\bf p}_i}{m_i} t} \right\rangle=\left\langle e^{-i{\bf Q} \cdot \frac{\bf p}{m} t} \right\rangle \label{fqt_free_class}$$ evaluating the thermal average one easily gets $$\begin{aligned} F(Q,t)=\int e^{-\frac{p^2}{2mk_BT}} e^{-i Q p m t} dp = e^{-\frac{k_B T Q^2 t^2}{2m}}\end{aligned}$$ and the corresponding dynamic structure factor is $$S(Q,\omega)=\sqrt{\frac{m}{2\pi k_B T Q^2}}e^{-\frac{m\omega^2}{2 k_B T Q^2}} \label{S_fp_cl}$$ By virtue of the previously mentioned considerations, the Van-Hove self correlation function reads: $$G_s(r,t)=\left ( \frac{m}{2\pi k_B T t^2} \right )^{3/2} e^{-\frac{mr^2}{2 k_B T t^2}} \label{Gs_single}$$ In the quantum case the correlation function (\[fqt\_free\_class\]) can be evaluated treating ${\bf r}_i$ and ${\bf p}_i$ as operators, and paying attention to the fact that, in this case, the product of the exponential in Eq.(\[fqt\]) can not be reduced to a single exponential, as in the classical treatment. Invoking the identity $e^{\hat{A}} e^{\hat {B}}=e^{\hat{A} + \hat{B} + \frac{1}{2} \left [ \hat{A},\hat{B} \right ]}$, holding when ,as in the present case, $\left [ \hat{A},\hat{B} \right ]$ is a number, one can write: $$\begin{aligned} &&\left \langle e^{-i{\bf Q}\cdot {\bf r}_i(0)}e^{i{\bf Q}\cdot {\bf r}_i(t)}\right\rangle = \nonumber \\ &&\;\;\;\left\langle e^{i{\bf Q}\cdot \left ({\bf r}_i(t)-{\bf r}_i(0) \right )+\frac{{\bf Q}^2}{2} \left [{\bf r}_i(0),{\bf r}_i(t) \right ]}\right\rangle = \;\;\;\;\;\;\;\; \nonumber \\ &&\;\;\;\left\langle e^{i{\bf Q} \cdot \frac{{\bf p}_i}{m} t-\frac{{\bf Q}^2}{2} \left [\frac{{\bf p}_i}{m} t,{\bf r}_i(t) \right ]} \right\rangle = \nonumber \\ &&\;\;\; e^{i\hbar\frac{Q^2t}{2m}}\left\langle e^{-i{\bf Q} \cdot \frac{\bf p}{m} t} \right\rangle \nonumber\end{aligned}$$ and, using the result of Eq.(\[fqt\_free\_class\]), one obtains: $$\begin{aligned} F(Q,t)=e^{-\frac{Q^2}{2m} \left(k_B T t^2-i \hbar t \right ) }\end{aligned}$$ the correspondent dynamic structure factor: $$S(Q,\omega)=\sqrt{\frac{m}{2\pi k_BTQ^2}}e^{-\frac{m}{2k_BTQ^2} \left(\omega-\frac{\hbar Q^2}{2m} \right )^2 } \label{sqw_free_quant}$$ Summing up, the quantum dynamic structure factor for a free moving particle is a gaussian with recoil energy $\omega_R(Q)=\frac{\hbar Q^2}{2m}$ and linewidth $\sigma=\sqrt {\frac{k_BT}{M}}Q$. It is worth to point out that Eq.(\[sqw\_free\_quant\]) satisfies the detailed balance condition $S(Q,\omega)=e^{\frac{\hbar \omega}{k_B T}}S(Q,-\omega)$ and coincides with the classical case for $\hbar \rightarrow 0$ or, equivalently, for $T \rightarrow \infty$. As far as the sound velocity is concerned, one can still define the apparent frequencies $\omega_l(Q)=\frac{1}{2}(\omega_R \pm \sqrt{8\sigma ^2 + \omega_R^2})$ as the positive and negative maxima of the longitudinal current $C_L(Q,\omega)=\frac{\omega^2 S(Q,\omega)}{Q^2}$. In the classical case, the two values coincides and are $\omega_l(Q)=\sqrt{2}\sigma$: The non-hydrodynamic region: single particle {#sec_selfnonhydro} -------------------------------------------- ### The gaussian approximation \[sec\_gaussapp\] As can be easily noticed looking at the expressions (\[Gs\_hydro\]) and (\[Gs\_single\]), both the hydrodynamic and the ideal gas limit end up with the Van Hove correlation functions that are gaussian in $r$. On the basis of this observation seems natural to assume the gaussian dependence as valid in the whole dynamical range. In terms of the second moment of $G_s(r,t)$ one can write the following expression: $$G_s(r,t)=\sqrt{\frac{3}{2\langle r^2(t) \rangle}} e^{-\frac{3r^2}{2\langle r^2(t) \rangle}} \label{Gs_gauss}$$ where $\langle r^2(t) \rangle$ is the mean square displacement which, in the hydrodynamic and single particle approximations reads $\langle r^2(t) \rangle=6Dt$ and $\langle r^2(t) \rangle=\frac{3KT}{m}t^2$, respectively. In the gaussian approximation, therefore, the self scattering function is related to the mean square displacement which can be, for instance, inferred by molecular dynamics simulations. ### The jump diffusion model \[sec\_jd\] The jump diffusion model was firstly introduced by Chudley and Elliot [@chud_jd]. The particle is thought to live for a residence time $\tau_0$ in the cage of its neighbors, and at some point to change cage. In some sense, therefore, it is the opposite of collisional models, where the free diffusion of a particle is sometimes interrupted by collisional events. The jump diffusion model sets a rate equation for the Van-Hove self scattering function of the kind: $$\begin{aligned} \frac{\partial G_s(\mathbf r,t)}{\partial t}=\frac{1}{\mathcal{N} \tau_0} \sum_{\mathbf l} G_s(\mathbf {r+l},t)-G_s(\mathbf r,t) \nonumber\end{aligned}$$ with $\mathcal{N}$ the number of available residence sites. By fourier transform in space and time one immediately gets: $$S_s(Q,\omega)=\frac{1}{\pi}\frac{f(Q)}{\omega^2+f(Q)^2} \label{jd}$$ which is a lorenzian function with $Q-$ dependent damping $$f(Q)=-\frac{1}{n\tau_0}\sum_{\mathbf l}(e^{i\mathbf Q \cdot \mathbf l}-1) \label{jd_width}$$ This latter can be conveniently estimated supposing that the vectors $\mathbf {l}$ have random and continuous orientations and distributions. In this case one can average Eq.(\[jd\_width\]): $$\begin{aligned} f(Q)=-\frac{1}{\tau_0}\left [ 1-\frac{1}{1+Q^2l_0^2} \right ] \nonumber\end{aligned}$$ It can be easily shown that Eq.(\[jd\]) tends to the Fick’s free diffusion expression (\[fick\]) [@EGELSTAFF] ### The mode coupling theory \[sec\_mct\] The Laplace transform of the intermediate scattering function can be generally written as: $$\begin{aligned} \tilde{S}_s(Q,s)=\frac{1}{s+Q^2\tilde{U}(Q,s)} \nonumber\end{aligned}$$ being $\tilde{U}(Q,s)$ a generalized frequency and wavevector dependent diffusion coefficient. The mode coupling theory provides a self consistent expression for $\tilde{U}_s(Q,s)$ [@desh_mc], and the resulting self dynamic structure factor reads: $$S_s(Q,\omega)\approx \frac{1}{\pi} \frac{DQ^2}{\omega^2+(DQ^2)^2} +\frac{1}{\pi D Q Q^*}\mathrm{Re}G\left( \frac{i\omega+DQ^2}{\delta DQ^2} \right) \label{mct1}$$ with $$G(s)=arctan \left(\frac{1}{\sqrt{s-1}} \right) - \frac{(s-2)\sqrt{s-1}}{s^2} \label{mct2}$$ where $Q^*=16\pi m \rho D^2 / k_BT$ and $\delta=D/(D+\nu)$, being $D$ the diffusion coefficient and $\nu$ the kinematic viscosity. An estimate of the FWHM can be numerically evaluated [@dej_phd], yielding: $$\Delta \omega \approx \left [1-\frac{Q}{Q^*}H(\delta)+O(Q^{3/2})\right ]DQ^2 \label{mct3}$$ with $$H(\delta) \approx 1.45\delta^{3/2}\left [1-0.73\delta-0.15\delta^2-O(\delta ^3)\right ] \label{mct4}$$ ### The Nelkin-Ghatak model \[sec\_ng\] Nelkin and Ghatak have considered a dilute gas in which the atomic motion is dominated by binary collisions, with a distribution function obeying a linearized Boltzman’s equation, valid in a small disturbance limit, i.e. for arbitrary large fluctuations compared to the mean collision time. In terms of the reduced variables $x=-\omega / Qv_0$ and $y=\alpha / Qv_0$, with $v_0=\sqrt{2KT/m}$ and $\alpha$ an adjustable parameter, and by introducing the real ($u(x,y)$) and imaginary ($v(x,y)$) parts of the probability integral for complex argument $z(x+iy)=\int_{-\infty}^{+\infty} e^{-t^2}(z-t)^{-1}dt$ $$\begin{aligned} U(x,y)=\sqrt \pi y u(x,y) \\ V(x,y)=\sqrt \pi y v(x,y)\end{aligned}$$ one gets the following expression for the incoherent scattering function [@nelkin_inco]: $$S^{NG}_s(Q,\omega)=\frac{1}{\pi \alpha}\frac{U(1-U)-V^2}{(1-U^2)+V^2} \label{NG}$$ it can be easily noticed that Eq.(\[NG\]) has the correct low $Q$ (lorenzian) and high Q (gaussian) limits: in the first case it is sufficient to pose $\alpha=v_0^2/2D$, while at high $Q$’s one has that $y\rightarrow 0$ and the familiar gaussian shape is recovered. ### The memory function formalism \[sec\_mfs\] The easiest way to abandon the hydrodynamic region is to assume the frequency dependence of the transport coefficients, entering the so called generalized hydrodynamics. The natural playground for performing such step is the memory function framework [@mori_mf]: we will recall here the basic formalism while a detailed treatment can be found in specialized books [@BALUCANI; @HANSEN]. Let ${\bf M}^{(0)}$ be the correlation matrix of a given set of dynamical variables $\bf A$ ($M^{(0)}_{\nu \sigma}=\langle A_\nu ^* A_\sigma (t) \rangle $). The equation of motion of ${\bf M}^{(0)}(t)$ can be conveniently expressed in terms of a chain of arbitrary order $n$ of integro-differential equation involving appropriate memory functions ${\bf M}^{(i)}(t)$ for $i=1...n$: $$\frac{d{\bf M}^{(i-1)}}{dt}-i {\bf \Omega}^{(i-1)} {\bf M}^{(i-1)} + \int_0^t {\bf M}^{(i)}(t-t') {\bf M}^{(i-1)}(t') dt'=0 \nonumber \label{memory}$$ with $$i{\bf \Omega}^{(i-1)}={\bf \dot M}^{(i-1)}(0) \cdot \left [{\bf M}^{(i-1)}(0)\right ]^{-1} \label{frequency}$$ here ${\bf \Omega}^{(i-1)}$ is a set of generalized frequency matrixes, while the memory kernels ${\bf M}^i(t)$ rule the dynamical evolution of the observables correlation matrix ${\bf M}^{(0)}$. In the specific case of self dynamics, as we have seen in section (\[sec\_selfhydro\]) the relevant set of variable is given by the self density only, therefore: $M^{(0)}(t)=\phi_s (t)$. The equation of motion for the density correlation function will be: $$\frac{d\phi_s(Q,t)}{dt} + \int_0^t M^{(1)}(Q,t-t') \phi_s(Q,t') dt'=0 \label{self_memory} \nonumber$$ being $\Omega^{(0)}=0$ due to the orthogonality of $\rho_s$ and $\dot{\rho_s}$. A good description of the evolution of $\phi_s(Q,t)$ is normally gained utilizing the first two equations of the chain (\[memory\]). In terms of Laplace trasform it holds: $${\tilde \phi}_s(Q,t)=\left [s+{\tilde M}^{(1)}(Q,s) \right]^{-1} =\left [s+\frac{M^{(1)}(Q,t=0)}{s+{\tilde M^{(2)}}(Q,s)} \right ]^{-1} \label{phis_laplace}$$ The initial values ${\bf M}^{(i)}(t=0)$, can be easily obtained from the general relation: $${\bf M}^{(i)}(0)=-{\bf \ddot M}^{(i-1)}(0) \cdot \left [{\bf M}^{(i-1)}(0)\right ]^{-1}-\left [{\bf \Omega}^{(i-1)} \right]^2$$ which, in turns, is obtained deriving Eq.(\[memory\]) and exploiting Eq.(\[frequency\]). For the first two memory function it holds therefore: $$\begin{aligned} &&M^{(1)}(Q,0)=\frac{k_BTQ^2}{m}=\langle \omega^2(Q) \rangle _{S_{s}} \nonumber \\ &&M^{(2)}(Q,0)=2\langle \omega^2(Q) \rangle _{S_{s}}+\Omega_0^2=\frac{\langle \omega^4(Q) \rangle _{S_{s}}}{\langle \omega^2(Q) \rangle _{S_{s}}}-\langle \omega^2(Q) \rangle _{S_{s}} \nonumber \\ \label{initial_Ms}\end{aligned}$$ Here $\langle \omega^n(Q) \rangle _{S_{s}}$ are the frequency moments of $S_{s}(Q,\omega)$ and the quantity $\Omega_0^2$ is related to the mean squared force $\langle |{\bf F}|^2 \rangle$ acting on the diffusing particle and, for a system of identical particle interacting via pairwise interactions potential $V(r)$, it holds: $$\Omega_0^2=\frac{\langle |{\bf F}|^2 \rangle }{3mk_BT} =\frac{\rho}{3m}\int \nabla^2 V(r)g(r)d{\bf r} \label{mom4_s}$$ where $g(r)$ is the pair distribution function. It is worth to stress that Eq. (\[phis\_laplace\]) is the exact solution of motion in which all the dynamics is detailed by the shape of the second order memory function $M^{(2)}(Q,t)$. The most common way of solving the equation is making a guess on the shape of the memory function. A useful approximation is provided by the exponential shape, which has the advantage of being easily Laplace transformed: $$M^{(2)}(Q,t)=\Delta^2_s(Q)e^{-t/\tau_s(Q)}=\left [ 2\langle \omega^2(Q) \rangle _{S_{s}}+\Omega_0^2 \right] e^{-t/\tau_s(Q)}$$ With such a choice it follows straightforward: $$S_s(Q,\omega )=\frac{1}\pi \frac{\langle \omega^2 \rangle _{S_{s}} ( 2\langle \omega^2 \rangle _{S_{s}}+\Omega_0^2) \tau_s} {\omega^2 \tau_s (\omega ^2-3\langle \omega^2 \rangle _{S_{s}}-\Omega_0^2) ^2+( \omega^2-\langle \omega^2 \rangle _{S_{s}})^2} \label{sqw_s_1t}$$ It is interesting now to look at the FWHM $\Gamma_s(Q)$ of Eq.(\[sqw\_s\_1t\]). In the small $Q$ limit it is easy to show that $$\frac{\Gamma_s(Q)}{DQ^2}=\frac{1}{\sqrt {1+\frac{2k_BT}{m\Omega_0^2}Q^2}}$$ The memory function approach with exponential kernel, therefore, predicts a quasielastic lineshape narrower then the hydrodynamic one, a result which is in agreement with several experimental data. Contrarily, in the gaussian approximation the linewidth is always larger than the hydrodynamic value. Theoretical expressions have been proposed in the past for the memory function in terms of kinetic theory, splitting the memory function in a contribution due to uncorrelated binary collision, obtained by a Fokker-Plank equation, and a long time contribution representing the coupling of a tagged particle to the collective motion of the surrounding particles. This approach have been tested in hard spheres, Lennard-Jones and alkali metals [@sjog_kin1; @sjog_sw; @beng_kinhs] The non-hydrodynamic region: collective motion \[sec\_collnonhydro\] -------------------------------------------------------------------- ### The Langevin equation \[sec\_le\] The most reliable approach to the study of collective dynamics at finite wavevectors parallels the one adopted for the single particle in section \[sec\_selfnonhydro\], i.e. an extension of the classical hydrodynamics assisted by the formalism of the memory function ruling the Langevin equation of motion of the density fluctuations. We will deal, therefore, with a $3 \times 3$ correlation matrix ${\bf M}^{(0)}$, and a set of $n$ (with $n$ arbitrary large) memory matrixes ${\bf M}^{(i)}$ with the same dimensionality and coupled by the chain of $n$ equations (\[memory\]). Actually, in order to work with a orthogonal set of variables (and energy and density are not), one normally prefers to replace the microscopic energy with the microscopic temperature $T(Q)=\frac{1}{m \rho c_V(Q)}\left [ E(Q)-\frac{\left \langle E ^*(Q) \rho(Q) \right \rangle }{\left \langle \rho ^*(Q) \rho(Q) \right \rangle }\rho(Q) \right ]$. With this choice, solving for $\phi(Q,t)=M^{(0)}_{\rho \rho}(Q,t) / M^{(0)}_{\rho \rho}(Q,0)=F(Q,t)/S(Q)$ in terms of Laplace transform one has: $${\tilde \phi}(Q,s)=\frac{1}{s+\frac{\omega_0^2(Q)}{s+{\tilde M^{(1)}}_{JJ}(Q,s)-\frac{({\tilde M^{(1)}}_{JT}(Q,s)-i \Omega^{(0)}_{JT})({\tilde M^{(1)}}_{TJ}(Q,s)-i \Omega^{(0)}_{TJ})}{s+{\tilde M^{(1)}}_{TT}(Q,s)}}} \label{coll_contfrac2}$$ where $$\omega _0^2(Q)=k_BTQ^2/mS(Q) \label{a2}$$ Recalling the definition of the isothermal sound velocity $c_t=\sqrt{1/\rho m \chi _T}$, and the expression for the low $Q\rightarrow 0$ limit of the static structure factor $S(0)=\rho \chi _T k_B T$ it seems natural to introduce a finite $Q$ generalization of the isothermal sound velocity as $c_t(Q)=\omega_0(Q)/Q$. It is easy to recognize in Eq. (\[coll\_contfrac2\]) the same structure of Eq.(\[phis\_laplace\]), once the following identification is made: $$\begin{aligned} \tilde{M}^{(eff)}_{\rho \rho}(Q,s)&=&{\tilde M^{(1)}}_{JJ}(Q,s) \nonumber \\ &-& \frac{({\tilde M^{(1)}}_{JT}(Q,s)-i \Omega^{(0)}_{JT}(Q))({\tilde M^{(1)}}_{TJ}(Q,s)-i \Omega^{(0)}_{TJ}(Q))}{s+{\tilde M^{(1)}}_{TT}(Q,s)} \nonumber\end{aligned}$$ The effective memory function $\tilde{M}^{(eff)}_{\rho \rho}(Q,t)$, therefore, can be formally considered as the second order memory function of a chain of two equations for the single variable $\rho(Q,t)$: $$\begin{aligned} &&\frac{dF(Q,t)}{dt} + \int_0^t M^{(1)}(Q,t-t') F(Q,t') dt'=0 \\ &&\frac{dM^{(1)}(Q,t)}{dt} + \int_0^t M^{(2)}(Q,t-t') M^{(1)}(Q,t') dt'=0 \label{collective_chain2} \nonumber\end{aligned}$$ which, as can be easily verified, corresponds to the single second order integro-differential equation: $$\begin{aligned} \stackrel{..}{\phi}(Q,t)&&+\omega _0^2(Q)\phi(Q,t) \\ &&+\int_0^tM(Q,t-t^{\prime })\stackrel{.}{\phi}(Q,t^{\prime })dt^{\prime }=0 \label{langevin} \nonumber\end{aligned}$$ From here on, to save writing, we define $M(Q,t)=M^{(2)}(Q,t)=\tilde{M}^{(eff)}_{\rho \rho}(Q,t)$. From the knowledge of $\widetilde{\phi}(Q,s)$ one straightforwardly obtains $S(Q,\omega )=[S(Q)/\pi] \Re \{ \widetilde{\phi}(Q,s=i\omega ) \}$ in terms of the real ($M^{\prime }$) and imaginary ($-M^{\prime \prime }$) parts of the Fourier-Laplace transform of the memory function: $$S(Q,\omega )=\frac{S(Q)}\pi \frac{\omega _0^2(Q)M^{\prime }(Q,\omega )}{\left[ \omega ^2-\omega _0^2-\omega M^{\prime \prime }(Q,\omega )\right] ^2+\left[ \omega M^{\prime }(Q,\omega )\right] ^2} \label{sqwgenerale}$$ The spectral features of the dynamic structure factor can be characterized by its frequency moments $\langle \omega^n (Q) \rangle _S\equiv \int \omega ^nS(Q,\omega )d\omega $, where, for a classical system, only the even frequency moments (such as $\langle \omega^0 (Q) \rangle _S=S(Q)$ and $\langle \omega^2 (Q) \rangle _S=\frac{k_B T}{m}Q^2 $ are different from zero. It can be easily proven that the dynamic structure factor is related to the longitudinal current spectrum through the relation $C_{L}(Q,\omega )=\left( \omega ^{2}/Q^{2}\right) S(Q,\omega )$. The presence of the factor $\omega ^{2}$ wipes out the low frequency portion of the dynamic structure factor, and consequently emphasizes the genuine inelastic features of $S(Q,\omega )$. After its definition and Eq.(\[phis\_laplace\]), it is readily seen that the Laplace transform $\widetilde{C}_{L}(Q,s)$ satisfies $$\begin{aligned} \widetilde{C}_{L}(Q,s) &=&-s[s\widetilde{F}(Q,s)-S(Q)] \label{cqz} \\ \ &=&\frac{k_BT}{m}\left\{ {s+[\omega _0^2(Q) / s]\ +\ \widetilde{M}(Q,s)}\right\} ^{-1} \nonumber\end{aligned}$$ Again, the spectrum $C_L(Q,\omega )$ can be expressed as $(1/\pi )\Re \{ \widetilde{C}_{L}(Q,s=i\omega ) \}$. Then the position and the width of the inelastic peaks in $C_L(Q,\omega )$ are determined by the poles of $\widetilde{C}_{L}(Q,s)$. ### Collective memory function and hydrodynamics\[sec\_mf\] The effective, second order memory function $M(Q,t)$ accounts for all the relaxation mechanisms affecting collective dynamics and, consequently, is the central quantity in most theoretical approaches. In analogy with the single particle case (Eq. (\[initial\_Ms\])), the initial value of $M(Q,t)$ is related to the spectral moments of $S(Q,\omega )$ by: $$\begin{aligned} M(Q,0)=\frac{\langle \omega^4(Q) \rangle _S}{\langle \omega^2(Q) \rangle _S}-\langle \omega^2(Q) \rangle _S \doteq \Delta^2(Q) \label{initial_M}\end{aligned}$$ Along the same line, relations similar to (\[initial\_Ms\]) and (\[mom4\_s\]) holds, and an explicit expression for $\langle \omega^4(Q) \rangle _S$ can be given, involving both the derivatives of the interparticle potential and the pair distribution function: $$\frac{\langle \omega^4(Q) \rangle _S}{\langle \omega^2(Q) \rangle _S}=\frac{3K_{B}TQ^{2}}{m}+\frac{\rho}{m}\int \frac{\partial^{2}V(r)}{\partial z^{2}} (1-e^{-iQz}) g(r) d^3 r \label{winf}$$ The second and fourth frequency moments are particularly significant, as they rule the sound velocity in the whole $Q-\omega$ domain. For sufficiently large $s$, indeed, $\widetilde{M}(Q,s)\simeq M(Q,t=0)/s$ and Eq. (\[cqz\]) is seen to have poles at $s=\pm i\sqrt{\omega _0^2(Q)+\Delta ^2(Q)} \doteq \pm i\omega_\infty(Q)$. This latter relation defines the frequency $\omega_\infty(Q)$ which characterizes the instantaneous collective response of the liquid at the wavevector $Q$ and, in turns, defines the unrelaxed sound velocity as $c_\infty (Q)\equiv \omega_\infty(Q)/Q$. In the opposite limit, i.e. for small $s$, one easily verifies that $\widetilde{M}(Q,s)\simeq \int M(Q,t) dt$. In this (relaxed) regime, therefore, the poles of the longitudinal current are located at $s=\pm i \omega _0^2(Q)$, i.e. the longitudinal modes propagate with the isothermal sound velocity $c_t(Q)=\omega_0(Q)/Q$. Summing up, whatever the details of the memory function are, in presence of a relaxation process one observes a transition of the longitudinal sound velocity between two different regime, associated to the evolution of the longitudinal current correlation maxima. It is interesting to study in a parallel way the evolution of $S(Q,\omega)$. In the low frequency limit, Eq. \[sqwgenerale\] reduces to the spectrum of a damped harmonic oscillator (DHO) of characteristic frequency $\omega_0$ (which, in general, does not coincides with the position of the inelastic maximum), with damping $\Gamma = \int M(Q,t) dt$. $$\frac{S(Q,\omega)}{S(Q)}=\frac{1}{\pi}\frac{\omega_0^2(Q)\Gamma(Q)}{(\omega^2-\omega_0^2)+\omega^2\Gamma^2}$$ In the opposite, high frequency, limit, again Eq. (\[sqwgenerale\]) tends to an harmonic oscillator with no damping and with an higher characteristic frequency $\pm \omega_\infty(Q)$. Additionally, it appears an elastic peak of area $\frac{\Delta^2(Q)}{\omega_0^2(Q) + \Delta^2(Q)}$. The same results can be directly retrieved in the time domain from Eq. (\[langevin\]), substituting the high frequency limit (constant) and low frequency limit (delta shaped) of the memory function, in turn, and taking the Fourier transform. The memory function formalism leads, in the small wavevectors limit, to the hydrodynamic prediction Of Eq.(\[S\_hidro\]). It can be easily proven, indeed, that the diagonal terms of the memory matrix appearing in Eq. (\[coll\_contfrac2\]) have a $Q^2$ dependence, while the cross terms follow a $Q^4$ dependence. For $Q\rightarrow 0$, therefore, these latter can be neglected. Moreover, from the continuity equations (\[3idro\]) it follows that in the same limit the time dependence of the conserved quantities and their associated currents becomes increasingly slow. Consequently, one can model the decay of the terms $M^{1}_{JJ}(Q\rightarrow 0,t)$ and $M^{1}_{TT}(Q\rightarrow 0,t)$ as instantaneous, or equivalently, constant in the Laplace domain. By making the identifications ${\tilde M}^{1}_{JJ}(Q\rightarrow 0,s)=D_V Q^2$ and ${\tilde M}^{1}_{TT}(Q\rightarrow 0,s)=\gamma D_T Q^2$, and computing $\Omega^{(0)}_{JT}(Q) \Omega^{(0)}_{TJ}(Q)=(\gamma-1)\omega_0^2(Q)$, from Eq.(\[coll\_contfrac2\]) one gets for the memory function: $$\begin{aligned} \tilde{M}(Q\rightarrow 0,s)&=&D_VQ^2+\frac{(\gamma-1)\omega_0^2(Q)}{s+\gamma D_TQ^2} \nonumber \\ M(Q\rightarrow 0,t)&=&2D_VQ^2\delta (t)+\omega_0^2(Q)(\gamma-1)e^{-\gamma D_TQ^2t} \nonumber \\ \label{mem_hydro}\end{aligned}$$ The dynamic structure factor can be obtained by substituting Eq. (\[mem\_hydro\]) in the general expression (\[sqwgenerale\]). In the $Q\rightarrow 0$ limit, moreover, one has $\omega_0=c_tQ>>\gamma D_TQ^2$. In this limit, $S(Q,\omega)$ is: i) a DHO function around the Brillouin peaks, which are located at $\omega_s=\sqrt \gamma c_t Q$ (adiabatic sound propagation) and have a linewidth $D_VQ^2+(\gamma -1) D_T Q^2$ ii) a lorenzian function around $\omega=0$, whose linewidth is $2 D_TQ^2$. In the small damping limit, the DHO is well approximated by two symmetrically shifted lorenzian, and the hydrodynamic limit of Eq.(\[S\_hidro\]) is finally recovered. The advantage of the memory function approach is, however, in providing a way to generalize the hydrodynamic result for wavevector and frequency dependent transport coefficients. To this purpose, from the very start it is convenient to separate in $M(Q,t)$ the decay channels which explicitly involve couplings to thermal fluctuations ( $M_{th}(Q,t)$ ) from those directly associated with longitudinal density modes ( $M_{L}(Q,t)$ ). ### Finite wavelengths generalization: beyond hydrodynamics A straightforward generalization of ordinary hydrodynamics at finite wavevectors suggests for the thermal contribution the following form $$M_{th}(Q,t)\approx (\gamma(Q)-1)\omega_0^2(Q)\exp[-\gamma D_T(Q)Q^{2}t] \label{memoryth}$$ where $D_T(Q)$ and $\gamma(Q)$ can be regarded as a finite $Q$ [*generalization*]{} of the quantities $D_{T}=\kappa /nC_{P},$ being $\kappa $ the thermal conductivity, and $\gamma=c_P / c_V$. It must be stressed, however, how the extension to finite wavevectors, in the special case of highly conductive systems, requires attention to the physics behind this model. As pointed out by Faber [@FABER], in a liquid metal, owing to the high thermal conductivity, the quantity $\gamma D_T(Q)Q^2$ may easily become larger than the Brillouin frequency as soon as $Q\approx \frac{c_t(Q)}{D_T(Q)}$ [^4]. Actually, both $c_t(Q)$ and $D_T(Q)$ are expected to decrease on approaching values comparable to the inverse mean interparticle distance, i.e. in coincidence with the first sharp rising edge of $S(Q)$. But assuming that the transition occurs below this edge, one finds a crossover condition of $Q=\frac{c_t}{D_T}$ which, considering typical values of sound speed and thermal diffusivity of metals (a few thousands meters/second and $\approx 50$ nm$^{-2}$/ps, respectively), lies at wavevectors around $0.1\div 0.5$ nm$^{-1}$, which is indeed consistent with the initial assumption. In other words, on increasing $Q's$, the thermal peak broadens ultimately overlapping with the Brillouin lines, the sound propagation turns from adiabatic to isothermal, and independent thermal fluctuations become impossible. By the memory function point of view, at wavevectors $Q>>\frac{c}{D_T}$ (i.e. when the condition $\omega \tau_{th} << 1$ holds), $M_{th}$ decays instantaneously in a similar fashion to $M_L$. In this condition, the dynamic structure factor is a DHO with isothermal characteristic frequency, while the brillouin damping is given by the area under the memory function which is $D_VQ^2 + \frac{(\gamma-1)c_t^2}{\gamma D_T}$. The existence of this adiabatic to isothermal crossover, is expected just below the lower accessible $Q's$ of an inelastic scattering experiment ($\approx 1$ nm$^{-1}$ at present) and therefore has never been observed directly. Moreover, the magnitude of this effect is ruled by $\gamma-1$, which is normally very small. A remarkable exception to this latter condition is constituted by Nickel [@ber_ni] ($\gamma=1.88$), which has been studied by INS, as it will be discussed in the next section. Beside the thermal contribution, also the viscous term $D_V$ is expected to exhibit a $Q$ dependence. For simplified Lennard-Jones pairwise interaction it has been shown a decay of $\eta_l(Q)$ of a factor ten up to the main peak of $S(Q)$, with $\eta_B$ going negative over the same region [@tank_visco]. ### Finite frequencies generalization: viscoelasticity A second important generalization of transport coefficients concerns their possible frequency dependence. This latter case stems when the frequencies of the observed density fluctuations are high enough that their timescale competes with the one ruling the decay of $M_L$. In this condition, one has to drop the hypothesis on the instantaneous (Markovian) nature of the viscous term, and has to introduce a finite timescale ($\tau$) for the decay of $M_L$. As discussed at the beginning of this section, the timescale of $M_L$ sets a new crossover between two different regimes, characterized by different sound velocities and attenuations. The simplest practical way to go beyond the hydrodynamic result (\[mem\_hydro\]) is to allow for an exponential decay of $M_L(Q,t)$: $$M_L(Q,t)=\Delta _L^2(Q)e^{-t/\tau (Q)}\label{memoryL1}$$ with $\Delta _L^2(Q)=\omega_L(Q)^2-\gamma \omega_0(Q)^2$, in order to have the correct normalization of the whole $M(Q,t=0)$. Although this has the advantage of analytical simplicity when dealing with Fourier transform, a drawback of this ansatz lies in the violation of some basic short-time features of the memory function (such as a zero derivative at $t=0$), causing the divergency of $\langle \omega^n \rangle _S$ for $n\geq 6.$ Neglecting thermal effects, Eq. (\[memoryL1\]) yields the so-called [*viscoelastic model*]{} [^5] for $S(Q,\omega )$ [@lov_visco]. Since as $Q\rightarrow 0$ $\widetilde{M}_L(Q,s=0)/Q^2$ can written as $[c_\infty ^2-c_0^2]\tau (Q\rightarrow 0)$, the requirement that this coincides with $\eta _L/nm$ shows that the time $\tau (Q)$ must be finite as $Q\rightarrow 0$. Such a connection with viscous effects justifies the physical interpretation of the rate $1/\tau (Q)$ as a parameter giving an overall account of all relaxation processes by which the longitudinal response of the liquid is affected by time-dependent disturbances. In particular, for slow perturbations developing over a timescale $t\gg \tau (Q)$ the system can adjust itself to attain local equilibrium and the usual viscous behavior holds. In contrast, for density fluctuations fast enough that $t\ll \tau (Q)$ the liquid responds instantaneously, with a solid-like (elastic) behavior. The crossover between these limiting situations marked by frequencies $\omega $ such that $\omega \tau (Q)\approx 1$) is ultimately responsible for the gradual changes often detected in the sound dispersion of several liquids at increasing wavevectors. Although appealing, the simplicity of the viscoelastic model can be deceptive. First of all, the model itself provides no clue for the physical origin of the decay mechanisms leading to the rate $1/\tau (Q)$. Actually, the situation is even more involved. In earlier MD studies of Lennard Jones fluids, it was soon realized that the viscous dynamics in the microscopic regime (i.e. at wavelength comparable with the inverse mean inter-particle separation) proceeds through two distinct processes, characterized by two well separate time scales [@lev_2t]. More recently, the advent of IXS provided the experimental evidence substantiating these speculations [@scop_prlli; @scop_prena; @scop_preal; @scop_prlga; @mon_k]. In view of these result, the obvious remedy is to modify the simple ansatz (\[memoryL1\]) by allowing a two step decay of $M_L(Q,t)$: $$M_{L}(Q,t)=\Delta _{L}^{2}(Q){\ }\left[ {(1-\alpha (Q))e}^{{-\gamma }_{1}{(Q)t}}{+\alpha (Q)e}^{{-\gamma }_{2}{(Q)t}}\right] \label{x2tempi}$$ where the rate ${\gamma }_{1}{(Q)}$ is chosen to be larger than ${\gamma }_{2}{(Q)}$, so that the dimensionless factor ${\alpha (Q)}$ measures the relative weight of the ”slow” decay channel. Besides being more flexible than the viscoelastic model, we shall see that the ansatz (\[x2tempi\]) has the much more important merit that the presence of two different timescale does in fact have a definite physical interpretation. A simplified version of the previous ansatz has been implicitly introduced in the viscoelastic analysis of Brillouin Light Scattering (BLS) spectra of glass forming materials (see for example [@cumm_visco]. In fact, in these works, the general expression of $M_L(Q,t)$ for a two times decay is always expressed as $$M_L(Q,t)={2}\Delta _\mu ^2(Q)\tau _\mu (Q){\delta (t)+}\Delta _\alpha ^2(Q){e}^{{-t/\tau }_\alpha (Q)}$$ with explicit reference to the so-called $\alpha $- (structural) relaxation process as responsible for the long lasting tail, and to the $\mu -$ microscopic process as additional, faster, relaxation dominant over a very short timescale (in the BLS window the condition $\omega \tau _\mu <<1$ holds). On the contrary, the above mentioned IXS works it have shown how this approximation is no longer tenable in the case of liquid metals at the IXS frequencies. In particular, it has been pointed out that the slower (-$\alpha$) relaxation time satisfies the condition $\omega_B(Q) \tau_\alpha(Q)>>1$, i.e. some part of the viscous flow is frozen. As a consequence, at the wavevectors typical of the IXS experiments ($Q=1\div 20$ nm$^{-1}$) the quasielastic spectrum acquires a component arising from this frozen structural relaxation. The origin, at the atomic level, of this fast decay channel is still an open issue: the rapidly decaying portion of $M_L(Q,t)$ is customarily attributed to largely uncorrelated collisional events, similar to those occurring in a dilute fluid. In addition, at the high densities typical of the liquid state, non-negligible correlations among the collisions can be expected, making no longer valid an interpretation only in terms of ”binary” collisions. Although the magnitude of the correlation effects is relatively small and their buildup slow, once established their decay is even slower, and for $t>1/{\gamma }_1{(Q)\equiv \tau }_\mu (Q)$ this relaxation channel dominates the decay of $M_L(Q,t)$, which consequently may exhibit a small but long-lasting ”tail”. The ansatz (\[x2tempi\]) can incorporate most of this physics: on the basis of the latter, one may reasonably anticipate that ${\alpha (Q)\ll }1$, and that the time $1/{\gamma }_2{(Q)\equiv \tau }_\alpha (Q)$ is distinctly longer than $1/{\gamma }_1{(Q)\equiv \tau }_\mu (Q)$. In this picture, the best fitted values of the viscoelastic rate $1/\tau (Q)$ clearly represent some sort of ”weighted average” between ${\gamma }_1{(Q)}$ and ${\gamma }_2{(Q)}$. Finally, we may argue that at increasing $Q$ (namely, over a shrinking length scale) the magnitude ${\alpha (Q)}$ of correlation effects should decrease, and that at higher temperatures the value of ${\alpha (Q)}$ at a given wavevector should equally decrease. On a general basis, the requirement that $\lim_{Q\rightarrow 0} \tilde{M}_L(Q,s=0)/Q^2\rightarrow \eta _L/nm$ now takes the form $$(c_\infty ^2-c_0^2)\left[ \frac{{(1-\alpha (Q\rightarrow 0))}}{{{\gamma }_1{(Q\rightarrow 0)}}}{\ +\ }\frac{{\alpha (Q\rightarrow 0)}}{{{\gamma }_2{(Q\rightarrow 0}\ )}}\right] \rightarrow \eta _L/nm \label{limite}$$ The refined model (\[x2tempi\]) yields a dynamic structure factor given by $$\begin{aligned} &&S(Q,\omega )=\frac{S(Q)}\pi \nonumber \\ &&\ \times Re\left\{ \frac{\omega _0^2(Q)}{i\omega +\frac{\Delta _\mu ^2(Q)}{i\omega +{\gamma }_1{(Q)}}+\frac{\Delta _\alpha ^2(Q)}{i\omega +{\gamma }_2{(Q)}}+\frac{\Delta _{th}^2(Q)}{i\omega +{\gamma (Q) D_T(Q)Q}^2}}\right\} ^{-1} \label{x3tempi}\end{aligned}$$ One of the major drawbacks of expression (\[x3tempi\]), is that one normally overestimates the relaxation strength of the faster viscous process. Such problem is somehow expected as one is trying to force an exponential dependence to reproduce the memory function at short times, which, instead, has a zero derivative in the $t \rightarrow 0$ limit. As a consequence, adjusting the characteristic time of the exponential memory on the experimental data one can reproduce the decay of the true memory function but will overestimate the short time limit, due to the cusp behavior of the exponential at $t=0$. An obvious remedy it is a better choice of the memory function model. An alternative possibility could be a gaussian shape. Also the $sech$ function has been proposed as solution of the Mori Equation [@tank_sech], but this latter case is of limited practical interest, as the fourier transform is related to the digamma function and, therefore, the expression (\[x3tempi\]) must be numerically evaluated. Kinetic theories: the hard sphere approximation \[sec\_kin\] ------------------------------------------------------------ Special attention has been devoted in the past to the theoretical and numerical study of the hard sphere model, as it conveniently mimics the behavior of more realistic simple liquids [@lebo_hs; @furt_hs]. In the eighthes, transport coefficients [@all_hs] and neutron scattering response [@all_hs1] have been evaluated by means of molecular dynamics. On the theoretical side, the dynamical properties have been investigated by a revisiting the so called “Enskog fluid”, i.e. by means of the spectral decomposition of the Enskog operator [@desh_hyd0; @desh_hs; @coh_hs1; @kag_hs; @mry_gcm]. This latter approach turned out to be particularly useful to describe INS experimental data [@coh_hs], and we will briefly recall the basics in this section. The main idea beyond the Enskog’s theory is to evaluate the correlation functions (\[cf\]) replacing the Liouville operator $\mathcal{L}$ and the dynamical variables $a_\alpha(Q)$, defined at the $N-$ particle ensemble level, with the one particle Enskog’s operator $L$, and appropriate one particle variables $\phi_\alpha(Q)$. It can be easily recognized, indeed, that $$\begin{aligned} a_\alpha(Q)=\frac{1}{\sqrt N}\sum_j \phi_\alpha(\mathbf{v}_j) e^{-i\mathbf{Q} \cdot \mathbf{r}_j} \nonumber\end{aligned}$$ where, for the first three dynamical variables (\[3dv\]), it holds $$\begin{aligned} &&\phi_1(\mathbf{v})=\frac{1}{S(Q)} \nonumber \\ &&\phi_2(\mathbf{v})=\sqrt{\frac{m}{k_BT}}\frac{\mathbf{Q}\cdot \mathbf{v}}{Q} \nonumber \\ &&\phi_3(\mathbf{v})=\frac{3-mv^2/k_BT}{\sqrt 6} \nonumber\end{aligned}$$ For an hard sphere system, a possible asymmetric representation of $L$ reads [@desh_hs]: $$L(\mathbf{Q},\mathbf{v}_1)=-i\mathbf{Q}\cdot\mathbf{v}_1+\rho g(\sigma) \Lambda(\mathbf{Q}) + \rho A(\mathbf{Q}) \label{enskog}$$ where $g(\sigma)$ is the pair distribution function evaluated at the contact point between two spheres. The first term appearing in (\[enskog\]) is a free streaming contribution. The second term accounts for binary collisions through the operator $\Lambda(\mathbf{Q})$, defined through its action over a generic function of the velocity $f(\mathbf{v}_1)$ $$\begin{aligned} &&\Lambda(\mathbf{Q})f(\mathbf{v}_1)=-\sigma^2 \int d\hat{\sigma} \int d\mathbf{v}_2 \xi(v_2) \mathbf{\delta} \theta(\mathbf{\delta}) \nonumber \\ && \{f(\mathbf{v}_1)-f(\mathbf{v}_1')+e^{-i\mathbf{Q} \cdot \hat {\sigma} \sigma} [f(\mathbf{v}_2)-f(\mathbf{v}_2')] \} \label{enskog_2}\end{aligned}$$ with $\xi(v)$ the normalized Maxwell distribution function, $\theta(x)$ the Heaviside step function, $\hat{\sigma}$ the unit vector, $\mathbf{\delta}=(\mathbf{v}_1-\mathbf{v}_2)\cdot \hat{\sigma}$ and $\mathbf{v}_{1,2}'=\mathbf{v}_{1,2}\mp \delta \hat{\sigma}$ the post-collision velocity. The third term, finally, is a mean field operator defined through $$\begin{aligned} &&\rho A(\mathbf{Q})f(\mathbf{v}_1)=[C(Q)-g(\sigma)C_0(Q)] \nonumber \\ && \int d\mathbf{v}_2 \xi(v_2) i \mathbf{Q} \cdot \mathbf{v}_2 f(\mathbf{v}_2) \label{enskog_3}\end{aligned}$$ where $C(Q)=[1-\frac{1}{S(Q)}]$ and $C_0(Q)=\lim_{\rho\rightarrow 0} C(Q)$. An explicit expression for the dynamic structure factor can be easily retrieved thought the spectral decomposition of $L$: $$\begin{aligned} L(\mathbf{Q},\mathbf{v}_1)=-\sum_j |\Psi_j(\mathbf{Q},\mathbf{v}_1 \rangle z_j(Q) \langle \Phi_j(\mathbf{Q},\mathbf{v}_1) | \nonumber\end{aligned}$$ in which $z_j$, $\Psi_j$ and $\Phi_j$ are eigenvalues, left and right eigenfunctions of $-L$, respectively. $S(Q,\omega)$ then reads $$\begin{aligned} \frac{\pi S(Q,\omega)}{S(Q)}&=&\mathrm{Re}\left \langle \frac{1}{i\omega-L(\mathbf{Q},\mathbf{v}_1)}\right \rangle_1 \nonumber \\ &=&\mathrm{Re} \sum_j \frac{B_j(Q)}{i\omega+z_j(Q)} \label{ehm}\end{aligned}$$ with $$B_j(Q)=\langle \Psi_j(\mathbf{Q},\mathbf{v}_1)\rangle_1 \langle \Phi_j^*(\mathbf{Q},\mathbf{v}_1) \rangle_1 \label{hscoeff}$$ The subscript $\langle ... \rangle_1$ explicitly indicates that we are now dealing with single particle averages over the Maxwell Bolzmann velocity distribution function. There are several approaches to determine the spectrum of $L$, but the main point is that different approximations can be performed according to the density and kinematic region of interest. These regions are marked by the values of the reduced density $V_0/V=\rho \sigma^3/\sqrt 2$, $V_0$ being the closed packed volume for spheres or radius $\sigma$, and the Enskog mean free path $l_E=l_0/\chi$ with $l_0$ the Bolzman mean free path $l_0=1/\sqrt 2 \rho \pi \sigma^2$ and $\chi=g(\sigma)$ the pair distribution function evaluated at the contact point between two spheres. The lower three eigenvalues of $L$ always go to zero with $Q\rightarrow 0$. In the same limit it can be shown that these latter eigenvalues are: $$\begin{aligned} &&z_1(Q)=z_h(Q)=D_{TE}Q^2 \nonumber \\ &&z_{2,3}(Q)=z_\pm (Q)=\pm i c_0Q+\Gamma_E Q^2 \label{coeff_hyd}\end{aligned}$$ and only the first three coefficients (\[hscoeff\]) are relevant, so that the eigenfunctions are linear combinations of the density, current and energy variables. In other words the hydrodynamic result of Eq. (\[S\_hidro\]) is recovered, with the dynamic structure factor composed of three Lorenzian functions: a diffusive heat mode and two propagating modes with the adiabatic sound velocity, with dampings $D_{TE}$ and $\Gamma_E$ as given within the Enskog’s transport theory. The relative intensities of the thermal and acoustic contributions are ruled by the Landau Placzek ratio. This limit is practically attained at low densities ($V_0/V <0.1$ and therefore $l_E\approx l_0$) and sufficiently small $Q$’s ($Q\sigma<<1$), when the contribution (\[enskog\_3\]) can be safely neglected and the term (\[enskog\_2\]) can be replaced with $\Lambda(Q\sigma=0)$. This normally occurs up to the case of light scattering of dilute gases ($Ql_0\approx 1$), and one speaks in terms of tree extended hydrodynamic modes. Still in the same low density-momentum range, but at wavevectors $1\le Ql_0 \le 3$ a description in terms of a few hydrodynamic modes is no longer allowed, while for $Ql_0> 3$ one can evaluate the first order corrections to the free particle result of Eq.(\[S\_fp\_cl\]) which reads: $$S(Q,\omega)=\sqrt{\frac{m}{2\pi k_B T Q^2}}\left[e^{-\frac{m\omega^2}{2 k_B T Q^2}}+\frac{S^B(\omega/Q)}{Ql_0}+O(1/Ql_0)^2\right ] \label{S_fp_hs}$$ with $S^B(\omega/Q)$ the leading correction to the free streaming term $-i\mathbf{Q}\cdot\mathbf{v}_1$ of Eq. (\[enskog\]) due to a single binary collision event, which can be numerically estimated [@kag_hs]. Conversely, for dense fluids ($V_0/V>0.35$), the hydrodynamic scheme breaks down at as soon as $Ql_E>0.05$. Above this value, only the extended heat mode is well separated from all the other modes. Still a description in terms of three *effective* hydrodynamic modes apply, and the low $Q$ limit of these modes is again coincident with the hydrodynamic result. For $0<Ql_E<1$ the free streaming and the mean field contributions of the Enskog operator can be treated as perturbation to the binary collision term (\[enskog\_2\]), and the following approximate expression for the extended heat mode can be given: $$z_h(Q)=\frac{D_EQ^2}{S(Q)}d(Q) \label{zh_hs}$$ in which $$\begin{aligned} D_E=-\left \langle v_{1x}\frac{1}{\rho \chi \Lambda(Q\rightarrow \infty)}v_{1x}\right \rangle \nonumber\end{aligned}$$ is the Enskog diffusion coefficient and $$\begin{aligned} d(Q)&=&-\frac{\left \langle v_{1x}\frac{1}{\rho \chi \Lambda(Q)}v_{1x}\right \rangle}{D_E}\approx \frac{\left \langle v_{1x}\Lambda(Q\rightarrow \infty)v_{1x}\right \rangle}{\left \langle v_{1x}\Lambda(Q)v_{1x}\right \rangle} \nonumber \\ &=& \frac{1}{1-j_0(Q\sigma)+2j_2(Q\sigma)}\end{aligned}$$ can be approximated in terms of the first two even spherical Bessel functions. Enskog’s diffusion coefficient is related to the Bolzman diffusion coefficient $$\begin{aligned} D_0=\frac{3}{8\rho \sigma^2}\sqrt{\frac{k_B T}{\pi m}} \approx \frac{0.216}{\rho \sigma^2}\sqrt{\frac{k_B T}{m}} \nonumber\end{aligned}$$ by the collision enhancing term $g(\sigma)$ as $D_E=D_0/g(\sigma)$. Using the analytic expression of $g(\sigma)$ for non attractive hard spheres one finally gets an expression for $D_E$ in terms of the packing fraction $\varphi=\pi \rho \sigma^3 / 6$: $$\begin{aligned} D_E=\frac{1}{16}\sqrt{\frac{\pi k_B T}{m}}\sqrt[3]{\frac{6}{\pi \rho \varphi ^2}} {\frac{(1-\varphi)^3}{1-\varphi/2}} \nonumber\end{aligned}$$ As in the low density case, for $Ql_E>3$ the the free streaming limit is recovered along with the binary collisions corrections. In this case, however, a different approximation holds, since the binary collision term (\[enskog\_2\]) is now replaced by the Lorentz-Boltzmann operator $\Lambda(Q\rightarrow \infty)$. One ultimately gets an expression similar to (\[S\_fp\_hs\]) in which $S^B(\omega/Q)$ is replaced by the different Lorentz-Boltzmann expression. At intermediate densities, finally, again the hydrodynamic result does not hold for $Ql_E\gtrsim 0.05$. Moreover, in this regime three effective modes do not suffice and one has to extend the description with two additional kinetic modes, i.e. including the kinetic part of the z-z component of the stress tensor and the z-component of the heat flux [@kag_hs]. Like in the high density case, for $1<Ql_E<3$ all the modes are closely interwoven, while at larger $Q$’s the free streaming limit with the Lorentz-Boltzmann correction is retrieved. According to kinetic theory, therefore, $S(Q,\omega)$ in simple liquids not too far from the melting temperature (such as Argon), should be described in terms of three lorenzians up to relatively large wavevectors ($Q\approx 30$nm$^{-1}$). Consequently, sound modes should exist even in a $Q$ region where side peaks are not distinctly observable. In this respects, Lovesey argued that the extended hydrodynamic picture should break up above $Q\approx 3$ nm$^{-1}$, while above a viscoelastic theory should be utilized [@lov_nokin]. Exploiting the previous results obtained for the coherent case, one can describe the incoherent dynamics via the Lorenz-Enskog operator $$L_s(\mathbf{Q},\mathbf{v}_1)=-i\mathbf{Q}\cdot\mathbf{v}_1+\rho g(\sigma) \Lambda(\mathbf{Q\rightarrow \infty}) \label{enskog_sp}$$ As already mentioned, at large $Q$’s the self and the collective dynamics coincide, and therefore the spectrum of $L$ will tend to the one of $L_s$ ($A(Q\rightarrow \infty)=0$). In this limit the extended heat mode $z_h$ will tend to the self-diffusion mode $z_D$. In the opposite, $Q\rightarrow 0$ limit, they both approach their hydrodynamic values: $z_h(Q\rightarrow 0)=D_{TE}Q^2$ and $z_D(Q\rightarrow 0)=D_{E}Q^2$, with $D_{TE}$ and $D_E$ the thermal diffusivity and the self diffusion coefficient of the Enskog’s theory. At intermediates $Q$ values, $z_h$ oscillates around $z_D$, with a periodicity dictated by $S(Q)$ and $d(Q)$ according to Eq. (\[zh\_hs\]). Sears has calculated the moments of the self part of the Van Hove scattering function al large wavevectors [@sears_inco]. Starting from the general case of a velocity independent central force field, he specialized the result to the hard sphere case, evaluating the leading correction to the impulse approximation due to final state interactions, which turns out to be $Q^{-1}$: $$\omega_H(Q)\approx \sqrt{\frac{2k_BTln2}{m}}Q\left( 1-\frac{0.27}{Ql_E}+O(Q^{-2})\right) \label{sears}$$ The transition from the Fickian to the Gaussian regime occurs in this case at $Ql_E\approx 1$. Finally, it is worth to recall here one the most significant achievement of the mode coupling theory applied to hard sphere fluids, but which has been shown to apply to the wider class of simple fluids. Ernst and Dorfman [@ern_pdis] have shown how Eq.(\[coeff\_hyd\]), which retrieves the hydrodynamic expression of the sound velocity and attenuation, is actually the leading term of an expansion of the kind: $$\begin{aligned} z_h(Q)&=&\alpha_h Q^2 - \beta_h Q^{5/2} + O(Q^{11/4}) \nonumber \\ z_\pm (Q)&=&\pm i c_0Q+\alpha_{{_\pm}} Q^2 + (\pm i-1)\beta_{{_\pm}} Q^{5/2} + O(Q^{11/4}) \nonumber \\ \label{coeff_hyd_MCT}\end{aligned}$$ The ionic plasma \[plasma\] --------------------------- The dynamical descriptions given up to this point are extensions of models holding for ordinary fluids which, in some cases, are modified *ad hoc* to account for the high thermal conductivity of liquid metals. A totally different approach is to look at liquid metals, from the very start, as a one component plasma, i.e. as an assembly of identical, point-like charged particles (ions) embedded in a uniform background (the electrons) which neutralizes the total charge [@Baus_ocp]. The long-ranged, Coulomb interaction active in this case give rises to peculiar phenomena such as screening effects and plasma oscillations. The ionic number density $\rho(\mathbf{r},t)$ and the current density $\mathbf j(\mathbf{r},t)$ are related through the continuity equation: $$\begin{aligned} \frac{\partial \rho(\mathbf{r},t)}{\partial t}=-\nabla \cdot \mathbf j(\mathbf{r},t) \nonumber\end{aligned}$$ The Poisson equation, for a fluid with $Ze$ charge, reads $$\begin{aligned} \nabla \cdot \mathbf E(\mathbf{r},t)= 4\pi Ze \delta \rho (\mathbf{r},t) \nonumber\end{aligned}$$ in which $\delta \rho (\mathbf{r},t)$ is the deviation of the ionic density from its average value $\rho$ (neutralized by the opposite uniform electronic density). Neglecting thermal conductivity effect (collisionless regime), from the Newton’s law for the equation of motion one can find a third equation and close the system. In the limit of large wavelengths fluctuations (compared to the ionic size) one can write $$\begin{aligned} m\frac{\partial \mathbf j(\mathbf{r},t)}{\partial t}=\rho Ze\mathbf E(\mathbf{r},t) \nonumber\end{aligned}$$ being $m$ the ionic mass. Combining the previous equations one easily gets: $$\begin{aligned} \frac{\partial^2 \rho(\mathbf{r},t)}{dt^2}=-\Omega^2_p \delta \rho (\mathbf{r},t) \nonumber\end{aligned}$$ which describes ionic plasma oscillation of characteristic $Q$ independent frequency $$\Omega^2_p=\frac{4\pi \rho Z^2e^2}{m} \label{oplas}$$ This simplified picture, therefore, contrasts with the experimental evidence of long wavelength excitations whose frequency vanishes in the $Q\rightarrow 0$ limit. The commonly used remedy is to “dress” the plasma frequency accounting for the electron screening effect, i.e. to take into account that the background electrons have their own dynamics that can be described introducing the dielectric response $\epsilon(Q)$. In such a way the coulomb interactions and, in turn, the plasma frequency are renormalized leading to the expression: $$\omega^2_p=\frac{\Omega^2_p}{\epsilon(Q)} \label{oplas_d}$$ In the small wavevector limit the Thomas-Fermi expression within the Random Phase Approximation (RPA, in with the system is assumed to have a free electron gas compressibility) yelds the useful expression [@MARCH] $$\epsilon(Q)=1+\frac{Q^2_{TF}}{Q^2} \label{TF}$$ in which $Q_{TF}=6\pi e^2\rho_e/E_F$, with $E_F=\hbar^2 (3\pi^2\rho_e)^{2/3} / 2m_e$, $\rho_e$ and $m_e$ the Fermi energy, the electronic density and the electronic mass, respectively. At small $Q$ values, Eq.(\[TF\]), together with Eqs.(\[oplas\]) and (\[oplas\_d\]), leads to the so called Bohm-Staver expression, i.e. a dispersive excitation with sound velocity $$c_{BS}=\frac{\omega_p}{Q}=\sqrt{\frac{m_eZ}{3m}}v_F \label{BS}$$ in which $v_F$ the electron velocity at the Fermi level. As we will show in the following, the experimental values for sound velocities in conductive liquids often contrast with the prediction of Eq.(\[BS\]), especially at small electron density or, equivalently, at large values of the reduced ionic radius $r_s=(\frac{3}{4\pi \rho_e a_0^3})^{1/3}$, where the Thomas-Fermi approximation is no longer valid. The interplay between plasma oscillations and sound waves can be better accounted for within a two component plasma description (TCP) in which nuclei and electrons are treated separately [@chih_tcp], eventually exploiting the framework of Mori-Zwanzig [@hans_tcp]. The details of the TCP are, however, beyond the scope of this review, the interested reader might want to consult more specialized literature. Experimental study of the microscopic dynamics \[sec\_exp\] =========================================================== The main experimental ways to study microscopic dynamics in liquid metals are acoustic spectroscopy and inelastic scattering experiments. These latter have to be necessarily performed with probes which can penetrate enough into the sample to give boundary free information, a requirement which restrict the choice to neutrons and X-rays only. Actually, a few attempts have been performed by means of visible light scattering [@dil_rev], for instance on liquid mercury and gallium [@dil_blsmet], but Brillouin scattering from opaque, liquid matter, presents several difficulties, mainly associated with the ill definition of the exchanged momentum in absorbing media. The basics of the neutron interaction with matter have been surveyed in many papers [@coplov_rev] and books [@MARSHALL; @LOVESEY; @EGELSTAFF] which provide exhaustive survey of this issue. Inelastic X-ray scattering is a relatively newer technique, and therefore we will detail the basics of an IXS experiment recalling from time to time INS features for comparison. The scattering problem ---------------------- The measured signal in an inelastic scattering experiment is determined by the double differential scattering cross-section. Within the linear response theory, where it is assumed that the coupling between the probe and the system is weak, this scattering differential cross-section can be written quite generally as the product of three terms: i) One term describes the intensity of the probe-sample coupling, and it is independent from the energy of the incident particle. ii) A second one is a kinematic term related to the phase-space volumes of the incident and scattered particles. iii) The third term is the space and time Fourier transform of the correlation function of the observable in the system that couples to the probe particle. This last quantity is the one related to the elementary excitations characteristic of the system. ### The photon-electron interaction Hamiltonian The actual expression for the scattering cross section can be derived by a perturbation expansion from the probe-system interaction Hamiltonian. In the case of the interaction of charges with the electromagnetic field, in the weak relativistic limit (i.e. to first order in $v^2/c^2$), neglecting the direct coupling of the field with the nuclei (i.e. to zero order in the electron-to-nuclei mass ratio $m_e /m$ ), and neglecting the magnetic terms (i.e. to the zero order in the electron spin) one gets: $$\begin{aligned} H_{INT}&=&\frac{e^2}{2m_ec^2} \sum_j \mathbf{A}(r_j)\cdot \mathbf{A}^*(r_j) + \frac{e}{2m_ec} \sum_j \{ \mathbf{A}(r_j), \mathbf{p}_j \} \nonumber \\ &\doteq& H^{(1)}_{INT} + H^{(2)}_{INT} \nonumber\end{aligned}$$ where the symbol $\{\}$ denotes the anti-commutator operator. The two “electric” terms, $H^{(1)}_{INT}$ and $H^{(2)}_{INT}$ contain, respectively, two and one field operators $\mathbf{A}(r)$. It is clear, therefore, that - in a perturbation expansion treatment of the interaction Hamiltonian - the term $H^{(1)}_{INT}$ will give rise to two-photons processes at first order while the term $H^{(2)}_{INT}$ will give rise to one-photon processes at first order. To have the two-photon processes from the latter (the so called $\mathbf{p}\cdot \mathbf{A}$ contribution), necessary to describe the scattering events, one must consider the second order in the perturbation expansion, which is consequently completely negligible in the off-resonance case. In the following, therefore, we will consider only the first charge scattering term. ### The X-ray scattering cross-section. The double differential cross-section, $\frac{\partial^2\sigma}{\partial E \partial \Omega}$, is proportional to the probability that an incident particle is scattered with a given energy and momentum variation within an energy range $\Delta E$ and a solid angle $\Delta \Omega$. In the process, a photon of energy $E_i$, wavevector $\mathbf{k}_i$, and polarization $\mathbf{\epsilon}_i$, is scattered into a final state of energy $E_f$, wavevector $\mathbf{k}_f$, and polarization $\mathbf{\epsilon}_f$, and the electron system goes from the initial state $|I\rangle$ to the final state $|F\rangle$ (states with energies $E_I$ and $E_F$, respectively). According to this definition, the double differential cross section can be related to the quantity $\frac{dP_{i\rightarrow f}}{dt}$ which is the probability rate per sample and probe unit that a probe particle makes the transition from the initial state to the final state: $$\begin{aligned} \frac{\partial^2\sigma}{\partial E \partial \Omega}=\frac{dP_{i\rightarrow f}}{dt} \frac{1}{j} \frac{\partial^2 n}{\partial E \partial \Omega} \nonumber\end{aligned}$$ In this equation $j$ is the incident particle current density ($j=\rho v$, being $\rho$ the particle density and $v$ its velocity) and $\frac{\partial^2 n}{\partial E \partial \Omega}$ the density of states of the scattered particle. For zero mass particles, the latter two quantities can be written as: $$\begin{aligned} j&=&\frac{c}{V_0} \\ \frac{\partial^2 n}{\partial E \partial \Omega}&=&\frac{V_0}{8\pi^3} \frac{k_f^2}{\hbar c}\end{aligned}$$ Therefore, the double differential cross section becomes $$\frac{\partial^2\sigma}{\partial E \partial \Omega}=\frac{V_0^2}{8\pi^3} \frac{k_f^2}{\hbar c^2} \frac{dP_{i\rightarrow f}}{dt} \label{csec}$$ The transition of the incident particles between states $i$ and $f$ involves, in general, different possible elementary excitations in the sample. This implies that, indicating with $\frac{dP_{i,I \rightarrow f,F}}{dt}$, the scattering probability involving the transition in the sample from the state $|I\rangle$ to the final state $|F\rangle$, the total probability $\frac{dP_{i\rightarrow f}}{dt}$ can be expressed as: $$\begin{aligned} \frac{dP_{i\rightarrow f}}{dt}=\sum_{F,I} \frac{dP_{i,I\rightarrow f,F}}{dt} \nonumber\end{aligned}$$ Equation (\[csec\]) is particularly useful, as the transition probability per unit time $\frac{dP_{i,I\rightarrow f,F}}{dt}$ can be calculated from the perturbation theory. To first order this quantity is written as (Fermi golden rule): $$\begin{aligned} \frac{dP_{i,I\rightarrow f,F}}{dt} &=& \frac{2\pi}{\hbar} \left | \langle i,I | H_{INT} |f,F \rangle \right |^2 \delta(E_i+E_I-E_f-E_F) \nonumber \\ \label{fgr} $$ Inserting the term $H^{(1)}_{INT}$ into Eq.(\[fgr\]), using Eq.(\[csec\]), and considering the initial and final photon states as plane waves one gets: $$\begin{aligned} \frac{\partial^2\sigma^{(1)}}{\partial E \partial \Omega}&=&\left (\frac{e^2}{m_ec^2}\right )^2 \frac{k_f}{k_i} (\mathbf{\epsilon}_i \cdot \mathbf{\epsilon}_f)^2 \nonumber \\ & \times &\sum_{F,I} P_I \delta(E-(E_F-E_I)) \left | \langle F | \sum_j e^{i \mathbf{Q} \cdot \mathbf{r_j}} |I \rangle \right |^2 \nonumber \\ \label{csec_1}\end{aligned}$$ where $Q=\mathbf{k}_i-\mathbf{k}_f$ $(E=E_f-E_i)$ is the momentum (energy) transferred from the photons to the system. The sum over the initial and final states is the thermodynamic average, and $P_I$ corresponds to the equilibrium population of the initial state. Apart from the sum over the phase factors of the photons scattered from the different particles, whose interference give rise to a truly $Q$ dependent scattering signal, the energy- and angle- integrated cross section is of the order of the square of the classical electron radius $r_0=\frac{e^2}{m_e c^2}$. ### The adiabatic approximation and the dynamic structure factor From Eq. (\[csec\_1\]), which implicitly contains the correlation function of the electron density, one arrives at the correlation function of the atomic density on the basis of the following considerations: i) One assumes the validity of the adiabatic approximation, and this allows to separate the system quantum state $|S\rangle$ into the product of an electronic part, $|S_e\rangle$, which depends only parametrically from the nuclear coordinates, and a nuclear part, $|S_n\rangle$: $|S\rangle =|S_e\rangle |S_n\rangle$. This approximation is particularly good for exchanged energies that are small with respect to the excitations energies of electrons in bound core states: this is indeed the case in basically any atomic species when considering values in the range of phonon energies. In metals we neglect the small portion of the total electron density in proximity of the Fermi level. ii) One limits to consider the case in which the electronic part of the total wavefunction is not changed by the scattering process, and therefore the difference between the initial state $|I\rangle =|I_e\rangle |I_n\rangle$ and the final state $|F\rangle =|I_e\rangle |F_n\rangle$ is due only to excitations associated with atomic density fluctuations. Using these two hypothesis we then obtain: $$\begin{aligned} \frac{\partial^2\sigma}{\partial E \partial \Omega}&=&\left (\frac{e^2}{m_e c^2}\right )^2 \frac{k_f}{k_i} (\mathbf{\epsilon}_i \cdot \mathbf{\epsilon}_f)^2 \sum_{F_{n},I_{n}} P_{I_{n}} \delta(E-(E_F-E_I)) \nonumber \\ &\times & \left | \langle F_n | \sum_j f_j(Q) e^{i \mathbf{Q} \cdot \mathbf{R}_j} |I_n \rangle \right |^2\end{aligned}$$ where $f_j(Q)$ is the atomic form factor of the $j$-th atom at $R_j$ and the sum is now extended to all the atoms (molecules) of the systems. Assuming that all the scattering units in the system are equal, this expression can be further simplified by the factorization of the form factor of these scattering units, and by the introduction of the dynamic structure factor $S(Q,E)$ defined as: $$\begin{aligned} S(Q,E)=\sum_{F_{n},I_{n}} P_{I_{n}} \delta(E-E_F+E_I)) \left | \langle F_n | \sum_j e^{i \mathbf{Q} \cdot \mathbf{R}_j} |I_n \rangle \right |^2 \nonumber\end{aligned}$$ By representing the $\delta$-function above as a time integral, indicating by $\langle ... \rangle$ the thermal average $\langle o \rangle=\sum_{I} P_{I}\langle I |\hat o |I\rangle$ and using the completeness operator $\sum_{F_n} |F_n \rangle \langle F_n| =1$ the dynamic structure factor can be also written in the more familiar form: $$\begin{aligned} S(Q,E)=\frac{1}{2\pi N} \int dt e^{\frac{iEt}{\hbar}} \sum_{j,k} \langle e^{i \mathbf{Q} \cdot \mathbf{R}_j(t)} e^{-i \mathbf{Q} \cdot \mathbf{R}_j(0)} \rangle \nonumber\end{aligned}$$ where $N$ is the number of particles in the system and the sum over $(j,k)$ extend over these $N$ particles. The double differential cross-section can then finally be re-written as: $$\frac{\partial^2\sigma}{\partial E \partial \Omega}= (\frac{e^2}{m_e c^2})^2 \frac{k_f}{k_i} (\mathbf{\epsilon}_i \cdot \mathbf{\epsilon}_f)^2 \left | f(Q) \right |^2 S(Q,E) \label{csx}$$ In the limit $Q\rightarrow 0$, the form factor $f(Q)$ is equal to the number of electrons in the scattering atom, i.e. $f(Q)=Z$. For increasing values of $Q$, the form factor decays almost exponentially with a decay constants determined by the size of the radial distributions of the electrons in the atomic shells of the considered atom. At $Q$-values large with respect to the inverse of these dimensions, therefore, the inelastic X-ray scattering from density fluctuations is strongly reduced. The cross-section derived so far is valid for a system composed of a single atomic species, This derivation, however, can be easily generalized to molecular or crystalline systems by substituting the atomic form factor with either the molecular form factor, or the elementary cell form factor respectively. The situation becomes more involved if the system is multi-component and disordered. In this case the factorization of the form factor is still possible only assuming some specific distribution among the different atoms. In the limiting case that such distribution is completely random, an incoherent contribution appears in the scattering cross-section. ### From cross section to count rate. \[crate\] The $Z$-dependence of the Thomson scattering cross-section seems to imply a facilitation in studying systems with high $Z$. In reality, this is no longer true when the effect of photoelectric absorption is taken into consideration. Indeed, neglecting multiple scattering events, the signal detected in an IXS experiment from an infinitesimal slab of thickness $\delta x$ orthogonal to $k_i$ can be written as: $$dN=N_0 (\frac{\partial^2\sigma}{\partial E \partial \Omega}) \Delta E \Delta \Omega \rho dx \label{scatt_inf}$$ where $N_0$ is the flux of the incident photons, $N$ is the flux of scattered photons in an energy interval $\Delta E$ and in a solid angle $\Delta \Omega$, $\rho$ is the number of scattering units per unit volume and $\mu$ is the total absorption coefficient. Dealing with a macroscopic sample of length $L$, in the relevant case of a nearly forward scattering geometry, Eq.(\[scatt\_inf\]) becomes $$dN(x)=N_0 e^{-\mu x}(\frac{\partial^2\sigma}{\partial E \partial \Omega}) \Delta E \Delta \Omega \rho dx e^{-\mu (L-x)} \label{scatt_1ph_diff}$$ which, integrated over the whole sample length, yields: $$N=N_0 (\frac{\partial^2\sigma}{\partial E \partial \Omega}) \Delta E \Delta \Omega \rho L e^{-\mu L} \label{scatt_1ph}$$ Let us discuss the L-dependence of this function. It is obvious that N attains a maximum (the optimal sample length) when $L=\mu^{-1}$, and that the value of N at this maximum point is proportional to $\mu^{-1}$. Considering an X-ray energy of approximately 20 keV and $Z>3$, $\mu$ is almost completely determined by the photoelectric absorption process. This process gives approximately $\mu \approx Z^4$ with modifications at energies close to electron absorption thresholds. Consequently the effective scattering volume is very much reduced in materials with a high $Z$ (as $Z^4$), while the cross section increases as $Z^2$, making the study of these materials by all means more difficult than for those with low $Z$. The behavior of the optimal signal intensity as a function of atomic number in monatomic systems with sample length $L=\frac{1}{\mu}$ can be deduced from the data reported in Fig. \[thom\_cross\]. There, we show the quantity $\frac{\sigma_c \rho}{\mu}$, with $\sigma_c=(r_0 Z)^2$, which gives directly a measure of the efficiency of the method at the considered photon energy: in this example we took an incident photon energy of 22 keV. The quantity $\frac{\sigma_c \rho}{\mu}$ is by definition the ratio between $\sigma_c$ and $\sigma_t$, where $\sigma_t$ is the (measured) total X-ray cross-section of the considered atom. This analysis is useful however only when it is possible to study samples of optimal length. In cases where the sample size is limited either by its availability or by the sample environment (extreme pressure, high/low temperature, high magnetic field...) it is obvious that one has great advantages in studying high $Z$ materials. \[h\] ![Ratio between the total number of photons scattered by the Thomson process and those lost through all the other processes, among which photoelectric absorption, in a sample of length $\mu^{-1}$ calculated as a function of the atomic number Z for photons of incident energy of $\approx 22$ keV[]{data-label="thom_cross"}](./figure/csec.eps "fig:"){width=".4\textwidth"} Equation (\[scatt\_1ph\]) accounts for one scattering events only. An estimate for the two scattering process intensity can be obtained by invoking the forward scattering approximation again, indeed, the ratio of the two over the one scattering rates $N^{(2)}/N^{(1)}$ reads: $$\begin{aligned} \frac{N^{(2)}}{N^{(1)}} = \frac{\pi \rho h r_0^2 \int_0^\pi \left [ f(\theta)S(\theta)\right ]^2 d\theta }{Z^2 S(0)} \nonumber\end{aligned}$$ in which $h$ is sample transverse dimension traversed by the incident beam. The integral accounts for all the possible two scattering process leading to a final forward scattering. This expression shows how to suppress multiple scattering one has to reduce the transverse beam dimension. A similar estimate for neutron scattering is prevented by the much more complicated scattering paths, since, within a similar $Q$ range, the scattering angle in INS is normally much lager. However, it can be noted that in the case of neutrons the typical transverse beam size is much larger ($\approx 10\div 100 $ mm) than the IXS ones ($\approx 100$ $\mu$m), thus resulting in a more important contribution requiring accurate corrections. ### Kinematics of the scattering processes. \[sec\_klim\] Another important difference between X-rays and neutrons scattering lies in the kinematics of the scattering processes. The momentum and energy conservation laws impose that: $$\begin{aligned} &&\mathbf{Q} = \mathbf{k}_i - \mathbf{k}_f \nonumber \\ &&E = E_f-E_i \nonumber \\ &&Q^2 = k^2_i + k^2_f -2 k_i k_f cos \theta \nonumber\end{aligned}$$ where $\theta$ is the scattering angle between the incident and scattered particles. The relation between momentum and energy in the case of photons is given by: $$\begin{aligned} E(k)=hck \hspace{2.0cm} \nonumber\end{aligned}$$ and therefore one obtains: $$\left (\frac{Q}{k_i} \right )^2 = 1 + \left ( 1 - \frac{E}{E_i} \right )^2 -2 \left (1 - \frac{E}{E_i} \right ) cos \theta \hspace{1cm} \label{cin_x}$$ Considering that the energy losses or gains associated to phonon-like excitations are always much smaller than the energy of the incident photon ($E << E_i$) this relation reduces to: $$\begin{aligned} \left (\frac{Q}{k_i} \right )= 2 sin \frac{\theta}{2} \hspace{1cm} (E<<E_i) \nonumber\end{aligned}$$ This last relation shows that, in the limit of small energy transfers, the ratio between the exchanged momentum and the incident photon momentum is completely determined by the scattering angle. Therefore, in inelastic X-ray scattering, there are basically no limitations in the energy transfer at a given momentum transfer for phonon-like excitations. \[h\] ![Kinematic region accessible to IXS in reduced $E/E_i$ and $Q/k_i$ units (left panel). The right panel shows the realistic cases of molten lithium (open circles) and sodium (full circles) for an incident energy of 25 keV.[]{data-label="kin_ixs"}](./figure/kin1.eps "fig:"){width=".5\textwidth"} ### X-rays vs. Neutrons. At variance with the previous equation, if the probe particles have mass $m_p$ $$\begin{aligned} E(k)=\frac{\hbar ^2 k^2}{2m_p} \hspace{1cm} \nonumber\end{aligned}$$ and, therefore: $$\left (\frac{Q}{k_i} \right )^2 = 1 + \left ( 1 - \frac{E}{E_i} \right )-2 \sqrt{1 - \frac{E}{E_i}} cos \theta \hspace{1cm} \label{cin_n}$$ In this case of thermal neutron scattering, the approximation $E<<E_i$ no longer holds, and the kinematics of the scattering experiments is determined by Eq. (\[cin\_n\]). As an example, in Fig. \[kin\_neu\], we report the accessible kinematics regions in the $\frac{E}{E_i}$ vs $\frac{Q}{k_i}$ plane for two different incident energies, indicating paths at constant scattering angles. In the same figure, similarly to Fig. \[kin\_ixs\], we also report the approximate dispersion curves for liquid lithium and sodium. In the best situation, i.e. in forward scattering where the accessible region is as wide as possible, the limiting curve is linear around $Q=0 (E=0)$, and its tangent is $\frac{E}{E_i} = 2 \frac{Q}{k_i}$. Recalling that $E_i = \frac{\hbar ^2k_i^2}{2m_p}$, one gets $E = v_N \hbar Q$, with $v_N$ the velocity of the incoming neutron. As the dispersion relation for acoustic phonon is linear, $E = v_s\hbar Q$, with $v_s$ the velocity of sound, it is clear that whenever $v_s$ is larger than $v_N$ the excitations peaks lie outside the accessible region and, therefore, when $v_s > v_N$ the neutron technique cannot be applied to study the acoustic branch. This limitation does not apply to the case of X-rays, as, according to Eq. (\[cin\_x\]), there are basically no limits to the energy region accessible at a given scattering angle. \[h\] ![Kinematic region accessible to neutron scattering experiments for incident energies $E_i=50$ meV (left panel) and $E_i=500$ meV (right panel), reported for different scattering angles. Open and full circles are the (approximated) sound dispersion of molten lithium and sodium, respectively.[]{data-label="kin_neu"}](./figure/kin2.eps "fig:"){width=".7\textwidth"} As discussed before, the presence of relevant absorption phenomena is the main effect that determine the scattering volume in an IXS experiment. This implies, therefore that the probability that a photon is scattered from the sample is small, and this strongly suppresses the multiple scattering processes. In IXS experiments, indeed, the multiple scattering can be disregarded, thus avoiding the use correction procedures. This is therefore an important advantage with respect to the neutrons case, where, on the contrary, the sample length is determined by the scattering (rather than absorption) length. In Fig. \[ixs-neu\_cross\] are reported for comparison the (coherent) scattering lengths of the elements for the X-rays and neutrons cases. Finally, it is worth to compare the double differential scattering cross section for X-rays obtained before (Eq. (\[csx\])) with the similar quantity derived for neutron scattering (in the hypothesis of fully coherent scattering). The latter quantity is not derived here for brevity and can be found in many textbooks [@LOVESEY; @MARSHALL; @EGELSTAFF]. The two cross-sections read as: $$\frac{\partial^2}{\partial E \partial \Omega}= \left \{ \begin{array}{l} r_0^2 \frac{k_f}{k_i} \left ( \mathbf{\epsilon}_f \mathbf{\epsilon}_i \right )^2 |f(Q)|^2 S(Q,E) \hspace{.2cm} xray\\ \hspace{.4cm} b^2 \hspace{.6cm} \frac{k_f}{k_i} \hspace{2.25cm} S(Q,E) \hspace{.2cm} neutron \\ \end{array} \right.$$ Beside to the proportionality of both the cross sections to the dynamics structure factor, it is worth to underline that: 1. The two cross section are proportional to a characteristic scattering length squared ($r_0$ in the case of X-ray and $b$ in the case of neutrons) that are comparable in magnitude (see Fig. \[ixs-neu\_cross\]). The further factor $|f(Q)|^2$ ($\propto Z^2$ for small $Q$’s) in the case of X-ray does not bring an increase of the actual signal in the experiment because, as discussed before, increasing $Z$ also limit the optimal scattering volume due to the increase of the photoelectric absorption. 2. In both cases the phase space of the incident and final plane waves gives rise to the factor $\frac{k_f}{k_i}$, however while in the X-rays case $k_f\approx k_i$ , and this factor is very close to 1, in the neutron case this term give rise to a $Q$ dependence of the scattered intensity. 3. No polarization terms are present in the cross section for neutron, while in the case of X-rays the term tells us that the Thomson scattering arises from a scalar interaction and therefore the polarizations of the incident and scattered photons must be parallel. 4. Finally, the X-ray scattering cross section contains the form factor $f(Q)$, i.e. the Fourier transform of the charge density spatial distribution. As the charge density is localized around the nuclei in a space region of typical dimension of few tenth to few hundredth of nm, the function $f(Q)$ decreases appreciably on a $Q$ range of several inverse nm$^{-1}$, thus it does not depresses too much the scattering cross section in the mesoscopic region of interest. In the case of neutrons this form factor is not present (actually it is equal to 1) as the neutrons interact with the nuclear matter, localized in typical dimension of $10^{-6}$ nm. The neutron form factor is therefore constant in the whole accessible $Q$ region. From the discussion made so far, it should be now quite easy to understand how important is the development of the X-ray method, which can access, in principle, an extremely large region of the $E-Q$ plane. Particularly important is the small $Q$ region, where the acoustic excitations have energies which are not of easy access to the neutron spectroscopies. \[h\] ![The coherent scattering cross section of the elements for X-ray (open circles) and for neutrons (full circles) reported as a function of the atomic number Z.[]{data-label="ixs-neu_cross"}](./figure/csec1.eps "fig:"){width=".4\textwidth"} From the experimental data to the dynamical quantities \[sec\_fetdq\] --------------------------------------------------------------------- In order to extract quantitative information from the experimental intensity, i.e. to perform measurements of $S_q(Q,\omega )$ on an absolute scale, the most direct way is to use a reference scatterer and this is customarily done in neutron experiments. In IXS, for instance, such a procedure can be quite difficult because of the $Q$-dependence of the form factor and of the analyzers efficiencies. For these reasons indirect method are always preferred. One possibility is to exploit the lowest order sum rules of $S_q(Q,\omega )$ [@scop_jpc]: in particular for the first two frequency moments one has the $$\begin{aligned} \langle \omega ^0 \rangle _{S_q} &=&\int S_q(Q,\omega )d\omega =S(Q), \\ \langle \omega ^1 \rangle _{S_q} &=&\int \omega S_q(Q,\omega )d\omega =\hbar Q^2/2m.\end{aligned}$$ where the second equality follows from Eq. (\[dispari\]) applied for $n=1$. The measured raw intensity is related to the dynamic structure factor through $$I(Q,\omega )=A(Q)\int d\omega ^{\prime }S_q(Q,\omega ^{\prime })R(\omega -\omega ^{\prime }) \label{convo}$$ where $R(\omega )$ is the experimental resolution function and $A(Q)$ is a factor taking into account the scattering geometries, the experimental setup and the atomic form factor. The first moments of the experimental data, $\langle \omega ^0 \rangle _I$ and $\langle \omega ^1 \rangle _I$, and those of the resolution function, $\langle \omega ^0 \rangle _R$ and $\langle \omega ^1 \rangle _R$, are related to $\langle \omega ^0 \rangle _S $ and $\langle \omega ^1 \rangle _S $ by: $$\begin{aligned} \langle \omega ^0 \rangle _I &=&A(Q)\langle \omega ^0 \rangle _{S_q}\langle \omega ^0 \rangle _R, \\ \langle \omega ^1 \rangle _I &=&A(Q)(\langle \omega ^0 \rangle _{S_q}\langle \omega ^1 \rangle _R+\langle \omega ^1 \rangle _{S_q}\langle \omega ^0 \rangle _R).\end{aligned}$$ From the previous equation one derives that $$S_{q}(Q)=\frac{\hbar Q^{2}}{2M}(\langle \omega ^1 \rangle _I/\langle \omega ^0 \rangle _I-\langle \omega ^1 \rangle _R/\langle \omega ^0 \rangle _R)^{-1}. \label{norma}$$ This procedure, therefore, can been adopted to establish an absolute scale for $S_q(Q,\omega )$ using the experimentally determined $I(Q,\omega )$ and $R(\omega )$. Handling liquid metals ---------------------- Working with liquid metals poses several practical problems. In particular alkali metals are highly reactive and need to be kept under a protective atmosphere. A relatively small impurity (less than 100 ppm) in Ar or nitrogen will cause a film to form on the surface of the liquid metal. In addition, glass is often limited in its use as a container for most liquid metals. Liquid metals are often strongly reducing. Glass, composed chiefly of silicon dioxide, is penetrated by the metal atoms, which can reduce the silicon by forming a metal oxide. As a result, the glass becomes discolored and brittle. For these reasons Pyrex can not be used above $600$ K and pure quartz above $900$ K. Preferred materials for working with liquid metals are the refractory metals. This refers to the titanium group (Ti, Zr) as well as the vanadium and chromium groups. These transition metals are much less likely to undergo reduction and be solvated by liquid metals. The disadvantage in addition to cost is that the refractory metals have certain properties that make fabrication difficult. A trade-off is to use iron plated with chromium. Other less reactive transition metals, such as the noble metals, are soluble in many liquid metal solutions. Austenitic stainless steel can be suitable up to $1000$ K Liquid metals also share a common chemistry. The increasing electropositivity of the metals composing the liquid metal solution will determine the liquid’s reactivity. Mercury, which is not very electropositive, is stable in air. Alkali metals, which are the most electropositive group of elements, are air-sensitive. Li reacts slowly with air, yet dissolves and reacts quite readily with nitrogen. The other alkalis are insensitive to nitrogen but react with other gasses. All the alkali metals violently react with water on this basis: $Cs>Rb>K>Na>>Li$ [@OSE]. The hydrogen generated in the water-alkali reaction can, in turn, react explosively with oxygen. Reactivities reflect those of the solid material. The more reactive liquid metals, usually alkali ones, often develop a film if exposed to air. The high surface tensions of many liquid metals enable this film to remain in place. However, rates are greatly increased if turbulence is introduced because the protective coating is often dissolved into the liquid metal. Products of reaction, such as hydrides and oxides, are often redissolved into the liquid metal solution. By virtue of the previously mentioned difficulties, experiments with alkali metals are often very hard to perform. In the case of Inelastic X ray Scattering, in particular, such difficulties are enhanced by sample dimension requirement. As we have seen in sec. \[crate\], indeed, in an optimal IXS experiment the sample length has to be comparable with the absorption length. With a few exceptions (Li and Na), this typically means sub-millimeter sample thickness. Common choices are, therefore, sample cells made of compatible metals provided with sealed sapphire or diamond windows. \[h\] ![Coupling IXS with levitation techniques: Constant $Q$ spectra of liquid alumina. From [@sin_al2o3][]{data-label="al2o3"}](./figure/al2o3.eps "fig:"){width=".5\textwidth"} Large efforts have been done recently to overcome the difficulty of performing X-ray experiments on liquid metals, the most remarkable example being the so called Tamura-type cells made with a single crystal sapphire with Be windows pressurized under He [@tam_cell1], performing up to $1900$ K and $2$ Kbars. More recently, a new sample environment especially tailored for alkali metals has been proposed [@tam_cell2]. In this case the cell is entirely made of molybdenum, with the windows made by single- crystal disks of controlled orientation electrolytically thinned at $\approx 40$ $\mu m$. A totally different approach is the one of contact-less techniques. In this case the sample is levitated either electrostatically or by means of a controlled gas jet. The main difficulty in this technique is related to the sample stability in the x-ray beam, but recent impressive advancement have been done in this field. A new beamline for electrostatic levitation (BESL) has been developed at the Advanced Photon Source (APS), and the relevance of icosahedral ordering in the supercooling capabilities of liquid metals has been investigated [@kel_lev]. Another example is a recent X-ray scattering experiment performed on liquid Al$_2$O$_3$ in which alumina droplet of 3-4 mm diameter have been levitated by gas jet flow on the inelastic scattering beamline 3ID-C [@sin_al2o3]. Experimental results \[sec\_res\] ================================= In this section we review, to the best of our knowledge, the experimental results reported so far, ordered according to the sample group in the periodic table. No results are available so far for elements belonging to group II. Alkali metals ------------- Alkali metals do not occur freely in nature, they are very reactive and can explode if exposed to water. These metals have only one electron in their outer shell and, as with all metals, they are malleable, ductile, and good conductors of heat and electricity. Alkali metals are softer than most other metals. Among the metallic elements they share the simplest pairwise interaction potential, which is also the closest to the Lennard Jones one. As a consequence, their structural properties are also particulary simple, with a structure factor resembling the one of hard spheres. Also the dynamics, therefore, is expected to mimic the theoretical and numerical results achieved for Lennard Jones and hard sphere systems. ### Lithium Liquid lithium is probably the system which better reveals the complementarity of neutrons and X-rays as far as inelastic scattering is concerned. Due to the high absorption cross section of the $^6$Li isotope ($\sigma_a=940$ b) neutron scattering experiments must necessarily be performed on $^7$Li enriched samples, which is the dominant specie in the natural abundance. The high sound velocity ($c_t \approx 4500$ m/s), and the almost equivalent neutron scattering cross sections ($\sigma_i=0.68$ b; $\sigma_c=0.62$ b for $^7$Li), pose severe limitations to the use of INS aiming at the determination of collective properties, while this technique turns out to be extremely useful for the investigation of the single particle motion. The first INS studies on this system can be traced back to the work of De Jong and Verkerk [@dej_phd; @ver_li; @dej_li], who showed the presence of collective modes with a series of accurate experiments. Though they had to face the above mentioned drawbacks, indeed, they were able to point out some significant issues: by modelling the coherent contribution with the extended hydrodynamic model (see Eq. (\[ehm\]) of section \[sec\_kin\]) they measured the dispersion curve above $Q_m/2$ (see Fig. \[bur2\]) and they reported deviations from the Landau-Plazek ratio, which is expected to hold in the hydrodynamic regime (see Eq. (\[lp\]) of section \[sec\_collhydro\]). On the other side they shed light on the single particle motion, accurately determining the incoherent contribution to the dynamic structure factor within the framework of section \[sec\_kin\] (Eqs (\[mct1\]-\[mct4\])). They corroborated the Mode Coupling predictions [@desh_mc], extracting values of the diffusion coefficient and determining its temperature dependence. \[h\] ![Pioneering (1991) low resolution IXS determination of the dynamic structure factor in liquid lithium with INELAX. From [@bur].[]{data-label="bur1"}](./figure/bur1.eps "fig:"){width=".4\textwidth"} \[h\] ![Dispersion curve of liquid lithium achieved with IXS at INELAX. Theoretical predictions and INS results at higher $Q$’s are also reported. From [@bur].[]{data-label="bur2"}](./figure/bur2.eps "fig:"){width=".3\textwidth"} An exhaustive characterization of the coherent dynamics was provided by the advent of Inelastic X ray Scattering developed in the early nineties, and liquid lithium has been the benchmark of such development. Being the lightest of the liquid metals, indeed, lithium played a privileged role in IXS, for the favorable signal to noise ratio and for the high sound velocity which allowed to resolve the inelastic spectral component minimizing the initial difficulty of achieving energy resolutions comparable to neutrons. Since the pioneering work of Burkel [@BURKEL] with the INELAX instrument (see fig. \[bur1\] and \[bur2\]), a decisive step forward achieved with the advent of the third generation sources which, combined to a brilliant technique for manufacturing silicon crystal analyzers, allowed to exploit IXS to gather insight into the microscopic dynamics of disordered systems. \[h\] ![First IXS measurements on liquid lithium performed on a third generation source (ESRF) [@sinn]. Energy resolution is here $\delta E=11$ meV. The continuous line is the best fit according to the extended hydrodynamic model of Eq. (\[ehm\]). From [@sinn][]{data-label="li_sinn"}](./figure/li_sinn.eps "fig:"){width=".5\textwidth"} The first remarkable result on a third generation facility (ESRF) was provided by Sinn et al. [@sinn] who, measuring energy spectra at fixed wavevectors, reported clear evidence of collective modes, being able to give significant hints for the choice of the most appropriate pseudopotential to describe liquid metals in numerical simulations [@can_lit]. In the same work, following the extended hydrodynamic model outlined in section \[sec\_kin\] [@desh_hyd0; @desh_hyd], it was also reported evidence for positive dispersion, i.e. for a sound velocity value exceeding the hydrodynamic one. This phenomenon was ascribed to a transition from a liquid to a solid-like response. Following the development of the IXS technique, new experiments have been more recently performed on liquid Li in the extended region $1.4$ to $110$ nm$ ^{-1} $, corresponding to $Q/Q_{m}\approx 5\cdot 10^{-2}\div 5$, which are reported in Fig. \[all\_li\]. \[h\] ![IXS measurement of liquid lithium in a wide energy momentum-region [@scop_epl]. The transition from hydrodynamic to gaussian-like response (continuous line in the right panel) can be clearly noticed. Energy resolutions are here $1.5$, $3.0$ and $7.0$ meV, increasing with the exchanged momentum[]{data-label="all_li"}](./figure/all_li_small.eps "fig:"){width=".5\textwidth"} In Fig. \[li\_fulldisp\] is reported the dispersion relation determined in the same energy-wavevector region, and the transition between the two distinct dynamical regime is here evidenced by the sound velocity behavior. Beyond the first quasi-hydrodynamic region (an initial nearly linear dispersion), structural effects take place suppressing the sound propagation around $Q_{m}/2$ due to strong negative interference. With increasing $Q$ values, the points in Fig. \[li\_fulldisp\] show a second pseudo-BZ, followed by a series of oscillations that damp out with increasing $Q$ - here, $\omega _{l}(Q)$ is approaching the single particle behavior. These oscillations are in anti-phase with those of $S(Q)$ and are therefore associated with the local order in the liquid. \[h\] ![Sound velocity as deduced by the maxima of the current correlation spectra, from the best fit with quantum corrected and resolution convoluted models.[]{data-label="li_fulldisp"}](./figure/li_fulldisp.eps "fig:"){width=".5\textwidth"} \[h\] ![Memory function at work: refined lineshape analysis of high resolution IXS spectra. Both thermal and viscous channel are taken into account, mimicking this latter with one (Eq. (\[memoryL1\]), dotted line) or two (Eq. (\[x3tempi\]), continuous line) exponential processes.[]{data-label="li_2times"}](./figure/li_2times.eps "fig:"){width=".4\textwidth"} While at low $Q$’s the dynamic structure factor is qualitatively described by an extended hydrodynamic treatment (Eq. (\[ehm\] of section \[sec\_kin\]), at wavevectors distinctly larger than $Q_m$ the single particle response is attained through the mechanism described in section \[sec\_swl\], ultimately leading to the expressions well accounted by a combination of two Eq. (\[sqw\_free\_quant\]), accounting for each of the two isotopes $^6$Li and $^7$Li. Thanks to the improvement in the energy resolution, which is nowadays comparable to the one of INS spectrometers in the same energy-wavevectors domain (1.5 meV at present), an approach based on the generalized hydrodynamics has been developed, which allowed to point out the presence and the role of relaxation processes driving the collective dynamic at the microscopic probed wavelenghts [@scop_jpc; @scop_prlli]. Within a memory function framework [@mori_mf], it has been ascertained the presence of two distinct viscous relaxation channel (see fig. \[li\_2times\]) beyond the thermal relaxation, clarifying the origin and the nature of sound dispersion and attenuation properties in simple fluids. Of the two processes, active over well separated timescales, one is related to the well known transition between a low frequency, liquid-like response to an high frequency, solid-like response. The second mechanism is instead a general relaxation process peculiar of the vibrational dynamics which is present regardless the thermodynamic state of the system. In this context, the positive dispersion has been shown to be strongly related to this latter process, being the solid like response already attained over the wavevectors range probed in IXS (or INS) experiments. \[h\] ![Constant energy slices of the dynamic structure factor determined by IXS. Umklapp modes are visible on the sides of the main structure factor peak[]{data-label="li_umk"}](./figure/li_umk.eps "fig:"){width=".5\textwidth"} An alternative route to the investigation of collective dynamics, which is dual to the one followed in the above mentioned experiments and which easily achievable through IXS, is the determination of the dynamic structure factor performing Q-scan for fixed values of the energy transfer, reported in fig. \[li\_umk\] for the case of lithium [@scop_prbumk]. In this way, one is able to have a direct sight over the so called umklapp modes, i.e. excitations characterized by wavevectors which differ by multiple of the reciprocal lattice spacing, which have been early reported by means of INS in liquid lead [@rand_umk; @coc_umk; @dor_umk]. ### Sodium Pioneering experimental determinations of the scattering law in liquid sodium can be traced back to the time of the IAEA symposium held in Chalk River [@coc_na; @rand_na]. In this system the ratio between the incoherent to coherent cross section is very close to one (see table \[table\]), therefore the separation between the two contributions is of crucial importance. Soon after Randolph’s experiment, his data were analyzed in terms of mean square displacement of an atom [@des_napb]. This framework (described in section \[sec\_gaussapp\]) poses on the Gaussian assumption for the incoherent cross section, while the coherent contribution was evaluated according to the effective mass approximation [@dej_ema]: $$\begin{aligned} S_s(Q,\omega)&=&\frac{1}{\pi} \int_0^\infty dt cos(\omega t) exp \left [-\frac{Q^2 \langle r^2(t)\rangle}{6} \right ] \nonumber \\ S(Q,\omega)&=&\frac{1}{\pi} \int_0^\infty dt cos(\omega t) exp \left [-\frac{Q^2 \langle r^2(t)\rangle}{6S(Q)} \right ] \nonumber\end{aligned}$$ The mean square displacement was then determined describing the atomic motion in terms of independent harmonic oscillators of frequency $\omega_0$ and lifetime $\tau_0$, which, in turns, are related to the spectral density of the velocity autocorrelation function $f(\omega)$: $$\begin{aligned} \omega_0^2&=&\int_0^\infty d\omega \omega^2 f(\omega)=\frac{\langle (\overrightarrow{\nabla} U)^2 \rangle}{3m} \nonumber \\ \tau_0&=&\frac{T}{mD\omega_0^2} \nonumber \\ f(\omega)&=&\frac{2}{\pi}\frac{\omega_0^2 / \tau_0}{(\omega^2-\omega_0^2)^2+(\omega^2 / \tau_0)^2} \nonumber\end{aligned}$$ The basic ingredients of this approach are, therefore, the knowledge of the static structure factor, of the macroscopic diffusion coefficient and of the mean squared force $\langle (\overrightarrow{\nabla} U)^2 \rangle$. The results of this description are tested against the experimental data in fig. \[nady\]. From the same figure, it emerges how the single particle regime is already attained at the lowest reported $Q$ value, i.e. $Q=12$ nm$^{-1}$, while the low frequency discrepancy has been tentatively ascribed to finite instrumental resolution and to multiple scattering effect. \[h\] ![Randolph’s measurement [@rand_na] of the dynamic structure factor of liquid sodium in reduced units: $\alpha=\hbar^2Q^2/2mT$ and $\beta=\hbar \omega / T$ for four different values of momentum transfer. Lineshape analysis according to different models [@des_napb] Continuous line: EMA+calculated MSD. Short dashed line: EMA+computer simulation computation of the MSD. Long dashed line: hydrodynamic prediction. Dotted-dashed line: free streaming limit[]{data-label="nady"}](./figure/dey_na.eps "fig:"){width=".53\textwidth"} Twenty years later, new INS data were reported [@mor_na; @sod_na], addressing in more detail the incoherent scattering contribution and showing how the diffusion process is actually more complex. Morkel and Gl[ä]{}ser, following for the coherent contribution the Lovesey’s prescription [@lov_visco], and adopting for the incoherent part the Nelkin-Gatak model [@nelkin_inco] described in section \[sec\_ng\], extracted the reduced halfwidth $\omega_{1/2}$ of the incoherent contribution finding a crossover between the hydrodynamic (lorenzian) and single particle (gaussian) regimes. In fig. \[gammana\] the linewidth $\omega_{1/2}/DQ^2$ is reported, and it clearly emerges how the single particle limit ($\omega_{1/2}/DQ^2\propto 1/Q$) is not yet attained even at $Q\approx 40$ nm$^{-1}$, which contrast the earlier assumptions of Desai and Yip. After a low $Q$ diffusion retardation the mobility increases in the transition region and finally tends to the free gas limit. The whole $Q$ dependency is well described within the Enskog’s hard sphere gas [@sears_inco], in terms of the expression (\[sears\]). An alternative description of the single particle dynamics can be recovered within the memory function approach, though it fails in the high $Q$ region [@got_na]. \[h\] ![Reduced quasielastic linewidth $\omega_{1/2}/DQ^2$ in liquid sodium at three different temperatures [@mor_na]. The dotted line is the Fickian limit while dash dotted line is the perfect gas behavior ($\propto 1/Q$). The dashed line is the hard sphere prediction [@coh_hs], while the continuous line is the result obtained within mode coupling theory [@got_na][]{data-label="gammana"}](./figure/gamma_na.eps "fig:"){width=".42\textwidth"} The first IXS determination of the collective dynamics in liquid sodium is due to Pilgrim and collaborators [@pil_na]. In this work, the coherent dynamic structure factor was measured at several temperatures, and analyzed according to the extended hydrodynamic model previously applied in liquid lithium [@sinn]. The take-home message is the presence of a positive dispersion effect which does not show significant temperature dependence. This result seems to rule out an interpretation of the positive dispersion in terms of an activated process, as is the case in hydrogen bonding systems [@monaco_water]. \[h\] ![Left panel: IXS determination of the $S(Q,\omega)$ in liquid sodium for selected temperatures. Right panel: dispersion curves at different temperature.[]{data-label="napilgrim"}](./figure/na_pilgrim.eps "fig:"){width=".5\textwidth"} IXS experiments on liquid lithium were then repeated with increased energy resolution [@scop_prena], and analyzed within the same two viscous relaxation processes proposed for liquid lithium [@scop_prlli]. The same data have been also interpreted within the framework of the scale invariance of relaxation processes [@yul_na], a theory originally developed for liquid cesium [@yulm_cs], which has been recently shown to be equivalent to the memory function approach in the sense that one solves the chain of equations \[memory\] with some *ad hoc* closure relation. ### Potassium The first experimental data on liquid potassium appeared surprisingly late relatively to the other liquid metals [@nov_k; @nov_k1; @nov_k2]. Moreover, the kinematic region $Q-E$ spanned in this experiment was quite narrow ($10<Q<13$ nm$^{-1}$) and only partial information on the microscopic dynamics could be obtained. Very recently, two sets of INS experiments have been reported in molten K just above the melting temperature, one at the ISIS source [@cab_k], and the other at the ILL [@bov_k]. In the first case, two time of flight spectrometers were utilized (IRIS and MARI) aiming at a combined study of the dynamic structure factor with different energy resolutions for the narrow quasielastic and the broader inelastic component. The experiment of Bove et al. has been instead performed on the triple axis spectrometer IN1 optimized to access a broader kinematic region, as shown in fig.\[kin\_k\] where a detail of the energy-momentum region accessed in the two experiments is shown. \[h\] ![Sketch of the kinematic regions accessed in the experiments of Bove et al. (dashed line) and Cabrillo et al. (continuous lines). The linear sound dispersion is also reported.[]{data-label="kin_k"}](./figure/k_kinematic.eps "fig:"){width=".4\textwidth"} The data taken on IRIS allowed for an accurate, high resolution, determination of the diffusive processes underlying the incoherent dynamics. The results support the hydrodynamics predictions corrected by the mode coupling terms (see eqs. (\[mct1\]-\[mct4\]) of section \[sec\_mct\]), as reported in fig. \[diff\_k\]. Beyond the diffusive mode, Cabrillo et al. identify a second contribution, coherent in nature according to the authors, which is weaker, broader and almost $Q$ independent up to $Q\approx Q_m$, while becomes narrower above $Q_m$. The $Q$ dependence of this coherent mode is rationalized in terms of extended heat mode (Eq. (\[zh\_hs\]) of section \[sec\_kin\]) but its ultimate origin is rather ambiguous, especially in view of the INS measurements taken at IN1 and MARI. \[h\] ![Quasielastic linewidth according to recent INS measurements. Full and open circles are the coherent and incoherent contributions, respectively, determined with TOF [@cab_k]. Open triangles are the incoherent linewidth as measured with TAS [@bov_k], the observed $2$ meV coherent contribution is also indicated. The lower dotted line indicates the fickian approximation.[]{data-label="diff_k"}](./figure/k1.eps "fig:"){width=".4\textwidth"} The results of these two experiments are reported in fig. \[kins\] for similar fixed Q values [@cab_k; @bov_k]. As can be easily noticed the possibility (offered by IN1) of extending the INS measurements at low Q is paid in terms of resolution. In both cases, however, evidence for inelastic coherent scattering is reported, though the incoherent scattering largely dominates in the region where collective modes are more visible. The two sets of data have been analyzed according to different approaches by the respective authors. Cabrillo et al utilized a memory function approach truncating the continued fraction at $n=2$, motivating this assumption as necessary to account for the nearly $Q$ independence of the coherent quasielastic contribution reported in fig. \[diff\_k\]. Odd enough, as evinced from Fig. \[kins\] (left panel), neither the inverse relaxation time, nor the raw quasielastic width that they extract with this model favourably compare with the coherent linewidth reported in fig. \[diff\_k\]. Bove et al., on the other side, utilize a Damped Harmonic Oscillator for the *purely* inelastic term and two lorentian for the quasielastic coherent and incoherent contributions, respectively. The results for the incoherent part, achieved within the jump diffusion model described in section \[sec\_jd\], are consistent with the high resolution measurements (IRIS) of Cabrillo et al. (see fig.\[diff\_k\]). On the other side, the coherent contribution turns out to be much broader (FWHM$\approx 4$ meV, i.e. a relaxation time $\tau\approx 0.32$ ps), in contrast with the Cabrillo’s data reported in fig.\[diff\_k\] (full circles), but in qualitative agreement with the quasielastic linewidth and the relaxation time of the same author’s measurements reported in fig.\[kins\]. \[h\] ![Left panel: TOF determination of the DSF in liquid potassium (circles) [@cab_k]. The dash-dotted line depicts the coherent contribution. Right panel: TAS measurements in a similar momentum transfer region (circles) [@bov_k]. The continuous line is the inelastic contribution to the collective dynamics[]{data-label="kins"}](./figure/k_ins.eps "fig:"){width=".4\textwidth"} Both the experiments extract the dispersion curves, in one case following the exact hydrodynamic prescription as the maxima of the current correlation function [@cab_k] and, in the other [@bov_k] as the DHO frequency, which coincides with the current correlation maximum if the presence of the quasielastic coherent term is neglected. The two independent determinations are indeed in good agreement, except at large wavevectors where the data of Bove et al. are systematically higher though with some scattering. The sound velocity values exhibit the usual excess in respect to the hydrodynamic value. This high frequency sound is ascribed by both the studies as a reminiscence of solid like behavior, i.e. as the upper edge of a transition occurring from the low $Q$, hydrodynamic domain to the high frequency regime. This claim stems on the basis of the similarity of the sound velocity value of molten potassium with the value for the crystalline acoustic phonons along the \[1 0 0\] direction. \[h\] ![High resolution IXS measurements in liquid potassium (open circles) [@mon_k]. The continuous line is the lineshape description according to a multiple relaxation memory function model (see text).[]{data-label="kixs"}](./figure/k_ixs.eps "fig:"){width=".4\textwidth"} \[h\] ![Viscous relaxation times as measured by means of IXS [@mon_k]. Circles: structural relaxation time. Triangles: microscopic relaxation time. The relaxation time obtained by means of INS is also reported [@cab_k], showing how it averages between the two mechanisms reported by IXS.[]{data-label="tau_k"}](./figure/k2.eps "fig:"){width=".4\textwidth"} \[h\] ![Dispersion curves (maxima of the current correlation function) measured by INS (open circles [@bov_k], stars [@cab_k]) and IXS (full circles [@mon_k])[]{data-label="kdisp"}](./figure/k_disp.eps "fig:"){width=".4\textwidth"} A recent IXS experiment on molten K [@mon_k] contributed to shed some light on some aspects of the collective dynamics, giving a coherent picture in terms of relaxation processes which is common to several other simple fluids and, more generally, to glass forming materials and molecular liquids (see fig \[kixs\]). First, it has been shown how the coherent dynamics is driven by thermal and viscous processes. These latter, which are dominant, proceeds over two different timescales. Consequently, the FWHM of the quasielastic (coherent) contribution is [*p*er se]{} not directly associated to any relevant timescale. The thermal process, indeed, is characterized by a timescale largely exceeding the Brillouin frequency, while both the viscous processes controls the quasielastic width. The corresponding relaxation times can be instead determined within the memory function formalism of Eq. (\[x3tempi\]), obtaining the results reported in fig. \[tau\_k\]. In the same plot, the relaxation time obtained by Cabrillo et al. (consistent with the determination of Bove et al.) is also reported. As one might expect, this value is somehow averaging between the two distinct viscous processes. More specifically, at low Q the one time approximation of Cabrillo et al. seem to mimic the slower process, while at higher Q is describing the faster process. This hypothesis is consistent with the observation, reported in other alkali metals, of a decreasing weight of the slow relaxation process on increasing the wavevector [@scop_jpc]. As far as the sound propagation properties are concerned, the IXS experiment analyzed in terms of generalized hydrodynamics suggest a minor role of the structural process, which accounts for approximatively $10\%$ of the whole positive dispersion effect, which is dominated by the faster process (see fig. \[kvelo\]). This observation, already reported for many other simple liquids, poses against the commonly invoked explanation of the positive dispersion in terms of transition from liquid to solid like regime (structural relaxation). \[h\] ![Sound velocities as determined by IXS: apparent (circles, from the maxima of $C_L(Q,\omega)$, isothermal (dotted line) and infinite frequency limit (continuous line) determined from Eqs.(\[a2\]) and (\[winf\]), respectively. The unrelaxed sound velocity values, due to the structural relaxation only (down triangle) and to the whole relaxation process (uptriangle) are also reported as estimated by the fitting.[]{data-label="kvelo"}](./figure/k_velo.eps "fig:"){width=".4\textwidth"} ### Rubidium Liquid Rubidium has been the first of the alkali metals to be addressed by a very famous neutron scattering experiment [@cop_rb], immediately substantiated by molecular dynamic simulations [@rahman_sim]. The reason for such interest lies in the possibility of extracting information on the collective dynamics, given the almost negligible incoherent cross section ($\sigma_i / \sigma_c \approx 10^{-4}$) and the relatively low sound velocity value. The result of this experiment contributed to open the route to the understanding of the collective dynamics in simple liquids, showing that the presence of an high frequency inelastic mode is an intrinsic property of the alkali metals not related to quantum properties or critical thermal population effect as suggested by earlier works on liquid hydrogen [@carn_h2]. Though the experiment was affected by an elaborated multiple scattering subtraction (due to the lack of an absolute normalization), which lead to a possibly unreliable quasielastic spectral component, some other chords of interest where hit. At variance with earlier results on liquid lead [@dor_pb], no evidence of secondary modes of transverse nature was reported in this system. Finally, a mild positive dispersion effect was observed (though Copley and Rowe looked at the maximum of $S(Q,\omega)$ rather than to the maximum of $J(Q,\omega)$) which was tentatively ascribed to the distinction of zero sound and first sound as discussed by Egelstaff [@EGELSTAFF]. \[h\] ![The experimental determination of constant $Q$ slices of the DSF by means of INS scattering in liquid Rubidium. A new era for the study of collective properties in simple liquids. From [@cop_rb][]{data-label="rbins"}](./figure/rb_ins.eps "fig:"){width=".4\textwidth"} More recently, an inelastic scattering experiment was performed with cold neutrons [@chi_rb] aiming at the determination of the dynamic scattering law in an extended temperature region beyond the one explored by Copley and Rowe. The kinematic accessed region is above $Q=9$ nm$^{-1}$, and therefore the observed excitations lies beyond the linear dispersion region. This work has the merit to stress the importance of the choice of the appropriate dynamical variable and of the fitting model to determine the dispersion curve. The old data of Copley and Rowe have been more recently reanalyzed in terms of generalized hydrodynamics [@mor_rbcs], comparing the results to the ones obtained in molten Cesium, which are discussed in section \[sec\_Cs\]. To our knowledge no IXS measurements have been reported on liquid Rubidium. The main difficulties for such an experiment would be the very small absorption length (about $200$ $\mu$m) and the quite low sound velocity value ($\approx 1400$ m/s) which would confine the elastic modes on the tail of the resolution function. ### Cesium \[sec\_Cs\] The experimental determination of the dynamic scattering law in liquid cesium is dated to the early nineties [@gla_cs; @bod_cs]. Due to the relatively small incoherent cross section and to the low sound velocity value, after Rubidium liquid Cs is the most favorable alkali metal aiming at the study of collective dynamics by means of INS. Despite its late outlet, the work of Gl[ä]{}ser and Bodensteiner reports an impressive state-of-the-art triple axis experiment, and a robust data reduction performed with innovative algorithms [@bod_phd]. \[h\] ![Dynamic structure factor of Cesium at the melting point (circles). The continuous line is the viscoelastic approximation. From [@bod_cs][]{data-label="csins"}](./figure/cs_ins.eps "fig:"){width=".4\textwidth"} The experiment is focused on the determination of the collective properties and, though a careful subtraction of the incoherent contribution is performed, once more it emerges the intrinsic difficulty of determining the quasielastic part of the coherent spectrum. This notwithstanding, the data clearly show the departure of the collective dynamics from the strict hydrodynamic region and the evolution toward the free streaming limit. This effect is quantified by the behavior of the FWHM of the quasielastic line reported in Fig.\[cswidth\]: while the hydrodynamic prediction, based on purely adiabatic thermal fluctuations, predicts $\omega_{1/2}=D_TQ^2$, the actual linewidth is always below this limit in the whole explored region, indicating the dominant presence of viscous processes in the quasielastic spectrum, as recently pointed out in other alkali metals and liquid aluminium [@scop_comm] against an opposite interpretation in terms of linearized hydrodynamic models [@sing_pre; @sing_rep]. The De Gennes narrowing was also observed [@dej_ema], and the $Q$ dependence of the linewidth was described within the hard sphere extended mode approximation (Eq.\[zh\_hs\], section \[sec\_kin\]). The free streaming limit is not yet attained at wavevectors as large as twice the position of the main peak of the static structure factor [@bod_cs]. The lineshape analysis was performed with several approaches, within the extended hydrodynamic model [@desh_hyd0], with the viscoelastic model [@lov_visco] and with two relaxation times accounting for both viscous and thermal processes. In this latter case, it was found a negligible role of the thermal process on approaching the first maximum of the structure factor, though the fitted values of the thermal relaxation time were in significant disagreement with the expected values $1/D_T Q^2$ (fig 9 of ref. [@gla_cs]). It was then pointed out the impossibility of discriminating the different models, given the s/n ratio of the available data. The $Q-$ dependence of the longitudinal viscosity, extracted from $S(Q,\omega=0)$ value (according to the prescription of generalized hydrodynamics), showed a decreasing behavior previously observed in Lennard-Jones systems [@aila_visco; @tank_visco] and more recently in several other liquid metals [@scop_prlga; @mon_k]. Finally, the maxima of current correlation spectra showed the usual positive dispersion effect, which was interpreted as precursor of the solidification according to the usual idea of a transition between a liquid and solid-like regime. \[h\] ![Linewidth of the coherent quasielastic spectral component in molten Cesium (circles). The continuous line is the thermal value $D_TQ^2$, while the dashed line is the hard sphere prediction from Eq.(\[zh\_hs\]). From [@bod_cs][]{data-label="cswidth"}](./figure/cs_width.eps "fig:"){width=".28\textwidth"} Neutron scattering data on liquid cesium have also been used as a benchmark to develop an approach based on the idea of timescale invariance of the relaxation processes [@yulm_cs]. Within the Zwanzig Mori projectors formalism, one can construct an infinite, non Markovian, set of interconnected kinetic equations relating each memory function with the one of higher order [@mori_mf]: $$\begin{aligned} \frac{dF(Q,t)}{dt}&=&-\Omega_{1}^{2}\int_{0}^{t}d\tau M_{1}(Q,\tau)F(Q,t-\tau) \nonumber \\ \frac{dM_{1}(Q,t)}{dt}&=&-\Omega_{2}^{2}\int_{0}^{t}d\tau M_{2}(Q,\tau)M_{1}(Q,t-\tau) \nonumber \\ ..............&.&................................................ \nonumber \\ \frac{dM_{i}(Q,t)}{dt}&=&-\Omega_{i+1}^{2}\int_{0}^{t}d\tau M_{i+1}(Q,\tau) M_{i}(Q,t-\tau). \label{chain} \end{aligned}$$ where $F(Q,t)$ is the normalized density correlation function and $\Omega_i$ are characteristic frequencies of the process. Following the Bogoliugov approach of the reduced description, one hypotizes the time scale invariance of the relaxation processes beyond a certain level, defining a closure level $M_{i+1}(t) \approx M_{i}(t)$ and thus getting an explicit expression for the DSF in terms of the spectral moments. To our knowledge no IXS experiments on liquid Cs have been reported, for the same kind of difficulties as in liquid Rb. Alkaline earth elements ----------------------- ### Magnesium Magnesium is one of the simplest divalent elements. The coherent dynamic structure factor has been recently determined at the SPring8 IXS beamline [@kaw_mg]. The dispersion curve shows a $8 \%$ deviation from the adiabatic sound velocity, with a maximum value lying halfway the hydrodynamic and the $c_\infty$ value. An average relaxation time was determined ($\tau=0.094$ ps), which is about one third of the one of the neighboring alkali element liquid Na [@pil_na]. The analysis of the quasielastic line revealed a $Q^2$ broadening in the quasi-hydrodynamic regime, while around the De Gennes narrowing region the linewidth was successfully reproduced by the de Schepper and Cohen model [@desh_hs], i.e. through Eq. (\[zh\_hs\]). Molecular dynamics simulations have been recently performed in this system by both classical molecular dynamics and orbital free *ab initio* simulations [@alem_mg; @gonz_potalc]. The two approaches give very similar results as far as the phonon dispersion is concerned, while the quasielastic contribution is less pronounced in the *ab initio* calculation. In this respect, a comparison with the experimental data [@kaw_mg] would be extremely interesting, though one should first convolute the calculated $S(Q,\omega)$ with the instrumental IXS resolution and take into account for the quantum correction arising from detailed balance condition. Group III elements ------------------ ### Aluminium One of the most puzzling results of early neutron spectroscopy is the striking similarity between the spectra of polycrystalline and liquid Aluminium observed in Stockholm in 1959 and published in the final form a few years later [@lar_al]. In this experiment TOF data were collected on a cold neutron spectrometer, but at those times multiple scattering corrections were almost impossible and therefore the spectra did not obtain a detailed explanation. A second TOF experiment was performed in the same period, but again the results were laking a detailed interpretation [@coc_al]. Fifteen years later the original experiment of Larsson was revisited, in an effort to re-analyze the results in the light of up to date theoretical developments. More specifically, the experimental results were used to test the old convolution approximation [@vine_convo], mean field approaches [@sing_meanf] and kinetic theory [@sjog_kin; @sjogr_al]. Nevertheless, these data have been collected at very large energy and wavevectors value and always presented as time of flight scan, and therefore they are not very helpful to establish the existence of collective modes in this system. Aluminium, indeed, is an almost purely coherent scatterer, but the high sound velocity value prevents the study of acoustic modes by means of INS (see table \[table\]). This is testified by a more recent INS experiment performed at IN4 (ILL) [@eder_al]. Although multiple scattering correction and constant -$Q$ cuts reduction have been performed in this case, no evidence of collective modes could be reported due to the restricted kinematic region corresponding to the slow neutrons utilized ($\approx 55$ meV). \[h\] ![Dynamic structure factor of molten aluminium. Comparison between the OF-AIMD calculation (continuous line) [@gonz_al] and the experimental IXS values with their fitting based on generalized hydrodynamics [@scop_preal]. From [@gonz_potalc][]{data-label="al"}](./figure/al_ixs_md "fig:"){width=".45\textwidth"} Much of the knowledge about the microscopic dynamics in liquid Al relies, in fact, on the numerical work of Ebbsj[ö]{} [@ebb_al], who calculated the dynamic structure factor utilizing two different local pseudopotentials and the local pseudopotential originally developed by Ashcroft [@ashc_potalk]. The dynamic structure factor has been shown to be somehow reminiscent of the viscoelastic model with the addition of a gaussian term, accounting for the approach to the high $Q$, free streaming limit. He was then able to predict the existence of collective modes for $Q<10$ nm$^{-1}$, though he reported sound velocity values distinctly larger then the adiabatic value over the whole explored range ($Q>3$ nm$^{-1}$). Triggered by this observation, a modified version of the viscoelastic approach was developed [@gas_al] and tested against the data of Ebbsj[ö]{}. At the same time, it has been proposed a single relaxation process scenario based on a $sech$ memory function shape within the Mori-Zwanzig scheme [@tank_al]. \[h\] ![Sound dispersion of liquid aluminium from the maxima of the current correlation function: open circles, OF-AIMD calculation [@gonz_al], full circles IXS experimental values [@scop_preal]. The dispersion from the maxima of the dynamic structure factor numerically evaluated is also reported (open triangles), as well as the hydrodynamic value (continuous line). From [@gonz_al][]{data-label="aldisp"}](./figure/al_disp "fig:"){width=".45\textwidth"} \[h\] ![Resolution deconvoluted, classical lineshape utilized to described the IXS spectra of molten Li [@scop_prlli; @scop_prena; @scop_preal], Na and Al, reported on relative momentum and energy scale (see text)[]{data-label="linaal"}](./figure/li_na_al_deco "fig:"){width=".45\textwidth"} The first experimental observation of collective modes in liquid Al has been reported much more recently by means of IXS [@scop_preal]. An high frequency regime has been observed for $Q>5$ nm$^{-1}$, while below this value the dynamics approaches the hydrodynamic limit, though the transition is not fully accomplished at the lowest investigated wavevector, $Q=1$ nm$^{-1}$. The scenario arising from the IXS study is much similar to the one characterizing alkali metals, though with significant quantitative differences such as a more intense quasielasic component testifying a more important role of the structural relaxation in this system. The IXS data have been recently used to test orbital free *ab initio* calculations (OF-AIMD) [@gonz_al; @gonz_potalc]. The overall agreement is quite satisfactory though the numerical calculations show somehow lower sound velocity value and tend to overemphasize the inelastic components. From Fig.\[aldisp\] one can argue the importance of the dynamical variable representing the sound velocity. The presence of positive dispersion, in particular, is strongly affected by the choice of the maxima of $C_L(Q,\omega)$ rather than the ones of $S(Q,\omega)$. In Fig. \[linaal\], finally, we report a comparison of the lineshape obtained from the resolution-deconvoluted, classical lineshape utilized to fit the IXS spectra of liquid lithium, sodium and aluminium at the same reduced values of momentum ($Q/Q_M$) and energy ($\omega t_0=\omega / \omega_0$ with $t_0=\sqrt{m/k_BT_m}/Q_M$) transfer. As can be clearly evinced, the attitude of alkali metals to sustain density fluctuation is much more pronounced than in other simple liquid metals. ### Gallium Among simple liquid metals, Ga is endowed with peculiar structural and electronic properties. In addition to the low melting point ($T_m=303$ K), it shows an extended polymorphism in the solid phase with complex crystal structures where a competition between metallic and covalent bindings takes place. Despite the nearly free electron electronic DOS anomalies associated with some covalency residue are present. Moreover, the first peak of the $S(q)$ presents a hump characteristics of non close-packed liquid structures [@bf]. Early inelastic scattering on liquid Ga were performed at the beginning of the seventies with neutrons [@loff_phd; @pag_ga; @bos_ga; @gla_ga]. Due to the quite large sound velocity value, compared to the available kinematic range, these studies were mainly addressed to the investigation of the quasielastic part of the dynamic structure factor. Twenty years later, another series of INS experiments were performed on a triple axis instrument [@ber_ga1] with the aim of ascertaining the presence of low $Q$ collective modes just above the melting temperature. Indeed, although by virtue of the above mentioned anomalies liquid Ga seems to elude the picture of the high frequency dynamics emerging in all the monatomic liquids, on the basis of the shape of the interaction potential evidence for collective modes should be expected below $Q_m/2$. Quite surprisingly, no evidence of inelastic signal was reported in the region were longitudinal modes were expected on the basis of the hydrodynamic sound velocity. This result was interpreted as an overdamping effect traced back to the high value of longitudinal viscosity ($\approx 10$ cP). Additionally, by comparing constant energy scan (see Fig.\[gaberqscan\]) to the expression of Buchenau for acoustic-like plane wave excitations in amorphous solids [@buch_Q], an excess of scattering was reported for frequency distinctly larger then the maximum frequency of the acoustic branch. This result was interpreted as the fingerprint of the presence of high energy optic-like modes. \[h\] ![INS constant energy scans (circles) compared to the model of Buchenau for plane wave excitations in solids [@buch_Q]. From [@ber_ga1].[]{data-label="gaberqscan"}](./figure/ga_ber_qscan "fig:"){width=".45\textwidth"} A few years later, then, a new set of experiments were performed at higher temperatures by the same authors [@ber_ga2], and a contrasting behavior with the previous findings was reported. More specifically, the appearance of non-overdamped sound modes was reported, accompanied by a second, higher frequency mode of presumably optical origin. The discrepancy between the low and high temperature experiment was ascribed to a viscosity drop of a factor $\approx 7$ and therefore to a narrowing of the acoustic mode according to the hydrodynamic expression \[gidro\]. The presence of an higher frequency mode appearing at in the constant $Q$ scan for wavevectors larger then $Q_m$ seems to corroborate the presence of an optic-like excitation suggested by the previously mentioned constant energy scans [@ber_ga1]. \[h\] ![Comparison between the INS (stars [@ber_ga1] and triangles [@bov_ga]) and IXS (open circles with error bars [@scop_prlga]) determinations of $S(Q,\omega)$ in gallium at the melting temperature, for two different values of the (constant) momentum transfer. The dotted line is the viscoelastic prediction [@ber_ga1] (convoluted with the INS instrumental resolution and accounting for the detailed balance condition), while the continuous line is the best fit to the IXS data utilizing a memory function accounting for the thermal relaxation and two viscous processes (see text). Molecular dynamics symulations for the coherent (dotted line) and total (dash-dotted line) $S(Q,\omega)$, convoluted to the INS resolution, are also reported [@bov_ga].[]{data-label="gaixsins"}](./figure/ga_ixs_vs_ins "fig:"){width=".45\textwidth" height="11"} A recent IXS experiment performed on liquid Gallium just above the melting point [@scop_prlga] portraits the collective dynamics in a much similar fashion to the one of Alkali metals and of liquid Al, thus removing the anomaly suggested by the neutron experiments. Collective modes, in fact, have been unambiguously observed in the low temperature region where neutrons suggested an overdamped regime. This result suggests the inadequacy of the expression \[gidro\] to estimate Brillouin linewidths, which can be easily understood in terms of the generalized hydrodynamics results reported for alkali metals [@scop_jpc]: outside the truly hydrodynamic region, the viscous relaxation dynamics proceeds over two distinct physical mechanisms, the structural relaxation and the short-lived rattling dynamics. On the high frequency region of interest, the structural relaxation is frozen (the system is responding as a solid) and therefore the viscosity associated to this process does not contribute to the sound damping. Moreover, the thermal contribution in Eq. (\[gidro\]) might not be correct at wavevectors as large as a few nm$^{-1}$, since the adiabatic regime could be replaced by as isothermal one, as already pointed out [@scop_comm]. Consequently, Eq.\[gidro\] is an overestimate of the actual linewidth (which in the case of liquid lithium has been quantified as a factor two [@scop_jpc]). In Fig. \[gaixsins\] we report the two experiments for similar values of the momentum transfer. The viscoelastic prediction [@lov_visco; @ber_ga1] is also reported, showing how it clearly underestimates the quasielastic contribution, though it provides a reasonable estimate of the genuine inelastic mode. The IXS findings have been recently corroborated by a new accurate INS experiment, used to test the reliability of a model interaction potential by comparing the dynamic structure factors [@bov_ga]. Summing up, the lack of low temperature collective excitations reported in this system with neutrons is probably due to the difficulty of a reliable extraction of the coherent part of the scattering function. On the other side, the interesting observation of possibly optic-like high frequency modes certainly calls for further investigation although, in our opinion, does not justify *per se* a description of the acoustic dynamics in terms of crystalline reminiscent dynamics. The analogy with several IXS results on different systems suggests indeed a prominent role of the topological disorder in characterizing the acoustic branch. \[h\] ![Full width at half maximum of the dynamic structure factor as determined by IXS in the kinetic regime (open circles). The prediction according to Enskog’s theory is also shown (continuous line) [@scop_prlga2]. []{data-label="gahs"}](./figure/ga_hs "fig:"){width=".45\textwidth"} Very recently, an IXS experiment explored the high $Q$ region ($20<Q<70$ nm$^{-1}$, i.e. lengthscales smaller than the size of the first coordination shell [@scop_prlga2]. While generalized hydrodynamics provides a coherent picture of the dynamics in the lower $Q$ region, not much is known about collective dynamics at such short lenghtscales. For hard spheres, Enskog’s kinetic theory predicts in this region the dominant effect of a generalized heat mode. Liquid Gallium, however, by no means can be modelled as an hard sphere fluid, for the above mentioned structural and electronic properties. Surprisingly, it turned out that a description in terms of heat mode (Eq. (\[zh\_hs\])) still applies, at the price of introducing an *effective* hard sphere diameter (larger than the one associated to the first $S(Q)$ maximum), which probably accounts for the associative tendency of this liquid (dimer-like structures). Group IV elements ----------------- ### Silicon Due to its several unusual properties, liquid Si is always classified as a non simple liquid. While in the crystalline phase Si is a diamond structure semiconductor, it undergoes a semiconductor-metal transition upon melting, which is accompanied by a density increases of about $10\%$, and by significant structural changes (the coordination number grows from four in the solid state to about seven in the liquid). Similarly to Gallium, the static structure factor, S(Q), exhibits a shoulder on the high-Q side of the first peak [@waseda_sn], a feature that cannot be reproduced using a simple hard-sphere model, appropriate for alkali metals. No neutron scattering data exist to the best of our knowledge, while very recently two Inelastic X ray Scattering experiments have been performed both at the ESRF [@hos_si1] and SPring8 [@hos_si2]. \[h\] ![Sound velocity and attenuation in molten silicon [@hos_si2][]{data-label="si"}](./figure/si "fig:"){width=".45\textwidth"} In the first of the two above referenced experiments, a positive dispersion of $15 \%$ has been found. In the second experiment, an higher resolution and more accurate study has been carried out, allowing to follow the transition from the high frequency to the low frequency regimes. The data were analyzed in terms of DHO model for the inelastic component and Lorenzian for the quasielastic, and no significant quantitative differences were detected utilizing the same memory function scheme proposed for other IXS studies on liquid metals [@scop_jpc]. In the vicinity of the main peak of the static structure factor, the lorenzian shape for the quasielastic component turned out to be inadequate, and has been replaced by a combination of Lorenzian and gaussian contributions (pseudo-Voigt). ### Germanium Liquid Ge shares the same peculiarities as liquid Silicon, though with some slight quantitative differences. IXS data on liquid Ge have been recently obtained [@hos_ge], and they show evidence for collective propagating modes. An analysis based on a Lorenzian shape for the quasi-elastic and a DHO for the inelastic modes, revealed the absence of positive dispersion effects in the investigated Q range ($2\div 28$ nm$^{-1}$). On our opinion, this result calls for further investigations, as this is an almost unique feature in respect to the other monoatomic liquid metals investigated sofar (especially Silicon, which has very similar structural properties). The De Gennes narrowing has been analyzed in terms of extended hydrodynamic heat mode, utilizing the analytical expression obtained within hard-sphere approximation [@coh_hs], but the quite large error bars and the limited spanned $Q$ range prevents to draw a final conclusion. ### Tin Tin is the heaviest of 4B element. Its structural properties are quite similar to the one of alkali metals, with a coordination number close to 12 but with the typical shoulder on the high Q side of the main $S(Q)$ peak [@waseda_sn] which is typical of Si, Ge and Ga. The first INS experiments in liquid Tin have to be traced back to the early sixties. Similarly to other early neutron scattering experiment no clear picture of the microscopic dynamics could be outlined. In some case [@pal_tin]the vineyard approximation [@vine_convo] was used to analyze the data, while in another study experimental strategy for suppressing multiple scattering were tested [@broc_tin]. Although a wavevector-energy plot was obtained from raw TOF data [@coc_tin], according to Copley and Lovesey no side peaks should be observed trasforming TOF data to $S(Q,\omega)$ on constant $Q$ slices [@coplov_rev]. Constant $Q$ IXS spectra of liquid tin have been very recently reported for low ($T=593 K$) and high ($T=1273 K$) temperatures at the ESRF [@hos_sn]. The sound velocity seems to exceed the hydrodynamic value at both temperatures of $6\%$ and$12\%$, respectively. This notwithstanding, these quantitative estimates must be taken with care, due to the quite large error bars. Moreover, the dispersion curves have been determined from the DHO frequency, neglecting the effect of the quasielastic component. For $Q$ values close to the main peak of the structure factor, as in the case of liquid Si [@hos_si2], the lineshape of $S(Q,\omega)$ turned out to be empirically described by a combination of Lorentian and Gaussian contributions or, equivalently, by a memory function analysis similar to the one reported for alkali metals [@scop_prl]. As a matter of fact, at such large $Q$ values, as previously observed for liquid lithium [@scop_epl; @scop_jpc], the microscopic dynamics undergoes a transition from the (generalized) hydrodynamic behavior to the free streaming limit and a detailed description of such transition is still missing. ### Lead Molten lead has been one of the first metals to be investigated by Inelastic Neutron Scattering, as the first experiment can be traced back to the fifties [@egel_pb; @brock_pb]. Details of the experiments performed up to 1975 [@dor_umk; @rand_umk; @dor_pb; @coc_pb] have been exhaustively reviewed by Copley and Rowe [@coplov_rev]. One of the most interesting results has been the evidence of both a longitudinal and a transverse branch in the dynamic structure factor, though this result was presented with some caution as the transverse mode could be an artifact arising from multiple scattering effects. In the early eighties new INS studies performed with both TAS and TOF spectrometers appeared [@sod_pb; @sod_pb1], aiming to validate the presence of a dispersion relation and of a transverse branch. A longitudinal mode, compatible with the higher frequency excitation previously reported by Dorner [@dor_pb] was reported, whose sound velocity is consistent with hydrodynamic value. The accessible kinematic range is too limited to asses any evidence of positive dispersion effect. No evidence for a lower frequency excitation was instead reported, corroborating the hypothesis that such feature is an artifact stemming from multiple scattering. Liquid lead has been recently in focus of molecular dynamic simulations aiming to describe collective dynamics in terms of generalized kinetic modes [@bry_pb1; @bry_pb2]. Beyond the hydrodynamic region, different branches corresponding to sound and heat waves have been identified, and their nature has been extensively discussed. Group V elements ---------------- ### Bismuth The first inelastic scattering investigation of molten bismuth was originally reported by Cocking [@coc_bi], who reported a dispersion curve extracted by TOF neutron spectra. At the IAEA symposium of 1968 two experiments Bi were presented: in one case two dispersion branches were obtained from TOF data [@tun_bi], though the low frequency excitation was probably due to an artefact of a missing multiple scattering correction. In the second study data were converted to constant $Q$ spectra [@mat_bi]. \[h\] ![FWHM of molten Bi compared to other simple liquids. It can be noted the presence of an intermediate minimum between the first two maxima which is not present in the other simple fluids. From [@dal_bi].[]{data-label="bi_ins"}](./figure/bi.eps "fig:"){width=".4\textwidth"} Liquid Bi has been recently the subject of new INS investigations [@dal_bi]. Measurements were performed just above melting at $T=578$ K, in a wavevector region spanning from just below the first maximium in $S(Q)$ up to wavevector as high as $70$ nm$^{-1}$. This kinematic region lies above the region were collective modes could be expected, so the study mainly deals with the quasielastic spectral component. Generalized hydrodynamics models based upon a single relaxation time were tested against the experimental data utilizing different approximation for the memory function shape [@lov_visco; @aila_visco]. Experimental data were then unfolded by instrumental resolution modelling the quasielastic shape as Lorenzian and the resolution as gaussian. The main point of the paper is the determination of the $Q$ dependence of the spectral FWHM, which is also compared to other systems. In particular, it is pointed out how, beside the expected De Gennes narrowing occurring at $Q_m$, the FWHM shows a minimum rather then the expected maximum at $Q \approx 1.5 Q_m$ (which characterize the dynamics in several simple liquids such as Pb, Rb and Ar). This anomaly is related by the authors to the shoulder observed in the static structure factor just above $Q_m$, and therefore identified as a non-simple nature of liquid Bi. The kinetic region has been investigated in liquid Bi within the generalized collective mode approach [@bry_bi1]. The presence of high frequency kinetic branches has been ascertained, and it has been pointed out that their weight is too small to make them visible in the dynamic structure factor. This result seems to be in agreement with recent IXS findings on a very similar system, namely liquid Ga, in which only acoustic modes were reported [@scop_prlga]. Transition metals ----------------- ### Mercury Experimental studies of microscopic dynamics in liquid mercury are very recent compared to the systems reviewed so far, and they were presented at the LAM XI conference (Yokohama, Japan). The first investigation was obtained by means of INS at the IN1 facility of the ILL [@bov_hg; @bov_hg_lam]. In this work, an detailed investigation of the dynamic structure factor is undertaken at room temperature, and presented as constant $Q$ cuts in the range $2.5 \div 12$ nm$^{-1}$ with a high energy resolution of $\delta E \approx 1$ meV. The data are analyzed with an empirical model consisting of a DHO for the purely inelastic part and the sum of two lorentian functions accounting for the quasielastic contribution. While the inelastic component is no doubt of coherent nature, the narrower of the two lorentians is ascribed to incoherent scattering, and modelled as a simple diffusive term of linewidth $DQ^2$. Given the values of $D$, this results in a quasi elastic contribution which could not be resolved by the much broader resolution function. The linewidth of the broader lorentian is almost $Q$ independent, and its origin is ascribed by the authors to an incoherent process, on the basis of the coincidence of the DHO area with independent (static) structure factor determinations. Turning our attention to the collective dynamics, the extrapolated high frequency value of the sound velocity ($c_\infty(Q\rightarrow 0)=2100 \pm 80$ m/s) , obtained by the DHO frequency parameter (therefore approximately equal to the maximum of the current correlation function, the difference being due to potential quasielastic coherent contribution), turns out to largely exceed the hydrodynamic value ($c_s=1451$ m/s), suggesting a huge positive dispersion effect close to $50\%$ largely exceeding similar effects reported in other metals. This result is rationalized in terms of Bohm-Staver model, which provides the expression of Eq.(\[BS\]) for the sound velocity yielding, for molten Hg, $c_l(Q\rightarrow 0)\approx2090$. \[h\] ![DSF of molten Hg measured by means of INS at the indicated $Q$ values. The inelastic and the quasielastic componens, modeled with two lorenzians and a DHO, respectively, are also shown. From [@bov_hg][]{data-label="hg_ins"}](./figure/hg_ins.eps "fig:"){width=".4\textwidth"} Nearly in coincidence with the neutron experiment of Bove *et al.*, an IXS study of the coherent dynamics in liquid Hg appeared [@hos_hg], in which the dynamic structure factor is examined in a wavevector region extended up to $Q_m \approx 25$ nm$^{-1}$, with a factor 2 worse resolution, but in a significantly larger energy region. As in the work of Bove *et al.*, the genuine inelastic features of the data are modelled with a DHO function, but it appears clearly from the raw data that a coherent quasielastic term dominates the $\omega \approx 0$ region. This latter contribution is modelled with a lorenzian shape. Although the authors neglect the presence of this quasielastic in the calculation of the sound velocity value, taking the DHO frequency as the relevant parameter they obtain a value of $c_\infty(Q\rightarrow 0)=1840$ m/s, an estimate which is directly comparable with the corresponding INS determination. This discrepancy, which may be due to the limited energy range at the low $Q$’s of the INS experiment, as well as to the non negligible resolution effect on the lowest $Q$’s of the IXS experiment, calls in our opinion for further investigations and suggests to take with some care any interpretations in terms of Bohm Staver model of the positive dispersion at least by the quantitative point ov view. Hosokawa *et al.*, on the other hand, cast the anomalous dispersion in the usual framework of the shear relaxation. The IXS work also confirms the non negligible presence of quasielastic signal in the coherent dynamic structure factor, suggesting that it should be taken into account also in INS data treatment. We expect that the broader lorenzian contribution reported by Bove *et al.*, for example, could be at least partially coherent in nature. \[h\] ![Left panel: IXS determination of the DSF in Hg near the melting temperature. Right upper panel: dispersion relation and sound attenuation properties as deduced by a DHO + one Lorenzian fit. The low and high frequency limits are also reported. Right lower panel: Corresponding sound velocities. From [@hos_hg].[]{data-label="hg_ixs"}](./figure/hg_ixs.eps "fig:"){width=".4\textwidth"} A more recent INS study contributed to shed some light on the possible origin of the quasielastic spectral components [@bad_hg]. The experiment has been performed on the TOF spectrometer MARI, optimized to access a restricted kinematic region ($-6<\omega<6$ meV at the lowest accessed momentum transfer $Q\approx 4$ nm$^{-1}$) with an energy resolution sufficient to study the diffusive dynamics ($\delta E=0.4$ and $\delta=0.8$ meV at the two incident energy utilized). The narrower incoherent contribution (self diffusion), resolution limited in the experiment of Bove *et al.*, was now detected with a procedure similar to the one applied in liquid potassium [@cab_k], i.e. fitting the data with two lorenzian in a restricted wavevector region where the two quasielastic features are well separated, determining the analytical $Q$ dependence of the FWHM of the diffusive term, and finally focusing on the broader component over the whole momentum transfer region keeping all the parameters of the diffusive term fixed. With this procedure, the narrower lorenzian is confirmed to be incoherent in nature, and well described by Fick’s law properly modified according to the revised Enskog’s theory [@kag_hs]. As far as the broader component is concerned, the authors point out that more than one lorenzian is needed to describe it at increasing wavevector, then they discuss its possible origin. First, they point out that the thermal origin of this broad component is ruled out by its low $Q$ intensity, which is by far larger than the one expected by the Landau Plazeck ratio. Moreover, the linewidth reaches a constant value on decreasing wavevector, instead of following the $Q^2$ dependency of the heat mode. Second, they examine the possibility of a cage diffusion mechanism, as proposed in MD simulations [@bov_hgsim]. In this respect, they point out how the experimentally observed mode intensity is too large than expected, but they propose an enhancement mechanism based on valence fluctuations which could be active at low wavevectors amplifying the expected intensity. Very recently, new state-of-the-art IXS experiments have been reported in expanded mercury near the critical point ($T_c=1751$ K, $P_c=1673$ bars and $\rho _c=5.8$ g cm$^{-3}$), aiming at the investigation of collective dynamics at the metal-non metal transition [@inui_sn]. Despite extremely difficult experimental conditions, the speed of sound has been accurately measured and no significant changes have been observed in the transition from the metallic ($m\rho _c=13.6$ g cm$^{-3}$) to the insulating ($m\rho _c=9.0$ g cm$^{-3}$) phase, while the ultrasonic sound velocity exhibits a significant drop across the same thermodynamic point [@yao_sn]. Only upon further expansion in the insulating phase the high frequency sound velocity ultimately drops reaching the adiabatic value. This result seems to indicate the presence of very large positive dispersion as peculiar of the metal-non metal transition, opening a new experimental route to the investigation of the interplay between acoustic and transport properties. Summing up, though the microscopic dynamics in molten Hg has been the subject of deep investigations in the last few years, some aspects still remain controversial. The sound velocity as determined by INS and IXS are not fully consistent with each other, though both techniques show positive dispersion effect the INS result show a very large effect never observed sofar. On the one side it has been emphasized the role of electronic properties [@bov_hg], while, on the other side, the collective dynamics as determined by IXS closely resemble the one of several other simple fluids [@hos_hg]. The most intriguing aspect concerns, however, the interpretation of the quasielastic component of $S(Q,\omega)$. Neutron scattering data suggest a negligible effect of thermal fluctuations [@bad_hg], adding a piece of information to a recently debated issue [@scop_comm; @sing_pre; @sing_rep]. On the other side, the IXS data unambiguously show the presence of a coherent quasielastic dynamics, which no doubt has to show up in neutron experiment as well. The broad quasielastic component as observed with two different experiments show however opposite $Q$ trends, monotonically increasing [@bov_hg] and decreasing [@bad_hg], respectively. Badyal *et al.* suggest a cage diffusion process, enabled by valence fluctuations mechanism. We believe that the broad mode observed in INS could be closely related to the coherent quasielastic scattering reported in the IXS data. In this case, the similarity with several other investigated systems would suggest an interpretation in terms of a high frequency structural relaxation process [@scop_prlli; @scop_preal; @scop_prena; @scop_prlga; @mon_k]. Very recent IXS investigations, however, suggest an enhancement of positive dispersion at the metal non metal transition, pointing out how changes in electronic transport properties dramatically affect acoustic properties [@inui_sn]. ### Nickel Early investigations in liquid Nickel have been reported in 1977 with TOF technique [@joh_ni]. Two different isotopic concentrations, one with the natural abundance ratio and the other a wholly incoherent mixture, were chosen in order to study separately the coherent and incoherent scattering contributions. These latter, in turns, have been related each other through the Vineyard approximation [@vine_convo]. The spectral density $g(\omega)$ was then extracted from the low $Q$ limit of the self dynamic law. The coherent $S(Q,\omega=0)$ were reported for wavevectors above $Q=22$ nm$^{-1}$, therefore beyond the first brillouin pseudozone. Consequently, this study was not able to ascertain the existence of low $Q$ collective modes. A neutron scattering experiment has been recently performed at the IN1 facility of the ILL [@ber_ni]. Constant $Q$ spectra have been collected from $Q=8$ nm$^{-1}$ all the way up to wavevectors as high as twice the main peak in the static structure factor. The data have been analyzed utilizing two lorentian terms for the quasielastic (coherent and incoherent) contribution and one DHO function for the purely inelastic spectral features. Evidence of collective modes has been reported in the whole investigated Q domain, despite the relatively high viscosity value ($\eta_s \approx 5.7$ mPa s, i.e. one order of magnitude larger than the typical values for alkali metals). The reason for this apparent oddness can be traced back to arguments similar to the one applying to the case of gallium, i.e. to the freezing of the diffusional motion over the probed high frequency regime, which reduces the effective Brillouin damping in respect to the hydrodynamic prediction of Eq. (\[gidro\]). \[h\] ![Sound velocity in liquid Nickel determined by INS (full circles) and MD (open circles). The isothermal ($c_t$) and adiabatic ($c_s$) values are also indicated.[]{data-label="ni_disp"}](./figure/ni_disp.eps "fig:"){width=".7\textwidth"} More interestingly, the low $Q$ limit of the sound velocity seems to approach the isothermal and not the adiabatic value, as shown in Fig. \[ni\_disp\]. It is worth to stress how, given the large value of $\gamma$ (and therefore the large differences between $c_t$ and $c_s$), this observation strikingly holds beyond the reported experimental error. If confirmed, this result would substantiate the hypothesis of an intermediate isothermal domain bridging the hydrodynamic limit and the high frequency regime, which has been recently matter of debate [@sing_pre; @scop_comm; @sing_rep]. On the other side, previous molecular dynamics simulations indicate higher values of the sound velocity which agree quite well with the adiabatic response [@alem_sim]. Summing up, liquid Ni seems to be an ideal system to test the evolution of the thermal relaxation once the hydrodynamic limit is abandoned. An IXS investigation would be helpful to clarify this issue, though the small absorption length, the high melting temperature and the reactivity of Ni poses severe experimental challenges. ### Copper Time of flight neutron scattering data have been reported for this system long time ago in solid and liquid phase. The accessed kinematic range was quite broad ($Q>10nm^{-1}$ and $E<30$ meV) and the data have been analyzed within pioneering models [@egel_model]. More recently, an IXS experiment has been performed at the ESRF, though this work is still in progress a very preliminary estimate of the sound velocity gives a value of $4230 \pm 70$m/s, well above the hydrodynamic value. Solutions of metals ------------------- Alkali metals easily dissolve in water, molten alkali halides and ammonia, resulting in a free electron and a positively charged ion. Given the relatively low electronic concentration of the saturated solutions, these are ideal systems to challenge the validity extents of plasma-based theories introduced in section \[plasma\], and the relative approximations for the dielectric function. Models like the RPA, indeed, are expected to hold in systems with low $r_s$ (or, equivalently, high electronic density) such as bulk metals, while they reach their limits on increasing the $r_s$ value. A second interesting aspect concerns the presence in liquid metals of the so called Kohn anomaly, i.e. a kink in the dispersion curve occurring at $Q=2k_F$ and reflecting a singularity of the dielectric function which is observed in metallic crystals, but which is not yet established in the disordered phase. The high frequency dynamics of metal-ammonia systems, in particular, have been recently investigated by means of X-rays and neutrons. High resolution IXS performed in lithium-ammonia solutions allowed to detect high frequency excitations, softening at twice the Fermi wavevector $k_f$ [@burns_prl1]. Unfortunately, at the investigated concentrations $2k_F$ is close to $Q=Q_m$, i.e. the main peak of the static structure factor. This coincidence generates an ambiguity in the interpretation of the observed dip in the sound dispersion, as it is not clear whether this feature is related to the structural periodicity as in non ordinary fluids or has to do with a purely electronic effect. The reported sound velocity values are intermediate between the one of the bare ions and the one appropriate for the Li(NH$_3$)$_4$ vibrating network, though close to this latter. In a later study [@burns_prb], collective excitations are rationalized in terms of ionic plasma oscillations, and the sound velocity values measured at several metal concentrations are compared with the prediction of the Bohm-Staver expression of Eq.(\[BS\]), taking as relevant mass either the bare ionic value or the metal-ammonia unit. In both cases the predictions do not agree with the measured values, and this seems to be a signature of the well known failure of the RPA approximation in the low-density regime, where the electron-electron interactions are relevant. A contemporary INS study on deuterated ammonia [@sacc_ammonia] also addresses the deviations from the BS model. In this case an improvement is achieved accounting for both the finite ionic size and the electron-electron interactions. As far as the first aspect is concerned, an alternative renormalization of the free ionic plasma frequency is undertaken, while an expression for the dielectric screening function, going beyond the RPA approximation of Eq.(\[TF\]) and accounting for local fields effects, is proposed. As far as the Kohn anomaly is concerned, this study suggests that the value of $2k_f$ ($\approx 10$ nm$^{-1}$ in saturated lithium-ammonia solutions) has to be compared with $Q\approx 20$nm$^{-1}$, which is actually the second peak of $S(Q)$, related to the $N-N$ periodicity, rather than with the first peak ($Q_m \approx 20$nm$^{-1}$) related to the Li(NH$_3$)$_4$ periodicity. This might arise from the different X ray and neutrons cross sections: while in the first case the two peaks have similar intensities, the neutron diffraction strongly enhance the $N-N$ peak. A later study has shown how an unambiguous separation between $2K_F$ and $Q_m$ occurs in low density solutions, although in this case the kink observed in the dispersion curve is almost within the error bars [@giura_liamm]. Further studies, therefore, seem to be necessary to draw a conclusive picture about the presence of the Kohn anomaly in metallic fluids. \[h\] ![Random phase approximation at work: data taken from Table \[table\] and [@bov_ga] for pure elements and from [@giura_liamm; @burns_prb] for alkali-ammonia solutions, with the BS values estimated through Eq. (\[BS\]). Full circles: the measured sound velocity, $c$, is the maximum high frequency value determined over the $Q$ range probed by inelastic scattering techniques, i.e. includes the positive dispersion effect. Open circles: $c$ is here the adiabatic value. The subscript of Li and Na indicates the concentrations in ammonia solutions. The Na group is relative to several different temperatures.[]{data-label="rpa"}](./figure/rpa.eps "fig:"){width=".5\textwidth"} The validity extent of the RPA approximation for the determination of the sound velocity via the Bohm-Staver expression (\[BS\]) is depicted in fig. \[rpa\], in which we report the relative deviations of the BS calculated values ($c_{RPA}$) from the experimental ones, for systems of different $r_s$, ranging from pure metals to alkali-ammonia solutions. In the latter case ($r_s>6$) the BS predictions underestimate more than $50 \%$ the calculated values. The deviations, however, show a trend which monotonically decreases towards low $r_s$ elements and finally get negative for $r_s$ values close to 2. A final remark concerns the choice of the dynamical variable to calculate the sound velocity when one is looking at subtle effects as in the present case. First, according to the hydrodynamic definition of sound velocity in liquids, one should look at the maxima of the current correlation spectra. While the difference with the maxima of $S(Q,\omega)$ is usually negligible in crystals, indeed, there can be a significant discrepancy in strongly overdamped cases such as the one metallic solutions. In this respect, the choice of the DHO to describe $S(Q,\omega)$ implicitly overcame this problem, as the characteristic frequency of this model coincides indeed with the maxima of $C_L(Q,\omega)$. Second, in all the reported studies, the “phonon” velocity is extracted through ad hoc models (DHO, extended hydrodynamic model etc.) looking only at the genuine inelastic component. Again, in liquids, the full $C_L(Q,\omega)$ spectrum should be considered, as when $S(Q,\omega)$ has a significant quasielastic contribution this latter can affect the peak positions of $C_L(Q,\omega)$ (see, for instance, Fig. (\[aldisp\]) for the aluminum case). Inelastic X-ray Scattering, with lower energy resolution (a few hundreds of $meV$) and in a broader energy range (up to a few eV), allows to study electronic excitations (plasmons). In this case the plasma oscillation is brought about by free electrons, while the background is constituted by the ionic network. Some recent studies [@burns_prl2; @burns_prl3] point out a decrease of the plasmon dispersion at low metal concentrations, which, in turn, is ascribed to the failure of the RPA approximation. Summary and perspectives \[sec\_sum\] ===================================== In this section we will try to summarize the scenario arising from the measurements reported so far. In respect to the collective properties, it seems useful to discuss the different dynamical regimes probed at different wavevectors and frequencies. Although the two domains are in principle independent (as testified, for instance, by the two separate generalization of the classical hydrodynamics introduced in sec. \[sec\_collnonhydro\]), the existence of a dispersion relation ultimately allows one to think in terms of wavelenghts only. In our point of view, though precise boundaries can not be traced, one can identify in liquid metals the following dynamical regimes: \[t\] ![Sketch of the different dynamical regimes on decreasing the wavelength.[]{data-label="forse"}](./figure/sq_small.eps "fig:"){width=".5\textwidth"} - The hydrodynamic, $Q\rightarrow 0$ limit, that, in liquid metals, basically means $Q\lesssim 0.1$ nm$^{-1}$. In this region simple hydrodynamic treatment based on three microscopic dynamical variables (density, current, energy) provides an exhaustive description of the main features. Although not accessible by neutron and X-ray spectroscopic techniques, the hydrodynamic predictions are in very good agreement with acoustic measurements of sound velocity and attenuation properties. Moreover, the strict analogy of long wavelength fluctuations in conductive and ordinary liquids (accessible via light scattering) corroborates the validity of the simple hydrodynamic theory. This region is characterized by *adiabatic* sound propagation, and the whole dynamical features are ruled by macroscopic transport parameters (viscosity, thermal diffusivity, specific heats). In this regime, other approaches tailored for conductive fluids such as plasma oscillation theories provides alternative descriptions, which turns out to be increasingly accurate for low $r_s$ systems. - An (hypothetic) isothermal region, which should be observable in the $0.2 \lesssim Q \lesssim 3$ nm$^{-1}$ momentum range. Upon increasing the wavevector outside the strict hydrodynamic limit, indeed, the lifetime of the entropy fluctuations becomes increasingly shorter ($\tau=(\gamma D_TQ^2)^{-1}$). Since the frequency of the corresponding density fluctuation increases almost linearly ($\omega\approx cQ$), one should expect a transition at $Q^*\approx c/D_T$. Given the typical thermal diffusivity values of liquid metals this crossover should be located at a few fractions of nm$^{-1}$. Since with both INS and IXS one normally access momentum transfers above $1$ nm$^{-1}$, this region can be hardly explored, and no direct and convincing indication for its existence are available at present. Some old INS data on liquid lead [@FABER] as well as a more recent experiment on liquid nickel [@ber_ni], however, seems to suggest indication for such existence. The case of Nickel, in particular, is an interesting one and would deserve new investigations since in this system the specific heat ratio is particularly high and therefore the difference in sound velocities between an ordinary adiabatic regime and an isothermal one would be of the order of $40\%$. It is worth to stress that the prediction for the existence of an isothermal regime poses on: i) a negligible $Q$ dependence of the thermodynamic quantities and ii) the validity of a one component effective description in which the effective thermal diffusivity is well described by the sum of the electronic and ionic contributions. Both this points have been recently discussed in the analysis of alkali metals and liquid aluminium IXS spectra [@scop_comm; @sing_pre; @sing_rep]. - A generalized hydrodynamic region, probed above $Q\approx 3$ nm$^{-1}$, in which the frequency-wavevector dependence of the transport properties heavily affects the sound mode. The upper limit of validity of this region is rather system dependent, normally higher for alkali metals (say up to $0.7 Q_m$). The natural framework is here the memory function formalism, which can be developed at different levels of accuracy, ranging from the celebrated Lovesey’s model [@lov_visco] (accounting for a single average relaxation time for the second order memory function) up to more refined memory function models based on multiple relaxation phenomena which, firstly introduced for Lennard-Jones systems [@lev_2t], have been more recently adapted and tested against IXS investigations of liquid metals [@scop_prlga; @scop_prlli; @mon_k; @scop_jpc] and numerical simulations of model undercooled and glassy alkali [@scop_presim]. In this respect, it is worth to point out how the high points of a memory function approach are not solely in a better agreement with experimental data, which, in general, heavily depends on the quality of the experimental data and, of course, on the number of the model parameters [@scop_prlli]. In most cases, indeed, simplified phenomenological models such as the damped harmonic oscillator and a lorenzian function for the inelastic and quasielastic feature, respectively, provide satisfactory agreements. On the contrary, the memory function framework allows to grasp an insight behind the mechanisms ruling the relaxation dynamics, extracting relevant information about quantities which are not directly related to any spectral features, such as the $Q$ dependencies of the relaxation time(s) and of the longitudinal viscosity. \[h\] ![Generalized longitudinal viscosities for a collection of liquid metals, extracted by the experimental data throughout a memory function approach. The long wavelenght limit, determined by ultrasonic measurements, is also reported for K, Na, Ga (bottom to top).[]{data-label="allvisco"}](./figure/visco_all_c.eps "fig:"){width=".45\textwidth"} Following the prescriptions illustrated in section \[sec\_mf\], indeed, the longitudinal viscosity is related to the total area under the viscous contribution to the memory function. In Fig.(\[allvisco\]), for instance, we report the (generalized) longitudinal viscosities extracted in this way for several investigated systems. As can be seen, the low $Q$ extrapolation of the experimental values compares well with independent acoustic determinations (when available), but also allows to determined the $Q$ generalization of this important transport properties, which can be directly estimated only by numerical simulations approaches. In this $Q$ region, hard sphere-based theories provide alternative descriptions in terms of extended hydrodynamic models [@kag_hs] but they still miss a convincing explanation of one of the most important points: the physical interpretation beyond the relaxation of the sound mode, which is now a firmly established evidence supported by uncountable experimental investigations. \[h\] ![Sound dispersion in several kind of liquids. Alkali metals (Li [@scop_epl] and K [@mon_k], dashed line is the isothermal sound velocity); Hydrogen bonding systems (Water [@monaco_water] and hydrogen Fluoride [@ange_HF]) low and high frequency sound velocities are also indicated by the dotted lines); liquid neon [@cun_ar]: adiabatic (circles and line) apparent (open triangles) and high frequency (open diamonds) sound velocities; liquid silica, molecular dynamics simulations [@kob_si02posdisp], in this case also the transverse branch is reported.[]{data-label="pdisp1"}](./figure/pdisp_1.eps "fig:"){width=".5\textwidth"} This latter aspect, which usually emerges in terms of a speed up of the sound velocity taking place in the $1\div 10$ nm$^{-1}$ region, is one of the most interesting aspects which is still lacking an explanation. Broadly speaking, it is always referred to as a shear relaxation, but the ultimate nature of the involved physical processes still have to be clarified. Actually, Mode Coupling Theory [@ern_pdis] provides a description of the acoustic dispersion curve in terms of even powers of $Q$ [@ern_pdis], but its interpretation is restricted to monoatomic systems in the liquid phase, while the observed phenomenology seems to be a more general feature of the disordered systems. Interestingly, indeed, a qualitatively similar phenomenon (see Fig. \[pdisp1\]) has also been reported numerically in fused silica [@kob_si02posdisp]) and experimentally in Lennard jones fluids [@cun_ar] and hydrogen bonding systems ([@monaco_water; @ange_HF]). While in this latter case the positive dispersion has been shown to be an activated process, related to the structural relaxation, a different scenario seems to characterize the other systems. Interestingly, indeed, (see Fig. \[pdisp2\]) the same behavior of the sound velocity also appear in *glasses* either in experimental IXS measurements in g-Se and vitreous silica [@scop_se; @ruz_sio2], in numerical works on model glasses of Lennard Jones systems [@gcr_prlsim], vitreous silica [@kob_si02posdisp] and alkali metals [@scop_presim] and, finally, in theoretical calculations for an hard sphere glass [@got_hs]. In the case of alkali metals, in particular, it has been shown how the speed up of the sound velocity persists upon cooling well below the glass transition, thus ruling out the possible role of the structural relaxation in this effect. The presence of positive dispersion at THz frequencies in glasses, quantitatively comparable to the one observed in liquids, therefore, challenges the interpretation of the positive dispersion in terms of a transition from a liquid like to a solid like behavior, an effect which seems to be quantitatively negligible (with the remarkable exception of hydrogen bonding systems). Accepting a description of the collective dynamics proceedings over two distinct viscous relaxations, therefore, the ultimate responsible for the bending up of the dispersion curve seems to be the faster of the two observed processes, which turns out to have a mild temperature dependence. In this respect, the physical nature of this faster process calls for deeper understanding. It is worth to point out however, that in Lennard Jones systems the positive dispersion is recovered within an *harmonic* description of the dynamic structure factor in terms of eigenvalues and eigenvectors, a result which relate the positive dispersion to the properties of the dynamical matrix and, ultimately, to the topological disorder of the inherent equilibrium position, being the interaction potentials comparable in glasses and crystals. \[h\] ![Sound velocities in several glassy systems. Hard spheres glass, exact solution within MCT [@got_hs]; Metallic glass obtained quenching a model system interacting via Price-Tosi pseudopotential [@scop_presim], the crystalline counterpart is also reported; Lennard Jones glass obtained in a similar way [@gcr_prlsim], reported with the low and high frequency sound velocities; SiO2, MD as in Fig.\[pdisp1\] but in the glassy state [@kob_si02posdisp] and experimentally determined by means of IXS scattering [@ruz_sio2]. In both cases the transverse branch is also reported; glassy selenium [@scop_se], again the sound velocity exceeds the adiabatic value.[]{data-label="pdisp2"}](./figure/pdisp_2.eps "fig:"){width=".5\textwidth"} - A kinetic regime, valid from around $Q_m$, up to a few oscillations of the structure factor. Here the hard sphere description provides remarkably accurate predictions, in terms of a quasielastic “extended heat mode” whose linewidth is described in terms of the Enskog’s diffusion coefficient and of an equivalent hard sphere diameter. The extent of validity of kinetic theory in this momentum range has been widely tested [@kag_hs; @coh_hs; @desh_hs; @coh_hs1] in several hard sphere-like systems (like alkali metals and lennard-jones fluids), and it also apply in colloidal systems. It would be interesting to challenge such theory in less simple liquid metals. Despite the success of hard sphere model, there is still an obscure point concerning the real origin of such extended heat mode: while in the low $Q$ limit it tends to the entropy fluctuation mode, indeed, at finite $Q$’s it certainly involves mass diffusion processes. Looking at things from the constant energy point of view, umklapp modes resembling crystalline phonons in Brillouin zones higher than the first seems to be still poorly understood [@scop_prbumk; @rand_umk; @coc_umk; @dor_umk]. - An high $Q$ region, probed as soon as the Van Hove distinct function vanishes. In this limit, both IXS and INS experiments provide the same information, about the atomic motion over timescales shorter than the interparticle collision time. An almost exact description for this regime is available, which can also account for quantum aspects such as recoil energy an quantum corrections to the spectral moments. The single particle dynamics seems to be better understood if compared to the collective motion, at least by a coarse grained point of view of an hydrodynamic diffusive mode with finite $Q$ corrections, evolving towards a ballistic regime. This is probably due to the intrinsic difficulty of isolating a (wide enough) constant $Q$ *coherent* energy spectrum from an INS experiment. Since the study of the strictly coherent spectrum became possible only in the last decade, indeed, a lot of efforts have been devoted in the past to the single particle case. Though all the approaches described in this review describe, on average, equally well the available experimental data, a memory function formalism paralleling the one for the collective case could provide the route to relate the single particle motion and the collective dynamics in the microscopic regime. Raising the level of detail of the description of the single particle dynamics, however, a major experimental challenge seems to be the identification of the different processes giving rise to the quasielastic incoherent scattering. Recent INS results, indeed, suggest the presence of two distinct physical mechanisms, active over different timescales, underlying the diffusive motion [@bov_hg; @bov_k; @bad_hg]. The combined presence of coherent and incoherent scattering, however, makes such identification still unclear although, in principle, the IXS signal might be used to subtract the coherent contribution from the INS spectra, thus extracting the purely incoherent dynamics. In this respect, the synergy of combined IXS and INS studies on a same sample seems us imperative and could help to shed light on this point. The IXS signal indeed, might be used to subtract the coherent contribution from the INS spectra, thus extracting the purely incoherent dynamics. Acknowledgements ================ J.-P. Hansen made extensive comments on this manuscript for which we are most grateful. We thank L.E. Bove and T. Bryk for several discussions and interesting comments on the preprint, and S. Cazzato for his help in compiling the data reported in table I. T.S. gratefully acknowledges his debt to U. Balucani for the vivifying influence exerted on his outset in the field of simple liquids. Most of our IXS activity greatly benefited from the support of the staff of the beamlines ID16-ID28 at the ESRF and BL35XU at SPring8. The assistance of the technical staffs of the ESRF (D. Gambetti, B. Gorges and C. Henriquet), of the University of L’Aquila (O. Consorte) and of the University of Rome “La Sapienza” (I. Deen, M. Pallagrossi, C. Piacenti and A. Salvati) is also acknowledged. Last but not least, thanks are due to all authors, editors, and publishers who granted us permission to include in this review previously published illustrations, images, and figures. Sample $T [K]$ $\gamma$ $c_s [m/s]$ $c_t [m/s]$ $max\{c_{l}\} [m/s]$ $\sigma_{inc}/\sigma_{coh}$ $D_T$ $[nm^2/ps]$ --------------- --------- --------------- ----------------- ------------- ---------------------- ----------------------------- ------------------- $\mathrm{Li}$ 453 1.08$^,$1.065 4554 4466 5762 19.1 488 5423$^,$ 500 20.3$^,$ 600 1.092 4356 5560 0.991.1 $\mathrm{Na}$ 371 1.121.091 2531 68.8 388 1.11$^,$ 2514 3160 390 2930 500 68.4 773 2310 2881 1073 2150 2577 1173 2093 2492 0.841.0060.976 $\mathrm{Mg}$ 923 1.29 4070 37 973 4038 4380 1000 39.8 0.06 $\mathrm{Al}$ 933 1.4 4750 35.2 1000 4670 7075 36.4 $\mathrm{Si}$ 1683 1.57 3977 9.4$^,$ 1753 3952 4597 0.05 $\mathrm{K}$ 336.7 1.111.102 1880 81.4 343 1.105 1877 16051710 23522260 350 2360 0.200.16 $\mathrm{Fe}$ 1808 1.8 4000$\div$4400 7.3 $\mathrm{Co}$ 1700 7.8$^,$ 1765 1.8 4033$\div$4090 $\mathrm{Ni}$ 1500 16$^,$ 1728 1.98 4036$\div 4045$ 9.6 1763 1.88 4280 3121 3855 0.350.30 $\mathrm{Cu}$ 1356 1.33 3440$\div$3485 4230 42.1 0.06 $\mathrm{Zn}$ 693 1.251.26 2835$\div$2850 15.7 $\mathrm{Ga}$ 303 1.08 11.6 315 2930 2600 3050 326 1.08 3240 350 13.6 0.070.02 $\mathrm{Ge}$ 1253 1.18$^,$ 2682 2682 1063 8$\div$9 0.006 $\mathrm{Rb}$ 312 1.151.097 1260 61.5 320 1370 1420$^,$ 0.00055 $\mathrm{Ag}$ 1233 1.32 2710$\div$2770 66.5 0.125 $\mathrm{Cd}$ 594 1.251.25 2235$\div$2255 39.8$^,$ 2.3 $\mathrm{Sn}$ 505 1.11 17.3 593 1.09$^,$$^,$ 2443 2736 1273 2228 2362 0.010.007 $\mathrm{Sb}$ 904 1.21 1893$\div$1900 15.5 0.046 $\mathrm{Te}$ 723 1.033 889 0.8$\div$1.3 0.05 Sample $T [K]$ $\gamma$ $c_s [m/s]$ $c_t [m/s]$ $max\{c_{l}\} [m/s]$ $\sigma_{inc}/\sigma_{coh}$ $D_T nm^2/ps$ --------------- --------- ------------ ------------- ------------- ---------------------- ----------------------------- --------------- $\mathrm{Cs}$ 302 967 44.6 308 1.1021.099 965 1140 0.0596 $\mathrm{Au}$ 1336 1.28 2560 40.4 0.06 $\mathrm{Hg}$ 234 3.62 293 1.14 1451 21001800 4.41 300 4.41 0.3240.31 $\mathrm{Tl}$ 576 1.143 1665 25.2 0.025 $\mathrm{Pb}$ 600 9.89 623 1.19$^,$ 1770 9.89 700 11.4 0.000088 $\mathrm{Bi}$ 544 1.15 8.09 [^1]: From here on we will implicitly assume that we are not dealing with microscopic quantities but rather with the hydrodynamic quantities resulting from their averages [^2]: The exact hydrodynamics expression contains a small additional contribution which makes the Brillouin components asymmetric, as emphasized in [@nic_hyd; @verk_rev]. The resulting lineshape can be easily recognized as the already mentioned Damped Harmonic Oscillator, originally proposed within a solid-like picture [@fak_dho], which can be actually retrieved by a liquid-like point of view within the memory function formalism, shown in section \[sec\_mf\]. [^3]: In the case of copper, for instance, the distance to the Fermi surface is 2.3 eV. Thus, electrons are promoted by energies associated with the blue-green end of the spectrum. As a result, red and orange light at the opposite end of the spectrum is reflected back and gives copper its characteristic color. With the alkali metals, the *s* electron is involved in promotion to the Fermi level. There is little overlap to the empty *3p* and *3d* orbitals that contribute to the conduction band. Therefore, only radiation close to the ultraviolet region is absorbed and visible light is reflected, hence the silver-like appearance of the alkali liquid-metals. [^4]: More specifically, the discrepancy observed in liquid lead between the sound velocity measured with ultrasound (adiabatic) and with inelastic neutron scattering has been tentatively assigned to the isothermal nature of the sound propagation at the wavevectors probed with neutrons. [^5]: Actually, the viscoelastic model stems from the approximation (\[memoryL1\]), with the additional condition $\gamma=1$ ($M_{th}(Q,t)=0$). Within the viscoelastic framework, indeed thermal effects are neglected, in the sense that the hydrodynamic limit is isothermal (i.e. $\Delta _L^2(Q)=\omega_L(Q)^2-\omega_0(Q)^2$)

--- abstract: 'A commercial single laser line Raman spectrometer is modified to accommodate multiline and tunable dye lasers, thus combining the high sensitivity of such single monochromator systems with broadband operation. Such instruments rely on high-throughput interference filters that perform both beam alignment and Rayleigh filtering. Our setup separates the dual task of the built-in monochromator into two independent elements: a beam splitter and a long pass filter. Filter rotation shifts the transmission passband, effectively expanding the range of operation. Rotation of the filters has a negligible effect on the optical path, allowing broadband operation and stray light rejection down to 70-150 cm$^{-1}$. Operation is demonstrated on single-walled carbon nanotubes, for which the setup was optimized.' author: - Gábor Fábián - Christian Kramberger - Alexander Friedrich - Ferenc Simon - Thomas Pichler title: Adaptation of a commercial Raman spectrometer for multiline and broadband laser operation --- Introduction ============ Raman spectroscopy is a widespread and important tool in various fields of science from biology to physics. Commercial Raman spectrometers are usually equipped with a built-in laser and a setup optimized for this single laser line, resulting in stable operation but inherently narrow-band characteristics. The electronic, optical, and vibrational characterization of certain materials, such as single-wall carbon nanotubes (SWCNTs)[@DresselhausCNTRamanReview] requires measurements with a large number of laser lines [@KuzmanyEPJB] or with a tunable laser system [@FantiniPRL2004; @TelgPRL2004]. Raman spectroscopy relies on the efficient suppression of “stray light” photons with wavelengths close to that of the exciting laser (e.g. from Rayleigh scattering) which dominate over the Raman signal by several orders of magnitude. Operation down to Raman shifts of 100 cm$^{-1}$ is made possible in modern spectrometers with the use of interference Rayleigh filters (often referred to as notch filters). The transmission of these filters typically exceeds $80\,\%$ for the passband, this is significantly higher than for a classical subtractive double monochromator system. Although interference filters are manufactured for the most common laser lines only, rotation extends the range of filter operation. Thus the narrow-band constraint could be circumvented to allow broadband operation. However in most spectrometers, the interference filter has a dual role: it reflects the laser light to the sample and it functions as a Rayleigh filter. Filter rotation changes the optical path of the excitation that can be corrected for with tedious and time consuming readjustment only, effectively nullifying the advantage of the higher sensitivity. In particular for the radial breathing mode of SWCNTs, the presence of the low energy ($\geq 100-150\,\text{cm}^{-1}$) [@RaoCNTRamanScience] Raman modes and the narrow (FWHM $\sim 30 \,\text{meV}$) optical transition energies [@FantiniPRL2004] pose several challenges to the instrumentation. A proper energy dependent Raman measurement requires a broadband spectrometer with efficient stray light rejection. Herein, we describe the modification of a commercial Raman spectrometer with interference Rayleigh filters, which enable broadband operation with relative ease. The improvement is based on replacing the built-in interference filter with a beam splitter and a separate interference filter. Thus the two functions of the filter are performed independently with no observable on influence the direction of the transmitted light. The different behavior of the filter passband for the $S$ and $P$ [^1] polarizations under rotation is overcome by the application of polarization filters on the spectrometer input. The setup operates with polarizations which are optimized when the so-called antenna effect of SWCNTs is taken into account, i.e. that the Raman light is polarized predominantly along the polarization of the excitation [@SunAntennaEffect; @JorioPhysRevLett85]. Spectrometer setup ================== A high sensitivity, confocal single monochromator Raman system with a interference Rayleigh filter—such as described in the previous section—can be modified to enable broadband measurements with multiple laser lines or even with a tunable laser, which we demonstrate for a LabRAM commercial spectrometer (Horiba Jobin-Yvon Inc.) as an example. The key step in achieving the broadband operation was replacing the built-in interference filter, which acts as a beam splitter and a Rayleigh filter at the same time, with a combination of a simple beam splitter and a serarate interference filter. We note herein that this modification also enables a cost effective operation with usual laser wavelengths (such as e.g. the lines of an Ar/Kr laser) since no complicated filter realignments are required. We have to emphasize that the use of standard optical elements, which are non-specific to the spectrometer allows economic implementation for most spectrometer designs. ![Schematic diagram of the broadband configuration of the LabRAM spectrometer. V and H denote vertical and horizontal polarizations, respectively. The tunable source is a dye laser pumped with a 532 nm solid state laser. The laser light is aligned with the spectrometer using the periscope element, which also rotates the polarization to horizontal, if needed. The laser outputs are cleaned with a filter. The sample emits a nominally horizontally polarized light and the unwanted vertical polarization is filtered with the polarizer.[]{data-label="schem"}](Fig1_setup.eps){width="0.98\columnwidth"} The setup for the modified LabRAM spectrometer is shown in Fig. \[schem\]. A multiline Ar/Kr laser (Coherent Inc., Innova C70C-Spectrum) and a dye laser (Coherent Inc., CR-590) pumped by a 532 nm 5 W solid state laser (Coherent Inc., Verdi G5) serve as excitation light sources. The former operates at multiple, well defined wavelengths while the latter allows fully tunable application. In our case, the dye laser is operated in the 545-580 nm, 580-610 nm, and 610-660 nm wavelength ranges with three dyes: Rhodamin 110, Rhodamin 6G, and DCM Special, respectively. The periscope allows beam alignment and sets the polarization of the excitation light to horizontal. In the case of the dye laser, the spurious fluorescent background of the laser output is filtered with short pass (“3rd Millennium filters” for 580 and 610 nm from Omega Optical Inc.) and band pass (“RazorEdge” for 568, 633, and 647 nm from Semrock Inc.) filters. For the clean-up of the multiline laser excitation band pass filters are used at the appropriate wavelengths (“RazorEdge” for 458, 488, 515, 532, 568, 633, and 647 nm from Semrock Inc.) The light is directed toward the sample with a broadband beam splitter plate (Edmund Optics Inc., NT47-240) with 30 % reflection and 70 % transmission. For both excitation sources a single, long pass interference edge filter (“RazorEdge” for 458, 488, 515, 532, 568, 633, and 647 nm from Semrock Inc.) performs stray light rejection. The use of a short pass filter for laser clean-up and long pass filters for Rayleigh photon supression limits operation for the Stokes Raman range. The long pass filter has double function in the original spectrometer: it mirrors the laser excitation to the sample and acts as a Rayleigh filter, quenching the stray light. In our construction, these two tasks are performed independently by a beam splitter and a long pass filter, respectively. The broadband beam splitter plate has 30 % reflection and 70 % transmission, thus only a small fraction of the Raman light is lost. The 70 % excitation power loss on the beam splitter can be compensated by reducing the attenuation of the intensive laser beam, maintaining a constant irradiation density on the sample. The application of an anti-reflective coating to the back side of the plate prohibited the emergence of higher order reflections and standing waves (whose effect is known as ghosts) within the plate. The beam splitter plate is mounted on a finely adjustable 2-axis holder (Thorlabs Inc., VM1) with a home made mounting. The fine adjustment is required to set the light alignment properly with the spectrometer. Final fine adjustment is performed with the holder to maximize the Raman signal. ![Transmittance of the 633 nm long pass filter using unpolarized white light; a.) at normal incidence and b.) rotated by $30^{\circ}$. When polarization filters are used, the two parts of the double step feature (solid black line) are separated according to the $S$- and $P$-polarization (dashed black and solid gray lines, respectively). Note the broadening of filter transition width upon rotation.[]{data-label="LP"}](Fig2_LongPass.eps){width="0.98\columnwidth"} Increasing the incidence angle of the light changes the range of filter operation of the interference filters without the misalignment of the light. Thus filter rotation enables broadband operation. In Fig. \[LP\]., we show the behavior of a 633 nm long pass filter at different incidence angles. The edge of transmission blue shifts upon rotation with respect to normal incidence. However, the shift is smaller for the $S$ than for the $P$ polarization; i.e. the shift is larger for the horizontally polarized light when the filter is rotated around a vertical axis. Vertical rotation of the long pass filter is more practical, meaning that the setup prefers horizontally polarized scattered (Raman) light as it is of the $P$ polarization, for which the edge shift is larger. For 1 inch apertures short and long pass filters rotation angles up to $30^{\circ}$ were used, yielding a blue shift of about 10 %; the 0.5 inch aperture of band pass “Razor edge” filters limited the blue shift to about 5 %. The width of the filter transition edge also broadens for larger incidence angles. This is defined as the maximum difference between the laser wavelength at which the attenuation exceeds optical density 6 and the filter edge-wavelength at the 50 % transmission point. For the $30^{\circ}$ incidence, a fivefold increase in the transition width is observed when compared to the normal incidence, allowing operation down to $70-140\,\text{cm}^{-1}$. For SWCNTs, the Raman light is polarized preferentially along the polarization of the excitation, this is due to a phenomenon called the antenna effect [@SunAntennaEffect; @JorioPhysRevLett85]. We also verified that the LabRAM spectrometer itself is not polarization-sensitive in contrast to an older triple monochromator system. Therefore a horizontally polarized laser excitation is preferred which explains the polarizations used in our design. The less shifted $S$ (in our construction vertically) polarized stray light is removed with a polarization filter before the spectrometer input. Test measurements ================= Test measurements of the broadband setup were carried out with the tunable dye laser on a HiPCO SWCNT sample (Carbon Nanotechnologies Inc., Houston, Texas), suspended in a 2 weight% solution of SDBS (Sodium dodecyl benzene sulfonate) and water using sonication. We focused on the radial breathing mode (RBM) Raman range located below 400 $\text{cm}^{-1}$, which is commonly studied to characterize the diameter distribution in SWCNTs [@DresselhausTubes]. Carbon-tetrachloride was used for Raman shift correction and Raman intensity normalization such as in Ref. [@FantiniPRL2004]. The suspended HiPCO sample was placed in a glass cuvette under the objective of the built-in microscope (Olympus LMPlan 50x/0.50, inf./0/NN26.5, $\sim 1 \times 1 \,\mu \text{m}^2$ spot size) and the CCl$_4$ reference sample was placed into the macro cuvette holder. The LabRAM spectrometer allows to change between a macro and micro mode with a mirror resulting in stable and robust spectral shift calibration and intensity normalization as there is no need for further adjustments nor for sample exchange. Laser excitation energies between 1.92 eV (648 nm) and 2.27 eV (545 nm) were covered with an energy resolution of about 12 meV ($\sim 100\,\text{cm}^{-1}$). The spectrometer was operated with a 600 grooves/mm grating and a liquid nitrogen cooled CCD with 1024 pixels along the spectral direction. This configuration yields a $\sim 1.3$ cm$^{-1}$ Raman shift resolution and $\sim 1800$ cm$^{-1}$ spectral range for 600 nm (both are wavelength dependent). Typical laser powers of 1-5 mW were used with no observable heating effects, due to the liquid nature of both samples. ![Main plot: Raman map for the RBM range of a HiPCO/SDBS suspension measured with the broadband Raman setup. Logarithmic scale shows the Raman intensity normalized to the maximum observed intensity. Full circles denote data published in Ref. [@FantiniPRL2004]. Inset A: A spectrum of the Raman map (horizontal line) at 592 nm (2.1 eV). Inset B: Energy cross section of the Raman map (vertical line). The black diamonds correspond to the cross section at $310$ cm$^{-1}$ Raman shift, solid curve refers to a resonance Raman fit. []{data-label="map"}](Fig3_RamanMap2.eps){width="0.98\columnwidth"} An approximately 8 minute long measurement cycle consists of changing the dye laser wavelength, rotating of the laser clean-up and the long pass filters to the appropriate positions, shifting the spectrometer grating, nulling the spectrometer. Typical measurement times of 4 minutes for the sample and a few seconds for the reference yield an acceptable signal to noise ratio of about $300$. The sample and reference measurements followed each other immediately without moving the grating position, which led to an accurate Raman shift measurement. Additional time is needed for the filter exchange (a few minutes) and to change the laser dye and to readjust the beam alignment (about 1 hour). We note that once the dye laser is set and the light path is properly aligned with the spectrometer, no further realignment is required when the wavelength is changed, even though the filters are rotated and the optical path is only minutely modified. In total, 9 hours were required to complete the energy dependent Raman experiments with the 29 laser lines including 2 laser dye exchanges. Measurements of a similar scale such as published previously using a triple monochromator spectrometer [@SimonPRB2006] last for about 2 weeks, mainly due to the approximately 50 times smaller S/N ratio and the need for a spectrometer realignment upon wavelength change. Raman shift correction, intensity normalization, and the Raman map preparation was performed using a home-made software. Linear baseline correction was sufficient since no fluorescent response was encountered. Raman shift was corrected for with the carbon-tetrachloride Raman modes at 218 and 314 cm$^{-1}$. Intensity normalization using CCl$_4$ is required as it accounts for instrumental uncertainties such as a slight misalignment of the scattered light upon dye exchange. We checked the consistency of the normalization by measuring the same wavelength with different laser dyes after spectrometer realignment. Fig. \[map\]. shows the 2D contour plot of the Raman map, compiled from spectra such as shown in Fig. \[map\].A. The normalized Raman intensity is displayed on a logarithmic scale. A good agreement is observed between our data and the measurements in Ref. [@FantiniPRL2004], whose resonance transition energies and Raman shifts are shown for the different SWCNT chiral indexes, $(n,m)$, with full circles. We do not observe the (8,4) SWCNT in our measurement, due the low intensity of the resonant Raman process species [@Jorio2006APL]. Points corresponding to the same SWCNT families, i.e. when $2n+m$ is constant [@DresselhausTubes], are connected by solid lines. Vertical, i.e. energy cross section of the Raman map were obtained by averaging around a given Raman shift. Representative energy cross section data are shown in Fig. \[map\].B, along with fits using the resonance Raman theory [@FantiniPRL2004; @SimonPRB2006]. Fits yield transition energies and quasiparticle scattering rates in good agreement with typical literature values, especially considering that the different solvent environment slightly modifies the Raman transition energies [@FantiniPRL2004]. We note that no further corrections were made to obtain the energy cross section data apart from the normalization by the reference. The result is therefore remarkably smooth in comparison with similar data published in Refs. [@FantiniPRL2004; @TelgPRL2004; @DoornPRB2008]. This is due to the robust and reproducible measurement of the reference sample and possibility of measuring Raman spectra at different wavelengths without spectrometer readjustment in between. The agreement shows the utility of the broadband arrangement with a clear advantage over previous results in terms of acquisition time. Conclusions =========== In conclusion, we presented the broadband modification of a high sensitivity commercial Raman spectrometer. The improvement allows the use of both multiline and tunable dye lasers. The spectrometer performance is demonstrated on SWCNTs where such broadband measurements are inevitable to obtain meaningful insight into vibrational and electronic properties. Work supported by the Austrian Science Funds (FWF) project Nr. P21333-N20, by the European Research Council Grant Nr. ERC-259374-Sylo, and by the New Hungary Development Plan Nr. TÁMOP-4.2.1/B-09/1/KMR-2010-0002. CK acknowledges an APART fellowship (Nr. 11456) of the Austrian Academy of Science. The authors acknowledge fruitful discussions with Dr. Neil Anderson from Semrock Inc. about the interference filters. merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked [12]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase DOI: 10.1016/j.physrep.2004.10.006) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase DOI: 10.1016/S0038-1098(98)00588-2) @noop [****,  ()]{} @noop [**]{} (, ) @noop [****,  ()]{} [****,  ()](\doibase 10.1063/1.2162688) [****,  ()](\doibase 10.1103/PhysRevB.78.165408) [^1]: The S and P refer to polarizations which are perpendicular and parallel to the plane of incidence, respectively.

--- abstract: | Most online platforms strive to learn from interactions with consumers, and many engage in *exploration*: making potentially suboptimal choices for the sake of acquiring new information. We initiate a study of the interplay between *exploration* and *competition*: how such platforms balance the exploration for learning and the competition for consumers. Here consumers play three distinct roles: they are customers that generate revenue, they are sources of data for learning, and they are self-interested agents which choose among the competing platforms. We consider a stylized duopoly model in which two firms face the same multi-armed bandit instance. Users arrive one by one and choose between the two firms, so that each firm makes progress on its bandit instance only if it is chosen. We study whether and to what extent competition incentivizes the adoption of better bandit algorithms, and whether it leads to welfare increases for consumers. We find that stark competition induces firms to commit to a “greedy" bandit algorithm that leads to low consumer welfare. However, we find that weakening competition by providing firms with some “free" consumers incentivizes better exploration strategies and increases consumer welfare. We investigate two channels for weakening the competition: relaxing the rationality of consumers and giving one firm a first-mover advantage. We provide a mix of theoretical results and numerical simulations. Our findings are closely related to the “competition vs. innovation" relationship, a well-studied theme in economics. They also elucidate the first-mover advantage in the digital economy by exploring the role that data can play as a barrier to entry in online markets. author: - 'Guy Aridor[^1]' - 'Yishay Mansour[^2]' - 'Aleksandrs Slivkins[^3]' - 'Zhiwei Steven Wu[^4]' bibliography: - 'bib-abbrv.bib' - 'bib-ML.bib' - 'refs.bib' - 'bib-bandits.bib' - 'bib-AGT.bib' - 'bib-slivkins.bib' - 'bib-random.bib' date: July 2020 title: | Competing Bandits:\ The Perils of Exploration under Competition[^5] --- Introduction {#sec:intro} ============ Related work {#sec:related-work} ============ Our model in detail {#sec:model} =================== Theoretical results: the [Bayesian-choice model]{} {#sec:theory} ================================================== Numerical simulations: the [reputation-choice model]{} {#sec:sim} ====================================================== Background for non-specialists: multi-armed bandits {#app:bg} =================================================== Monotone MAB algorithms {#app:examples} ======================= Non-degeneracy via a random perturbation {#app:perturb} ======================================== Full proofs for Section \[sec:theory\] {#sec:theory-proofs} ====================================== Full experimental results {#app:expts} ========================= [^1]: Columbia University, Department of Economics. Email: g.aridor@columbia.edu [^2]: Google and Tel Aviv University, Department of Computer Science. Email: mansour.yishay@gmail.com [^3]: Microsoft Research New York City. Email: slivkins@microsoft.com [^4]: University of Minnesota - Twin Cities, Department of Computer Science. Email: zsw@umn.eduPart of the research was done when Z.S. Wu was an intern and a postdoc at Microsoft Research NYC. [^5]: This is a merged and final version of two conference papers, @CompetingBandits-itcs18 and @CompetingBandits-ec19, with a unified and streamlined presentation and expanded background materials. All theoretical results are from @CompetingBandits-itcs18, and all experiments are from @CompetingBandits-ec19. Appendices \[app:bg\],\[app:examples\] are completely new compared to the conference versions.

--- abstract: 'The vicinity of the unidentified EGRET source 3EG J1420–6038 has undergone extensive study in the search for counterparts, revealing the energetic young pulsar PSR J1420-6048 and its surrounding wind nebula as a likely candidate for at least part of the emission from this bright and extended gamma-ray source. We report on new Suzaku observations of PSR J1420–6048, along with analysis of archival XMM Newton data. The low background of Suzaku permits mapping of the extended X-ray nebula, indicating a tail stretching $\sim 8 \arcmin$ north of the pulsar. The X-ray data, along with archival radio and VHE data, hint at a pulsar birthsite to the North, and yield insights into its evolution and the properties of the ambient medium. We further explore such properties by modeling the spectral energy distribution (SED) of the extended nebula.' author: - 'Adam Van Etten, Roger W. Romani' title: 'The Extended X-ray Nebula of PSR J1420–6048' --- Introduction ============ The campaign to identify 3EG J1420–6038 has revealed sources across the electromagnetic spectrum from radio to VHE $\gamma$-rays. The complex of compact and extended radio sources in this region is referred to as the Kookaburra [@robertsetal99], and covers nearly a square degree along the Galactic plane. Within a Northeasterly excess in this complex labeled “K3” @d'amicoetal01 discovered PSR J1420–6048 (hereafter J1420), a young energetic pulsar with period 68 ms, characteristic age $\rm \tau_c = 13$ kyr, and spin down energy $\rm \dot E = 1.0 \times 10^{37} \, erg \, s^{-1}$. The NE2001 dispersion measure model [@candl02] of this pulsar places it 5.6 kpc distant. Subsequent ASCA observations by @robertsetal01b revealed extended X-ray emission around this pulsar, and @ngetal05 further examined the K3 pulsar wind nebula (PWN) with Chandra and XMM-Newton, resolving a bright inner nebula along with fainter emission extending $\sim 2\arcmin$ from the pulsar. @aharonianetal06b report on the discovery of two bright VHE $\gamma$-ray sources coincident with the Kookaburra complex. HESS J1420-607 is centered just north of J1420, with best fit extension overlapping the pulsar position. The other H.E.S.S. source appears to correspond to the Rabbit nebula half a degree southwest, which is also observed in the radio [@robertsetal99] and X-ray [@robertsetal01a]. Most recently, PSR J1420-6048 was detected by the Fermi Large Area Telescope (LAT) [@abdoetal09]. This crowded region clearly merits further study, and we report on new X-ray results obtained with Suzaku and XMM-Newton, as well as SED modeling of the K3 nebula. Data Analysis ============= The Suzaku pointing (obsID 503110010) occurred on January 11-12 2009 for a total of 50.3 ks. We utilize the standard pipeline screened events, and analyze the XIS front side (XIS0 and XIS3) and back side (XIS1) illuminated chips with XSelect version 2.4. We also obtained recent archival XMM data to augment the Suzaku data; observation 0505840101 occurred on February 15 2008, for 35.0 ks, while observation 0505840201 added 5.6 ks. The second data set has a slightly different CCD placement, and suffers from high background, so we only use the 35.0 ks of data. We apply the standard data processing, utilizing SAS version 9.0. After screening the data for periods of high background 19.9 ks remain with the MOS chips. The PN chip suffers greatly from flaring, and we discard this data. Spectral fits are accomplished with XSPEC version 12.5. Broadband Morphology and Point Sources -------------------------------------- Suzaku X-ray emission is peaked in the vicinity of the pulsar, with a bright halo extending $\sim3 \arcmin$ and a fainter tail extending north $\sim 8 \arcmin$. A number of other excesses of emission correspond to point sources, as discussed below. Figure 1 shows the Suzaku data in the 2–10 keV band, which highlights the extended PWN emission to the north. Also depicted is the XMM exposure, clearly showing a number of point sources, though no obvious extended emission is apparent. To identify X-ray point sources we use the SAS source detection function edetect$\_$chain on the XMM MOS chips and search in two energy bands of 0.3–2 keV and 2–10 keV for sources with a probability $P < 10^{-13}$ of arising from random Poissonian fluctuations. The source detection algorithm also attempts to determine source extension via a Gaussian model, though all detections are consistent with a point source. Counts are therefore extracted from a 15 pixel ($16.5 \arcsec$) radius circle. A total of 8 sources pass this test, 4 of which also appear in @ngetal05: PSR J142–6048 (source 5 in our dataset), the X-ray sources denoted star 1 (source 1) and star 3 (source 2), and another point source to the southeast (source 3) unlabeled by [@ngetal05] but visible in their XMM exposure. This source to the southeast is also a field star, as it appears quite bright in DSS2 red images. Of the four remaining sources, only one, a hard bright source $8.5 \arcmin$ north of the J1420 labeled source 7, lacks an optical counterpart. Source 7 also overlaps a radio hotspot to the north. Below we list the properties of these sources, defining the hardness ratio as: HR=$ \rm (C_{hi}-C_{lo})/(C_{hi} + C_{lo})$ where $\rm C_{lo}$ and $\rm C_{hi}$ are MOS counts in the 0.3–2 keV and 2–10 keV bands, respectively. It is worth noting that PSR J1420–6048 is only the fifth brightest point source in the XMM field and that all 8 XMM sources appear as excesses in the soft band Suzaku data as well. All point sources are quite soft, save for J1420 and source 7. No. ($\tablenotemark{*}$) R.A. Dec. Pos. Err.$\arcsec$ Counts HR --------------------------- --------------- ---------------- -------------------- -------------- -------------------- 1 (Star 1) $14:19:11.52$ $-60:49:34.00$ $0.26$ $462 \pm 27$ $-0.99 \pm 0.026$ 2 (Star 3) $14:19:31.48$ $-60:46:20.29$ $0.39$ $146 \pm 18$ $-0.85 \pm 0.12 $ 3 (Unlabeled) $14:20:22.72$ $-60:53:21.47$ $0.45$ $164 \pm 19$ $-1.00 \pm 0.069$ 4 $14:19:17.61$ $-60:45:23.45$ $0.61$ $123 \pm 17$ $-0.81 \pm 0.12 $ 5 (PSR J1420–6048) $14:20:08.19$ $-60:48:14.85$ $0.63$ $150 \pm 20$ $ 0.95 \pm 0.072$ 6 $14:20:40.75$ $-60:41:20.22$ $0.79$ $40 \pm 11$ $-0.79 \pm 0.30 $ 7 $14:20:09.78$ $-60:39:42.86$ $0.85$ $62 \pm 14$ $ 0.79 \pm 0.18 $ 8 $14:19:35.85$ $-60:42:11.39$ $1.16$ $34 \pm 11$ $-0.68 \pm 0.36 $ : XMM Source Properties \[srcprop\] On a larger scale, extended emission is observed in all wavebands. Australia Compact Telescope Array (ATCA) observations within the error ellipse of 3EG J1420–6038 (which is broad enough to encompass both the K3 wing and the Rabbit nebula) by @robertsetal99 revealed the “K3” excess, a resolved knot of emission surrounding the pulsar of flux density 20 mJy at 20 cm with index $\alpha = -0.4 \pm 0.5$. Adjacent is the “K2 wing,” with 1 Jy at 20 cm and index of $-0.2 \pm 0.2$. Closer inspection of the both the 13 cm and 20 cm continuum maps reveal that J1420 lies on the southeastern rim of an apparent radio shell $\approx 3\arcmin$ in radius. This shell is also apparent in the SUMSS 843 MHz map of the region. The center of this shell coincides with a dearth of emission in Spitzer $8 \, \mu \rm m$ maps. We place extended PWN upper limits in the radio by measuring the 843 MHz, 20 cm and 13 cm flux densities from the entire shell (which is significantly larger than the X-ray extension), finding 0.61, 0.75 Janksy and 0.58 Jansky, respectively. We deem these flux densities to be upper limits since the poor spatial resolution of Suzaku prevents ascertaining how the radio and X-ray emitting regions relate. We also remeasure the K3 excess and find flux densities of 15 mJy, 19 mJy and 17 mJy at 843 MHz, 20 cm and 13 cm, respectively, consistent with the result of @robertsetal99). At higher energies HESS J1420–607 shines at 13% of the Crab [@aharonianetal06b], with photon index of 2.2 and extent of $3.3\arcmin$ centered $2.6\arcmin$ north of the pulsar. The H.E.S.S. spectrum is extracted from a $9.6\arcmin$ circle to minimize contamination from HESS J1418-609 (The Rabbit) $33\arcmin$ to the southwest. Spectrum -------- The XMM point source identification described above is valuable since the broad Suzaku PSF hinders the disentanglement of point sources from extended emission. We are accordingly free to define extended regions for spectral analysis while steering clear of the X-ray point sources; we label these “Inner,” “Halo,” and “Tail”. All spectral extraction regions lie external to the Suzaku $1\arcmin$ half power radius of the 8 point sources (except the very faint and soft source 8, which slightly overlaps the “Tail”). The Northernmost emission is encompassed by the “Tail” region which extends north of the pulsar from $3\arcmin \sim 8\arcmin$. Suzaku does best with large regions, and accordingly we capture the pulsar and surrounding PWN emission with a circular region of radius $3\arcmin$ (“Halo”). To best isolate the pulsar and inner PWN we also use a $1.7\arcmin$ circle we call “Inner,” which is the minimum size recommended for Suzaku spectral analysis. The Suzaku background region occupies the northern and eastern edges of the field of view. We extract Suzaku XIS spectra with XSelect from all three active XIS chips, while response files (both ARF and RMF) are generated with the xisresp script. The ASCA FTOOL addascaspec is utilized to combine the spectra and responses of the XIS front side illuminated chips (XIS0 and XIS3). The XIS1 back side illuminated chip possesses a markedly different response function from the front side illuminated chips, and so is analyzed in parallel rather than added to the other two chips. Finally, spectra are binned to a minimum of 30 counts per bin. The high XMM background prevents an adequate fit to the extended emission (flux errors of $\sim100$%). We are able, however, to extract the spectrum from the “Inner” region encompassing the pulsar. To minimize calibration errors due to differing chips and distance from the pointing axis, we select an XMM background region on the northeast corner of the central chip containing PSR J1420-6048. We extract XMM spectral data with the SAS function evselect, create responses with the functions rmfgen and arfgen, and group to 30 counts per bin. To best probe spectral index variations we make a global estimate for the best fit hydrogen column density. To maximize the signal to noise, we simultaneously fit $n_H$ for the Suzaku “Halo” spectrum and the “Inner” XMM spectrum. For an absorbed power law model we find a $n_H$ of $4.1_{-0.4}^{+0.6} \times 10^{22} \rm \, cm^{-2}$ (90 % single parameter error). An individual fit to the inner PWN region indicates that an absorbed power law model provides the best fit to the data; addition of a blackbody, neutron star atmosphere, or thermal plasma (mekal) component does not improve the fit. Below 2 keV the fit to all regions significantly underestimates the flux. This feature is not well fit with a thermal plasma, neutron star atmosphere, or blackbody, and might simply constitute excess soft emission from the myriad faint sources in this crowded region. The $4.1_{-0.4}^{+0.6} \times 10^{22} \rm \, cm^{-2}$ column is marginally consistent with the value measured by @robertsetal01b of $2.2 \pm 0.7 \times 10^{22} \rm \, cm^{-2}$ using ASCA GIS data ($3\arcmin$ radius), and matches well with the short Chandra ACIS exposure fit by @ngetal05 ($2\arcmin$ radius aperture) of $5.4_{-1.7}^{+2.2} \times 10^{22} \rm \, cm^{-2}$ (1$\sigma$ error). The nominal total Galactic H column in this direction is estimated as $1.6 \times 10^{22} \rm \, cm^{-2}$ [@kalberlaetal05] and $2.1 \times 10^{22} \rm \, cm^{-2}$ [@dandl90]. With $n_H$ fixed at $4.1 \times 10^{22} \rm \, cm^{-2}$, we extract spectra from the three regions described above. Table 1 lists the results of fitting an absorbed power law to these regions in the 2-10 keV range; errors are 90% single parameter values. We quote 2–10 keV fluxes to compare to previous work, minimize sensitivity to the hydrogen column density fit, and mitigate soft X-ray contamination in this crowded region. In the inner $100\arcsec$ region we fit simultaneously the XMM and Suzaku data, and find a power law index of $1.8 \pm 0.1$. The entire $3 \arcmin$ “Halo” is fit with an index of $2.0 \pm 0.1$, implying a steeper power law near the edge of the nebula. To further explore this, we extract the spectrum from an annulus of radius $1.7\arcmin - 3\arcmin$ from J1420. For such annular regions the Suzaku response normalization proves unreliable, though the spectral shape is still of value. We find a power law index of $1.9 \pm 0.1$; this is indeed softer (though within errors) of the inner PWN. Further spectral softening is hinted at in the outer nebula “Tail,” which boasts an index of $2.1 \pm 0.1$. --------------------------------- -------------------------- ----------------- ------------------------ ------------------------ -------------- Region $N_{\rm H}$ $\Gamma$ abs. flux unabs. flux $\chi^2$/dof $\times10^{22}$cm$^{-2}$ $f_{2-10}$ $f_{2-10}$ Inner$\tablenotemark{\Diamond}$ $4.1$ $1.82 \pm 0.13$ $1.19_{-0.20}^{+0.24}$ $1.62_{-0.28}^{+0.32}$ 333/345 Halo $4.1$ $2.00\pm0.10$ $2.45_{-0.31}^{+0.35}$ $3.40_{-0.43}^{+0.48}$ 219/309 Tail $4.1$ $2.14\pm0.16$ $1.26_{-0.25}^{+0.30}$ $1.79_{-0.35}^{+0.42}$ 138/224 --------------------------------- -------------------------- ----------------- ------------------------ ------------------------ -------------- : Spectral Fits \[pwnspec\] These spectral values are largely consistent (though with smaller errors) with previous data fits. @ngetal05 extract the spectrum of the inner nebula from a 2’ circle, which gives (for $n_H$ fixed at $5.4 \times 10^{22} \rm \, cm^{-2}$) $\Gamma = 2.3_{-0.8}^{+0.9}$ (projected multidimensional 1 sigma error), unabsorbed 2-10 keV flux $1.3 \pm0.14 \times 10^{-12} \rm \,erg\,cm^{-2}\,s^{-1}$ (1 sigma single parameter error). @robertsetal01b measure a power law index of $1.6\pm0.4$ and a 2-10 keV flux of $4.7 \times 10^{-12} \rm \,erg\,cm^{-2}\,s^{-1}$ for an extraction region of size $3\arcmin$. SED Fitting =========== Matching SED data points can help constrain physical properties of the source. To this end we we apply a two-zone time-dependent numerical model with constant injection luminosity. We inject a power law spectrum of relativistic particles (either electrons or protons) with a high energy exponential cutoff into zone 1, and then evolve this spectrum over time according to radiation losses from synchrotron and IC (Klein-Nishina effects included). The resultant spectrum is subsequently injected into zone 2 and further evolved. Adiabatic cooling is ignored, given that offsetting heating effects of the SNR reverse shock interaction with the PWN may significantly alter particle energetics. The typical timescale for the reverse shock to reach the PWN is $\sim7$ kyr [@randc84], while the characteristic age of the pulsar is only 13 kyr, so this perturbation may still be felt. The complex nature of this interaction is also why we use a constant injection luminosity rather than allowing injection luminosity to vary with pulsar spin down power. Injection (and evolution) occurs in time steps much smaller than the assumed age. We consider a low density environment with density $n$ = 1 cm$^{-3}$ and photon fields comprised of CMBR, near IR, and far IR (though starlight is unimportant due to Klein-Nishina suppression of IC at high photon energies). Our two zones are defined as the inner $100\arcsec$ region surrounding the pulsar (zone 1) and the broader extended nebula (zone 2). Figures 3 and 4 show the SED of J1420. While we must select spectral extraction regions such that interference from stellar sources is minimized and region surface brightness remains significantly above background, the nebular flux extends beyond these regions. An estimate of the total nebular flux is essential if we wish to compare the X-ray flux with the VHE flux, which is extracted from a large $\approx 10 \arcmin$ circle. To estimate the total extended nebula flux, we extract 2-10 keV counts from a large ellipse $5 \arcmin \times 6 \arcmin$ in radius which seems to encompass the majority of the X-ray flux (excluding point sources within this field) and find 40% more counts in this region than the combined “Halo” plus “Tail” minus “Inner” region. We therefore extrapolate that the total extended nebular flux is 40% greater than the combined “Halo” plus “Tail” minus “Inner” fluxes. Accordingly, for SED modeling of the extended nebula we plot the spectral points from the “Tail” region, scaled up to represent this estimated 2-10 keV flux of $5.0 \times 10^{-12} \, \rm erg \, cm^{-2} \, s^{-1}$. wn Figures 3 and 4 We plot the gamma-ray flux from HESS J1420–607, along with radio and X-ray data. Also indicated are the Fermi LAT one-year and ten-year $5\sigma$ flux limit for a point source residing in a background $10\times$ greater than the extragalactic background, which is reasonable for the Galactic plane. ### Leptonic gamma-ray emission model For an X-ray spectral index approximately flat on the SED plot one cannot easily match the X-ray data to the H.E.S.S. data. A single electron injection component works, but only after significant fine tuning of injection and ambient medium parameters. A dual component injection comprises the other alternative, though the attentive reader might note that the northern X-ray nebula spatially corresponds quite well to the H.E.S.S. position and size, hinting at a single electron population responsible for both types of emission. If one assumes all particles evolve in the same spatial region (and hence same magnetic field and ambient photon field) one cannot independently fit the X-ray data to one component and VHE data to another significantly different component. Therefore, we refrain from displaying the unsatisfactory dual component fits, and focus on a single component model. A single electron component demands that the H.E.S.S. detection arise from IC scattering of far IR photons, as the CMB provides insufficient seed photons to account for the VHE peak given the constraints imposed by the X-ray data. For the Galactic radius of J1420, far IR photons typically peak at $\approx T = 25$ K with a density $\approx 1$ eV$\rm \, cm^{-3}$ [@pms06]. We find a better fit with a denser far IR photon field (2 eV$\rm \, cm^{-3}$), which is reasonable given the large scatter in ambient photon fields throughout the Galaxy. The starlight photon field is taken to have a temperature of 3500 K with density 1 eV$\rm \, cm^{-3}$. The innermost $100\arcsec$ region is populated by young electrons emitting in a relatively strong magnetic field. In the ideal case of a magnetic dipole, the pulsar surface magnetic field can be estimated as $B = 3.2 \times 10^{19} \, (P \, \dot P)^{1/2} \, \rm G$ where $P$ is in seconds. This gives a surface magnetic field of $2.4 \times 10^{12}$ G for PSR J1420–6048. If one assumes the magnetic field is dipolar out to the light cylinder, and then falls off as the inverse of radius past the light cylinder, the volume averaged magnetic field out to $50 \arcsec$ from the pulsar is calculated as $11 \, \mu \rm G$. Without knowledge of where the bulk of the synchrotron emission stems from (given the poor spatial resolution of Suzaku) we therefore adopt a mean magnetic field of of $12 \, \mu \rm G$ for the innermost region. Electrons responsible for the hard X-rays can diffuse out of this region in as little as 400 years, as shall be seen in Section 4. We therefore evolve the inner zone 1 electrons over 500 years. A power-law distribution of electrons with the standard index of 2 is injected, with a high energy cutoff at 400 TeV. For these parameters, we require an energy in the innermost region of $9 \times 10^{45} \, d_{5.6}^2$ erg in the form of leptons, where distance $d = d_{5.6}\times 5.6 \, {\rm kpc}$. The electron index of 2 underpredicts the K3 radio flux excess by a factor of $\sim4$. A slightly softer electron index allows us to fit the radio points, but increases the energy requirements by a factor of $\sim 3$ and results in an outer nebula radio flux butting up against the upper limits established above. Better constraints on the non-thermal radio emission from this region would aid greatly in differentiating between models. In a magnetic field of $12 \, \mu \rm G$ evolving over 500 years results in a lepton spectral break at $\sim 100$ TeV. This broken spectrum is subsequently injected into zone 2, the outer nebula. With this electron spectrum and for the photon field described above the outer nebula requires a total lepton energy of $9 \times 10^{47} \, d_{5.6}^2$ erg in order to fit the H.E.S.S. data. This energy requirement is reasonable for an extended PWN, and can be compared to the Vela X nebula possessing $\sim 10^{48}$ erg in the form of leptons [@dJ07]. Evolving the electrons over 12.5 kyr places the X-ray and VHE emitting electrons in the cooled regime for the $8 \, \mu$G field we select to match the X-ray data. ### Hybrid hadronic and leptonic gamma-ray emission model One can also model the VHE gamma-rays as the product of pion decay from proton proton interactions. The timescale for pion production via p-p interactions is given by $\rm \tau_{pp} \approx 1.5 \times 10^{8} \, (n/1 cm^{-3})^{-1}$ years [@b70]; this timescale is significantly greater than the expected age of the system, so the proton spectrum is treated as static. For an injected power law proton spectrum, we calculate the photons from proton-proton interactions and subsequent $\pi^0$ and $\eta$-meson decay following @ket06. Proton-proton interactions also yield $\pi^{\pm}$ mesons which decay into secondary electrons, which we evolve over 13 kyr. IC and synchrotron emission from the resultant secondary electron spectrum are subsequently calculated. Secondary electrons evolved over 13 kyr cannot possibly account for the radio or X-ray data from the extended nebula for plausible magnetic fields. We nevertheless indicate on Figure 4 the synchrotron radiation from secondary electrons. The X-ray data must therefore be attributed to synchrotron radiation from a primary electron population. We adopt the same two-zone lepton model described above, though with the outer nebula parameters tweaked slightly. We adopt the typical far IR field density of $1$ eV$\rm \, cm^{-3}$, reduce lepton injection energy by $\sim 15$ % in the outer nebula, and adopt a magnetic field of $9 \, \mu$G for this region. A magnetic field any greater than this is precluded both by the radio data and by increased synchrotron cooling, which steepens the synchrotron slope in the Suzaku range. With these parameters gammas from pion decay can account approximately equally with IC emission for the VHE data. The hadronic contribution to the H.E.S.S. detection can be modeled by a a proton power law of index 2, cutoff at 200 TeV, and energy $\rm E = 7\times10^{50} \, (n/1 \, cm^{-3}) \, d_{5.6}^2$ erg. The SED from this hybrid hadronic plus leptonic scenario is shown in Fig 4. We omit the inconsequential IC radiation from secondary electrons. Discussion ========== X-ray flux extending $8\arcmin$ north of the pulsar indicates a spatial extent of $\rm R = 13 \, d_{5.6}$ pc. Though this requires an average flow speed of under 1% the speed of light for a 13 kyr age, diffusion across magnetic field lines can be an extremely slow process. For a cooling limited electron source with Bohm diffusion, the diffusion timescale varies with size ($\theta$), magnetic field ($B$), and electron energy ($E_e$) as: $$t_{diff} \approx 68 \, (\theta/8\arcmin)^2 \,d_{5.6}^2 \, (B/10 \, \mu {\rm G}) \, (E_e/100 \, {\rm TeV})^{-1} \, \, {\rm kyr} \eqno (1)$$ Therefore over the $\sim 10$ kyr lifetime of J1420, the inferred presence of distant VHE electrons is likely due to a different process than simple diffusion, as discussed in the next section. The observation of hard synchrotron X-rays requires an efficient particle accelerator, since in a transverse magnetic field the mean synchrotron photon energy ($E_{\gamma}$) scales as: $$E_{\gamma} \approx 2.2 (E_e/100 \, {\rm TeV})^2 (B/10 \, \mu {\rm G}) \, \, {\rm keV} \eqno (2)$$ Such energetic electrons cool rapidly, however, on a timescale of: $$\tau_{sync} \approx 820 \, (E_e/100 \, {\rm TeV})^{-1} (B/10 \, \mu {\rm G})^{-2} \, \, {\rm year} \eqno (3)$$ Therefore the cooling timescale for electrons radiating photons $E_{\gamma}$ at mean energy is: $$\tau_{sync} \approx 1200 \, (E_{\gamma}/1 \, {\rm keV})^{-1/2} (B/10 \, \mu {\rm G})^{-3/2} \, \, {\rm year} \eqno (4)$$ Synchrotron cooling dominates over IC cooling for electron energies above 5 TeV for the external photon and magnetic fields chosen, so we consider only synchrotron cooling in the following discussion. The electron cooling break results in a similar break in the emitted photon spectrum, which scales with magnetic field and age ($\tau$) as: $$E_{\gamma,Break} \approx 26 \, (\tau / 13 \, {\rm kyr})^{-2} (B/10 \, \mu {\rm G})^{-3} \, {\rm eV} \eqno (5)$$ This break is apparent in the models plotted in Figures 3 (break at $\sim 50$ eV) and 4 (break at $\sim$ 30 eV). The break location is also of key importance to IC modeling of the H.E.S.S. detection. Electrons upscatter blackbody photons (at temperature $T$) to a mean energy of: $$E_{\gamma} \approx 2.2 \, (E_e/10 \, {\rm TeV})^{2} \, (T/ 25 \, {\rm K}) \, \, {\rm TeV} \eqno (6)$$ where we scale $T$ to the value adopted for the far IR field. HESS J1420–607 appears to peak in flux at $\approx 0.5-2.5$ TeV; we expect this peak to result from a cooling break rather than an exponential cutoff in the electron spectrum due to the need for significant numbers of VHE electrons to synchrotron radiate in the X-ray regime. Inserting the value of $E_{\gamma} = 0.5-2.5$ TeV into Equation 6 we see that at the break: $$E_{e,Break} \approx (5-11) \, (T/25 \, {\rm K})^{-1/2} \, \, {\rm TeV} \eqno (7)$$ This VHE break is primarily due to synchrotron cooling, so substituting Equation 7 into Equation 3 yields an estimate for the mean magnetic field strength in the nebula: $$B \approx (8-11) \, (T/25 \, {\rm K})^{-1/4} \, (\tau/ 13 \, {\rm kyr})^{-1/2} \, \, \mu {\rm G} %6.95-10.2 \eqno (8)$$ While this range of magnetic field (which ignores IC cooling) should not be taken too seriously, the fact that the 8 $\mu$G field we selected for single component modeling also matches the X-ray data lends some credence to this estimate. Comparing the merits of each model, we find that the leptonic model provides a slightly better fit to the data, without the cost of the additional spectral components of the hybrid model. In addition, in order for pion decay to account for the VHE emission completely the energy in protons exceeds $10^{51}$ erg for a typical density of $n=1 \, \rm cm^{-3}$, and the fit to the lower energy H.E.S.S. data points is unimpressive. Even with $ > 10^{51}$ erg in hadrons, a primary electron population required to match the X-ray data contributes significantly to the VHE flux via IC, as seen in Figure 4. Only by raising the ambient magnetic field to $> 20 \mu$G and lowering the total energy budget of this electron population can the IC from these electrons be rendered insignificant. Yet for a magnetic field of this magnitude synchrotron cooling results in a 2–10 keV photon spectrum far steeper than our measured data, plus such a high magnetic field over such a large region would be surprising. The high energy requirement in hadrons, as well as the need for two, rather than one, highly tuned injection components makes the hybrid model a less appealing explanation for the observed extended nebula fluxes than a single electron population. In addition, further support for the leptonic model stems from its ability to account for the innermost X-ray flux with a younger population of similar electrons radiating in the stronger magnetic field near the pulsar. Early detection by the Fermi LAT of the K3 nebula would undermine this assumption of a single electron component, however, as in this model very little flux is expected in the GeV range. Furthermore, a photon index of $\alpha = (p+1)/2$ (where $p$ is the electron index) is expected, which for our injection would imply a hard LAT index of 1.5. Detection by the LAT is much more likely in the hadronic scenario, and one would expect a GeV index of $\approx 2$. A LAT index steeper than 2 cannot be matched by either model, and would imply the existence of another particle population, likely electrons. Indeed, @dj08 suggested the Vela X nebula could be modeled with two populations of electrons. Such a situation for the K3 nebula would allow for detection by the LAT, and might also explain the radio excess. LAT detection of the K3 nebula would also greatly aid in distinguishing between scenarios, though even LAT flux upper limits might preclude the hybrid model. Birthsite --------- The radio, X-ray, and VHE data all hint at a pulsar birthsite to the north. The possible radio filament extending to the hotspot $8.5\arcmin$ to the North might mark the synchrotron trail of J1420’s proper motion. The nature of the X-ray (source 7) and radio point source at the end of this tail remains mysterious, though an AGN provides a likely candidate. The radio “filament” might also mark the eastern boundary of the apparent radio shell mentioned in Section 2. Under this assumption, the pulsar could have originated $3 \arcmin$ northwest at the center of this shell. With this assumption, electrons spewed off the pulsar and left behind by its motion should also synchrotron radiate in the X-rays and emit VHE photons via IC scattering off field photons. The observed X-ray tail to the North, and the H.E.S.S. position offset $\sim 3 \arcmin$ in this direction nicely support this hypothesis. If the pulsar indeed originated $3\arcmin$ distant this implies a reasonable velocity of $$v = 370 \, (\theta/3 \arcmin) \, d_{5.6} \, (t/ 13 \, {\rm kyr})^{-2} \rm \, km \, s^{-1} \eqno (9)$$ The northern tail could alternatively indicate an asymmetric reverse shock returning from the South has crushed and displaced the K3 PWN, much like the situation observed in Vela X [@blondinetal01]. The pulsar birthsite would remain unconstrained by our X-ray observations in this scenario. Indeed, @jandw06 measured the polarization angle of the polarization sweep, which they argue correlates with the velocity. The perpendicular of this line (marking the pulsar spin and likely proper motion axis if it is emitting in the orthogonal mode) points southwest back to the shell body of the Kookaburra $\sim11 \arcmin$ away. Association would require a very high space velocity of $1400 \, d_{5.6} \, (t/ 13 \, {\rm kyr})^{-1} \rm \, km \, s^{-1}$ . Conclusions =========== The Suzaku data provides the first deep mapping of the asymmetric extended X-ray emission surrounding J1420. The spectral index appears to soften in the outer nebula, as expected in a leptonic model with synchrotron cooling of electrons. SED studies of this nebula allow for a hybrid leptonic $+$ hadronic model, yet favor a pure leptonic model if one takes into account energy requirements and the degree of tuning required to fit the hybrid model. The outer nebula also yields information about the pulsar birthsite, hinting at an origin to to the North, though admittedly the northern X-ray tail could be due to reverse shock interaction with the PWN and therefore not correspond to the pulsar velocity. Further investigation of the apparent radio shell along with deeper Chandra imaging of the compact nebula might also help elucidate the origin of PSR J1420–6048. [*Acknowledgments: We thank Stefan Funk for useful comments on this paper.*]{} Abdo, A. A., & for the Fermi LAT collaboration 2009, arXiv:0910.1608 Aharonian, F., et al. 2006, , 448, L43 Aharonian, F., et al. 2006, , 456, 245 Blondin, J. M., Chevalier, R. A., & Frierson, D. M. 2001, , 563, 806 Blumenthal, G. R. 1970, , 1, 1596 Cordes, J. M., & Lazio, T. J. W. 2002, arXiv:astro-ph/0207156 D’Amico, N., et al.  2001, , 552, L45 de Jager, O. C. 2007, , 658, 1177 de Jager, O. C., Slane, P. O., & LaMassa, S. 2008, , 689, L125 Dickey, J. M., & Lockman, F. J. 1990, , 28, 215 Johnston, S., & Weisberg, J. M. 2006, , 368, 1856 Kalberla, P. M. W., Burton, W. B., Hartmann, D., Arnal, E. M., Bajaja, E., Morras, R., [ P&ouml]{}ppel, W. G. L. 2005, , 440, 775 Kelner, S. R., Aharonian, F. A., & Bugayov, V. V. 2006, , 74, 034018 Ng, C.-Y., Roberts, M. S. E., & Romani, R. W. 2005, , 627, 904 Porter, T. A., Moskalenko, I. V., & Strong, A. W. 2006, , 648, L29 Reynolds, S. P., & Chevalier, R. A. 1984, , 278, 630 Roberts, M. S. E., Romani, R. W., Johnston, S., & Green, A. J. 1999, , 515, 712 Roberts, M. S. E., Romani, R. W., & Kawai, N. 2001, , 133, 451 Roberts, M. S. E., Romani, R. W., & Johnston, S. 2001, , 561, L187

--- abstract: 'Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros of $L_f(s)$ and the derivatives of $L_f(s)$ using Berndt’s method, and an estimate of zero-density of the derivatives of $L_f(s)$ based on Littlewood’s method.' author: - | Yoshikatsu Yashiro\ Graduate School of Mathematics, Nagoya University,\ 464-8602  Chikusa-ku, Nagoya, Japan\ E-mail: m09050b@math.nagoya-u.ac.jp title: '**Distribution of zeros and zero-density estimates for the derivatives of *L*-functions attached to cusp forms**' --- [^1] [^2] Introduction ============ Let $f$ be a cusp form of weight $k$ for $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform. Let $a_f(n)$ be the $n$-th Fourier coefficient of $f$ and set $\lambda_f(n)=a_f(n)/n^{(k-1)/2}$. Rankin showed that $\sum_{n\leq x}|\lambda_f(n)|^2=C_fx+O(x^{3/5})$ for $x\in\mathbb{R}_{>0}$, where $C_f$ is a positive constant depending on $f$ (see [@RAN (4.2.3), p.364]). The $L$-function attached to $f$ is defined by $$\begin{aligned} L_f(s)=\sum_{n=1}^\infty\frac{\lambda_f(n)}{n^s}=\prod_{p\text{:prime}}\left(1-\frac{\alpha_f(p)}{p^s}\right)^{-1}\left(1-\frac{\beta_f(p)}{p^s}\right)^{-1} \quad (\text{Re }s>1), \label{4LD}\end{aligned}$$ where $\alpha_f(p)$ and $\beta_f(p)$ satisfy $\alpha_f(p)+\beta_f(p)=\lambda_f(p)$ and $\alpha_f(p)\beta_f(p)=1$. By Hecke’s work ([@HEC]), the function $L_f(s)$ is analytically continued to the whole $s$-plane by $$\begin{aligned} (2\pi)^{-s-\frac{k-1}{2}}\Gamma(s+\tfrac{k-1}{2})L_f(s)=\int_0^\infty f(iy)y^{s+\frac{k-1}{2}-1}dy, \label{4AC}\end{aligned}$$ and has a functional equation $$\begin{aligned} L_f(s)=\chi_f(s)L_f(1-s)\end{aligned}$$ where $\chi_f(s)$ is given by $$\begin{aligned} \chi_f(s)=&(-1)^{-\frac{k}{2}}(2\pi)^{2s-1}\frac{\Gamma(1-s+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})} \notag\\ =&2(2\pi)^{-2(1-s)}\Gamma(s+\tfrac{k-1}{2})\Gamma(s-\tfrac{k-1}{2})\cos\pi(1-s). \label{4XFE}\end{aligned}$$ The second equality is deduced from the fact $\Gamma(s)\Gamma(1-s)=\pi/\sin(\pi s)$ and $\sin\pi(s+(k-1)/2)=(-1)^{k/2}\cos\pi(1-s)$. Similarly to the case of the Riemann zeta function $\zeta(s)$, it is conjectured that all complex zeros of $L_f(s)$ lie on the critical line $\text{Re }s= 1/2$, namely, the Generalized Riemann Hypothesis (GRH). In order to support the truth of the GRH, the distribution and the density of complex zeros of $L_f(s)$ are studied without assuming the GRH. Lekkerkerker [@LEK] proved the approximate formula of a number of complex zeros of $L_f(s)$: $$\begin{aligned} N_f(T)=\frac{T}{\pi}\log\frac{T}{2\pi e}+O(\log T), \label{ML0}\end{aligned}$$ where $T>0$ is sufficiently large, and $N_f(T)$ denotes the number of complex zeros of $L_f(s)$ in $0<{\rm Im\;}s\leq T$. The formula (\[ML0\]) is an analogy of $N(T)$ which denotes the number of complex zeros of $\zeta(s)$ in $0<{\rm Im\;}s\leq T$. Riemann [@RIE] showed that $$\begin{aligned} N(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+O(\log T). \label{Z0A}\end{aligned}$$ (Later von Mangoldt [@VOM] proved (\[Z0A\]) rigorously.) In the Riemann zeta function, the zeros of derivative of $\zeta(s)$ have a connection with RH. Speiser [@SPE] showed that the Riemann Hypothesis (RH) is equivalent to the non-existence of complex zero of $\zeta'(s)$ in $\text{Re }s<1/2$, where $\zeta'(s)$ denotes the derivative function of $\zeta(s)$. Levinson and Montgomery proved that if RH is true, then $\zeta^{(m)}(s)$ has at most finitely many complex zeros in $0<\text{Re }s<1/2$ for any $m\in\mathbb{Z}_{\geq0}$. There are many studies of the zeros of $\zeta^{(m)}(s)$ without assuming RH. Spira [@SP1], [@SP2] showed that there exist $\sigma_{m}\geq(7m+8)/4$ and $\alpha_{m}<0$ such that $\zeta^{(m)}(s)$ has no zero for ${\rm Re\;}s\leq\sigma_{m}$ and ${\rm Re\;}s\leq\alpha_m$, and exactly one real zero in each open interval $(-1-2n,1-2n)$ for $1-2n\leq\alpha_m$. Later, Y[i]{}ld[i]{}r[i]{}m [@YIL] showed that $\zeta''(s)$ and $\zeta'''(s)$ have no zeros in the strip $0\leq\text{Re\;}s<1/2$. Berndt [@BER] gave the relation of the number of complex zeros of $\zeta(s)$ and $\zeta^{(m)}(s)$: $$\begin{aligned} N_m(T)=N(T)-\frac{T\log 2}{2\pi}+O(\log T), \label{ZMA}\end{aligned}$$ where $m\in\mathbb{Z}_{\geq1}$ is fixed and $N_m(T)$ denotes the number of complex zeros of $\zeta^{(m)}(s)$ in $0<{\rm Im\;}s\leq T$. Recently, Aoki and Minamide studied the density of zeros of $\zeta^{(m)}(s)$ in the right hand side of critical line ${\rm Re\;}s=1/2$ by using Littlewood’s method. Let $N_m(\sigma,T)$ be the number of zeros of $\zeta^{(m)}(s)$ in $\text{Re\;}s\geq\sigma$ and $0<\text{Im\;}s\leq T$. They showed that $$\begin{aligned} N_m(\sigma,T)=O\left(\frac{T}{\sigma-1/2}\log\frac{1}{\sigma-1/2}\right), \label{MME}\end{aligned}$$ uniformly for $\sigma>1/2$. From (\[ZMA\]) and (\[MME\]), we see that almost all complex zeros of $\zeta^{(m)}(s)$ lie in the neighbourhood of the critical line. The purpose of this paper is to study the corresponding results of Berndt, Aoki and Minamide for the derivatives of $L_f(s)$, namely, the relation between the number of complex zeros of $L_f(s)$ and that of $L_f^{(m)}(s)$, and the density of zeros of $L_f^{(m)}{(s)}$ in the right half plane ${\rm Re\;}s>1/2$. Let $n_f$ be the smallest integer greater than 1 such that $\lambda_f(n_f)\ne0$. Here $L^{(m)}_f(s)$ denotes the $m$-th derivative of $L_f(s)$ given by $$\begin{aligned} L_f^{(m)}(s)=\sum_{n=1}^\infty\frac{\lambda_f(n)(-\log n)^m}{n^s}=\sum_{n=n_f}^\infty\frac{\lambda_f(n)(-\log n)^m}{n^s} \quad (\text{Re }s>1),\end{aligned}$$ Differentiating both sides of (\[4AC\]), we find that $L^{(m)}_f(s)$ is holomorphic in the whole $s$-plane and has the functional equation: $$\begin{aligned} L^{(m)}_f(s)=\sum_{r=0}^m\binom{m}{r}(-1)^{r}\chi_f^{(m-r)}(s)L_f^{(r)}(1-s). \label{4DFE}\end{aligned}$$ First in order to achieve the above purpose, we shall show the zero free regions for $L_f^{(m)}(s)$ by following Berndt’s method (see [@BER]) and Spira’s method (see [@SP1], [@SP2]). \[THM0\] The following assertions hold for any $m\in\mathbb{Z}_{\geq0}$. (i) \[LRZF\] There exists $\sigma_{f,m}\in\mathbb{R}_{>1}$ such that $L_f^{(m)}(s)$ has no zero for ${\rm Re\;}s\geq\sigma_{f,m}$. (ii) \[LLZF\] For any $\varepsilon\in\mathbb{R}_{>0}$, there exists $\delta_{f,m,\varepsilon}\in\mathbb{R}_{>(k-1)/2+1}$ such that $L_f^{(m)}(s)$ has no zero for $|s|\geq\delta_{f,m,\varepsilon}$ satisfying ${\rm Re\;}s\leq -\varepsilon$ and $|{\rm Im\;}s|\geq \varepsilon$. (iii) \[LOZF\] There exists $\alpha_{f,m}\in\mathbb{R}_{<-(k-1)/2-1}$ such that $L_f^{(m)}(s)$ has only real zeros for ${\rm Re\;}s\leq\alpha_{f,m}$, and one real zero in each interval $(n-1,n)$ for $n\in\mathbb{Z}_{\leq \alpha_{f,m}}$. Next, based on Berndt’s proof, we can obtain the following formula of the numbers of complex zeros of $L_f^{(m)}(s)$: \[THM1\] For any fixed $m\in\mathbb{Z}_{\geq1}$, let $N_{f,m}(T)$ be the number of complex zeros of $L_f^{(m)}(s)$ in $0<{\rm Im\;}s\leq T$. Then for any large $T>0$, we have $$\begin{aligned} N_{f,m}(T)=\frac{T}{\pi}\log\frac{T}{2\pi e}-\frac{T}{2\pi}\log{n_f}+O(\log T).\end{aligned}$$ Moreover the relation between $N_f(T)$ and $N_{f,m}(T)$ are given by $$\begin{aligned} N_{f,m}(T)=N_{f}(T)-\frac{T}{2\pi}\log{n_f}+O(\log T).\end{aligned}$$ Finally using the mean value formula for $L_f^{(m)}(s)$ obtained in [@YY3] and Littlewood’s method, we obtain the estimate of density of zeros: \[THM2\] For any $m\in\mathbb{Z}_{\geq0}$, let $N_{f,m}(\sigma,T)$ be the number of complex zeros of $L_f^{(m)}(s)$ in ${\rm Re\;}s\geq\sigma$ and $0<{\rm Im\;}s\leq T$. For any large $T>0$, we have $$\begin{aligned} N_{f,m}(\sigma,T)=O\left(\frac{T}{\sigma-1/2}\log\frac{1}{\sigma-1/2}\right) \label{ZD2}\end{aligned}$$ uniformly for $1/2<\sigma\leq 1$. More precisely we have $$\begin{aligned} &\hspace{-12pt}N_{f,m}(\sigma,T)\notag\\ \leq& \displaystyle\frac{2m+1}{2\pi}\frac{T}{\sigma-1/2}\log\frac{1}{\sigma-1/2}+\frac{1}{2\pi}\frac{T}{\sigma-1/2}\log\frac{(2m)!n_fC_f}{|\lambda_f(n_f)|^2(\log n_f)^{2m}}+ & \notag\\ &+O(\log T)+\frac{1}{2\pi}\frac{T}{\sigma-1/2}\times & \notag\\[9pt] &\times\begin{cases} \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}(\log T)^{2m}}{T^{2\sigma-1}}\right)\right), & 1/2<\sigma<1, \\ \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}(\log T)^{2m+2}}{T}\right)\right), \hspace{6pt} & \sigma=1,\\ \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}}{T}\right)\right), & 1<\sigma<\sigma_{f,m}, \end{cases} \label{ZD1}\end{aligned}$$ where $\sigma_{f,m}$ is given by (\[LRZF\]) of Theorem \[THM0\]. Proof of Theorem \[THM0\] ------------------------- First in order to show , we write $L_f^{(m)}(s)=\lambda_f(n_f)(-\log n_f)^m F(s)n_f^{-s}$ where $$\begin{aligned} F(s)=1+\sum_{n=n_f+1}^\infty\frac{\lambda_f(n)}{\lambda_f(n_f)}\left(\frac{\log n}{\log n_f}\right)^m\left(\frac{n_f}{n}\right)^s \quad ({\rm Re\;}s>1). \label{FFF}\end{aligned}$$ Deligne’s result $|\lambda_f(n)|\leq d(n)\ll n^\varepsilon$ gives that there exist $c_f\in\mathbb{R}_{>0}$ and $\sigma_{f,m}\in\mathbb{R}_{>1}$ depending on $f$ and $m$ such that $$\begin{aligned} |F(\sigma+it)-1| \leq&\sum_{n=n_f+1}^\infty\left|\frac{\lambda_f(n)}{\lambda_f(n_f)}\right|\left(\frac{\log n}{\log n_f}\right)^m\left(\frac{n_f}{n}\right)^\sigma\notag\\ \leq& c_f\sum_{n=n_f+1}^\infty\frac{(\log{n}/\log{n_f})^m}{(n/n_f)^{\sigma-\varepsilon}}\leq\frac{1}{2}\label{FMZ}\end{aligned}$$ for $\sigma\in\mathbb{R}_{\geq\sigma_{f,m}}$ and $t\in\mathbb{R}$, where $\varepsilon$ is an arbitrary positive number. Hence $L_f^{(m)}(s)$ has no zeros for ${\rm Re\;}s\geq\sigma_{f,m}$, that is, is showed. Next we shall show and . Replacing $s$ to $1-s$ in and , we have $$\begin{aligned} &(-1)^mL_f^{(m)}(1-s)\notag\\ &=\sum_{r=0}^m\binom{m}{r}L_f^{(m-r)}(s)\left(2(2\pi)^{-2s}\cos(\pi s)\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2})\right)^{(r)}. \label{CLZ1}\end{aligned}$$ By the facts $(\cos(\pi s))^{(r)}=\pi^r(a_r\cos(\pi s)+b_r\sin(\pi s))$ where $a_r, b_r \in\{0,\pm 1\}$ and $((2\pi)^{-2s})^{(r)}=(-2\log 2\pi)^r(2\pi)^{-2s}$ for $r\in\mathbb{Z}_{\geq 0}$, the formula is written as $$\begin{aligned} (-1)^mL_f^{(m)}(1-s) =2(2\pi)^{-2s}\sum_{r=0}^mR_{m-r}(s)(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))^{(r)}, \label{CLZ2}\end{aligned}$$ where $$\begin{aligned} R_{m-r}(s)=\cos(\pi s)\sum_{j=0}^{m-r}a_j'L_f^{(j)}(s)+\sin(\pi s)\sum_{j=0}^{m-r}b_j'L_f^{(j)}(s)\end{aligned}$$ and $a_r', b_r'\in\mathbb{R}$. It is clear that $a_0'=1$, $b_0'=0$ and $R_0(s)=L_f(s)\cos(\pi s)$. Moreover we write as $$\begin{aligned} \frac{(-1)^mL_f^{(m)}(1-s)}{2(2\pi)^{-2s}}=f(s)+g(s) \label{CLZ3}\end{aligned}$$ where $$\begin{aligned} f(s)=&R_0(s)(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))^{(m)},\\ g(s)=&\sum_{r=0}^{m-1}R_{m-r}(s)(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))^{(r)}. \end{aligned}$$ The formula implies that if $|f(s)|>|g(s)|$ in a some region then $L_f^{(m)}(s)$ has no zero in this region. In order to investigate the behavior of $f(s)$ and $g(s)$, we shall consider the approximate formula for $(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))^{(r)}/(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))$. By Stirling’s formula, it is known that $$\begin{aligned} \frac{\Gamma'}{\Gamma}(s)=\log s-\frac{1}{2s}+\int_0^\infty\frac{\{u\}-1/2}{(u+s)^2}du $$ for $s\in\mathbb{C}$ such that $|\arg s|\leq\pi-\delta$ where $\delta\in\mathbb{R}_{>0}$ is fixed (see ). Writing the right-hand side of the above formula to $G^{(1)}(s)$ and putting $G^{(j)}(s)=(d^{j-1}/ds)G^{(1)}(s)$ for $j\in\mathbb{Z}_{\geq2}$, we shall use the following lemma: \[LIB\] Let $F$ and $G$ be holomorphic function in the region $D$ such that $F(s)\ne0$ and $\log F(s)=G(s)$ for $s\in D$. Then for any fixed $r\in\mathbb{Z}_{\geq1}$, there exist $l_1,\cdots,l_r\in\mathbb{Z}_{\geq0}$ and $C_{(l_1,\cdots,l_r)}\in\mathbb{Z}_{\geq0}$ such that $$\begin{aligned} \frac{F^{(r)}}{F}(s) =\sum_{1l_1+\cdots+rl_r=r}C_{(l_1,\cdots,l_r)}(G^{(1)}(s))^{l_1}\cdots(G^{(r)}(s))^{l_r} $$ for $s\in D$. Especially $C_{(r,0,\cdots,0)}=1$. The estimates $$\begin{aligned} |u+s|^2=&u^2+|s|^2+2u|s|\cos\arg s\\\geq&\begin{cases} |s|^2, & u\leq|s|, \; |\arg s|\leq\pi/2, \\ |s|^2(\sin\arg s)^2, & u\leq|s|,\; \pi/2\leq|\arg s|\leq\pi-\delta, \\ u^2, & u\geq|s|, \;|\arg s|\leq\pi/2, \\ 4(1+\cos\arg s)u^2, & u\geq|s|,\; \pi/2\leq|\arg s|\leq\pi-\delta \end{cases}\end{aligned}$$ give $$\begin{aligned} \int_{0}^{\infty}\frac{\{u\}-1/2}{(u+s)^{j+1}}du \ll&\int_0^{|s|}\frac{du}{|s|^{j+1}}+\int_{|s|}^\infty\frac{du}{u^{j+1}}\ll\frac{1}{|s|^{j}}\end{aligned}$$ for $|\arg s|\leq\pi-\delta$ and $j\in\mathbb{Z}_{\geq1}$. Then $G^{(j)}(s)$ is calculated as $$\begin{aligned} G^{(1)}(s)=&\log s+O\left(\frac{1}{|s|}\right),\\ G^{(j)}(s)=&\frac{(-1)^{j-1}(j-2)!}{s^{j-1}}+\frac{(-1)^j(j-1)!}{2s^j}+(-1)^{j+1}j!\int_0^\infty\frac{\{u\}-1/2}{(u+s)^{j+1}}du\\ =&O\left(\frac{1}{|s|^{j-1}}\right)\end{aligned}$$ for $j\in\mathbb{Z}_{\geq2}$. Hence the approximate formula for $(\Gamma^{(r)}/\Gamma)(s)$ is written as $$\begin{aligned} \frac{\Gamma^{(r)}}{\Gamma}(s) =&(G^{(1)}(s))^r+O\left(\sum_{1q_1+\cdots+rq_r=r, \ q_1\ne r \;}\prod_{j=1}^r|G^{(j)}(s)|^{q_j}\right)\notag\\ =&\left(\log s+O\left(\frac{1}{|s|}\right)\right)^r+O\left(\frac{|\log s|^{r-1}}{|s|}\right)=(\log s)^r\sum_{j=0}^r\frac{M_j(s)}{(\log s)^{j}} \label{CLZ4}\end{aligned}$$ for $|\arg s|\leq\pi-\delta$ and $r\in\mathbb{Z}_{\geq0}$, where $M_j(s)$ is given by $M_j(s)=O(1/|s|^j)$ for $j\in\mathbb{Z}_{\geq 1}$ and $M_0(s)=1$. Using and the trivial estimates $|1\pm({k-1})/{2s}|<2<|s|^{1/2}$, $$\begin{aligned} &|s\pm\tfrac{k-1}{2}|=|s|\times|1\pm\tfrac{k-1}{2s}|\gg|s|,\notag\\ &\log|s\pm\tfrac{k-1}{2}|=(\log|s|)\times\left(1+\frac{\log|1\pm\tfrac{k-1}{2s}|}{\log|s|}\right)\gg\log|s|, \label{CLE2}\\ &\log(s\pm\tfrac{k-1}{2})\ll\sqrt{(\log|s\pm\tfrac{k-1}{2}|)^2+(\arg(s\pm\tfrac{k-1}{2}))^2}\ll\log|s|\notag\end{aligned}$$ for $|s|>(k-1)/2$, we obtain a desired formula: $$\begin{aligned} &\frac{(\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2}))^{(l)}}{\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2})}\notag\\ =&\sum_{j=0}^l\binom{l}{j}\frac{\Gamma^{(j)}}{\Gamma}(s-\tfrac{k-1}{2})\frac{\Gamma^{(l-j)}}{\Gamma}(s+\tfrac{k-1}{2})\notag\\ =&\sum_{j=0}^l\binom{l}{j}(\log(s-\tfrac{k-1}{2}))^j(\log(s+\tfrac{k-1}{2}))^{l-j}\times\notag\\ &\times\sum_{\begin{subarray}{c} 0\leq j_1+j_2\leq l,\\ 0\leq j_1\leq j,\; 0\leq j_2\leq l-j \end{subarray}}\frac{M_{j_1}(s-\tfrac{k-1}{2})}{(\log(s-\tfrac{k-1}{2}))^{j_1}}\frac{M_{j_2}(s+\tfrac{k-1}{2})}{(\log(s+\tfrac{k-1}{2}))^{j_2}}\notag\\ =&S_l(s)+T_l(s) \label{CLZ5}\end{aligned}$$ for $l\in\mathbb{Z}_{\geq0}$ and $s\in\mathbb{C}$ such that $|s|>(k-1)/2$ and $|\arg s|\leq\pi-\delta$ , where $S_l(s)$, $T_l(s)$ are given by $$\begin{aligned} S_l(s)=&(\log(s-\tfrac{k-1}{2})+\log(s+\tfrac{k-1}{2}))^l,\\ T_l(s)=&O\left(\frac{1}{|s|\log|s|}\sum_{j=0}^l(\log|s|)^j(\log|s|)^{l-j}\right)=O\left(\frac{(\log |s|)^{l-1}}{|s|}\right)\end{aligned}$$ respectively for $l\in\mathbb{Z}_{\geq 1}$, especially $S_0(s)=1$ and $T_0(s)=0$. Next using $R_r(s)$, $S_r(s)$ and $T_r(s)$, we shall write a condition which gives $|f(s)|>|g(s)|$ for some region. From and , the inequality $|f(s)|>|g(s)|$ is equivalent to $$\begin{aligned} |S_m(s)+T_m(s)|>\left|\sum_{r=0}^{m-1}\frac{R_r}{R_0}(s)(S_r(s)+T_r(s))\right|. \end{aligned}$$ Dividing the both sides of the above formula by $S_{m-1}(s)$ and applying the triangle inequality, we see that if $$\begin{aligned} |S_1(s)|>\left|\frac{T_m}{S_{m-1}}(s)\right|+\left|\sum_{r=0}^{m-1}\frac{R_r}{R_0}(s)\left(\frac{1}{S_{m-1-r}}(s)+\frac{T_r}{S_{m-1}}(s)\right)\right| \label{CLZ6}\end{aligned}$$ is true, then $|f(s)|>|g(s)|$ is true for $|s|>(k-1)/2$ and $|\arg s|\leq\pi-\delta$. To show the truth of , we shall consider an upper bound of $(1/S_r)(s)$, $T_r(s)$ and $(R_r/R_0)(s)$. The estimates and $|\log z|\geq\log|z|$ for $z\in\mathbb{C}$ give $$\begin{aligned} \left|\frac{1}{S_{r}}(s)\right|\leq\frac{1}{(\log|s-\tfrac{k-1}{2}|+\log|s+\tfrac{k-1}{2}|)^r}\leq\frac{C_1}{(\log|s|)^r} \label{CLZ7}\end{aligned}$$ for the above $s$ and $r\in\mathbb{Z}_{\geq 0}$, here and later $C_1, C_2, \dots$ denote positive constants depending on $f$, $r$ and $\delta$. Since $L_f^{(j)}(s)$ and $(1/L_f)(s)$ are absolutely convergent for ${\rm Re\;}s>1$, it follows that $$\begin{aligned} \left|\frac{R_r}{R_0}(s)\right|=\left|\sum_{j=0}^ra'_j\frac{L_f^{(j)}}{L_f}(s)+\tan(\pi s)\sum_{j=0}^rb'_j\frac{L_f^{(j)}}{L_f}(s)\right|\leq C_2+C_3|\tan(\pi s)| \label{CLZ9}\end{aligned}$$ for ${\rm Re\;}s\geq1+\varepsilon$. Here $\tan(\pi s)$ is estimated as $$\begin{aligned} |\tan\pi(\sigma+it)|=\left|\frac{e^{-2t}e^{2\pi i\sigma}-1}{e^{-2t}e^{2\pi i\sigma}+1}\right|\leq\begin{cases} 2/(1-e^{-2\varepsilon}), & \text{if } |t|\geq\varepsilon, \\ 3, & \text{if } \sigma\in\mathbb{Z} \end{cases} \label{CLZ10}\end{aligned}$$ where $\varepsilon$ is a fixed positive number. Combining –, we see that the right-hand side of is estimated as $$\begin{aligned} &\left|\frac{T_m}{S_{m-1}}(s)\right|+\left|\sum_{r=0}^{m-1}\frac{R_r}{R_0}(s)\left(\frac{1}{S_{m-1-r}}(s)+\frac{T_r}{S_{m-1}}(s)\right)\right|\notag\\ &\leq\frac{C_4}{|s|}+C_5|\tan(\pi s)|\sum_{r=0}^{m-1}\left(\frac{1}{(\log|s|)^{m-1-r}}+\frac{1}{|s|(\log|s|)^{m-r}}\right)\leq C_{f,m,\delta,\varepsilon} \label{CLZ11} \end{aligned}$$ for $|s|>(k-1)/2$ and ${\rm Re\;}s\geq1+\varepsilon$ provided $|{\rm Im\;}s|\geq\varepsilon$ or ${\rm Re\;}s\in\mathbb{Z}$, where $C_{f,m,\delta,\varepsilon}$ is a positive constant depending on $f$, $m$, $\delta$ and $\varepsilon$. Fix $\delta=\tan^{-1}(2\varepsilon/(k-1))$ and choose $r_{f,m}\in\mathbb{R}_{>(k-1)/2}$ such that $C_{f,m,\delta,\varepsilon}<(\log r_{f,m})/C_1$. The inequalities and imply that is true, that is, $L_f^{(m)}(1-s)$ has no zero for $s\in\mathbb{C}$ such that $|s|\geq r_{f,m}$, ${\rm Re\;}s\geq1+\varepsilon$ and $|{\rm Im\;}s|\geq\varepsilon$. Therefore, we conclude that for any $\varepsilon\in\mathbb{R}_{>0}$ there exists $\delta_{f,m,\varepsilon}\in\mathbb{R}_{>(k-1)/2+1}$ such that $L_f^{(m)}(s)$ has no zero in the region $|s|\geq\delta_{f,m,\varepsilon}$, ${\rm Re\;}s\leq-\varepsilon$ and $|{\rm Im\;}s|>\varepsilon$, that is, the proof of is completed. Finally we shall show applying Rouché’s theorem to $f(s)$ and $g(s)$. For $n\in\mathbb{Z}_{\geq1}$ let $D_n$ be the region $n\leq{\rm Re\;}s\leq n+1$ and $|{\rm Im\;}s|\leq 1/2$. By and , we see that there exists $\delta_{f,m,1/2}\in\mathbb{R}_{>(k-1)/2}$ such that $|f(s)|>|g(s)|$ is true in the boundary of $D_n$ and the region $|s|>\delta_{f,m,1/2}$ and ${\rm Re\;}s\geq1+1/2$. Then the number of zeros of $f(s)$ is equal to that of $f(s)+g(s)$ in the interior of $D_n$. From , and , the function $f(s)$ is written as $$\begin{aligned} f(s)=L_f(s)\cos(\pi s)\Gamma(s-\tfrac{k-1}{2})\Gamma(s+\tfrac{k-1}{2})(S_m(s)+T_m(s)). \label{FST}\end{aligned}$$ When $R_{f,m}$ is chosen such that $R_{f,m}\geq\delta_{f,m,1/2}$ and $C_6/(R_{f,m}\log R_{f,m})<1$, the formula gives $$\begin{aligned} \left|\frac{T_m}{S_m}(s)\right|\leq \frac{C_6}{|s|\log|s|}<1, \label{TS1}\end{aligned}$$ that is, $S_m(s)+T_m(s)$ has no zero for $|s|\geq R_{f,m}$. Hence $f(s)$ has the only real zero $s=n+1/2$ in $D_n$. It is clear to show $\overline{f(s)}=f(\overline{s})$ and $\overline{g(s)}=g(\overline{s})$ for $s\in\mathbb{C}$, which imply that $L_f^{(m)}(1-s)$ has a only real zero in the interior of $D_n$. Replacing $1-s$ to $s$, we obtain the fact that there exists $\alpha_{f,m}\in\mathbb{R}_{<-(k-1)/2-1}$ such that $L_f^{(m)}(s)$ has no complex zero for ${\rm Re\;}s<\alpha_{f,m}$ and one real zero in each open interval $(n-1,n)$ for $n\in\mathbb{Z}_{\leq\alpha_{f,m}}$. The proof of is completed. Proof of Theorem \[THM1\] ------------------------- Using Theorem \[THM0\], we can choose $\alpha_{f,m}\in\mathbb{R}_{<-(k-1)/2}$ and $\sigma_{f,m}\in\mathbb{R}_{>1}$ such that $L_f^{(m)}(s)$ has no zeros in the region ${\rm Re\;}s\leq\alpha_{f,m}$ and ${\rm Re\;}s\geq\sigma_{f,m}$. Moreover, choose $\tau_{f,m}\in\mathbb{R}_{>2}$ and $T\in\mathbb{R}_{>0}$ such that $L_f^{(m)}(s)$ has no zeros for $0<{\rm Im\;}s\leq\tau_{f,m}$ and ${\rm Im\;}s=T$. Using the residue theorem in the region $\alpha_{f,m}\leq{\rm Re\;}s\leq\sigma_{f,m}$ and $\tau_{f,m}\leq{\rm Im\;}s\leq T$, we get $$\begin{aligned} N_{f,m}(T)=&\frac{1}{2\pi i}\left(\int_{\alpha_{f,m}+i\tau_{f,m}}^{\sigma_{f,m}+i\tau_{f,m}}+\int_{\sigma_{f,m}+i\tau_{f,m}}^{\sigma_{f,m}+iT}+\int_{\sigma_{f,m}+iT}^{\alpha_{f,m}+iT}+\right.\notag\\ &\left.+\int_{\alpha_{f,m}+iT}^{\alpha_{f,m}+i\tau_{f,m}}\right)(\log L_f^{(m)}(s))'ds=:I_1+I_2+I_3+I_4. \label{LDB}\end{aligned}$$ First, it is clear that $$\begin{aligned} I_1=\frac{\log L_f^{(m)}(\sigma_{f,m}+i\tau_{f,m})-\log L_f^{(m)}(\alpha_{f,m}+i\tau_{f,m})}{2\pi i} =O(1). \label{LDC}\end{aligned}$$ To approximate $I_2$, we write $L_f^{(m)}(s)=\lambda_f(n_f)(-\log{n_f})^mF(s)n_f^{-s}$ where $F(s)$ is given by . Using (\[FMZ\]) we find that $1/2\leq|F(s)|\leq3/2$, ${\rm Re\;}F(s)\geq1/2$ and $|\arg F(s)|<\pi/2$ for $s=\sigma_{f,m}+it\;(t\in\mathbb{R})$. Hence $I_2$ is approximated as $$\begin{aligned} I_2=&\frac{1}{2\pi i}\left[\log\frac{\lambda_f(n_f)(-\log{n_f})^m}{n_f^s}+\log F(s)\right]_{\sigma_{f,m}+i\tau_{f,m}}^{\sigma_{f,m}+iT}\notag\\ =&\frac{-(\sigma_{f,m}+iT)\log n_f}{2\pi i}+O(1) =-\frac{T}{2\pi}\log{n_f}+O(1). \label{LDE}\end{aligned}$$ Next we shall estimate $I_3$. The formula (\[4DFE\]), the approximate functional equation for $L_f^{(m)}(s)$ (see [@YY3 Theorem 1.2]) and Rankin’s result, there exists $A\in\mathbb{R}_{\geq0}$ such that $L_f^{(m)}(\sigma+it)=O(|t|^A)$ uniformly for $\sigma\in[\alpha_{f,m},\sigma_{f,m}]$. It implies that $$\begin{aligned} I_3=&\frac{\log L_f^{(m)}(\alpha_{f,m}+iT)-\log L_f^{(m)}(\sigma_{f,m}+iT)}{2\pi i}\notag\\ =&\frac{\arg L_f^{(m)}(\alpha_{f,m}+iT)-\arg L_f^{(m)}(\sigma_{f,m}+iT)}{2\pi}+O(\log T). \label{LDF}\end{aligned}$$ To estimate the first term of the right-hand side of (\[LDF\]), we write $L_f^{(m)}(\sigma+iT)=(-1)^me^{-iT\log n_f}\lambda_f(n_f)G(\sigma+iT)$ where $$\begin{aligned} G(\sigma+iT)=\frac{(\log{n_f})^m}{n_f^{\sigma}}+\frac{1}{\lambda_f(n_f)}\sum_{n=n_f+1}^\infty\frac{\lambda_f(n)(\log n)^m}{n^{\sigma}}e^{iT\log\frac{n_f}{n}} $$ for $\sigma\in\mathbb{R}_{>1}$. Let $Q$ be the number of zeros of ${\rm Re\;}G(s)$ on the line segment $(\alpha_{f,m}+iT,\sigma_{f,m}+iT)$. Divide this line into $Q+1$ subintervals by these zeros. Then the sign of ${\rm Re\;}G(s)$ is constant, and the variation of $\arg G(s)$ is at most $\pi$ on each subinterval. Hence, there exists constant $C$ such that $\arg G(s)=\arg L_f^{(m)}(s)+C$ on the divided line, it follows that $$\begin{aligned} |\arg L_f^{(m)}(\alpha_{f,m}+iT)-\arg L_f^{(m)}(\sigma_{f,m}+iT)|\leq (Q+1)\pi. \label{LDH}\end{aligned}$$ In order to estimate $Q$, let $H(z)=(G(z+iT)+\overline{G(\overline{z}+iT)})/2$. Then we find that $$\begin{aligned} H(\sigma)=&{\rm Re\;}G(\sigma+iT)\notag\\ =&\frac{(\log n_f)^m}{n_f^\sigma}\left(1+\sum_{n=n_f+1}^\infty\frac{\lambda_f(n)}{\lambda_f(n_f)}\left(\frac{\log n}{\log n_f}\right)^m\left(\frac{n_f}{n}\right)^\sigma\cos\left(T\log\frac{n_f}{n}\right)\right) \label{LDG}\end{aligned}$$ for $\sigma\in\mathbb{R}_{>1}$. The formulas (\[FMZ\]) and (\[LDG\]) give $$\begin{aligned} \frac{1}{2}\frac{(\log n_f)^m}{{n_f}^{\sigma_{f,m}}}\leq H(\sigma_{f,m})\leq\frac{3}{2}\frac{(\log n_f)^m}{{n_f}^{\sigma_{f,m}}}. \label{HEE}\end{aligned}$$ Take $T$ sufficiently large such that $T-\tau_{f,m}>2(\sigma_{f,m}-\alpha_{f,m})$ if necessary. Since ${\rm Im}(z+iT)\geq T-(T-\tau_{f,m})>0$ for $z\in\mathbb{C}$ such that $|z-\sigma_{f,m}|<T-\tau_{f,m}$, it follows that $H(z)$ is analytic in the circle $|z-\sigma_{f,m}|<T-\tau_{f,m}$. Note that there exists a positive constant $B$ such that $H(z)=O(T^B)$ in this circle because of the fact that $L_f(\sigma+it)=O(|t|^A)$. For $u\in\mathbb{R}_{\geq0}$, let $P(u)$ be the number of zeros of $H(z)$ in $|z-\sigma_{f,m}|\leq u$. Then using the trivial estimate $$\begin{aligned} P(\sigma_{f,m}-\alpha_{f,m})&\leq\frac{1}{\log 2}\int_{\sigma_{f,m}-\alpha_{f,m}}^{2(\sigma_{f,m}-\alpha_{f,m})}\frac{P(u)}{u}du\\ &\leq\frac{1}{\log 2}\int_{0}^{2(\sigma_{f,m}-\alpha_{f,m})}\frac{P(u)}{u}du,\end{aligned}$$ Jensen’s formula (see [@TID Chapter 3.61]), the above note and (\[HEE\]), we have $$\begin{aligned} &P(\sigma_{f,m}-\alpha_{f,m})\\ &\ll\int_0^{2(\sigma_{f,m}-\alpha_{f,m})}\frac{P(u)}{u}du\\ &=\frac{1}{2\pi}\int_0^{2\pi}\log|H(\sigma_{f,m}+2(\sigma_{f,m}-\alpha_{f,m})e^{i\theta})|d\theta-\log|H(\sigma_{f,m})|\\ &\ll\int_0^{2\pi}\log T^Bd\theta+1\ll \log T,\end{aligned}$$ Therefore $Q$ is estimated as $$\begin{aligned} Q=\#\{\sigma\in(\alpha_{f,m},\sigma_{f,m})\mid F(\sigma)=0\}\ll P(\sigma_{f,m}-\alpha_{f,m})\ll \log T. \label{LDI}\end{aligned}$$ Combining (\[LDF\]), (\[LDH\]), (\[LDI\]), we obtain the estimate of $I_3$: $$\begin{aligned} I_3=O(\log T). \label{LD3}\end{aligned}$$ Finally in order to approximate $I_4$, we shall obtain the approximate formula for $\log L_f^{(m)}(\alpha_{f,m}+iT)$ as $T\to\infty$. By the proof of Theorem \[THM0\], there exists $\delta_{f,m}\in\mathbb{R}_{>0}$ such that $$\begin{aligned} \left|\frac{g}{f}(1-s)\right|< 1, \quad \left|\frac{T_m}{S_m}(1-s)\right|<1 \label{FGST}\end{aligned}$$ for $s\in\mathbb{C}$ in the region $|s-(1-(k-1)/2)|>\delta_{f,m}$, ${\rm Re\;}s<1-(k-1)/2$ and $|{\rm Im\;}s|>1/2$. Here choose $\alpha_{f,m}\in\mathbb{R}_{<0}$ such that $\alpha_{f,m}<1-(k-1)/2-\delta_{f,m}$ if necessary. Then the path of $I_4$ is contained in the above region. Replacing $s$ to $1-s$ and taking logarithmic function in the both sides of , we obtain $$\begin{aligned} \log L_f^{(m)}(\alpha_{f,m}+iT) =&-2(1-\alpha_{f,m}-iT)\log 2\pi+\log f(1-\alpha_{f,m}-iT)+\notag\\ &+\log\left(1+\frac{g}{f}(1-\alpha_{f,m}-iT)\right)+O(1). \label{I4A}\end{aligned}$$ The first formula of gives $|\arg(1+(g/f)(1-\alpha_{f,m}-iT))|<\pi/2$ and $$\begin{aligned} &\log\left(1+\frac{g}{f}(1-\alpha_{f,m}-iT)\right)\notag\\ &\ll\sqrt{\left|1+\frac{g}{f}(1-\alpha_{f,m}-iT)\right|^2+\left(\arg\left(1+\frac{g}{f}(1-\alpha_{f,m}-iT)\right)\right)^2}\ll 1. \label{LAFG}\end{aligned}$$ By , the second term of the right-hand side of is written as $$\begin{aligned} &\hspace{-12pt}\log f(1-\alpha_{f,m}-iT)\notag\\ =&\log\Gamma(1-\alpha_{f,m}-\tfrac{k-1}{2}-iT)+\log\Gamma(1-\alpha_{f,m}+\tfrac{k-1}{2}-iT)+\notag\\ &+\log S_m(1-\alpha_{f,m}-iT)+\log\left(1+\frac{T_m}{S_m}(1-\alpha_{f,m}-iT)\right)+\notag\\ &+\log L_f(1-\alpha_{f,m}-iT)+\log\cos\pi(1-\alpha_{f,m}-iT). \label{I4B}\end{aligned}$$ Now it is clear that $$\begin{aligned} &\cos\pi(1-\alpha_{f,m}-iT)=e^{\pi T}e^{i(1-\alpha_{f,m})-\log 2}(1+{e^{-2\pi(1-\alpha_{f,m})i}}/{e^{2\pi T}}),\\ &\log L_f(1-\alpha_{f,m}-iT)=\sum_{n=1}^\infty\frac{b_f(n)}{n^{1-\alpha_{f,m}-iT}}\end{aligned}$$ where $b_f(n)$ is given by $$\begin{aligned} b_f(n)=\begin{cases} (\alpha_f(p)^r+\beta_f(p)^r)/r, & n=p^r, \\ 0 , & \text{otherwise} \end{cases}\end{aligned}$$ and $\alpha_f(p)$, $\beta_f(p)$ are given by . Hence the fifth and sixth terms of the right-hand sides of are approximated as $$\begin{aligned} \log L_f(1-\alpha_{f,m}-iT)+\log\cos\pi(1-\alpha_{f,m}-iT)=\pi T+O(1).\end{aligned}$$ By the similar discussion of , the fourth term of the right-hand sides of is estimated as $$\begin{aligned} \log\left(1+\frac{T_m}{S_m}(1-\alpha_{f,m}-iT)\right)\ll 1.\end{aligned}$$ The trivial approximate formula $$\begin{aligned} \log(1-\alpha_{f,m}\pm\tfrac{k-1}{2}-iT)=\log T-(\pi/2)i+O(1/T)\end{aligned}$$ gives that the third term of the right-hand sides of is approximated as $$\begin{aligned} &\log S_m(1-\alpha_{f,m}-iT)\notag\\ &=m\log\left(\log(1-\alpha_{f,m}-\tfrac{k-1}{2}-iT)+\log(1-\alpha_{f,m}+\tfrac{k-1}{2}-iT)\right)\notag\\ &=m\log\log T+O(1).\end{aligned}$$ Using Stirling’s formula $$\log\Gamma(s)=(s-1/2)\log s-s+\log\sqrt{2\pi}+O(1/|s|)$$ and the approximate formula of $\log(1-\alpha_{f,m}\pm({k-1})/{2}-iT)$, we approximate the first and second terms of the right-hand sides of as $$\begin{aligned} &\log\Gamma(1-\alpha_{f,m}+\tfrac{k-1}{2}-iT)+\log\Gamma(1-\alpha_{f,m}-\tfrac{k-1}{2}-iT)\notag\\ &=(1-2\alpha_{f,m}-2iT)(\log T-(\pi/2)i+O(1/T))-2(1-\alpha_{f,m}-iT)+O(1)\notag\\ &=-2iT\log(T/e)-\pi T+(1-2\alpha_{f,m})\log T+O(1). \label{I4B12}\end{aligned}$$ Combining (\[I4A\])–(\[I4B12\]), we obtain a desired approximate formula as $$\log L_f^{(m)}(\alpha_{f,m}+iT)=-2iT\log\frac{T}{2\pi e}+O(\log T),$$ which implies that $$\begin{aligned} I_4=\frac{T}{\pi}\log\frac{T}{2\pi e}+O(\log T). \label{LD4}\end{aligned}$$ From (\[LDC\]), (\[LDE\]), (\[LD3\]) and (\[LD4\]), the proof of Theorem \[THM1\] is completed. Proof of Theorem \[THM2\] ------------------------- Write $L_f^{(m)}(s)=\lambda_f(n_f)(-\log n_f)^mF(s)/n_f^s$ where $F(s)$ is given by (\[FFF\]). By the proof of Theorem \[THM0\], we can choose $\sigma_{f,m}\in\mathbb{R}_{>1}$ such that $L_f(s)$ has no zero for ${\rm Re\;}s>\sigma_{f,m}$ and $$\begin{aligned} \sum_{n=n_f+1}^\infty\left|\frac{\lambda_f(n)}{\lambda_f(n_f)}\right|\left(\frac{\log n}{\log n_f}\right)^m\left(\frac{n_f}{n}\right)^{\sigma_{f,m}/2}\leq\frac{1}{2}.\end{aligned}$$ Note that and the above inequality give $$\begin{aligned} |F(s)-1|& \leq\sum_{n=n_f+1}^\infty\left|\frac{\lambda_f(n)}{\lambda_f(n_f)}\right|\left(\frac{\log n}{\log n_f}\right)^m\left(\frac{n_f}{n}\right)^{\sigma_{f,m}/2+\sigma/2} \leq\frac{1}{2}\left(\frac{n_f}{n_f+1}\right)^{\sigma/2}. \label{LAS}\end{aligned}$$ for ${\rm Re\;}s\geq\sigma_{f,m}$. Applying Littlewood’s formula (see [@TID chapter 3.8]) to $F(s)$, we obtain $$\begin{aligned} 2\pi\sum_{\begin{subarray}{c}F(\rho)=0, \\ \sigma\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\rho\leq T \end{subarray}}({\rm Re\;}\rho-\sigma) =&\int_1^T\log|F(\sigma+it)|dt-\int_1^T\log|F(\sigma_{f,m}+it)|dt+\notag\\[-24pt] &+\int_\sigma^{\sigma_{f,m}}\arg{F(u+iT)}dt-\int_\sigma^{\sigma_{f,m}}\arg{F(u+i)}dt\notag\\ =&:I_1+I_2+I_3+I_4 \label{LA1}\end{aligned}$$ for $\sigma\in\mathbb{R}_{>1/2}$. Here we shall estimate $I_2$. Cauchy’s theorem gives that $$\begin{aligned} I_2=\int_1^T\log|F(v+it)|dt+\int_{\sigma_{f,m}}^{v}\log|F(u+i)|du-\int_{\sigma_{f,m}}^{v}\log |F(u+iT)|du\label{L2A}\end{aligned}$$ for all $v>\sigma_{f,m}$. The fact that $\log{|X|}\leq |X-1|$ for $X\in\mathbb{C}$ and $-\log|Y|\leq 2|Y-1|$ for $Y\in\mathbb{C}$ satisfying $|Y|\geq1/2$, and (\[LAS\]) imply that $$\begin{aligned} \int_1^T\log|F(v+it)|dt\leq\frac{(T-1)}{2}\left(\frac{n_f}{n_f+1}\right)^{{v}/{2}} \label{L2B}\end{aligned}$$ and $$\begin{aligned} &\int_{\sigma_{f,m}}^{v}\log|F(u+i)|du-\int_{\sigma_{f,m}}^{v}\log |F(u+iT)|du\notag\\ &\ll\int_{\sigma_{f,m}}^{v}\left(\frac{n_f}{n_f+1}\right)^{{u}/{2}}du\ll 1. \label{L2C}\end{aligned}$$ Combining (\[L2A\])–(\[L2C\]) we get $$\begin{aligned} I_2=O(1). \label{LI2}\end{aligned}$$ By the same discussion of an estimate of $I_3$ in proof of Theorem \[THM1\], we can obtain $$\begin{aligned} I_3+I_4=O(\log T). \label{LA3}\end{aligned}$$ To estimate $I_1$, we calculate $$\begin{aligned} I_1=\frac{T-1}{2}\log\frac{n_f^{2\sigma}}{|\lambda_f(n_f)|^2(\log n_f)^{2m}}+\frac{1}{2}\int_1^T\log|L_f^{(m)}(\sigma+it)|^2dt. \label{LA4}\end{aligned}$$ Jensen’s inequality gives $$\begin{aligned} \int_1^T\log|L_f^{(m)}(\sigma+it)|^2dt\leq(T-1)\log\left(\frac{1}{T-1}\int_1^T|L_f^{(m)}(\sigma+it)|^2dt\right). \label{LA5}\end{aligned}$$ Combining (\[LA1\]), (\[LI2\])–(\[LA5\]), we obtain $$\begin{aligned} \sum_{\begin{subarray}{c}F(\rho)=0, \\ \sigma\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\rho\leq T \end{subarray}}({\rm Re\;}\rho-\sigma)\leq&\frac{T-1}{4\pi}\log\left(\frac{1}{T-1}\int_1^T|L_f^{(m)}(\sigma+it)|^2dt\right)+\notag\\[-24pt] &+\frac{T-1}{4\pi}\log\frac{n_f^{2\sigma}}{|\lambda_f(n_f)|^2(\log n_f)^{2m}}+O(\log{T}). \label{LAA}\end{aligned}$$ First, we consider the mean square of $L_f^{(m)}(s)$ for ${\rm Re\;}s>1$. Then we can calculate as follows: $$\begin{aligned} &\hspace{-12pt}\int_1^T|L_f^{(m)}(\sigma+it)|^2dt\notag\\ =&\sum_{n_1, n_2=1}^\infty\frac{\overline{\lambda_f(n_1)}\lambda_f(n_2)(\log n_1)^m(\log n_2)^m}{(n_1n_2)^\sigma}\int_{\max\{n_1,n_2\}}^T \left(\frac{n_1}{n_2}\right)^{it}dt\notag\\ =&(T-1)\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+\notag\\ &+\frac{1}{i}\sum_{\begin{subarray}{c} n_1, n_2=1,\\ n_1\ne n_2 \end{subarray}}^\infty\frac{\overline{\lambda_f(n_1)n_1^{-iT}}\lambda_f(n_2)n_2^{-iT}(\log n_1)^m(\log n_2)^m}{(n_1n_2)^\sigma\log(n_1/n_2)}-\notag\\ &-\frac{1}{i}\sum_{\begin{subarray}{c} n_1, n_2=1,\\ n_1\ne n_2 \end{subarray}}^\infty\frac{\overline{\lambda_f(n_1)n_1^{-i\max\{n_1,n_2\}}}\lambda_f(n_2)n_2^{-i\max\{n_1,n_2\}}(\log n_1)^m(\log n_2)^m}{(n_1n_2)^\sigma\log(n_1/n_2)}. \label{L2I}\end{aligned}$$ By the same discussion for $U_\sigma(x)$ with $$\begin{aligned} (\alpha_{n_1},\beta_{n_2})=&(\lambda_f(n_1)n_1^{-iT}(\log n_1)^m,\lambda_f(n_2)n_2^{-iT}(\log n_2)^m), \\ &(\lambda_f(n_1)n_1^{-i\max\{n_1,n_2\}}(\log n_1)^m, \lambda_f(n_2)n_2^{-i\max\{n_1,n_2\}}(\log n_2)^m)\end{aligned}$$ in [@GD1 p.348, LEMMA 6], we find that the second and third terms on the right-hand side of are $=O(1)$ uniformly for $\sigma>1$. Hence, the mean square of $L_f^{(m)}(s)$ for ${\rm Re\;}s>1$ is obtained as $$\begin{aligned} \int_1^T|L_f^{(m)}(\sigma+it)|^2dt=(T-1)\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+O(1). \label{LAB}\end{aligned}$$ Next the mean square of $L_f^{(m)}(s)$ for $1/2<{\rm Re\;}s\leq1$ is obtained as follows: For any $m\in\mathbb{Z}_{\geq0}$ and $T>0$, we have $$\begin{aligned} &\int_1^T|L_f^{(m)}(\sigma+it)|^2dt\notag\\ &=\begin{cases} \displaystyle (T-1)\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+O(T^{2(1-\sigma)}(\log T)^{2m}), & 1/2<\sigma<1, \\ \displaystyle (T-1)\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}+O((\log T)^{2m+2}), & \sigma=1. \end{cases}\label{LAC}\end{aligned}$$ Using Rankin’s result mentioned in Introduction and the following fact $$\begin{aligned} \int_{n_f}^\infty\frac{(\log u)^{2m}}{u^{2\sigma}}du=\frac{(2m)!n_f^{1-2\sigma}}{(2\sigma-1)^{2m+1}}\sum_{j=0}^{2m}\frac{(\log{n_f})^j(2\sigma-1)^j}{j!},\end{aligned}$$ which is obtained by induction, we find that the series of main term of (\[LAC\]) is approximated as $$\begin{aligned} &\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2(\log n)^{2m}}{n^{2\sigma}}\notag\\ &=-\int_{n_f}^\infty\left(\frac{(\log u)^{2m}}{u^{2\sigma}}\right)'\sum_{n_f<n\leq u}|\lambda_f(n)|^2du\notag\\ &=-\frac{C_f(\log n_f)^{2m}}{n_f^{2\sigma-1}}+C_f\int_{n_f}^{\infty}\frac{(\log u)^{2m}}{u^{2\sigma}}du+O\left(\int_{n_f}^\infty\frac{(\log u)^{2m}}{u^{2\sigma+2/5}}du\right)\notag\\ &=\frac{(2m)!n_fC_f}{n_f^{2\sigma}}\frac{1}{(2\sigma-1)^{2m+1}}+O\left(\frac{1}{(2\sigma-1)^{2m}}\right) \label{LAD}\end{aligned}$$ as $\sigma\to1/2+0$. From (\[LAA\])–(\[LAD\]), the following approximate formula is obtained: $$\begin{aligned} &\hspace{-12pt}\sum_{\begin{subarray}{c}F(\rho)=0, \\ \sigma\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\leq T \end{subarray}}({\rm Re\;}\rho-\sigma)\notag\\ \leq& \displaystyle\frac{(2m+1)(T-1)}{4\pi}\log\frac{1}{2\sigma-1}+\frac{T-1}{4\pi}\log\frac{(2m)!n_fC_f}{|\lambda_f(n_f)|^2(\log n_f)^{2m}}+\notag\\ &+O(\log T)+\frac{T-1}{4\pi}\times \notag\\ &\times\begin{cases} \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}(\log T)^{2m}}{T^{2\sigma-1}}\right)\right), & 1/2<\sigma<1, \\ \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}(\log T)^{2m+2}}{T}\right)\right), \hspace{-6pt} & \sigma=1, \\ \displaystyle\log\left(1+O\left(\frac{(2\sigma-1)^{2m+1}}{T}\right)\right), & \sigma{>1}. \end{cases} \label{LBA}\end{aligned}$$ Finally, we shall give an upper bound of $N_{f,m}(\sigma,T)$. Since $N_{f,m}(\sigma,T)$ is monotonically decreasing function with respect to $\sigma$, it follows that $$\begin{aligned} N_{f,m}(\sigma,T)=&N_{f,m}(\sigma,T)-N_{f,m}(\sigma,1)+C\notag\\ \leq& \frac{1}{\sigma-\sigma_1}\int_{\sigma_1}^{\sigma_{f,m}}(N_{f,m}(u,T)-N_{f,m}(u,1))du+C, \label{LBB}\end{aligned}$$ where we put $\sigma_1=1/2+(\sigma-1/2)/2$. Note that $\sigma-\sigma_1=(\sigma-1/2)/2$, $2\sigma_1-1=\sigma-1/2$. Since the numbers of zeros of $F_m(s)$ is equal to that of $L_f^{(m)}(s)$, it follows that $$\begin{aligned} &\int_{\sigma_1}^{\sigma_{f,m}}(N_{f,m}(u,T)-N_{f,m}(u,1))du\notag\\ &=\int_{\sigma_1}^{\sigma_{f,m}}\sum_{\begin{subarray}{c} F(\rho)=0,\\ u\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\leq T \end{subarray}}1du =\sum_{\begin{subarray}{c} F(\rho)=0,\\ \sigma_1\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\rho\leq T \end{subarray}}\int_{\sigma_1}^{{\rm Re\;}\rho}1du\notag\\ &=\sum_{\begin{subarray}{c} F(\rho)=0,\\ \sigma_1\leq{\rm Re\;}\rho\leq\sigma_{f,m},\\ 1\leq{\rm Im\;}\rho\leq T \end{subarray}}({\rm Re\;}\rho-\sigma_1). \label{LBC}\end{aligned}$$ Combining (\[LBA\])–(\[LBC\]) we obtain (\[ZD1\]) and (\[ZD2\]). Hence the proof of Theorem \[THM2\] is completed. [20]{} M. Aoki and M. Minamide, *A zero density estimate for the derivatives of the Riemann zeta function*, Journal for Algebra and Number Theory Academia **2** (2012), 361–365. B. C. Berndt, *The number of zeros for $\zeta^{(k)}(s)$*, J. London Math. Soc. (2) **2** (1970), 577–580. A. Good, *Approximative Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die Spitzenformen assoziiert sind*, Comment. Math. Helv. **50** (1975), 327–361. E. Hecke, *Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I*, Math. Ann. **114** (1937), 1–28. A. A. Karatsuba and S. M. Voronin, *The Riemann Zeta-Function*, Walter de Gruyter de Gruyter Explosions in Mathematics 5, 1992. C. G. Lekkerkerker, *On the zeros of a class of Dirichlet series*, Dissertation, Utrecht, 1955. N. Levinson and H. L. Montgomery, *Zeros of the derivatives of the Riemann zeta-function*, Acta Math. **133** (1974), 49–65. R. A. Rankin, *Contributions to the theory of Ramanujan’s function $\tau(n)$ and similar functions. II. The order of the Fourier coefficients of integral modular forms*, Proc. Cambridge Phil. Soc. **35** (1939), 357–373. B. Riemann, *Über die Anzahl der Primzahlen unterhalb einer gegebenen Grösse*, Monatsber. Königl. Preuss. Akad. Wiss. Berlin (1859), 671–680. A. Speiser, *Geometrisches zur Riemannschen Zetafunktion*, Math. Ann. **110** (1935), 514–521. R. Spira, *Zero-free regions of $\zeta^{(k)}(s)$*, J. London Math. Soc. **40** (1965), 677–682. R. Spira, *Another zero-free region for $\zeta^{(k)}(s)$*, Proc. Amer. Math. Soc. **26** (1970), 246–247. E. C. Titchmarsh, *The Theory of Functions*, Oxford Univ. Press, 1939. H. von Mangoldt, *Zu Riemann’s Abhandlung ‘Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse*, J. Reine Angew. Math. **114** (1895), 255–305. Y. Yashiro, *Approximate functional equation and mean value formula for the derivatives of $L$-functions attached to cusp forms*, preprint; arXiv:1402.3716. C. Y. Y[i]{}ld[i]{}r[i]{}m, *Zeros of $\zeta''(s)$ & $\zeta'''(s)$ in $\sigma<1/2$*, Turk. J. Math. **24** (2000): 89–108. [^1]: 2010 *Mathematics Subject Classification*: Primary 11M26; Secondary 11N75. [^2]: *Key words and phrases*: cusp forms, $L$-functions, derivative, zeros.

--- abstract: 'One finding of cognitive research is that people do not automatically acquire usable knowledge by spending lots of time on task. Because students’ knowledge hierarchy is more fragmented, “knowledge chunks” are smaller than those of experts. The limited capacity of short term memory makes the cognitive load high during problem solving tasks, leaving few cognitive resources available for meta-cognition. The abstract nature of the laws of physics and the chain of reasoning required to draw meaningful inferences makes these issues critical. In order to help students, it is crucial to consider the difficulty of a problem from the perspective of students. We are developing and evaluating interactive problem-solving tutorials to help students in the introductory physics courses learn effective problem-solving strategies while solidifying physics concepts. The self-paced tutorials can provide guidance and support for a variety of problem solving techniques, and opportunity for knowledge and skill acquisition.' author: - Chandralekha Singh title: Problem Solving and Learning --- [ address=[Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania, 15260]{} ]{} Cognitive Research and Problem Solving ====================================== Cognitive research deals with how people learn and solve problems [@nrc1; @joe]. At a coarse-grained level, there are three components of cognitive research: how do people acquire knowledge, how do they organize and retain the knowledge in memory (brain) and how do they retrieve this knowledge from memory in appropriate situations including to solve problems. These three components are strongly coupled, e.g., how knowledge was organized and retained in memory during acquisition will determine how effectively it can be retrieved in different situations to solve problems. We can define problem solving as any purposeful activity where one must devise and perform a sequence of steps to achieve a set goal when presented with a novel situation. A problem can be quantitative or conceptual in nature. Using the findings of cognitive research, human memory can be broadly divided into two components: the working memory or the short term memory (STM) and the long term memory (LTM). The long term memory is where the prior knowledge is stored. Appropriate connections between prior knowledge in LTM and new knowledge that is being acquired at a given time can help an individual organize his/her knowledge hierarchically. Such hierarchical organization can provide indexing of knowledge where more fundamental concepts are at the top of the hierarchy and the ancillary concepts are below them. Similar to an index in a book, such indexing of knowledge in memory can be useful for accessing relevant knowledge while solving problems in diverse situations. It can also be useful for inferential recall when specific details may not be remembered. The working memory or STM is where information presented to an individual is processed. It is the conscious system that receives input from memory buffers associated with various sensory systems and can also receive input from the LTM. Conscious human thought and problem solving involves rearranging and synthesizing ideas in STM using input from the sensory systems and LTM. One of the major initial findings of the cognitive revolution is related to Miller’s magic numbers 7$\pm$2 (5 to 9), i.e., how much information can STM hold at one time. [@miller] Miller’s research found that STM can only hold 5 to 9 pieces of information regardless of the IQ of an individual. Here is an easy way to illustrate this. If an individual is asked to memorize the following sequence of 25 numbers and letters in that order after staring at it for 30 seconds, it is a difficult task: 6829-1835-47DR-LPCF-OGB-TWC-PVN. An individual typically only remembers between 5 to 9 things in this case. However, later research shows that people can extend the limits of their working memory by organizing disparate bits of information into chunks or patterns. [@chunk] Using chunks, STM can evoke from LTM, highly complex information. An easy way to illustrate it is by asking an individual to memorize the following sequence of 25 numbers and letters: 1492-1776-1865-1945-AOL-IBM-USA. This task is much easier if one recognizes that each of the four digit number is an important year in history and each of the three letters grouped together is a familiar acronym. Thus, an individual only has to remember 7 separate chunks rather than 25 disparate bits. This chunking mechanism is supported by research in knowledge rich fields such as chess and physics where experts in a field have well organized knowledge. [@chess] For example, research shows that if experts in chess are shown a very good chess board that corresponds to the game of a world-class chess player, they are able to assemble the board after it is disassembled because they are able to chunk the information on the board and remember the position of one piece with respect to another. If chess novices are shown the same board, they are only able to retrieve 5-9 pieces after it is jumbled up because they are not able to chunk large pieces of information present on the chess board. On the other hand, both chess experts and novices are poor at assembling a board on which the chess pieces are randomly placed before it was jumbled up. In this latter case, chess experts are unable to chunk the random information due to lack of pattern. A crucial difference between expert and novice problem solving is the manner in which knowledge is represented in their memory and the way it is retrieved to solve problems. Experts in a field have well organized knowledge. They have large chunks of “compiled" knowledge in LTM and several pieces of knowledge can be accessed together as a chunk [@automatic]. For example, for an expert in physics, vector addition, vector subtraction, displacement, velocity, speed, acceleration, force etc. can be accessed as one chunk while solving problems while they can be seven separate pieces of information for beginning students. If a problem involves all of these concepts, it may cause a cognitive overload if students’ STM can only hold 5 or 6 pieces of information. Experts are comfortable going between different knowledge representations, e.g., verbal, diagrammatic/pictorial, tabular etc. and employ representations that make problem solving easier. [@rep] Experts categorize problems based upon deep features unlike novices who can get distracted by context dependent features. For example, when physics professors and introductory physics students are asked to group together problems based upon similarity of solution, professors group them based upon physics concepts while students can choose categories that are dependent on contexts such as ramp problems, pulley problems, spring problems etc [@chi; @hardiman; @reif2; @larkin]. Of course, an important goal of most physics courses is to help students develop expertise in problem solving and improve their reasoning skills. In order to help students, instructors must realize that the cognitive load, which is the amount of mental resources needed to solve a problem, is subjective [@cogload]. The complexity of a problem not only depends on its inherent complexity but also on the expertise, experience and intuition of an individual [@intuition]. It has been said that problems are either “impossible" or “trivial". A ballistic pendulum problem that may be trivial for a physics professor may be very difficult for a beginning student [@rosengrant]. Cognitive load is higher when the context is abstract as opposed to concrete. The following Wason tasks [@wason] are examples of abstract and concrete problems which are conceptually similar, but the abstract problem turns out to be cognitively more demanding. - You will lose your job unless you enforce the following rule: “If a person is rated K, then his/her document must be marked with a 3".\ Each card on the table for a person has a letter on one side and a number on the other side. Indicate only the card(s) shown in Figure 1 that you definitely need to turn over to see if the document of any of these people violates this rule.\ - You are serving behind the bar of a city centre pub and will lose your job unless you enforce the following rule: “If a person is drinking beer, then he/she must be over 18 years old".\ Each person has a card on the table which has his/her age on one side and the name of his/her drink on the other side. Indicate only the card(s) shown in Figure 2 that you definitely need to turn over to see if any of these people are breaking this rule. The correct answer for the abstract case is that you must turn the cards with K and 7 (to make sure that there is no K on the other side). Please note that the logic presented in the task is one sided in that it is ok for a document with a 3 to have anything on the other side. The correct answer for the concrete case is “beer" and “16 years old", and it is much easier to identify these correct answers than the correct answers for the abstract case. A major reason for why the cognitive load is high during problem solving in physics is because the laws of physics are abstract. It is important to realize that it is not easy to internalize them unless concrete contexts are provided to the students. Another difficulty is that, once the instructor has built an intuition about a problem, it may not appear difficult to him/her even if it is abstract. In such situations the instructor may overlook the cognitive complexity of the problem for a beginning student unless the instructor puts himself/herself in the students’ shoes. An important lesson from cognitive research is that new knowledge that an individual acquires builds on prior knowledge. This idea is commensurate with Piaget’s notion of “optimal mismatch" [@piaget] or Vygotsky’s idea of “zone of proximal development" (ZPD) [@vygotsky]. ZPD is the zone defined by what a student can do on his/her own vs. with the help of a guide who is familiar with the student’s initial knowledge and targets instruction somewhat above it continuously for effective learning. This is analogous to the impedance matching of a transformer in which the power transfer can be maximized if the input and output impedances are matched. Another analogy is with light passing through two polarizers placed perpendicular to each other vs. having several polarizers stacked one after another where the transmission axes of adjacent polarizers are slightly different from each other. In the first case of crossed polarizer, no light passes through whereas in the second case most of the light passes through if the angle $\theta$ between the transmission axes of the adjacent polarizers is small enough. Similarly, if the instruction is targeted significantly above students’ prior knowledge, learning won’t be meaningful and even if the students make an effort to store some hap-hazardous information in their brain till the final exam, it will get “shampooed out" soon after that. On the other hand, if the instructional design takes into account students’ initial knowledge and builds on it, learning will be meaningful. Another important lesson from cognitive research is that students must construct their own understanding. This implies that we should give students an opportunity to reconstruct, extend, and organize their knowledge. Such opportunities will come from ensuring that the students are actively engaged in the learning process and take advantage of their own knowledge resources and also benefit from interactions with their peers. Computer-based Interactive Tutorials ==================================== We now describe computer-based interactive problem solving tutorials that we have been developing that build on introductory physics students’ prior knowledge and keep them actively engaged in the learning process. The tutorials combine quantitative and conceptual problem solving. They focus on helping students develop a functional understanding of physics while learning useful skills [@singh]. It is worthwhile thinking about why quantitative problem solving alone often fails to help most students extend and organize their physics knowledge. Without guidance, most students do not exploit the problem solving opportunity to reflect upon what they have actually learned and build a more robust knowledge structure. If only quantitative problems are asked, students often view them as “plug-and-chug" exercises, while conceptual problems alone are often viewed as guessing tasks with little connection to physics content. The interactive tutorials we have been developing combine quantitative and conceptual problem solving and provide guidance and support for knowledge and skill acquisition. They provide a structured approach to problem solving and promote active engagement while helping students develop self reliance. Other computer-based tutorials are also being developed [@tutor]. Our tutorials are unique in that they focus on helping students learn effective problem solving strategies and the conceptual questions are developed based upon the common difficulties found via research on students’ difficulties in learning a particular topic in physics. Development of Problem Solving Skills in Introductory Physics ------------------------------------------------------------- A major goal of an introductory physics course for science and engineering majors is to enable students to develop complex reasoning and problem solving skills to explain and predict diverse phenomena in everyday experience. However, numerous studies show that students do not acquire these skills from a [*traditional*]{} course [@chi; @hardiman; @reif2; @hake]. The problem can partly be attributed to the fact that the kind of reasoning that is usually learned and employed in everyday life is not systematic or rigorous. Although such hap-hazardous reasoning may have little measurable negative consequences in an individual’s personal life, it is insufficient to deal with the complex chain of reasoning that is required in rigorous scientific field such as physics [@reif2]. Educational research suggests that many introductory physics students solve problems using superficial clues and cues, applying concepts at random without thinking whether they are applicable or not [@chi; @hardiman; @reif2]. Also, most traditional courses do not [*explicitly*]{} teach students effective problem solving strategies. Rather, they may reward inferior problem solving strategies in which many students engage. Instructors often implicitly assume that students know that the analysis, planning, evaluation, and reflection phases of problem solving are as important as the implementation phase. Consequently, they may not discuss these strategies explicitly while solving problems during the lecture. There is no mechanism in place to ensure that students make a conscious effort to interpret the concepts, make qualitative inferences from the quantitative problem solving tasks, or relate the new concepts to their prior knowledge. In order to develop scientific reasoning by solving quantitative problems, students must learn to exploit problem solving as an opportunity for knowledge and skill acquisition. Thus, students should not treat quantitative problem solving merely as a mathematical exercise but as a learning opportunity and they should engage in effective problem solving strategies. Effective Problem Solving Strategies ------------------------------------ Effective problem solving begins with a conceptual analysis of the problem, followed by planning of the problem solution, implementation and evaluation of the plan, and last but not least reflection upon the problem solving process. As the complexity of a physics problem increases, it becomes increasingly important to employ a systematic approach. In the qualitative or conceptual analysis stage, a student should draw a picture or a diagram and get a visual understanding of the problem. At this stage, a student should convert the problem to a representation that makes further analysis easier. After getting some sense of the situation, labeling all known and unknown numerical quantities is helpful in making reasonable physical assumptions. Making predictions about the solution is useful at this level of analysis and it can help to structure the decision making at the next stage. The prediction made at this stage can be compared with the problem solution in the reflection phase and can help repair, extend and organize the student’s knowledge structure. Planning or decision making about the applicable physics principles is the next problem solving heuristic. This is the stage where the student brings everything together to come up with a reasonable solution. If the student performed good qualitative analysis and planning, the implementation of the plan becomes easy if the student possesses the necessary algebraic manipulation and mathematical skills. After implementation of the plan, a student must evaluate his/her solution, e.g., by checking the dimension or the order of magnitude, or by checking whether the initial prediction made during the initial analysis stage matches the actual solution. One can also ask whether the solution is sensible and, possibly, consistent with experience. The reflection phase of problem solving is critical for learning and developing expertise. Research indicates that this is one of the most neglected phase of problem solving [@chi; @hardiman; @reif2]. Without guidance, once a student has an answer, he/she typically moves on to the next problem. At the reflection stage, the problem solver must try to distill what he or she has learned from solving the problem. This stage of problem solving should be used as an opportunity for reflecting upon why a particular principle of physics is applicable to the problem at hand and how one can determine in the future that the same principle should be applicable even if the problem has a new context. Description of the Tutorials ---------------------------- The development of the computer-based tutorials to help students learn effective problem solving strategies is guided by a learning paradigm which involves three essential components: modeling, coaching, and weaning [@cog]. In this approach, “modeling" means that the instructor demonstrates and exemplifies the skills that students should learn (e.g., how to solve physics problems systematically). “Coaching" means providing students opportunity, guidance and practice so that they are actively engaged in learning the skills necessary for good performance. “Weaning" means reducing the support and feedback gradually so as to help students develop self-reliance. Each of the tutorials starts with an overarching problem which is quantitative in nature. Before using a tutorial, students use a pre-tutorial worksheet which divides each quantitative problem given to them into different stages involved in problem solving. For example, in the conceptual analysis stage of problem solving, the worksheet explicitly asks students to draw a diagram, write down the given physical quantities, determine the target quantity, and predict some features of the solution. After attempting the problem on the worksheet to the best of their ability, students access the tutorial on the computer (or use a paper version for evaluation purposes as discussed in the evaluation section below). The tutorial divides an overarching problem into several sub-problems, which are research-guided conceptual multiple-choice questions related to each stage of problem solving. The alternative choices in these multiple-choice questions elicit common difficulties students have with relevant concepts as determined by research in physics education. Incorrect responses direct students to appropriate help sessions where students have the choice of video, audio or only written help with suitable explanations, diagrams, and equations. Correct responses to the multiple-choice questions give students a choice of either advancing to the next sub-problem or directs them to the help session with the reasoning and explanation as to why the alternative choices are incorrect. While some reasonings are problem-specific, others focus on more general ideas. After students work on the implementation and assessment phase sub-problems posed in the multiple-choice format, they answer reflection sub-problems. These sub-problems focus on helping students reflect upon what they have learned and apply the concepts learned in different contexts. If students have difficulty answering these sub-problems, the tutorial provides further help and feedback. Thus, the tutorials not only model or exemplify a systematic approach to problem solving, they also engage students actively in the learning process and provide feedback and guidance based upon their need. Each tutorial problem is matched with other problems (which we call paired problems) that use similar physics principles but which are somewhat different in context. Students can be given these paired problems as quizzes so that they learn to de-contextualize the problem solving approach and concepts learned from the tutorial. The paired problems play an important role in the weaning part of the learning model and ensure that students develop self-reliance and are able to solve problems based upon the same principle without help. These paired problems can also be assigned as homework problems and instructors can inform students that they can use the tutorials as a self-paced study tool if they have difficulty in solving the paired problems assigned as homework related to a particular topic. We have developed computer-based tutorials related to introductory mechanics, electricity, and magnetism. Topics in mechanics include linear and rotational kinematics, Newton’s laws, work and energy, and momentum. Topics in electricity and magnetism include Coulomb’s law, Gauss’s law, potential and potential energy, motion of charged particles in an electric field, motion of charged particles in a magnetic field, Faraday’s law, and Lenz’s law. Figures 3-6 show screen captures from a computer-based tutorial which starts with a quantitative problem in which two blocks with masses $m_1$ and $m_2$ are in contact on a frictionless horizontal surface and a horizontal force $F_H$ is applied to the block with mass $m_1$. Students are asked to find the magnitude of force exerted by the block with mass $m_2$ on $m_1$. We have found that this problem is sufficiently challenging for students in both algebra and calculus-based introductory physics courses that most students are unable to solve it without help. In the tutorial, the quantitative problem is broken down into several conceptual problems in the multiple-choice format that students have to answer. For example, one of the conceptual questions related to the initial analysis of the problem is shown in Figure 3 along with a screen capture of a help session that a student is directed to if he/she chooses an incorrect response. Figure 4 is a multiple-choice question about the free body diagram and figure 5 is a screen capture of a help screen in which an instructor explains relevant concepts to the students related to difficulty with the question asked in Figure 4. Figure 6 is a help screen related to a reflection question in which students are asked about the force exerted by the block of mass $m_1$ on $m_2$ if the force of the hand $F_H$ was applied to the block of mass $m_2$ in the opposite direction (instead of being applied to the block of mass $m_1$ as in the tutorial). Case-Study for Evaluating the Computer-based Tutorials ------------------------------------------------------ Below, we describe a case-study to evaluate the tutorials. In one case study, we compared three different groups who were given different aid tools: - Group (1) consists of students who used the tutorials as aid tool. - Group (2) consists of students who were given the solved solutions for the tutorial problems which were similar to the solutions in the textbook’s solutions manual. However, the solutions were not broken down into the multiple-choice questions with alternative choices targeting common misconceptions as was done in the tutorials. - Group (3) consists of students who were given the textbook sections that dealt with the relevant concepts as the aid tool and were asked to brush up on the material for a quiz on a related topic. Fifteen students were recruited and divided into two pools based upon their prior knowledge. Then, the students from each of these pools were randomly assigned to one of the three groups discussed above. During the interview session, students in each group initially answered a pre-questionnaire to determine their level of preparation, their views about problem solving in physics, and their perception of physics instruction. It was interesting to note that a majority of the students (regardless of the group to which they belonged) disagreed with the statement “When confronted with a physics problem, I first spend a reasonable amount of time planning how to solve the problem before actually solving it". Half of the students also agreed with the statement “Physics problem solving is all about matching given quantities in the problem to the formula in the book". Half of the students thought that the pace of their introductory physics courses was very fast. After this pre-questionnaire, all students were given the following problem on a worksheet and were asked to solve it to the best of their ability before using their aid tools: - [*An insulating sphere of radius $b$ has a spherical cavity of radius $a$ located within its volume and centered a distance $R$ from the center of the sphere. A cross section of the sphere is shown in Figure 7. The solid part of the insulating sphere has a uniform volume charge density $\rho$. Find the electric field $\vec E$ at a point inside the cavity.*]{} All students identified the correct principle to use (Gauss’s law) when asked to solve the problem to the best of their ability on the worksheet. The above problem is challenging enough that none of the interviewed students could solve it without help. Most students initially thought that the problem was relatively easy. Except for one student who came up with a different incorrect answer, the rest of the students came up with the same incorrect answer: the electric field is zero everywhere inside the cavity. All of them invoked Gauss’s law, which states that the total electric flux through a closed surface is equal to the net charge inside the surface divided by $\epsilon_0$. Their reasoning for zero electric field in the cavity was based on the incorrect interpretation that whenever the electric flux through a closed surface is zero, the electric field at every point inside the surface must be zero too regardless of whether there was a symmetric charge distribution to justify such claims. Thus, students drew a Gaussian sphere inside the hole and concluded that the electric field must be zero everywhere inside since the enclosed charge is zero. Students ignored the asymmetric charge distribution surrounding the cavity, which is a common difficulty. [@gauss] Among many of the difficulties the interviewed students faced, one difficulty was not recognizing that the charge distribution was not symmetric enough and therefore the net electric field at a point inside the cavity cannot be zero. When asked to show why the electric field should be zero everywhere inside, most students drew a spherical Gaussian surface inside the hole, wrote down Gauss’s law in the integral form and pulled out the electric field from inside the integral. Interviews show that many students believed that the electric field can always be pulled out from the surface integral regardless of the symmetry of the charge distribution. When pressed harder about why the electric field should be equal everywhere on the Gaussian surface and why it can be pulled out of the integral, some students noted that one should not worry about this issue at least in this case since the zero charge enclosed implies zero electric field everywhere inside anyway. Their convoluted reasoning showed that many students have difficulty in organizing different knowledge resources and applying them at the same time. After the students tried to solve the problem on the worksheet on their own to the best of their ability, students in Group 1 were given the corresponding tutorial to work on, those in Group 2 were given a textbook-style solution for the problem (similar to the solutions in a textbook solution manual), and those in Group 3 were given the section in the textbook, University Physics by Young and Freedman, which deals with this topic. Each student used his/her respective aid tool for the same amount of time (20 minutes). All students were told that they would have to solve a paired problem involving similar concepts after help from the tools they were provided (tutorial, textbook-style solution, relevant chapter from textbook). All students were informed that aid tool was only for learning the material and could not be used while working on the paired problem they will be given later. The paired problem that followed tested whether students could transfer relevant knowledge acquired from the aid tools to the paired problem. [@transfer] For example, for the Gauss’s law problem discussed above, the paired problem was similar to the tutorial problem but was for an infinite solid insulating cylinder with a uniform volume charge and an asymmetric cylindrical cavity inside it. We used a rubric to grade students. The average performance of the tutorial group was approximately $85\%$. All of them made the correct assumption that the electric field is not zero inside the cavity due to the asymmetry of the charge distribution and explained how Gauss’s law cannot be used in such cases to conclude that the electric field is zero in the cavity. During the interviews, these students were able to explain verbally their thought processes and how they solved the problem to find the electric field. By analyzing the students’ thought processes during the interviews, it appears that the pattern of reasoning employed by these students was significantly better on an average than the reasoning of students from the other two groups. The other two groups didn’t show as much improvement as the tutorial group when graded on a rubric after using the aids. Between Groups 2 and 3, the students who used the textbook-style solution as a guide did better on the paired problem than those who used the relevant textbook section. The average performance of Group 2 was approximately $60\%$ and Group 3 was less than $30\%$. Students who used the textbook-style solution still had difficulties in solving the paired problem and four of them did not solve the entire problem correctly. The solution of the problem involves breaking the problem into subproblems each of which has a spherical symmetry and can be solved by known methods using Gauss’s law. Most of the students in the second group (those who were given a solution of the type given in solutions manual) realized that they had to combine or superpose two electric fields. However, their most common difficulty was in using vectors in order to relate the final solution to the solutions to the two subproblems which have a spherical symmetry. The textbook-type solution showed them that calculating the electric field at a point in the cavity involved subtracting from the electric field due to the full insulating sphere as though there was no cavity, the electric field due to an insulating sphere of the size of the cavity. From the surveys given after the paired problem, some students mentioned that the textbook-style solutions didn’t explain in words the steps used thoroughly. Since the misconceptions students had at the beginning were not explicitly targeted by asking explicit questions in the textbook-type solution (as was done explicitly in the multiple-choice questions which were part of the tutorials), students did not transfer relevant knowledge from the solved example to the paired problem as well as the tutorial group did. All of the students who made use of the textbook as an aid for learning (rather than the tutorial or the solved example) did poorly on the paired problem. This finding is consistent with another study that shows that unless introductory physics students have seen solved examples. [@iso2] Students in this group realized that the problem solution could not be that the magnitude of the electric field inside the cavity is $\vert \vec E \vert=0$ because otherwise they would not be given 20 minutes to browse over the section of the textbook trying to formulate a solution. But the responses they provided after browsing over the book were often difficult to understand and dimensionally incorrect. When asked to explain what they had done, students noted that they were not very sure about how to solve the problem. They added that the relevant section of the textbook did not help because it did not have a solved example exactly like the problem that was asked. Summary ======= People do not automatically acquire usable knowledge by spending lots of time on task. Limited capacity of STM can make cognitive load high for beginning students. For learning to be meaningful, students should be actively engaged in the learning process. Moreover, it is important to consider the difficulty of a problem from students’ perspective and build on their prior knowledge. We are developing computer-based interactive tutorials for introductory mechanics and electricity and magnetism that are suited for a wide variety of students. The self-paced tutorials combine quantitative and conceptual problem solving. They engage students actively in the learning process and provide feedback based upon their needs. They focus on helping students learn problem solving and reasoning skills while helping them build a more coherent knowledge structure related to physics. They can be used as a self-study tool by students. The paired problems can be incorporated into regular quizzes or assigned as homework problems. We thank Daniel Haileselassie and Josh Bilak for help in the development and evaluation of the tutorials and thank F. Reif, J. Levy, and R. 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--- abstract: 'We use freeness assumptions of random matrix theory to analyze the dynamical behavior of inference algorithms for probabilistic models with dense coupling matrices in the limit of large systems. For a toy Ising model, we are able to recover previous results such as the property of vanishing effective memories and the analytical convergence rate of the algorithm.' address: | Department of Artificial Intelligence, Technische Universität Berlin,\ Berlin 10587, Germany author: - Manfred Opper and Burak Çakmak bibliography: - 'mybib.bib' title: 'Understanding the dynamics of message passing algorithms: a free probability heuristics [^1]' --- \#1\#2[[ ]{}]{} Introduction ============ Probabilistic inference plays an important role in statistics, signal processing and machine learning. A major task is to compute statistics of unobserved random variables using distributions of these variables conditioned on observed data. An exact computation of the corresponding expectations in the multivariate case is usually not possible except for simple cases. Hence, one has to resort to methods which approximate the necessary high-dimensional sums or integrals and which are often based on ideas of statistical physics [@mezard2009information]. A class of such approximation algorithms is often termed [*message passing*]{}. Prominent examples are [*belief propagation*]{} [@pearl2014probabilistic] which was developed for inference in probabilistic Bayesian networks with sparse couplings and [*expectation propagation*]{} (EP) which is also applicable for networks with dense coupling matrices [@Minka1]. Both types of algorithms make assumptions on weak dependencies between random variables which motivate the approximation of certain expectations by Gaussian random variables invoking central limit theorem arguments [@Adatap]. Using ideas of the statistical physics of disordered systems, such arguments can be justified for the [*fixed points*]{} of such algorithms for large network models where couplings are drawn from random, rotation invariant matrix distributions. This extra assumption of randomness allows for further simplifications of message passing approaches [@ccakmak2016self; @CakmakOpper18], leading e.g. to the [*approximate message passing*]{} AMP or VAMP algorithms, see [@Ma; @rangan2019vector; @takeuchi]. Surprisingly, random matrix assumptions also facilitate the analysis the [*dynamical*]{} properties of such algorithms [@rangan2019vector; @takeuchi; @CakmakOpper19] allowing e.g. for exact computations of convergence rates [@CakmakOpper19; @ccakmak2020analysis]. This result might not be expected, because mathematically the updates of message passing algorithms somewhat resemble the dynamical equations of spin-glass models or of recurrent neural networks which often show a complex behavior in the large system limit [@Mezard]. This manifests itself e.g. in a slow relaxation towards equilibrium [@cugliandolo1993analytical] with a possible long-time memory on initial conditions [@Eisfeller]. Such properties would definitely not be ideal to the design of a numerical algorithm. So a natural question is: which properties of the dynamics enable both their analytical treatment and guarantee fast convergence? In this paper, we give a partial answer to this question by interpreting recent results on the dynamics of algorithms for a toy inference problem for an Ising network. We develop a heuristics based on freeness assumptions on random matrices which lead to an understanding of the simplifications in the analytical treatment and provide a simple way for predicting the convergence rate of the algorithm. The paper is organized as follows: In Section 2 we introduce the motivating Ising model and provide a brief presentation on the TAP mean-field equations. In Section 3 and Section 4 we present the message passing algorithm of [@CakmakOpper19] (to solve the TAP equations) and provide a brief discussion on its dynamical properties in the thermodynamic limit, respectively. In Section 5 and Section 6 we recover the property of vanishing-memories and analytical convergence speed of the messaging passing algorithm using a free probability heuristic. Comparisons of our results with simulations are given in Section 7. Section 8 presents a summary and outlook. Motivation: Ising models with random couplings and TAP mean field equations =========================================================================== We consider a model of a multivariate distribution of binary units. This is given by an Ising model with pairwise interactions of the spins ${\mathlette{\boldmath}{s}}=(s_1,\ldots,s_N)^\top\in\{-1,1\}^{N}$ described by the Gibbs distribution $$p({\mathlette{\boldmath}{s}}\vert {\mathlette{\boldmath}{J}},{\mathlette{\boldmath}{h}})\doteq \frac{1}{Z}\exp\left(\frac{1}{2}{\mathlette{\boldmath}{s}}^\top{\mathlette{\boldmath}{J}}{\mathlette{\boldmath}{s}}+{\mathlette{\boldmath}{s}}^\top{\mathlette{\boldmath}{h}}\right)\label{Gibbs}$$ where $Z$ stands for the normalizing partition function. While such models have been used for data modeling where the couplings ${\mathlette{\boldmath}{J}}$ and fields ${\mathlette{\boldmath}{h}}$ are adapted to data sets [@hinton2007boltzmann], we will restrict ourselves to a toy model where all external fields are equal $$h_{i}=h\neq 0,~\forall i.$$ The coupling matrix ${\mathlette{\boldmath}{J}}={\mathlette{\boldmath}{J}}^{\top}$ is assumed to be drawn at random from a rotation invariant matrix ensemble, in order to allow for nontrivial and rich classes of models. This means that ${\mathlette{\boldmath}{J}}$ and ${\mathlette{\boldmath}{V}}{\mathlette{\boldmath}{J}}{\mathlette{\boldmath}{V}}^\top$ have the same probability distributions for any orthogonal matrix ${\mathlette{\boldmath}{V}}$ independent of ${\mathlette{\boldmath}{J}}$. Equivalently, ${\mathlette{\boldmath}{J}}$ has the spectral decomposition [@Collins14] $${\mathlette{\boldmath}{J}}={\mathlette{\boldmath}{O}}^ \top{\mathlette{\boldmath}{D}}{\mathlette{\boldmath}{O}} \label{decom}$$ where ${\mathlette{\boldmath}{O}}$ is a random Haar (orthogonal) matrix that is independent of a diagonal matrix ${\mathlette{\boldmath}{D}}$. This class of models generalizes the well known SK (Sherrington–Kirkpatrick) model [@SK] of spin glasses for which ${\mathlette{\boldmath}{J}}$ is a symmetric Gaussian random matrix. The simplest goal of probabilistic inference would reduce to the computation of the magnetizations $${\mathlette{\boldmath}{m}}=\mathbb E[{\mathlette{\boldmath}{s}}]$$ where the expectation is taken over the Gibbs distribution. For random matrix ensembles, the so–called TAP equations [@SK] were developed in statistical physics to provide approximate solutions to ${\mathlette{\boldmath}{m}}.$ Moreover, these equations can be assumed (under certain conditions) to give exact results (for a rigorous analysis in case of the SK model, see [@chatterjee2010spin]) for the magnetizations in the thermodynamic limit [@Mezard] $N\to\infty$ for models with random couplings. For general rotation invariant random coupling matrices, the TAP equations are given by \[tap\] $$\begin{aligned} {\mathlette{\boldmath}{m}}&={\rm Th}({\mathlette{\boldmath}{\gamma}})\\ {\mathlette{\boldmath}{\gamma}}&={\mathlette{\boldmath}{J}}{\mathlette{\boldmath}{m}}-{\rm R}(\chi){\mathlette{\boldmath}{m}}\\ \chi &= \mathbb E[{\rm Th}'(\sqrt{(1-\chi){\rm R}'(\chi)} u)] \label{chi}. \end{aligned}$$ Here $u$ denotes the normal Gaussian random variable and for convenience we define the function $${\rm Th}(x)\doteq\tanh(h+x) .$$ Equation (\[tap\]) provides corrections to the simpler naive mean-field method. The latter, ignoring statistical dependencies between spins, would retain only the term ${\mathlette{\boldmath}{J}}{\mathlette{\boldmath}{m}}$ as the “mean field” acting on spin $i$. The so-called [*Onsager reaction term*]{} $-{\rm R}(\chi){\mathlette{\boldmath}{m}}$ models the coherent small changes of the magnetisations of the other spins due to the presence of spin $i$. Furthermore, $\chi$ coincides with static susceptibility computed by the replica-symmetric ansatz. The Onsager term for a Gaussian matrix ensemble was developed in [@TAP] and later generalized to general ensembles of rotation invariant coupling matrices in [@Parisi] using a free energy approach. For alternative derivations, see [@Adatap] and [@CakmakOpper18]. The only dependency on the random matrix ensemble in is via the R-transform ${\rm R}(\chi)$ and its derivative ${\rm R}'(\chi)$. The R-transform is defined as [@Hiai] $${\rm R}(\omega)={\rm G}^{-1}(\omega)-\frac{1}{\omega}, \label{Rtrans}$$ where ${\rm G}^{-1}$ is the functional inverse of the Green-function $${\rm G}(z)\doteq {\rm Tr}((z{\bf I}-{\mathlette{\boldmath}{J}})^{-1}). \label{Greens}$$ Here, for an $N\times N$ matrix ${\mathlette{\boldmath}{X}}$ we define its limiting (averaged) normalized-trace by $${\rm Tr}({\mathlette{\boldmath}{X}})\doteq \lim_{N\to\infty}\frac{1}{N}\mathbb E_{{\mathlette{\boldmath}{X}}}{\rm tr}({\mathlette{\boldmath}{X}}).$$ From a practical point of view, for a concrete $N$ dimensional coupling matrix ${\mathlette{\boldmath}{J}}$, the R-transform term can be approximated using the spectral decomposition . The Green function is then replaced by its empirical approximation as $$\begin{aligned} {\rm G}(z)&\simeq \frac{1}{N} {\rm tr}((z{\bf I}-{\mathlette{\boldmath}{D}})^{-1}). \label{Green}\end{aligned}$$ The R-transform ${\rm R}\doteq{\rm R}(\chi)$ (for short) and its derivative ${\rm R}'\doteq {\rm R}'(\chi)$ are then obtained by solving the fixed-point equations \[alg1\] $$\begin{aligned} \lambda &={\rm R}+\frac{1}{\chi}\\ {\rm R}&=\lambda-\frac{1}{{\rm G}(\lambda)}\\ {\rm R}'&=\frac{1}{{\rm G}(\lambda)^2}+\frac{1}{{\rm G}'(\lambda)}\\ \chi &= \mathbb E[{\rm Th}'(\sqrt{(1-{\rm G}(\lambda)){\rm R}'} u)]. \end{aligned}$$ Approximate message passing algorithm for TAP equations ======================================================= In this section we reconsider an iterative algorithm for solving the TAP equations which was introduced in [@CakmakOpper19] and was motivated by the so–called VAMP algorithms of [@rangan2019vector; @takeuchi]. We introduce a vector of auxiliary variables ${\mathlette{\boldmath}{\gamma}}(t)$, where $t$ denotes the discrete time index of the iteration. We then proceed by iterating a nonlinear dynamics which is of the simple form $$\label{dynamics} {\mathlette{\boldmath}{\gamma}}(t) ={\mathlette{\boldmath}{A}} f({\mathlette{\boldmath}{\gamma}}(t-1))$$ for $t=1,2, 3, \ldots$. Here $f$ is a nonlinear function which is applied component wise to the vector ${\mathlette{\boldmath}{\gamma}}(t-1)$ and ${\mathlette{\boldmath}{A}}$ is a fixed $N\times N$ matrix. Before we specify the dynamical system (\[dynamics\]) for the TAP equations and its parameters, we should mention that the point wise nonlinear operation followed by a matrix multiplication is typical of the dynamics of a (single layer) [*recurrent neural network*]{} [@Goodfellow-et-al-2016]. Hence, the analysis of (\[dynamics\]) could also be of interest to these types of models. For the current application to the TAP equations, we specialize to the function $$f(x)\doteq \frac{1}{\chi}{\rm Th}(x) -x \label{alg2}$$ where $\chi$ was defined in (\[chi\]). The *time-independent* random matrix is given by $${\mathlette{\boldmath}{A}}\doteq\frac 1\chi\left[\left(\frac 1 \chi+{\rm R}(\chi)\right){\bf I}-{\mathlette{\boldmath}{J}}\right]^{-1} -\bf I. \label{def_matrixA}$$ The initialization of the dynamics is given by ${\mathlette{\boldmath}{\gamma}}(0)=\sqrt{(1-\chi){\rm R}'(\chi)} {\mathlette{\boldmath}{u}}$ where ${\mathlette{\boldmath}{u}}$ is a vector of independent normal Gaussian random variables. It is easy to see that the fixed points of ${\mathlette{\boldmath}{\gamma}}(t)$ coincide with the solution of the TAP equations for ${\mathlette{\boldmath}{\gamma}}$, , if we identify the corresponding magnetizations by ${\mathlette{\boldmath}{m}} = \chi({{\mathlette{\boldmath}{\gamma}}} + f({\mathlette{\boldmath}{\gamma}}))$. We have the following important properties of the dynamics $${\rm Tr}({\mathlette{\boldmath}{A}})=0~~\text{and}~~ {\rm Tr}({\mathlette{\boldmath}{E}}(t))=0 ~~\text{with}~~[{\mathlette{\boldmath}{E}}(t)]_{ij}\doteq f'(\gamma_i(t))\delta_{ij}, \forall t.\label{vtrace}$$ Here, the first and second equalities follow by the constructions of the random matrix ${\mathlette{\boldmath}{A}}$ and random initialization ${\mathlette{\boldmath}{\gamma}}(0)$, respectively [@CakmakOpper19]. It is also worth mentioning that we have the freedom to replace the function $f$ with an appropriate sequence of function, say $f_t$, in such a way that the conditions ${\rm Tr}[{\rm diag}(f_t'({\mathlette{\boldmath}{\gamma}}(t)))]=0$ and $f_t\to f$ as $t\to \infty$ are fulfilled, see [@CakmakOpper19 Section VIII.B]. Dynamics in the thermodynamic limit =================================== Dynamical properties of fully connected disordered systems can be analyzed by a discrete time version of the dynamical functional theory (DFT) of statistical physics originally developed by Martin, Siggia and Rose [@Martin] and later used for the study of spin-glass dynamics, see e.g. [@sompo82; @Eisfeller; @Opper16], and neural network models [@sompolinsky1988chaos]. Using this approach, it is possible to perform the average over the random matrix ensemble of ${\mathlette{\boldmath}{A}}$ and initial conditions for $N\to\infty$ and marginalize out all degrees of freedom $\gamma_j(t)$ for $j\neq i$ and all times $t$ to obtain the statistical properties of trajectories of length $T$ for an arbitrary single node $\{\gamma_i(t)\}_{t=1}^T$. Since the nodes are exchangeable random variables under the random matrix assumption, one can obtain the convergence properties of the algorithm by studying a single node. For a rotation-invariant matrix ${\mathlette{\boldmath}{A}}$ and an arbitrary function $f$, the DFT yields an “effective” stochastic dynamics for $\gamma_i(t)$ which is of the universal form (we skip the index $i$, since it is the same for all nodes) $$\label{esp} \gamma(t) = \sum_{s < t}{\hat {\mathcal G}(t,s)}f(\gamma(s-1)) + \phi(t), \quad t\leq T.$$ Here $\phi(t)$ is a colored Gaussian noise term. This dynamics is of a “mean field” type because the statistics of the noise must be computed from averages over the process itself which involves the function $f$ and the ${\rm R}$ transform [@Opper16]. In general, the explicit analysis of the the single node statistics becomes complicated by the presence of the additional memory terms ${\hat {\mathcal G}(t,s)}$ which can be explicitly represented as a function of the $T\times T$ order parameter matrix $$\mathcal G(t,s)\doteq\mathbb E\left[\frac{\partial f(\gamma(t-1))}{\partial \phi (s)}\right], \quad t,s\leq T \label{resp}$$ which again must be computed from the entire ensemble of trajectories of $\gamma(t)$. $\mathcal G(t,s)$ represents the average (linear) response of the variable $f(\gamma(t-1))$ to a small perturbation of the driving force $\phi(s)$ at previous times. Hence, by causality $\mathcal G$ is an upper triangular matrix (i.e. $\mathcal G(t,s)=0$ for $s \geq t$). Also, the case of zero response matrix $\mathcal G={\mathlette{\boldmath}{0}}$ leads to ${\hat {\mathcal G}}={\mathlette{\boldmath}{0}}$. The combination of the Gaussian noise and the response function in the dynamics has an intuitive meaning: The Gaussian can be understood as a representation of the incoherent addition of random variables arising from the multiplication of the vector $f({\mathlette{\boldmath}{\gamma}}(t-1))$ with the random matrix ${\mathlette{\boldmath}{A}}$. On the other hand, by treating the typically small matrix elements $A_{ij}$ in a perturbative way [@Mezard Chapter 6], one can estimate the influence of a node $i$ (using a linear response argument) on the $N-1$ neighboring nodes $j\neq i$, which by the symmetry of the matrix, will lead to a coherent, retarded influence of all nodes $j$ back on node $i$ at later times. This explains, why memory terms were found to be absent for neural network dynamics with i.i.d. [*non symmetric*]{} random couplings [@sompolinsky1988chaos]. This has made a complete analytical treatment of the effective dynamics in such a case possible. Surprisingly, for the non-linear function $f$ given in eq. and the [*symmetric*]{} matrix ${\mathlette{\boldmath}{A}}$, we have shown in [@CakmakOpper19] that the response functions (\[resp\]) vanish, i.e. $\mathcal G(t,s) = 0$ for all $t,s$. As a result also the memory terms vanish; $\gamma (t)$ in simply becomes a Gaussian field. Hence an analytical treatment is possible as was also shown in the previous studies [@rangan2019vector; @takeuchi]. In the following section we will use the freeness argument of random matrix theory to explain this result. Absence of memory terms and asymptotic freeness {#gmfc} =============================================== To analyze the average response (\[resp\]) for a single node, we use the chain rule in the dynamical susceptibility for the original $N$ node dynamics (see ) $$G_{ij}(t,s) \doteq \frac{\partial{f(\gamma_i(t-1))}}{\partial\gamma_j(s)} = \left[({\mathlette{\boldmath}{E}}(s){{\mathlette{\boldmath}{A}}}{\mathlette{\boldmath}{E}}(s+1){\mathlette{\boldmath}{A}}\cdots{\mathlette{\boldmath}{E}}(t-2){{\mathlette{\boldmath}{A}}}{{\mathlette{\boldmath}{E}}(t-1)})\right]_{ij} \label{jacob},\quad s<t.$$ By its construction, we can argue that the derivative w.r.t. $\gamma_i(s)$ acts in the same way as the derivative w.r.t. $\phi(s)$ and thus we will have (as $N\to \infty$) $$\mathbb E\left[G_{ii}(t,s)\right] \to \mathcal G(t,s).$$ Here $\{G_{ii}(t,s)\}_{i\leq N}$ are random w.r.t. the random matrix ${\mathlette{\boldmath}{A}}$ and random initialization ${\mathlette{\boldmath}{\gamma}}(0)$. By ex-changeability $G_{ii}(t,s)\sim G_{jj}(t,s),j\neq i$, the condition ${\rm Tr}({\mathlette{\boldmath}{E}}(t))=0$ (see ) implies vanishing single-step memories, i.e. $\mathbb E[G_{ii}(t,t-1)]\to 0$. We next argue that for further time-lags the memories do vanish *in a stronger sense*. Specifically, we will show that $$\epsilon(t,s)\doteq \lim_{N\to \infty}\mathbb E[G_{ii}(t,s)^2]=0,\quad s<t-1.\label{epsilon}$$ To this end, we introduce an auxiliary random diagonal $N\times N$ matrix ${\mathlette{\boldmath}{Z}}$ which is independent of ${\mathlette{\boldmath}{A}}$ and $\{{\mathlette{\boldmath}{E}}(t)\}$. The diagonal entries of ${\mathlette{\boldmath}{Z}}$ are independent and composed of $\pm 1$ with equal probabilities. Note that ${\mathbb E}[Z_{nn}Z_{kk}]=\delta_{nk}$. Hence, we can write $$\begin{aligned} \frac{1}{N}\mathbb E\left[{\rm tr}(({\mathlette{\boldmath}{Z}}{\mathlette{\boldmath}{G}}(t,s))^2)\right]&=\frac{1}{N}\sum_{i,j\leq N}{\mathbb E}[Z_{ii}Z_{jj}]\mathbb E[G_{ij}(t,s)G_{ji}(t,s)]\\ &=\frac 1 N\sum_{j\leq N} \mathbb E[G_{jj}(t,s)^2]=\mathbb E[G_{ii}(t,s)^2].\end{aligned}$$ Then, we have $$\begin{aligned} \epsilon(t,s)&={\rm Tr} ({\mathlette{\boldmath}{Z}}{\mathlette{\boldmath}{E}}(s){\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(s+1)\cdots {\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(t-1){\mathlette{\boldmath}{Z}}{\mathlette{\boldmath}{E}}(s){\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(s+1)\cdots{\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(t-1))\nonumber \\ &={\rm Tr} ({\mathlette{\boldmath}{E}}_Z(t,s){\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(s+1)\cdots {\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}_Z(t,s){\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}(s+1)\cdots{\mathlette{\boldmath}{A}}). \label{prod}\end{aligned}$$ Here, we have defined the diagonal matrix ${\mathlette{\boldmath}{E}}_Z(t,s)\doteq{\mathlette{\boldmath}{E}}(t-1){\mathlette{\boldmath}{Z}}{\mathlette{\boldmath}{E}}(s)$. To simplify we will make us of the concept of *asymptotic freeness* of random matrices. [@Hiai] For the two families of matrices ${\mathcal A\doteq \{{\mathlette{\boldmath}{A}}_1,{\mathlette{\boldmath}{A}}_2,\ldots,{\mathlette{\boldmath}{A}}_a\}}$ and ${\mathcal E\doteq \{{\mathlette{\boldmath}{E}}_1,{\mathlette{\boldmath}{E}}_2,\ldots,{\mathlette{\boldmath}{E}}_e\}}$ let ${{\mathlette{\boldmath}{P}}_i(\mathcal A)}$ and ${{\mathlette{\boldmath}{Q}}_i(\mathcal E)}$ stand for (non-commutative) polynomials of the matrices in ${\mathcal A}$ and the matrices in ${\mathcal E}$, respectively. Then, we say the families ${\mathcal A}$ and ${\mathcal E}$ are asymptotically free if for all $i\in[1,K]$ and for all polynomials ${{\mathlette{\boldmath}{P}}_i(\mathcal A)}$ and ${{\mathlette{\boldmath}{Q}}_i(\mathcal E)}$ we have $${\rm Tr}({{\mathlette{\boldmath}{P}}_1(\mathcal A)}{{\mathlette{\boldmath}{Q}}_1(\mathcal E)}{{\mathlette{\boldmath}{P}}_2(\mathcal Q)}{{\mathlette{\boldmath}{Q}}_2(\mathcal E)}\cdots {{\mathlette{\boldmath}{P}}_K(\mathcal A)}{{\mathlette{\boldmath}{Q}}_K(\mathcal E)})=0 \label{key}$$ given that all polynomials in are centered around their limiting normalized-traces, i.e. $${\rm Tr}({{\mathlette{\boldmath}{P}}_i(\mathcal A)})={\rm Tr}({{\mathlette{\boldmath}{Q}}_i(\mathcal E)})=0, \quad \forall i.$$ Namely, the limiting normalized-trace of any adjacent product of powers of matrices—which belong to different free families and are centered around their limiting normalized-traces—vanishes. In the product the matrices belong to two families: rotation invariant and diagonal. Under certain technical conditions—which includes the independence of matrix families—these two matrix families can be treated as asymptotically free [@Hiai]. E.g. ${\mathlette{\boldmath}{A}}$ is asymptotically free of ${\mathlette{\boldmath}{Z}}$. Our [**heuristic assumption**]{} is that ${\mathlette{\boldmath}{A}}$ is also free of the diagonals $\{{\mathlette{\boldmath}{E}}(t)\}$. A subtle point should be noted here: Being outcomes of the dynamical system, the diagonal matrices $\{{\mathlette{\boldmath}{E}}(t)\}$ are not independent from ${\mathlette{\boldmath}{A}}$. Nevertheless, since we expect that the diagonals ${\mathlette{\boldmath}{E}}(t)$ have limiting spectral distributions, we consider that asymptotic freeness is a fair heuristic here. The result follows immediately from the asymptotic freeness assumption: we have that all adjacent factors in the product are polynomials belonging to the different free families and all matrices in the product are centered around their limiting normalized-traces. Asymptotic of the local convergence =================================== We will analyze the convergence rate of the dynamics in terms of the following measure $$\begin{aligned} \mu_\gamma &\doteq \lim_{t\to\infty} \lim_{N\to \infty}\frac{\mathbb E\Vert {\mathlette{\boldmath}{\gamma}}(t+1)-{\mathlette{\boldmath}{\gamma}}(t)\Vert^2}{\mathbb E{\Vert {\mathlette{\boldmath}{\gamma}}(t)-{\mathlette{\boldmath}{\gamma}}(t-1)\Vert^2}}.\end{aligned}$$ To this end, we will assume that one starts the iterations at a point which is close enough to the fixed point of ${\mathlette{\boldmath}{\gamma}}(t)$, denoted by ${\mathlette{\boldmath}{\gamma}}^*$ such that a linearization of the dynamics is justified. We conjecture (in accordance with our simulations) that the initialization does not affect the asymptotic rates. This means that we can substitute ${\mathlette{\boldmath}{\gamma}}(t)$ by the following “effective” dynamics $${\mathlette{\boldmath}{\gamma}}(t) = {\mathlette{\boldmath}{\gamma}}^*+ {\mathlette{\boldmath}{\epsilon}}(t)$$ with ${\mathlette{\boldmath}{\epsilon}}(t)$ small enough to justify the linearised dynamics $$\begin{aligned} {\mathlette{\boldmath}{\epsilon}}(t) &\simeq {\mathlette{\boldmath}{A}} {\mathlette{\boldmath}{E}} {\mathlette{\boldmath}{\epsilon}}(t-1)=({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}})^t{\mathlette{\boldmath}{\epsilon}}(0)~~ \text{with}~~[{\mathlette{\boldmath}{E}}]_{ij}\doteq f'(\gamma_i^{*})\delta_{ij}.\end{aligned}$$ Moreover, we consider a random initialization ${\mathlette{\boldmath}{\epsilon}}(0)$ with $\mathbb E[{\mathlette{\boldmath}{\epsilon}}(0){\mathlette{\boldmath}{\epsilon}}(0)^\top]=\sigma^2{\bf I}$. Then, one can write $$\begin{aligned} \mu_\gamma &= \lim_{t\to\infty}\frac{{\rm Tr}[({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}}-{\bf I})({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}})^t({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}-{\bf I})({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}})^t]}{{\rm Tr}[({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}}-{\bf I})({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}})^{t-1}({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}}-{\bf I})({\mathlette{\boldmath}{A}} {\mathlette{\boldmath}{E}})^{t-1}]}.\end{aligned}$$ Similar to the response function we encounter the same product of two (asymptotic) trace free matrices. We then assume that ${\mathlette{\boldmath}{A}}$ and ${\mathlette{\boldmath}{E}}$ can be treated as free matrices. Doing so leads to $$\begin{aligned} {\rm Tr}[({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}})^{t\mp 1}({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}})^t]=0\quad \text{and}\quad {\rm Tr}[({\mathlette{\boldmath}{E}}{\mathlette{\boldmath}{A}})^t({\mathlette{\boldmath}{A}}{\mathlette{\boldmath}{E}})^t]={\rm Tr}({\mathlette{\boldmath}{A}}^2)^t{\rm Tr}({\mathlette{\boldmath}{E}}^2)^t.\end{aligned}$$ So that we get the simple expression for the convergence rate as $$\begin{aligned} \mu_\gamma&={\rm Tr}({\mathlette{\boldmath}{A}}^2){\rm Tr}({\mathlette{\boldmath}{E}}^2).\label{sres}\end{aligned}$$ This shows that when ${\rm Tr}({\mathlette{\boldmath}{A}}^2){\rm Tr}({\mathlette{\boldmath}{E}}^2)< 1$ we obtain local convergence of the algorithm towards the fixed point. Moreover, a straightforward calculation shows that $${\rm Tr}({\mathlette{\boldmath}{A}}^2){\rm Tr}({\mathlette{\boldmath}{E}}^2)=1-\frac{1-{{\rm Tr}({\mathlette{\boldmath}{E}}^2)}{\rm R}'(\chi)}{1-\chi^2{\rm R}'(\chi)}$$ which exactly agrees with the result of the more complex DFT calculation [@CakmakOpper19]. In the following section, we will support our heuristics by simulations on two instances of random matrices. Simulations =========== In the sequel we illustrate the results of the free probability heuristics, i.e. and . Since we expect that these results are self-averaging in the large-system limit, our simulations are based on single instances of a large random matrix ${\mathlette{\boldmath}{A}}$ and random initialization ${\mathlette{\boldmath}{\gamma}}(0)$. In particular, we consider the empirical approximation of the limit as $$\epsilon_{N}(t,s)\doteq\frac{1}{N}\sum_{i=1}^{N}G_{ii}(t,s)^2.\label{neweps}$$ In Fig. 1(a) and 1(b), we illustrate the vanishing memory property and the convergence rate of the dynamics for the SK model $${\mathlette{\boldmath}{J}}=\beta {\mathlette{\boldmath}{G}}$$ where $G_{ij}$, $ 1 ² i < j ² N$, are i.i.d. centered Gaussian random variables with variance $1/N$. Second, motivated by a recent study [@Greg] in random matrix theory, we consider a non-rotation invariant random coupling matrix model. The model is related to the random orthogonal model discussed by Parisi and Potters [@Parisi] which is defined as $${\mathlette{\boldmath}{J}}=\beta{\mathlette{\boldmath}{O}}^\top{\mathlette{\boldmath}{D}}{\mathlette{\boldmath}{O}}$$ where ${\mathlette{\boldmath}{O}}$ is a Haar matrix and ${\mathlette{\boldmath}{D}}={\rm diag}(d_1,\cdots,d_N)$ has random binary elements $d_i=\mp 1$ with $\vert\{d_i=1\}\vert=N/2$. Specifically, we substitute the Haar basis of the random orthogonal model with a randomly-signed DCT (discrete-cosine-transform) matrix as $${\mathlette{\boldmath}{J}}=\beta\tilde{{\mathlette{\boldmath}{O}}}^\top{\mathlette{\boldmath}{D}}\tilde{{\mathlette{\boldmath}{O}}}~~ \text{with}~~\tilde{{\mathlette{\boldmath}{O}}}\doteq {\mathlette{\boldmath}{\Theta}}_{N}{\mathlette{\boldmath}{Z}}. \label{rsh}$$ Here, ${\mathlette{\boldmath}{Z}}$ is an $N\times N$ diagonal matrix whose diagonal entries are independent and composed of binary $\mp1$ random variables with equal probabilities and ${\mathlette{\boldmath}{\Theta}}$ is $N\times N$ (deterministic) DCT matrix. The simulation results for the latter model are illustrated in Figure 2. They indicate that the free probability heuristics are also very accurate for randomly signed (deterministic) DCT matrix (which contains considerably less randomness compared to the rotation invariant case). As a mater of fact, this is not surprising because for a random permutation matrix ${\mathlette{\boldmath}{P}}$ and diagonal matrices ${\mathlette{\boldmath}{D}}_1$ and ${\mathlette{\boldmath}{D}}_2$ such that all matrices are mutually independent, it is proved that the matrices ${\mathlette{\boldmath}{P}}^\top\tilde{{\mathlette{\boldmath}{O}}}^\top{\mathlette{\boldmath}{D}}_1\tilde{{\mathlette{\boldmath}{O}}}{\mathlette{\boldmath}{P}}$ and ${\mathlette{\boldmath}{D}}_2$ are asymptotically free [@Greg]. Summary and Outlook =================== In this paper we have presented a free probability heuristics for understanding and recovering analytical results for the dynamical behavior of so-called message passing algorithms for probabilistic inference. Such algorithms have the form of a discrete time, recurrent neural network dynamics. We were able to show for a toy Ising model with random couplings, that parts of previous results which were obtained by more complicated techniques can be understood and re-derived under the heuristic hypothesis of asymptotic freeness of two matrix families. Under this condition, and if matrices are trace free, the diagonal elements of the response function which determines the effective memory in the dynamics vanish. This property also yields an analytical result for the exponential convergence of the algorithm towards its fixed point. We have tested these predictions successfully on two types of random matrix ensembles. We expect that similar arguments can be applied to the analysis of more general types of inference algorithms of the expectation propagation type. It would also be interesting to design novel algorithms that can be analyzed assuming the freeness heuristics. Of course, the heuristics should eventually be replaced by more rigorous arguments. While our results indicate that message passing algorithms could be analyzed under somewhat weaker conditions on random matrices (compared to explicit assumptions on rotational invariant ensembles) the applicability of these concepts to real data needs to be shown. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank Yue M. Lu for inspiring discussions. This work was supported by the German Research Foundation, Deutsche Forschungsgemeinschaft (DFG), under Grant No. OP 45/9-1 and BMBF (German ministry of education and research) joint project 01 IS18037 A : BZML- Berlin Center for Machine Learning. [^1]: Presented at the conference “Random Matrix Theory: Applications in the Information Era” 2019 Kraków.

--- abstract: 'We discuss the dependence of pure Yang-Mills equation of state on the choice of gauge algebra. In the confined phase, we generalize to an arbitrary simple gauge algebra Meyer’s proposal of modelling the Yang-Mills matter by an ideal glueball gas in which the high-lying glueball spectrum is approximated by a Hagedorn spectrum of closed-bosonic-string type. Such a formalism is undefined above the Hagedorn temperature, corresponding to the phase transition toward a deconfined state of matter in which gluons are the relevant degrees of freedom. Under the assumption that the renormalization scale of the running coupling is gauge-algebra independent, we discuss about how the behavior of thermodynamical quantities such as the trace anomaly should depend on the gauge algebra in both the confined and deconfined phase. The obtained results compare favourably with recent and accurate lattice data in the $\mathfrak{su}(3)$ case and support the idea that the more the gauge algebra has generators, the more the phase transition is of first-order type.' author: - Fabien - Gwendolyn title: 'Comments on Yang-Mills thermodynamics, the Hagedorn spectrum and the gluon gas' --- Introduction ============ The existence of a critical temperature, $T_c$, in QCD, is of particular phenomenological interest since it signals a transition from a confined phase of hadronic matter to a deconfined one. When $T<T_c$, a successful effective description of QCD is the hadron resonance gas model, in which the hadronic matter is seen as an ideal gas of hadrons. It compares well with current lattice data when the meson and baryon resonances below 2.5 GeV are included [@borsanyi2010]. A problem is that experimental information about resonances above 3 GeV is still lacking. To describe the high-lying hadronic spectrum, Hagedorn [@hage65] proposed a model in which the number of hadrons with mass $m$ is found to increase as $\rho(m)\propto m^a \, {\rm e}^{m/T_h}$ ($a$ is real): the so-called Hagedorn spectrum. Thermodynamical quantities, computed using hadronic degrees of freedom, are then undefined for $T>T_h$. Other degrees of freedom are then needed at higher temperatures, so it is tempting to guess that $T_h\approx T_c$, the new degrees of freedom being deconfined quarks and gluons. Although the current lattice studies agree on a value of $T_c$ in the range $(150-200)$ MeV when $2+1$ light quark flavours are present [@borsanyi2010; @tcd], there is currently no consensus concerning the value of $T_h$. Indeed, to reach values of $T_h$ as low as 200 GeV demands an ad hoc modification of $\rho(m)$: By introducting an extra parameter $m_0$ and setting $\rho(m)\propto (m^2+m^2_0)^{a/2} \, {\rm e}^{m/T_h}$, one can reach values of $T_h$ in the range $(160-174)$ MeV, that agree with lattice computations, see *e.g.* [@hage68; @cley]. However, by taking the original form $m_0=0$, one rather ends up with values of $T_h$ around $(300-360)$ MeV, see [@cudell0; @bronio]. Moreover, it has been observed in some pure gauge lattice simulations with the gauge algebra $\mathfrak{su}(N)$ that $T_c\lesssim T_h$ [@TcTh0; @TcTh] as intuitively expected. It has to be said that the value of $T_h$ and its relation to $T_c$ are still a matter of debate. Open strings as well as closed strings naturally lead to a Hagedorn spectrum, see *e.g.* [@zwie]. Modelling mesons as open strings is a way to make appear a Hagedorn spectrum in QCD [@cudell]. The question of showing that a Hagedorn spectrum arises from QCD itself is still open but, under reasonable technical assumptions, it has recently been found in the large-$N$ limit of QCD [@cohen] (glueballs and mesons have a zero width in this limit). In the pure gauge sector, the $\mathfrak{su}(3)$ equation of state computed on the lattice has been shown to be compatible with a glueball gas model in which the high-lying spectrum is modelled by a gas of closed bosonic strings [@meyer]. Besides QCD, pure Yang-Mills (YM) thermodynamics is challenging too, in particular because it can be formulated for any gauge algebra. A clearly relevant case is the one of $\mathfrak{su}(N)$-type gauge algebras, linked to the large-$N$ limit of QCD. Moreover, a change of gauge algebra may lead to various checks of the hypothesis underlying any approach describing $\mathfrak{su}(3)$ YM theory. To illustrate this, let us recall the pioneering work [@sve], suggesting that the phase transition of YM theory with gauge algebra $\mathfrak{g}$ is driven by a spontaneous breaking of a global symmetry related to the center of $\mathfrak{g}$. Effective $Z_3$-symmetric models are indeed able to describe the first-order phase transition of $\mathfrak{su}(3)$ YM thermodynamics [@Z3]. However, a similar phase transition has also been observed in lattice simulations of G$_2$ YM theory [@G2] even though the center of G$_2$ is trivial, meaning that the breaking of center symmetry is not the only mechanism responsible for deconfinement. For example, it is argued in [@diakonov] that the YM phase transition for any gauge group is rather driven by dyons contributions. In this case, still under active investigation, studying different gauge algebras helps to better understand the general mechanisms of (de)confinement in YM theory. For completeness, we mention that the structure of the gluon propagator at low momentum as well as the Dyson-Schwinger equations in scalar-Yang-Mills systems have recently started to be studied for generic gauge algebra [@maas; @maas2]. The main goal of the present work is to give predictions for the equation of state of YM theory with an arbitrary simple gauge algebra. This topic has, to our knowledge, never been investigated before and will be studied within two well-established different frameworks: A glueball gas with a high-lying Hagedorn spectrum in the confined phase (Sec. \[conf\]) and a gluon gas above the critical one (Sec. \[deconf\]). Some phenomenological consequences of the obtained results will then be discussed in Sec. \[conclu\]. More specifically, our results apply to the following gauge algebras : A$_{r\geq 1}$ related to $\mathfrak{su}$ algebras, B$_{r\geq 3}$ and D$_{r\geq 4}$ related to $\mathfrak{so}$ algebras, C$_{r\geq 2}$ related to $\mathfrak{sp}$ algebras, and the exceptional algebras E$_{6}$, E$_7$, F$_4$ and G$_2$. The case of E$_8$ is beyond the scope of the present paper as it will be explained below. Glueball gas and the Hagedorn spectrum {#conf} ====================================== The model --------- In the confined phase, glueballs, *i.e.* colour singlet bound states of pure YM theory, are the relevant degrees of freedom of YM matter. Hence it can be modelled in a first approximation by an ideal gas of glueballs, assuming that the residual interactions between these colour singlet states are weak enough to be neglected [@dashen]. Note that the glueball gas picture emerges from a strong coupling expansion in the case of large-$N$ $\mathfrak{su}(N)$ YM theory [@langelage10], where glueballs are exactly noninteracting since their scattering amplitude scales as $1/N^2$ [@witten]. The glueball gas picture implies that, for example, the total pressure should be given by $\sum_{J^{PC}}p_0(2J+1,T,M_{J^{PC}})$, where the sum runs on all the glueball states of the YM theory with a given gauge algebra, and where $$p_0(d,T,M)=\frac{d}{2\pi^2}M^2T^2\sum_{j=1}^\infty\frac{1}{j^2}K_2(j\, M/T)$$ is the pressure associated with a single bosonic species with mass $M$ and $d$ degrees of freedom. Performing the sum $\sum_{J^{PC}}$ demands the explicit knowledge of all the glueball states, not only the lowest-lying ones that can be known from lattice computations or from effective approaches. To face this problem, it has been proposed in [@meyer] to express the total pressure of $\mathfrak{su}(3)$ YM theory as $$\label{preh} p= \hspace{-0.3cm}\sum_{M_{J^{PC}}<2M_{0^{++}}}\hspace{-0.65cm} p_0(2J+1,T,M_{J^{PC}})+\int^\infty_{2M_{0^{++}}}\hspace{-0.5cm}dM\ p_0(\rho(M),T, M),$$ where the high-lying glueball spectrum (above the two-glueball threshold $2M_{0^{++}}$) is approximated by a closed-string Hagedorn density of states reading, in 4 dimensions [@zwie; @meyer], $$\label{rhoh} \rho(M)=\frac{(2\pi)^3}{27 T_h}\left( \frac{T_h}{M} \right)^4{\rm e}^{M/T_h}.$$ The idea of modelling glueballs as closed fundamental strings was actually already present in the celebrated Isgur and Paton’s flux-tube model, inspired from the Hamiltonian formulation of lattice QCD at strong coupling [@isgur]. Moreover, it has also been shown within a constituent picture that, in the $\mathfrak{su}(3)$ case, a many-gluon state (typically more than three gluons in a Fock-space expansion) tends to form a closed gluon chain [@gluphen]. In Eq. (\[rhoh\]), $T_h$ is the Hagedorn temperature, which reads in this case $$T^2_h=\frac{3}{2\pi}\sigma^{(f)},$$ where $\sigma^{(f)}$ is the fundamental string tension, here defined as the slope of the static energy between two sources in the fundamental representation of a given gauge algebra. The Casimir scaling of the string tension, which is an analytic prediction from the strong coupling expansion of the Wilson loop, says that the string tension is given by [@casi; @casi2] $$\sigma^{(r)}=C_2^{(r)}\, \Theta ,$$ where the colour sources are in a given representation $r$ of the gauge algebra, and where $\Theta$ reads, in a lattice formulation of the theory [@casi] $$\Theta=\frac{g^2(a\Lambda)}{2a}.$$ $a$ is the lattice size and $g(a\Lambda)$ is the running coupling with the renormalization scale $\Lambda$. Following well-known two-loop calculations, one can extract the explicit gauge-algebra dependence in the running coupling as follows: $g^2(a\Lambda)=\lambda(a\Lambda)/C_2^{(ad\hspace{0.1pt}j)}$ [@g2r], where $\lambda$ is nothing else than the ’t Hooft coupling when the gauge algebra is $\mathfrak{su}(N)$. One can finally define $$\sigma^{(r)}=\frac{C_2^{(r)}}{C_2^{(ad\hspace{0.1pt}j)}}\, \sigma_0,$$ where $\sigma_0$, that can be interpreted as the adjoint string tension, does not depend explicitly on the gauge algebra. However, an implicit dependence in the renormalization scale $\Lambda$ may be present. Throughout this work we consider a gauge-algebra independent value for $\Lambda$. The structure of the low-lying glueball spectrum for an arbitrary simple gauge algebra has been discussed in detail in [@buiss11] within a constituent picture, although the results which are useful for our purpose could be recovered in a more model-independent way by studying *e.g.* the structure of glueball-generating field-strength correlators. Let us recall those results: - The lightest glueballs are the scalar, pseudoscalar and tensor ones, whose masses are ordered as $M_{0^{++}}<M_{2^{++}}$, $M_{0^{-+}}$ in agreement with lattice results in the $\mathfrak{su}(N)$ case [@glueb1; @luciN]. Those states are found to be lighter than $2\, M_{0^{++}}$ in these last works. Note that it has been proved in  [@west] that the $0^{++}$ glueball is always the lightest one in YM theory. - At masses typically around (3/2)$M_{0^{++}}$, states that can be seen as mainly three-gluon ones in a Fock-space expansion appear: They can have $C=+$ for any gauge algebra, but $C=-$ for A$_{r\geq2}$ ($\mathfrak{su}(N\geq 3)$) only. In this last case, the $1^{+-}$ glueball is still lighter than $2\, M_{0^{++}}$ [@glueb1; @luciN]. - Higher-lying states (containing more than three gluons in a Fock space expansion) obviously exist, but their exhaustive study cannot be performed explicitly, eventually justifying the use of a Hagedorn spectrum. An important remark has nevertheless to be done: If all the representations of a given gauge algebra are real, the gluonic field $A_\mu $ is its own charge-conjugate, eventually forbidding $C=-$ glueball states. This happens for the algebras A$_1$, B$_{r\geq2}$, C$_r$, D$_{{\rm even}-r\geq4}$, E$_7$, E$_8$, F$_4$, and G$_2$. It is worth noticing that a closed-string picture for high-lying glueballs is not only a consequence of Isgur and Paton’s flux-tube-like approaches but may also be compatible with constituent approaches such as the one used in [@buiss11]: An excited closed string is then alternatively viewed as a closed chain of quasigluons where the quasigluons are linked by fundamental strings. From a string theory point of view, the Nambu-Goto string can be coherently quantized within both pictures using *e.g.* the Gupta-Bleuler method [@gershun10]. Moreover, since $ad\hspace{0.1pt}j\in f\otimes f$ or $f\otimes\bar f$, with $f$ ($\bar f$) the fundamental (conjugate) representation for any simple gauge algebra, a gluon can always generate two fundamental strings, with $\sigma^{(f)}=\sigma^{(\bar f)}$ in virtue of the Casimir scaling, instead of one adjoint string. In the case of E$_8$, the lowest-dimensional representation, that we have called fundamental before, is the adjoint one, so the closed-string picture seems less justified by comparison to a constituent picture. We therefore prefer not to investigate further the case of E$_8$ in the following. Linking $T_h$ to $T_c$ ---------------------- As a first step, the link between $T_h$ and $T_c$ has to be fixed. A straightforward way to do it is to briefly recall Meyer’s results in the pure gauge $\mathfrak{su}(3)$ case [@meyer], where the lattice entropy density $s=\partial_T p$ computed below $T_c$ has been fitted by using the present model. It appears that the best agreement is reached for $T_h/T_c=1.069(5)$. Finding $T_h>T_c$ is actually an indication that a metastable, superheated, hadronic phase of matter exists at temperatures between $T_c$ and $T_h$; this phase has actually been studied on the lattice in [@TcTh], where, for example, $T_h/T_c=1.116(9)$ has been found for the gauge algebra $\mathfrak{su}(12)$, and discussed within the framework of an open-string model in [@cudell]. As seen from the above discussion, an accurate determination of the ratio $T_h/T_c$ is of great phenomenological interest. However, such a study is not the main purpose of the present paper, where we aim at giving reliable predictions for the equation of state of YM theory with an arbitrary gauge algebra. As observed in [@meyer], typical values $T_h\approx T_c$ give very good results in fitting the lattice data. Setting $T_c=T_h$, as we will do in the rest of this work, means that the deconfinement temperature may be identified with the maximal allowed temperature for the confined hadronic phase. This assumption has two advantages. First, it will reproduce accurately the latest $\mathfrak{su}(3)$ lattice data of [@borsa] (see next section), and it is not in strong disagreement with current $\mathfrak{su}(N)$ results, where $T_h/T_c$ is at most around $10 \%$ [@TcTh0; @TcTh]. Second, it is applicable to any gauge algebra without having to guess a value for $T_h/T_c$, that cannot be fitted on lattice results since no equation of state is available for gauge algebras different than $\mathfrak{su}(N)$ so far. The drawback of this choice is that it forbids any discussion about a superheated hadronic phase in generic YM theories. Such a refinement of the model will rather be the topic of a separate study. For completeness, we notice that the somewhat surprising value $T_h=2.8\, T_c\gg T_c$ has been found in [@Megias] by using a Hagedorn picture too. The difference with our approach comes from the fact that, in  [@Megias], $T_h$ is fitted by assuming that the low-lying glueballs currently known from lattice simulations should exhibit a Hagedorn-type spectrum. On the contrary, we think here that the Hagedorn-like behavior only appears in the high-lying sector, that mostly concerns the glueballs that are not known so far by lattice calculations, see Eq. (\[preh\]). Numerical results ----------------- According to standard $\mathfrak{su}(3)$ studies, it is relevant to set $\sigma_0\approx(9/4)\ 0.2$ GeV$^2$, leading to $T_h=$309 MeV. The masses of the lightest glueballs are proportional to $\sqrt{\sigma_0}$ [@buiss11], so they can be thought as constant with respect to a change of gauge algebra in our approach. Consequently, the sum $\sum_{M_{J^{PC}}<2M_{0^{++}}}$ should run on all the states below $3.46$ GeV found in the $\mathfrak{su}(3)$ lattice work [@glueb1]. There is an exception however: The $1^{+-}$ glueball, whose mass is below the two-glueball threshold, only exists when the gauge algebra is A$_{r\geq2}$ [@buiss11]; hence its contribution will be omitted in the other cases. Concerning the Hagedorn spectrum, it is worth recalling that the density (\[rhoh\]) is able to reproduce the $\mathfrak{su}(3)$ lattice equation of state with $T_c\approx T_h$ [@meyer]. But $\rho(M)$ accounts for both the $C=+$ and $C=-$ glueballs. When the gauge algebra has only real representations, the $C=-$ sector is absent as said before. So in such cases, the substitution $\rho(M)\rightarrow\rho(M)/2$ will be done. The validity of this prescription has been explicitly checked in [@case] by computing the equation of state of $2+1$-dimensional YM theory below $T_c$ with $\mathfrak{su}(N)$ gauge algebras: $\rho(M)$ correctly describes the data for $\mathfrak{su}(3-6)$, while $\rho(M)/2$ must be used for $\mathfrak{su}(2)$ in order to compensate for the absence of $C=-$ states in the theory. We are now in position of explicitly computing the pressure (\[preh\]) for any gauge algebra, E$_8$ excepted. We actually compute from $p$ the trace anomaly, using $$\label{taoh} \Delta=T^5\partial_T \left(\frac{p}{T^4}\right),$$ so that our results can be compared to the recent and accurate $\mathfrak{su}(3)$ lattice data of [@borsa], displayed in Fig. \[fig1\]. ![(Color online) Trace anomaly below $T_c$, computed using Eqs. (\[preh\]) and (\[taoh\]) with $T_h=T_c$ and $\sigma_0=(9/4)0.2$ GeV$^2$, for the gauge algebras A$_2$ (solid line), A$_{N\rightarrow\infty}$ and D$_{N\rightarrow\infty}$ (dashed line), G$_2$ and C$_{N\rightarrow\infty}$ (dotted line). All the possible cases are located within the grey area, whose upper and lower borders are E$_6$ and A$_1$ respectively. $\mathfrak{su}(3)$ lattice data from [@borsa] are plotted for comparison (orange points and area). The orange points correspond to $N_t=8$ data.[]{data-label="fig1"}](Hagedorn.pdf){width="10.0cm"} As a first check, we can see that the proposed model compares well with the $\mathfrak{su}(3)$ lattice data of [@borsa]. In a first approximation, the choice $T_c=T_h$ thus gives good results. A generic feature of $p$ and $\Delta$ is that they are finite in $T_h$, and mostly located below the E$_6$ and A$_1$ cases at any $T$. This finiteness is due to the $M^{-4}$ factor in (\[rhoh\]) [@frautschi], which is a consequence of the closed-string picture used here. Note that this finiteness is present in $2+1$ dimensions too [@case]. An interesting feature is that the large-$N$ limits of the A$_N$ and D$_N$ (when $N$ is odd) cases are equivalent, in agreement with the large-$N$ orbifold equivalence between $\mathfrak{su}(N)$ and $\mathfrak{so}(2N)$ YM theories, see *e.g.* [@cher]. The large-$N$ limit of the C$_N$ ($\mathfrak{sp}(N)$) case is however inequivalent to the A$_N$ one, but equal to the G$_2$ case. The observed significant numerical differences between the gauge algebras are moreover relevant from a physical point of view since they come from changes in the structure of the glueball spectrum, mainly at the level of the allowed quantum numbers. It is worth mentioning that an alternative to the Hagedorn spectrum has been proposed in [@strange], *i.e.* to consider that a Hagedorn spectrum is not present but that the glueball masses actually decrease near the critical temperature. This scenario can also lead to an agreement with the data of [@borsa] as checked by the authors of this last work. Only the lightest glueballs ($0^{\pm+}$ and $2^{++}$) will then give relevant thermodynamical contributions for any gauge algebra, and the corresponding equation of state might depend even less on the gauge algebra than within the Hagedorn picture. However, checking the dependence on $T$ of the glueball masses for different gauge algebras would demand detailed lattice computations or effective models that are currently unavailable, thus this topic is out of the scope of the present paper. Gluon gas and the deconfined phase {#deconf} ================================== As already mentioned, the Hagedorn temperature can be interpreted as a limiting temperature above which confined matter ceases to exist. In the deconfined phase, the relevant degrees of freedom are expected to be the ${\rm dim}(ad\hspace{0.1pt}j)$ gluons of the considered YM theory. When the temperature tends toward infinity, the Stefan-Boltzmann limit should thus be reached, that is *e.g.* the pressure $$p_{SB}={\rm dim}(ad\hspace{0.1pt}j)\, \frac{\pi^2}{45}T^4,$$ corresponding to the pressure an ideal gas of massless transverse gluons with ${\rm dim}(ad\hspace{0.1pt}j)$ colour degrees of freedom in $3+1$ dimensions. Corrections to this ideal-gas picture are nevertheless worth to study since it is known from $\mathfrak{su}(3)$ lattice simulation that one has to reach temperatures of about $(10^7-10^8)$ $T_c$ to get pressures compatible with the Stefan-Boltzmann limit up to the error bars [@highT]. The YM pressure (as well as cases with $N_f\neq 0$) can be systematically computed by performing expansions in the coupling constant $g$; terms of order $g^6 \ln(1/g)$ [@Laine] and parts of the full $g^6$ terms [@Laine2] are known so far. Hard-thermal-loop (HTL) resummation techniques also allow for a determination of YM pressure; results at next-to-next-to leading order (NNLO) are nowadays available [@HTL]. Recalling the scaling $ g^2\propto 1/C_2^{(ad\hspace{0.1pt}j)}$, the observation of the formulas obtained in [@Laine; @HTL] lead to the conclusion that the pressure behaves schematically as $$\label{phT} \frac{p}{p_{SB}}\equiv 1- \phi(\Lambda ,T)$$ where $\Lambda$ is a renormalization scale, that we assume to be gauge-independent as before, and where $\phi$ is a positive function that decreases when $T$ increases so that the SB limit is asymptotically reached. Once the a priori unknown parameters are fitted, both the $O(g^6 \ln(1/g))$ and the NNLO HTL formulae compares very well with the latest $\mathfrak{su}(3)$ lattice data of [@borsa], the best agreement being reached with the $O(g^6 \ln(1/g))$ formula. In particular, the trace anomaly $\Delta$, given by $$\label{delta} \frac{\Delta}{p_{SB}}=T\, \partial_T\left(\frac{p}{p_{SB}}\right),$$ is accurately reproduced above 10 $T_c$ (plots range from 1 to 100 $T_c$ in [@borsa]). One is straightforwardly led to the conclusion that the pressure (\[phT\]) is gauge-algebra independent; hence the high-temperature regime of YM thermodynamics should not depend on the considered gauge algebra once the equation of state is normalized to ${\rm dim}(ad\hspace{0.1pt}j)$. For example, the normalized trace anomaly (\[delta\]) should be gauge-algebra independent. This feature has already been checked on the lattice in the $\mathfrak{su}(N)$ case, where it appears that the pure YM equation of state normalized to $(N^2-1)$ is indeed universal above $T_c$ up to the error bars [@case; @panero; @gupta]. Just above $T_c$, where HTL or perturbative methods cannot give reliable information so far because of convergence problems, gluon-gluon interactions are expected to be quite strong although not confining. One would then speak of strongly coupled YM plasma. Those interactions, typically of one-gluon-exchange form, should be proportional to the color factor $(C_2^{(r)}-2C_2^{(ad\hspace{0.1pt}j)})g^2/2$, where $r$ is the color representation of the gluon pair. The universality of static colour interactions, once normalized to this last colour factor, has been checked on the lattice in the $\mathfrak{su}(3)$ case [@scalT]. For any algebra, one has $ad\hspace{0.1pt}j\otimes ad\hspace{0.1pt}j=\bullet\oplus ad\hspace{0.1pt}j\oplus \dots$. The singlet ($\bullet$) and adjoint channels will lead to attractive interactions that should not depend on the gauge algebra since $g^2\propto1/C_2^{(ad\hspace{0.1pt}j)}$. Other representations appearing in this tensor product will have larger values of $C_2^{(r)}$ and will lead to either weakly attractive, vanishing, or repulsive interactions that may eventually be gauge-algebra dependent. The interesting point is that the most attractive channel is that of a colour-singlet gluon pair, which should not depend on the considered gauge algebra and which is eventually able to form glueballs. So the glueball formation (or not) above deconfinement might well be a universal feature of YM theory; arguments favoring the existence of glueballs beyond $T_c$ have been given for example in [@brau]. We mention finally that, in the case of $\mathfrak{su}(N)$ gauge algebras, each channel of the tensor product $ad\hspace{0.1pt}j\otimes ad\hspace{0.1pt}j$ has been explicitly computed in [@buiss10]. Two channels lead to weak $N$-dependent interactions (with a $1/N$ colour factor) that may lead to some subleading $N$-dependent corrections. Summary and discussion {#conclu} ====================== To summarize, we have discussed two pictures of YM matter that allow to compute its thermodynamical properties for any gauge algebra. In the confined phase, the relevant degrees of freedom are glueballs, whose low-lying states can be separately described, while the high-lying states are modelled by a closed bosonic string Hagedorn spectrum. Such a spectrum exhibits a Hagedorn temperature, above which hadronic matter ceases to exist: The partition function of a glueball gas with Hagedorn spectrum is not defined above $T_h$, suggesting a phase transition to a deconfined regime. In the deconfined phase, YM thermodynamics should be the one of an interacting gluon gas. In the confined phase, the present model compares favorably with the recent pure gauge $\mathfrak{su}(3)$ lattice data of [@borsa] with a standard value $(9/4)\, 0.2$ GeV$^2$ for the adjoint string tension and the assumption $T_c=T_h$. This does not excludes that a better fit can be found with $T_h\gtrsim T_c$ as in [@meyer], or that the value $T_c=T_h$ is an artifact due to the simplicity of the model, especially near $T_c$. But, the success of equating $T_c$ and $T_h$ also suggests that the temperature range in which a metastable hadronic phase exists is quite small with the gauge algebra $\mathfrak{su}(3)$. Keeping the relation $T_c=T_h$ as well as the value of the adjoint string tension unchanged, predictions for the equation of state of YM theory with arbitrary gauge algebras have been given; it can be hoped that future lattice simulations will be able to confirm them (or not), at least is some cases of current interest like YM theory with $G_2$ gauge algebra. It is worth saying that identifying the critical temperature to the Hagedorn temperature leads to the possibility of estimating the gauge-algebra dependence of $T_c$. A relevant example is that, in the case of $\mathfrak{su}(N)$ gauge algebras, we are led to the prediction that $T_c[\mathfrak{su}(2)]/T_c[\mathfrak{su}(\infty)]=\sqrt{3}/2=0.866$, which can be favourably compared to the Polyakov-loop based approach [@braun] finding the value 0.898 for this last ratio. For a $\mathfrak{sp}(2)$ gauge algebra, we find $T_c[\mathfrak{sp}(2)]/T_c[\mathfrak{su}(\infty)]=\sqrt{5/6}=0.913$ while a comparable ratio of $0.969$ is found in [@braun]. Our framework implies that the thermodynamical observables are of $O((d-1)\times{\rm dim}(ad\hspace{0.1pt}j))$ above $T_c$ for a Yang-Mills theory in $d+1$ dimensions and a gauge algebra having ${\rm dim}(ad\hspace{0.1pt}j)$ generators. Consequently, these observables should of $O(1)$ when both $C=+$ and $-$ glueballs are present, *i.e.* for A$_{r\geq2}$, D$_{{\rm odd}-r\geq5}$, and E$_6$, and of $O(1/2)$ in the other cases. The pressure ratio $$\delta=\lim_{\eta\rightarrow 0}\frac{p(T_c+\eta)}{p(T_c-\eta)},$$ where $\eta$ is positive, is then generally of order $2(d-1){\rm dim}(ad\hspace{0.1pt}j)$, but of order $(d-1){\rm dim}(ad\hspace{0.1pt}j)$ for A$_{r\geq2}$, D$_{{\rm odd}-r\geq5}$, and E$_6$[^1]. More explicitly, $\delta=16$ for $\mathfrak{su}(3)$ in $3+1$ dimensions, a case for which the phase transition is known to be weakly first order. Some cases can be mentioned for which $\delta\ll 16$: $\mathfrak{su}(2)$ in $3+1$ dimensions and $\mathfrak{su}(2,3)$ in $2+1$ dimensions. It is tempting to say that such small gaps should lead to a second order phase transition. Although the argument seems quite naive, this is indeed the case: It is known from lattice simulations that the phase transition is of second order in those cases [@case]. Moreover, $\delta=15\approx16$ for $\mathfrak{su}(4)$ in $2+1$ dimensions, presumably leading to a (very) weakly first-order phase transition, as observed in [@case]. Moreover, $\delta\gg 16$ for $\mathfrak{su}(N>3)$ in $3+1$ dimensions, corresponding to a phase trantision more and more of first-order type for $\mathfrak{su}(N)$ when $N$ increases, in agreement with previous lattice results [@TcTh0]. It seems thus that our picture eventually leads to criterion allowing to guess the strength of the deconfining phase transition in YM theories. Note that, according to this criterion, any gauge algebra for Yang-Mills theory in $2+1$ and $3+1$ dimensions should lead to a first-order phase transition, $\mathfrak{su}(2)$ ($\mathfrak{su}(2,3)$) in $3+1$ $(2+1)$ dimensions excepted. Finally, these results can be linked to an already proposed argument, saying that the mismatch of the number of degrees of freedom above and below the phase transition is responsible for the weakly or strongly first-order character of the deconfinement phase transition [@G2; @holl; @braun]. 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--- abstract: 'Let $E$ be a rank $2$, degree $d$ vector bundle over a genus $g$ curve $C$. The loci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action are expressed as products of ${\operatorname{Quot}}$ schemes. Using virtual localization, the stable pairs invariants of $E$ are related to the virtual intersection theory of ${\operatorname{Quot}}E$. The latter theory is extensively discussed for an $E$ of arbitrary rank; the tautological ring of ${\operatorname{Quot}}E$ is defined and is computed on the locus parameterizing rank one subsheaves. In case $E$ has rank $2$, $d$ and $g$ have opposite parity, and $E$ is sufficiently generic, it is known that $E$ has exactly $2^g$ line subbundles of maximal degree. Doubling the zero section along such a subbundle gives a curve in the total space of $E$ in class $2[C]$. We relate this count of maximal subbundles with stable pairs/Donaldson-Thomas theory on the total space of $E$. This endows the residue invariants of $E$ with enumerative significance: they actually *count* curves in $E$.' address: 'Department of Mathematics, Brown University, Providence, RI 02912' author: - 'W. D. Gillam' title: | Maximal subbundles, Quot schemes,\ and curve counting --- Introduction {#section:introduction} ============ This note is concerned with the Gromov-Witten (GW), Donaldson-Thomas (DT), and Pandharipande-Thomas stable pairs (PT) residue invariants of the total space of a rank $2$ bundle $E$ over a smooth proper curve $C$ (see §\[section:curvecounting\] for a brief review). The latter invariants are well-understood from a computational perspective. In this paper I attempt to shed some light on the enumerative significance of such invariants and to explain the relationship between sheaf theoretic curve counting on $E$ and the virtual intersection theory of ${\operatorname{Quot}}$ schemes of symmetric products of $E$. In particular, in §\[section:maximalsubbundles\], I explain how to relate the “count" of maximal subbundles of $E$ (which belongs properly to the theory of stable bundles on curves) with the DT/PT theory of $E$. Recall that the ${\operatorname{Quot}}$ scheme of a trivial rank $n$ bundle on a smooth proper curve $C$ may be viewed as a compactification of the space of maps from $C$ to a Grassmannian. The relationship between the virtual intersection theory of this ${\operatorname{Quot}}$ scheme and various “Gromov invariants" has been studied by many authors [@PR], [@Ber], [@BDW], [@MO], culminating in the theory of *stable quotients* [@MOP] where the curve $C$ is also allowed to vary, as it is in GW theory. The relationship between GW invariants of Grassmannians and counts of subbundles of maximal degree has also been studied by several authors [@Hol], [@LN], [@OT]. These connections, however, are rather difficult to make, and one must take a circuitous route to link maximal subbundle counts to GW invariants. Since counting maximal subbundles is an inherently sheaf-theoretic problem, it is reasonable to suspect that the most direct connections with curve counting should be made through the sheaf theoretic curve counting theories of Donaldson-Thomas [@Tho], [@MNOP] and Pandharipande-Thomas [@PT]. In general, the relationship between the PT theory of $E$ and virtual intersection theory on the various ${\operatorname{Quot}}{\operatorname{Sym}}^n E$ is quite subtle; we will treat the general case in a separate paper [@Gil2]. In the present article, we will focus on the case of PT residue invariants in homology class $2[C]$ (twice the class of the zero section of $E$), where the most complete results can be obtained. In particular, we will see that PT residue invariants of $E$ in class $2[C]$ are completely determined by virtual intersection numbers on ${\operatorname{Quot}}{\mathcal O}_C$ (symmetric products of $C$) and on the ${\operatorname{Quot}}$ scheme ${\operatorname{Quot}}^1 E$ parameterizing rank one subsheaves of $E$. Our methods could even be used to compute the *full* PT theory of $E$ (including descendent invariants involving odd cohomology classes from $C$) in class $2[C]$, though we do not provide the details here. Once we have in hand the relationship between PT theory and virtual intersection theory of ${\operatorname{Quot}}$ schemes described above, we will be all the more interested in the latter. In §\[section:tautologicalclasses\], we suggest packaging this theory into a “tautological ring." In the course of proving the Vafa-Intriligator formula, Marian and Oprea [@MO] explained that this entire theory can be reduced, via virtual localization, to intersection theory on symmetric products of $C$, hence it can be treated as “known." On the other hand, it is quite painful in practice to write down manageable formulas for such invariants and it seemed to me to be overkill to appeal to the general results of [@MO] for the invariants actually needed in our study. Instead, we give (§\[section:quotschemes\]) a direct computation of the virtual intersection theory of the ${\operatorname{Quot}}$ scheme ${\operatorname{Quot}}^1 V$ parameterizing rank one subsheaves of a vector bundle $V$ on $C$ as follows. The universal such subbundle $S$ is a line bundle on ${\operatorname{Quot}}^1 V \times C$, hence its dual yields a map $S^\lor : {\operatorname{Quot}}^1 V \to {\operatorname{Pic}}C$. In case $V={\mathcal O}_C$ this is the “usual" map ${\operatorname{Sym}}^d C \to {\operatorname{Pic}}^d C$. Just as in the case of the “usual" map, the map $S^\lor$ is a projective space bundle when $S^\lor$ has sufficiently large degree $d$, and, as in the “usual" case, one can compute the desired (virtual) intersection theory by descending induction on $d$. The only new ingredient is the use of the virtual class; otherwise the computation is not significantly different from what MacDonald does in [@Mac]. We include a review of the DT/GW/PT correspondence for the residue invariants of $E$ in §\[section:correspondence\]. In principle, our computations could be used to verify this explicitly in degree $2[C]$, though we content ourselves with checking some special cases in §\[section:computations\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- Much of this note is based on conversations with Matt Deland and Joe Ross which took place at Columbia in the spring of 2009. I thank Ben Weiland for helpful discussions and Davesh Maulik for spurring my interest in PT theory. This research was partially supported by an NSF Postdoctoral Fellowship. Conventions {#conventions .unnumbered} ----------- We work over the complex numbers throughout. All schemes considered are disjoint unions of schemes of finite type over ${\mathbb{C}}$. We write ${{\bf Sch}}$ for the category of such schemes. Set $T := \mathbb{G}_m = {\operatorname{Spec}}{\mathbb{C}}[t,t^{-1}]$. We will often consider an affine morphism $f : Y \to X$, in which case a $T$ action on $Y$ making $f$ equivariant for the trivial $T$ action on $X$ is a ${\mathbb{Z}}$-grading on $f_* {\mathcal O}_Y$ as an ${\mathcal O}_X$ algebra (so the corresponding direct sum decomposition is in the category of ${\mathcal O}_X$ modules). A $T$ equivariant ${\mathcal O}_Y$ module is then the same thing as a graded $f_* {\mathcal O}_Y$ module. Throughout, if $\pi : E \to X$ is a vector bundle on a scheme $X$, we use the same letter to denote its (locally free coherent) sheaf of sections, so $E = {\operatorname{Spec}}_X {\operatorname{Sym}}^* E^\lor$. The sheaf ${\operatorname{Sym}}^* E^\lor$ has an obvious ${\mathbb{Z}}$-grading supported in nonnegative degrees; we call the corresponding $T$ action the *scaling action*. If $Z \subseteq E$ is a subscheme, then by abuse of notation, we will also use $\pi$ to denote the restriction of $\pi$ to $Z$. When $T$ acts trivially on $X$, the $T$ equivariant cohomology ${\operatorname{H}}^*_T(X)$ is identified with ${\operatorname{H}}^*(X)[t]$; a $T$ equivariant vector bundle $V$ on $X$ decomposes into eigensubbundles $V=\oplus_n V_n$ where $T$ acts on $V_n$ through the composition of the character $\lambda \mapsto \lambda^n$ and the scaling action. For $n \neq 0$, the $T$ equivariant Euler class $e_T(V_n)$ is invertible in the localized equivariant cohomology ${\operatorname{H}}^*_T(X)_t = {\operatorname{H}}^*(X)[t,t^{-1}]$. The eigensubbundle decomposition induces an analogous decomposition $K_T(X)=\oplus_{{\mathbb{Z}}} K(X)$ of the $T$ equivariant vector bundle Grothendieck group of $X$ and $e_T$ descends to a group homomorphism $$\begin{aligned} e_T : \oplus_{{\mathbb{Z}}\setminus \{ 0 \} } K(X) & \to & ({\operatorname{H}}^*_T(X)^\times, \cdot) \\ V-W & \mapsto & e_T(V)e_T(W)^{-1} \end{aligned}$$ which plays a fundamental role in (virtual) localization. Tensor products are usually denoted by juxtaposition and exponents. Notation for pullback of sheaves is often omitted, so $E$ sometimes means $f^* E$ for a morphism $f$ which should be clear from context. Quot schemes {#section:quotschemes} ============ Let $C$ be a smooth projective curve, $E$ a vector bundle on $C$. Recall [@Gro] that the *quotient scheme* ${\operatorname{Quot}}E \in {{\bf Sch}}$ represents the presheaf on ${{\bf Sch}}$ which associates to $Y$ the set of exact sequences $$0 \to S \to \pi_2^*E \to Q \to 0$$ on $Y \times C$ with $Q$ (hence $S$) flat over $Y$ (up to isomorphism). In particular, a point of ${\operatorname{Quot}}E$ is a SES $$0 \to S \to E \to Q \to 0$$ of sheaves on $C$. The restriction maps for this presheaf along $f : Y \to Y'$ are defined by pulling back via $f \times {\operatorname{Id}}_C$. The identity map of ${\operatorname{Quot}}E$ corresponds to an exact sequence $$0 \to S \to \pi_2^*E \to Q \to 0$$ on ${\operatorname{Quot}}E \times C$ called the *universal exact sequence*. Let ${\operatorname{Quot}}^{r,e} E$ denote the component of ${\operatorname{Quot}}E$ where $S$ has rank $r$ and degree $e$. Note that $r,e$ determine the Hilbert polynomial of $Q$ via Riemann-Roch. ${\operatorname{Quot}}^{r,e}$ is a projective scheme. When $r=1$ we drop it from the notation and refer to ${\operatorname{Quot}}^e E$ as the *rank one Quot scheme*. This will be the main object of interest in the later sections of the paper. A subsheaf $S$ of a vector bundle on a curve is torsion free since it is contained in a torsion free sheaf, hence it is locally free because ${\mathcal O}_C$ is a sheaf of PIDs. By flatness, the universal $S$ is a locally free sheaf on ${\operatorname{Quot}}E \times C$. Abel-Jacobi maps {#section:abeljacobimaps} ---------------- Let $E$ be a vector bundle on a smooth proper curve $C$. The determinant $\land^r S$ of the universal $S$ is an invertible sheaf on ${\operatorname{Quot}}^{r,e} E \times C$ and hence defines an Abel-Jacobi map $${a}_e : {\operatorname{Quot}}^{r,e} E \to {\operatorname{Pic}}^e C$$ to the Picard scheme of degree $e$ line bundles on $C$. Indeed, recall that ${\operatorname{Pic}}C = \coprod_e {\operatorname{Pic}}^e C$ represents the presheaf (of abelian groups under tensor product) $$\begin{aligned} {{\bf Sch}}^{\rm op} & \to & {{\bf Ab}}\\ Y & \mapsto & \frac{ \{ \; {\rm line \; bundles \; on } \; Y \times C \; \} }{ \{ \; {\rm line \; bundles \; pulled \; back \; from} \; Y \; \} } . \end{aligned}$$ Fix a point $x \in C$. As in [@ACGH] IV.2, we fix, once and for all, a *Poincaré* line bundle $L$ on ${\operatorname{Pic}}C \times C$ such that the restriction of $L$ to ${\operatorname{Pic}}C \times \{ x \}$ is trivial. The pair $({\operatorname{Pic}}C, L)$ has the following universal property, which characterizes it up to tensoring $L$ with a line bundle pulled back from ${\operatorname{Pic}}C$: for any scheme $Y$ and any line bundle $L'$ on $Y \times C$, there is a unique morphism $f: Y \to {\operatorname{Pic}}C$ such that $L' = (f \times {\operatorname{Id}}_C)^* L \otimes \pi_1^* M$ for some line bundle $M$ on $Y$. Let $L_e$ denote the restriction of the Poincaré bundle $L$ to ${\operatorname{Pic}}^e C$; by definition of ${\operatorname{Pic}}^e C$, $L_e|_{\{ P \} \times C}$ is a degree $e$ line bundle on $C$ for each point $P$ of ${\operatorname{Pic}}^e C$. Each of the components ${\operatorname{Pic}}^e C$ is non-canonically isomorphic to the torus $${\operatorname{H}}^1(C,{\mathcal O}_C)/{\operatorname{H}}^1(C,{\mathbb{Z}}) = {\operatorname{Ker}}( c_1 : {\operatorname{H}}^1(C,{\mathcal O}_C^*) \to {\operatorname{H}}^2(C,{\mathbb{Z}}) ),$$ so its cohomology is an exterior algebra on ${\operatorname{H}}^1$: $$\begin{aligned} {\operatorname{H}}^*({\operatorname{Pic}}^e C,{\mathbb{Z}}) & = & \land^* {\operatorname{H}}^1({\operatorname{Pic}}^e C,{\mathbb{Z}}) \\ & \cong & \land^* {\mathbb{Z}}^{2g}. \end{aligned}$$ If $E={\mathcal O}_C$, then ${a}_{-n}$ is just the usual Abel-Jacobi map $$\begin{aligned} {\operatorname{Sym}}^n C & \to & {\operatorname{Pic}}^n C \\ D & \mapsto & {\mathcal O}(D),\end{aligned}$$ once we identify ${\operatorname{Sym}}^n C$ with ${\operatorname{Quot}}^{-n} E$ via $$D \mapsto ({\mathcal O}_C(-D) \hookrightarrow {\mathcal O}_C).$$ The situation is very similar when $E$ is any invertible sheaf. In the case of the *rank one* Quot scheme ${\operatorname{Quot}}^e E$, the fiber of ${a}_e$ over a line bundle $L \in {\operatorname{Pic}}^e C$ is the projective space $${\mathbb{P}}{\operatorname{Hom}}(L,E)$$ because any nonzero map $L \to E$ must be injective. Under the inclusion $${\mathbb{P}}{\operatorname{Hom}}(L,E) \hookrightarrow {\operatorname{Quot}}^e E,$$ the universal subbundle $S \to \pi_2^* E$ on ${\operatorname{Quot}}^e E \times C$ restricts to $$\begin{aligned} {\mathcal O}(-1) \boxtimes L & \to & \pi_2^* E \\ f \otimes l & \mapsto & f(l) \end{aligned}$$ on ${\mathbb{P}}{\operatorname{Hom}}(L,E) \times C$. It is well-known that the Abel-Jacobi map ${a}_n : {\operatorname{Sym}}^n C \to {\operatorname{Pic}}^n C$ is a ${\mathbb{P}}^{n-g}$ bundle for $n \geq 2g-2$. In fact, a similar statement holds for ${\operatorname{Quot}}^e E$, as we now prove. For any coherent sheaf $E$ on a smooth proper curve $C$, there is an integer $d_0$ such that ${\operatorname{H}}^1(C, E \otimes L)=0$ for every line bundle $L$ on $C$ of degree at least $d_0$. This is an easy consequence of semicontinuity and quasi-compactness of the Jacobian ${\operatorname{Pic}}^0 C$. Fix a degree one (hence ample) invertible sheaf ${\mathcal O}_C(1)$ on $C$. Then for any fixed degree zero line bundle $L_0$ on $C$, there is a $d_0$ such that ${\operatorname{H}}^1(C,E \otimes L_0(n))=0$ for all $n \geq d_0$. By semicontinuity, ${\operatorname{H}}^1(C,E \otimes L(n))=0$ for all $L$ in a neighborhood of $L_0$ in ${\operatorname{Pic}}^0 C$, so by quasi-compactness of ${\operatorname{Pic}}^0 C$ we can choose $d_0$ large enough that ${\operatorname{H}}^1(C,E \otimes L(n))=0$ for all $n \geq d_0$ for all degree zero line bundles $L$. As $L$ ranges over all degree zero line bundles, $L(n)$ ranges over all degree $n$ line bundles. \[thm:quotforsmalle\] Let $E$ be a rank $N$, degree $d$ vector bundle on a smooth proper curve $C$, ${\operatorname{Quot}}^e E$ the Quot scheme of rank one, degree $e$ subsheaves of $E$. For all sufficiently negative $e$, the Abel-Jacobi map ${a}_e : {\operatorname{Quot}}^e E \to {\operatorname{Pic}}^e C$ is a ${\mathbb{P}}^{m-1}$ bundle for $m = (1-g)N+d - eN$. Let $L_e$ the universal degree $e$ line bundle on ${\operatorname{Pic}}^e E \times C$. By flatness, cohomology and base change, the lemma, and Grauert’s Criterion, ${\operatorname{R}}^1 \pi_{1*} (L_e^\lor \otimes \pi_2^* E)=0$ and $$V := \pi_{1*} (L_e^\lor \otimes \pi_2^* E)$$ is a vector bundle on ${\operatorname{Pic}}^e C$ of rank $m$ whenever $e$ is sufficiently negative. Let $p : {\mathbb{P}}V \to {\operatorname{Pic}}^e C$ be the projection, $i : {\mathcal O}_{{\mathbb{P}}V}(-1) \to p^*V$ the tautological inclusion of bundles on ${\mathbb{P}}V$. The “evaluation" map $$\begin{aligned} e : \pi_1^* {\mathcal O}_{{\mathbb{P}}V}(-1) \otimes (p \times {\operatorname{Id}}_C)^* L_e & \to & \pi_2^* E \\ f \otimes l & \mapsto & \iota f(l) \end{aligned}$$ is a map of bundles on ${\mathbb{P}}V \times C$ with the correct discrete invariants, hence it defines a morphism $f : {\mathbb{P}}V \to {\operatorname{Quot}}^e E$ with $(f \times {\operatorname{Id}}_C)^*j=e$, where $j : S \to \pi_2^*E$ is the universal subsheaf on ${\operatorname{Quot}}^e E \times C$. I claim $f$ is an isomorphism. To construct its inverse, recall that, by the universal property of ${\mathbb{P}}V$, lifting the Abel-Jacobi map ${a}_e : {\operatorname{Quot}}^e E \to {\operatorname{Pic}}^e E$ to a map $g : {\operatorname{Quot}}^e \to {\mathbb{P}}V$ is the same thing as giving a rank one subbundle of the bundle ${a}_e^* V$ on ${\operatorname{Quot}}^e E$ (under the correspondence, this rank one subbundle will then be $g^*i$). By flatness and cohomology and base change in the diagram $$\xymatrix@C+15pt{ {\operatorname{Quot}}^e E \times C \ar[r]^{{a}_e \times {\operatorname{Id}}_C} \ar[d]_{\pi_1} & {\operatorname{Pic}}^e C \times C \ar[d]^{\pi_1} \\ {\operatorname{Quot}}^e E \ar[r]^{{a}_e} & {\operatorname{Pic}}^e C, }$$ we have $$\begin{aligned} {a}_e^* V & = & {a}_e^* \pi_{1*}(L_e^\lor \otimes \pi_2^* E) \\ &=& \pi_{1*}( ({a}_e \times {\operatorname{Id}}_C)^* (L_e^\lor \otimes \pi_2^*E)) .\end{aligned}$$ By the universal property of ${\operatorname{Pic}}^e C$, there is a line bundle $M$ on ${\operatorname{Quot}}^e E$ such that $$\begin{aligned} \label{lb} ({a}_e \times {\operatorname{Id}}_C)^*L_e \otimes \pi_1^* M = S.\end{aligned}$$ We then have $$M^\lor \otimes \pi_{1*}(({a}_e \times {\operatorname{Id}}_C)^*L_e^\lor \otimes \pi_2^* E) = \pi_{1*}(S^\lor \otimes \pi_2^*E)$$ by the projection formula. Tensoring the universal inclusion $j:S \hookrightarrow \pi_2^*E$ on ${\operatorname{Quot}}^e E \times C$ with $S^\lor$ and applying $\pi_{1*}$, yields a rank one subsheaf $${\mathcal O}_{{\operatorname{Quot}}^e E} \hookrightarrow \pi_{1*}(S^\lor \otimes \pi_2^* E).$$ In fact it is a sub*bundle*, because otherwise $\iota$ would fail to be injective at some point of ${\operatorname{Quot}}^e E$, which is absurd. Tensoring with $M$, we obtain a rank one subbundle $$k := \pi_{1*}(j \otimes {\operatorname{Id}}_{S^\lor}) \otimes {\operatorname{Id}}_M : M \hookrightarrow a_e^* V,$$ and hence a morphism $g : {\operatorname{Quot}}^e E \to {\mathbb{P}}V$ with the property $g^* i = k$. The proof is completed by checking that $fg$ is the identity of ${\operatorname{Quot}}^e E$ and $gf$ is the identity of ${\mathbb{P}}V$, which is straightforward. For example, checking that $fg={\operatorname{Id}}$ amounts to showing that $(g \times {\operatorname{Id}}_C)^* j=j$. We have $$\begin{aligned} (g \times {\operatorname{Id}}_C)^* j &=& (g \times {\operatorname{Id}}_C)^*(f \times {\operatorname{Id}}_C)^* j \\ &=& (g \times {\operatorname{Id}}_C)^* e, \end{aligned}$$ and $$\begin{aligned} (g \times {\operatorname{Id}}_C)^* (\pi_1^* {\mathcal O}_{{\mathbb{P}}V}(-1) \otimes (p \times {\operatorname{Id}}_C)^* L_e) &=& g^* {\mathcal O}_{{\mathbb{P}}V}(-1) \otimes (a_e \times {\operatorname{Id}}_C)^* L_e \\ & = & \pi_1^*M \otimes (a_e \times {\operatorname{Id}}_C)^* L_e \\ & = & S.\end{aligned}$$ Furthermore, if $s$ is a local section of $S$ corresponding to a local section $m \otimes l$ under the last isomorphism, then by definition of the evaluation morphism $e$, we compute $$\begin{aligned} (g \times {\operatorname{Id}}_C)^*e(s) & = & (g^*i)(m)(l) \\ &=& k(m)(l) \\ &=& j(s), \end{aligned}$$ where the last equality is just unravelling the definition of $k$. Although the rank one Quot scheme ${\operatorname{Quot}}^e E$ is always smooth in genus zero (it is just the projective space ${\mathbb{P}}{\operatorname{Hom}}({\mathcal O}_{{\mathbb{P}}^1}(e),E)$), it can be singular in general. For example, if $C$ has genus $2$ and $E= {\mathcal O}_C \oplus {\mathcal O}_C$, then the fiber of the Abel-Jacobi map $$a_{-2} : {\operatorname{Quot}}^{-2} E \to {\operatorname{Pic}}^{-2} C$$ over the dual $L^\lor$ of the $g^1_2$ on $C$ is $${\mathbb{P}}{\operatorname{Hom}}_C(L^\lor,E) = {\mathbb{P}}( {\operatorname{H}}^0(C,L)^{\oplus 2}) \cong {\mathbb{P}}^3$$ (the other fibers are ${\mathbb{P}}^1$’s). The singular locus of ${\operatorname{Quot}}^{-2} E$ is a quadric ${\mathbb{P}}^1 \times {\mathbb{P}}^1$ in this ${\mathbb{P}}^3$ fiber corresponding to subsheaves $L^\lor \hookrightarrow E$ whose inclusion factors through an inclusion ${\mathcal O}_C \hookrightarrow E$. The quotient of such a subsheaf is of the form ${\mathcal O}_C \oplus {\mathcal O}_{P+Q}$ and the tangent space at such a point is $4$ dimensional, with one dimensional obstruction space $${\operatorname{Ext}}^1(L^\lor,{\mathcal O}_C \oplus {\mathcal O}_{P+Q}) = {\operatorname{H}}^1(C,L)$$ (the obstruction space is easily seen to vanish away from this locus). Stratification {#section:stratification} -------------- Let $E$ be a vector bundle on a smooth proper curve $C$. The ${\operatorname{Quot}}$ scheme ${\operatorname{Quot}}^{r,e} E$ can be stratified as follows. Let ${\operatorname{Quot}}^{r,e}_0 E$ be the open subscheme of ${\operatorname{Quot}}^{r,e} E$ parameterizing sub*bundles*. Define a map $$\begin{aligned} \iota_n : {\operatorname{Quot}}^{r,e+rn} E \times {\operatorname{Sym}}^n C & \to & {\operatorname{Quot}}^{r,e} E \\ (S \hookrightarrow E, D) & \mapsto & (S(-D) \hookrightarrow S \hookrightarrow E) \end{aligned}$$ (this obviously makes sense on the level of universal families). Each map $\iota_n$ is a closed embedding. We obtain a stratification $${\operatorname{Quot}}^{r,e} E = \coprod_{n \geq 0} \iota_n[ {\operatorname{Quot}}^{r,e+rn}_0 E \times {\operatorname{Sym}}^n C]$$ by locally closed subschemes. It is finite since ${\operatorname{Quot}}^{r,e+rn} E$ is empty for large $n$. In the rank one case, this stratification respects the Abel-Jacobi maps of in the sense that $a_e \iota_n = a_{e+n} \otimes a_{-n},$ where $\otimes$ is multiplication for the abelian variety ${\operatorname{Pic}}C$. That is, the diagram $$\xymatrix{ {\operatorname{Quot}}^{e+n} E \times {\operatorname{Quot}}^{-n} {\mathcal O}_C \ar[d]_{a_{e+n} \times a_{-n}} \ar[r]^-{\iota_n} & {\operatorname{Quot}}^e E \ar[d]^{a_e} \\ {\operatorname{Pic}}^{e+n} C \times {\operatorname{Pic}}^{-n} C \ar[r]^-{\otimes} & {\operatorname{Pic}}^e C }$$ commutes. Obstruction theory {#section:virtualclass} ------------------ Let $E$ be a vector bundle on a smooth proper curve $C$ as usual and let $\pi: {\operatorname{Quot}}E \times C \to {\operatorname{Quot}}E$ be the projection. The goal of this section is to show that there is a map $$\begin{aligned} \label{POTonQuot} {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}\pi_* {\curly Hom}(S,Q), {\mathcal O}_{{\operatorname{Quot}}E}) \to {\mathbb{L}}_{{\operatorname{Quot}}E} \end{aligned}$$ to the cotangent complex of ${\operatorname{Quot}}E$ (see [@Ill]) defining a perfect obstruction theory (POT) on ${\operatorname{Quot}}E$ in the sense of Behrend-Fantechi [@BF]. Our proof basically follows the proof of Theorem 1 in [@MO], but we work with the Behrend-Fantechi formalism as opposed to that of Li-Tian.[^1] Following [@MO], we produce an embedding from ${\operatorname{Quot}}E$ into a smooth scheme[^2] so that ${\operatorname{Quot}}E$ is realized as the zero section of a vector bundle with the expected rank. We then simply identify the paradigm POT on this zero locus (whose virtual class is cap product with the Euler class of the bundle; see Section 6 in [@BF]) with the map . Before carrying this out, I should mention that this is probably not “the best" way to construct this POT. A more intrinsic construction of this POT appears in my paper [@Gil1]. The idea there is to use the “reduced Atiyah class" of the universal quotient, which is a map $$S \to \pi_1^* {\mathbb{L}}_{{\operatorname{Quot}}E} \otimes Q$$ in the derived category $D({\operatorname{Quot}}E \times C)$. Tensoring with the derived dual $Q^\lor$ of (the perfect complex) $Q$ and tracing, this yields a $D({\operatorname{Quot}}E \times C)$ morphism $$Q^\lor \otimes^{{\operatorname{\bf L}}} S \to \pi_1^* {\mathbb{L}}_{{\operatorname{Quot}}E}.$$ Using Serre duality for $\pi_1$, one can show that this is the same thing as a map . It is shown in [@Gil1] that this is a POT. It can be shown that the POT we will construct here coincides with the POT constructed in [@Gil1], though we will omit this argument in the interest of brevity. We have chosen to present this version of the construction of the POT to keep the paper self-contained. Let $j : D \hookrightarrow C$ be an effective divisor in $C$ of degree $n$ and consider the embedding $$\begin{aligned} \iota_D : {\operatorname{Quot}}^{r,e} E & \hookrightarrow & {\operatorname{Quot}}^{r,e-rn} E \\ (S \hookrightarrow E) & \mapsto & (S(-D) \hookrightarrow E) \end{aligned}$$ obtained by restricting the map $\iota_n$ from the previous section to $${\operatorname{Quot}}^{r,e} E \times \{ D \} \cong {\operatorname{Quot}}^{r,e} E.$$ To save notation, set $$\begin{aligned} Z & := & {\operatorname{Quot}}^{r,e} E \\ X & := & {\operatorname{Quot}}^{r,e-rn} E \end{aligned}$$ and write $0 \to {\overline}{S} \to \pi_2^* E \to {\overline}{Q} \to 0$ for the universal SES on $X \times C$. By definition of $\iota_D$, we have a commutative diagram with exact rows $$\xymatrix{ 0 \ar[r] & S(-D) \ar@{^(->}[d] \ar[r] & \pi_2^* E \ar@{=}[d] \ar[r] & {\overline}{Q} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & S \ar[r] & \pi_2^* E \ar[r] & Q \ar[r] & 0}$$ on $Z \times C$, where the top row is the pullback of the universal sequence on $X \times C$ via $\iota_D \times {\operatorname{Id}}_C$ (we should probably write ${\overline}{Q}|_{Z \times C}$ instead of ${\overline}{Q}$) and the bottom row is the universal sequence on $Z \times C$. By abuse of notation, we also write $j$ for $j$ times the identity map of either ${\operatorname{Quot}}$ scheme, so the cokernel of the left vertical map can be written $j_* j^* S$. By the Snake Lemma, the right vertical arrow fits into an exact sequence $$\begin{aligned} \label{quotientSES} 0 \to j_* j^* S \to {\overline}{Q} \to Q \to 0. \end{aligned}$$ \[lem:smoothness\] For a divisor $D \subset C$ of sufficiently large degree $n$, the embedding $\iota_D$ factors through the smooth locus of $X = {\operatorname{Quot}}^{r,e-rn} E$. By compactness of $Z = {\operatorname{Quot}}^{r,e} E$ and semicontinuity, we can choose a $D$ with degree large enough that $$\begin{aligned} \label{firstvanishing} {\operatorname{H}}^1(C, {\curly Hom}(S,Q)|_z (D))=0 \end{aligned}$$ for every point $z$ of $Z$. We will use the formal criterion for smoothness, so we must show that we can solve the lifting problem $$\xymatrix{ Y_0 \ar@{^(->}[d] \ar[r] & Z \ar@{^(->}[r]^{\iota_D} & X \ar[d] \\ Y \ar[rr] \ar@{.>}[rru] & & {\operatorname{Spec}}{\mathbb{C}}}$$ whenever $Y_0 \hookrightarrow Y$ is a closed embedding of affine schemes with square zero ideal $I$. By basic obstruction theory of quotients, it is equivalent to show that the obstruction to such a lift in $${\operatorname{Ext}}^1_{Y_0 \times C}({\overline}{S}, {\overline}{Q} \otimes \pi_1^* I)$$ vanishes, so it suffices to prove that this whole vector space is zero. The functor ${\operatorname{Hom}}_{Y_0 \times C}$ is the composition of three functors: $${\operatorname{Hom}}_{Y_0 \times C} = \Gamma \circ \pi_{1*} \circ {\curly Hom}_{Y_0 \times C},$$ so to prove the vanishing of $${\operatorname{Ext}}^1_{Y_0 \times C}({\overline}{S},{\overline}{Q} \otimes \pi_1^* I) = {\operatorname{R}}^1 {\operatorname{Hom}}_{Y_0 \times C}({\overline}{S},{\overline}{Q} \otimes \pi_1^* I),$$ it suffices (by the Grothendieck spectral sequence) to prove the vanishing of the three vector spaces $$\begin{array}{l} {\operatorname{H}}^1(Y_0, \pi_{1*} {\curly Hom}({\overline}{S},{\overline}{Q} \otimes \pi_1^* I) \\ {\operatorname{H}}^0(Y_0, \pi_{1*} {\curly Ext}^1({\overline}{S}, {\overline}{Q} \otimes \pi_1^* I)) \\ {\operatorname{H}}^0(Y_0, {\operatorname{R}}^1 \pi_{1*} {\curly Hom}({\overline}{S},{\overline}{Q}) \otimes I)). \end{array}$$ The first vanishes because $Y_0$ is affine and the second vanishes because ${\overline}{S}$ is locally free. For the third vanishing, we will show that ${\operatorname{R}}^1 \pi_{1*} {\curly Hom}({\overline}{S},{\overline}{Q})=0$. By cohomology and base change and the assumption that $Y_0 \to X$ factors through $\iota_D$, we might as well assume $Y_0=X$, so ${\overline}{S}=S(-D)$. Using the vanishing together with cohomology and base change (and flatness of ${\curly Hom}(S,Q)$), we first observe that $${\operatorname{R}}^1 \pi_{1*} {\curly Hom}({\overline}{S},Q) = {\operatorname{R}}^1 \pi_{1*} {\curly Hom}(S,Q)(D) = 0.$$ Next, we tensor the sequence $\eqref{quotientSES}$ with $S^\lor(D)$ and apply $\pi_{1*}$ to get an exact sequence $$\cdots \to {\operatorname{R}}^1 \pi_{1*} j_*j^*(S \otimes S^\lor(D)) \to {\operatorname{R}}^1 \pi_{1*} {\curly Hom}({\overline}{S},{\overline}{Q}) \to {\operatorname{R}}^1 \pi_{1*} {\curly Hom}({\overline}{S},Q) \to 0.$$ The term on the left vanishes because $\pi_1 j$ has relative dimension zero, so we obtain the desired vanishing. The embedding $\iota_D$ identifies $Z$ with the closed subscheme of $X$ defined by the degeneracy locus where the map $$\pi_{1*} {\overline}{S}|_{X \times D} \to \pi_{1*} \pi_2^* E|_{X \times D}$$ of vector bundles on $X$ is zero. Indeed, $\iota_D$ identifies the presheaf ${\operatorname{Quot}}^{r,e} E$ with the subpresheaf of ${\operatorname{Quot}}^{r,e-rn} E$ which associates to a scheme $Y$ the set of $$0 \to {\overline}{S} \to \pi_2^* E \to {\overline}{Q} \to 0$$ in $({\operatorname{Quot}}^{r,e-rn} E)(Y)$ such that the inclusion $j : {\overline}{S} \to \pi_2^* E$ factors (necessarily uniquely) as below. $$\begin{aligned} \label{factorization} \xymatrix{ {\overline}{S} \ar@{^(->}[r]^-j \ar[rd] & \pi_2^* E \\ & {\overline}{S}(Y \times D) \ar@{.>}[u]} \end{aligned}$$ If ${\operatorname{Spec}}(A \otimes B) \hookrightarrow X \times C$ is a basic affine open on which ${\overline}{S}$ and $\pi_2^* E$ are trivialized and $D \subset C$ is given by $f \in B$, then $j$ can be viewed as a monomorphism $$j : (A \otimes B)^r \to (A \otimes B)^N$$ of free $(A \otimes B)$ modules. For any $A$ algebra $\phi : A \to C$, the morphism ${\operatorname{Spec}}C \to X$ factors through $\iota_D$ iff there is a factorization with $Y={\operatorname{Spec}}C$ iff there is a factorization $$\xymatrix{ (C \otimes B)^r \ar@{^(->}[r]^-{{\overline}{j}} \ar[rd]_{(1 \otimes f) \cdot {\operatorname{Id}}_r} & (C \otimes B)^N \\ & (C \otimes B)^r \ar@{.>}[u]}$$ of $C \otimes B$ module maps, where ${\overline}{j} = ({\operatorname{Spec}}\phi \times {\operatorname{Id}}_B)^*j.$ On the other hand, such a factorization exists iff $${\overline}{j}[(C \otimes B)^r] \subseteq (1 \otimes f) \cdot (C \otimes B)^N$$ iff ${\overline}{j} \otimes_{C \otimes B} C \otimes B/f=0$. But $$\begin{aligned} {\overline}{j} \otimes_{C \otimes B} C \otimes B/f &=& {\overline}{j}|_{{\operatorname{Spec}}C \times D} \\ &=& ({\operatorname{Spec}}\phi \times {\operatorname{Id}}_D)^*(j|_{X \times D}),\end{aligned}$$ which vanishes iff ${\operatorname{Spec}}\phi \times {\operatorname{Id}}_D$ factors through the zero locus of $j|_{X \times D}$ iff ${\operatorname{Spec}}C \to X$ factors through the zero locus of $\pi_{1*} (j|_{X \times D})$. If $E$ has rank one and degree $d$, then for any $n \in {\mathbb{Z}}_{\geq 0}$, there is an the identification ${\operatorname{Sym}}^n C \cong {\operatorname{Quot}}^{d-n} E$ which identifies $S^\lor$ with ${\mathcal O}(D)$, where $D \subseteq {\operatorname{Sym}}^n C \times C$ is the universal divisor. When $D=Q$ is a point of $C$, the embedding $$\begin{aligned} \iota_Q : {\operatorname{Sym}}^{n-1} C & \hookrightarrow & {\operatorname{Sym}}^n C \\ P_1+ \cdots+P_{n-1} & \to & P_1+\cdots+P_{n-1}+Q \end{aligned}$$ identifies ${\operatorname{Sym}}^{n-1} C \subset {\operatorname{Sym}}^n C$ with the zero locus of a section of the restriction of ${\mathcal O}(D)$ to ${\operatorname{Sym}}^n C \times \{Q\} = {\operatorname{Sym}}^n C$. Of course, $\pi_2^* E|_{X \times D} = \pi_2^* E|_D$ is a trivial bundle of rank $N$ equal to the rank of $E$, so $\pi_{1*} \pi_2^* E|_D$ is a trivial bundle of rank $nN$ ($\pi_{1*} {\mathcal O}_{X \times D}$ is a free ${\mathcal O}_X$ module of rank $n = \deg D$). The degeneracy locus $Z$ may therefore be viewed as the zero locus of a section of the bundle $V := (\pi_{1*} {\overline}{S}^\lor|_{X \times D})^N$ on $X$. In general, if $Z$ is the zero locus of a vector bundle $V$ on a smooth scheme $X$ (smooth in a neighborhood of $Z$ is good enough), then there is a map $$[V^\lor|_Z \to \Omega_X|_Z] \to {\mathbb{L}}_Z$$ (the complex on the left is supported in degrees $-1,0$) defining a POT on $Z$ whose associated virtual class is just the Chow class $e(V) \cap [X]$ supported on $Z$. This is a special case of the discussion in Section 6 of [@BF]. In the situation of this section, we have: The complexes $[V^\lor|_Z \to \Omega_X|_Z]$ and ${\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}\pi_{1*} S^\lor Q, {\mathcal O}_Z)$ on $Z$ are naturally quasi-isomorphic. It is equivalent to prove that the dual complex $[TX|_Z \to V|_Z]$ is naturally quasi-isomorphic to ${\operatorname{\bf R}}\pi_{1*} S^\lor Q$. By basic obstruction theory, we have an identification $TX = \pi_{1*} {\overline}{S}^\lor {\overline}{Q}$. Following through this identification and the definition of the boundary map in the paradigm POT, we see that the boundary map $TX|_Z \to V$ is identified with $\pi_{1*}$ of the adjunction map $${\overline}{S}^\lor {\overline}{Q} \to {\overline}{S}^\lor {\overline}{Q}|_{Z \times D}.$$ If we show that the vertical arrows in the natural diagram $$\xymatrix{ & {\overline}{S}^\lor {\overline}{Q} \ar[r] & {\overline}{S}^\lor {\overline}{Q}|_{Z \times D} \\ S^\lor S|_{Z \times D} \ar[r] & S^\lor {\overline}{Q} \ar[d] \ar[u] \\ & S^\lor Q }$$ define quasi-isomorphisms between the complexes (on $Z \times C$) given by the rows, then the proof is completed by applying the functor ${\operatorname{\bf R}}\pi_{1*}$. To see that the bottom vertical arrow is a quasi-isomorphism, just tensor with $S^\lor$. To say that the top vertical arrow is a quasi-isomorphism is equivalent to saying that the sequence $$\begin{aligned} \label{exactseq} 0 \to S^\lor S|_{Z \times D} \to S^\lor {\overline}{Q} \to {\overline}{S}^\lor {\overline}{Q} \to {\overline}{S}^\lor {\overline}{Q}|_{Z \times D} \to 0\end{aligned}$$ is exact. Tensoring the exact sequence $$0 \to S^\lor \to {\overline}{S}^\lor \to {\overline}{S}^\lor|_{Z \times D} \to 0$$ with ${\overline}{Q}$ gives an exact sequence $$0 \to {\curly Tor}_1^{Z \times C}( {\overline}{S}^\lor |_{Z \times D} , {\overline}{Q}) \to S^\lor {\overline}{Q} \to {\overline}{S}^\lor {\overline}{Q} \to {\overline}{S}^\lor {\overline}{Q}|_{Z \times D} \to 0,$$ so we want to show that $S^\lor S|_{Z \times D}$ is identified with ${\curly Tor}_1^{Z \times C}( {\overline}{S}^\lor|_{Z \times D}, {\overline}{Q})$. Tensoring the locally free resolution $$0 \to {\overline}{S} \to \pi_2^*E \to {\overline}{Q} \to 0$$ (restricted from $X \times C$ to $Z \times C$) of ${\overline}{Q}$ with ${\mathcal O}_{Z \times D}$ gives an exact sequence $$0 \to {\curly Tor}_1^{Z \times C}({\mathcal O}_{Z \times D}, {\overline}{Q}) \to {\overline}{S}|_{Z \times D} \to \pi_2^* E|_{Z \times D} \to {\overline}{Q}|_{Z \times D} \to 0$$ of sheaves on $Z \times C$. But by the definition of $Z$, the the map in the middle is zero, so the left and right maps are isomorphisms and we conclude ${\curly Tor}_1^{Z \times C}({\mathcal O}_{Z \times D},{\overline}{Q})= {\overline}{S}|_{Z \times D}.$ Since ${\overline}{S}^\lor|_{Z \times D}$ is a locally free ${\mathcal O}_{Z \times D}$ module, we have $$\begin{aligned} {\curly Tor}_1^{Z \times C}({\overline}{S}^\lor|_{Z \times D}, {\overline}{Q}) & = & {\curly Tor}_1^{Z \times C}({\mathcal O}_{Z \times D}, {\overline}{Q}) \otimes {\overline}{S}^\lor|_{Z \times D} \\ &=& {\overline}{S}^\lor {\overline}{S}|_{Z \times D} \\ & = & S^\lor S|_{Z \times D}. \end{aligned}$$ This proves that is exact, hence completes the proof. One can show directly that this POT is independent of the choice of “sufficiently positive" $j : D \hookrightarrow C$ by comparing two such $D_1,D_2$ with the inclusion of the sum $D_1+D_2$. Tautological classses {#section:tautologicalclasses} --------------------- Let $\eta \in {\operatorname{H}}^2(C,{\mathbb{Z}})$ be the fundamental class and fix a basis $\delta_1,\dots,\delta_{2g}$ for ${\operatorname{H}}^1(C,{\mathbb{Z}})$ such that $\delta_i \delta_{g+i}=-\delta_{g+i}\delta_i=\eta$ for $i \in \{1,\dots,g \}$ and all other $\delta_i \delta_j$ are zero. Let $S$ be the universal subbundle on ${\operatorname{Quot}}E \times C$. Let $$c_i(S^\lor) = a_i \otimes 1 + \sum_{j=1}^{2g} b_{i,j} \otimes \delta_j + f_i \otimes \eta$$ be the Künneth decomposition of $c_i(S^\lor)$. In the argot of [@Mar], [@MO], etc. the classes $$\begin{aligned} a_i & \in & {\operatorname{H}}^{2i}({\operatorname{Quot}}^{r,e} E) \\ b_{i,j} & \in & {\operatorname{H}}^{2i-1}({\operatorname{Quot}}^{r,e} E) \\ f_i & \in & {\operatorname{H}}^{2i-2}({\operatorname{Quot}}^{r,e} E)\end{aligned}$$ are called $a,b$ and $f$ *classes*. A *tautological class* is an element of the subring ${\overline}{{\operatorname{R}}}^*$ of ${\operatorname{H}}^*({\operatorname{Quot}}E)$ generated by the $a$, $b$, and $f$ classes. The *tautological ring* ${\operatorname{R}}^*$ is the quotient of ${\overline}{{\operatorname{R}}}^*$ by the ideal $$\{ \alpha \in {\overline}{{\operatorname{R}}}^* : \int_{[{\operatorname{Quot}}E]^{\rm vir}} \alpha \beta =0 \; {\rm for \; all} \; \beta \in {\overline}{{\operatorname{R}}}^* \}.$$ Unlike the actual cohomology ring ${\operatorname{H}}^*({\operatorname{Quot}}E)$, the tautological ring is deformation invariant: it does not depend on the curve $C$ or the bundle $E$, except through their discrete invariants. This follows from deformation-invariance properties of the virtual class and the fact that the $a$, $b$, and $f$ classes have globally defined analogues as $C,E$ vary in families which restrict to the given classes at points of the family. The virtual intersection theory of ${\operatorname{Quot}}E$ is the study of the tautological ring, usually through calculations of integrals of tautological classes over the virtual class. A typical result is the Vafa-Intriligator formula for integrals of polynomials in the $a$ classes [@MO] (for the case $E={\mathcal O}_C^N$). For later use, let us record some facts about the first Chern class $c_1(L_e) \in {\operatorname{H}}^2({\operatorname{Pic}}^e C \times C)$ of the Poincaré bundle discussed on pages 334-335 of [@ACGH]. Its Künneth decomposition looks like $$\begin{aligned} \label{c1ofL} c_1(L_e) &=& 0 \otimes 1 + \sum_{i=1}^{2g} b_i \otimes \delta_j + e \otimes \eta \end{aligned}$$ (the $(2,0)$ Künneth component is zero because $L_e$ restricts trivially to ${\operatorname{Pic}}^e C \times \{ Q \}$). Note that, by the universal property, any two components of ${\operatorname{Pic}}C$ are identified by tensoring $L$ with a line bundle pulled back from $C$; this only changes the $(0,2)$ Künneth component of $c_1$, so the $(1,1)$ Künneth components of $L_e$ and $L_{d}$ are identified under this isomorphism, so the $b_i$ are “independent of $e$". Recall that the $b_i$ form a basis for ${\operatorname{H}}^1({\operatorname{Pic}}^e C, {\mathbb{Z}}) \cong {\mathbb{Z}}^{2g}$ which is oriented so that $$\begin{aligned} \label{topeven} \int_{{\operatorname{Pic}}^e C} b_1b_{g+1} b_2b_{g+2} \cdots b_g b_{2g} = 1.\end{aligned}$$ To prove this (following [@ACGH]), one may assume $e$ is large. Consider the divisor $\Delta + (e-1) C \times \{ Q \}$ in $C \times C$, where $\Delta$ is the diagonal. By the universal property of ${\operatorname{Pic}}^e C$, the associated line bundle $$L' := {\mathcal O}_{C \times C}(\Delta+(e-1) C \times \{ Q \})$$ determines a morphism $f : C \to {\operatorname{Pic}}^e C$ so that $(f \times {\operatorname{Id}}_C)^*L_e$ differs from $L'$ only by tensoring with a line bundle pulled back from $C$ via $\pi_1$; this doesn’t change the $(1,1)$ Künneth component of $c_1$, so we see that the $(1,1)$ Künneth component of $L_e$ pulls back via $f \times {\operatorname{Id}}_C$ to the $(1,1)$ Künneth component of $c_1(L')$, which is just the $(1,1)$ Künneth component of the diagonal $\Delta$: $$\begin{aligned} \sum_{1=1}^{2g} f^* b_i \otimes \delta_i & = & \Delta_{1,1} \\ & = & \sum_{i=1}^g -\delta_{g+i} \otimes \delta_i + \sum_{i=1}^g \delta_i \otimes \delta_{g+i}. \end{aligned}$$ To see that $f^*$ induces an isomorphism on ${\operatorname{H}}^1$, notice that it factors as a sequence of stratification maps (c.f. , ) $$\begin{aligned} \iota_Q : {\operatorname{Sym}}^n C & \to & {\operatorname{Sym}}^{n+1} C \\ P_1+\cdots+P_n & \mapsto & P_1+\cdots +P_n +Q \end{aligned}$$ (starting from $C={\operatorname{Sym}}^1 C$) followed by the Abel-Jacobi map ${a}_e : {\operatorname{Sym}}^e C \to {\operatorname{Pic}}^e C$. Since $e$ is large, ${a}_e$ is a ${\mathbb{P}}^{e-g}$ bundle, hence ${a}_e^*$ is an isomorphism on ${\operatorname{H}}^1$, and each $\iota_Q$ has affine complement and is an isomorphism on ${\operatorname{H}}^1$ by the Lefschetz Hyperplane Theorem (c.f. (12.2) in [@Mac]). Set $\theta := \sum_{i=1}^g b_i b_{g+1} \in {\operatorname{H}}^2({\operatorname{Pic}}^e C)$, and let $\gamma = \sum_{i=1}^{2g} b_i \otimes \delta_i$ be the $(1,1)$ Künneth component of $c_1(L_e)$ as in [@ACGH]. Then $\gamma^2 = - 2 \eta \theta$, $$\begin{aligned} \label{thetatotheg} \int_{{\operatorname{Pic}}^e C} \theta^g = g! \int_{{\operatorname{Pic}}^e C} b_1b_{g+1}b_2b_{2+g} \cdots b_g b_{2g} = g!,\end{aligned}$$ and $$\begin{aligned} \label{chL} {\operatorname{ch}}L_e = 1+e \eta + \gamma-\eta \theta \end{aligned}$$ (dropping notation for pullbacks). Let us now focus on $c_1(S^\lor)=c_1(\land^r S^\lor)$, whose Künneth decomposition we will simply write as $$c_1(S^\lor) = a \otimes 1 - \sum_{i=1}^{2g} b_i \otimes \delta_i - e \otimes \eta$$ for the sake of this discussion. As the notation intimates, $b_i$ is the pullback of $b_i \in {\operatorname{H}}^1({\operatorname{Pic}}^e C)$ via the Abel-Jacobi map ${a}_e : {\operatorname{Quot}}^{r,e} E \to {\operatorname{Pic}}^e E$. Indeed, by definition of ${a}_e$ (Section \[section:abeljacobimaps\]) via the universal property of ${\operatorname{Pic}}^e C$, $({a}_e \times {\operatorname{Id}}_C)^*L_e$ is equal to $\land^r S$ up to tensoring with a line bundle pulled back from ${\operatorname{Quot}}^e E$; tensoring with such a line bundle does not effect the $(1,1)$ Künneth component of $c_1$. Since ${\operatorname{H}}^*({\operatorname{Pic}}C)$ is generated by ${\operatorname{H}}^1({\operatorname{Pic}}C)$, all classes in ${\operatorname{H}}^*({\operatorname{Quot}}E)$ pulled back via the Abel-Jacobi map are tautological. We will usually drop ${a}_e^*$ from the notation, and simply write, e.g. $\theta \in {\overline}{{\operatorname{R}}}^2({\operatorname{Quot}}^{r,e} E)$ for ${a}_e^* \theta$. The rest of this section is devoted to determining the tautological ring of the rank one Quot scheme. The necessary intersection number formulae will be obtained as sequelae of the structure theory of the rank one Quot scheme from and the basic properties of the virtual class in . A monomial in the $b_i$ equal to $b_{i_1}b_{g+i_1} \cdots b_{i_k}b_{g+i_k}$ for some distinct $i_1, \dots, i_k \in \{ 1, \dots, g \}$ in ${\operatorname{H}}^*({\operatorname{Pic}}C)$ will be called *even*, while a monomial in the $b_i$ not equal to (plus or minus) an even monomial will be called *odd*. The possible “parities" of a (non-zero) product of monomials of given parity are: $$\begin{aligned} ({\rm odd }) ({\rm even }) &=& {\rm \; odd \; } \\ ({\rm even }) ({\rm even }) &=& {\rm \; even } \\ ({\rm odd }) ({\rm odd }) &=& {\rm \; odd \; or \; even } \end{aligned}$$ \[thm:virtualintersectionnumbers\] Let $E$ be a vector bundle of rank $N$ and degree $d$ on a smooth proper curve $C$. Set $m := (1-g)N+d-Ne$ (as in Theorem \[thm:quotforsmalle\]). For any $k \in \{0,\dots,g\}$ with $m-1+k \geq 0$, and any even monomial $b \in {\operatorname{H}}^{2g-2k}({\operatorname{Quot}}^e E)$, $$\int_{[{\operatorname{Quot}}^e E]^{\rm vir}} a^{m-1+k} b = N^k,$$ while this integral vanishes for any odd $b$. We first prove the theorem when $e$ is sufficiently small. Let $\pi_i$ be the projections to the factors of ${\operatorname{Pic}}^e C \times C$. Since $e$ is small, ${\operatorname{\bf R}}^1 \pi_{1*} (L_e^\lor \otimes \pi_2^* E)=0$, $V := \pi_{1*} (L_e^\lor \otimes \pi_2^* E)$ is a vector bundle of rank $m$ on ${\operatorname{Pic}}^e C$, and ${\operatorname{Quot}}^e E = {\mathbb{P}}V $ (see the proof of Theorem \[thm:quotforsmalle\]). By GRR and we find ${\operatorname{ch}}V = m-N \theta$. By the proof of Theorem \[thm:quotforsmalle\], we have $S={\mathcal O}_{{\mathbb{P}}V}(-1) \otimes (p \times {\operatorname{Id}}_C)^* L_e$, where $p : {\mathbb{P}}V \to {\operatorname{Pic}}^e C$ is the projection. Since the $(2,0)$ Künneth component of $L_e$ is zero, $a=c_1({\mathcal O}_{{\mathbb{P}}V}(1))$. The Chern character formula for $V$ implies $c(V)=e^{-N \theta}$ (see page 336 in [@ACGH]), so by definition of Chern classes, we have the relation $$\begin{aligned} \label{relation} a^m &=& a^{m-1}(N\theta)-a^{m-2}\frac{(N \theta)^2}{2!}+ a^{m-3}\frac{(N \theta)^3}{3!} - \cdots \end{aligned}$$ in ${\operatorname{H}}^*({\mathbb{P}}V)={\overline}{R}^*({\operatorname{Quot}}^e E)$. The proof in the small $e$ case can now be completed by induction on $k$. When $k=0$, we compute $$\begin{aligned} \int_{[{\operatorname{Quot}}^e E]^{\rm vir}} a^{m-1}b & = & \int_{{\mathbb{P}}V} c_1({\mathcal O}_{{\mathbb{P}}V}(1))^{{\operatorname{rk}}V -1} b \\ &=& \int_{{\operatorname{Pic}}^e C} b, \end{aligned}$$ which, according to , is indeed given by $N^0=1$ when $b$ is the unique even monomial of degree $2g$ and certainly vanishes when $b$ is odd (hence zero). For the induction step, since $k>0$, we can use to write $$\begin{aligned} \label{step1} \int_{{\mathbb{P}}V} a^{m-1+k} b &=& \int_{{\mathbb{P}}V} a^{m-1+(k-1)}(N\theta)b - a^{m-1+(k-2)} \frac{(N \theta)^2}{2!} b + \cdots .\end{aligned}$$ Note $$\theta^j = j! \sum_{1 \leq i_1 < \cdots < i_j \leq g} b_{i_1}b_{g+i_1} \cdots b_{i_j} b_{g+i_j}$$ is $j!$ times the sum of all even monomials of degree $2j$. If $b$ is odd, then by the rules for the parity of a product, $\theta^j b$ is a sum of odd monomials, hence vanishes by the induction hypothesis. On the other hand, if $b$ is an even monomial of degree $2g-2k$, then $(\theta^j / j!)b$ is a sum of $\begin{pmatrix} k \\ j \end{pmatrix}$ even monomials of degree $2g-2(k-j)$ and we have $$\int_{{\mathbb{P}}V} a^{m-1+(k-j)} \frac{\theta^j}{j!}b = \begin{pmatrix} k \\ j \end{pmatrix} r^{k-j}$$ by the induction hypothesis, hence $$\begin{aligned} \eqref{step1} & = & N^k \left ( \begin{pmatrix} k \\ 1 \end{pmatrix} - \begin{pmatrix} k \\ 2 \end{pmatrix} + \begin{pmatrix} k \\ 3 \end{pmatrix} - \cdots \right ) \\ &=& N^k, \end{aligned}$$ as claimed. For a general $e$, choose a divisor $D \subset C$ of sufficiently large degree $n$ and consider the embedding $$\iota_D : {\operatorname{Quot}}^e E \hookrightarrow {\operatorname{Quot}}^{e-n} E$$ of Section \[section:virtualclass\]. Following the notation of Section \[section:virtualclass\], let $Z = {\operatorname{Quot}}^e E$, $X = {\operatorname{Quot}}^{e-n} E$, and let $S$, ${\overline}{S}$ be the universal subsheaves on $Z \times C$ and $X \times C$, respectively. As in Section \[section:virtualclass\], we have $S = {\overline}{S}(D)|_{Z \times C}$. The twist by $\pi_2^* {\mathcal O}_C(D)$ doesn’t change the $(2,0)$ Künneth component of $c_1$, so we have $a = {\overline}{a}|_Z$. Let ${\overline}{S}_D^\lor := \pi_{1*} ({\overline}{S}^\lor|_{X \times D})$. Evidently ${\overline}{S}_D^\lor$ is a vector bundle on $X$ with rank $n$ and $c_n( {\overline}{S}_D^\lor ) = {\overline}{a}^n$. We saw in Section \[section:virtualclass\] that $$[Z]^{\rm vir} = c_n( {\overline}{S}_D^\lor )^r \cap [X]$$ (we are assuming that $n$ is large enough that $X$ is a ${\mathbb{P}}^{{\overline}{m}-1}$ bundle over ${\operatorname{Pic}}^e C$, so in particular smooth of the expected dimension). Note that $$\begin{aligned} {\overline}{m} & = & (1-g)N+d-(e-n)N \\ & = & m+nN. \end{aligned}$$ Since we know the desired formula holds on $X$, the proof is completed by the following computation: $$\begin{aligned} \int_{[Z]^{\rm vir}} a^{m-1+i} b &=& \int_{X} {\overline}{a}^{m-1+i} b c_n( {\overline}{S}_D^\lor )^r \\ & = & \int_{X} {\overline}{a}^{{\overline}{m}-1+i} b. \end{aligned}$$ The formula $$\begin{aligned} \label{powersofatheta} \int_{[{\operatorname{Quot}}^e E]^{\rm vir}} a^{m-1+k} \theta^{g-k} & = & \frac{N^k g!}{k!} \end{aligned}$$ is an immediate consequence. The following corollary is also an important special case of the theorem. \[corollary:zerodimensionalvirtualclass\] Let $E$ be a vector bundle of rank $N$ and degree $d$ on a smooth proper curve $C$ and suppose $e \in {\mathbb{Z}}$ satisfies $$(1-g)(N-1)+d-Ne=0$$ (i.e. the expected dimension of ${\operatorname{Quot}}^e E$ is zero). Then $$\int_{[{\operatorname{Quot}}^e E]^{\rm vir}} 1 = N^g.$$ Notice that the virtual intersection theory of ${\operatorname{Quot}}^e E$ has little dependence on the rank of $E$. In case $N=1$, ${\operatorname{Quot}}^e E={\operatorname{Sym}}^n C$ (for an appropriate $n$) and the tautological ring $R^*({\operatorname{Quot}}^e E)$ is just the usual cohomology ring of the symmetric product. Presumably the tautological ring of ${\operatorname{Quot}}^e E$ admits a presentation similar to that of ${\operatorname{H}}^*({\operatorname{Sym}}^n C)$ given in [@Mac].[^3] We leave the details to the reader. In case $E = {\mathcal O}_C^N$ is the trivial rank $N$ bundle, our ${\operatorname{Quot}}^{r,-d} {\mathcal O}_C^N$ is denoted ${\operatorname{Quot}}_d {\mathcal O}_C^N$ in [@Ber] and [@MO]. If $P(X_1,\dots,X_r)$ is a polynomial in $r$ variables, our $$\int_{[{\operatorname{Quot}}^{r,-d} {\mathcal O}_C^N]^{\rm vir}} P(a_1,\dots,a_r)$$ is Bertram’s $$N_d(P(X_1,\dots,X_r),g).$$ In this case, our Theorem \[thm:virtualintersectionnumbers\] is a special case of Proposition 2 in [@MO]. Indeed, fix $s \in \{0,\dots,g \}$ and indices $1 \leq j_1 \leq \cdots \leq j_s \leq g$, so $b := b_{j_1}b_{g+j_1} \cdots b_{j_s}b_{g+j_s} \in {\operatorname{H}}^{2s}({\operatorname{Quot}}{\mathcal O}_C^N)$. Let $m := N(1-g+d)$ as usual. Using their notation, let $R(x)=P(x) = x^{m-1+g-s}$, $J(x) = N/x$. Let $\zeta_N$ be a primitive $N^{\rm th}$ root of unity. A very special case of their formula yields $$\begin{aligned} \int_{ [{\operatorname{Quot}}^{-d} {\mathcal O}_C^N]^{\rm vir} } a^{m-1+g-s} b &=& N^{-s} \sum_{i=1}^N (\zeta_N^i)^s R(\zeta_N^i) J(\zeta_N^i)^{g-1} \\ &=& N^{-s} \sum_{i=1}^N N^{g-1}(\zeta_N^i)^{N(1-g+d)} \\ &=& N^{g-s}, \end{aligned}$$ in agreement with our formula. Curve Counting {#section:curvecounting} ============== We begin this section with a brief review of various curve counting theories, then in §\[section:residueinvariants\] we explain how to define the “residue" invariants of a rank two bundle $E$ on a curve $C$ for each of these theories. Finally, in §\[section:correspondence\] we recall the (somewhat conjectural) correspondence between these residue theories. DT theory {#section:DTtheory} --------- For a smooth $3$-fold $X$ and $\beta \in {\operatorname{H}}_2(X)$, let $I_n(X,\beta)$ denote the Hilbert scheme of ideal sheaves $I \subset {\mathcal O}_X$ of $1$-dimensional subschemes $Z \subset X$ with $[Z] = \beta$ and $\chi({\mathcal O}_Z) = n$. Viewing this as the moduli space of rank one torsion free sheaves with trivial determinant (and the appropriate discrete invariants), the fixed determinant deformation theory of sheaves endows this space with a perfect obstruction theory (POT) and hence a virtual fundamental class $$[I_n(X,\beta)]^{\rm vir} \in A_e(I_n(X,\beta)).$$ The tangent and obstruction spaces at a point $I \in I_n(X,\beta)$ are given by the traceless ${\operatorname{Ext}}$ groups $${\operatorname{Ext}}^1(I,I)_0, \; {\operatorname{Ext}}^2(I,I)_0,$$ respectively and the *expected dimension* $e$ is given by $$\dim {\operatorname{Ext}}^1(I,I)_0 - \dim {\operatorname{Ext}}^2(I,I)_0 = \int_\beta c_1(TX)$$ ([@MNOP2], Lemma 1). The *Donaldson-Thomas (DT)* invariants of $X$ are defined by pairing various cohomology classes on $I_n(X,\beta)$ with the virtual class. PT theory {#section:PTtheory} --------- Let $P_n(X,\beta)$ denote the moduli space of (flat families of) pairs $(F,s)$ consisting of a sheaf $F$ on $X$ and a section $s \in {\operatorname{H}}^0(X,F)$ satisfying: 1. The support of $F$ is one dimensional and $[{\operatorname{Supp}}F]=\beta$. 2. $\chi(F)=n$ 3. $F$ is pure: any subsheaf $G \subset F$ with zero dimensional support is zero. 4. ${\operatorname{Cok}}s$ has zero dimensional support. The purity of $F$ ensures that ${\operatorname{Supp}}F$ is a Cohen-Macaulay (CM) curve of pure dimension one (no embedded points). This is the *stable pairs* moduli space of Pandharipande-Thomas. It also carries a POT and a virtual class $$[P_n(X,\beta)]^{\rm vir} \in A_e(P_n(X,\beta))$$ reflecting the fixed determinant deformation theory of the two term complex $$I^\bullet := [ s: {\mathcal O}_X \to F ]$$ (sitting in degrees $0,1$) in the derived category of $X$; the deformation and obstruction spaces at a point $I^\bullet$ are again given by traceless ${\operatorname{Ext}}$ groups: $${\operatorname{Ext}}^1(I^\bullet,I^\bullet)_0, \, {\operatorname{Ext}}^2(I^\bullet,I^\bullet)_0.$$ The expected dimension is again given by $$\dim {\operatorname{Ext}}^1(I^\bullet,I^\bullet)_0 - \dim {\operatorname{Ext}}^2(I^\bullet,I^\bullet)_0 = \int_\beta c_1(TX)$$ (the proof is the same as that of Lemma 1 in [@MNOP2] and is hence omitted in [@PT]). It is a standard fact ([@PT], 1.7) that the cokernel of $s$ is the ideal sheaf $I_Z$ of $Z := {\operatorname{Supp}}F$ in $X$, so we have an exact sequence $$0 \to I_Z \to {\mathcal O}_X \to F \to {\operatorname{Cok}}s \to 0,$$ hence $\chi(F) = \chi({\mathcal O}_Z)+\chi( {\operatorname{Cok}}s)$. Since ${\operatorname{Cok}}s$ has zero dimensional support, the second term is nonnegative and is just the length of ${\operatorname{Cok}}s$. The *Pandharipande-Thomas (PT)* invariants of $X$ are also defined by pairing various cohomology classes on $P_n(X,\beta)$ with the virtual class. DT=PT for minimal $n$ {#section:DTequalsPT} --------------------- Fix $\beta$ and consider the minimum $n$ so that $P_n(X,\beta)$ is non-empty. Then we have an isomorphism $P_n(X,\beta) = I_n(X,\beta)$ identifying the POTs on the two spaces. This is because minimality of $n$ ensures that every stable pair is just the natural surjection ${\mathcal O}_X \to {\mathcal O}_Z$ onto the structure sheaf of a CM curve $Z \subset X$ of pure dimension one. Similarly, minimality of $n$ ensures that every curve $Z \in I_n(X,\beta)$ is CM of pure dimension one, else we could pass to the subcurve $Z' \subseteq Z$ defined by the largest ideal $I \subset {\mathcal O}_Z$ with zero dimensional support to obtain a curve with smaller Euler characteristic. The structure sheaf of such a CM curve is pure, hence it defines a stable pair ${\mathcal O}_X \to {\mathcal O}_Z$. The POTs are identified since the complex $I^\bullet = [{\mathcal O}_X \to {\mathcal O}_Z]$ associated to a stable pair of minimal Euler characteristic is quasi-isomorphic to the ideal sheaf of $Z$. Residue invariants {#section:residueinvariants} ------------------ Let $E$ be a rank two vector bundle over a smooth proper curve $C$. The total space of $E$ is a smooth $3$-fold with ${\operatorname{H}}_2(E) = {\mathbb{Z}}[C]$ generated by the class of the zero section; we will often just write $b$ for $b[C]$. The GW, DT, and PT “residue" invariants of $E$ are defined by formally applying the virtual localization formula using the natural $T$ action on $E$ by scaling (which induces a $T$-action on each moduli space in question making the perfect obstruction theory $T$-equivariant). For example, the GW invariants of $E$ are defined by $$M_{g,b}(E) := \int_{ [{\overline}{M}^\bullet_{g,0}(C,b)]^{\rm vir} } e_T( - \pi_! f^* E) ,$$ where $\pi$ is the universal domain curve and $f$ is the universal map. The integral is equivariant pushforward to a point and takes values in the localized equivariant cohomology ring of a point (i.e. the ring of rational functions of the generator $t$ of the equivariant cohomology ring of a point). The superscript $\bullet$ indicates that the domain curve $\Sigma$ of a stable map may be disconnected (note $g := 1 - \chi({\mathcal O}_\Sigma)$ can be negative) though no connected component can be contracted to a point. Note that the $T$-fixed stable maps are just the stable maps to the zero section and the $T$-fixed part of the GW POT is nothing but the POT on stable maps to the zero section. Similarly, the PT invariants of $E$ are defined by $$P_{n,b}(E) := \int_{ [P_n(E,b)^T]^{\rm vir} } e_T(- N^{\rm vir} ),$$ where the integral is again equivariant pushforward to a point. Here $[P_n(E,b)^T]^{\rm vir}$ is the virtual class associated to the POT on $P_n(E,b)^T$ obtained from the $T$ fixed part of the POT on $P_n(E,b)$ and $N^{\rm vir}$ is the moving part of this POT (we will examine this closely in Section \[section:obstructiontheory\]), viewed as an element of (perfect, $T$-equivariant) $K$ ring $K_T(P_n(E,b)^T)$. The DT invariants $I_{n,b}(E)$ are defined similarly. It is not obvious that these invariants have any enumerative meaning, though our goal is to show that they do. GW/DT/PT residue correspondence {#section:correspondence} ------------------------------- The GW, DT, and PT residue invariants are all $T$-equivariant-deformation invariant, so they are independent of the choice of degree $d$ bundle $E$, hence we may write $M_{g,b}(d), I_{n,b}(d), P_{n,b}(d)$ in lieu of $M_{g,b}(E), I_{n,b}(E), P_{n,b}(E)$. Let $$\begin{aligned} Z^{\rm GW}_b(d)(u) & := & \sum_g M_{g,b}(d) u^{2g-2} \in {\mathbb{Q}}(t)((u)) \\ Z^{\rm DT}_b(d)(q) & := & \sum_n I_{n,b}(d) q^n \in {\mathbb{Z}}(t)((q)) \\ Z^{\rm PT}_b(d)(q) & := & \sum_n P_{n,b}(d) q^n \in {\mathbb{Z}}(t)((q)) \end{aligned}$$ be the generating functions for these invariants. Let $$M(q) := \prod_{n=1}^\infty \frac{1}{(1-q^n)^n}$$ denote the MacMahon function (the generating function for plane partitions). The GW/DT correspondence proved in [@OP] says that $$\begin{aligned} \label{ZDT0} Z^{\rm DT}_0(d) &=& M(-q)^{8g-8-d}\end{aligned}$$ (this is a special case of Conjecture 1 in [@BP2]), that the *reduced DT partition function* $Z^{\rm red \; DT}_b(d) := Z^{\rm DT}_b(d) / Z^{\rm DT}_0(d)$ is a rational function of $q$, and that $$\begin{aligned} \label{GWDT} (-iu)^{b(2-2g+d)}Z_b^{\rm GW}(d) &=& (-q)^{(-b/2)(2-2g+d)} Z^{\rm red \; DT}_b(d).\end{aligned}$$ (These are the “absolute" special cases of Theorems 1,2, and 3 in [@OP].) The conjectural DT/PT correspondence of [@PT] asserts (for any $3$-fold $X$) that $Z^{\rm PT}_\beta(X)=Z^{\rm red \; DT}_\beta(X)$ for any $\beta \in {\operatorname{H}}_2(X)$. In particular, this would imply $$\begin{aligned} \label{GWPT} (-iu)^{b(2-2g+d)}Z_b^{\rm GW}(d) &=& (-q)^{(-b/2)(2-2g+d)} Z^{\rm PT}_b(d).\end{aligned}$$ In fact, it is expected that the same TQFT method used to compute the GW [@BP1], [@BP2] and DT [@OP] invariants of $E$ can also be used to compute the PT invariants of $E$, thus establishing . Let us now focus on the case $b=2$, which we will study throughout. The explicit formula $$\begin{aligned} \label{ZGW2} Z^{\rm GW}_2(d) & = & (ut)^{4g-4-2d}4^{g-1}(2 \sin \frac{u}{2})^{2d} \left ( (1+\sin \frac{u}{2})^{d+1-g} + (1- \sin \frac{u}{2})^{d+1-g} \right ) \\ \nonumber & = & (ut)^{4g-4-2d}2^{2g-2}(2 \sin \frac{u}{2})^{2d} \sum_{i} \begin{pmatrix} d+1-g \\ 2i \end{pmatrix} 2 (\sin \frac{u}{2})^{2i} \\ \nonumber &=& (ut)^{4g-4-2d} \sum_{i} \begin{pmatrix} d+1-g \\ 2i \end{pmatrix} 2^{2g-1-2i} (2\sin \frac{u}{2})^{2i+2d} \end{aligned}$$ can be found in Section 8 of [@BP2].[^4] (The binomial coefficient should be defined as in [@ACGH] so that the usual binomial expansion holds for negative exponents.) When $b=2$, specializes to $$\begin{aligned} \label{GWPTdegree2} u^{4-4g+2d} Z^{\rm GW}_2(d) &=& q^{2g-2-d} Z^{\rm PT}_2(d). \end{aligned}$$ Under the change of variables $ -q = e^{iu}$, we have $(2 \sin u/2)^2=q^{-1}(1+q)^2$, so by combining with we arrive at the (conjectural) formula $$\begin{aligned} \label{ZPT} Z^{\rm PT}_2(d) &=& t^{4g-4-2d} \sum_i \begin{pmatrix} d+1-g \\ 2i \end{pmatrix} 2^{2g-1-2i} q^{2-2g-i}(1+q)^{2d+2i} \end{aligned}$$ for the residue PT invariants in degree $2[C]$. Maximal Subbundles {#section:maximalsubbundles} ================== Let $E$ be a rank $2$, degree $d$ vector bundle over a genus $g$ curve $C$. Define integers $\epsilon \in \{ 0,1 \}$ and $e$ by the formula $$g-1 + \epsilon = d-2e.$$ Let $f$ be the largest integer such that a generic (rank $2$ degree $d$) stable bundle contains a line subbundle of degree $f$. We wish to argue by dimension counting that $f=e$. Recall that the dimension of the moduli space of rank $2$ stable bundles with fixed determinant is $3g-3$ ([@NS], Theorem 1(iv)). By definition of $f$, a generic stable bundle $E$ fits into a SES $$0 \to S \to E \to Q \to 0,$$ so the dimension of the moduli of stable bundles is bounded above by $$\dim {\operatorname{Pic}}C + \dim {\mathbb{P}}{\operatorname{Ext}}^1(Q,S)$$ since $Q$ is determined by $S$ because $\det E$ is fixed. On the other hand, stability of $E$ implies ${\operatorname{Hom}}(Q,S)=0$, so the dimension of the ${\operatorname{Ext}}$ group can be read off from Riemann-Roch and we obtain an inequality $$3g-3 \leq g - (1-g+2f-d) -1$$ implying $f \leq e$. For the other inequality $e \leq f$ it is enough to show that ${\operatorname{Hom}}(S,E) \neq 0$ for some $S$ of degree $e$, which will follow from the construction (below) of the locus of such $S$ as a determinantal locus in ${\operatorname{Pic}}^e C$ together with the fact that the class of expected dimension $0$ supported on this locus is nonzero in homology. By similar dimension counting arguments, one can show that the dimension of ${\operatorname{Quot}}^e E$ is $\epsilon$ for generic $E$, and that ${\operatorname{Quot}}^e E$ is generically smooth (so it is smooth when $\epsilon=0$). In case $\epsilon=0$ and $E$ is sufficiently generic, the smooth space ${\operatorname{Quot}}^e E$ is just a finite number of points; the fact that this number is $2^g$ was apparently known to Segre (in some form) in 1889 [@Seg]. A “modern" proof via Grothendieck-Riemann-Roch and the Thom-Porteous Formula can be given fairly easily. We sketch the argument (following the proof of Theorem 3.1 in [@LN]) since it is relevant to our discussion of the geometry of the ${\operatorname{Quot}}$ scheme.[^5] Choose a smooth canonical divisor $j : D \hookrightarrow C$ (so $D$ is just $2g-2$ points of $C$) corresponding to a section ${\mathcal O}_C \to \omega_C$ vanishing on $D$. Then we have an exact sequence $$\begin{aligned} \label{sequence1} 0 \to {\mathcal O}_C \to \omega_C \to j_* j^* \omega_C \to 0\end{aligned}$$ on $C$. Let $$\iota = ({\operatorname{Id}}\times j): {\operatorname{Pic}}^e C \times D \to {\operatorname{Pic}}^e C \times C$$ be the inclusion, $\pi_1, \pi_2$ the projections from ${\operatorname{Pic}}^e C \times C$. Tensor with $E$, pull back to ${\operatorname{Pic}}^e C \times C$ and tensor with the dual of the universal degree $e$ line bundle $L_e$ to get an exact sequence $$0 \to L_e^\lor \otimes \pi_2^*E \to L_e^\lor \otimes \pi_2^* (E \otimes \omega_C) \to \iota_* \iota^* L_e^\lor \otimes \iota_* \iota^* \pi_2^* (E \otimes \omega_C) \to 0.$$ To save notation, set $$\begin{aligned} F & := & L_e^\lor \otimes \pi_2^* (E \otimes \omega_C) \\ G & := & \iota_* \iota^* L_e^\lor \otimes \iota_* \iota^* \pi_2^* (E \otimes \omega_C).\end{aligned}$$ For a sufficiently generic $E$ one can argue that $R^1 \pi_{1*} F=0$ and that the number of maximal subbundles is the degeneracy locus of the map $\pi_{1*}F \to \pi_{1*} G$ of vector bundles on ${\operatorname{Pic}}^e C$. (We will see in a moment that the difference in ranks here is $${\operatorname{ch}}_0(\pi_{1*}F-\pi_{1*}G) = g - 1 = \dim {\operatorname{Pic}}^e C-1,$$ so one expects this degeneracy locus to be zero dimensional.) By the Thom-Porteous formula, if this degeneracy locus is smooth and zero dimensional (which it will be for generic $E$), then the number of points in it is $$\int_{{\operatorname{Pic}}^e C} c_g( \pi_{1*} G - \pi_{1*} F ).$$ We adopt the notation of Section \[section:quotschemes\]. The formula for $c_1(L_e)$ implies that it is topologically trivial on ${\operatorname{Pic}}^e C \times D$. Since $\iota^* L_e^\lor \otimes \iota^* \pi_2^* (E \otimes \omega_C)$ differs from $\iota^*L_e^\lor$ only by tensoring with a bundle pulled back from $D$ (hence trivial), it is also topologically trivial, hence so is $\pi_{1*} G$ and we have ${\operatorname{ch}}\pi_{1*}G = 4g-4.$ Using formula , we have $$\begin{aligned} {\operatorname{ch}}F & = & ({\operatorname{ch}}L_e^\lor )( \pi_2^* {\operatorname{ch}}E )(\pi_2^* {\operatorname{ch}}\omega_C) \\ & = & (1-e \eta-\gamma - \eta \theta)(2+d\eta)(1+(2g-2)\eta) \\ & = & 2-2\gamma+(5g-5)\eta-2 \eta \theta, \end{aligned}$$ so by GRR we compute $$\begin{aligned} {\operatorname{ch}}\pi_{1*}F & = & {\operatorname{ch}}\pi_{1!} F \\ & = & \pi_{1*}( ({\operatorname{ch}}F )({\operatorname{td}}\pi_1)) \\ &=& \pi_{1*} ( (2-2\gamma+(5g-5)\eta-2 \eta \theta)(1+(1-g)\eta) )\\ & = & 5g-5-2 \theta. \end{aligned}$$ Since ${\operatorname{ch}}(\pi_{1*}G-\pi_{1*}F) = 1-g + 2 \theta$, it follows from an exercise with symmetric functions (c.f. page 336 in [@ACGH]) that $c(\pi_{1*}G-\pi_{1*}F) = \exp (2 \theta)$, so finally we compute $$\int_{{\operatorname{Pic}}^e C} c_g( \pi_{1*}G- \pi_{1*}F) = \int_{{\operatorname{Pic}}^e C} \frac{(2 \theta)^g}{g!} = 2^g$$ using . Let us see how to intepret the $2^g$ maximal subbundle count as a DT=PT invariant. If we expand as a Laurent series in $q$, the smallest power of $q$ with nonzero coefficient occurs when $i$ takes the value $e$ determined by the equation $$g-1+\epsilon = d-2e$$ ($\epsilon \in \{0,1 \}$ as usual). When $\epsilon=0$ (i.e. when $d$ and $g$ have opposite parity), this lowest order term is $$t^{4g-4-2d} 2^{3g-2-d} q^{2-2g-e}$$ so the DT=PT invariant in minimal Euler characteristic is given by $$\begin{aligned} \label{PT} P_{2-2g-e,2}(d) & = & t^{4g-4-2d} 2^{3g-2-d}. \end{aligned}$$ The equality of DT and PT invariants in minimal Euler characteristic as discussed in \[section:DTequalsPT\], together with the known equivalence of DT and GW mentioned above, ensures that this is actually the correct PT invariant even though the coefficients of higher powers of $q$ are technically only conjectural. Assume we are in the $\epsilon=0$ case so $g-1 = d-2e$. Let $Y$ be the space (${\operatorname{Quot}}$ scheme) of such minimal (i.e. degree $e$) line subbundles of $E$. As mentioned above, if $E$ is sufficiently generic, $Y$ is just a finite number of points. On the other hand, we can identify $Y$ with the $T$-fixed subscheme of $$P := I_{2g-2-e}(E,2) = P_{2g-2-e}(E,2)$$ using Theorem \[thm:Tfixedstablepairs\] below (or by using Proposition \[prop:CMcurvesinE\] and the general remarks about this common moduli space in minimal Euler characteristic). Since $Y$ is smooth and zero dimensional, the tangent space to $Y$ at a point $L \hookrightarrow E$ is zero. On the other hand, this tangent space is given by ${\operatorname{Hom}}(L, Q) = {\operatorname{H}}^0(C,L^\lor Q),$ where $Q = E/L$ as usual. The degree of $L^\lor Q$ is $-e+d-e = g-1$, so by Riemann-Roch, we also have ${\operatorname{H}}^1(C,L^\lor Q)=0$. Let $Z = {\operatorname{Spec}}_C {\mathcal O}_C[L^\lor]$ be the degree $2[C]$ curve in $E$ corresponding to a point $L \in Y$. The DT=PT deformation and obstruction spaces at this point are given by $${\operatorname{H}}^0(Z,N_{Z/E}), \, {\operatorname{H}}^1(Z,N_{Z/E})$$ respectively. This is a consequence of the results of Section \[section:obstructiontheory\], as mentioned in Remark \[rem:simplifiedvnb\]. In fact, the statement about the deformation space can be seen directly, since the moduli space in question is a Hilbert scheme. The identification of the obstruction space can also be derived from Proposition 4.4 in [@PT] (or its proof). The point is that the stable pairs in question are of the ideal form: they are structure sheaves of embedded CM curves. We will see in the next section (Equation \[normalbundleofZ\]) that the normal bundle fits into a SES $$0 \to \pi^* L^{\otimes 2} \to N_{Z/E} \to \pi^*Q \to 0.$$ Since $\pi$ is an affine morphism, we can compute global sections (and the higher direct images of pushforward to a point) by first pushing forward to $C$ via $\pi$. Note $\pi_* \pi^* G = G \oplus G L^\lor$ for any sheaf $G$ on $C$, so pushing forward this exact sequence to $C$ we have a SES $$0 \to L_t \oplus L_{2t}^2 \to \pi_* N_{Z/E} \to Q_t \oplus (QL^\lor)_0 \to 0$$ on $C$, where the subscripts indicate the weight of the natural $T$ action. We’ve already observed that $${\operatorname{H}}^0(C,QL^\lor)={\operatorname{H}}^1(C,QL^\lor)=0,$$ so the fixed part of the $T$-equivariant POT on $Y$ is trivial, and the virtual fundamental class on $Y$ is its usual fundamental class. We conclude that the entire POT is moving, so the virtual normal bundle (at a point $(L \hookrightarrow E) \in Y$) is then $$\begin{aligned} N^{\rm vir}_{ \{ L \} / P} &=& {\operatorname{H}}^0(Z,N_{Z/E})-{\operatorname{H}}^1(Z,N_{Z/E}) \\ & = & {\operatorname{H}}^0(C,Q)_t + {\operatorname{H}}^0(C,L)_t + {\operatorname{H}}^0(C,L^2)_{2t} \\ & & -{\operatorname{H}}^1(C,Q)_t - {\operatorname{H}}^1(C,L)_t - {\operatorname{H}}^1(C,L^2)_{2t} \end{aligned}$$ (viewed as an element of the $T$ equivariant $K$ group of a point). By Riemann-Roch we have $$\begin{aligned} \chi(Q) & = & 1-g+d-i \\ \chi(L) & = & 1-g+e \\ \chi(L^2) & = & 1-g+2e, \end{aligned}$$ so we compute $$e_T(-N^{\rm vir}_{ \{ L \} / P})= t^{g-1+e-d}t^{g-1-e}(2t)^{g-1-2e} = 2^{2g-2-d}t^{3g-3-d-2e}.$$ Since the DT=PT invariant is given by $$\begin{aligned} P_{2g-2-e,2}(E) & = & \sum_{L \in Y} e_T ( -N^{\rm vir}_{ \{ L \} / P } ) \\ &=& (\# Y) 2^{2g-2-d}t^{3g-3-d-2e} \\ & = & (\# Y) 2^{2g-2-d}t^{4g-4-2d},\end{aligned}$$ we see that $\#Y = 2^g$ by using the known value of this DT=PT invariant given in . Notice that $P_{2g-2-e}(E,2)$ actually *counts* the number of $T$-invariant CM curves in class $2[C]$ in a generic $E$. To my knowledge, there is no such reasonable enumerative interpretation of the residue Gromov-Witten invariants. Of course, if one wishes to compute this minimal Euler characteristic PT invariant without using deformation invariance and the choice of a generic $E$, then one simply replaces the $2^g$ maximal subbundle *count* with the virtual class formula of Corollary \[corollary:zerodimensionalvirtualclass\] and the general formula for $e_T(-N^{\rm vir})$ which we will derive later. $T$ fixed curves and pairs ========================== The main goal of this section is to identify the CM curves and stable pairs on $E$ fixed by the scaling action. As mentioned in the Introduction, we will only treat the case of homology class $2[C] \in {\operatorname{H}}_2(E,{\mathbb{Z}})$ here, leaving the general case for [@Gil2]. Though it is not used elsewhere, we have also included a description of the fixed locus of the zero dimensional Hilbert scheme of the total space of a line bundle $L$ on $C$ (Theorem \[thm:TfixedHilbertscheme\]). \[prop:CMcurvesinE\] Let $\pi : E \to C$ be a vector bundle over a smooth curve. Suppose $Z \subseteq E$ is a proper CM curve in $E$ invariant under the scaling action and in homology (or Chow) class $2[C]$. Then $Z$ is the doubling of the zero section along a line subbundle $L \subset E$. Certainly such a $Z$ must be supported on the zero section, so ${\mathcal O}_Z = \pi_* {\mathcal O}_Z$ is a sheaf of ${\mathcal O}_C$-algebras. Since $Z$ is $T$-invariant, ${\mathcal O}_Z$ has a grading making ${\operatorname{Sym}}^* E^\lor \to {\mathcal O}_Z$ a surjection of graded ${\mathcal O}_C$-algebras. Since $Z$ is CM, each graded piece of ${\mathcal O}_Z$ must be a locally free ${\mathcal O}_C$-module, else its torsion submodule violates purity. Since ${\mathcal O}_Z$ has length $2$ at the generic point of $C$, ${\mathcal O}_Z$ is nonzero only in grading zero (where it must be ${\mathcal O}_C$ since ${\mathcal O}_C = {\operatorname{Sym}}^0 E^\lor$ surjects onto it) and one other grading. Since ${\operatorname{Sym}}^* E^\lor$ is generated in grading $1$, this other grading must be $1$ and we have a surjection $E^\lor \to ({\mathcal O}_Z)_1$ in grading $1$. The locally free sheaf sheaf $L := ({\mathcal O}_Z)_1^\lor \subset E$ must be a line bundle because of the homology class requirement. Evidently ${\mathcal O}_Z = {\mathcal O}_C[L^\lor]$ is the trivial square zero extension of ${\mathcal O}_C$ by $L^\lor$. Note that the $T$-fixed subscheme of such a $Z$ is nothing but the zero section $C$. The scheme $Z$ is l.c.i. and hence $Z$ is a Cartier divisor when $E$ has rank $2$. The map $\pi : Z \to C$ is flat and finite of degree $2$. We can compute $\chi({\mathcal O}_Z)$ by first pushing forward to $C$: $$\chi({\mathcal O}_Z) = \chi({\mathcal O}_C \oplus L^\lor) = 2g-2-\deg L.$$ Since $Z$ is the first infinitesimal neighborhood of $C$ in $L$, we also have a closed embedding $\iota : C \to Z$ corresponding to the surjection ${\mathcal O}_C[L^\lor] \to {\mathcal O}_C$ with kernel $L^\lor$. Evidently $\pi \iota = {\operatorname{Id}}_C$. We find that $$\begin{aligned} \iota^* \pi^* &=& {\operatorname{Id}}\\ \pi_* \iota_* &=& {\operatorname{Id}}\\ \pi_* \pi^* G &=& G \oplus L^\lor G. \end{aligned}$$ We now assume $E$ has rank $2$. \[rem:otherTfixedcurves\] One can show similarly that such a $Z$ in class $3[C]$ is either ${\operatorname{Spec}}{\mathcal O}_C[E^\lor]$ (the first infinitesimal neighborhood of the zero section in $E$) or the second infinitesimal neighborhood $${\operatorname{Spec}}_C ( {\mathcal O}_C \oplus L^\lor \oplus L^{\lor \otimes 2} )$$ of the zero section in a line subbundle $L \subset E$. The latter is l.c.i. in $E$ while the former is not even Gorenstein. Which of the two has smaller Euler characteristic depends on the relationship between $d$ and $g$. One can again describe and filter the normal bundle as in the above proof. Let $Z := {\operatorname{Spec}}{\mathcal O}_C[L^\lor]$ be the doubling of $C$ along $L \hookrightarrow E$ as in the proposition, and let $I$ be the ideal sheaf of $Z$ in $E$ and $Q:=E/L$ the locally free quotient. Then we have a short exact sequence $$\begin{aligned} \label{normalbundle} 0 \to \pi^* Q^\lor \to I/I^2 \to \pi^* L^{\lor 2} \to 0 \end{aligned}$$ of sheaves on $Z$. To see this, note that $I$ is defined by the SES $$0 \to I \to {\operatorname{Sym}}^* E^\lor \to {\mathcal O}_Z \to 0,$$ which we can analyse grading by grading. In grading zero the second map is an isomorphism, so $I_0 = 0$, and in grading $1$ it is $E^\lor \to L^\lor$, so $I_1 = Q^\lor$. In higher gradings the second map is zero, so $I_n = {\operatorname{Sym}}^n E^\lor$ when $n \geq 2$. Now we analyse the inclusion $I^2$ and the inclusion $I^2 \subset I$ in each grading; we will write $I^2_n$ for the degree $n$ part $(I^2)_n$ of $I^2$. We have $I^2_1 = 0$, so $(I/I^2)_1 = Q^\lor$. We have $I^2_2 = {\operatorname{Sym}}^2 Q^\lor$, so $$(I/I^2)_2 = {\operatorname{Sym}}^2 E^\lor / {\operatorname{Sym}}^2 Q^\lor = E^\lor \otimes L^\lor$$ by linear algebra. In grading $3$, $I^2_3 \hookrightarrow I_3$ is $$Q^\lor \otimes {\operatorname{Sym}}^2 E^\lor \hookrightarrow {\operatorname{Sym}}^3 E^\lor,$$ and the quotient $(I/I^2)_3$ is $L^{\lor 3}$ by linear algebra. In grading $n>3$ we have $I_n = I^2_n$. Thus, as a graded ${\mathcal O}_C$ module we have $$\pi_*(I/I^2) = Q^\lor \oplus L^\lor E^\lor \oplus L^{\lor 3},$$ and this has the “obvious" structure of a locally free ${\mathcal O}_C[L^\lor]$-module of rank $2$ (the direct sum decompositions here and elsewhere are only as ${\mathcal O}_C$ modules, not as ${\mathcal O}_Z$ modules). We have $\pi^* G = G \oplus L^\lor G$ for any ${\mathcal O}_C$-module $G$, so $\pi^* Q^\lor = Q^\lor \oplus L^\lor Q^\lor$ and the natural injection $$Q^\lor \oplus L^\lor Q^\lor \to Q^\lor \oplus L^\lor E^\lor \oplus L^{\lor 3}$$ of ${\mathcal O}_C$-modules is in fact easily seen to be an ${\mathcal O}_Z$-module map with quotient $\pi^* L^{\lor 2} = L^{\lor 2} \oplus L^{\lor 3}$. In fact, the dual $$\begin{aligned} \label{normalbundleofZ} 0 \to \pi^* L^2 \to N_{Z/L} \to \pi^*Q \to 0 \end{aligned}$$ of the exact sequence is nothing but the normal bundle sequence $$0 \to N_{Z/L} \to N_{Z/E} \to N_{L/E}|_Z \to 0.$$ \[thm:TfixedHilbertscheme\] Let $\pi : L \to C$ be a line bundle over smooth curve endowed with the scaling action. Then there is an isomorphism of schemes $$\coprod_{\lambda} C^{l(\lambda)} / {\operatorname{Aut}}\lambda \to ({\operatorname{Hilb}}^n L)^T,$$ where the coproduct is over partitions $\lambda$ of $n$, $l(\lambda)$ denotes the length of $\lambda$, and ${\operatorname{Aut}}(\lambda)$ is the automorphism of group of $\lambda$. It is interesting that $({\operatorname{Hilb}}^n L)^T$ does not depend on $L$, though, for example, its normal bundle in ${\operatorname{Hilb}}^n L$ does depend on $L$. We begin by constructing a map from the LHS to the RHS. Let $\lambda = (1)^{m_1} \cdots (k)^{m_k}$ be a typical partition of $n$, so $$1m_1+2m_2+\cdots +km_k=n.$$ Let $$\begin{aligned} X & := & C^{l(\lambda)} / {\operatorname{Aut}}\lambda \\ & = & {\operatorname{Sym}}^{m_1} C \times \cdots \times {\operatorname{Sym}}^{m_k} C \end{aligned}$$ and let $\pi_i : X \to {\operatorname{Sym}}^{m_i} C$ be the projection. Let $Z_i \subset {\operatorname{Sym}}^{m_i} C \times C$ be the universal Cartier divisor, let $W_i := (\pi_i \times \pi)^*Z_i$ be its pullback to $X \times L$, and let $I_i \subset {\mathcal O}_{X \times L}$ be the ideal of $W_i$ in $X \times L$. Let $$J := {\operatorname{Sym}}^{\geq 1} \pi_{C}^* L^\lor$$ be the ideal of $X \times C$ in $X \times L$. Consider the ideal $$K := I_1 \cdots I_k + I_2 \cdots I_k J + I_3 \cdots I_k J^2 + \cdots + I_k J^{k-1} + J^k \subset {\mathcal O}_{X \times L}.$$ Notice that $K$ is generated by “monomials in $J$" so the subscheme $Z \subset X \times L$ that it defines is certainly $T$ invariant. I claim that $Z$ has length $n$ at every point of $X$. Since $J^k \subset K$, certainly $Z$ is topologically supported on $X \times C \subset X \times L$; in fact it is scheme theoretically supported on the $(k-1)$-st infinitesimal neighborhood of $X \times C$ in $X \times L$ and is relatively zero dimensional over $X$. Let $D=(D_1,\dots,D_k)$ be a point of $X$ and let $$Z_D \subset \{ D \} \times L = L$$ be the zero dimensional subscheme of $L$ determined by $Z$ at $D$. For a point $P \in C$ and a zero dimensional $W \subset C$, write $\ell(W,P)$ for the length of $W$ at $P$. Then by definition of $X$ we have $$\sum_{P \in C} \ell(D_i,P) = m_i$$ and we want to prove $$\sum_{P \in C} \ell(Z_D,P) = m_1+2m_2+\cdots+km_k=n,$$ so it suffices to prove $$\ell(Z_D,P) = \ell(D_1,P)+2 \ell(D_2,P)+\cdots + k \ell(D_k,P)$$ for each $P \in C$ for then we get the desired equality by summing over $P$. Set $\ell_i := \ell(D_i,P)$ to ease notation. Let $x$ be a local coordinate on $C$ centered at $P$ (a generator of ${\mathfrak{m}}_P \subset {\mathcal O}_{C,P}$ if you like) and let $y$ be a local coordinate in the $L$ direction corresponding to a trivialization of $L$ near $P$. Then, at $P$, the ideal $I_i$ is given by $(x^{\ell_i})$ and the ideal $J$ is given by $(y)$, so at $P$, the ideal $K$ is given by $$(x^{\ell_1+\cdots+\ell_k}, x^{\ell_2+\cdots+\ell_k}y, \dots , x^{\ell_k} y^{k-1}, y^k),$$ and it is easy to see, by drawing a picture of the corresponding Young diagram, that this monomial ideal has length $\ell_1+2\ell_2+\cdots+k \ell_k$ at the origin $P$. We have proved that $Z$ is a flat family of $T$-invariant length $n$ subschemes of $L$ parameterized by $X$, hence we obtain a map $X \to ({\operatorname{Hilb}}^n L)^T.$ Taking the coproduct of these maps over all partitions gives the purported isomorphism. To see that this is in fact an isomorphism, recall that ${\operatorname{Hilb}}^n L$ is smooth, hence so is the $T$-fixed locus $({\operatorname{Hilb}}^n L)^T$, so by general nonsense we may conclude that our map is an isomorphism of schemes simply by checking that it is bijective on points. Suppose $Z \subset L$ is a zero dimensional $T$ invariant subscheme of $L$ corresponding to a point of $({\operatorname{Hilb}}^n L)^T$. Then at each point $P$ if we choose local coordinates $x,y$ on $L$ near $P$ as above, then the ideal $I$ of $Z$ at $P$ can be generated by monomials in $y$ since $Z$ is $T$ invariant. The coefficient $f(x)$ of a monomial $y^\ell$ is an arbitrary polynomial in $x$, but if we factor out the largest power of $x$ dividing $f(x)$, then what is left over is a unit near $P$. We find that $I$ is a monomial ideal at $P$, so we can write it uniquely as $$(x^{\ell_1(P)+\cdots+\ell_k(P)}, x^{\ell_2(P)+\cdots+\ell_k(P)}y, \dots , x^{\ell_k(P)} y^{k-1}, y^k),$$ for some nonnegative integers $\ell_1(P),\dots,\ell_k(P)$ (independent of the choice of coordinates $x,y$). Set $m_i := \sum_{P \in C} \ell_i(P)$. Then the fact that $Z$ has length $n$ easily implies $$m_1+2m_2+\cdots+km_k = n$$ and it is easy to see that $Z$ is in the image of the component on the LHS corresponding to this partition of $n$. \[thm:Tfixedstablepairs\] There is an isomorphism of schemes $$\coprod_{2n-e=m} {\operatorname{Quot}}^e E \times {\operatorname{Sym}}^n C \cong P_{2-2g+m}(E,2)^T.$$ First we define a map from the LHS to the RHS. Let $Y := {\operatorname{Quot}}^e E \times {\operatorname{Sym}}^n C$ to ease notation. Let $$0 \to S \to \pi_2^* E \to Q \to 0$$ be the universal sequence on $Y \times C$ pulled back from ${\operatorname{Quot}}^e E \times C$ and let $$s_D : {\mathcal O}_{Y \times E} \to {\mathcal O}_{Y \times E}(D)$$ be the pullback of the universal section of the universal Cartier divisor on ${\operatorname{Sym}}^n C \times C$ via the projection $Y \times E \to {\operatorname{Sym}}^n C \times C$. Let $$t : {\operatorname{Sym}}^*_{{\mathcal O}_{Y \times C}} \pi_2^* E^\lor \to {\mathcal O}_{Y \times C}[S^\lor]$$ be the induced map on symmetric algebras obtained by quotienting ${\operatorname{Sym}}^* S^\lor$ by the ideal ${\operatorname{Sym}}^{\geq 2} S^\lor$, viewed simply as a map of graded ${\operatorname{Sym}}^* \pi_2^* E^\lor$ modules. In degree $1$, $$t_1 : \pi_2^* E^\lor \to S^\lor$$ may not be surjective since $Q$ may not be locally free. However, the sheaf $F := {\mathcal O}_{Y \times C}[S^{\lor}]$ is certainly a pure sheaf, topologically supported on $C$ at each point of $Y$. Composing $$t : {\mathcal O}_{Y \times E} \to F$$ with the natural map $1 \otimes s_D : F \to F(D)$ defines a map $$s : {\mathcal O}_{Y \times E} \to F(D),$$ which is evidently a family of stable pairs over $Y$ with the correct discrete invariants. Furthermore, $s$ is clearly a graded map of graded sheaves (with $F(D)_0 = {\mathcal O}_C(D)$ and $F(D)_1 = S^\lor(D)$), hence it defines a $T$ invariant stable pair, that is, a morphism $$Y \to P_{2-2g+m}(E,2)^T.$$ Now we define a map from the RHS to the LHS. Suppose $s : {\mathcal O}_E \to F$ is in $P := P_{2-2g+n}(E,2)^T$. Then, pushing forward to $C$, we may view $s$ as a graded map of graded ${\operatorname{Sym}}^* E^\lor$ modules $$s : {\operatorname{Sym}}^* E^\lor \to F.$$ By purity of $F$, each graded piece $F_d$ must be a locally free ${\mathcal O}_C$-module. Since $s$ has zero dimensional cokernel, $s_0 : {\mathcal O}_C \to F_0$ must be a map of locally free sheaves on $C$ with zero dimensional cokernel, so it must be $s_D : {\mathcal O}_C \to {\mathcal O}_C(D)$ for some Cartier divisor $D \subset C$. Since $F_0$ varies flatly over the universal family, so does $D$, and we obtain a map $P \to {\operatorname{Sym}}^d C$ for some $d$. In degree $1$, $s$ is a map $s_1 : E^\lor \to F_1$. The condition $[{\operatorname{Supp}}F]=2[C]$ and the fact that ${\operatorname{Sym}}^* E^\lor$ is generated in degree $1$ ensure that the locally free sheaf $F_1$ has rank $1$, and $F_i=0$ for $i>1$. The fact that $s$ is a map of graded ${\operatorname{Sym}}^* E^\lor$ modules means we have a commutative diagram $$\xymatrix@C+10pt{ E^\lor \otimes {\mathcal O}_C \ar[d]_m^{\cong} \ar[r]^{1 \otimes s_0} & E^\lor \otimes F_0 \ar[d]^a \\ E^\lor \ar[r]^{s_1} & F_1 }$$ expressing the compatibility (with $s$) of the action $a$ of ${\operatorname{Sym}}^* E^\lor$ on $F$ and the action of ${\operatorname{Sym}}^* E^\lor$ on itself by multiplication. Using our identification of $s_0 = s_D$, we may view this diagram as providing a factorization of $s_1$ through a map $t : E^\lor \to F_1(-D)$. Since ${\operatorname{Cok}}t$ has zero dimensional support, we have ${\curly Hom}({\operatorname{Cok}}t, {\mathcal O}_C)=0$, so $t^\lor : F_1^\lor(D) \to E$ is monic. Noting that $F_1^\lor(D)$ varies flatly over $P$, the dual of $t$ therefore yields a map $P \to {\operatorname{Quot}}^e E$ for the unique $e$ satisfying $2n-e=m$. Taking the product of this map and the previous map we get a map $P \to Y$. Since the two maps are easily seen to be inverse, the isomorphism is established. \[rem:Fasacurve\] Observe that the sheaf $F = {\mathcal O}_C[S^\lor]$ in a stable pair as in the above theorem is itself the structure sheaf of a (Cohen-Macaulay) curve with the same topological space as $C$ (namely the first infinitesimal neighborhood of the zero section $C$ in $S$). The section $s$ may be viewed as a (finite) map of schemes $s : {\operatorname{Spec}}_C F \to E$ (over $C$) which factors through the inclusion $$Z = {\operatorname{Spec}}_C {\mathcal O}_C[L^\lor] \hookrightarrow E$$ and which is generically an isomorphism onto $Z$. Locally, where $S$ and $L$ can be trivialized, $s : {\operatorname{Spec}}_C \to Z$ looks like ${\operatorname{Spec}}_C$ of the map of ${\mathcal O}_C$-algebras $$\begin{aligned} {\mathcal O}_C[y]/y^2 & \to & {\mathcal O}_C[z]/z^2 \\ y & \mapsto & fz \end{aligned}$$ for some $f \in {\mathcal O}_C$. In particular, $(F,s)$ defines a point of Honsen’s moduli space of finite maps from CM curves to $E$, which is in keeping with the original motivation for considering the stable pairs space. It might be interesting to compare the deformation theory of the stable pair $(F,s)$ with the deformation theory of the map $s : {\operatorname{Spec}}_C F \to E$, possibly viewing the latter in some moduli space where the domain of the map is also allowed to vary. Obstruction Theory {#section:obstructiontheory} ================== In this section, we will identify the POT on $$Y := {\operatorname{Quot}}^e E \times {\operatorname{Sym}}^n C$$ inherited from the fixed part of the stable pairs POT via the isomorphism of Theorem \[thm:Tfixedstablepairs\] as well as the virtual normal bundle of $Y$ in the stable pairs space. It will be helpful if the reader is familiar with virtual localization [@GP]. A more general study of this POT can be found in [@Gil2]. In general, if $P = P_n(X,\beta)$ is a stable pairs space and ${\mathbb{I}}^\bullet$ is the universal stable pair, then an important fact ([@PT], 2.2) subtly alluded to above is that $P$ can also be viewed as a moduli space of derived category objects. In particular, if $j: U \to P$ is determined by a stable pair $I^\bullet = [{\mathcal O}_{U \times X} \to F]$ parameterized by $U$, then $I^\bullet = {\operatorname{\bf L}}(j \times {\operatorname{Id}}_X)^* {\mathbb{I}}^\bullet$ in $D(U \times X)$. We will use this when $j$ is the inclusion $Y \hookrightarrow P$ of a component of the $T$ fixed locus. The POT on $P$ is given by a morphism $${\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)_0 \otimes \omega_\pi [2] \to {\mathbb{L}}_P$$ in $D(P)$ to the cotangent complex of $P$ defined using the Atiyah class of ${\mathbb{I}}^\bullet$ (see [@PT], 2.3). The virtual normal bundle of $Y$ in $P$ is, by definition, the image of the moving part of $$\begin{aligned} \label{virtualnormalbundle} {\operatorname{\bf L}}j^* {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)_0 \otimes \omega_\pi [2] , {\mathcal O}_P) \end{aligned}$$ under the map $D_T(Y) \to K_T(Y)$ from the (perfect $T$ equivariant) derived category of $Y$ to its Grothendieck ring. The $T$ fixed part of defines a POT on $Y$ (using the isomorphism $({\operatorname{\bf L}}j^* {\mathbb{L}}_P)^T \to {\mathbb{L}}_Y$) which we will also have to identify. Using Serre duality for $\pi$, cohomology and base change for the square $$\xymatrix@C+15pt{ Y \times E \ar[r]^{j \times {\operatorname{Id}}_E} \ar[d]_\pi & P \times E \ar[d]^\pi \\ Y \ar[r]^j & P}$$ and the (obvious) “self duality" $${\operatorname{\bf R}}{\curly Hom}( {\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)_0 = {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}{\curly Hom}( {\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)_0 , {\mathcal O}_{P \times E}),$$ we can rewrite : $$\begin{aligned} & & {\operatorname{\bf L}}j^* {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)_0 \otimes \omega_\pi [2] , {\mathcal O}_P) \\ & = & {\operatorname{\bf L}}j^* {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet,{\mathbb{I}}^\bullet)_0 \otimes \omega_\pi [2], \omega_\pi[3]) \\ & = & {\operatorname{\bf R}}\pi_* {\operatorname{\bf L}}(j \times {\operatorname{Id}}_E)^* {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)[1] \\ & = & {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf L}}(j \times {\operatorname{Id}}_E)^* {\mathbb{I}}^\bullet, {\operatorname{\bf L}}(j \times {\operatorname{Id}}_E)^* {\mathbb{I}}^\bullet)_0[1] \\ & = & {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}(I^\bullet,I^\bullet)_0[1].\end{aligned}$$ Notation -------- Let $p : Y \times C \to Y$ and $\pi : Y \times E \to Y$ be the projections, $\iota : Y \times C \to Y \times E$ the zero section, so $p=\pi \iota$. We drop all notation for pullbacks, so we will just write $$0 \to S \to E \to Q \to 0$$ instead of $$0 \to \pi_{13}^* S \to \pi_3^* E \to \pi_{13}^* Q \to 0$$ for the pullback to $Y \times C$ of the universal sequence on ${\operatorname{Quot}}^e E \times C$. Similarly, we write $D \subset Y \times C$ instead of $\pi_{23}^* D$ for the pullback of the universal Cartier divisor, we write ${\mathcal O}$ for ${\mathcal O}_{Y \times C}$ and sometimes we just write $D$ for the invertible sheaf ${\mathcal O}(D)$. Let $F'={\mathcal O}[S^\lor]$, $F = D[S^\lor D]$, and let $I^\bullet = [ {\mathcal O}\to F]$ be the universal stable pair on $Y \times E$ as described in the proof of . The pair $I^\bullet$ corresponds to the map $j : Y \to P$ giving an isomorphism onto a component of the $T$ fixed subscheme. \[lem:virtualnormalbundle\] The virtual normal bundle of $Y$ in $P$ is given by $$\begin{aligned} \label{Nvir} N^{\rm vir}_{Y/P} &=& p_!(S^\lor D + D^\lor \land^2 E + D^\lor S \land^2 E + 2E + SE) \\ \nonumber & & \quad -p_!( S^\lor+2 \land^2 E +S \land^2 E + S + S^\lor \land^2 E) \quad \in K_T(Y). \end{aligned}$$ This is easy to prove because we can calculate everything on the level of the ($T$ equivariant) Grothendieck rings $K_T=K_T(Y \times E)$, $K_T(Y \times C)$, and $K_T(Y)$. We just need the classes of various derived duals—these are easier to calculate than actual duals or ${\operatorname{Ext}}$ groups. We seek a formula for the moving part of $$\begin{aligned} -\pi_! {\operatorname{\bf R}}{\curly Hom}(I^\bullet,I^\bullet)_0 &=& \pi_! {\mathcal O}_{Y \times E} - \pi_! {\operatorname{\bf R}}{\curly Hom}(I^\bullet, I^\bullet) \end{aligned}$$ in $K_T(Y)$ (the minus sign results from the shift $[1]$). The filtration of $F = D[S^\lor D]$ by gradings $$0 \to F^{\geq 1} \to F \to F / F^{\geq 1} \to 0$$ is identified with $$0 \to \iota_* S^\lor D \to F \to \iota_* D \to 0,$$ so we have $F = \iota_* D + \iota_* S^\lor D$ in $K_T(Y \times E)$. The Koszul complex $$\begin{aligned} \label{Koszulresn} \land^2 E^\lor \to E^\lor \to {\mathcal O}_{Y \times E} \to \iota_* {\mathcal O}_{Y \times C} \end{aligned}$$ is a ($T$-equivariant for the obvious actions) resolution of $\iota_* $ by vector bundles on $Y \times E$. Similarly, if $G$ is any vector bundle on $Y \times C$, we can resolve $\iota_* G$ by tensoring with $\pi^* G$, so we have $$\iota_* G = G( \land^\bullet E^\lor)$$ in $K_T$. Here we have denoted multiplication in $K_T$ by juxtaposition (so we often just write $1$ for a structure sheaf) and we use the shorthand $\land^\bullet E := \sum_i (-1)^i \land^i E$. (We are just reproducing the proof of GRR for $\iota$.) For the rest of the proof we use $F^\lor, (F')^\lor$ for the classes of the derived duals in $K_T$. Putting the Koszul resolution together with the filtration of $F$, we find that $$\begin{aligned} F & = & \iota_* D + \iota_* S^\lor D \\ &=& ( D + S^\lor D)(\land^\bullet E^\lor) \\ F^\lor & = & (D^\lor \land^2 E + D^\lor S \land^2 E)(\land^\bullet E^\lor) \\ &=& \iota_*(D^\lor \land^2 E + D^\lor S \land^2 E) \\ F F^\lor & = & (F')(F')^\lor \\ & = & (1 + S)( 1 + S^\lor)(\land^\bullet E)(\land^\bullet E^\lor)) \\ & = & \iota_*(2-2E+2\land^2 E + S -ES + S \land^2 E + S^\lor - S^\lor E + S^\lor \land^2 E) \end{aligned}$$ in $K_T$. Here we have used the “Koszul duality" identity $(\land^\bullet E )(\land^2 E^\lor) = \land^\bullet E^\lor$, which is basic linear algebra. Since $I^\bullet = {\mathcal O}_{Y \times E} - F$ in $K_T$, we have $$\begin{aligned} {\operatorname{\bf R}}{\curly Hom}(I^\bullet,I^\bullet) & = & ({\mathcal O}_{Y \times E}-F)({\mathcal O}_{Y \times E}-F^\lor) \\ & = & {\mathcal O}_{Y \times E} + \iota_*(2+2\land^2 E + S + S \land^2 E + S^\lor + S^\lor \land^2 E) \\ & & - \iota_*(D+S^\lor D + D^\lor \land^2 E + D^\lor S \land^2 E + 2E + SE) \end{aligned}$$ in $K_T(Y \times E)$. We can subtract off the first term to get the traceless part. The $T$-fixed part of this expression is $$\iota_*(2-D-S^\lor E ) ,$$ which we can discard to get the moving part. The result now follows because $\pi \iota = p,$ so $\pi_! \iota_* = p_!.$ \[rem:simplifiedvnb\] There are several situations where the virtual normal bundle formula simplifies dramatically. If $n=0$, then $D=1$ in the Grothendieck ring and we have $$\begin{aligned} \label{simplifiedNvir} N^{\rm vir}_{Y/P} &=& p_!(2E + SE)-p_!(\land^2 E +S + S^\lor \land^2 E) \quad \in K_T(Y). \end{aligned}$$ If, furthermore, we restrict to the locus $U \subset Y$ where $S$ is a subbundle (i.e. where $Q$ is locally free), then we have $\land^2 E=SQ$ and $E=S+Q$. The formula becomes $$\begin{aligned} \label{bestNvir} N^{\rm vir}_{Y/P} &=& p_!(S+Q+S^2) \quad \in K_T(Y). \end{aligned}$$ On this locus, the line subbundle $L$ generated by $S$ coincides with $S$, which is to say: the universal stable pair is nothing but the structure sheaf of an embedded curve $Z \subset E$ (really: a flat family of embedded curves $Z \subset U \times E$) . Using the formula , we recognize $S+Q+S^2$ as the class of the moving part of the normal bundle $N_{Z/U \times E}$ in the Grothendieck group (the fixed part of the normal bundle is our old friend $S^\lor Q$). In the course of the proof we saw that the $T$-fixed part of was given by $$\begin{aligned} p_!(D-2+S^\lor E ) & = & p_!(D-1+S^\lor Q) \\ & = & {\operatorname{\bf R}}p_*({\mathcal O}_D(D) \oplus {\curly Hom}(S, Q) \end{aligned}$$ in the Grothendieck ring. Our next step is to lift this equality to the actual derived category. We begin with the case $n=0$. \[lem:vfc1\] When $n=0$, the $T$-fixed part of is isomorphic to ${\operatorname{\bf R}}p_* {\curly Hom}(S,Q)$. The $T$-fixed stable pairs perfect obstruction theory gives rise to a virtual class $$[ {\operatorname{Quot}}^e E ]^{\rm vir} \in A_{1-g+d-2e}({\operatorname{Quot}}^e E).$$ The sheaf $F = {\mathcal O}_{Y \times C}[S^\lor]$ on $Y \times E$ admits a $T$-equivariant (i.e. graded) locally free resolution $$\pi^* S^\lor \land^2 E^\lor \oplus \pi^* \land^2 E^\lor \to \pi^* E^\lor S^\lor \oplus \pi^* E^\lor \to \pi^* S^\lor \oplus \pi^* {\mathcal O}_{Y \times C}$$ whose first few graded pieces look like: $$\begin{array}{rcccl} & & & & {\mathcal O}_{Y \times C} \\ & & E^\lor & \to & S^\lor \oplus E^\lor \\ \land^2 E^\lor & \to & E^\lor S^\lor \oplus E^\lor E^\lor & \to & E^\lor S^\lor \oplus {\operatorname{Sym}}^2 E^\lor \\ S^\lor \land^2 E^\lor \oplus E^\lor \land^2 E^\lor & \to & {\operatorname{Sym}}^2 E^\lor S^\lor \oplus {\operatorname{Sym}}^2 E^\lor E^\lor & \to & {\operatorname{Sym}}^2 E^\lor S^\lor \oplus {\operatorname{Sym}}^3 E^\lor \end{array}$$ (here we write, e.g. ${\operatorname{Sym}}^2 E^\lor S^\lor$ for $({\operatorname{Sym}}^2 E^\lor) \otimes S^\lor$ as opposed to ${\operatorname{Sym}}^2 (E^\lor \otimes S^\lor)$). That is, all the rows except the first are exact except the cokernel of the rightmost map in the second row is $S^\lor$. Appyling ${\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, F)$ to this, we get a complex quasi-isomorphic to ${\operatorname{\bf R}}{\curly Hom}(F,F).$ Its graded pieces look like: $$\begin{array}{rcccl} & & & & S \land^2 E \\ & & SE & \to & \land^2 E \oplus \land^2 E \\ S & \to & E \oplus E & \to & S^\lor \land^2 E \\ {\mathcal O}_{Y \times C} \oplus {\mathcal O}_{Y \times C} & \to & S^\lor E \\ S^\lor \end{array}$$ From a brief inspection of the boundary maps, two things are clear: (1) The kernel of the leftmost map is just $F$, so ${\curly Hom}(F,F)=F$ (we could see this directly anyway) and (2) In grading zero, the $T$-fixed subcomplex (the penultimate row above) is cohomologically formal (quasi-isomorphic to the direct sum of its shifted cohomology sheaves). That is, we have $$\begin{aligned} {\operatorname{\bf R}}{\curly Hom}(F,F)^T & = & {\mathcal O}_{C \times Y} \oplus S^\lor Q [-1] \\ & = & {\curly Hom}(F,F)^T \oplus \iota_* {\curly Hom}(S,Q)[-1] \end{aligned}$$ in $D(Y \times E)$. Similarly, if we apply ${\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, {\mathcal O}_{Y \times E})$ to the above resolution of $F$, we get a complex quasi-isomorphic to ${\operatorname{\bf R}}{\curly Hom}(F, {\mathcal O}_X)$. Its first few graded pieces look like $$\begin{array}{rcccl} & & & & S \land^2 E \\ & & SE & \to & \land^2 E \oplus E^\lor S \land^2 E \\ S & \to & E \oplus E^\lor S E & \to & E^\lor \land^2 E \oplus {\operatorname{Sym}}^2 E^\lor S \land^2 E \\ {\mathcal O}_{Y \times C} \oplus E^\lor S & \to & E^\lor E \oplus {\operatorname{Sym}}^2 E^\lor SE & \to & {\operatorname{Sym}}^2 E^\lor \land^2 E \oplus {\operatorname{Sym}}^3 E^\lor S \land^2 E. \end{array}$$ Examining the boundary maps and doing a little linear algebra, we can see that all the rows are exact except the first two. In particular, the $T$-fixed subcomplex (the bottom row) is exact, hence quasi-isomorphic to zero. The map in the second row is monic with cokernel $\land^2 E$, and we see that $${\operatorname{\bf R}}{\curly Hom}(F,{\mathcal O}_{Y \times E}) = F \otimes \pi^* S \land^2 E [-2].$$ In fact, for any stable pair ${\mathcal O}_X \to F$, we always have $${\operatorname{\bf R}}{\curly Hom}(F,{\mathcal O}_X) = {\curly Ext}^2(F,{\mathcal O}_X)[-2]$$ because the sheaves ${\curly Ext}^i(F,{\mathcal O}_X)$ vanish for $i \neq 2$ because $F$ is supported in codimension $2$ and has cohomological dimension $\leq 2$ (c.f. Page 10 in [@PT3]). The rest of the argument is quite formal, but we include it for completeness. Applying ${\operatorname{\bf R}}{\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, {\mathcal O}_{Y \times E})$ to the triangle $$F[-1] \to I^\bullet \to {\mathcal O}_{Y \times E},$$ we get a triangle $${\mathcal O}_{Y \times E} \to {\operatorname{\bf R}}{\curly Hom}(I^\bullet , {\mathcal O}_X) \to {\operatorname{\bf R}}{\curly Hom}(F[-1] , {\mathcal O}_{Y \times E}) .$$ Examining the associated long exact cohomology sequence and using the established vanishings, we find: $$\begin{aligned} {\curly Hom}(I^\bullet, {\mathcal O}_{Y \times E}) &=& {\mathcal O}_{Y \times E} \\ {\curly Ext}^1(I^\bullet, {\mathcal O}_{Y \times E}) & = & {\curly Ext}^2(F,{\mathcal O}_X) \\ {\curly Ext}^i(I^\bullet, {\mathcal O}_{Y \times E}) & = & 0, \quad \quad i \neq 0,1. \end{aligned}$$ Applying ${\operatorname{\bf R}}{\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, {\mathcal O}_{Y \times E})$ to the same triangle, we get a triangle $$F[-1] \to {\operatorname{\bf R}}{\curly Hom}(I^\bullet, F)[-1] \to {\operatorname{\bf R}}{\curly Hom}(F,F)$$ whose long exact cohomology sequence looks like $$\begin{array}{rccccl} & 0 & \to & {\curly Ext}^{-1}(I^\bullet,F) & \to & {\curly Hom}(F,F) \\ \to & F & \to & {\curly Hom}(I^\bullet,F) & \to & {\curly Ext}^1(F,F) \\ \to & 0 & \to & {\curly Ext}^1(I^\bullet,F) & \to & {\curly Ext}^2(F,F). \end{array}$$ The map ${\curly Hom}(F,F) \to F$ is the one induced by $s : {\mathcal O}_{Y \times E} \to F$ and is given by $\phi \mapsto \phi(s(1))$. It is easily seen to be an isomorphism since this $F$ is in fact a sheaf of ${\mathcal O}_{Y \times E}$ *algebras* (Remark \[rem:Fasacurve\]), hence we can use the multiplication by any local section. We conclude: $$\begin{aligned} {\curly Hom}(I^\bullet,F) &=& {\curly Ext}^1(F,F) \\ {\curly Ext}^1(I^\bullet,F) &=& {\curly Ext}^2(F,F) \\ {\curly Ext}^i(I^\bullet,F) &=& 0, \quad \quad i \neq 0,1. \end{aligned}$$ Finally we apply ${\operatorname{\bf R}}{\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, I^\bullet)$ to the same triangle to get the following long exact sequence: $$\begin{array}{rccccl} & 0 & \to & {\mathcal O}_{Y \times E} & \to & {\mathcal O}_{Y \times E} \\ \to & {\curly Ext}^1(F,F) & \to & {\curly Ext}^1(I^\bullet,I^\bullet) & \to & {\curly Ext}^1(I^\bullet,{\mathcal O}_{Y \times E}) \\ \to & {\curly Ext}^2(F,F) & \to & {\curly Ext}^2(I^\bullet,I^\bullet) & \to & 0 \end{array}$$ The map in the first row is an isomorphism (c.f. Lemma 1.20 in [@PT]) and we have already established the vanishings $${\curly Ext}^1(I^\bullet,{\mathcal O}_{Y \times E})^T = {\curly Ext}^2(F,{\mathcal O}_{Y \times E})^T = 0$$ and ${\curly Ext}^2(F,F)^T=0$, so we have: $$\iota_* {\curly Hom}(S,Q) = {\curly Ext}^1(F,F)^T = {\curly Ext}^1(I^\bullet,I^\bullet)^T .$$ Since scalar multiplication ${\mathcal O}_{Y \times E} \to {\curly Hom}(I^\bullet,I^\bullet)$ is an isomorphism and the trace map splits this off of ${\operatorname{\bf R}}{\curly Hom}(I^\bullet,I^\bullet)$ we may in fact view the above isomorphism as an isomorphism $$\iota_* {\curly Hom}(S,Q) = {\operatorname{\bf R}}{\curly Hom}(I^\bullet,I^\bullet)^T_0[1]$$ in $D(Y \times E)$. Applying ${\operatorname{\bf R}}\pi_*$ and noting that ${\operatorname{\bf R}}\pi_* \iota_* = {\operatorname{\bf R}}p_*$ completes the proof. One can check that this virtual class on ${\operatorname{Quot}}^e E$ is the “well known" [@CFK], [@MO] class discussed in Section \[section:virtualclass\]. We can use the same technique to handle the general case. \[thm:vfc\] The $T$-fixed part of the stable pairs POT on $Y$ is given by $${\operatorname{\bf R}}p_* {\curly Hom}(S,Q) \oplus T {\operatorname{Sym}}^d C,$$ hence the corresponding virtual fundamental class is $$[ {\operatorname{Quot}}^e E ]^{\rm vir} \times [{\operatorname{Sym}}^d C],$$ where $[{\operatorname{Quot}}^e E]^{\rm vir}$ is the virtual class on ${\operatorname{Quot}}^e E$ as in Lemma \[lem:vfc1\]. According to the proof of , the universal stable pair on $Y \times E$ is $$I^\bullet = [{\mathcal O}_{Y \times E} \to F ],$$ where $F=F'(D)$ is obtained from $F'={\mathcal O}_{Y \times C}[S^\lor]$ by tensoring with the section ${\mathcal O}_{Y \times E} \to {\mathcal O}_{Y \times E}(D)$ determined by the Cartier divisor $D \subset Y \times E$. Note $F'$ is pulled back from ${\operatorname{Quot}}^e E \times C$ and $D$ is pulled back from ${\operatorname{Sym}}^d C \times C.$ In particular, ${\mathcal O}_{Y \times E}(D)$ is locally free, so we have equalities: $$\begin{aligned} {\operatorname{\bf R}}{\curly Hom}(F'(D),F'(D)) &=& {\operatorname{\bf R}}{\curly Hom}(F',F') \\ {\operatorname{\bf R}}{\curly Hom}( F'(D), {\mathcal O}_{Y \times E} ) &=& {\operatorname{\bf R}}{\curly Hom}(F', {\mathcal O}_{Y \times E})(-D), \end{aligned}$$ and so forth (we have used the Grothendieck groups analogues in the proof of Lemma \[lem:virtualnormalbundle\]). In particular, we can see that $${\operatorname{\bf R}}{\curly Hom}(F'(D), {\mathcal O}_{Y \times E})^T = 0,$$ just as we did in the proof of Lemma \[lem:vfc1\]. The only departure from the proof of Lemma \[lem:vfc1\] occurs when we apply ${\operatorname{\bf R}}{\curly Hom}( { \hspace{0.05in} {\rm \_} \hspace{0.05in} }, {\mathcal O}_{Y \times E})$ to the triangle $$F'(D)[-1] \to I^\bullet \to {\mathcal O}_{Y \times E}.$$ This time the long exact cohomology sequence looks like $$\begin{array}{rccccl} & 0 & \to & {\curly Ext}^{-1}(I^\bullet,F'(D)) & \to & {\curly Hom}(F',F') \\ \to & F'(D) & \to & {\curly Hom}(I^\bullet,F'(D)) & \to & {\curly Ext}^1(F',F') \\ \to & 0 & \to & {\curly Ext}^1(I^\bullet,F'(D)) & \to & {\curly Ext}^2(F',F'). \end{array}$$ As in the proof of Lemma \[lem:vfc1\], we have ${\curly Hom}(F',F')=F',$ and the map ${\curly Hom}(F',F') \to F'(D)$ is the obvious one. The $T$ fixed part of this map is the usual map ${\mathcal O}_{Y \times C} \to {\mathcal O}_{Y \times C}(D)$ whose cokernel is ${\mathcal O}_D(D)$, so the $T$-fixed part of the middle row yields a SES $$0 \to {\mathcal O}_D(D) \to {\curly Ext}^1(I^\bullet, F'(D))^T \to {\curly Ext}^1(F',F')^T \to 0.$$ This sequence splits using the identification ${\curly Ext}^1(F',F')^T = {\curly Ext}^1(I^\bullet, F')^T$ from the proof of and the natural inclusion $${\curly Ext}^1(I^\bullet, F')^T \to {\curly Ext}^1(I^\bullet, F')^T(D) = {\curly Ext}^1(I^\bullet, F'(D))^T.$$ Exactly as in the proof of we obtain an identification $$\iota_* {\curly Hom}(S,Q) \oplus {\mathcal O}_D(D) = {\operatorname{\bf R}}{\curly Hom}(I^\bullet, I^\bullet)^T_0[1].$$ Finally, we use the standard formula $$\begin{aligned} T {\operatorname{Sym}}^d C & = & p_* {\curly Hom}(I_D, {\mathcal O}_D) \\ &=& p_* {\curly Hom}( {\mathcal O}_{{\operatorname{Sym}}^d C \times C}(-D), D) \\ & = & {\operatorname{\bf R}}p_* {\mathcal O}_D(D) \end{aligned}$$ for the tangent space of the (smooth) Hilbert scheme ${\operatorname{Sym}}^d C$. Computations {#section:computations} ============ Consider the expression for the virtual normal bundle of $Y={\operatorname{Quot}}^e E \times {\operatorname{Sym}}^n C$ in $P=P_{2-2g-e+2n}(E,2)$ given in Lemma \[lem:virtualnormalbundle\]. It is clear from GRR that $e_T(-N_{Y/P}^{\rm vir})$ can be expressed in terms of tautological classes on $Y$; therefore $$\begin{aligned} \label{localcontribution} \int_{[Y]^{\rm vir}} e_T(-N^{\rm vir}_{Y/P}) & \in & {\mathbb{Z}}[t,t^{-1}] \end{aligned}$$ can be computed using Theorem \[thm:virtualintersectionnumbers\]. The purpose of this section is to write out an explicit formula for $e_T(-N_{Y/P}^{\rm vir})$ and calculate in various cases. A full reconciliation of our computations with the prediction of Equation \[ZPT\] poses combinatorial difficulties not addressed herein. On the other hand, the formula derived below is simple enough that it is feasible to compute any particular invariant $P_{n,2}(d)$ (with minor computer assistance). Notation {#notation-1 .unnumbered} -------- Let $\pi_1,\pi_2$ be the projections from $Y$ to ${\operatorname{Quot}}^e E$, ${\operatorname{Sym}}^n C$, respectively. Let $a_1 \in {\operatorname{H}}^2(Y)$ be the pullback of the $a$ class on ${\operatorname{Quot}}^e E$ via $\pi_1$ (Section \[section:tautologicalclasses\]) and let $b_{1,i} \in {\operatorname{H}}^1(Y)$ be the pullback of the $b_i$ class on ${\operatorname{Quot}}^e E$ via $\pi_1$. Similarly, let $a_2 \in {\operatorname{H}}^2(Y)$ be the pullback of the $a$ class (the coefficient of $\eta$ in the Künneth decomposition of the universal divisor $D \subset {\operatorname{Sym}}^n C \times C$) on ${\operatorname{Sym}}^n C$ via $\pi_2$ and let $b_{2,i} \in {\operatorname{H}}^1(Y)$ be the pullback of the $b_i$ class on ${\operatorname{Sym}}^n C$. Set $\theta_j := \sum_{i=1}^g b_{j,i}b_{j,g+i} \in {\operatorname{H}}^2(Y)$, so $\theta_j$ is the pullback of the $\theta$ class on the Jacobian under the composition of $\pi_j$ and the Abel-Jacobi map. It is convenient to set $$B := \sum_{i=1}^g b_{1,i} b_{2,g+i}-b_{1,g+i} b_{2,i}$$ so that $$\begin{aligned} \label{crossterm} \left ( \sum_{i=1}^{2g} b_{1,i} \delta_i \right ) \left ( \sum_{i=1}^{2g} b_{2,i} \delta_i \right ) &=& -B \eta \end{aligned}$$ in ${\operatorname{H}}^*(Y \times C)$. Write $\approx$ for equality in ${\operatorname{H}}^*({\operatorname{Pic}}C)$ modulo odd monomials (c.f. Section \[section:tautologicalclasses\]). Then we have $$B^{2l} \approx (-1)^l \begin{pmatrix} 2l \\ l \end{pmatrix} l! \sum_{1 \leq i_1 < \cdots < i_l \leq g} b_{1,i_1}b_{1,g+i_1} \cdots b_{1,i_l}b_{1,g+i_l} b_{2,i_1} b_{2,g+i_1} \cdots b_{2,i_l} b_{2,g+i_l}.$$ For distinct $i_1,\dots,i_m \in \{ 1 ,\dots, g \}$, Theorem \[thm:virtualintersectionnumbers\] yields $$\begin{aligned} \label{formula1} \int_{[{\operatorname{Quot}}^e E]^{\rm vir}} a^{1-g+d-2e-m} b_{i_1}b_{g+i_1} \cdots b_{i_m}b_{g+i_m} &=& 2^{g-m} \\ \nonumber \int_{{\operatorname{Sym}}^n C} a^{n-m} b_{i_1} b_{g+i_1} \cdots b_{i_m} b_{g+i_m} &=& 1.\end{aligned}$$ Since $$\begin{aligned} \theta^j &=& j! \sum_{1\leq i_1 < \cdots < i_j \leq g} b_{i_1}b_{g+i_1} \cdots b_{i_j}b_{g+i_j} \end{aligned}$$ and the product of an even and an odd monomial is odd it follows that $\theta_1^j \theta_2^k B^{2l}$ is equal (modulo odd monomials in the $b_{1,i}$ and $b_{2,i}$) to $$(-1)^l \begin{pmatrix} 2l \\ l \end{pmatrix} j!k!l!$$ times a sum of $$\begin{pmatrix} g \\ l \end{pmatrix} \begin{pmatrix} g-l \\ j \end{pmatrix} \begin{pmatrix} g-l \\ k \end{pmatrix}$$ terms of the form $$b_{1,i_1}b_{1,g+i_1} \cdots b_{1,i_{j+l}} b_{1,g+i_{j+l}} b_{2,i_1} b_{2,g+i_1} \cdots b_{2,i_{k+l}} b_{2,g+i_{k+l}} .$$ Since $[Y]^{\rm vir} = [{\operatorname{Quot}}^e E]^{\rm vir} \times [{\operatorname{Sym}}^n C]$, the formula $$\begin{aligned} \label{mainformula} \int_{[Y]^{\rm vir}} a_1^{1-g+d-2e-j-l} a_2^{n-k-l} \theta_1^j \theta_2^k B^{2l} &=& (-1)^l \begin{pmatrix} 2l \\ l \end{pmatrix} \frac{g!(g-l)!2^{g-j-l}}{(g-j-l)!(g-k-l)!} \end{aligned}$$ therefore follows from after slight simplifications. In this notation, we have $$\begin{aligned} {\operatorname{ch}}S^\lor &=& e^{a_1-t}(1-\sum_{i=1}^{2g}b_{1,i} \delta_i - e \eta - \theta_1 \eta) \\ {\operatorname{ch}}D &=& e^{a_2}(1-\sum_{i=1}^{2g} b_{2,i} \delta_i + n \eta - \theta_2 \eta) \end{aligned}$$ in ${\operatorname{H}}^*_T(Y \times C)$. To calculate, for example, $e_T(p_!S^\lor D)$, we first calculate $${\operatorname{ch}}S^\lor D = e^{a_1+a_2-t} \left ( 1+\sum_{i=1}^{2g}(b_{1,i}+b_{2,i})\delta_i +(n-e-\theta_1-\theta_2-B)\eta \right )$$ using , then by GRR for $p :Y \times C \to Y$, we compute $${\operatorname{ch}}p_!S^\lor D = e^{a_1+a_2-t} ( 1-g+n-e-\theta_1-\theta_2-B).$$ Evidently $p_!S^\lor D$ has the same Chern character as the product of a line bundle with $c_1=a_1+a_2-t$ and a (virtual) vector bundle of rank $1-g+n-e$ and total Chern class $e^{-\theta_1-\theta_2-B}$, so from the usual formula for the Euler class of such a product, we obtain: $$e_T(p_! S^\lor D) = (a_1+a_2-t)^{1-g+n-e} \exp \left ( \frac{-\theta_1-\theta_2-B}{a_1+a_2-t} \right ).$$ Similar calculations yield $$\begin{aligned} \label{eTNvir} e_T(-N_{Y/P}^{\rm vir}) &=& \frac{e_T(p_!(S^\lor+2\land^2 E+S \land^2 E+S+S^\lor \land^2 E))}{e_T(p_!(S^\lor D+D^\lor \land^2 E+D^\lor S \land^2 E+2E +SE))} \\ &=& \frac{(a_1-t)^{1-g-e}(2t)^{2-2g+2d}(3t-a_1)^{1-g+e+d}}{(a_1+a_2-t)^{1-g+n-e}(2t-a_2)^{1-g+d-n}(3t-a_1-a_2)^{1-g+d+e-n}} \\ & & \cdot \frac{(t-a_1)^{1-g+e}(t+a_1)^{1-g+d-e}}{(t)^{4-4g+2d}(2t-a_1)^{2-2g+d+2e}} \\ & & \cdot \exp \left ( \frac{-\theta_1}{a_1-t} \right ) \exp \left ( \frac{-\theta_1}{3t-a_1} \right ) \exp \left ( \frac{-\theta_1}{t-a_1} \right ) \exp \left ( \frac{-\theta_1}{t+a_1} \right ) \\ & & \cdot \exp \left ( \frac{\theta_1+\theta_2+B}{a_1+a_2-t} \right ) \exp \left ( \frac{\theta_2}{2t-a_2} \right ) \exp \left ( \frac{\theta_1+\theta_2+B}{3t-a_1-a_2} \right ) \exp \left ( \frac{2 \theta_1}{2t-a_1} \right ). \end{aligned}$$ Expanding this out and integrating over $[Y]^{\rm vir}$ using presents no particular difficulty in any given case, though a simple general formula is difficult to obtain. When $n=0$, $Y={\operatorname{Quot}}^e E$, and this formula simplifies to $$\begin{aligned} \label{eTNvir} e_T(-N^{\rm vir}_{Y/P}) &=& \frac{e_T(p_!(\land^2 E +S+S^\lor \land^2 E))}{e_T(p_!(2E+SE))} \\ \nonumber &=& \frac{(2t)^{1-g+d}(t-a)^{1-g+e}(t+a)^{1-g+d-e}}{(t)^{4-4g+2d}(2t-a)^{2-2g+d+2e}} \\ \nonumber & & \cdot \exp \left ( \frac{-\theta}{t-a} \right ) \exp \left ( \frac{-\theta}{t+a} \right ) \exp \left ( \frac{2 \theta}{2t-a} \right ) \\ \nonumber &=& 2^{g-2e-1}t^{3g-3-d-2e} -2^{g-2e-1}t^{3g-4-d-2e}\theta \\ \nonumber & & +2^{g-2e-2}t^{3g-4-d-2e}(2-2g+3d-2e)a + \dots \end{aligned}$$ where the $\dots$ are terms in $R^{>2}({\operatorname{Quot}}^e E)$. Expected dimension one ---------------------- Suppose $E$ is a rank $2$ bundle on $C$ whose degree $d$ has the same parity as the genus $g$ of $C$. Define an integer $e$ by setting $d-g=2e.$ Then the smallest power of $q$ appearing in the Laurent expansion of $Z_2^{\rm PT}(d)$ occurs when $i=e$ in . It is given by $t^{4g-4-2d}2^{3g-1-d}(d+1-g)q^{2-d-g},$ so the DT=PT invariant of $E$ in minimal Euler characteristic is $$\begin{aligned} \label{example1} P_{2-d-g,2}(d) &=& t^{4g-4-2d}2^{3g-1-d}.\end{aligned}$$ On the other hand, Theorem \[thm:Tfixedstablepairs\] provides an identification $$P_{2-d-g}(E,2)^T = {\operatorname{Quot}}^e E$$ (after possibly throwing away fixed components with negative expected dimension). The formulae $$\begin{aligned} \int_{[{\operatorname{Quot}}^e E]^{\rm vir}} a &=& 2^g \\ \int_{[{\operatorname{Quot}}^e E]^{\rm vir}} \theta &=& g2^{g-1} \end{aligned}$$ are obtained by taking $k=g,g-1$ (respectively) in . Using , we then compute $$\begin{aligned} P_{2-d-g,2}(d) &=& \int_{[P_{2-d-g}(E,2)^T]^{\rm vir}} e_T(-N^{\rm vir}) \\ &=& -2^{g-2e-1}t^{3g-4-d-2e} \int_{ [{\operatorname{Quot}}^e E]^{\rm vir} } \theta \\ & & +2^{g-2e-2}t^{3g-4-d-2e}(2-2g+3d-2e) \int_{ [{\operatorname{Quot}}^e E]^{\rm vir} } a \\ &=& -2^{2g-d-1}t^{4g-4-2d}(g2^{g-1}) +2^{2g-d-2}t^{4g-4-2d}(2-g+2d)(2^g) \\ &=& t^{4g-4-2d}2^{3g-1-d} \end{aligned}$$ in agreement with the formula obtained from the GW/PT correspondence and the equivalence of DT and PT invariants in minimal Euler characteristic. Target genus zero ----------------- When $C={\mathbb{P}}^1$ the Jacobian is trivial so the ${\operatorname{Quot}}$ schemes are just projective spaces. We can identify the virtual fundamental class on the ${\operatorname{Quot}}$ scheme and explicitly compute the PT invariants from first principles (i.e. without using Theorem \[thm:virtualintersectionnumbers\] or the GRR calculation above). Consider the case where $C={\mathbb{P}}^1$ and $E= {\mathcal O}(d_1) \oplus {\mathcal O}(d_2)$, with $d=d_1+d_2$. If we make the identification $${\operatorname{Quot}}^e E \times {\operatorname{Sym}}^n {\mathbb{P}}^1 \times {\mathbb{P}}^1 \cong {\mathbb{P}}^N \times {\mathbb{P}}^n \times {\mathbb{P}}^1$$ (so $p$ is the projection on the first two factors), then the bundles appearing in the virtual normal bundle formula of Lemma \[lem:virtualnormalbundle\] are tensor products and duals of the following bundles: $$\begin{aligned} S &=& {\mathcal O}(-1,0,e)_t \\ E &=& {\mathcal O}(0,0,d_1)_t \oplus {\mathcal O}(0,0,d_2)_t \\ \land^2 E & = & {\mathcal O}(0,0,d)_{2t} \\ D &=& {\mathcal O}(0,1,n)_0. \end{aligned}$$ We have used subscripts to keep track of the weight of the $T$ action under these identifications. (The description of the universal bundle on ${\operatorname{Sym}}^n {\mathbb{P}}^1 \times {\mathbb{P}}^1$ is elementary and we discussed the universal bundle $S$ on ${\operatorname{Quot}}^e E \times {\mathbb{P}}^1$ in Section \[section:quotschemes\].) Assume $1+d-2e \geq 0$. If $E$ is balanced, the virtual fundamental class on ${\operatorname{Quot}}^e E$ is its usual fundamental class. In any case, ${\operatorname{Quot}}^e E$ is a projective space of dimension at least $1+d-2e$ and the virtual class is just the fundamental class of a linearly embedded projective space of the expected dimension $1+d-2e$. As discussed in , $${\operatorname{Quot}}^e E = {\mathbb{P}}( {\operatorname{H}}^0({\mathbb{P}}^1,{\mathcal O}(d_1-e)) \oplus {\operatorname{H}}^0({\mathbb{P}}^1,{\mathcal O}(d_2-e))).$$ We have $h^0({\mathbb{P}}^1,{\mathcal O}(d_1-e)=1+d_1-e$ and $h^0({\mathbb{P}}^1,{\mathcal O}(d_2-e))=1+d_2-e$, so the moduli space is smooth of the expected dimension unless one of the $1+d_i-e$ is negative and the formula is invalid. They can’t *both* be negative because we are assuming $$d_1-e+d_2-e \geq -1,$$ and balancing will also ensure both expressions are nonnegative. Say $1+d_1-e$ is negative, so we have: --------------------------------------------------- --- ---------------- Actual Dimension of ${\operatorname{Quot}}^e E$ = $d_2-e$ Expected Dimension of ${\operatorname{Quot}}^e E$ = $1+d_1+d_2-2e$ Excess Dimension = $e-d_1-1.$ --------------------------------------------------- --- ---------------- In this situation, ${\mathcal O}(e)$ admits no nonzero map to ${\mathcal O}(d_1)$, so every point of the ${\operatorname{Quot}}$ scheme will look like $$0 \to {\mathcal O}(e) \to {\mathcal O}(d_1) \oplus {\mathcal O}(d_2) \to {\mathcal O}(d_1) \oplus T \to 0,$$ where $T$ is a torsion sheaf of degree $d_2-e$. The torsion sheaf doesn’t contribute to the higher direct image under $$p : {\operatorname{Quot}}^e E \times {\mathbb{P}}^1 \to {\operatorname{Quot}}^e E,$$ and we compute $$\begin{aligned} {\operatorname{R}}^1 p_* S^\lor Q & = & {\operatorname{R}}^1p_*( {\mathcal O}(-1,e)^\lor \otimes {\mathcal O}(0,d_1) ) \\ & = & {\operatorname{R}}^1 p_* {\mathcal O}(1,d_1-e) \\ & = & {\mathcal O}(1) \otimes_{\mathbb{C}}{\operatorname{H}}^1({\mathbb{P}}^1, {\mathcal O}(d_1-e)) \\ & = & {\mathcal O}(1)^{e-d_1-1} \end{aligned}$$ using the projection formula. Since the virtual class is $e( {\mathcal O}(1)^{e-d_1-1}) \cap [{\operatorname{Quot}}^e E]$ the proof is complete. Let $a_i \in {\operatorname{H}}^2({\mathbb{P}}^N \times {\mathbb{P}}^n)$ be the pullback of the hyperplane class along $\pi_i$ (notice that the hyperplane class is the $a$ class when the rank one Quot scheme is identified with a projective space). Because of the lemma, the actual value of $$N = \dim {\operatorname{Hom}}({\mathcal O}(e),{\mathcal O}(d_1)\oplus {\mathcal O}(d_2)) - 1$$ will be irrelevant. When we integrate over $[{\mathbb{P}}^N]^{\rm vir}$ it will be as if $$N = 1+d-2e.$$ Pushing forward over the ${\mathbb{P}}^1$ factor is easy; the projection formula implies $$p_! {\mathcal O}(a,b,c) = (c+1) {\mathcal O}(a,b)$$ in the Grothendieck ring. For example, we find $$e_T(p_! S) = (t-a_1)^{e+1}.$$ Since we now know how to work out the $T$ equivariant Euler class of the virtual normal bundle, the computation of the PT invariant is is now just a matter of putting the pieces together: $$\begin{aligned} P_{2+m,2}(d) & = & \int_{P_{2+m}(E,2)^T} e_T(-N^{\rm vir}) \\ & = & \sum_{\stackrel{2n-e=m}{1+d-2e \geq 0}} \int_{[{\mathbb{P}}^N]^{\rm vir} \times {\mathbb{P}}^n} \frac{e_T(p_!(S^\lor+2 \land^2 E +S \land^2 E + S + S^\lor \land^2 E))}{e_T(p_!(S^\lor D + D^\lor \land^2 E + D^\lor S \land^2 E + 2E + SE))} \\ & = & \sum_{\stackrel{2n-e=m}{1+d-2e \geq 0}} C(d,e,n), \end{aligned}$$ where we define $C(d,e,n)$ to be the coefficient of $a_1^{1+d-2e}a_2^n$ in the expression $$\frac{(a_1-t)^{1-e}(2t)^{2d+2}(3t-a_1)^{d+e+1}(t-a_1)^{e+1}(a_1+t)^{1-d-e}}{(a_1+a_2+t)^{1+n-e}(2t-a_2)^{1+d-n}(3t-a_1-a_2)^{1+d+e-n}(t)^{2d+4}(2t-a_1)^{d+2e+2}}.$$ Presumably it is possible to completely reconcile this with Equation \[ZPT\], but we will content ourselves here with a few explicit computations. Local ${\mathbb{P}}^1$ {#section:localP1} ---------------------- Consider the bundle $E = {\mathcal O}(-1)^{\oplus 2}$ on ${\mathbb{P}}^1$. The stable pairs spaces $P_m(E,2)$ are easily seen to be compact, and each has expected dimension zero, so the residue invariants are bonafide integrals taking values in ${\mathbb{Z}}$. The total space $E$ is Calabi-Yau, so the invariants can be computed as weighted Euler characteristics of the moduli space, weighted by Behrend’s constructible function (Lemma 1.3 in [@PT3], [@Beh]). The pairs space $P_4(E,2)$, for example, is discussed extensively in Section 4.1 of [@PT]. It can be described as the closed subscheme $$P_4(E,2) \hookrightarrow {\operatorname{Spec}}_{{\mathbb{P}}^3} {\mathcal O}_{{\mathbb{P}}^3}[{\mathcal O}_{{\mathbb{P}}^3}(2)]$$ of the first infinitesimal neighborhood of ${\mathbb{P}}^3$ in ${\mathcal O}(-2)$ determined by the ideal generated by sections of ${\mathcal O}_{{\mathbb{P}}^3}(2)$ vanishing along a quadric ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \subseteq {\mathbb{P}}^3$. Locally, it looks like a product of $$Z = {\operatorname{Spec}}{\mathbb{C}}[x,y]/(y^2,xy)$$ and ${\mathbb{A}}^2$. The normal cone of $Z \subset {\mathbb{A}}^2_{x,y}$ has two irreducible components, one lying over the origin with length $2$ at its generic point and one generically smooth component surjecting to the topological space $|Z|=|{\mathbb{A}}^1|$, so the characteristic cycle ([@Beh], 1.1) of $Z$ is $\mathfrak{c}_Z=-[{\mathbb{A}}^1]+2[0]$. These cycles are smooth, so the local Euler obstruction construction of Section 1.2 in [@Beh] simply yields their characteristic functions, so Behrend’s function is $-1$ on the smooth locus and $+1$ at the origin. Since Behrend’s function $\nu$ is local and respects products, we conclude that it is given by $1$ on the quadric ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \subset {\mathbb{P}}^3$ and $-1$ away from the quadric, so the pairs invariant is $$P_{4,2}(-2) = -1 \cdot \chi({\mathbb{P}}^3 \setminus {\mathbb{P}}^1 \times {\mathbb{P}}^1) + 1 \cdot \chi({\mathbb{P}}^1 \times {\mathbb{P}}^1) = -1 \cdot 0+1 \cdot 4=4.$$ From our point of view, we have $$P_4(E,2)^T = {\operatorname{Quot}}^{-2} E = {\mathbb{P}}{\operatorname{Hom}}( {\mathcal O}(-2), E) \cong {\mathbb{P}}^3.$$ The universal sequence $$0 \to S \to \pi_2^* E \to Q \to 0$$ on ${\operatorname{Quot}}^{-2} E \times {\mathbb{P}}^1$ is identified with the sequence $$0 \to {\mathcal O}(-1,-2) \to {\mathcal O}(0,-1)^{\oplus 2} \to Q \to 0$$ on ${\mathbb{P}}^3 \times {\mathbb{P}}^1$. The quotient sheaf $Q$ restricts to ${\mathcal O}$ on $\{ P \} \times {\mathbb{P}}^1$ for a generic $P \in {\mathbb{P}}^3$, but $Q$ restricts to the direct sum of ${\mathcal O}(-1)$ and the structure sheaf of a point when $P$ is in the stratum $${\operatorname{Quot}}^{-1} E \times {\operatorname{Sym}}^1 {\mathbb{P}}^1 \hookrightarrow {\operatorname{Quot}}^{-2} E$$ discussed in Section \[section:quotschemes\]. Of course, ${\operatorname{Quot}}^{-1} E = {\mathbb{P}}{\operatorname{Hom}}( {\mathcal O}(-1),E) \cong {\mathbb{P}}^1$ and this stratum is exactly the quadric ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \hookrightarrow {\mathbb{P}}^3$ along which the full stable pairs space has a $Z \times {\mathbb{A}}^2$ singularity. We can compute the PT invariant explicity using the method explained at the beginning of this section: $$\begin{aligned} P_{4,2}(-2) & = & \int_{[P_4(E,2)^T]^{\rm vir}} e_T(-N^{\rm vir}) \\ & = & \int_{[{\operatorname{Quot}}^{-2} E]^{\rm vir}} e_T(-N^{\rm vir}) \\ & = & \int_{{\mathbb{P}}^3} e_T(p_!(S^\lor \land^2 E - 2E + S + \land^2 E - SE)) \\ & = & \int_{{\mathbb{P}}^3} \frac{e_T(p_* S^\lor \land^2 E)e_T(R^1 p_* SE)}{e_T(R^1 p_*S)e_T(R^1p_* \land^2 E)} \\ & = & \int_{{\mathbb{P}}^3} \frac{e_T( {\mathcal O}(1)_t)e_T({\mathcal O}(-1)_{2t})^4}{e_T({\mathcal O}(-1)_t) e_T({\mathcal O}_{2t})} \\ & = & \int_{{\mathbb{P}}^3} \frac{(a+t)(2t-a)^4}{(t-a)(2t)} \\ & = & \int_{{\mathbb{P}}^3} 4a^3 - 4a^2t + 8 t^3 \\ & = & 4. \end{aligned}$$ According to , the generating function for degree $2$ stable pairs invariants of local ${\mathbb{P}}^1$ is $$\begin{aligned} Z_2^{\rm PT}(-2) &=& \sum_m P_{m,2}({\mathcal O}(-1)^{\oplus 2},2)q^m \\ &=& -2q^3(1+q)^{-4}(1-q)^{-2} \\ &=& -2q^3(1-(-q)^2)^{-2}(1-q^2)^{-2} \\ & = & -2q^3 \left( \sum_{d=0}^{\infty} (d+1)(-q)^d \right ) \left ( \sum_{e=0}^{\infty} (e+1)(-q)^{2e} \right ) \\ & = & 2 \sum_{m=3}^{\infty} \; \sum_{d+2e=m-3} (d+1)(e+1)(-q)^m \\ &=& -2q^3+4q^4-10q^5+16q^6-28q^7+\cdots \end{aligned}$$ Our computations yield $$\begin{aligned} Z_2^{\rm PT}(-2) &=& C(-2,-1,0)q^3 + C(-2,-2,0)q^4 \\ & & + (C(-2,-3,-0)+C(-2,-1,-1))q^5 \\ & & +(C(-2,-4,0)+C(-2,-2,1))q^6 \\ & & +(C(-2,-5,0)+C(-2,-3,1)+C(-2,-1,2))q^7 +\cdots \\ & = & -2q^3+4q^4+(18-28)q^5+(424-408)q^6 +(7750-8404+626)q^7 + \cdots \end{aligned}$$ in complete agreement. 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[^4]: It is a matter of definitions that, for integers $k_1,k_2$ satisfying $d=k_1+k_2$, the partition function ${\rm GW}_b(g|k_1,k_2)$ defined in [@BP2] is related to our $Z^{\rm GW}_b(d)$ by $$Z^{\rm GW}_b(d) = u^{b(2g-2-d)} {\rm GW}_b(g|k_1,k_2)_{t=t_1=t_2} .$$ [^5]: All of these statements make sense when $C$ is ${\mathbb{P}}^1$, as long as one understands “generic" to mean “the splitting type is as balanced as possible". By semicontinuity, this is an open condition in families.

--- abstract: | =0.6 cm [**Abstract**]{} We have studied numerically the shadows of Bonnor black dihole through the technique of backward ray-tracing. The presence of magnetic dipole yields non-integrable photon motion, which affects sharply the shadow of the compact object. Our results show that there exists a critical value for the shadow. As the magnetic dipole parameter is less than the critical value, the shadow is a black disk, but as the magnetic dipole parameter is larger than the critical one, the shadow becomes a concave disk with eyebrows possessing a self-similar fractal structure. These behavior are very similar to those of the equal-mass and non-spinning Majumdar-Papapetrou binary black holes. However, we find that the two larger shadows and the smaller eyebrow-like shadows are joined together by the middle black zone for the Bonnor black dihole, which is different from that in the Majumdar-Papapetrou binary black holes spacetime where they are disconnected. With the increase of magnetic dipole parameter, the middle black zone connecting the main shadows and the eyebrow-like shadows becomes narrow. Our result show that the spacetime properties arising from the magnetic dipole yields the interesting patterns for the shadow casted by Bonnor black dihole. author: - 'Mingzhi Wang$^{1}$, Songbai Chen$^{1,2,3}$[^1], Jiliang Jing$^{1,2,3}$ [^2]' title: Shadows of Bonnor black dihole by chaotic lensing --- =0.8 cm Introduction ============ A shadow is a two-dimensional dark region in the observer’s sky corresponding to light rays that fall into an event horizon when propagated backwards in time. It is shown that the shape and size of the shadow carry the characteristic information of the geometry around the celestial body [@sha1; @sha2; @sha3], which means that the shadow can be regarded as a useful tool to probe the nature of the celestial body and to check further various theories of gravity. The investigation [@sha2; @sha3] indicate that the shadow is a perfect disk for a Schwarzschild black hole and it changes into an elongated silhouette for a rotating black hole due to its dragging effect. The cusp silhouette of shadow is found in the spacetime of a Kerr black hole with Proca hair [@fpos2] and of a Konoplya-Zhidenko rotating non-Kerr black hole [@sb10] as the black hole parameters lie in a certain range. Moreover, the shadow of a black hole with other characterizing parameters have been studied recently [@sha4; @sha5; @sha6; @sha7; @sha9; @sha10; @sha11; @sha12; @sha13; @sha14; @sha14a; @sha15; @sha16; @sb1; @sha17; @sha19; @shan1] (for details, see also a review [@shan1add]), which indicate that these parameters bring the richer silhouettes for the shadows casted by black holes. However, most of the above investigation have been focused only on the cases where the null geodesics are variable-separable and the corresponding dynamical systems are integrable. As the dynamical systems are non-integrable, the motion of photons could be chaotic, which could lead to some novel features for the black hole shadow. Recently, it is shown that due to such chaotic lensing the multi-disconnect shadows with fractal structures emerge for a Kerr black hole with scalar hair [@sw; @swo; @astro; @chaotic] or a binary black hole system [@binary; @sha18]. The further analysis show that these novel patterns with fractal structures in shadows are determined by the non-planar bound orbits [@fpos2] and the invariant phase space structures [@BI] for the photon motion in the black hole spacetimes. The similar analysis have also been done for the cases with ultra-compact object [@bstar1; @bstar2]. It is well known that there exist enormous magnetic fields around large astrophysical black holes, especially in the nucleus of galaxies [@Bm1; @Bm2; @Bm3; @Bm4]. These strong magnetic fields could be induced by currents in accretion disks near the supermassive galactic black holes. On the base of strong magnetic fields, there are substantial current theoretical models accounted for black hole jets, which are one of the most spectacular astronomical events in the sky [@Blandford1; @Blandford2; @Punsly]. In general relativity, one of the most important solutions with magnetic fields is Ernst solution [@Ernst], which describes the gravity of a black hole immersed in an external magnetic field. Interestingly, for an Ernst black hole, the polar circumference for the event horizon increases with the magnetic field, while the equatorial circumference decreases. Bonnor’s metric [@mmd1] is another important solution of the Einstein field equations in the vacuum, which describes a static massive object with a dipole magnetic field in which two static extremal magnetic black holes with charges of opposite signs are situated symmetrically on the symmetry axis. For Bonnor black dihole spacetime, the area of the horizon is finite, but the proper circumference of the horizon surface is zero. Especially, it is not a member of the Weyl electromagnetic class and it can not reduce to Schwarzschild spacetime in the limit without magnetic dipole. The new properties of spacetime structure originating from magnetic dipole will lead to chaos in motion of particles [@mmd; @mmd10; @bbon1]. Since the shadow of black hole is determined by the propagation of light ray in the spacetime, it is expectable that the chaotic lensing caused by the new spacetime structure will yields some new effects on the black hole shadow. Therefore, in this paper, we focus on studying the shadow of Bonnor black dihole [@mmd1] and probe the effect of magnetic dipole parameter on the black hole shadow. The paper is organized as follows. In Sec. II, we review briefly the metric of Bonnor black dihole and then analyze the propagation of light ray in this background. In Sec. III, we investigate the shadows casted by Bonnor black dihole. In Sec. IV, we discuss invariant phase space structures of photon motion and formation of the shadow casted by Bonnor black dihole. Finally, we present a summary. Spacetime of Bonnor black dihole and null geodesics =================================================== Let us now to review briefly the spacetime of Bonnor black dihole. In 1960s, Bonnor obtained an exact solution [@mmd1] of Einstein-Maxwell equations which describes a static massive source carrying a magnetic dipole. In the standard coordinates, the metric of this spacetime has a form [@mmd1] $$\begin{aligned} \label{xy} ds^{2}= -\bigg(\frac{P}{Y}\bigg)^{2}dt^{2}+\frac{P^{2}Y^{2}}{Q^{3}Z}(dr^{2}+Zd\theta^{2}) +\frac{Y^{2}Z\sin^{2}\theta}{P^{2}}d\phi^{2},\end{aligned}$$ where $$P=r^{2}-2mr-b^{2}\cos^{2}\theta,\;\;Q=(r-m)^{2}-(m^{2}+b^{2})\cos^{2}\theta, \;\;Y=r^{2}-b^{2}\cos^{2}\theta,\;\;Z=r^{2}-2mr-b^{2}.$$ The corresponding vector potential $A_{\mu}$ is given by $$\begin{aligned} A_{\mu}= (0,0,0,\frac{2mbr\sin^{2}\theta}{P}),\end{aligned}$$ where $\mu=0,1,2,3$ correspond to the element of $A_{\mu}$ associated with the coordinates $t, r, \theta, \phi$, respectively. It is a static axially-symmetric solution characterized by two independent parameters $m$ and $b$, which are related to the total mass of Bonnor black dihole $M$ as $M=2m$ and to the magnetic dipole moment $\mu$ as $\mu=2mb$. Obviously, this spacetime is asymptotically flat since as the polar coordinate $r$ approaches to infinity the metric tends to the Minkowski one. The event horizon of the spacetime (\[xy\]) is the null hypersurface $f$ satisfied $$\begin{aligned} g^{\mu\nu}\frac{\partial f}{\partial x^{\mu}}\frac{\partial f}{\partial x^{\nu}}=0,\end{aligned}$$ which yields $$\begin{aligned} r^{2}-2mr-b^{2}=0.\end{aligned}$$ It is obvious that there exists only a horizon and the corresponding horizon radius is $r_h=m+\sqrt{m^2+b^2}$. The area of the horizon is $\mathcal{A}=16\pi m^2r^2_h/(m^2+b^2)$, but the proper circumference of the horizon surface is zero since $g_{\phi\phi}=0$ on the horizon. This implies that the $Z=0$ surface is not a regular horizon since there exists conical singularities at $r=r_h$. The singularity along the segment $r=r_h$ can be eliminated by selecting a proper period $\Delta\phi=2\pi[b^2/(m^2+b^2)]^2$, but such a choice yields a conical deficit running along the axes $\theta=0, \;\pi$, from the endpoints of the dipole to infinity [@mmd101; @mmd102]. The defects outside the dipole can be treated as open cosmic strings and then Bonnor black dihole is held apart by the cosmic strings that pull from its endpoints. Since the angular coordinate $\phi$ is periodic, an azimuthal curve $\gamma=\{t=Constant, r=Constant, \theta=Constant\}$ is a closed curve with invariant length $s^2_{\gamma}=g_{\phi\phi}(2\pi)^2$. And then the integral curve with $(t, r, \theta)$ fixed is closed timelike curve as $g_{\phi\phi}<0$. Thus, there exist closed timelike curves inside the horizon. However, the region outside the horizon is regular and there is no closed timelike curves. Moreover, the spacetime (\[xy\]) possesses the complicated singular behaviour at $P=0$, $Q=0$ and $Y=0$, but there is no singularity outside the horizon. As $b=0$, one can find that it does not reduce to Schwarzschild spacetime, but to the Zipoy-Voorhees one with $\delta=2$ [@mmd12; @mmd11], which describes a monopole of mass $2m$ together with higher mass multipoles depended on the parameter $m$. These special spacetime properties affect the propagation of photon and further changes shadow of Bonnor black dihole(\[xy\]). The Hamiltonian of a photon motion along null geodesics in the spacetime (\[xy\]) can be expressed as $$\label{hami} H(x,p)=\frac{1}{2}g^{\mu\nu}(x)p_{\mu}p_{\nu}=0.$$ Since the metric functions in the spacetime (\[xy\]) are independent of the coordinates $t$ and $\phi$, it is easy to obtain two conserved quantities $E$ and $L_{z}$ with the following forms $$\begin{aligned} \label{EL} E=-p_{t}=-g_{00}\dot{t},\;\;\;\;\;\;\;\;\;\;\;\;\;\; L_{z}=p_{\phi}=g_{33}\dot{\phi},\end{aligned}$$ which correspond to the energy and the $z$-component of the angular momentum of photon moving in the background spacetime. With these two conserved quantities, we can obtain the equations of a photon motion along null geodesics $$\begin{aligned} \label{cdx} \ddot{r}&=&-\frac{1}{2}\frac{\partial }{\partial r}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3Z}\bigg)\bigg]\dot{r}^{2}-\frac{\partial }{\partial \theta}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3Z}\bigg)\bigg]\dot{r}\dot{\theta} +\frac{Z}{2}\frac{\partial }{\partial r}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3}\bigg)\bigg]\dot{\theta}^{2}\nonumber\\ &&-\frac{Q^3Z}{2}\bigg[\frac{E^2}{P^4}\frac{\partial}{\partial r}\ln\bigg(\frac{P^2}{Y^2}\bigg)- \frac{L^2_z}{Y^4Z\sin\theta}\frac{\partial}{\partial r}\ln\bigg(\frac{Y^2Z\sin\theta}{P^2}\bigg)\bigg], \nonumber\\ \ddot{\theta}&=&\frac{1}{2Z}\frac{\partial }{\partial \theta}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3}\bigg)\bigg]\dot{r}^{2}-\frac{\partial }{\partial r}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3Z}\bigg)\bigg]\dot{r}\dot{\theta} +\frac{1}{2}\frac{\partial }{\partial \theta}\bigg[\ln\bigg(\frac{P^2Y^2}{Q^3}\bigg)\bigg]\dot{\theta}^{2}\nonumber\\ &&-\frac{Q^3}{2}\bigg[\frac{E^2}{P^4}\frac{\partial}{\partial \theta}\ln\bigg(\frac{P^2}{Y^2}\bigg)- \frac{L^2_z}{Y^4Z\sin\theta}\frac{\partial}{\partial \theta}\ln\bigg(\frac{Y^2Z\sin\theta}{P^2}\bigg)\bigg],\end{aligned}$$ with the constraint condition $$\begin{aligned} \label{lglr} H=\frac{1}{2}\bigg(\frac{Q^{3}Z}{P^{2}Y^{2}}p_{r}^{2}+\frac{Q^{3}}{P^{2}Y^{2}}p_{\theta}^{2} +V\bigg)=0,\end{aligned}$$ where $p_{r}$ and $p_{\theta}$ are the components of momentum of the photon $p_{r}=g_{11}\dot{r}$ and $p_{\theta}=g_{22}\dot{\theta}$. $V$ is the effective potential with a form $$\begin{aligned} \label{vv} V=-(\frac{Y}{P})^{2}E^{2}+\frac{P^{2}}{Y^{2}Z\sin^{2}\theta}L_{z}^{2}.\end{aligned}$$ Obviously, in the case with magnetic dipole (i.e.,$b\neq0$), we find that the equations of motion (\[cdx\]) and (\[lglr\]) can not be variable-separable and the corresponding dynamical system is non-integrable because it admits only two integrals of motion $E$ and $L_z$. This implies that the motion of the photon could be chaotic in the spacetime (\[xy\]), which will bring some new features for the shadow casted by Bonnor black dihole. Shadow casted by Bonnor black dihole ==================================== In this section, we will study the shadow casted by Bonnor black dihole with the method called “backward ray-tracing" [@sw; @swo; @astro; @chaotic] in which the light rays are assumed to evolve from the observer backward in time. In this method, we must solve numerically the null geodesic equations (\[EL\]) and (\[cdx\]) for each pixel in the final image with the corresponding initial condition. The image of shadow in observer’s sky is composed of the pixels corresponding to the light rays falling down into the horizon of black hole. Since the spacetime of Bonnor black dihole (\[xy\]) is asymptotic flat, we can define the same observer’s sky at spatial infinite as in the usual static cases. The observer basis $\{e_{\hat{t}},e_{\hat{r}},e_{\hat{\theta}},e_{\hat{\phi}}\}$ can be expanded in the coordinate basis $\{ \partial_t,\partial_r,\partial_{ \theta},\partial_{\phi} \}$ as a form [@sw; @swo; @astro; @chaotic] $$\begin{aligned} \label{zbbh} e_{\hat{\mu}}=e^{\nu}_{\hat{\mu}} \partial_{\nu},\end{aligned}$$ where $e^{\nu}_{\hat{\mu}}$ satisfies $g_{\mu\nu}e^{\mu}_{\hat{\alpha}}e^{\nu}_{\hat{\beta}} =\eta_{\hat{\alpha}\hat{\beta}}$, and $\eta_{\hat{\alpha}\hat{\beta}}$ is the usual Minkowski metric. For a static spacetime, it is convenient to choice a decomposition $$\begin{aligned} \label{zbbh1} e^{\nu}_{\hat{\mu}}=\left(\begin{array}{cccc} \zeta&0&0&0\\ 0&A^r&0&0\\ 0&0&A^{\theta}&0\\ 0&0&0&A^{\phi} \end{array}\right),\end{aligned}$$ where $\zeta$, $A^r$, $A^{\theta}$, and $A^{\phi}$ are real coefficients. From the Minkowski normalization, one can find that the observer basis obey $$\begin{aligned} e_{\hat{\mu}}e^{\hat{\nu}}=\delta_{\hat{\mu}}^{\hat{\nu}}.\end{aligned}$$ Therefore, we have $$\begin{aligned} \label{xs} \zeta=\frac{1}{\sqrt{-g_{00}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\; A^r=\frac{1}{\sqrt{g_{11}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; A^{\theta}=\frac{1}{\sqrt{g_{22}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; A^{\phi}=\frac{1}{\sqrt{g_{33}}},\end{aligned}$$ and then the locally measured four-momentum $p^{\hat{\mu}}$ of a photon can be obtained by the projection of its four-momentum $p^{\mu}$ onto $e_{\hat{\mu}}$, $$\begin{aligned} \label{dl} p^{\hat{t}}=-p_{\hat{t}}=-e^{\nu}_{\hat{t}} p_{\nu},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;p^{\hat{i}}=p_{\hat{i}}=e^{\nu}_{\hat{i}} p_{\nu},\end{aligned}$$ In the spacetime of Bonnor black dihole (\[xy\]), the locally measured four-momentum $p^{\hat{\mu}}$ can be further written as $$\begin{aligned} \label{smjt} p^{\hat{t}}&=&\frac{1}{\sqrt{-g_{00}}}E,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; p^{\hat{r}}=\frac{1}{\sqrt{g_{11}}}p_{r} ,\nonumber\\ p^{\hat{\theta}}&=&\frac{1}{\sqrt{g_{22}}}p_{\theta}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; p^{\hat{\phi}}=\frac{1}{\sqrt{g_{33}}}L_z.\end{aligned}$$ After some similar operations in Refs.[@sw; @swo; @astro; @chaotic], we can obtain the position of photon’s image in observer’s sky [@sb10] $$\begin{aligned} \label{xd1} x&=&-r_{obs}\frac{p^{\hat{\phi}}}{p^{\hat{r}}}=-r_{obs}\frac{L_{z}}{\sqrt{g_{11} g_{33}}\dot{r}}, \nonumber\\ y&=&r_{obs}\frac{p^{\hat{\theta}}}{p^{\hat{r}}}= r_{obs}\frac{\sqrt{g_{22}}\dot{\theta}}{\sqrt{g_{11}}\dot{r}}.\end{aligned}$$ Following the way done in [@sw; @swo; @astro; @chaotic; @binary; @sha18], one can divide celestial sphere into four quadrants marked with a different color (green, blue, red and yellow as shown in FIG.\[gy\]). The grid of longitude and latitude lines are marked with adjacent brown lines separated by $10^\circ$. The observer is placed off-centre within the celestial sphere at some a real radius $r_{obs}$. For the sake of simplify, it is placed at the intersection of the four colored quadrants on the celestial sphere, i.e., $r_{obs}=r_{sphere}$, which is not shown in Fig.\[gy\]. The white reference spot in Fig.\[gy\] lies at the other intersection of the four colored quadrants, which could provide a direct demonstration of Einstein ring [@sw; @swo; @astro; @chaotic; @binary; @sha18]. We can integrate these null geodesics with different initial conditions until they either reach a point on the celestial sphere or they fall into the horizon of the compact object and the latter defines the shadow. ![The fractal structure in the shadow of Bonnor black dihole (\[xy\]) for fixed $b=1.0$. Here we set $m=1$ and the observer is set at $r_{obs}=30m$ with the inclination angle $\theta_{0}=90{^\circ}$. []{data-label="fx"}](sfig3.eps){width="14cm"} In Fig. \[shb\], we present the shadow casted by Bonnor black dihole (\[xy\]) with different $b$. Here we set $m=1$ and the observer is set at $r_{obs}=30m$ with the inclination angle $\theta_{0}=90{^\circ}$. Our numerical results show that there exists a critical value $b_c\sim 0.404$ for the shadow. As $b<b_c$, we find that the shadow is a black disk, which is similar to those in the usual static compact object spacetimes with horizon. Moreover, we find that the size of the shadow decreases with the parameter $b$ in this case. However, for the case $b>b_c$, there exist two anchor-like bright zones imbedded symmetrically in the black disk shadow so that the shadow looks like a concave disk with four larger eyebrows, which are shown in Fig. \[shb\] (c)-(d). The eyebrow-like features of shadow are also found in Refs.[@sw; @swo; @astro; @chaotic; @binary; @sha18]. Actually, many other smaller eyebrow-like shadows can be detected in two anchor-like bright zones as shown in Fig.\[fx\]. This hints that the shadow possess a self-similar fractal structure, which is caused by chaotic lensing. It is an interesting property of shadows, which is qualitatively different from those in the spacetimes where the equations of motion are variable-separable and the corresponding dynamical system is integrable. With the increase of magnetic dipole parameter $b$, the eyebrows becomes long and the fractal structure become more rich. Moreover, we find that the two anchor-like bright zones increase with the parameter $b$, but for arbitrary $b$, two anchor-like bright zones are disconnected since they are always separated by a black region. In other words, for Bonnor black dihole, the two larger shadows and the smaller eyebrow-like shadows are joined together by the middle black zone. Moreover, the white circle in Figs. \[shb\] and \[fx\] denote Einstein ring, which are consistent with the prediction of multiple images of a source due to gravitational lensing. Invariant phase space structures and formation of shadow casted by Bonnor black dihole ====================================================================================== In this section, we will discuss the formation of the shadow casted by Bonnor black dihole through analysing the invariant phase space structures as in Ref. [@BI]. The invariant phase space structures including fixed points, periodic orbits and invariant manifolds, are one of important features for dynamical systems, which are applied extensively in the design of space trajectory for various of spacecrafts, such as, a low energy transfer from the Earth to the Moon and a “Petit Grand Tour" of the moons of Jupiter [@BI17; @BI18; @BI19; @BI20; @BI22]. Recent investigations [@BI] show that these invariant structures play an important role in the emergence of black hole shadows. For the spacetime of Bonnor black dihole (\[xy\]), the fixed point $x_0=(r_{0},\theta_{0},0,0)$ in phase space $(r,\theta,p_r,p_{\theta})$ satisfies the condition $$\begin{aligned} \label{bdd} \dot{x}^{\mu}=\frac{\partial H}{\partial p_{\mu}}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\; \dot{p}_{\mu}=-\frac{\partial H}{\partial x^{\mu}}=0,\end{aligned}$$ which means $$\begin{aligned} \label{bdd1} V\bigg|_{r_{0},\theta_{0}}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{\partial V}{\partial r}\bigg|_{r_{0},\theta_{0}}=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{\partial V}{\partial \theta}\bigg|_{r_{0},\theta_{0}}=0.\end{aligned}$$ The local stability of the fixed point $x_0=$($r_{0},\theta_{0},0,0$) can be obtained by linearizing the equations (\[bdd\]) $$\begin{aligned} \label{xxh} \mathbf{\dot{X}}=J\mathbf{X},\end{aligned}$$ where $\mathbf{X}=(\tilde{x}^{\mu},\tilde{p}_{\mu})$ and $J$ is the Jacobian. The circular photon orbits in the equatorial plane named light rings are fixed points of the dynamics for the photon motion [@BI; @fpos2]. After linearizing the equations (\[bdd\]) near the fixed point $(r_{0},\pi/2,0,0)$ and setting $m=1$, we obtain the Jacobian $$\label{jjj} J=\left[ \begin{array}{cccc} 0 & 0 & 2A & 0 \\ 0 & 0 & 0 & 2B \\ -2C & 0 & 0 & 0 \\ 0 & -2D & 0 & 0 \end{array} \right],$$ with $$\begin{aligned} \label{jjt} A&=&\frac{(r_{0}-1)^{6}(r_{0}^{2}-2r_{0}-b^{2})}{r_{0}^{6}(r_{0}-2)^{2}},\\ \nonumber B&=&\frac{(r_{0}-1)^{6}}{r_{0}^{6}(r_{0}-2)^{2}},\\ \nonumber C&=&\frac{\eta^{2}[3r_{0}^{2}(r_{0}-4)(r_{0}-2)^{3}+b^{2}r_{0} (r_{0}-2)^{2}(16+r_{0})-4b^{4}(r_{0}-3)]}{r_{0}^{4} (r_{0}^{2}-2r_{0}-b^{2})^3}-4\frac{r_{0}+1}{(r_{0}-2)^{4}},\\ \nonumber D&=&\frac{\eta^{2}(r_{0}-2)(r_{0}^{3}-2r_{0}^{2}-4b^{2})}{r_{0}^{4} (r_{0}^{2}-2r_{0}-b^{2})}-\frac{4b^{2}}{(r_{0}-2)^{3}},\\ \nonumber r_{0}&=&\frac{1}{3}\bigg[(3\sqrt{3}\sqrt{108b^4-112b^2-225}-54b^2+28)^{1/3}+7+ \frac{19}{(3\sqrt{3}\sqrt{108b^4-112b^2-225}-54b^2+28)^{1/3}}\bigg],\end{aligned}$$ where $\eta\equiv L_z/E$. Let us now adopt the case $m=1$ and $b=1.4$ as an example to analyse the formation of the shadow of Bonnor black dihole (\[xy\]) which is shown in Fig.\[shb\] (d). In this special case, we find that there exist two fixed points. Their positions in phase space are overlapped at ($4.07,\pi/2,0,0$), but their impact parameters are $\eta_1=-9.83$ and $\eta_2=9.83$, respectively. The special distribution of two fixed points is attributed to that the considered magnetic dipole spacetime (\[xy\]) is a non-rotating spacetime. The eigenvalues of the Jacobian (\[jjj\]) are $\pm \lambda$, $\pm \nu i$, where $\lambda=0.46$ and $\nu=0.60$. According to Lyapunov central theorem, we know that each purely imaginary eigenvalue gives rise to a one parameter family $\gamma_{\epsilon}$ of periodic orbits, which is the so-called Lyapunov family [@BI] and the orbit $\gamma_{\epsilon}$ collapses into the fixed point as $\epsilon\rightarrow0$. We show Lyapunov family for the above fixed points (light rings) in Fig. \[zqt\]. The two thick dots represent the two light rings, and the solid lines denote a family of periodic Lyapunov orbits arising from these two light rings. These periodic orbits can be parameterized by impact parameter $\eta$ in an interval $[-9.83,9.83]$. All of these periodic Lyapunov orbits are nearly spherical orbits with radius $r=4.07$, which are responsible for determining the boundary of shadow of Bonnor black dihole as in Refs. [@fpos2; @BI]. The positive ( negative ) real eigenvalue $\pm \lambda$ suggests that there is a unstable ( stable ) invariant manifold, in which points exponentially approach the fixed point in backward ( forward ) time. For each Lyapunov orbit, its corresponding invariant manifolds are two dimensional surfaces forming tubes in the three dimensional reduced phase space $(r; \theta; p_{\theta})$. In Fig. \[sn\], we show a projection of the unstable invariant manifolds associated with the periodic orbits for $\eta=-6$ in the plane ($X,\theta$), where $X$ is a compactified radial coordinate defined as $X=\sqrt{r^{2}-r_{h}^{2}}/(1+\sqrt{r^{2}-r_{h}^{2}})$ [@BI]. The orbits inside the unstable invariant manifold tube can reach the horizon of Bonnor black dihole. Moreover, we note the periodic orbit touched the boundary of the black region approaches perpendicularly to the boundary $V(r,\theta)= 0$ as in Ref.[@binary]. In order to probe the shape of the invariant manifolds as in Ref.[@BI], we present in Fig.\[pjl\] the Poincaré section in the plane ($\theta, p_{\theta}$) for the unstable manifolds of Lyapunov orbits at the observers radial position with $\eta=-6$ and $\eta=0$. ![The Poincaré section at $r=r_{obs}$ for the unstable manifolds (green) of Lyapunov orbits in the spacetime of Bonnor black dihole (\[xy\]) with $b=1.4$. The figures (a)-(c) show the fractal-like structure for $\eta=-6$ and the figures (d) is $\eta=0$. Here we set $m=1$. []{data-label="pjl"}](sfig6.eps){width="13cm"} All photons starting within the green regions always move only in the unstable manifold tube. Moreover, we also note that there exist some white regions which corresponds to that photons move outside the unstable manifolds. In Fig.\[pjl\], the intersection of the dashed line $\theta=\frac{\pi}{2}$ with these manifolds denotes the trajectories which can be detected by the observer on the equatorial plane. This can be generalized to the cases with other values of $\theta$. Actually, these intersection points also determine the positions of the photons with a certain angular momentum on the image plane. In Fig. \[jx\], we present the lensing image marking the intersection points for fixed $\eta=-9.83$, $\eta=-6$, and $\eta=0$. The boundary of the shadow of Bonnor black dihole are determined entirely by the intersection points deriving from these fixed points. The anchor-like bright zones in Fig. \[shb\] (d) are originated from the top, middle and bottom parts of the $S-$shape white region in the Poincaré section ( see in Fig.\[pjl\] (a)) and the fractal-like structure shown in Fig.\[pjl\] (a)-(c) is responsible for the fractal shadow structure in Fig.3. For the case $\eta=0$, there is no white region in the Poincaré section ( see in Fig.\[pjl\] (d)), which is responsible for that two anchor-like bright zones are separated by the black shadow in the middle regions in Fig. \[shb\] (d). In order to make a comparison, in Fig. \[jx04\], we also plot the Poincaré section and the intersections of the unstable manifolds with the image plane for $\eta=-6$ in the spacetime of Bonnor black dihole (\[xy\]) with $b=0.4$. Obviously, there is no white region in the Poincaré section, which is consistent with that the shadow of Bonnor black dihole is a black disk and there exist no bright zones in the shadow in this case. Finally, we make a comparison between the shadows casted by the equal-mass and non-spinning Majumdar-Papapetrou binary black holes [@binary; @sha18] and by Bonnor black dihole (\[xy\]). In Fig.9, we present the shadow for the Majumdar-Papapetrou binary black holes [@binary; @sha18] with two equal-mass black holes separated by the parameter $a=0.5$, $a=1$ and $a=2$ ( see figures (a)-(c)) and for the cases of Bonnor black dihole (\[xy\]) separated by the parameter $b=0.5$, $b=1$ and $b=2$ ( see figures (d)-(e)). From Fig.9, one can find that the shadows of Bonnor black dihole possess some properties closely resembling those of Majumdar-Papapetrou binary black holes, which is understandable since there exists the similar black hole configurations in both cases. However, there exists the essential difference in the shadows for the chosen parameter in these two cases. From Fig.9, we find that the two larger shadows and the smaller eyebrow-like shadows are joined together by the middle black zone for Bonnor black dihole, but they are disconnected in the case of the equal-mass and non-spinning Majumdar-Papapetrou binary black holes [@binary; @sha18]. ![The comparison between the shadows of Majumdar-Papapetrou binary black holes and of Bonnor black dihole (\[xy\]). Figures (a), (b) and (c) correspond to the Majumdar-Papapetrou binary case [@binary; @sha18] with two equal-mass black holes separated by the parameter $a=0.5$, $a=1$ and $a=2$, respectively. Figures (d),(e) and (f) denote the shadow for the cases of Bonnor black dihole (\[xy\]) separated by the parameter $b=0.5$, $b=1$ and $b=2$, respectively. Here we set the inclination angle of observer $\theta_{0}=90{^\circ}$ and $m=1$. ](sfig99.eps){width="10cm"} Moreover, with the increase of magnetic dipole parameter, we find that the middle black zone connecting the main shadows and the eyebrow-like shadows becomes narrow for Bonnor black dihole. From the previous discussion, we know that due to the existence of singularity on the symmetric axis, Bonnor black dihole is held apart by the cosmic string with tension $\mu=\frac{1}{4}[1-b^4/(m^2+b^2)^2]$ [@mmd101; @mmd102], which decreases with the parameter $b$. Therefore, we can obtain that the middle black zone increases with the tension of the cosmic string. This behavior is consistent with that in the case of a Kerr black hole pierced by a cosmic string in which the size of black hole shadow increases with the string tension [@sha14a]. Therefore, the appearance of the middle black zone in the shadow of Bonnor black dihole can be attributed to the existence of the conical singularity on the symmetric axis in the background spacetime. In the case of Majumdar-Papapetrou binary black holes [@binary; @sha18], there is no such conical singularity since the configuration is supported by the balance between the gravitational force and the Coulomb force. Thus, the difference of the shadow shape in these two spacetimes is caused by the existence of singularity on the symmetric axis in Bonnor’s spacetime. Summary ======= In this paper we have studied the shadows of black dihole described by Bonnor’s exact solution of Einstein-Maxwell equations. The presence of magnetic dipole yields that the equation of photon motion can not be variable-separable and the corresponding dynamical system is non-integrable. With the technique of backward ray-tracing, we present numerically the shadow of Bonnor black dihole. For the smaller magnetic dipole parameter $b$, the shadow is a black disk as in the usual static compact object spacetimes with horizon. The size of shadow decreases with the parameter $b$. For the larger magnetic dipole parameter $b$, we find that there exist two anchor-like bright zones imbedded symmetrically in the black disk shadow so that the shadow looks like a concave disk with four large eyebrows. The anchor-like bright zones increase and the eyebrows become long with the increase of $b$. Moreover, many other smaller eyebrow-like shadows can be detected in two anchor-like bright zones and the shadow possess a self-similar fractal structure, which is caused by chaotic lensing. This interesting property of shadows is qualitatively different from those in the spacetimes in which the equations of motion are variable-separable and the corresponding dynamical system is integrable. Finally, we analyse the invariant manifolds of certain Lyapunov orbits near the fixed point and discuss further the formation of the shadow of Bonnor black dihole, which indicates that all of the structures in the shadow originate naturally from the dynamics near fixed points. Our result show that the spacetime properties arising from the magnetic dipole yields some interesting patterns for the shadow casted by Bonnor black dihole. Comparing with that in the case of Majumdar-Papapetrou binary black holes, we find that the two larger shadows and the smaller eyebrow-like shadows are joined together by the middle black zone for Bonnor black dihole, but they are disconnected in the Majumdar-Papapetrou one. The appearance of the middle black zone in the shadow of Bonnor black dihole can be attributed to the existence of the conical singularity on the symmetric axis in the background spacetime. 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--- abstract: 'We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite dimensional General Probabilistic Theory. The conditions are on the geometry of the state space; they require existence of two pairs of parallel exposed faces with additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet non-existing) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.' author: - Anna Jenčová - Martin Plávala bibliography: - 'citations.bib' title: 'Conditions on the existence of maximally incompatible two-outcome measurements in General Probabilistic Theory' --- Introduction ============ General Probabilistic Theories have recently gained a lot of attention. It was identified that several non-classical effects that we know from Quantum Mechanics, such as steering and Bell nonlocality [@WisemanJonesDoherty-nonlocal], can be found in most General Probabilistic Theories. Moreover, it was shown that one can violate even the bounds we know from Quantum Mechanics. In finite dimensional Quantum Mechanics the minimal degree of compatibility of measurements is bounded below by a dimension-dependent constant [@HeinosaariSchultzToigoZiman-maxInc], while a General Probabilistic Theory may admit pairs of maximally incompatible two-outcome measurements [@BuschHeinosaariSchultzStevens-compatibility], i.e. two-outcome measurements such that their degree of compatibility attains the minimal value $\frac{1}{2}$. In the present article we show necessary and sufficient conditions for a pair of maximally incompatible measurements to exist in a given General Probabilistic Theory. The conditions restrain the possible geometry of the state space. We also introduce the notion of discrimination two-outcome measurement and show how the concept of discrimination measurements is connected to maximally incompatible measurements. Our results are demonstrated on some examples. In particular, it is shown that maximally incompatible measurements exist for quantum channels. A somewhat different notion of compatibility of measurements on quantum channels and combs has been recently researched in [@SedlakReitznerChiribellaZiman-compatibility], where similar results were found. The article is organized as follows: in Sec. \[sec:preliminary\] we provide a quick review of General Probabilistic Theory and of the notation we will use, in Sec. \[sec:meas\] we introduce the two-outcome measurements, in Sec. \[sec:degcom\] we introduce the degree of compatibility and the linear program for compatibility of two-outcome measurements. In Sec. \[sec:maxInc\] we formulate and prove the necessary and sufficient conditions for maximally incompatible two-outcome measurements to exist. In Sec. \[sec:disc\] we introduce the concept of discrimination measurement and we show that two-outcome measurements are maximally incompatible if and only if their joint measurement would have to discriminate affinely dependent points, which is impossible. Structure of General Probabilistic Theory {#sec:preliminary} ========================================= General Probabilistic Theories form a general framework that provides a unified description of all physical systems known today. We will present the standard definition of a finite dimensional General Probabilistic Theory in a quick review just to settle the notation. The central notion is that of a state space, that is a compact convex subset $K\subset \mathbb{R}^n$, representing the set of states of some system. The convex combinations are interpreted operationally, see e.g. [@HeinosaariZiman-MLQT Part 2]. Let $A(K)$ denote the ordered linear space of affine functions $f:K \to \mathbb{R}$. The order on $A(K)$ is introduced in a natural way; let $f, g \in A(K)$ then $f \geq g$ if and only if $f(x) \geq g(x)$ for all $x \in K$. Let $A(K)^+$ be the positive cone, that is the generating, pointed and convex cone of positive affine functions on $K$. We denote the constant functions by the value they attain, i.e. $1(x)=1$ for all $x \in K$. Let $E(K) = \{ f \in A(K): 1 \geq f \geq 0 \}$ denote the set of effects on $K$. Let $A(K)^*$ be the dual to $A(K)$ and let $\< \psi, f \>$ denote the value of the functional $\psi\in A(K)^*$ on $f \in A(K)$. Using the cone $A(K)^+$ we define the dual order on $A(K)^*$ as follows: let $\psi_1, \psi_2 \in A(K)^*$, then $\psi_1 \geq \psi_2$ if and only if $\< \psi_1, f \> \geq \< \psi_2, f \>$ for every $f \in A(K)^+$. The dual positive cone is $A(K)^{*+} = \{ \psi \in A(K)^*: \psi \geq 0 \}$ where $0$ denotes the zero functional, $\<0, f \> = 0$ for all $f \in A(K)$. Let $x \in K$, then $\phi_x$ will denote the positive and normed functional such that $\< \phi_{x}, f \> = f(x)$. It can be seen that for every functional $\psi \in A(K)^{*+}$ such that $\< \psi, 1 \> = 1$ there is some $x \in K$ such that $\psi = \phi_x$, see [@AsimowEllis Theorem 4.3]. This implies that the set $\states_K = \{ \phi_x: x \in K \}$ is a base of the cone $A(K)^{*+}$, i.e. for every $\psi \in A(K)^{*+}$, $\psi \neq 0$ there is a unique $x \in K$ and unique $\alpha \in \mathbb{R}$, $\alpha > 0$ such that $\psi = \alpha \phi_x$. For any $X \subset \mathbb{R}^n$, $\conv(X)$ will denote the convex hull of $X$ and $\aff(X)$ the affine hull of $X$. Measurements in General Probabilistic Theory {#sec:meas} ============================================ Let $K\subset \mathbb{R}^n$ be a state space. A measurement on $K$ is an affine map $m: K \to \Pe(\Omega)$, where $\Omega$ is the sample space, that is a measurable space representing the set of all possible measurement outcomes, and $\Pe(\Omega)$ is the set of all probability measures on $\Omega$. We will be mostly interested in two-outcome measurements, i.e. measurements with the sample space $\Omega = \{ \omega_1, \omega_2 \}$. Let $\mu \in \Pe ( \Omega )$, then $\mu = \lambda \delta_1 + (1-\lambda) \delta_2$ for some $\lambda \in [0, 1] \subset \mathbb{R}$, where $\delta_1=\delta_{\omega_1}$, $\delta_2=\delta_{\omega_2} $ are the Dirac measures. This shows that the general form of two-outcome measurement $m_f$ is $$m_f = f \delta_1 + (1-f) \delta_2$$ for some $f \in E(K)$. Strictly speaking, this should be written as $m_f = f \otimes \delta_1 + (1-f) \otimes \delta_2$, since any map $m_f: K \to \Pe(\Omega)$ can be identified with a point of $A(K)^+ \otimes \Pe(\Omega)$, see e.g. [@Ryan-tensProd]. The interpretation is that a point $x \in K$ is mapped to the probability measure $m_f(x) = f(x) \delta_1 + (1-f(x)) \delta_2$, i.e. $f(x)$ corresponds to the probability of measuring the outcome $\omega_1$. Let $f, g \in E(K)$ and let $m_f$, $m_g$ be the corresponding two-outcome measurements. We will keep this notation throughout the paper. The two-outcome measurements $m_f$, $m_g$ are compatible if and only if there exists a function $p \in E(K)^+$ such that $$\begin{aligned} f &\geq p, \label{eq:meas-cond-1} \\ g &\geq p, \label{eq:meas-cond-2} \\ 1 + p &\geq f + g, \label{eq:meas-cond-3}\end{aligned}$$ see [@Plavala-simplex] for a derivation of these conditions from the standard conditions that can be found e.g. in [@Holevo-QT Chapter 2]. \[prop:meas-postProc\] $m_f$, $m_g$ are compatible if and only if $m_{(1-f)}$, $m_g$ are compatible. Assume that $m_f$, $m_g$ are compatible and let $p\in E(K)$ satisfy - . Let $p' = g - p$, then Eq. implies $p' \geq 0$, Eq. implies $1-f \geq p'$, $p \geq 0$ implies $g \geq p'$ and implies $1 + p' \geq (1-f) + g$. Since $1 - (1-f) = f$ it is clear that the compatibility of $m_{(1-f)}$, $m_g$ implies compatibility of $m_f$, $m_g$ in the same manner. Degree of compatibility {#sec:degcom} ======================= A coin-toss measurement on $K$ is a constant map $$\tau (x) = \mu \in \Pe(\Omega),\qquad x\in K.$$ It is straightforward that a coin-toss measurement is compatible with any other measurement. In the following we state the usual definition of the degree of compatibility. Let $m_f, m_g$ be two-outcome measurement on $K$ with sample space $\Omega$. The degree of compatibility of $m_f, m_g$ is defined as $$\begin{aligned} \degcom (m_f, m_g) =& \sup_{\substack{0 \leq \lambda \leq 1 \\ \tau_1, \tau_2}} \{ \lambda : \lambda m_f + (1-\lambda) \tau_1, \\ & \lambda m_g + (1-\lambda) \tau_2 \; \text{are compatible} \}\end{aligned}$$ where the supremum is taken over all coin-toss measurements $\tau_1, \tau_2$. It is easy to see that for every two measurements $m_f$, $m_g$ we always have $\degcom(m_f, m_g) \geq \frac{1}{2}$, see e.g. [@HeinosaariMiyaderaZiman-compatibility]. The following immediately follows from Prop. \[prop:meas-postProc\]. \[coro:degcom-postProc\] Let $m_f$, $m_g$ be two-outcome measurements, then $$\begin{aligned} \degcom(m_f, m_g) &= \degcom(m_{(1-f)}, m_g) \\ &= \degcom(m_f, m_{(1-g)}) \\ &= \degcom(m_{(1-f)}, m_{(1-g)}).\end{aligned}$$ We will say that two measurements are maximally incompatible if $\degcom(m_f, m_g) = \frac{1}{2}$. It is known that such measurements exist for some state spaces [@BuschHeinosaariSchultzStevens-compatibility], but they do not exist in finite dimensional Quantum Mechanics [@HeinosaariSchultzToigoZiman-maxInc]. Let $\tau = \frac{1}{2}(\delta_1 + \delta_2)$, then we define $$\begin{aligned} \degcom_{\frac{1}{2}} (m_f, m_g) =& \sup_{0 \leq \lambda \leq 1} \{ \lambda : \lambda m_f + (1-\lambda) \tau, \\ & \lambda m_g + (1-\lambda) \tau \; \text{are compatible} \}\end{aligned}$$ as the degree of compatibility provided only by mixing the measurements $m_f$, $m_g$ with the coin-toss measurement $\tau$. Clearly we have $$\degcom_{\frac{1}{2}} (m_1, m_2) \leq \degcom (m_f, m_g)$$ so $\degcom_{\frac{1}{2}} (m_f, m_g) = 1$ implies $\degcom (m_f, m_g) = 1$ and $\degcom (m_f, m_g) = \frac{1}{2}$ implies $\degcom_{\frac{1}{2}} (m_f, m_g) = \frac{1}{2}$. In [@Plavala-simplex] it was shown that the dual linear program for the compatibility of the measurements $m_f$, $m_g$ is of the form $$\begin{aligned} &\sup \big( a_3 (f(z_3) + g(z_3) - 1) - a_1 f(z_1) - a_2 g(z_2) \big) \\ &a_1 + a_2 \leq 2 \\ &a_3 \phi_{z_3} \leq a_1 \phi_{z_1} + a_2 \phi_{z_2}\end{aligned}$$ where $z_1, z_2, z_3 \in K$ and $a_1, a_2, a_3$ are non-negative numbers. Let $\beta$ denote the supremum, then we have $$\beta = \dfrac{1 - \degcom_{\frac{1}{2}} (m_f, m_g)}{\degcom_{\frac{1}{2}} (m_f, m_g)},$$ i.e. $\beta = 0$ if the measurements are compatible and $\beta > 0$ implies that the measurements are incompatible. Assume that the measurements $m_f$, $m_g$ are incompatible. Then we have $\beta > 0$, which implies $a_1 + a_2 > 0$. We will show that if the supremum is reached, we must have $a_1 + a_2 = 2$. Assume that the supremum is reached for some $a_1, a_2, a_3$ and $z_1, z_2, z_3$ such that $a_1 + a_2 < 2$ and define $$\begin{aligned} a_1' &= \dfrac{2}{a_1 + a_2} a_1, \\ a_2' &= \dfrac{2}{a_1 + a_2} a_2, \\ a_3' &= \dfrac{2}{a_1 + a_2} a_3. \\\end{aligned}$$ It is straightforward to see that $a_3' \phi_{z_3} \leq a_1' \phi_{z_1} + a_2' \phi_{z_2}$, moreover $$\begin{aligned} \beta < a_3' (f(z_3) + g(z_3) - 1) - a_1' f(z_1) - a_2' g(z_2)\end{aligned}$$ which is a contradiction. It follows that in the case when the measurements $m_f$, $m_g$ are incompatible we can write the linear program as $$\begin{aligned} &\sup 2 \big( \eta (f(z_3) + g(z_3) - 1) - \nu f(z_1) - (1-\nu) g(z_2) \big) \nonumber \\ &\eta \phi_{z_3} \leq \nu \phi_{z_1} + (1-\nu) \phi_{z_2} \label{eq:degcom-linProg-final}\end{aligned}$$ where we have set $a_1 + a_2 = 2$ and used a substitution $2 \nu = a_1$, $2 \eta = a_3$. Also note that $\eta \phi_{z_3} \leq \nu \phi_{z_1} + (1-\nu) \phi_{z_2}$ implies that there exists $z_4 \in K$ such that $$\nu z_1 + (1-\nu) z_2 = \eta z_3 + (1-\eta) z_4.$$ Maximally incompatible two-outcome measurements {#sec:maxInc} =============================================== In this section, we wish to find conditions on the state space, under which $\degcom(m_f, m_g) = \frac{1}{2}$ for a pair of two-outcome measurements $m_f, m_g$. A sufficient condition was proved in [@BuschHeinosaariSchultzStevens-compatibility]: a pair of maximally incompatible two-outcome measurements exists if the state space $K$ is a square, or more generally, there are two pairs of parallel hyperplanes tangent to $K$, such that the corresponding exposed faces contain the edges of a square. Here a square is defined as the convex hull of four points $x_1$, $x_2$, $x_3$, $x_4$ satisfying $x_1+x_2=x_3+x_4$. Besides the square, such state spaces include the cube, pyramid, double pyramid, cylinder etc. We will show that this condition is also necessary, so that it characterizes state spaces admitting a pair of maximally incompatible two-outcome measurements. Let us fix a pair of effects $f,g\in E(K)$. The following notation will be used throughout. $$\begin{aligned} F_0 &= \{ z \in K : f(z) = 0 \}, \\ F_1 &= \{ z \in K : f(z) = 1 \}, \\ G_0 &= \{ z \in K : g(z) = 0 \}, \\ G_1 &= \{ z \in K : g(z) = 1 \}.\end{aligned}$$ We begin by rephrasing the above sufficient condition. For completeness, we add a proof along the lines of [@BuschHeinosaariSchultzStevens-compatibility]. \[prop:maxInc-necessary\] Assume there are some points $x_{00} \in F_0 \cap G_0$, $x_{10} \in F_1 \cap G_0$, $x_{01} \in F_0 \cap G_1$, $x_{11} \in F_1 \cap G_1$ such that $$\dfrac{1}{2} ( x_{00} + x_{11} ) = \dfrac{1}{2} ( x_{10} + x_{01} ).$$ Then $\degcom(m_f, m_g) = \frac{1}{2}$. Let $p$ be any positive affine function on $K$, then we have $$p(x_{11}) + p(x_{00}) = p(x_{10}) + p(x_{01})$$ and $$p(x_{11}) \leq p(x_{10}) + p(x_{01})$$ follows. Let $\tau_1 = \mu_1 \delta_{\omega_1} + (1-\mu_1) \delta_{\omega_2}$ and $\tau_2 = \mu_2 \delta_{\omega_1} + (1-\mu_2) \delta_{\omega_2}$ be coin-toss measurements, then the conditions - for $\lambda m_f + (1-\lambda) \tau_1$ and $\lambda m_g + (1-\lambda) \tau_2$ take the form $$\begin{aligned} \lambda f + (1-\lambda) \mu_1 &\geq p, \\ \lambda g + (1-\lambda) \mu_2 &\geq p, \\ 1 + p &\geq \lambda (f + g) + (1-\lambda)(\mu_1 + \mu_2).\end{aligned}$$ Expressing some of these conditions at the points $x_{10}, x_{01}, x_{11}$ we get $$\begin{aligned} 1 + p(x_{11}) &\geq 2 \lambda + (1-\lambda)(\mu_1 + \mu_2), \label{eq:max-Inc-necessary-x11} \\ (1-\lambda) \mu_1 &\geq p(x_{01}), \label{eq:max-Inc-necessary-x01} \\ (1-\lambda) \mu_2 &\geq p(x_{10}). \label{eq:max-Inc-necessary-x10}\end{aligned}$$ From we obtain $$2 \lambda \leq 1 + p(x_{11}) - (1-\lambda)(\mu_1 + \mu_2)$$ and since from and we have $$p(x_{11}) \leq p(x_{10}) + p(x_{01}) \leq (1-\lambda)(\mu_1 + \mu_2)$$ it follows that $\lambda \leq \frac{1}{2}$. At this point we can demonstrate that maximally incompatible two-outcome measurements exist for the set of quantum channels. We will use the standard and well-know definitions that may be found in [@HeinosaariZiman-MLQT]. \[ex:maxInc-channels\] Let $\Ha$ denote a finite dimensional complex Hilbert space, let $B_h(\Ha)$ denote the set of self-adjoint operators on $\Ha$ and let $\I$ denote the identity operator. Let $A \in B_h(\Ha)$, then we write $A \geq 0$ if $A$ is positive semi-definite. Let $\Ha \otimes \Ha$ denote the tensor product of $\Ha$ with itself and let $\operatorname{Tr}_1$ denote the partial trace. Let $$\Ce_\Ha = \{ C \in B_h(\Ha \otimes \Ha): \operatorname{Tr}_1 (C) = \I, C \geq 0 \}$$ be the set of Choi matrices of all channels $B_h(\Ha) \to B_h(\Ha)$. This is clearly a finite dimensional state space. The effects $f\in E(\Ce_\Ha)$ have the form $f(C)=\operatorname{Tr}CM$, $C\in \Ce_\Ha$, where $M\in B_h(\Ha\otimes\Ha)$ is such that $$0\le M\le \I\otimes \sigma,$$ for some density operator $\sigma$ on $\Ha$, [@Jencova-genChannels; @HeinosaariMiyaderaZiman-compatibility], so that effects are given by two-outcome PPOVMs defined in [@Ziman-ppovm]. Let $\dim(\Ha) = 2$, let $|0\>, |1\>$ be an orthonormal basis of $\Ha$ and let $M,N \in B_h(\Ha \otimes \Ha)$ be given as $$\begin{aligned} M&= |0\>\<0| \otimes |0\>\<0|, \\ N &= |0\>\<0| \otimes |1\>\<1|. \\\end{aligned}$$ Then $0\le M\le \I \otimes |0\>\<0|$ and $0\le N\le \I \otimes |1\>\<1|$, so that $f(C)=\operatorname{Tr}CM$ and $g(C)=\operatorname{Tr}CN$ define effects on $\Ce_\Ha$. Let $$\begin{aligned} C_{00} &= |1\>\<1| \otimes \I\\ C_{10} &= |0\>\<0| \otimes |0\>\<0| + |1\>\<1| \otimes |1\>\<1|, \\ C_{01} &= |0\>\<0| \otimes |1\>\<1| + |1\>\<1| \otimes |0\>\<0|, \\ C_{11} &= |0\>\<0| \otimes \I.\end{aligned}$$ It is easy to check that $C_{00}, C_{10}, C_{01}, C_{11} \in \Ce_\Ha$. Moreover $$C_{00} + C_{11} = \I \otimes \I = C_{10} + C_{01}$$ and $$\begin{aligned} \operatorname{Tr}(C_{00} M) = \operatorname{Tr}(C_{00} N) &= 0, \\ \operatorname{Tr}(C_{10} M) =1,\ \operatorname{Tr}(C_{10} N) &= 0, \\ \operatorname{Tr}(C_{01} M) =0,\ \operatorname{Tr}(C_{01} N) &= 1, \\ \operatorname{Tr}(C_{11}M) = \operatorname{Tr}(C_{11} N) &= 1.\end{aligned}$$ In conclusion, $C_{00}$, $C_{10}$, $C_{01}$, $C_{11}$ satisfies the properties in Prop. \[prop:maxInc-necessary\], so that the two-outcome measurements $m_f$ and $m_g$ are maximally incompatible. Analogical fact was also observed in [@SedlakReitznerChiribellaZiman-compatibility; @HeinosaariMiyaderaZiman-compatibility] in different circumstances. We proceed to prove some necessary conditions. \[prop:maxInc-notEmpty\] $\degcom(m_f, m_g) = \frac{1}{2}$ only if $$\begin{aligned} F_0 \cap G_0 &\neq \emptyset, \\ F_0 \cap G_1 &\neq \emptyset, \\ F_1 \cap G_0 &\neq \emptyset, \\ F_1 \cap G_1 &\neq \emptyset.\end{aligned}$$ Let $F_1 \cap G_1 = \emptyset$, then $f+g < 2$. Let $\tau = \delta_{\omega_2}$ and consider the measurements $\lambda m_f + (1-\lambda)\tau= m_{\lambda f}$ and $\lambda m_g + (1-\lambda) \tau= m_{\lambda g}$, $\lambda \in [0, 1]$. Since $f+g < 2$, we can choose $\lambda > \frac{1}{2}$ such that $1 \geq \lambda(f+g)$, so that $p = 0$ satisfies equations Eq. - for $m_{\lambda f}$, $m_{\lambda g}$. The result for the other sets follows by using the Corollary \[coro:degcom-postProc\]. The conditions given by the Prop. \[prop:maxInc-notEmpty\] are not sufficient as we will demonstrate in the following example. Let $K$ be a simplex with the vertices $x_1, x_2, x_3, x_4$ and let $b_1, b_2, b_3, b_4$ denote positive affine functions such that $$b_i (x_j) = \delta_{ij}.$$ Such functions exist because $K$ is a simplex. Let $$\begin{aligned} f &= b_1 + b_2, \\ g &= b_1 + b_3,\end{aligned}$$ then we have $$\begin{aligned} F_1 \cap G_1 &= \{ x_1 \}, \\ F_1 \cap G_0 &= \{ x_2 \}, \\ F_0 \cap G_1 &= \{ x_3 \}, \\ F_0 \cap G_0 &= \{ x_4 \},\end{aligned}$$ but clearly the measurements $m_f$ and $m_g$ must be compatible as $K$ is a simplex. Matter of fact, the Eq. - are satisfied with $p = b_1$. \[prop:maxInc-sufficient\] $\degcom(m_f, m_g) = \frac{1}{2}$ if and only if there exist points $x_{00}, x_{01}, x_{10}, x_{11}$ such that $x_{00} \in F_0 \cap G_0$, $x_{10} \in F_1 \cap G_0$, $x_{01} \in F_0 \cap G_1$, $x_{11} \in F_1 \cap G_1$ and $$\dfrac{1}{2} ( x_{00} + x_{11} ) = \dfrac{1}{2} ( x_{10} + x_{01} ).$$ The ’if’ part is proved in Prop. \[prop:maxInc-necessary\]. Conversely, if $\degcom(m_f, m_g) = \frac{1}{2}$ then according to the results in Section \[sec:degcom\], the supremum in must be equal to 1, so that we must have $$\eta( f(z_3) + g(z_3) - 1) - \nu f(z_1) - (1-\nu) g(z_2) = \dfrac{1}{2} \label{eq:maxInc-functions}$$ for some $\eta, \nu \in [0, 1]$ and $z_1, z_2, z_3 \in K$, such that $$\nu \phi_{z_1} + (1-\nu) \phi_{z_2} \geq \eta \phi_{z_3}.$$ It follows that $$\begin{aligned} \nu \phi_{z_1} \geq \eta \phi_{z_3} - (1-\nu) \phi_{z_2}, \label{eq:maxInc-pointIneq-mod-1} \\ (1-\nu) \phi_{z_2} \geq \eta \phi_{z_3} - \nu \phi_{z_1}. \label{eq:maxInc-pointIneq-mod-2}\end{aligned}$$ Rewriting Eq. we get $$\<\eta \phi_{z_3} - \nu \phi_{z_1},f\> + \<\eta \phi_{z_3} - (1-\nu) \phi_{z_2},g\> - \eta = \dfrac{1}{2}.$$ We clearly have $\<\eta \phi_{z_3} - \nu \phi_{z_1},f\> \leq \eta$, but Eq. implies $\<\eta \phi_{z_3} - \nu \phi_{z_1},f\> \leq 1-\nu$ and thus we must have $\<\eta \phi_{z_3} - \nu \phi_{z_1},f\> \leq \min(\eta, 1-\nu)$. Similarly, we get $\<\eta \phi_{z_3} - (1-\nu) \phi_{z_2},g\> \leq \min(\eta, \nu)$ and $$\dfrac{1}{2} \leq \min(\eta, \nu) + \min(\eta, 1-\nu) - \eta = \min(\nu, 1-\nu, \eta, 1-\eta)$$ which implies $\nu = \eta = \frac{1}{2}$. Moreover, there must be some $z_4 \in K$ such that $$\dfrac{1}{2}( z_1 + z_2 ) = \dfrac{1}{2} ( z_3 + z_4 ). \label{eq:maxInc-squarePoints}$$ Eq. now becomes $$f(z_3) + g(z_3) - f(z_1) - g(z_2) = 2$$ which implies $f(z_3) = g(z_3) = 1$ and $f(z_1) = g(z_2) = 0$ as $f, g \in E(K)$. From Eq. we get $$f(z_2) = 1 + f(z_4),$$ which implies $f(z_2) = 1$, $f(z_4) = 0$ and $$g(z_1) = 1 + g(z_4),$$ which implies $g(z_1) = 1$, $g(z_4) = 0$. Together we get $$\begin{aligned} z_3 \in F_1 \cap G_1, \\ z_2 \in F_1 \cap G_0, \\ z_1 \in F_0 \cap G_1, \\ z_4 \in F_0 \cap G_0.\end{aligned}$$ In the remainder of this section, we aim to give some geometric interpretation of the condition in Prop. \[prop:maxInc-sufficient\]. \[coro:maxInc-parallelogram\] Let $S \subset \mathbb{R}^2$ be a state space, then maximally incompatible two-outcome measurements exist on $S$ if and only if $S$ is a parallelogram. Assume that maximally incompatible measurements exist on $S$, then it is clear that $S$ must contain 4 exposed faces, such that each of them has a nonempty intersection with other two and is disjoint from the third. It follows that the faces must be line segments and that $K$ is a polygon with 4 vertices corresponding to the intersections of the aforementioned exposed faces. The opposite edges of $S$ must be parallel as they are the intersections of $S$ with the hyperplanes $\{x \in \mathbb{R}^2: f(x) = 0 \}$ and $\{x \in \mathbb{R}^2: f(x) = 1 \}$ for some $f \in E(S)$. Assume that $S$ is a parallelogram, then its vertices $z_1, z_2, z_3, z_4$ must satisfy $$\frac{1}{2} ( z_1 + z_2 ) = \frac{1}{2} ( z_3 + z_4 )$$ by definition. It follows that maximally incompatible measurements on $S$ exist by Prop. \[prop:maxInc-sufficient\]. \[prop:maxInc-KcapVeqS\] Maximally incompatible two-outcome measurements on $K$ exists only if there is an affine subspace $V \subset \aff(K)$, $\dim(V) = 2$ such that $V \cap K = S$, where $S$ is a parallelogram. The principal idea is that $V = \aff(S)$ where $S$ is the parallelogram in question. Lets assume that there exist maximally incompatible measurements $m_f$, $m_g$ on $K$ and let $x_{00}, x_{01}, x_{10}, x_{11}$ be the four points as in Proposition \[prop:maxInc-sufficient\]. Let $V = \aff(x_{00}, x_{10}, x_{01})$ and let $S = V \cap K$, we will show that $S = \conv(x_{00}, x_{01}, x_{10}, x_{11})$. Let $y \in S$, then we must have $$y = \alpha_{10} x_{10} + \alpha_{01} x_{01} + (1 - \alpha_{10} - \alpha_{01}) x_{00}$$ for some $\alpha_{10}, \alpha_{01} \in \mathbb{R}$. Since $f, g \in E(S)$ we must have $\alpha_{10}, \alpha_{01} \in [0,1]$ which implies $(1 - \alpha_{10} - \alpha_{01}) \in [-1, 1]$. If $(1 - \alpha_{10} - \alpha_{01}) \in [0, 1]$ then $y \in \conv(x_{00}, x_{10}, x_{01})$. If $(1 - \alpha_{10} - \alpha_{01}) \in [-1, 0]$, then $$y = ( 1 - \alpha_{01}) x_{10} + (1 - \alpha_{10}) x_{01} - (1 - \alpha_{10} - \alpha_{01}) x_{11}$$ and $y \in \conv(x_{10}, x_{01}, x_{11})$. It follows that $S$ is a parallelogram by Corollary \[coro:maxInc-parallelogram\]. We will present an example to show that the condition in Prop. \[prop:maxInc-KcapVeqS\] is not sufficient, even if the parallelogram $S$ is an exposed face of $K$. \[ex:maxInc-cutOffPyramind\] Let $K \subset \mathbb{R}^3$ defined as $$\begin{aligned} K = \conv ( \{ &(0, 0, 0), (2, 0, 0), (0, 2, 0), \\ &(2, 1, 0), (1, 2, 0), (1, 1, 1), \\ &(1, 0, 1), (0, 1, 1), (0, 0, 1) \} ),\end{aligned}$$ see Fig. \[fig:cutOffPyramid\]. Let $$V = \left\lbrace (x_1, x_2, x_3) \in \mathbb{R}^3 : x_3 = 1 \right\rbrace$$ then $K \cap V = S$ where $$S = \conv ( \{ (1, 1, 1), (1, 0, 1), (0, 1, 1), (0, 0, 1) \} )$$ is an exposed face and a square. To see that there is not a pair of maximally incompatible measurements $m_f$, $m_g$, corresponding to $S$, it is enough to realize that the effects $f$, $g$ would have to reach the values $0$ and $1$ on maximal faces that are not parallel, i.e. we would have to have $\aff(F_0) \cap \aff(F_1) \neq \emptyset$ and $\aff(G_0) \cap \aff(G_1) \neq \emptyset$ which is impossible. ![The state space $K$ used in Example \[ex:maxInc-cutOffPyramind\].[]{data-label="fig:cutOffPyramid"}](cutOffPyramid.eps) On the other hand, the examples of double pyramid or a cylinder show that maximally incompatible two-outcome measurements may exist on $K$ even if the parallelogram $S$ described in Prop. \[prop:maxInc-KcapVeqS\] is not a face of $K$. Discrimination measurements {#sec:disc} =========================== To avoid the problems presented in Example \[ex:maxInc-cutOffPyramind\] we will introduce a new type of measurement that will allow us to formulate the conditions for existence of maximally incompatible two-outcome measurements in a clearer way. This will also clarify why the measurements are maximally incompatible. We say that a two-outcome measurement $m_f$ discriminates the sets $E_0, E_1 \subset K$, if it holds that $$\begin{aligned} E_0 &\subset \{ x \in K: f(x) = 0 \}, \\ E_1 &\subset \{ x \in K: f(x) = 1 \}.\end{aligned}$$ We call such measurement a discrimination measurement. The idea of the definition is simple: assume that a system is in an unknown state, but we know that it either belongs to $E_0$ or $E_1$. By performing the discrimination measurement $m_f$ we can tell with $100\%$ accuracy whether the state of the system belongs to $E_0$ or $E_1$. The definition can be generalized to general measurements that can discriminate more than two exposed faces. Most well-known discrimination measurements used in Quantum Mechanics are projective measurements consisting of rank-1 projections that discriminate the states corresponding to the projections. We are ready to formulate the necessary and sufficient condition for existence of a pair of maximally incompatible two-outcome measurements. \[prop:disc-discJointMeas\] The measurements $m_f$ and $m_g$ are maximally incompatible if and only if there is an affine subspace $V \subset \aff(K)$ such that $S=K \cap V$ is a parallelogram and $m_f$ and $m_g$ discriminate the opposite edges of $S$. First assume that there is an affine subspace $V$ such that $K \cap V = S$ is a parallelogram whose opposite edges can be discriminated by measurements $m_f$, $m_g$. Denote the vertices of $S$ as $x_{00}$, $x_{10}$, $x_{01}$, $x_{11}$, then it is clear that the requirements of Prop. \[prop:maxInc-necessary\] are satisfied and thus we must have $\degcom(m_f, m_g) = \frac{1}{2}$. Assume that for some two-outcome measurements $m_f$, $m_g$ we have $\degcom(m_f, m_g) = \frac{1}{2}$. As a result of Prop. \[prop:maxInc-KcapVeqS\] there must exist an affine subspace $V$ and a parallelogram $S$ such that $K \cap V = S$, moreover it can be seen that we must have $$\begin{aligned} E_1 &\subset F_0 \\ E_3 &\subset F_1 \\ E_2 &\subset G_0 \\ E_4 &\subset G_1\end{aligned}$$ for the edges $E_1$, $E_2$, $E_3$, $E_4$ of $S$. This implies that $m_f$ discriminates $E_1$ and $E_3$, $m_g$ discriminates $E_2$ and $E_4$. To give some insight into maximal incompatibility of two-outcome measurements, consider that in general the joint measurement $m$ of $m_f$ and $m_g$ is of the form $$m = p \delta_{(1,1)} + (f-p) \delta_{(1,2)} + (g-p) \delta_{(2,1)} + (1+p-f-g) \delta_{(2,2)},$$ for some $p \in E(K)$ as in Eq. - , see [@Plavala-simplex] for a more detailed calculation. Assume that $m_f$, $m_g$ satisfy the requirements of Prop. \[prop:maxInc-necessary\]. Inserting $x_{11} \in F_1 \cap G_1$ into Eq. , we get $$p(x_{11}) \geq 1$$ that together with $p \in E(K)$ implies $p(x_{11}) = 1$. Eq. , and the positivity of $p$ imply $$p(x_{00}) = p(x_{10}) = p(x_{01}) = 0.$$ Expressing also the functions $(f-p)$, $(g-p)$ and $(1+p-f-g)$ on the points $x_{00}$, $x_{10}$, $x_{01}$, $x_{11}$ we get $$\begin{aligned} (f-p)(x_{00}) = (f-p)(x_{01}) = (f-p)(x_{11}) &= 0, \\ (f-p)(x_{10}) &= 1, \\ (g-p)(x_{00}) = (g-p)(x_{10}) = (g-p)(x_{11}) &= 0, \\ (g-p)(x_{01}) &= 1, \\ (1+p-f-g)(x_{00}) = (1+p-f-g)(x_{10}) &= 0, \\ (1+p-f-g)(x_{01}) &= 0, \\ (1+p-f-g)(x_{11}) &= 1.\end{aligned}$$ This shows that the joint measurement $m$ would have to discriminate the points $x_{00}$, $x_{10}$, $x_{01}$, $x_{11}$, which is generally impossible as they are required to be affinely dependent. We have proved the following: Two-outcome measurements $m_f$, $m_g$ are maximally incompatible if and only if the Eq. - imply that the joint measurement would have to discriminate points $x_{00}$, $x_{10}$, $x_{01}$, $x_{11}$ such that $$\dfrac{1}{2} (x_{00} + x_{11}) = \dfrac{1}{2} (x_{10} + x_{01}).$$ This result may also be tied to the way how the joint measurement can be constructed as described also in [@HeinosaariMiyaderaZiman-compatibility]; we can toss a fair coin and implement one of the measurements based on the result while ignoring the other. This just roughly shows that maximally incompatible measurements are so incompatible, that the only way how to perform them both requires as much noise as randomly ignoring one of the measurements. Conclusions =========== We have shown necessary and sufficient condition for existence of maximally incompatible two outcome measurements. It turned out that the example of square state space in [@BuschHeinosaariSchultzStevens-compatibility] did cover the essence of why and how maximally incompatible measurements come to be in General Probabilistic Theory, but it was also demonstrated by Example \[ex:maxInc-cutOffPyramind\] that in general more conditions have to be required for more than two dimensional state spaces. The geometric interpretation of the minimal degree of compatibility that can be attained on a state space $K$ is an interesting question for future research. It would be also of interest whether the connection between discriminating certain sets and compatibility of measurements could be fruitful from information-theoretic viewpoint. This area of research might also yield some insight into why there exist maximally incompatible measurements on quantum channels as demonstrated by Example \[ex:maxInc-channels\], when they do not exist on quantum states. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are thankful Teiko Heinosaari as the idea for this calculation emerged during our conversation. This research was supported by grant VEGA 2/0069/16. MP acknowledges that this research was done during a PhD study at Faculty of Mathematics, Physics and Informatics of the Comenius University in Bratislava.

--- abstract: 'Between the launch of the *GGS Wind* spacecraft in 1994 November and the end of 2010, the Konus-*Wind* experiment detected 296 short-duration gamma-ray bursts (including 23 bursts which can be classified as short bursts with extended emission). During this period, the IPN consisted of up to eleven spacecraft, and using triangulation, the localizations of 271 bursts were obtained. We present the most comprehensive IPN localization data on these events. The short burst detection rate, $\sim$18 per year, exceeds that of many individual experiments.' author: - 'V. D. Pal’shin, K. Hurley, D. S. Svinkin, R. L. Aptekar, S. V. Golenetskii, D. D. Frederiks, E. P. Mazets, P. P. Oleynik, M. V. Ulanov, T. Cline, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, C. Fellows, K. Harshman, J. Trombka, T. McClanahan, R. Starr, J. Goldsten, R. Gold, A. Rau, A. von Kienlin, V. Savchenko, D. M. Smith, W. Hajdas, S. D. Barthelmy, J. Cummings, N. Gehrels, H. Krimm, D. Palmer, K. Yamaoka, M. Ohno, Y. Fukazawa, Y. Hanabata, T. Takahashi, M. Tashiro, Y. Terada, T. Murakami, K. Makishima, M. S. Briggs, R. M. Kippen, C. Kouveliotou, C. Meegan, G. Fishman, V. Connaughton, M. Boër, C. Guidorzi, F. Frontera, E. Montanari, F. Rossi, M. Feroci, L. Amati, L. Nicastro, M. Orlandini, E. Del Monte, E. Costa, I. Donnarumma, Y. Evangelista, I. Lapshov, F. Lazzarotto, L. Pacciani, M. Rapisarda, P. Soffitta, G. Di Cocco, F. Fuschino, M. Galli, C. Labanti, M. Marisaldi, J.-L. Atteia, R. Vanderspek, G. Ricker' title: 'IPN localizations of Konus short gamma-ray bursts' --- INTRODUCTION ============ Between 1994 November and 2010 December, the Konus gamma-ray spectrometer aboard the *Global Geospace Science Wind* spacecraft detected 1989 cosmic gamma-ray bursts (GRBs) in the triggered mode, 296 of which were classified as short-duration gamma-ray bursts or short bursts with extended emission (EE). The classification was made based on the duration distribution of an unbiased sample of 1168 Konus-*Wind* GRBs. The instrument trigger criteria cause undersampling of faint short bursts relative to faint long bursts, so this subsample of fairly bright (in terms of peak count rate in Konus-*Wind*’s trigger energy band) bursts has been chosen for the purpose of classification. Taking in account other characteristics of these short-duration bursts such as hardness ratio and spectral lag shows that about 16% of them can be in fact Type II (collapsar-origin), or at least their classification as Type I (merger-origin) is questionable (see @zhang09 for more information on the Type I/II classification scheme). Nevertheless we consider here all 296 Konus-*Wind* short-duration and possible short-duration with EE bursts (hereafter we refer to them simply as Konus short bursts). Full details of the Konus-*Wind* GRB classification are given in Svinkin et al. (2013, in preparation). Every short burst detected by Konus was searched for in the data of the spacecraft comprising the interplanetary network (IPN). We found that 271 ($\sim$92%) of the Konus-*Wind* short GRBs were observed by at least one other IPN spacecraft, enabling their localizations to be constrained by triangulation. The IPN contained between 3 and 11 spacecraft during this period. They were, in addition to Konus-*Wind*: *Ulysses* (the solar X-ray/cosmic gamma-ray burst instrument, GRB), in heliocentric orbit at distances between 670 and 3180 lt-s from Earth [@hurley92]; the *Near-Earth Asteroid Rendezvous* mission (NEAR) [the remote sensing X-ray/Gamma-Ray Spectrometer, XGRS; @trombka99], at distances up to 1300 lt-s from Earth; *Mars Odyssey* [the Gamma-Ray Spectrometer (GRS) that includes two detectors with GRB detection capabilities, the gamma sensor head (GSH), and the High Energy Neutron Detector (HEND); @boynton04; @hurley06], launched in 2001 April and in orbit around Mars starting in 2001 October, up to 1250 lt-s from Earth [@saunders04]; *Mercury Surface, Space Environment, Geochemistry, and Ranging* mission (*MESSENGER*) [the Gamma-Ray and Neutron Spectrometer, GRNS; @goldsten07], en route to Mercury (in Mercury orbit since March 2011), launched in 2004 August, but commencing full operation only in 2007, up to $\sim$700 lt-s from Earth [@gold01; @solomon07]; the *International Gamma-Ray Astrophysics Laboratory* (*INTEGRAL*) [the anti-coincidence shield ACS of the spectrometer SPI, SPI-ACS; @rau05], in an eccentric Earth orbit at up to 0.5 lt-s from Earth; and in low Earth orbits: the *Compton Gamma-Ray Observatory* [the Burst and Transient Source Experiment, BATSE; @fishman92]; *BeppoSAX* [the Gamma-Ray Burst Monitor, GRBM; @frontera97; @feroci97]; the *Ramaty High Energy Solar Spectroscopic Imager* [*RHESSI*; @lin02; @smith02]; the *High Energy Transient Explorer* (*HETE-2*) [the French Gamma-Ray Telescope, FREGATE; @ricker03; @atteia03]; the *Swift* mission [the Burst Alert Telescope, BAT; @barthelmy05; @gehrels04]; the *Suzaku* mission [the Wide-band All-sky Monitor, WAM; @yamaoka09; @takahashi07]; *AGILE* (the Mini-Calorimeter, MCAL, and Super-AGILE) [@tavani09]; the *Fermi* mission [the Gamma-Ray Burst Monitor, GBM; @meegan09], the *Coronas-F* solar observatory (Helicon) [@oraevskii02], the *Cosmos 2326* [Konus-A; @aptekar98], *Cosmos 2367* (Konus-A2), and *Cosmos 2421*(Konus-A3) spacecraft, and the *Coronas-Photon* solar observatory (Konus-RF). At least two other spacecraft detected GRBs during this period, although they were not used for triangulation and therefore were not, strictly speaking, part of the IPN. They are the *Defense Meteorological Satellite Program* (DMSP) [@terrell96; @terrell98; @terrell04] and the *Stretched Rohini Satellite Series* (SROSS) [@marar94]. Here we present the localization data obtained by the IPN for 271 Konus-*Wind* short bursts observed by at least one other IPN s/c. In a companion paper, we present the durations, energy spectra, peak fluxes, and fluences of these bursts. OBSERVATIONS ============ For each Konus short gamma-ray burst, a search was initiated in the data of the IPN spacecraft. For the near-Earth spacecraft and *INTEGRAL*, the search window was centered on the Konus-*Wind* trigger time, and its duration was somewhat greater than the *Wind* distance from Earth. For the spacecraft at interplanetary distances, the search window was twice the light-travel time to the spacecraft if the event arrival direction was unknown, which was the case for most events. If the arrival direction was known, even coarsely, the search window was defined by calculating the expected arrival time at the spacecraft, and searching in a window around it. The mission timelines and the number of Konus-*Wind* short GRBs observed by each mission/instrument are shown in Figure \[Fig\_TimeLines\]. In this study, the largest number of bursts detected by an IPN instrument, after Konus, was 139, detected by *INTEGRAL* (SPI-ACS). Table \[Table\_Basic\] lists the 271 Konus-*Wind* short GRBs observed by the IPN. The first column gives the burst designation, ‘`YYYYMMDD``sssss`’, where `YYYYMMDD` is the burst date, and `sssss` is the Konus-*Wind* trigger time (s UT) truncated to integer seconds (Note that, due to *Wind*’s large distance from Earth, this trigger time can differ by up to 6.1 seconds from the Earth-crossing time; see below). The next two columns give the burst date and Konus-*Wind* trigger time in the standard date and time formats. The ‘Type’ column specifies the burst type following the classification given in Svinkin et al. (2013, in preparation). The types are: I (merger-origin), II (collapsar-origin), I/II (the type is uncertain), Iee (type I which shows extended emission), and Iee/II (the type is uncertain: Iee or II). The ‘Time delay’ column gives the propagation time delay from *Wind* to the Earth center and its 3$\sigma$ uncertainty (calculated using the IPN localizations, presented in this catalog: see section \[Sec\_ErrorRegions\]). The ‘Observed by’ column lists the missions/instruments which observed the burst (detections by several instruments which are not part of the IPN have also been listed, namely COMPTEL on *CGRO*, DMSP, *Fermi* LAT, MAXI, (*Monitor of All-sky X-ray Image*), and SROSS (*Stretched Rohini Satellite Series*). The next two columns give the total number of IPN s/c and the number of the distant IPN s/c which observed the burst. The last column contains the comments. During the period of consideration, four interplanetary s/c participated in the IPN: *Ulysses*, *NEAR*, *Mars Odyssey*, and *MESSENGER*. Of the 271 bursts listed in Table \[Table\_Basic\], 30 bursts were observed by two distant s/c, 102 by one distant s/c, and 139 were not observed by any distant s/c. 17 Konus short bursts were precisely localized by instruments with imaging capabilities, namely, *Swift*-BAT, *HETE-2* (WXM and SXC), and *INTEGRAL* IBIS/ISGRI. For most of these bursts an X-ray afterglow has been detected; for some of them a redshift z has been determined based on the optical afterglow or host galaxy spectroscopy. We have used these bursts to verify our IPN triangulations (see section \[Sec\_AnnVerification\]). METHODOLOGY =========== When a GRB arrives at two spacecraft with a delay $\delta$T, it may be localized to an annulus whose half-angle $\theta$ with respect to the vector joining the two spacecraft is given by $$\cos \theta=\frac{c \delta T}{D}$$ where $c$ is the speed of light and $D$ is the distance between the two spacecraft. (This assumes that the burst is a plane wave, i.e., that its distance is much greater than $D$.) The measured time delay has an uncertainty which is generally not symmetrical $d_{\pm} (\delta T)$, i.e., the measured time delay can take values from $\delta T + d_{-}(\delta T)$ to $\delta T + d_{+} (\delta T)$ ($d_{-}(\delta T)$ is negative) at a given confidence level. The annulus half-widths $d \theta_\pm$, are $$\label{EqHW} % d \theta_\pm \equiv \theta_\pm - \theta = \cos^{-1} \left[\frac{c (\delta T + d_\mp (\delta T))}{D} \right] - \cos^{-1} \left[\frac{c \delta T }{D} \right]$$ It should be noted that even in case of symmetrical errors $|d_{-}(\delta T)|$ = $d_{+}(\delta T)$, the annulus can still be significantly asymmetrical if $c(\delta T + d_\pm (\delta T))/D \sim 1$ (i.e. the source is close to the vector joining the two spacecraft). For the case $d (\delta T) \ll D/c$ Eq. (\[EqHW\]) reduces to commonly used expression: $$\label{EqHWsimple} % d \theta_\pm = -\frac{c d_\mp(\delta T)}{D \sin \theta}$$ To derive the most probable time delay $\delta T$ and its uncertainty $d_\pm (\delta T)$ we have used the $\chi^2$ method described in @hurley99a for triangulations with distant s/c and this method with some modifications for Konus-*Wind*-near-Earth s/c (or *INTEGRAL*) triangulations. Given a burst time history recorded by two instruments, the most probable time lag $\tau$ and its uncertainty $d_\pm(\tau)$ can be estimated as follows. Let $n_{1,i} = n (t_{1,i})$, $n_{2,j} = n (t_{2,j})$ and $\sigma_{1,i}$, $\sigma_{2,j}$ denote background-subtracted counts and their uncertainties measured by two instruments at evenly spaced intervals $t_{1,i} = t_{01} + i \Delta_1$, $t_{2,j} = t_{02} + j \Delta_2$, where $i = 0,..,m_1$, $j=0,..,m_2$; and $\Delta_1$, $\Delta_2$ are the bin sizes and $t_{01}$, $t_{02}$ are the absolute reference times (UT). To make things simpler, suppose $\Delta_1$ = $\Delta_2$ = $\Delta$. Usually one assumes Poisson statistics, so $\sigma_{1(2),i} = n_{tot1(2),i}^{1/2}$, where $n_{tot1(2),i}$ is the total number of counts (source + background) in the $i$-th bin. Let us assume that both time histories contain the burst of interest along with some intervals before and after it (if they do not exist they can always be added and padded with zeros with the background variance), and $N+1$ bins from $i_{start}$ contain the burst (or the part of the burst we want to cross-correlate) in the second time history. Let us construct the statistic: $$R^2(\tau \equiv k \Delta) = \sum_{i=i_{start}}^{i=i_{start}+N} \frac{(n_{2,i} - s n_{1,i+k})^2}{(\sigma_{2,i}^2+s^2\sigma_{1,i+k}^2)}$$ Where $s$ is the scaling factor estimated as a ratio of the burst counts detected by the instruments $s = \sum_i n_{1,i}/ \sum_j n_{2,j}$. In the perfect case of identical detectors, with identical energy ranges and arrival angles, and Poisson statistics, $R^2$ is distributed as $\chi^2$ with $N$ degrees of freedom (dof). In practice, there are several complicating factors. The detectors have different energy ranges, different responses, and operate in different background environments. Fortunately, for short GRBs some of these complicating factors are less important: a) the background variation on short time scales is small, and b) different spectral evolutions which produce a significant lag between light curves measured in different energy bands, is almost absent in short GRBs [e.g., @norris01]. To account for all deviations from the perfect case we adopt the following approach: for a given $N$ (the number of bins used to construct $R^2$) we calculate $\chi^2(N)$ corresponding to the 3$\sigma$ confidence level (that is, $\chi^2$ for which the chi-square probability function $Q(\chi^2 | N) = 2.7 \times 10^{-3}$), and adopt the corresponding 3$\sigma$ level for the reduced $R^2_r (\equiv R^2/N)$ of $$\label{EqR2sigma3} % R^2_{r, 3\sigma} = \chi^2_{r, 3\sigma}(N) + R^2_{r,min} - 1$$ where $R^2_{r,min}$ is the minimum of $R^2_{r}(\tau)$; 1 is subtracted, since $R^2_{r,min} \sim 1$ for the perfect case (in practice it is often $>1$ and hence, $R^2_{r,3\sigma} > \chi^2_{r, 3\sigma}(N)$). To identify the 3$\sigma$ confidence interval for $\tau$, we use the nearest points of the $R^2_r(\tau)$ curve above the 3$\sigma$ level given by Eq. (\[EqR2sigma3\]) -– see the examples in Figure \[Fig\_R2examples\]. After the time lag $\tau$ and its errors $d_\pm(\tau)$ have been found, the time delay and its uncertainty can be calculated as $\delta T = t_{02} - t_{01} + \tau$; $d_\pm(\delta T)= d_\pm(\tau)$ (here we assume there is no error in the absolute times $t_{01}$ and $t_{02}$). For simplicity we will further refer to $R^2$ as $\chi^2$. LOCALIZATIONS: TRIANGULATION ANNULI =================================== Using the above methodology one or more triangulation annuli have been obtained for 271 Konus-*Wind* short bursts. Specific details on the time delay determination for different pairs of instruments are given in the subsections below. Annuli involving distant s/c ---------------------------- Distant (interplanetary) spacecraft play an important role in GRB triangulation. Their long baselines make it possible to derive very small error boxes for many bursts. However, the detectors aboard these missions tend to be smaller than ones in orbits closer to Earth, and in some cases, are not dedicated GRB detectors, but rather, are planetary experiments which have GRB detection modes. Thus, they may have coarser time resolution and less sensitivity. Also, the spacecraft clocks on these missions are not always calibrated to UTC as accurately as the ones on missions closer to Earth (or their calibration cannot be determined as accurately). In the present catalog, the data of four interplanetary missions have been used: *Ulysses*, *NEAR*, *Mars Odyssey*, and *MESSENGER*. Of the four, only *Ulysses* had a dedicated GRB experiment. The time resolutions of the four ranged from 32 ms (*Ulysses*, triggered mode) to 1 s (*MESSENGER*, *NEAR*). When a short GRB is detected by an experiment with time resolution much greater than the burst duration, the result is usually an increase in the count rate in a single time bin, which means that the timing uncertainty is approximately one-half of the larger time resolution. The accuracy of the spacecraft clocks have been verified in two ways. For *Ulysses*, commands were sent to the GRB experiment at accurately known times, and, taking light-travel time and delays aboard the spacecraft into account, the timing could be verified to between several milliseconds and 125 milliseconds. (Although we believe that the timing was accurate to several milliseconds, technical issues often prevented us from verifying it.) In addition, the timing of all the interplanetary missions can be verified using triangulation of transient sources whose positions are well known by other means: soft gamma-ray repeaters (SGRs) are one possibility, and GRBs localized by the *Swift* XRT or UVOT, for example, are another. For the purposes of this catalog, we have taken the extremely conservative approach that no 3$\sigma$ cross-correlation uncertainty is less than 125 ms. In total 132 Konus short bursts were observed by distant s/c: 30 by two distant s/c and 102 by one distant s/c. Among them 9 were precisely localized by instruments with imaging capabilities. We do not give here distant s/c annuli for these 9 bursts since they do not improve the precise localizations. As a result 150 annuli have been obtained using the distant s/c data (two distant annuli for 27 bursts and one distant annulus for 96 bursts). Some of them have already been presented in the IPN catalogs: for BATSE bursts in @hurley99a [@hurley99b; @hurley11a], for *BeppoSAX* bursts in @hurley10a, and for *HETE-2* bursts in @hurley11b. The histogram of Figure \[Fig\_DistAnnuli\] shows the distribution of 3$\sigma$ half-widths (HWs) of these 150 annuli. The smallest HW is 0$\fdg$0024 (0.14), the largest is 2$\fdg$21, the mean is 0$\fdg$099 (5.9), and the geometrical mean is 0$\fdg$028 (1.7). Annuli involving Konus-*Wind*, *INTEGRAL*, and a near-Earth s/c --------------------------------------------------------------- The Konus-*Wind* (hereafter *KW*) experiment plays a special role in the IPN thanks to its unique set of characteristics: continuous coverage of the full sky by two omnidirectional spectrometers, orbit in interplanetary space that provides an exceptionally stable background, wide energy range (10 keV – 10 MeV nominal; $\sim$20 keV – 15 MeV at the present time), and a rather high sensitivity of about $10^{-7}$ erg cm$^{-2}$. The *KW* duty cycle, defined as the time for data recovered divided by the total operational time of the experiment, is about 95%. It has observed most of the IPN events, providing an important vertex in the IPN at a distance of $\simeq$1–7 lt-s (see Figure \[Fig\_WindDistance\]). In the triggered mode *KW* records a burst time history in three energy ranges G1, G2, G3 with nominal bounds 10–50 keV, 50–200 keV, and 200–750 keV, with a variable time resolution from 2 ms up to 256 ms [for more details see @aptekar95]. The time interval with the finest time resolution of 2 ms runs from T$_0$-0.512 s to T$_0$+0.512 s (T$_0$ is the trigger time) and in most cases covers the whole short burst, or at least its most intense pulse, thereby allowing very accurate cross-correlation with light curves of other instruments. The *KW* clock is accurate to better than 1 ms, and this accuracy has been extensively verified by triangulation of many SGR bursts and GRBs. The best results for cross-correlation with *KW* (i.e., minimum uncertainties in derived time delays) are provided by near-Earth s/c with high effective areas, namely *CGRO* BATSE, *BeppoSAX* GRBM, *INTEGRAL*-SPI-ACS, *Suzaku*-WAM, *Swift*-BAT, and *Fermi*-GBM. At present, triangulation with *Fermi*-GBM usually provides the best result (the narrowest annulus) thanks to the similar designs of the *KW* and GBM detectors (NaI(Tl) scintillators), the high effective area of the GBM (several hundred cm$^2$ for the combination of several NaI(Tl) detectors), and photon time-tagging in 128 energy channels, that allows GBM light curves to be obtained in the same energy ranges as *KW* light curves with any desired time binning. Since the clocks of most of the near-Earth instruments are very accurate, high count statistics combined with fine time resolution often results in an uncertainty in time delays as low as several milliseconds. Therefore, despite the rather small distance between *KW* and near-Earth s/c of several lt-s, the resulting relative error in time delay, $d_{\pm}(\delta T)/D$, which determines the width of the annulus (see Eq. (\[EqHW\],\[EqHWsimple\])) can be comparable to or sometimes even smaller than for annuli involving distant s/c. Such small uncertainties of several ms, and hence narrow annuli, can often be derived for short bursts with sharp peaks or very fast rise and/or decay times. On the other hand, bursts with smooth single-pulse light curves usually give rather large cross-correlation uncertainties in time delays, and hence, rather wide triangulation annuli. For *Wind*, *INTEGRAL*, and near-Earth s/c, ephemeris uncertainties are negligible compared to uncertainties in time delays and we do not take them in account. In total 356 *KW*-near-Earth s/c and *KW-INTEGRAL* annuli have been obtained. The histograms of Figure \[Fig\_KWAnnuli\] show the distributions of uncertainties in time delays and 3$\sigma$ half-widths of these annuli. The smallest time delay uncertainty is 2 ms, the largest is 504 ms, the mean is 43 ms, and the geometrical mean is 23 ms. The smallest 3$\sigma$ half-width is 0027 (1.6), the largest is 322, the mean is 130, and the geometrical mean is 043. In the following subsections some details on triangulation involving *KW*, *INTEGRAL*, and the near-Earth s/c are given. ### *KW*-*CGRO* (BATSE) triangulations\[Sec\_KWBATSEann\] The Burst and Transient Source Experiment (BATSE) was a high-energy astrophysics experiment aboard the *Compton Gamma Ray Observatory (CGRO)* [@fishman92]. Its Large Area Detectors (LADs) measured burst time histories in four energy channels Ch1, Ch2, Ch3, Ch4, with approximate channel boundaries: 25-55 keV, 55-110 keV, 110-320 keV, and $>$320 keV. The clock on the *CGRO* is accurate to 100 $\mu$s, and this accuracy was verified through pulsar timing. Onboard software increases the uncertainty in the BATSE trigger times to $\simeq$1 ms. BATSE observed 52 *KW* short GRBs: 44 in the triggered mode, and 8 in the real-time mode. We derived *KW*-BATSE annuli for the 44 *KW* short bursts observed in the triggered mode and for 6 bursts observed by BATSE in real-time mode (these bursts were observed by only *KW* and BATSE). For cross-correlation with triggered BATSE bursts we utilized *KW* light curves in the G2+G3 or in the G2 band with 2 or 16-ms resolution and BATSE concatenated light curves (DISCLA, PREB, and DISCSC data types) in the Ch2+Ch3+Ch4 or in the Ch2+Ch3 band with a time resolution of 64 ms. For several GRBs such light curves are not available and we utilized other types of BATSE data. Usually we tried different energy bands to check the consistency of the derived time delay and finally chose those with minimum $\chi^2$. The cross-correlation curves for different bands may be slightly shifted relative each other (by several ms) but the 3$\sigma$ intervals for cross-correlation lag $\tau$ are always in good agreement. The resulting $\chi^2_{r,min}$ range from 0.06 to 4.51 with a mean of 0.81. The maximum $\chi^2_{r,min}$ of 4.51 (dof=6) is a clear outlier in the distribution of bursts over $\chi^2_{r, min}$. It corresponds to the exceptionally intense GRB19970704\_T04097 (BATSE \#6293) with a peak count rate of $1.8 \times 10^5$ counts s$^{-1}$ (*KW* 2-ms time scale) and $6.9 \times 10^5$ counts s$^{-1}$ (BATSE 64-ms time scale). Both light curves (*KW* and BATSE) must be significantly distorted due to dead-time and pile-up effects. The derived statistical uncertainty in the time delay is only 3 ms, so we added 6 ms systematic uncertainty to account for the distortions. The derived uncertainties in the time delays range from 5 ms to 84 ms with a mean of 24 ms, and a geometrical mean of 18 ms. The resulting annuli 3$\sigma$ half-widths range from 0082 to 110 with a mean of 114, and a geometrical mean of 060. The widest annulus with half-width of 110 was obtained for GRB19991001\_T04950 (BATSE \#7781) – at that time *Wind* was only 0.34 lt-s from Earth. The distances between the center lines of the *KW*-BATSE annuli and the centers of BATSE locations range from 0007 to 77 with a mean of 223. For 14 bursts the BATSE error circle does not intersect the *KW*-BATSE annulus and the distances from the closest boundary of the annulus range from 1.02$\sigma$ to 7.2$\sigma$ [^1]. Of the 52 bursts, 16 were observed by only *KW* and BATSE, and 12 were observed by only *KW*, BATSE, and *BeppoSAX*. For these bursts, we constrained the burst location to a segment of the *KW*-BATSE annulus using the following method. We took the center of the segment to be the point on the annulus center line nearest to the center of the BATSE position, and then we derived the corners of the segment from the intersection of the annulus and a circle centered at this point with a radius equal to the sum of twice the BATSE 1$\sigma$ error and a systematic error, taken to be the larger of 20 and the distance between the BATSE position and the center line of the annulus (a core systematic error of $\simeq 2^\circ$ was found for BATSE locations: see @briggs99). This is illustrated in Figure \[Fig\_KW\_BATSE\_loc\]. ### *KW*-*Fermi* (GBM) triangulations The Gamma-Ray Burst Monitor (GBM) aboard the *Fermi* observatory is primarily designed to study of GRBs by making observations in the $\sim$8 keV – 40 MeV band [@meegan09]. The GBM has the advantage of high effective area and time-tagged data. The absolute timing of the GBM clock has an accuracy better than 20$\mu$s. The GBM TTE data contain counts in 128 energy channels from $\sim$5 keV to 2 MeV, which enables the preparation of the GBM light curve in three energy channels which are nearly the same as those of *KW*. GBM observed 34 *KW* short GRBs. We derived *KW*-GBM triangulation annuli for all of them. For cross-correlation we utilized *KW* light curves in the G2+G3 or in the G2 band with 2 or 16-ms resolution and GBM light curves with 1 or 16 ms resolution made from the TTE data (only NaI data were used). The resulting $\chi^2_{r, min}$ range from 0.16 to 2.10 with a mean of 0.90. The derived uncertainties in the time delays range from 2.5 ms to 136 ms with a mean of 22 ms, and a geometrical mean of 15 ms. The resulting annuli 3$\sigma$ half-widths range from 0035 (2.1) to 165 with a mean of 035, and a geometrical mean of 023. ### *KW*-*INTEGRAL* (SPI-ACS) triangulations The Anti-Coincidence Shield (ACS) of the SPI instrument on-board *INTEGRAL*, besides serving to veto the background in the germanium spectrometer, is routinely used as a nearly omnidirectional detector for gamma-ray bursts [@kienlin03]. It measures burst light curves with a time resolution of 50 ms in a single energy range above $\sim$80 keV (for more details see @lichti00). A systematic error in the ACS timing of 125$\pm$10 ms has been found [@rau04] and all SPI-ACS lc have been corrected automatically for this error after 2004 April; corrections for it were applied by hand to the data prior to this date. This systematic uncertainty is related to the approximate nature of conversion from on-board time to UTC used in the producing SPI-ACS light curves in real time (within just seconds after the event at <ftp://isdcarc.unige.ch/arc/FTP/ibas/spiacs/>). On the other hand, the time conversion used in archived and near real-time data is precise. Recently it was shown that the drift of the ACS clock with respect to the germanium detector clock during the whole *INTEGRAL* mission is around 1 ms [@zhang10], thereby reducing the systematic uncertainty of ACS timing from 10 ms to 1 ms. SPI-ACS light curves corrected for systematic shifts and characterized by high timing precision (at least down to 1 ms) are available in the *INTEGRAL* data archive since revision 3. The archived and equally precise near real-time data (available within hours after the observation) are accessible through web services [http://isdc.unige.ch/\~{}savchenk/spiacs-online/](http://isdc.unige.ch/~{}savchenk/spiacs-online/) and <http://www.isdc.unige.ch/heavens/>. They are routinely used for near real-time triangulation. SPI-ACS observed 139 *KW* short GRBs. We derived *KW*-SPI-ACS annuli for 103 of them. For cross-correlation we utilized *KW* light curves in the G2+G3 or in the G3 band with 2 or 16-ms resolution. The resulting $\chi^2_{r,min}$ range from 0.04 to 3.96 with a mean of 1.02. The derived uncertainties in the time delays range from 4 ms to 175 ms with a mean of 24 ms, and a geometrical mean of 19 ms. The resulting annuli 3$\sigma$ half-widths range from 0047 (2.8) to 43 with a mean of 041, and a geometrical mean of 029. ### *KW*-*Suzaku* (WAM) triangulations The Wide-band All-sky Monitor (WAM) is the active shield of the Hard X-ray detector aboard the *Suzaku* mission [@yamaoka09]. In the triggered mode it measures light curves with a time resolution of 1/64 s in four energy channels which cover the $\simeq$50–5000 keV range. In the real-time mode the time resolution is 1 s. It was established that the *Suzaku*-WAM timing is consistent with negligible systematic uncertainties [@yamaoka09]. WAM observed 61 *KW* short GRBs: 51 in the triggered mode and 10 in the real-time mode. We derived *KW*-WAM annuli for 45 triggered bursts. For cross-correlation we utilized *KW* light curves in the G2+G3 or in the G3 band with 2 or 16-ms resolution and WAM light curves summed over four energy channels of 1 to 4 WAM detectors with the strongest response. The resulting $\chi^2_{r,min}$ range from 0.21 to 1.78 with a mean of 1.03. The derived uncertainties in the time delays range from 4 ms to 104 ms with a mean of 20 ms, and a geometrical mean of 14 ms. The resulting annuli 3$\sigma$ half-widths range from 0060 (3.6) to 244 with a mean of 030, and a geometrical mean of 021. ### *KW-BeppoSAX* (GRBM) triangulations The *BeppoSAX* Gamma-Ray Burst Monitor (GRBM) was the anticoincidence shield of the high energy experiment PDS [Phoswich Detection System; @frontera97]. In the triggered mode it measured light curves with a time resolution of 7.8125 ms in the 40–700 keV range; in the real-time mode the time-resolution was 1 s [for more details see @frontera09]. GRBM observed 50 *KW* short GRBs: 41 in the triggered mode and 9 in the real-time mode. We derived *KW*-GRBM annuli for 38 bursts observed in the triggered mode and for one burst observed in the real-time mode (this burst was observed by only *KW* and GRBM). For cross-correlation we utilized *KW* light curves in the G2 or in the G2+G3 band with 2 or 16-ms resolution and GRBM light curves rebinned to 32 ms. The resulting $\chi^2_{r,min}$ range from 0.25 to 12.1 with a mean of 1.43. The maximum $\chi^2_{r,min}$ of 12.1 (dof=6) is a clear outlier in the distribution of bursts over $\chi^2_{r, min}$. It corresponds to the exceptionally intense GRB19970704\_T04097 with a peak count rate of 1.8$\times 10^5$ counts s$^{-1}$ (*KW* 2-ms time scale) and 1.5$\times 10^5$ counts s$^{-1}$ (GRBM 32-ms time scale). Both light curves (*KW* and GRBM) must be significantly distorted due to dead-time and pile-up effects. The derived statistical uncertainty in the time delay is only 2 ms, so we increased this uncertainty to 6 ms to account for the distortions. The derived statistical uncertainties in the time delays range from 4.5 ms to 216 ms with a mean of 32 ms, and a geometrical mean of 18 ms. Comparison of the initially derived annuli with other available IPN annuli as well as comparison of the GRBM and BATSE light curves for common bursts has revealed a systematic error in the GRBM timing up to 100 ms. Since this error varies from burst to burst (both in value and in sign), we had to introduce 100 ms systematic error for *KW*-SAX triangulations. This leads to a significant broadening of the annuli, so their final 3$\sigma$ half-widths range from 123 to 322 with a mean of 530, and a geometrical mean of 387. ### *KW*-*Swift* (BAT) triangulations The *Swift* Burst Alert Telescope (BAT) is a highly sensitive, large field of view (FOV) coded aperture telescope that detects and localizes GRBs in real time [@barthelmy05]. When a burst occurs outside its FoV, it cannot be imaged, but the BAT light curve can be used for triangulation. For such bursts the 64 ms light curves in the four standard BAT energy channels (15–25 keV, 25–50 keV, 50–100 keV, and 100–350 keV) are always available. For some bursts, TTE data are available, enabling any desired energy and time binning. BAT observed 44 *KW* short bursts outside its FoV. We derived *KW*-BAT annuli for 23 of them. For cross-correlation we utilized *KW* light curves in the G2 or in the G2+G3 band with 2 or 16-ms resolution and BAT 64 ms light curves usually taken in the energy range above 50 keV (which often provides the best S/N and corresponds better to the *KW* energy band). The resulting $\chi^2_{r,min}$ range from 0.25 to 7.48 with a mean of 1.41. The maximum $\chi^2_{r,min}$ of 7.48 (dof=6) is a clear outlier in the distribution of bursts over $\chi^2_{r, min}$. It corresponds to the exceptionally intense GRB20060306\_T55358 with strong spectral evolution and a peak count rate of $1.9 \times 10^5$ counts s$^{-1}$ (*KW* 2-ms time scale). The derived statistical uncertainty in the time delay is only 5 ms, so we added 10 ms systematic uncertainty. The derived uncertainties in the time delays range from 5 ms to 64 ms with a mean of 22 ms, and a geometrical mean of 18 ms. The resulting annuli 3$\sigma$ half-widths range from 0059 (3.5) to 118 with a mean of 041, and a geometrical mean of 029. ### *KW*-*Coronas-F* (Helicon) triangulations The Helicon gamma-ray spectrometer was one of the instruments onboard the *Coronas-F* solar space observatory [@oraevskii02]. It was similar to the *KW* spectrometer in the characteristics of its two detectors and in the data presentation structure. The similar design of both instruments enabled good cross-correlations of burst light curves. Helicon observed 14 *KW* short GRBs. We derived *KW*-Helicon annuli for all of them. The resulting $\chi^2_{r,min}$ range from 0.25 to 2.67 with a mean of 1.02. The derived uncertainties in the time delays range from 4 ms to 80 ms with a mean of 25 ms, and a geometrical mean of 17 ms. The resulting annuli 3$\sigma$ half-widths range from 0045 (2.7) to 115 with a mean of 038, and a geometrical mean of 025. ### *KW*-*Cosmos* (Konus-A,A2,A3) triangulations Konus-A, Konus-A2, and Konus-A3 were gamma-ray spectrometers aboard the *Cosmos* spacecraft 2326, 2367, and 2421, respectively. A brief description of the Konus-A instrument is given in @aptekar98. Konus-A2 and Konus-A3 were similar to Konus-A in the characteristics of its detectors and in the data presentation structure. They observed 5 *KW* short GRBs in the triggered mode. We derived *KW*-*KA* annuli for 4 of them. The resulting $\chi^2_{r,min}$ range from 0.73 to 1.42 with a mean of 1.07. The derived uncertainties in the time delays range from 4 ms to 56 ms with a mean of 35 ms. The resulting annuli 3$\sigma$ half-widths range from 015 to 120 with a mean of 079. ### *KW-RHESSI* triangulations The *Reuven Ramaty High-Energy Solar Spectroscopic Imager* (*RHESSI*) is a high resolution spectrometer designed to study high-energy emission from solar flares over a broad energy range from 3 keV to 17 MeV [@lin02; @smith02]. The data are collected in the TTE mode enabling arbitrary energy and time binning. *RHESSI* observed 58 *KW* short GRBs. We derived *KW*-RHESSI annuli for 32 bursts. For cross-correlation we utilized *KW* light curve in the G2, in the G1+G2, or in the G2+G3 band with 2, 16, 64, and 256-ms resolution (depending on burst intensity). The resulting $\chi^2_{r,min}$ range from 0.36 to 2.62 with a mean of 1.07. The derived uncertainties in the time delays range from 2 ms to 184 ms with a mean of 36 ms, and a geometrical mean of 20 ms. The resulting annuli 3$\sigma$ half-widths range from 0027 (1.6) to 271 with a mean of 053, and a geometrical mean of 030. ### *KW-HETE-2* (FREGATE) triangulations The gamma-ray detector of *HETE-2*, called FREGATE, was designed to detect gamma-ray bursts in the energy range 8–-400 keV [@ricker03; @atteia03]. In the triggered mode it measures light curves with a time resolution of 1/32 s in the 8–400 keV range; in the real-time mode the time-resolution is 0.1638 s. FREGATE observed 16 *KW* short GRBs: 8 in the triggered mode and 8 in the real-time mode. In most cases the FREGATE response is significantly weaker than the responses of other instruments flying on low-Earth s/c, so we used the FREGATE data only for a few cases when no other low-Earth s/c detected the burst. We derived *KW-HETE* annuli for 4 bursts observed in the triggered mode. For cross-correlation we utilized *KW* light curve in the G2 or in the G2+G3 band with 2 or 16-ms resolution. The resulting $\chi^2_{r,min}$ range from 0.50 to 1.43 with a mean of 0.96. The derived uncertainties in the time delays range from 56 ms to 168 ms with a mean of 102 ms. The resulting annuli 3$\sigma$ half-widths range from 095 to 147 with a mean of 110. ### *KW-AGILE* (MCAL) triangulations The mini-calorimeter (MCAL) aboard the *AGILE* mission is a spectrometer sensitive to gamma-rays in the energy band $\simeq$0.35 – 100 MeV [@tavani09]. MCAL has the advantage of time-tagged data. MCAL observed 24 *KW* short GRBs: 22 in the triggered mode and 2 in the real-time mode. In many cases the MCAL response is rather weak due to its high energy threshold and strong attenuation by the GRID instrument, so we used the MCAL data only for several intense bursts. In total we derived 9 *KW*-MCAL annuli. For cross-correlation we utilized *KW* light curve in the G3 or in the G2+G3 band with 2 or 16-ms resolution. The resulting $\chi^2_{r,min}$ range from 0.29 to 2.26 with a mean of 1.08. The derived uncertainties in the time delays range from 5 ms to 21 ms with a mean of 13 ms. The resulting annuli 3$\sigma$ half-widths range from 0071 (4.3) to 060 with a mean of 021. ### INTEGRAL – near-Earth s/c triangulations Even without the planetary missions, the mini-network of low-Earth orbiters, plus *INTEGRAL* and Konus-*Wind*, often make it possible to obtain error boxes for many bursts. Since *INTEGRAL* orbits at distances $\lesssim$0.5 lt-s, which are much smaller than the *Wind*-to-Earth distance $\simeq$5 lt-s, *KW-INTEGRAL* and *KW*-near-Earth s/c annuli intersect at grazing incidence, resulting in one or two long boxes. In some cases, the intersection of *KW*-near-Earth s/c and *INTEGRAL*-near-Earth s/c results in smaller localization regions. In total 11 *INTEGRAL*-near-Earth s/c annuli have been obtained. The resulting annuli half-widths range from 10 to 140 with a mean of 59. Verifying triangulation annuli\[Sec\_AnnVerification\] ------------------------------------------------------ Of the 271 Konus short bursts localized by IPN, 17 were precisely localized by instruments with imaging capabilities: 15 by *Swift*-BAT (one of them, GRB 090510, was also localized by *Fermi*-LAT), 1 by *HETE-2* (WXM and SXC), and 1 by *INTEGRAL* IBIS/ISGRI. We utilized these bursts to verify our triangulations. For these 17 bursts, 21 *KW*-near-Earth s/c and 12 *KW*-*INTEGRAL* annuli were obtained (we have not used the light curves of the instruments which imaged the burst, since the instrument response for imaged bursts is different from those detected outside the FoV and used for IPN triangulations). In each case the triangulation annuli are in agreement with the known position of the source, thereby confirming the reliability of our triangulations. Indeed, such tests constitute “end-to-end" calibrations, as they confirm not only spacecraft timing and ephemeris information, but also the cross-correlations of the various time histories and derivations of the annuli. The histograms of Figure \[Fig\_AnnuliVerification\] show the distribution of relative source offsets from the center lines of the annuli. One can see that all offsets (in absolute values) are less than 2$\sigma$. The minimum offset of the precise position is -2.0$\sigma$, the maximum is 1.9$\sigma$, the average is 0.04$\sigma$, and the standard deviation is 1.1$\sigma$. Besides this verification, the consistency of several *KW*-near-Earth s/c annuli often obtained for a given burst with each other and with distant s/c annuli (when available) confirms the reliability of our *KW*-near-Earth triangulations, many of which have time delay uncertainties less than 10–20 ms (see Figure \[Fig\_KWAnnuli\]). LOCALIZATIONS: ADDITIONAL CONSTRAINS ==================================== In addition to triangulation annuli, several other types of localization information are included in this catalog. They are ecliptic latitude range, autonomous burst localizations obtained by *CGRO* BATSE, *BeppoSAX* GRBM, and *Fermi* GBM, and Earth- or Mars-blocking (MESSENGER is in an eccentric orbit around Mercury, so Mercury-blocking is quite rare). This additional information helps constrain the triangulation position, i.e., to choose one of two triangulation boxes, or to eliminate portions of a single annulus. Ecliptic latitudes ------------------ The ecliptic latitudes of the bursts are derived by comparing the count rates of the two *KW* detectors taken in the waiting mode with 1.472 s or 2.944 s time resolution. The axis of the detector S2 points towards the north ecliptic pole, and the axis of S1 points toward the south ecliptic pole. In addition to statistical uncertainties, the ecliptic latitude determination is subject to systematic uncertainties due to, among other things, time-variable cosmic X-ray sources and absorption by other instruments aboard the spin-stabilized *Wind* spacecraft. The estimated ecliptic latitudes can be taken to be at the 95% confidence level. The ecliptic latitude range, namely the best estimate of $b$, and the lower and upper limits $b_{min}$, $b_{max}$ can be considered to be an annulus centered at the north or south ecliptic pole, with a half-angle $\theta = 90^\circ-|b|$ and half-widths $d_{-}(\theta) = b_{min}-b$, $d_+(\theta) = b_{max} - b$. Planet-blocking --------------- Planet-blocking is specified by the right ascension and declination of the planet’s center and its radius. When a spacecraft in low Earth or Mars orbit observes a burst, the planet blocks up to $\approx$ 3.7 sr of the sky. The source position must be outside this occulted part of the sky. The allowed part of the sky can be described as a degenerate annulus centered at the direction opposite to the planet’s center, with a half-angle $\theta=0$ whose widths $d_{-}(\theta)=0$, $d_{+}(\theta) = \sin^{-1} (R_{planet}/R)$, where $R$ is the radius of the s/c orbit (here we neglect the oblateness of the planet and absorption in its atmosphere). Autonomous localizations ------------------------ A principle of autonomous burst localization using a system of detectors posessing anisotropic angular sensitivity was suggested by @golenetskii74 and first implemented in the KONUS instruments on the Venera 11 and 12 missions [@mazets81]. Similar localization systems consisting of different numbers of detectors have been placed on *CGRO* (BATSE), *BeppoSAX* (GRBM), and *Fermi* (GBM). These autonomous localizations, derived by comparing the count rates of various detectors, are affected by Earth albedo and absorption by spacecraft materials, among other things, and their shapes are in general complex. The error circles are approximations to these shapes. They are centered at the point which is the most likely arrival direction for the burst, and their radii are defined so that their areas are equal to the 1$\sigma$(BATSE, GBM) or 90% confidence (GRBM) statistical-only true error regions. All these localizations also have systematic errors of several degrees or more. These error circles can also be described as degenerate annuli centered at the most likely arrival direction for the burst, with a half-angle $\theta=0$ whose widths $d_{-}(\theta)=0$, $d_{+}(\theta) = r$, where $r$ is the positional error. LOCALIZATIONS: RESULTS ====================== Table \[Table\_Annuli\] summarizes localization information for 271 Konus short bursts. The first column gives the burst designation (see Table \[Table\_Basic\]). The second column gives the number of localization constraints (the number of rows with localization information for the burst). The six subsequent columns give localizations expressed as a set of annuli: the third column gives the source of the location: either sc1–sc2 (triangulation annulus derived using sc1 and sc2), or ‘Ecl.Band’ (range of ecliptic latitudes), or ‘Instr’ (name of the instrument which autonomously localized the bursts), or ‘Occ.sc’ (planet-blocking); columns 4–8 list the right ascension and declination of the annulus center (J2000), the annulus radius $\theta$, and the 3$\sigma$ uncertainties in the radius $d_{-}(\theta)$, $d_{+}(\theta)$. Planet-blocking is given only if it constrains the location. The ecliptic latitude range is given for all bursts. All available autonomous localizations are given. The *Swift*-BAT localizations are taken from the second *Swift* BAT catalog covering 2004 December 19 to 2009 December 21 [@sakamoto11], and for the latest bursts from the GCN Circulars with BAT refined positions. The *HETE-2* localizations for GRB 040924 (=GRB20040924\_T42735) is taken from @arimoto06. The IBIS/ISGRI localization for GRB 070707 (=GRB20070707\_T58122) is taken from @gotz07. The BATSE localizations are taken from the current catalog on the BATSE website [^2], as well as from the BATSE untriggered burst catalogs [@stern01; @kommers00][^3]. The *BeppoSAX* localizations are taken either from the GRBM catalog [@frontera09] or from the IAU and GCN Circulars. The GBM localizations are taken from the first *Fermi* GBM catalog covering 2008 July 12 to 2010 July 11 [@paciesas12], the GCN Circulars, or from the latest version of the corresponding ‘glg\_tcat\*.fit’ file in the GBM data archive [^4]. Boxes ----- For those bursts which were detected by three or more well separated s/c, a triangulation box can be derived. In general, the intersection of two annuli involving distant s/c gives a small box with an area as small as 1 arcmin$^2$. The intersection of two annuli derived from a distant s/c, Konus-*Wind*, and a near-Earth s/c usually gives an elongated box, which nevertheless in most cases has a small area of several hundred arcmin$^2$. In some cases the intersection of annuli derived from a single distant s/c, Konus-*Wind*, and a near-Earth s/c can give a smaller error box than annuli derived using two distant s/c. Long boxes were derived for bursts not observed by any distant s/c, but observed by *KW*, *INTEGRAL* SPI-ACS, and one or more near-Earth s/c. In such cases, the box is formed by a *KW*–near-Earth s/c annulus and an *INTEGRAL*–near-Earth s/c annulus, or by a *KW*–near-Earth s/c annulus and a *KW*–*INTEGRAL* annulus intersecting at grazing incidence. In all cases, if the three s/c which formed the box were nearly aligned, the annuli intersect at grazing incidence, resulting in a long box. In total we derived 162 error boxes for Konus short bursts: 27 for bursts observed by two distant s/c, 84 for bursts observed by one distant s/c and at least one near-Earth s/c, and 51 for bursts observed by only *KW*, *INTEGRAL*, and one or more near-Earth s/c. In some cases these error regions are actually long arcs rather than boxes (in particular this is a case when the burst was not observed by a distant s/c), but for simplicity we still refer to them as boxes since they are formed by intersection of two or more triangulation annuli. Segments -------- For those bursts which were detected only by *KW* and another s/c, or by *KW* and one or more near-Earth s/c, the resulting localization is formed by a triangulation annulus (the narrowest in the case of several *KW*-near-Earth s/c annuli) and additional constraints. These localizations consist of the entire annulus (in the case where it is entirely inside the allowed ecliptic latitude band and there are no other constraints) or one or two annulus segments, formed by the intersection of the annulus with the ecliptic latitude band, and/or by exclusion of the occulted part of the annulus, or by combination with the BATSE localization (see section \[Sec\_KWBATSEann\]). 114 bursts had this kind of localization: 20 were bursts observed by *KW* and a distant s/c (of these, 3 were also observed by a near-Earth s/c in real-time mode, but *KW*-near-Earth s/c annuli have not been derived for them), and 94 were bursts observed by only *KW* and one or more near-Earth s/c. Resulting error regions\[Sec\_ErrorRegions\] -------------------------------------------- Table \[Table\_Boxes\] gives the description of the final IPN error regions for 254 Konus short bursts (this sample does not include the 17 imaged bursts). The nine columns contain the following information: (1) the burst designation (see Table \[Table\_Basic\]), (2) the number of error regions for the burst, $N_{r}$: 1 or 2, (3) the number of corners of the region, $N_{c}$, (4) the region type: ‘B’ (box), ‘LB’ (long box: box with maximum dimension $>10^\circ$), ‘S’ (segment), or ‘A’ (annulus), (5) the area (for two regions the sum of their areas), (6) the maximum dimension of the region (that is the maximum angular distance between two points at the region boundary; for segments larger than half an annulus, this is just the outer diameter of the annulus), (7) the right ascension of the center of the error region, in the first row, and the right ascensions of the corners in the following $N_{c}$ rows, and (8) the declination of the center of the error region, in the first row, and the declinations of the corners in the following $N_{c}$ rows (if there are two error regions, additional $N_{c}$+1 rows are given: the center of the second region and its corners, so the total number of rows for such a burst is 2($N_{c}$+1)). All coordinates are J2000. In general, a simple, four-corner error region description is inaccurate and the curvature of the annuli should be taken into account. Only in cases where the maximum dimension of the error region is less than several degrees, can the box be reasonably well represented by its four corners. In other cases, especially when the region is a long arc or annulus segment, the given corners and center are intended to roughly indicate the position of the region on the sky. Figures showing the IPN localization (all derived annuli and the resulting error region(s)) can be found at the Ioffe Web site [^5]. A histogram of IPN error region areas is shown in Figure \[Fig\_AreaStat\]. For bursts observed by distant s/c the areas range from 2.40$\times 10^{-4}$ sq. deg (0.86 arcmin$^2$) to 142.1 sq. deg with a mean of 3.49 sq. deg, and a geometrical mean of 0.141 sq. deg. For bursts without distant s/c detections the areas range from 0.210 sq. deg to 4420 sq. deg with a mean of 242 sq. deg, and a geometrical mean of 46.2 sq. deg. COMMENTS ON SPECIFIC EVENTS =========================== GRB 051103 (=GRB20051103\_T33943) may in fact be a giant SGR flare in the nearby M81 group of interacting galaxies as was suggested by @frederiks07. The final IPN localization of this event along with further exploration of this possibility are given in @hurley10b. See also [@ofek06] for implications of the optical and radio followup observations and @abadie12 for implications of the gravitational-wave non-detection of this event. GRB 070201 (=GRB20070201\_T55390) is likely a giant SGR flare from the Andromeda galaxy [@mazets08]. See also @abbott08 for implications of the gravitational-wave non-detection of this event, and @ofek08 for implications of the optical afterglow and X-ray periodic source non-detections. GRB 000420 (=GRB20000420\_T42271): based on the *KW-NEAR* annulus it was suggested by @ofek07 that this burst might be associated with the nearby Sc-type galaxy M74 (NGC 628). The position of this galaxy lies well outside the wide *KW-SAX* annulus, thereby excluding it as a possible host for this short GRB – see Figure \[Fig\_GRB000420\]. GRB 990405 (=GRB19990405\_T30059): initially this event was classified as a burst from SGR 1900+14 since the narrow *SAX-Ulysses* annulus (3$\sigma$ half-width of 0$\fdg$035) passes through the position of this SGR. The derived wide *KW-SAX* annulus (3$\sigma$ half-width of 64) is also consistent with the SGR position. But this burst is substantially harder even than two unusually hard bursts from SGR 1900+14: 981022, 991001 [@woods99], making the possible association of this burst with the SGR doubtful. SUMMARY AND CONCLUSION ====================== This paper continues a series of catalogs of gamma-ray burst localizations obtained by arrival-time analysis, or “triangulation" between the spacecraft in the 3rd interplanetary network, as summarized in Table \[Table\_IPNcatalogs\]. We have presented the most comprehensive IPN localization data on 271 Konus-*Wind* short bursts. For 254 bursts IPN error regions were obtained and for 17 bursts precisely localized by instruments with imaging capability IPN triangulation annuli were derived for calibration purposes. In total we derived 517 triangulation annuli, including 150 annuli with distant s/c. It was shown that for many shorts bursts *KW*–near-Earth s/c (or *INTEGRAL*) triangulation yields a rather narrow annulus (with half-width sometimes comparable to or even better than the annuli using distant s/c data), thereby providing small error boxes with areas of several hundred arcmin$^2$ even for those *KW* short bursts which were detected by only one distant s/c (and one or more near-Earth s/c), and providing a long box in cases where the burst was detected by Konus-*Wind*, *INTEGRAL* SPI-ACS, and one or more near-Earth s/c. The localizations can be used for a wide variety of purposes, including, but not limited to, searches for a) gravitational wave and neutrino signals from merging compact objects b) very high energy photons from the burst sources c) giant SGR flares in nearby galaxies. The Konus-*Wind* experiment is supported by a Russian Space Agency contract and RFBR grants 12-02-00032a and 13-02-12017-ofi-m. 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M., Kouveliotou, C., van Paradijs, J., et al., 1999, , 527, L47 Zhang, B., Zhang, B.-B., Virgili, F., et al., 2009, 703, 1696 Zhang, X.-L., Rau, A., & von Kienlin, A. 2010, in proceeding of the 8th INTEGRAL Workshop The Restless Gamma-ray Universe, 2010 September 27–30, Dublin, Ireland, PoS (INTEGRAL 2010)161 [ccccrcccc]{} GRB19950210\_T08424 & 1995 Feb 10 & 02:20:24.148 & I & -2.602(-0.009,+0.007) & Uly(T),GRO(\#3410) & 3 & 1 &\ GRB19950211\_T08697 & 1995 Feb 11 & 02:24:57.749 & I & 0.003(-0.009,+0.005) & Uly(T),GRO(\#3412),SRS(T) & 4 & 1 &\ GRB19950414\_T40882 & 1995 Apr 14 & 11:21:22.798 & I & 0.350(-0.008,+0.006) & Uly(R) & 2 & 1 &\ GRB19950419\_T08628 & 1995 Apr 19 & 02:23:48.860 & I & 0.418(-0.030,+0.026) & Uly(T) & 2 & 1 &\ GRB19950523\_T31302 & 1995 May 23 & 08:41:42.284 & I & 0.436(-0.068,+0.040) & Uly(T) & 2 & 1 &\ [cclrrrrr]{} GRB19950210\_T08424 & 4 & Uly-GRO & 155.4443 & +25.7475 & 53.6317 & -0.0078 & +0.0078\ & & KW-GRO & 130.3580 & +18.8483 & 52.3229 & -0.1532 & +0.1189\ & & Ecl.Band & 90.000 & -66.561 & 43.3 & -19.5 & +46.0\ & & BATSE & 154.55 & -27.48 & 0 & 0 & 1.15\ GRB19950211\_T08697 & 4 & Uly-GRO & 335.8036 & -25.1240 & 85.8298 & -0.0063 & +0.0063\ & & KW-GRO & 311.5868 & -18.2357 & 89.9796 & -0.0678 & +0.1221\ & & Ecl.Band & 270.000 & 66.561 & 65.5 & -39.5 & +22.6\ & & BATSE & 9.51 & 52.65 & 0 & 0 & 1.08\ [ccccrrccc]{} GRB19950210\_T08424 & 1 & 4 & B & 5.65E-001 & 8.39E-003 & Uly-GRO,KW-GRO,Ecl.Band & 154.6820 & -27.8792\ & & & & & & & 154.3544 & -27.8662\ & & & & & & & 154.9633 & -27.8744\ & & & & & & & 154.9330 & -27.8897\ & & & & & & & 154.3240 & -27.8812\ GRB19950211\_T08697 & 1 & 4 & B & 5.13E-001 & 6.09E-003 & Uly-GRO,KW-GRO,Ecl.Band & 14.7526 & +53.8367\ & & & & & & & 14.5265 & +53.9214\ & & & & & & & 15.1974 & +53.6449\ & & & & & & & 15.1690 & +53.6722\ & & & & & & & 14.4975 & +53.9485\ [ccc]{} 1990–1992 & 16 & *Ulysses, Pioneer Venus Orbiter,* WATCH, SIGMA, PHEBUS GRBs\ 1990–1994 & 56 & *Granat-*WATCH supplement\ 1991–1992 & 37 & *Pioneer Venus Orbiter, Compton Gamma-Ray Observatory, Ulysses* GRBs\ 1991–1994 & 218 & BATSE 3B supplement\ 1991–2000 & 211 & BATSE untriggered burst supplement\ 1992–1993 & 9 & *Mars Observer* GRBs\ 1994–1996 & 147 & BATSE 4Br supplement\ 1994–2010 & 279 & Konus short bursts\ 1996–2000 & 343 & BATSE 5B supplement\ 1996–2002 & 475 & *BeppoSAX* supplement\ 2000–2006 & 226 & HETE-2 supplement\ 2008–2010 & 146 & GBM supplement ![*Left*: Timelines of the IPN missions since the launch of *Wind* in 1994, November (instrument names are given in the parentheses). *Right:* Number of Konus short bursts observed by each mission (for *Wind* (Konus) – the number of bursts observed by at least one other IPN s/c is given).[]{data-label="Fig_TimeLines"}](fig1.eps){width="\textwidth"} ![Examples of cross-correlation curves $\chi^2_r(\tau)$. Horizontal red lines denote 3$\sigma$ levels. Vertical blue lines show the best cross-correlation time lag $\tau$ (dashed-dotted line) and its 3$\sigma$ confidence interval (dashed lines). *Top Left:* GRB19971118\_T29008. Cross-correlation of the *KW* 2 ms light curve with the BATSE 64 ms light curve; $\tau = 0.060(-0.034,+0.012)$ s (dof=2). *Top Right:* GRB20070321\_T67937. Cross-correlation of the *KW* 2 ms light curve with the WAM 1/64 s light curve; $\tau = 0.075(-0.008,+0.016)$ s (dof=12). *Bottom Left:* GRB20090715\_T62736. Cross-correlation of the *KW* 2 ms light curve with the SPI-ACS 50 ms light curve; $\tau = 0.120(-0.016,+0.052)$ s (dof=7). *Bottom Right:* GRB20100206\_T48606. Cross-correlation of the GBM 1 ms light curve with the *KW* 16 ms light curve; $\tau = -0.033 \pm 0.012$ s (dof=9).[]{data-label="Fig_R2examples"}](fig2.eps){width="80.00000%"} ![Distribution of 3$\sigma$ half-widths (HWs) of the 150 triangulation annuli obtained using the distant s/c data. The smallest HW is 0$\fdg$0024 (0.14), the largest is 2$\fdg$21, the mean is 0$\fdg$099 (5.9), and the geometrical mean is 0$\fdg$028 (1.7).[]{data-label="Fig_DistAnnuli"}](fig3.eps){width="70.00000%"} ![*Wind* distance from Earth as a function of time. The maximum distance was $\simeq$7 lt-s in January and May 2002, when it was in a Distant Prograde Orbit (DPO). Since 2004 *Wind* has been in a Lissajous orbit at the L$_1$ libration point of the Sun-Earth system at a distance of $\simeq$5 lt-s.[]{data-label="Fig_WindDistance"}](fig4.eps){width="70.00000%"} ![Distributions of uncertainties in time delay $d(\delta T) \equiv (d_{+}(\delta T) + |d_{-}(\delta T)|)/2$ and 3$\sigma$ half-widths (HWs) of the 356 triangulation annuli obtained using the Konus-*Wind* and near-Earth (or INTEGRAL) s/c data. The smallest $d(\delta T)$ is 2 ms, the largest is 504 ms, the mean is 43 ms, and the geometrical mean is 23 ms. The smallest HW is 0027 (1.6), the largest is 322, the mean is 130, and the geometrical mean is 043.[]{data-label="Fig_KWAnnuli"}](fig5.eps){width="70.00000%"} ![IPN/BATSE localization of GRB19960420\_T16844 (BATSE \#5439). The center of the BATSE error circle (R.A., Decl.(J2000), Err = 23425, -2723, 237) lies 338 from the center line of the 167 wide *KW*-BATSE annulus. The resulting long box is shown by the solid black line and its center (that is, the nearest point to the BATSE center at the annulus center line) is indicated by the asterisk. The corners of the box are formed by the intersection of the circle centered at the asterisk with a radius of 8$\fdg$12, that is the sum of the 2$\sigma$ BATSE error radius and 338 systematics (dashed line), and the *KW*-BATSE annulus.[]{data-label="Fig_KW_BATSE_loc"}](fig6.eps){width="70.00000%"} ![Distribution of the offsets of the accurate GRB positions from the center lines of the 33 *KW*-near-Earth (or *INTEGRAL*) s/c annuli.[]{data-label="Fig_AnnuliVerification"}](fig7.eps){width="70.00000%"} ![Distributions of error region areas for 123 Konus short bursts observed by at least one distant s/c (red dashed line), 131 bursts not observed by any distant s/c (blue dashed line), and all 254 bursts (17 imaged bursts are not counted).[]{data-label="Fig_AreaStat"}](fig8.eps){width="70.00000%"} ![IPN localization of GRB 000420 (=GRB20000420\_T42714). The 1.7$\arcmin$ wide *SAX-NEAR* annulus passes through the nearby M74 galaxy, while the galaxy is well outside the wide 3$\sigma$ *KW-SAX* annulus.[]{data-label="Fig_GRB000420"}](fig9.eps){width="70.00000%"} [^1]: For GRB19961225\_T36436 (BATSE \#5725) the BATSE position given in the current BATSE catalog (R.A., Decl.(J2000), Err = 17131, -285, 15) is 252 away from the center line of the 16 wide *KW*-BATSE annulus. This GRB is a “mirror" case for BATSE, with a bimodal probability distribution for the location. The alternative solution is R.A., Decl.(J2000) = 1414, -55, with a statistical error of 18 (M. Briggs, private communication, 2011). This position is 73 away from the center line of the *KW*-BATSE annulus. We use this alternative BATSE position here. [^2]: <http://www.batse.msfc.nasa.gov/batse/grb/catalog/current/> [^3]: Since the catalog by @stern01 contains the localizations for all 8 Konus short bursts detected by BATSE in the real-time mode, and the catalog by @kommers00 misses some of them, the given localizations are solely from @stern01. [^4]: ftp://legacy.gsfc.nasa.gov/fermi/data/gbm/bursts [^5]: <http://www.ioffe.ru/LEA/ShortGRBs_IPN/>

--- abstract: 'Recently, some quantum algorithms have been implemented by quantum adiabatic evolutions. In this paper, we discuss the accurate relation between the running time and the distance of the initial state and the final state of a kind of quantum adiabatic evolutions. We show that this relation can be generalized to the case of mixed states.' author: - Zhaohui Wei and Mingsheng Ying title: A relation between fidelity and quantum adiabatic evolution --- Implementing quantum algorithms via quantum adiabatic evolutions is a novel paradigm for the design of quantum algorithms, which was proposed by Farhi et al. [@FGGS00]. In a quantum adiabatic algorithm, the evolution of the quantum register is governed by a hamiltonian that varies continuously and slowly. At the beginning, the state of the system is the ground state of the initial hamiltonian. If we encode the solution of the algorithm in the ground state of the final hamiltonian and if the hamiltonian of the system evolves slowly enough, the quantum adiabatic theorem guarantees that the final state of the system will differ from the ground state of the final hamiltonian by a negligible amount. Thus after the quantum adiabatic evolution we can get the solution with high probability by measuring the final state. For example, Quantum search algorithm proposed by Grover [@GROVER97] has been implemented by quantum adiabatic evolution in [@RC02]. Recently, the new paradigm for quantum computation has been tried to solve some other interesting and important problems [@TH03; @TDK01; @FGG01]. For example, T. D. Kieu has proposed a quantum adiabatic algorithm for Hilbert’s tenth problem [@TDK01] , while this problem is known to be mathematically noncomputable. Usually, after the design of a quantum adiabatic evolution, the estimation of the running time is not easy. In [@RC02], Roland et al. introduced a policy to design a class of quantum local adiabatic evolutions with a performance that can be estimated accurately. Using this policy Roland et al. reproduced quantum search algorithm, which is as good as Grover’s algorithm. For convenience of the readers, we briefly recall the local adiabatic algorithm. Suppose $H_0$ and $H_T$ are the initial and the final Hamiltonians of the system, we choose them as $$H_0=I-|\alpha\rangle\langle\alpha|,$$ and $$H_T=I-|\beta\rangle\langle\beta|,$$ where $|\alpha\rangle$ is the initial state of the system and $|\beta\rangle$ is the final state that encodes the solution. Then we let the system vary under the following time dependent Hamiltonian: $$H(t)=(1-s)H_0+sH_T,$$ where $s=s(t)$ is a monotonic function with $s(0)=0 $ and $s(T)=1$ ($T$ is the running time of the evolution). Let $|E_0,t\rangle$ and $|E_1,t\rangle$ be the ground state and the first excited state of the Hamiltonian at time t, and let $E_0(t)$ and $E_1(t)$ be the corresponding eigenvalues. The adiabatic theorem [@LIS55] shows that we have $$|\langle E_0,T|\psi(T)\rangle|^{2}\geq1-\varepsilon^2,$$ provided that $$\frac{D_{max}}{g_{min}^2}\leq\varepsilon,\ \ \ \ 0<\varepsilon\ll1,$$ where $g_{min}$ is the minimum gap between $E_0(t)$ and $E_1(t)$ $$g_{min}=\min_{0\leq t \leq T}[E_1(t)-E_0(t)],$$ and $D_{max}$ is a measurement of the evolving rate of the Hamiltonian $$D_{max}=\max_{0\leq t \leq T}|\langle\frac{dH}{dt}\rangle_{1,0}|=\max_{0\leq t \leq T}|\langle E_1,t|\frac{dH}{dt}|E_0,t\rangle|.$$ In the local adiabatic evolution of [@RC02], $$|\alpha\rangle=\frac{1}{\sqrt{N}}\sum\limits_{i=1}^{N}{|i\rangle}, \ |\beta\rangle=|m\rangle ,$$ where $N$ is the size of the database and $m$ is the solution of the search problem. To evaluate the running time of the adiabatic evolution, Roland and Cerf calculated accurately the gap $g_{min}$ in Eq. (2) and just estimated the quantity $D_{max}$ in Eq. (3) using the bound $$|\langle\frac{dH}{dt}\rangle_{1,0}|\leq |\frac{ds}{dt}|.$$ To evaluate the performance of this algorithms, this is enough, because calculating accurately the quantity $D_{max}$ in (7) can’t improve the result much. However, in this paper we will take into account all the related quantities. Later we will find that this will result in a simple and intrinsical relation between the running time of the adiabatic evolution and the distance of the initial and the final states. In this paper, we will choose fidelity, one of the most popular distance measures in the literature, as the measure of the hardness to evolve from one state to another using adiabatic evolutions. The fidelity of states $\rho$ and $\sigma$ is defined to be $$F(\rho,\sigma)=tr\sqrt{\rho^{1/2}\sigma\rho^{1/2}}.$$ Although fidelity is not a metric, its modified version $$A(\rho,\sigma)=\arccos{F(\rho,\sigma)}$$ is easily proved to be a metric [@Nielsen00]. Another important metric for the distance between quantum states we will use in this paper is the trace distance defined as $$D(\rho,\sigma)=\frac{1}{2}tr|\rho-\sigma|.$$ Now, we can represent the main result as the following theorem. Suppose $|\alpha\rangle$ and $|\beta\rangle$ are two states of a quantum system. We can make the system evolve from the initial state $|\alpha\rangle$ to the final state $|\beta\rangle$ by a quantum adiabatic evolution, if we set the initial Hamiltonian $H_0$ and the final Hamiltonian $H_T$ of the adiabatic evolution as follows: $$H_0=I-|\alpha\rangle\langle\alpha|,$$ $$H_T=I-|\beta\rangle\langle\beta|.$$ To success with a probability at least $1-\varepsilon^2$, the minimal running time that the adiabatic evolution requires is $$T(|\alpha\rangle,|\beta\rangle)=\frac{1}{\varepsilon}\cdot\tan{(\arccos{F(|\alpha\rangle,|\beta\rangle)})},$$ where $$F(|\alpha\rangle,|\beta\rangle)=|\langle\alpha|\beta\rangle|$$ is the fidelity between $|\alpha\rangle$ and $|\beta\rangle$. [*Proof.*]{} Let $$H(s)=(1-s)(I-|\alpha\rangle\langle\alpha|)+s(I-|\beta\rangle\langle\beta|),$$ where $s=s(t)$ is a function of $t$ as described above. It is not easy to calculate the eigenvalues of $H(s)$ in the computational basis. We use the following orthonormal basis $\{|i\rangle, 1\leq i\leq N\}$ to eliminate the difficulty: $$|1\rangle=|\alpha\rangle,$$ $$|2\rangle=\frac{1}{c}(|\beta\rangle-\langle\alpha|\beta\rangle|\alpha\rangle),$$ where $c=||\beta\rangle-\langle\alpha|\beta\rangle|\alpha\rangle|=\sqrt{1-|\langle\alpha|\beta\rangle|^{2}}$. We don’t need to care about $|\alpha_i\rangle$ for $i=3,4,...,N$. Then we have $$|\beta\rangle=c|2\rangle+\langle\alpha|\beta\rangle|1\rangle.$$ Now it is not difficult to check that, in the new orthonormal basis, $H(s)$ has a form of $$H(s)= \begin{pmatrix} -s|\langle\alpha|\beta\rangle|^2+s & -sc\langle\alpha|\beta\rangle \\ -sc\langle\alpha|\beta\rangle^{*} & -sc^2+1\\ & & I_{(N-2)\times(N-2)}\\ \end{pmatrix},$$ where the empty spaces of the matrix are all zeroes. Letting $a=|\langle \alpha|\beta\rangle|$, it is easy to get the two lowest eigenvalues of $H(s)$ $$E_i(t)=\frac{1}{2}(1\pm\sqrt{1-4(1-a^2)s(1-s)}), \ i=0,1,$$ and two corresponding eigenvectors $$|E_i,t\rangle=\frac{1}{\sqrt{1+y_i^2}}(|1\rangle+y_i|2\rangle), \ i=0,1,$$ where $$y_i=\frac{\sqrt{1-a^2}}{a}-\frac{E_i(t)}{sa\sqrt{1-a^2}} \ \ (s\neq0).$$ Thus, we get $g(s)$: $$g(s)=\sqrt{1-4(1-a^2)s(1-s)}.$$ On the other hand, it is easy to kown $$\frac{dH}{ds}=H_T-H_0=\frac{H(s)-H_0}{s} \ \ (s\neq0).$$ Because $|E_0,t\rangle$ and $|E_1,t\rangle$ are eigenvectors of $H(s)$, we have $$\langle E_0,t|E_1,t\rangle = 0,$$ and $$\langle E_0,t|H(s)|E_1,t\rangle = 0.$$ Then it can be shown that $$|\langle\frac{dH}{ds}\rangle_{0,1}|=|\langle E_0,t|\frac{H(s)-H_0}{s}|E_1,t\rangle|=|\frac{\langle E_0,t|1\rangle\langle 1|E_1,t\rangle}{s}|.$$ So $$|\langle\frac{dH}{dt}\rangle_{0,1}|=|\frac{ds}{dt}|\cdot|\langle\frac{dH}{ds}\rangle_{0,1}|=|\frac{ds}{dt}|\cdot\frac{1}{s\sqrt{(1+y_0^2)(1+y_1^2)}}.$$ Substituting Eq.(21) into Eq.(27) we have $$|\langle\frac{dH}{dt}\rangle_{0,1}|=|\frac{ds}{dt}|\cdot\frac{a\sqrt{1-a^2}}{\sqrt{1-4(1-a^2)s(1-s)}}.$$ In a local adiabatic evolution [@RC02], the adiabaticity condition (5) must be satisfied at any instant of time t, $$|\frac{ds}{dt}|\cdot\frac{a\sqrt{1-a^2}}{\sqrt{1-4(1-a^2)s(1-s)}}\leq \varepsilon(1-4(1-a^2)s(1-s)).$$ To make the evolution as fast as possible, we can let $s(t)$ satisfy the equation $$\frac{ds}{dt}=\varepsilon\frac{(1-4(1-a^2)s(1-s))^{\frac{3}{2}}}{a\sqrt{1-a^2}}.$$ By integration, we can get the lower bound of the running time of the whole evolution $$T(|\alpha\rangle, |\beta\rangle)=\frac{1}{\varepsilon}\cdot\frac{\sqrt{1-a^2}}{a}=\frac{1}{\varepsilon}\cdot\tan{(\arccos{F(|\alpha\rangle,|\beta\rangle)})}.$$ That completes the proof of this theorem. $\Box$ In fact, it is interesting to notice that we can rewrite the relation above as $$T(|\alpha\rangle, |\beta\rangle)=\frac{1}{\varepsilon}\cdot\frac{D(|\alpha\rangle,|\beta\rangle)}{F(|\alpha\rangle,|\beta\rangle)},$$ where $D(|\alpha\rangle,|\beta\rangle)$ is the trace distance between $|\alpha\rangle$ and $|\beta\rangle$. In [@RC02], the fidelity between the initial state and the final state of the local quantum adiabatic evolution is $\frac{1}{\sqrt{N}}$. According to Theorem 1 the running time is $O(\sqrt{N})$. This is consistent with the result of [@RC02]. Similarly, In [@DKK02] S. Das et al. implement Deutsch’s algorithm [@DD85; @DJ92] by an adiabatic evolution of the form discussed in Theorem 1. In that work, if the system has $n$ qubits, $|\alpha\rangle$ and $|\beta\rangle$ will be $(N=2^n)$ $$|\alpha\rangle=\frac{1}{\sqrt{N}}\sum\limits_{i=0}^{N-1}{|i\rangle},$$ $$|\beta\rangle=\mu|0\rangle+\frac{\nu}{\sqrt{N-1}}\sum\limits_{i=1}^{N-1}{|k\rangle},$$ with $$\mu=\frac{1}{N}|\sum\limits_{x\in\{0,1\}^n}^{}{(-1)^{f(x)}}|,$$ $$\nu=1-\mu.$$ Here, the function $f:\{0,1\}^n\rightarrow\{0,1\}$ is either constant (i.e., all outputs are identical) or balanced (i.e., has an equal number of 0’s and 1’s as outputs), and our task is to decide whether it is constant or not. It is not difficult to know that in this case $$F(|\alpha\rangle,|\beta\rangle) = |\langle\alpha|\beta\rangle| = \frac{1}{\sqrt{N}} \ or \ \sqrt{1-\frac{1}{N}}.$$ To make the algorithm success, we must let the running time of the adiabatic evolution be long enough. So the minimal running time should be $O(\sqrt{N})$. This result is consistent with [@DKK02]. Let’s try to explain the meaning of the theorem. As we know, $\arccos (F(\rho,\sigma))$ measures the distance of two quantum states $\rho$ and $\sigma$ [@Nielsen00]. Quantum adiabatic evolution, On the other hand, changes the state of a quantum system from the initial state $|\alpha\rangle$ to the final state $|\beta\rangle$. Our theorem says that if the precision of the evolution is fixed, the minimal running time $T(|\alpha\rangle,|\beta\rangle)$ will be direct proportional to the tangent of $\arccos (F(|\alpha\rangle,|\beta\rangle))$. The smaller the distance of the two states is, the shorter the running time of the adiabatic evolution will be. This is consistent with our intuition. However, we should notice that as the fidelity becomes smaller, the running time will increase very quickly. Fore example, when $F(|\alpha\rangle,|\beta\rangle)$ is 0.5, the running time $T(|\alpha\rangle,|\beta\rangle)$ is $\frac{\sqrt{3}}{\varepsilon}$. While as $F(|\alpha\rangle,|\beta\rangle)$ tends to 0, the running time tends to infinite. In the quantum adiabatic evolution, the initial and the final states are pure. Using Uhlmann’s theorem [@Nielsen00] we can generalize the relation to the case of mixed states. Suppose $\rho$ and $\sigma$ are two states of a quantum system $A$. Let $B$ is another system and $A$ is a part of $B$. Suppose an adiabatic evolution makes the state of $B$ evolve from $|\psi\rangle$ to $|\varphi\rangle$ and in the same evolution the state of $A$ evolves from $\rho$ to $\sigma$. We may ask — is there any relation between the running time of the adiabatic evolution and the fidelity of $\rho$ and $\sigma$? We say yes by the following theorem. Suppose $\rho$ and $\sigma$ are two mixed states, and let $$T(\rho,\sigma)=\min_{|\psi\rangle,|\varphi\rangle}T(|\psi\rangle,|\varphi\rangle),$$ where $|\psi\rangle$ is any purification of $\rho$ and and $|\varphi\rangle$ for $\sigma$. Then we have $$T(\rho,\sigma)=\frac{1}{\varepsilon}\cdot\tan{(\arccos{F(\rho,\sigma)})},$$ where $\varepsilon$ is the precision of the evolution. [*Proof.*]{} $$\begin{aligned} \aligned T(\rho,\sigma)=&\min_{|\psi\rangle,|\varphi\rangle}T(|\psi\rangle,|\varphi\rangle)\\ =&\min_{|\psi\rangle,|\varphi\rangle}\frac{1}{\varepsilon}\cdot\tan{(\arccos{F(|\psi\rangle,|\varphi\rangle)})}\\ =&\frac{1}{\varepsilon}\cdot\tan{(\arccos(\max_{|\psi\rangle,|\varphi\rangle}{F(|\psi\rangle,|\varphi\rangle)}))}. \endaligned\end{aligned}$$ Applying Uhlmann’s theorem [@Nielsen00] to the last equation, we can get $$T(\rho,\sigma)=\frac{1}{\varepsilon}\cdot\tan{(\arccos{F(\rho,\sigma)})}.$$ $\Box$ In conclusion, we have shown the accurate relation between the distance of the initial and the final states and the running time of a class of quantum adiabatic evolution applied in [@RC02]. We have pointed out that via this relation it is convenient to estimate the running times of some adiabatic algorithms. Furthermore, this relation can be generalized to the case of mixed states. This relation maybe can help to design quantum algorithms. We would like to thank Ji Zhengfeng for useful discussions. [9]{} E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, e-print quant-ph/0001106. L. K. Grover, Phys. Rev. Lett 79, 325(1997). J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308(2002). Tad Hogg, Phys. Rev. A 67, 022314(2003). T. D. Kieu, e-print quant-ph/0110136. E. Farhi et al. e-print quant-ph/0104129. L. I. Schiff, Quantum Mechanics (McGraw-Hill, Singapore, 1955). S. Das, R. Kobes, G. Kunstatter, Phys. Rev. A 65, 062310(2002). D. Deutsch, Proc. R. Soc. London, Ser, A 400, 97 (1985). D. Deutsch and R. Jozsa, Proc. R. Soc. London, Ser, A 439, 553 (1992). Michael A. Nielsen, Isaac L. Chuang Quantum Computation and Quantum Information, Cambridge University Press, 2000

--- abstract: | We investigate a double layer system with tight-binding hopping, intra-layer and inter-layer interactions, as well as a Josephson like coupling. We find that an antiferromagnetic spin polarization induces additional spin-triplet pairing (with $S_z =0$) to the singlet order parameter. This causes an undamped collective mode in the superconducting state below the particle-hole threshold, which is interpreted as a Goldstone excitation. author: - | Christian Helm, Franz Forsthofer, and Joachim Keller\ Institute for Theoretical Physics, University of Regensburg,\ D-93040 Regensburg, Germany date: 'submitted to J. of Low Temp. Phys.' title: Collective Spin Modes in Superconducting Double Layers --- PACS numbers: 71.45-d, 74.80.Dm, 74.50+r INTRODUCTION AND MODEL ====================== Collective density fluctuations in superconductors due to the breakdown of the global gauge invariance are well known theoretically [@WuGriffin; @wir]. However, since these modes couple to charge oscillations, the long range Coulomb force usually pushes up their energies to the plasma frequency. One possibility to avoid the Coulomb interaction completely, is to consider spin fluctuations instead of charge fluctuations between the layers. In the following we will show the existence of such a sharp, collective spin mode in the gap, which might have been observed[@Fong] in inelastic neutron scattering on Y-Ba-Cu-O. We consider an electronic double-layer system described by the Hamiltonian $H=H_0+H_S$: $$\begin{aligned} H_0 &=& \sum_{k\sigma} \epsilon_k (c^\dagger_{1k\sigma}c^{\phantom{\dagger}}_{1k\sigma} + c^\dagger_{2k\sigma}c^{\phantom{\dagger}}_{2k\sigma}) + t_k (c^\dagger_{2k\sigma} c^{\phantom{\dagger}}_{1k\sigma} + c^\dagger_{1k\sigma} c^{\phantom{\dagger}}_{2k\sigma}) \\ H_S &=& \small{\frac{1}{2}} \sum_{k k'q \sigma \sigma'} \sum_i {\phantom+} V_\parallel \, c^\dagger_{i k+q \sigma} c^\dagger_{i k'-q\sigma'} c^{\phantom\dagger}_{i k'\sigma'} c^{\phantom\dagger}_{i k \sigma} +V_\perp \, c^\dagger_{i k+q \sigma} c^\dagger_{j k'-q\sigma'} c^{\phantom\dagger}_{j k'\sigma'} c^{\phantom\dagger}_{i k \sigma} \nonumber \\ & &{\phantom{ {1\over 2} \sum_{k k'q \sigma \sigma'} } } +J\, ( c^\dagger_{i k+q \sigma} c^\dagger_{i k'-q\sigma'} c^{\phantom\dagger}_{j k'\sigma'} c^{\phantom\dagger}_{j k \sigma} +c^\dagger_{i k+q \sigma} c^\dagger_{j k'-q\sigma'} c^{\phantom\dagger}_{i k'\sigma'} c^{\phantom\dagger}_{j k \sigma} ) .\end{aligned}$$ Here $t_k$ describes a tight-binding coupling between the two layers $i=(1,2), j=3-i$, while $V_\parallel$ ($V_\perp$) are intra-(inter)-layer pairing interactions and the Josephson-like coupling $J$ describes the coherent transfer of two particles from one layer to the other. In a previous publication [@wir] we treated this model using the Nambu formalism including vertex corrections to calculate charge fluctuations between the layers. In this paper we are primarily interested in the calculation of correlation functions involving the operator $$S = \sum_{k} c^\dagger_{2k\uparrow} c^{\phantom\dagger}_{2k\uparrow} -c^\dagger_{2k\downarrow} c^{\phantom\dagger}_{2k\downarrow} -c^\dagger_{1k\uparrow} c^{\phantom\dagger}_{1k\uparrow} +c^\dagger_{1k\downarrow} c^{\phantom\dagger}_{1k\downarrow}$$ describing the difference of the spin polarization in the two layers and the operators coupling to it ($\Delta_{ij}^{\dagger} := c^{\dagger}_{i k \uparrow} c^{\dagger}_{j -k \downarrow}$) $$\begin{aligned} &\Phi_T = -i \sum_{k} \Delta_{21}^{\dagger} - \Delta_{12}^{\dagger} - \Delta_{21} + \Delta_{12}, \nonumber \\ &M = -i \sum_{k} c^\dagger_{2k\uparrow} c^{\phantom\dagger}_{1k\uparrow} -c^\dagger_{1k\uparrow} c^{\phantom\dagger}_{2k\uparrow} -c^\dagger_{2k\downarrow} c^{\phantom\dagger}_{1k\downarrow} +c^\dagger_{1k\downarrow} c^{\phantom\dagger}_{2k\downarrow} . %{\rm with} \Delta_{ij}^{\dagger} := c^{\dagger}_{i k \uparrow} % c^{\dagger}_{j -k \downarrow}. &\end{aligned}$$ The quantity $M$ corresponds to the spin current between the two layers. $\Phi_T$ and $A_T$ describe pairing in different layers in a spin-triplet state with total spin $S_z=0$ and are the real and imaginary part of the inter-layer triplet-pairing amplitude $\Delta_{\perp, T}:= \Delta_{12} - \Delta_{21}$ . To shorten the notation, we introduce $$P^{ij} := \sum_k \Psi_k^{\dagger} D^{ij} \Psi_k, \,\, D^{ij} := \sigma_i \otimes \tau_j, \,\,\, \Psi_k := ( c_{1 k \uparrow}, c_{1 -k \downarrow}^{\dagger}, c_{2 k \uparrow}, c_{2 -k \downarrow}^{\dagger} )^t$$ $\tau_i$ ($\sigma_i$) being the Pauli matrices in the Nambu or two-layer space, respectively (examples: $S = - P^{30}, A_T = - P^{22}, \Phi_T = -P^{21}$). ANALYTICAL RESULTS AND GOLDSTONE MODES ====================================== In general the correlation functions $\ll P^{ij},P^{lm}\gg$ have to be determined numerically. However, for constant hopping $t_k = t$ with $t, \omega \ll \Delta$ ($\Delta$ is the superconducting s-wave gap) and weak coupling the collective modes can be calculated analytically (for $\omega_S, \omega_0 \ll \Delta$) in the cases i (ii) of [*pure*]{} intra-(inter)-layer pairing. $$\label{correl} \renewcommand{\arraystretch}{1.7} \begin{array}{cccc} \mbox{We obtain}&\mbox{for} &\mbox{case (i)}& \mbox{case (ii)}\\ \ll S,S\gg &\approx& 4 N_0 \displaystyle\frac{(2t)^2}{\omega^2-\omega_S^2} &4 N_0 \displaystyle\frac{\omega_S^2}{\omega^2-\omega_S^2}\,, \\ \ll \Phi_T,S\gg &=& 0&4 i N_0 \displaystyle\frac{\omega_0^2}{\omega^2-\omega_S^2} \displaystyle \frac{\omega}{2\Delta} \, \displaystyle \frac{V_\perp -J}{2J} \,, \\ \ll A_T, S\gg &\approx& 4 N_0\displaystyle \frac{\omega_0^2}{\omega^2-\omega_S^2} \displaystyle\frac{t}{\Delta} \, \displaystyle \frac{V_\perp -J}{V_\parallel+V_\perp+2J} &0\,, \\ \ll M,S\gg &\approx& - 4i N_0 \displaystyle\frac{2t\omega}{\omega^2-\omega_S^2 } & - 4i N_0 \displaystyle\frac{2t\omega}{\omega^2-\omega_S^2}\,, \\ \omega_S^2 &=& (2t)^2+\omega_0^2&(2t)^2+\omega_0^2 \,, \\ \omega_0^2& = & \displaystyle\frac{-(V_\parallel-V_\perp+2J)} {(V_\perp-J)(V_\parallel+J)} \displaystyle \frac{(2\Delta)^2}{N_0} & \displaystyle\frac{-2J}{V_\perp^2-J^2}\frac{(2\Delta)^2}{N_0}\,. \end{array} \label{approx}$$ A spin polarization $S$ with opposite sign in the two layers obviously induces inter-layer triplet-amplitudes $A_T$ ($\Phi_T$). These results are closely connected to the collective modes discovered in density-density-correlation functions like $ \ll P, P \gg $ ($P := -P^{33} $) in our previous work.[@wir] For pure inter-(intra)-layer pairing one has the exact relation $$\label{intrainter} \ll P, P \gg_{\rm inter ( intra ) } = \ll S, S \gg_{\rm intra ( inter ) } ,$$ which follows from a unitary change of representation $\tilde{A} = U A U^{\dagger},\tilde{\mid \psi \rangle} = U \mbox{$\mid \psi \rangle$} $ with $U := \exp( - i \pi \sum_{k} ( c^{\phantom\dagger}_{1 k \downarrow} c_{2 k \downarrow}^{\dagger} + c^{\phantom\dagger}_{2 k \downarrow} c_{1 k \downarrow}^{\dagger})) $ The Goldstone theorem [@goldstone] predicts the existence of excitations with vanishing energy, if a continuous, dynamical symmetry $\Omega$ ($[ \Omega, H ] = 0$) is spontaneously broken, e.g. the groundstate $ \mid 0 \, \rangle$ is not an eigenstate of $\Omega$. Thereby it can help to classify the resonances found in the correlation functions (\[correl\]) as so-called Goldstone modes connected with certain symmetries of $H$. Assuming pure singlet pairing in equilibrium, the superconducting groundstate is given by $$\mid \theta_{\parallel} , \theta_{\perp} \, \rangle = \prod_{k} \left( 1 + \alpha_{k \parallel} e^{i 2 \theta_{\parallel}} \Delta_{k \parallel, S}^{\dagger} + \alpha_{k \perp} e^{i 2 \theta_{\perp}} \Delta_{k \perp,S}^{\dagger} ) \right) \mid 0 \, \rangle$$ with $\Delta_{\parallel, S} := \Delta_{11} + \Delta_{22}$ and $\Delta_{\perp, S} := \Delta_{12} + \Delta_{21}$ being the singlet-order parameters for intra- and inter-layer pairing, respectively. The analytical calculations of case i (ii) refer to pure intra-(inter-)layer pairing with $\alpha_{\perp}=0$ ($\alpha_{\parallel}=0 $). Table \[Tabellespontan\] shows (for different parameters $t, J , V_{\parallel}, V_{\perp}$) the symmetries $\Omega_{ij}(\phi) := \exp (i \phi P^{ij} )$, which are broken in the presence of intra-(inter)-layer pairing $\alpha_{\parallel} \neq 0$ ($\alpha_{\perp} \neq 0$). ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ case parameters $\alpha_{\parallel} \neq 0$ $\alpha_{\perp} \neq 0$ ------ --------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- 1 $t,J, V_{\parallel} - V_{\perp} = 0 $ $\Omega_{03},\,\,\Omega_{33},\,\,\Omega_{13},\,\,\Omega_{23}$ $\Omega_{03},\,\,\Omega_{13},\,\,\Omega_{30},\,\,\Omega_{20}$ 2 $t,J\ne0,\, V_{\parallel} = V_{\perp}$ $\Omega_{03},\,\,\Omega_{13}$ $\Omega_{03},\,\,\Omega_{13}$ 3 $t,J\to0,\, V_{\parallel} \neq V_{\perp} $ $\Omega_{03},\,\,\Omega_{33}$ $\Omega_{03},\,\,\Omega_{30}$ 4 $t,J, V_{\parallel} - V_{\perp} \neq 0$ $\Omega_{03}$ $\Omega_{03}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ : \[Tabellespontan\] Broken symmetries if $\alpha_{\parallel} \neq 0$ or $\alpha_{\perp} \neq 0$. The spontaneous breakdown $ \Omega_{03} \mid \theta_{\parallel} , \theta_{\perp} \, \rangle = \mbox{$\mid \theta_{\parallel} + \phi , \theta_{\perp} + \phi \, \rangle$} $ of the global gauge symmetry, which is generated by the total particle number, in $\mid \theta_{\parallel} , \theta_{\perp} \, \rangle$ is a defining property of the superconducting phase (case 4), as it is invariably connected with non-vanishing Cooper-pair amplitudes ($\langle \Delta_{ij} \rangle \neq 0 $). In Eq. \[correl\] Goldstone modes ($\omega_S = 0$) appear in case i (ii), if and only if $t, J, V_{\parallel} - V_{\perp} = 0$ ($t, J = 0$). We can identify these resonances in case i (ii) with the modes in table \[Tabellespontan\] in the cases 1 (3), which are connected with the symmetries $\Omega_{23}$ ($\Omega_{30}$). The transformations $$\begin{aligned} \scriptstyle \Omega_{23} (\phi) \mid \theta_{\parallel}, \theta_{\perp} \rangle &=\scriptstyle &\scriptstyle \prod_{k} \left( 1 + \alpha_{\parallel} e^{i 2 \theta_{\parallel}} \cos (2 \phi) \Delta_{\parallel, S}^{\dagger} + \alpha_{\perp} e^{i 2 \theta_{\perp}} ( \Delta_{\perp,S}^{\dagger} - \sin(2 \phi ) \Delta_{\perp, T}^{\dagger} ) \right) \mid 0 \, \rangle, \\ \scriptstyle \Omega_{30} (\phi) \mid \theta_{\parallel}, \theta_{\perp} \rangle &=\scriptstyle &\scriptstyle \prod_{k} \left( 1 + \alpha_{\parallel} e^{i 2 \theta_{\parallel}} \Delta_{\parallel, S}^{\dagger} + \alpha_{\perp} e^{i 2 \theta_{\perp}} (\cos (2 \phi) \Delta_{\perp,S}^{\dagger} - i\sin(2 \phi ) \Delta_{\perp, T}^{\dagger} ) \right) \mid 0 \, \rangle \nonumber\end{aligned}$$ show that in both cases the $S_z$=0-component $\Delta_{\perp, T}$ of the triplet-order parameter is excited, which for $\Omega_{23}$ ($\Omega_{30}$) also creates non-vanishing expectation values $ \langle A_T \rangle $ ($\langle \Phi_T \rangle $) and finite responses $\ll A_T, S \gg $ ($\ll \Phi_T , S \gg $) to an external spin polarization $S$. Thereby $\Omega_{23}$ mixes intra-layer pairs $\langle \Delta_{\parallel,S}\rangle$with triplet-inter-layer pairs $\langle \Delta_{\perp,T}\rangle$, which is connected with a spin transfer between the layers without breaking up Cooper pairs. On the other hand, $\Omega_{30}$ leaves the modulus of the pairing amplitudes invariant, but creates a phase difference between $\Delta_{12}$ and $\Delta_{21}$, which is the origin of a supercurrent of inter-layer pairs, the so-called [*spin Josephson-effect*]{}. This terminology is motivated by the close analogy to the usual Josephson effect, where a charge rather than a spin transfer is driven by a phase difference of intra- rather than inter-layer pairs. The density-modes $\Omega_{20}$ ($\Omega_{33}$), which correspond according to the relation (\[intrainter\]) to the spin modes $\Omega_{23}$ ($\Omega_{30}$), can be observed as poles in $\ll P, P \gg$ calculated in our previous work [@wir] rather than in $\ll S, S \gg$. According to Eq. \[correl\] all these modes cannot be excited in the absence of particle transfer ($t, J = 0$) between the layers. Finally, $\Omega_{13}$ connects groundstates with different ratios of inter- and intra-layer-pairing, which for $t, J = 0$ in case 1 and 2 are energetically degenerate. NUMERICAL RESULTS ================= We carried out numerical calculations for $\ll S,S\gg$ using two slightly different effective masses $ 2m_1 / \hbar^2 = 1\,\, {\rm eV^{-1}} \AA^{-2},\quad 2m_2 / \hbar^2 =1.2\,\, {\rm eV^{-1}} \AA^{-2} $ for the two bands $\epsilon_k \pm t_k= \hbar^2 k^2 / 2 m_{2/1}$ and parameters: $\mu = 0.3$ eV, $\omega_c = 0.25$ eV (cut-off in k-space), $ ( V_\parallel+V_\perp+ 2 J) N_0 = -0.44, \quad (V_\parallel-V_\perp) N_0 = \pm 0.2 $ ($N_0$: averaged density of states of the two bands). Then the pairing is mixed and for $(V_\parallel-V_\perp)N_0 < 0$ ($ > 0$) dominated by intra-(inter)-layer pairing. \[h\] =1. Fig. \[SSinterintra\] shows the imaginary part of the spin-polarization function for different $J$ in the case a (b) of dominant inter-(intra)-layer pairing at $T=0$. The collective modes appear as $\delta$-peaks below the onset of particle hole excitations around 60 meV. In the appropriate parameter range the resonances in case a (b) coincide with the poles calculated analytically in Eq. \[correl\] in case ii (i), but they exist in a much larger parameter range. For larger positive or negative $J$-values than given in the figures the mode frequencies pass zero indicating an instability of the system. The spectral weight of the phase mode decreases for dominant inter-layer pairing going from negative to positive $J$ as indicated by the spectral weight $\omega_S^2$ in the approximation formula (\[approx\]). For positive $J$ a further collective mode, the so-called amplitude mode, can be seen in both cases. It is inside the particle-hole spectrum for small $J$, but undamped for large $J$ (arrows (a) or small peaks below the particle-hole threshold (b)). Because of the mixing of intra-layer and inter-layer pairing the spin excitation couples to both the phase $\Phi_T$ and the amplitude $A_T$ of the triplet order-parameter. This causes two collective modes, which were already discussed in [@wir]. \[h\] =1. The temperature dependence of the spin polarization function in Fig. \[SSintertemp\] shows the broadening of the collective mode at about 50 meV with increasing $T$. In case b a further collective mode appears at $T_c$. To conclude, the anti-ferromagnetic spin polarization couples to the phase and amplitude of the triplet-order parameter with $S_z=0$. This causes a collective mode where spin-up and spin-down particle tunnel in opposite direction ([*spin Josephson-effect*]{}). This might be connected with a resonance found in magnetic neutron scattering on Y-Ba-Cu-O at finite $q$, which shows the same temperature dependence as our mode[@Fong]. This work has been supported by grants from the Studienstiftung des Deutschen Volkes (C.H.) and the Bayerische Forschungsstiftung within the research project FORSUPRA II (F.F.). [99]{} Wen-Chin Wu, A. Griffin, [ *Phys. Rev. Letters*]{} [**74**]{}, 158 (1995), [*Phys. Rev. B*]{} [**51**]{}, 15317 (1995); K. Kuboki, P. A. Lee, [*J. Phys. Soc. Jpn.*]{} [**64**]{}, 3179 (1995); D. van der Marel, [*Phys. Rev. B*]{} [**51**]{}, 1147 (1995) and references therein. F. Forsthofer, S. Kind, and J. Keller, [ *Phys. Rev. B* ]{} [**53**]{}, 14481 (1996). J. Goldstone, A. Salam, S. Weinberg, [ *Phys. Rev.*]{} 127, 965 (1962). Hung Fai Fong et al., [*Phys. Rev. Lett.*]{} [**75**]{}, 316 (1995); and references therein.

\ [**Theory of Gravity**]{} A. Barros\ [Departamento de Física, Universidade Federal de Roraima,\ 69310-270, Boa Vista, RR - Brazil.]{}\ and\ C. Romero[^1]\ Departamento de Física, Universidade Federal da Paraíba,\ Caixa Postal 5008, 58059-970, João Pessoa, PB - Brazil. [**Abstract**]{} [The gravitational field of a global monopole in the context of Brans-Dicke theory of gravity is investigated. The space-time and the scalar field generated by the monopole are obtained by solving the field equations in the weak field approximation. A comparison is made with the corresponding results predicted by General Relativity.]{} $ $ Monopoles resulting from the breaking of global $O(3)$ symmetry lie among those strange and exotic objects like cosmic strings and domain walls [@1], generally referred to as topological defects of space-time, which may have existed due to phase transitions in the early universe. Likewise cosmic strings, the most studied of these structures, the gravitational field of a monopole exhibits some interesting properties, particularly those concerning the appearance of nontrivial space-time topologies. The solutions corresponding to the metrics generated by strings [@2], domain walls [@2] and global monopoles [@3] in the context of General Relativity were all first obtained using the weak field approximation. In a similar approach, the gravitational fields of cosmic strings and domain walls have been obtained regarding Brans-Dicke theory of gravity and more general scalar-tensor theories of gravity [@4; @5]. In this paper we consider the global monopole and investigate its gravitational field by working out Brans-Dicke equations using once more the weak field approximation, essentially in the same way as in the previous works mentioned above. Let us consider Brans-Dicke field equations in the form $$\begin{aligned} \label{2.1} R_{\mu \nu} = {8\pi\over \phi}\left[T_{\mu \nu} - {g_{\mu \nu}\over 2}\left({2\omega +2\over 2\omega +3}\right)T\right] + {\omega \over \phi^2}\phi_{,\mu}\phi_{,\nu} + {1\over \phi}\phi_{;\mu;\nu}\hspace{.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.2} \Box \phi = {8\pi T\over 2\omega +3}\hspace{.2cm},\end{aligned}$$ where $\phi$ is the scalar field, $\omega$ is a dimensionless coupling constant and $T$ denotes the trace of $T^{\mu}_{\nu}$— the energy-momentum tensor of the matter fields. The energy-momentum tensor of a static global monopole can be approximated (outside the core) as [@3] $$\begin{aligned} \label{2.3} T^{\mu}_{\nu} = \hbox{diag}\left({\eta^2\over r^2}, {\eta^2\over r^2}, 0, 0\right),\end{aligned}$$ where $\eta$ is the energy scale of the symmetry breaking. Due to spherical symmetry we consider $\phi = \phi (r)$ and the line element $$\begin{aligned} \label{2.4} ds^2 = B(r)dt^2 - A(r)dr^2 - r^2(d\theta^2 + \sin^2\theta d\varphi^2).\end{aligned}$$ Substituting this into Eq. (\[2.1\]) and Eq. (\[2.2\]), and taking in account Eq. (\[2.3\]) we obtain the following set of equations: $$\begin{aligned} \label{2.5} {B''\over 2A} - {B'\over 4A}\left({A'\over A}+{B'\over B}\right)+{1\over r}{B'\over A} = {8\pi\over \phi}\left[{\eta^2B\over r^2(2\omega +3)}\right]-{B'\phi'\over 2A\phi}\hspace{.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.6} -{B''\over 2B} + {B'\over 4B}\left({A'\over A}+{B'\over B}\right)+{1\over r}{A'\over A}=&-&{8\pi\over \phi}\left[{\eta^2A\over r^2(2\omega +3)}\right]+{\omega\phi'^2\over \phi^2}\nonumber \\ &+&{1\over \phi}\left[\phi''-{A'\over 2A}\phi'\right]\hspace{.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.7} \phi'' + {1\over 2}\phi'\left[{B'\over B}-{A'\over A}+{4\over r}\right]= -{16\pi\over (2\omega +3)}\left({\eta^2\over r^2}\right)A\hspace{.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.8} 1 - {r\over 2A}\left({B'\over B}-{A'\over A}\right)-{1\over A}= {8\pi\over \phi}\left[\eta^2\left({2\omega +2\over 2\omega+3}\right)\right]+{r\phi'\over A\phi}\hspace{.2cm},\end{aligned}$$ where prime denotes differentiation with respect to $r$. Now, dividing (\[2.5\]) and (\[2.6\]) by $B$ and $A$, respectively, and adding we get $$\begin{aligned} \label{2.9} {\alpha\over r} = {\omega \phi'^2\over \phi^2} +{\phi''\over \phi}-{\phi'\over 2\phi}\alpha \hspace{.2cm},\end{aligned}$$ where we have put $$\begin{aligned} \label{2.10} \alpha = {A'\over A}+{B'\over B}.\end{aligned}$$ Then, equations (\[2.7\]) and (\[2.8\]) read $$\begin{aligned} \label{2.11} \phi'' + {\phi'\over 2}\left[\alpha - {2A'\over A} + {4\over r}\right] = -{16\pi\over 2\omega +3}\left({\eta^2\over r^2}\right)A\hspace{.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.12} 1-{r\over 2A}\left(\alpha - {2A'\over A}\right)-{1\over A} = {8\pi\over \phi}\left[\eta^2\left({2\omega +2\over 2\omega+3}\right)\right] +{r\over A}{\phi'\over \phi}.\end{aligned}$$ At this stage, let us consider the weak field approximation and assume that\ $A(r)=1+f(r),\qquad B(r)=1+g(r)$and $\phi(r)=\phi_o+\epsilon(r)$,\ where $\phi_o$ is a constant which may be identified to $G^{-1}$ when $\omega \rightarrow \infty$ ($G$ being the Newtonian gravitational constant), and the functions $f, g$ and ${\epsilon \over \phi_o}$ should be computed to first order in ${\eta^2\over \phi_o}$, with $|f(r)|,\hspace{.2cm}|g(r)|,\hspace{.2cm}\left|{\epsilon(r)\over \phi_o}\right| \ll 1$. In this approximation it is easy to see that $$\begin{aligned} {\phi'\over \phi}={\epsilon'\over \phi_o[1+\epsilon/\phi_o]}= {\epsilon'\over \phi_o}\hspace{.2cm}, \qquad{\phi''\over \phi} = {\epsilon''\over \phi_o[1+\epsilon/\phi_o]}= {\epsilon''\over \phi_o}\hspace{.2cm}, \nonumber\end{aligned}$$ $$\begin{aligned} {B'\over B} = {g'\over 1+g}= g', \qquad {A'\over A} = {f'\over 1+f} = f', \nonumber \end{aligned}$$ and so on. From equation (\[2.9\]) it follows that $$\begin{aligned} \label{2.13} {\alpha \over r} = {\epsilon''\over \phi_o}.\end{aligned}$$ And from (\[2.11\]) we have $$\begin{aligned} \label{2.14} \epsilon'' + {2\epsilon' \over r} = -{16\pi\over (2\omega +3)}{\eta^2\over r^2} \hspace{.2cm},\end{aligned}$$ the solution of which is given by $$\begin{aligned} \label{2.15} \epsilon = -{16\pi\over 2\omega +3}\eta^2 \ln {r\over r_o} - {\kappa\over r}\hspace{.2cm},\end{aligned}$$ $r_o$ and $\kappa$ being integration constants. On the other hand, considering Eq. (\[2.13\]) and Eq. (\[2.15\]), equation (\[2.12\]) becomes $$\begin{aligned} \label{2.16} f' + {f\over r} = {16\pi \eta^2\over \phi_o(2\omega +3)r}(\omega + {1\over 2}) \hspace{0.2cm},\end{aligned}$$ which yields the solution $$\begin{aligned} \label{2.17} f = {8\pi \eta^2 (2\omega +1)\over \phi_o(2\omega+3)}+{l\over r}\hspace{.2cm},\end{aligned}$$ where $l$ is an arbitrary constant. Therefore, $$\begin{aligned} \label{2.18} A = 1+ f = 1+ {8\pi \eta^2 (2\omega +1)\over \phi_o(2\omega +3)}+{l\over r}\end{aligned}$$ and $$\begin{aligned} A^{-1} = 1- {8\pi \eta^2 (2\omega +1)\over \phi_o(2\omega +3)}-{l\over r}.\end{aligned}$$ It is currently known that solutions of Brans-Dicke field equations do not always go over General Relativity solutions when $\omega \rightarrow \infty$ [@6]. However, as the term ${\omega \phi_{,\mu}\phi_{,\nu}\over \phi^2}$ in equation (\[2.1\]) is neglected in the weak field approximation we expect that in the limit $\omega \rightarrow \infty$ our solution reduces to Barriola-Vilenkin space-time, which is given by [@3] $$\begin{aligned} \label{2.20} ds^2 = & &\left(1-8\pi G\eta^2 - {2GM\over r}\right)dt^2 - \left(1-8\pi G\eta^2 - {2GM\over r}\right)^{-1}dr^2 \nonumber \\ & &- r^2(d\theta^2 + \sin^2 \theta d\varphi^2).\end{aligned}$$ Then, we should have $$\begin{aligned} \lim _{\omega \rightarrow \infty} l = 2GM,\end{aligned}$$ where $M$ is the mass of the monopole core. Indeed, if we take $\eta=0$ in a region outside the monopole core, then a simple comparison of the $r$-dependent term in (\[2.18\]) with the corresponding term of Brans-Dicke solution for a spherically symmetric matter distribution in the weak field approximation [@7], which may be written as $$\begin{aligned} \label{2.22} ds^2 = & &\left[1-{2M\over r\phi_o}\left(1 + {1\over 2\omega +3}\right)\right]dt^2 - \left[1+{2M\over r\phi_o}\left(1 - {1\over 2\omega +3}\right)\right]dr^2 \nonumber \\ & & -r^2(d\theta^2 + \sin^2 \theta d\varphi^2)\hspace{.2cm},\end{aligned}$$ gives $ l = {2M\over \phi_o}\left[1 - {1\over 2\omega +3}\right].$ The same argument concerning the scalar field leads us to $\kappa = -{2M\over 2\omega +3}$. Thus, we have $$\begin{aligned} \label{2.23} A = 1 + {8\pi \eta^2\over \phi_o}\left({2\omega +1\over 2\omega +3}\right) + {2M\over r\phi_o}\left(1 - {1\over 2\omega +3}\right),\end{aligned}$$ $$\begin{aligned} \label{2.24} \phi = \phi_o - {16\pi \eta^2\over 2\omega +3}\ln {r\over r_o} + {2M\over (2\omega +3)r}.\end{aligned}$$ From Eq. (\[2.10\]) it is straightforward to verify that $$\begin{aligned} \label{2.25} B = {a\over A}\left[1-{4M\over r\phi_o(2\omega +3)} + {16\pi \eta^2\over \phi_o(2\omega +3)}\ln {r\over r_o}\right],\end{aligned}$$ where $a$ is an integration constant. For convenience let us rescale the time by putting $a=1 -{16\pi \eta^2\over \phi_o(2\omega +3)}$. Then, $$\begin{aligned} \label{2.26} B = {1\over A}\left[1-{4M\over r\phi_o(2\omega +3)} + {16\pi \eta^2\over \phi_o(2\omega +3)}\ln {r\over r_o}\right]\left[1 - {16\pi \eta^2\over \phi_o(2\omega +3)}\right].\end{aligned}$$ Taking into account (\[2.23\]) we obtain $$\begin{aligned} \label{2.27} B = 1 - {8\pi \eta^2\over \phi_o} + {16\pi \eta^2\over \phi_o(2\omega +3)}\ln {r\over r_o} - {2M\over r\phi_o}\left[1+{1\over 2\omega +3}\right] .\end{aligned}$$ Following Barriola-Vilenkin’s reasoning we drop the mass term in (\[2.23\]), (\[2.24\]) and (\[2.27\]) as it is negligible on the astrophysical scale. Thus, we have finally $$\begin{aligned} \label{2.28} A(r) = 1 + {8\pi \eta^2 (2\omega +1)\over \phi_o(2\omega +3)}\hspace{0.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.29} B(r) = 1 - {8\pi \eta^2 \over \phi_o} + {16\pi \eta^2\over \phi_o(2\omega +3)}\ln {r\over r_o}\hspace{0.2cm},\end{aligned}$$ $$\begin{aligned} \label{2.30} \phi(r) = \phi_o - {16\pi \eta^2\over 2\omega +3}\ln {r\over r_o}.\end{aligned}$$ It is not difficult to show that the line element defined by the functions $A(r)$ and $B(r)$ above is conformally related to the Barriola-Vilenkin monopole solution. To do so, let us consider the coordinate transformation given by the equations $$\begin{aligned} \label{2.31} B(r) = h(r^*)\left(1-{8\pi \eta^2\over \phi_o}\right),\end{aligned}$$ $$\begin{aligned} \label{2.32} A(r)dr^2 = h(r^*)\left(1+{8\pi \eta^2\over \phi_o}\right)dr^{*2},\end{aligned}$$ $$\begin{aligned} \label{2.33} r = h^{1/2}(r^*)r^*,\end{aligned}$$ where $h(r^*)$ is to be calculated and $h(r^*) = 1 + q(r^*)$, with $|q(r^*)| \ll 1$. Differentiating (\[2.33\]) we obtain $$\begin{aligned} \label{2.34} dr^2 = (1 + \dot{q}r^* + q)dr^{*2},\end{aligned}$$ where dot stands for derivative with respect to $r^*$. Substituting (\[2.34\]) into (\[2.32\]) one gets $$\begin{aligned} \label{2.35} q(r^*)={16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r^*\over r_o}\hspace{0.2cm},\end{aligned}$$ whence $$\begin{aligned} \label{2.36} h(r^*)=1 + {16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r^*\over r_o}.\end{aligned}$$ In order to verify the consistency of this result with (\[2.29\]) let us calculate $B(r)$ directly from (\[2.31\]) and (\[2.36\]). Keeping only linear terms in ${\eta^2\over \phi_o}$ and using (\[2.33\]), we have then $$\begin{aligned} \label{2.37} B(r) &=& \left(1+{16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r^*\over r_o}\right)\left(1-{8\pi \eta^2\over \phi_o}\right)\nonumber \\&=& 1-{8\pi \eta^2\over \phi_o}+{16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r\over r_o}.\end{aligned}$$ Therefore, the line element (\[2.4\]) which represents the space-time generated by the monopole may be written in terms of the new coordinate $r^*$ as $$\begin{aligned} \label{2.38} ds^2 = & &\left(1+{16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r^*\over r_o}\right)\biggr[\left(1-{8\pi \eta^2\over \phi_o}\right)dt^2 - \left(1+{8\pi \eta^2\over \phi_o}\right) dr^{*2} \nonumber \\ & &-r^{*2}\left(d\theta^2 + \sin^2\theta d\varphi^2\right)\biggr] .\end{aligned}$$ Rescaling the time and defining a new radial coordinate $r=\left(1+{4\pi \eta^2\over \phi_o}\right)r^*$ we end up with $$\begin{aligned} \label{2.39} ds^2 = & &\left(1+{16\pi \eta^2\over \phi_o(2\omega +3)}\ln{r\over r_o}\right)\biggr[dt^2-dr^2-\left(1-{8\pi \eta^2\over \phi_o}\right)\nonumber \\ & & \times r^2(d\theta^2 + \sin^2\theta d\varphi^2)\biggr].\end{aligned}$$ In order to obtain the correct Newtonian limit from Brans-Dicke field equations the constant $\phi_o$ must be given by [@7] $\phi_o=\left({2\omega +4\over 2\omega +3}\right){1\over G}$. Then, the final form of (\[2.39\]) reads $$\begin{aligned} \label{2.40} ds^2 = & &\left(1+{16\pi \eta^2G\over (2\omega +4)}\ln{r\over r_o}\right)\biggr[dt^2 -dr^2-\biggr(1- 8\pi \eta^2G\left({2\omega +3\over 2\omega +4}\right)\biggr) \nonumber \\ & &\times r^2(d\theta^2 + \sin^2\theta d\varphi^2)\biggr].\end{aligned}$$ Thus, we have shown that in the weak field approximation equation (\[2.40\]) represents the space-time generated by a global monopole in Brans-Dicke theory of gravity. Analogously to the General Relativity case this curved space-time also presents a deficit solid angle in the hypersurfaces $t=const.$ The area of a sphere of radius $r$ in these spaces would be given by $$4\pi r^2\left[1- 8\pi \eta^2G\left({2\omega +3\over 2\omega +4}\right) + {16\pi \eta^2 G\over (2\omega +4)}\ln{r\over r_o}\right] \hspace{0.2cm},$$ rather than $4\pi r^2$. Also, a simple comparison of (\[2.40\]) with Barriola-Vilenkin solution shows that for large values of $\omega$ both space-times are related by a conformal transformation. In this case the motion of light rays is the same in the two space-times. For finite values of $\omega$, null geodesics in the space-time of Brans-Dicke global monopole are still closely related to their counterpart in General Relativity. Indeed, the only change predicted by Brans-Dicke theory reduces, in this case, to the replacement of the Newtonian gravitational constant $G$ by the $\omega$-dependent “effective” gravitational constant $G_{0}=\frac{2\omega +3}{2\omega+4}G$. For a value of $\omega$ consistent with solar system observations, say, $\omega\sim 500$ [@8], it would mean that massless particles travelling in the space-time described by (\[2.40\]) would experience a gravitational strength $G_{0}\sim 0,999 G$. In conclusion we see that in going from General Relativity to Brans-Dicke theory both space-time curvature and topology are affected by the presence of the scalar field. In particular the deficit solid angle becomes $\omega$ dependent. As a consequence, following Barriola-Vilenkin’s argument concerning light propagation in the gravitational field of a global monopole one can easily show that a light signal propagating from a source $S$ to an observer $O$ when $S,O$ and the monopole are perfectly aligned produces an image with the form of a ring of angular diameter given by $$\begin{aligned} \delta \Omega = 8\pi^2 \eta^2 \left({2\omega + 3\over 2\omega +4}\right)G{l\over l +d}\hspace{.5cm}, \nonumber\end{aligned}$$ where $d$ and $l$ are the distances from the monopole to the observer and to the source, respectively. Another interesting physical property in connection with Brans-Dicke’s global monopole involves the appearance of gravitational forces exerted by the monopole on the matter around it. This effect is absent in the case of General Relativity’s monopole as was shown in ref. [@3]. To see how this gravitational effect comes about one has to work out the Newtonian potential associated with (\[2.40\]). As is well known in Galilean coordinates the motion of a nonrelativistic test particle in a weak gravitational field is given by the equation [@9] $$\begin{aligned} \label{2.41} \mbox{\"x}^{i}=-\frac{1}{2}\frac{\partial h_{00}}{\partial x^{i}},\end{aligned}$$ where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ and $\eta_{\mu\nu}=\mbox{diag}(1,-1,-1,-1)$ is Minkowski metric tensor. In order to express (\[2.40\]) in Galilean coordinates let us consider the transformation $$\begin{aligned} t=\left[1-4\pi\eta^{2}G\left(\frac{2\omega+3}{2\omega +4}\right)\right]T, \nonumber\end{aligned}$$ $$\begin{aligned} r=\left[1+4\pi\eta^{2}G\left(\frac{2\omega+3}{2\omega +4}\right)- 4\pi\eta^{2}G\left(\frac{2\omega+3}{2\omega +4}\right)\ln{\frac{R}{R_{0}}}\right]R, \nonumber\end{aligned}$$ with $$\begin{aligned} R_{0}=\left[1-4\pi\eta^{2}G\left(\frac{2\omega+3}{2\omega +4}\right)\right]r_{0}. \nonumber\end{aligned}$$ Then, we have $$\begin{aligned} \label{2.42} ds^{2}&=&\left[1-8\pi\eta^{2}G\left(\frac{2\omega+3}{2\omega +4}\right)+ \frac{16\pi\eta^{2}G}{2\omega+4}\ln{\frac{R}{R_{0}}}\right]dT^{2} \nonumber \\ & &-\left[1-8\pi\eta^{2}G\left(\frac{2\omega+1}{2\omega +4}\right)\ln{\frac{R}{R_{0}}}\right] (dx^{2}+dy^{2}+dz^{2}),\end{aligned}$$ with $R=[x^{2}+y^{2}+z^{2}]^{1/2}$. Thus, (\[2.41\]) becomes, finally $$\begin{aligned} \label{2.43} \mbox{\"x}^{i}=-\frac{4\pi\eta^{2}G}{(\omega+2)}\frac{x^{i}}{R^{2}},\end{aligned}$$ which shows explicitly that particles around the monopole are subject to an attractive force exerted by it. Naturally, if it turns out to be that global monopoles possess any kind of physical reality then a number of other effects such as quantum particle creation [@10], vacuum polarization [@11] and gravitational scattering [@12] among others, which would be in principle amenable to observation may be investigated with the help of equation (\[2.40\]), thereby providing alternative ways for testing the predictable power of both General Relativity and Brans-Dicke theory. C. Romero was partially supported by CNPq (Brazil). [99]{} A. Vilenkin and E.P.S. Shellard, [*Cosmic Strings and other Topological Defects*]{} (Cambridge University Press, Cambridge, 1994). A. Vilenkin, Phys. Rev. D [**23**]{}, 852 (1981). M. Barriola and A. Vilenkin, Phys. Rev. Lett. [**63**]{}, 341 (1989). A. Barros and C. Romero, J. Math. Phys. [**36**]{}, 5800 (1995). M.E.X. Guimarães, Class. Quantum Grav. [**14**]{}, 435 (1997). C. Romero and A. Barros, Phys. Lett. A [**173**]{}, 243 (1993); M. A. Scheel, S.L. Shapiro and S.A. Teukolsky, Phys. Rev. D [**51**]{}, 4236 (1995). C. Brans and R.H. Dicke, Phys. Rev. [**124**]{}, 925 (1961). C. M. Will, [*Theory and Experiment in Gravitational Physics*]{}, revised edition (Cambridge University Press, Cambridge, 1993). L. D. Landau and E. M. Lifshitz, [*The Classical Theory of Fields*]{} (Per-gamon, Oxford, 1971). W. A. Hiscock, Phys. Lett. B [**234**]{}, 469 (1990). F. Mazzitelli and C. Lousto, Phys. Rev. D [**43**]{}, 468 (1991). P. O. Mazur and J. Papavassiliou, Phys. Rev. D [**44**]{}, 1317 (1991). [^1]: Electronic address: cromero@dfjp.ufpb.br

--- abstract: 'The dynamics of multiple scalar fields on a flat FLRW spacetime can be described entirely as a relational system in terms of the matter alone. The matter dynamics is an autonomous system from which the geometrical dynamics can be inferred, and this autonomous system remains deterministic at the point corresponding to the singularity of the cosmology. We show the continuation of this system corresponds to a parity inversion at the singularity, and that the singularity itself is a surface on which the space-time manifold becomes non-orientable.' author: - David Sloan bibliography: - 'FLRW+SF.bib' title: Scalar Fields and the FLRW Singularity --- Introduction ============ Gravitational fields are not measured directly, but rather inferred from observations of matter that evolves under their effects. This is illustrated clearly by a gedankenexperiment in which two test particles are allowed to fall freely. In this the presence of a gravitational field is felt through the reduction of their relative separation. In cosmology we find ourselves in a similar situation; the expansion of the universe is not directly observed. It is found through the interaction between gravity and matter which causes the redshift of photons. The recent successes of the LIGO mission [@LIGO] in observing gravitational waves arise as a result of interferometry wherein the photons experience a changing geometry and when brought together interfere either constructively or destructively as a result of the differences in the spacetime that they experienced. What is key to this is the relational measurement of the photons; are they in or out of phase? This relational behaviour informs our work. Here we will show how given simple matter fields in a cosmological setup, the dynamics of the system can be described entirely in relational terms. We see that this relational behaviour, being more directly related to physical observations, can be described without some of the structure of space-time. As such we treat general relativity as an operational theory; a means to the end of describing the relational dynamics of matter. This changes the ontological status of space-time. We do not treat the idea of scale in a four-dimensional pseudo-Riemannian geometry as absolutely fundamental to the description of physics, but rather as a tool through which dynamics can be calculated. This change of status is common to many approaches to fundamental physics; string theory [@Strings1; @Strings2; @Strings3; @Strings4] often posits the existence of very small compact extra dimensions. In Loop Quantum Gravity [@LQG1; @LQG2] the fundamental object is a spin network to which regular geometry is a an approximation at low curvatures, and in the cosmological sector this is responsible for removing the initial singularity [@LQC1; @LQC2]. A minimalist approach is taken in Causal Set theory [@CST1; @CST2] where the idea of geometry is rebuilt from causal relations between points, and a geometry is overlaid on top of these relations, and in Group Field Theory [@GFT] condensate states are interpreted as macroscopic geometries. In this approach we will differ in one key way from the aforementioned approaches to fundamental theories of gravity. Rather than positing the existence of a more fundamental object on which our theory is based, we instead simply note that we do not have empirical access to the volume of the universe, and instead consider the relational evolution of observables. Aspects of this are captured in the Shape Dynamics [@Shapes1; @Shapes2; @Shapes3; @Shapes4] program. At the heart of this is the idea that certain necessary factors in forming a space-time, such as the idea of an overall notion of scale, are not empirically measurable. As such, any choice of how dynamics is described in terms of these non-measurable quantities should not affect the evolution of measurable quantities. In previous work [@Through] we have shown that this leads to a unique continuation of Bianchi cosmologies (homogeneous, anisotropic solutions to Einstein’s equations) through the initial singularity, and this has recently been extended by Mercati to include inflationary potentials for the matter [@FlavNew]. There are two principal reasons why this is possible; the first is that the system exhibits “Dynamical Similarity" [@DynSim] and thus solutions can be evolved in terms of a smaller set of variables than are required to describe the full phase-space. The second is that the equations of motion for these variables are Lipschitz continuous even at the initial singularity. By the Picard-Lindelöf theorem, they can be uniquely continued beyond this point and reveal that there is a qualitatively similar, but quantitatively distinct, solution on the other side. In this paper we will show that the same results hold when working with scalar fields in a flat Friedmann-Lemaître-Roberston-Walker (FLRW) cosmology. This paper is laid out as follows. In section \[FLRWSec\] we recap the dynamics of flat FLRW cosmologies in the presence of scalar fields, and express the dynamics as a flow on the usual phase space. Then in section \[DynSimSec\] we show the role of dynamical similarity in these spacetimes, establishing a vector field on phase-space whose integral curves take solutions to those which are are indistinguishable. This allows us to formulate the more compact description of the system which is well-defined at and beyond the singularity. We then show some general features of such systems. In section \[FreeFieldsSec\] we show how massless noninteracting scalar fields provide this continuation, and in section \[ShapeSec\] we show the fully intrinsic form of the equations of motion when interactions are reintroduced. We examine how one can reconstruct a geometrical interpretation on the other side of the singularity in section \[NonOrientableSec\] and show that this would appear to be an orientation flip when viewed in this way. Finally in section \[BeyondSec\] we show how the results we have obtained extend beyond the isotropic case and make contact with prior results, and give some concluding thoughts in section \[SecDiscussion\]. FLRW Cosmology with Scalar Fields {#FLRWSec} ================================= We will examine the dynamics of scalar fields in a flat FLRW cosmology. To retain the homogeneity and isotropy of our solutions, we will assume the same holds for our scalar fields, and thus each field has only temporal variation. The metric takes the form: s\^2 = -t\^2 + a(t)\^2 (x\^2+y\^2+z\^2) It is important to note that there are two tetrad representations of this system which are compatible with the geometry corresponding to left-handed and right-handed orientations $g=\eta(\mathbf{e},\mathbf{e})$, with the choices $\mathbf{e}_L = (\d t,a\d x,a\d y,a\d z)$ and $\mathbf{e}_R = (dt,-a\d x,-a\d y,-a\d z)$. In fact, since the form $\eta$ is bilinear, we could have chosen to distribute the - signs with any of these components, however since we will be primarily interested in the behaviour of the spatial parts across the initial singularity, we choose to keep a time direction fixed and will only be interested in the relative signs of the one-forms across $t=0$. Dynamics are derived from the Einstein-Hilbert action for gravity minimally coupled to matter, which has the usual scalar field Lagrangian. Our spacetime is topologically $\R \times \Sigma$ where the spatial slice $\Sigma$ can be $\R^3$ or $\mathbb{T}^3$. In the case of $\R^3$ we choose a fiducial cell to capture the entire system since homogeneity means that the dynamics of the entire space can be determined by the dynamics of any chosen subregion, and thus we avoid infinities. S = (R - \_m) = \_\_a\^3(6( + ) - + V() ) In order to simplify the algebra, in the following we make the choice to work with the volume instead of the scale factor, $v=a^3$. The momentum conjugate to $v$ is proportional to the Hubble parameter, $h=p_v=\frac{4\dot{v}}{v}$ and that to $\phi$ is $p_i=v\dot{\phi}_i$, where we choose to denote this conjugate momentum $h$ to avoid notation clashes. The Hamiltonian and symplectic structure are given: = v(- + + V() ) = hv + The Hamiltonian vector field, $\X_\H$ describes the evolution of a solution in phase-space. It is determined uniquely through the global invertibility of the symplectic form (summing over repeated indices of the scalar field and its momentum): = \_[\_]{} \_= - + - v \[HVF\] The dynamics of the matter present is given by the Klein-Gordon equation, which corresponds to the usual Hamiltonian dynamics of the scalar fields given the above: \[KG\] + + = 0 and the dynamics of the geometry is given by the Friedmann equation: h\^2 = (+V()) In the case where there is no potential for the scalar field, we can solve these analytically to see $v=v_o t$, $\phi_i = A_i \log t + B_i$. The singularity of this system corresponds to the fact that along its orbit on phase space, $\X_\H$ reaches a point at which it is no longer integrable. From the Picard-Lindelöf theorem, this arises because uniqueness of solutions to the equations of motion fails when coefficients of the basis vectors ($\frac{\partial}{\partial h}$ etc) are not Lipschitz continuous. We see that this can occur in two ways; the first is that some of the phase space variables will tend to infinity. We will show that this can be solved through a compactification which is brought about by considering only relational variables. The second point at which Lipschitz continuity can fail is when $v=0$. However, it turns out that this system can be expressed in such a way that $v$ is not strictly required for evolution. This happens because the Hamiltonian has solutions which are dynamically similar, and thus the evolution can be described on a contact manifold in terms of relational variables. Dynamical Similarity in FLRW Cosmology {#DynSimSec} ====================================== Since the total Hamiltonian $\H=0$ is a constraint, we can use this to replace $h$ in equation (\[KG\]) and express the dynamics of the matter fields purely in a closed form. \[KG2\] + + = 0 On first glance this may appear surprising; we began with a $2n+2$ dimensional phase-space subject to a single constraint, and have arrived at a set of $n$ coupled differential equations which are second order in the fields alone. However, the reason for this reduction is that there is a somewhat hidden symmetry of this system; it exhibits dynamical similarity [@DynSim] under rescaling both the momenta of the matter fields and the volume (measured through to an appropriately chosen fiducial cell) of the spatial slice. The dynamical similarity that arises is an exact consequence of the fact that the dynamics of the system chosen should not be affected by the choice of a cell by which it is measured. In this system we have universal coordinates on phase space and thus we can define a non standard canonical transformation through the vector field = v + Following the procedure laid out in [@DynSim], we see that this system has an autonomous subsystem of dynamics of the invariants of $\V$, given by the contact Hamiltonian $H^c$ and contact form $\eta$ \^[c]{} = - + + V() = -h + It is thus clear that this is a system of $2n+1$ invariants of $\V$ subject to a single constraint, and thus the fact that we can describe the relational motion in terms of just the matter fields themselves is no longer surprising. The dynamics of contact systems is an interesting and active area of study [@Bravetti; @Leon]. The contact space is odd-dimensional, and thus differs from symplectic dynamics. The is one particular difference which is important in the cosmological case: contact systems are ‘frictional’. It is clear from the contact form that one of the variables is distinct from the rest, as it appears as a coordinate without a corresponding momentum. Thus there exists a vector field $\nu$ such that $\nu(\eta)=1, \nu(\d\eta)=0$. This is called the ‘Reeb vector field’, which in our case is $\nu=\frac{\partial}{\partial h}$, through which friction becomes apparent. The contact Hamiltonian vector field $\X_{\H^{\c}}$ is determined uniquely through \_[\_[\^[c]{}]{}]{} + (\_[\^[c]{}]{}) =\^[c]{} - ((\^[c]{})+\^[c]{})Thus our system has a contact Hamiltonian vector field given by \_[\^[c]{}]{} = -\_i\^2 + \_i - ( +) We see immediately that we have overcome one of the failures of Lipschitz continuity; since this system does not refer to $v$ at any point, its vanishing does not impede the integration of the contact Hamiltonian vector field. Thus the points at which we may encounter problems are where one or more of $\phi,\dot{\phi},h$ become infinite. However as we have previously noted, this is solved by the compactification induced in choosing relational variables. Since the volume $v$ is not a part of the relational system itself, and is never needed to find the equations of motion of our contact system, we will not consider it to be fundamental to the description of our system. Rather we will note that we can completely integrate our equations of motion to find the behaviour of $h$ for all times. We then create a geometrical model which corresponds to our system by choosing a value for $v$ at any given time and finding its time evolution by quadrature. In doing so we can reconstruct a space-time geometry which exactly reproduces the dynamics of the FLRW system. However since $v$ is not fundamental to our description, neither is its role as volume in this system; the singularity at $v=0$ is thus not a problem; we have simply reached the boundary of the set of solutions that can be reproduced using our chosen space-time geometry. As we will see this is still a regular point in the relational description, and thus we can create another space-time geometry beyond $v=0$ corresponding to the evolution of the relational system past this point. The singularity of GR is thus a failure of the description of physics in terms of an orientable manifold. The frictional nature of a contact system is manifested in two ways; the first is that the contact Hamiltonian, $\H^{\c}$ is not necessarily preserved along its integral curves: = -\^c however, we note that as this is a constraint in cosmology, $\H^{\c}=0$, and thus the system is conserved. The second is that Liouville’s theorem is modified for the system [@Bravetti2]. As the space is odd dimensional, we cannot make a volume form in the usual way by taking exterior products of the symplectic form. Instead, the canonical volume form on a contact space is given = \^n By evaluating the time derivative of this form, we see that the Hamiltonian flow on contact space has a divergence, and thus = -(n+1) which gives rise to attractors on the space of invariants. Since the Reeb vector field in our cosmological systems is along the direction of the Hubble parameter, its dual in the original symplectic system is the expansion volume; $\iota_\frac{\partial}{\partial h} \omega = \d v$. Thereby we see that the friction present in the matter systems arises as a consequence of the expansion of the universe. Free Fields {#FreeFieldsSec} =========== \[Free\] For clarity we will focus on a system with two fields present $\vec{\phi} = (\phi_1, \phi_2)$, though the procedure described is entirely general. A theorem due to Foster [@Foster] shows that on approach to a singularity in such a spacetime for a broad class of potentials the scalar fields asymptote to their massless ($V=0$) behaviour. Therefore I shall first work in the case of free fields. As such the dynamics are easy to express: =- h= = = Thus we see solutions in parameter time given by \_1 = A\_1 t + B\_1 \_2 = A\_2 t + B\_2 If $t$ is not a direct observable then we have our complete motion in the $(\phi_1,\phi_2)$ plane determined along the line $\phi_2 = \frac{A_2}{A_1} \phi_1 + \frac{A_1 B_2 - A_2 B_2}{A_1}$. This is simply the equation of a straight line. Note that there the solution is degenerate in choices of $A_i$ and $B_i$ since these four variables determine a single straight line in $\mathbf{R}^2$. This is unsurprising as the physics of the underlying system is unchanged under a shift of $B_1$ and $B_2$ – these leave $\dot{\phi_i}$ unchanged, and similarly we could reparametrize the time direction changing $A_1$ and $A_2$ but retaining their proportion. We can express the system on shape space by making the compactification of the system on a sphere: $$\label{SphericalShapeCoordinates} \left( \begin{array}{c}\phi_1\\\phi_2\end{array}\right) =|\tan\beta| \left( \begin{array}{c} \cos\alpha \\ \sin \alpha \end{array} \right)$$ This compactification can be visualized by considering taking a unit sphere tangent to the $(\phi_1 - \phi_2)$ plane touching at the north pole. The mapping takes a point $P$ on the plane onto the sphere by passing a line from the center of the sphere to $P$ and identifying this with the point of intersection with the sphere. Thus we have kinematical equation for $\alpha$ and $\beta$: &=& \^2( + )\ &=& ( - ) Therefore we can find the relational equation on a solution to the equations of motion: = This equation appears singular at the point where $\tan\alpha = -A_1/A_2$, however note that this is simply the point of closest approach of the line to the origin in the $(\phi_1,\phi_2)$ plane, and therefore would be a stationary point of $\beta$ on a solution. Which we can integrate to find: A\_1 - A\_2 = (A\_1 B\_2 - A\_2 B\_1) which is the equation of a geodesic on the sphere. The constant of integration is set to zero to match the solution we are representing here with the representation of the scalar fields in terms of time. The equator of the sphere is at $\beta=\pi/2$ which represents the singularity in the FLRW system, but is again a regular point in the relational shape system. We can thus recover the space that is covered by our solutions in the $(\phi_1,\phi_2)$ plane by noting that our solutions are parametrized by $\alpha$, which is monotonic in time. Hence $$\label{FreeShapeSolutiononPlane} \left( \begin{array}{c}\phi_1\\\phi_2\end{array}\right) =\left|\frac{A_1 B_2-A_2 B_1}{A_1 \sin\alpha - A_2 \cos\alpha} \right| \left( \begin{array}{c} \cos\alpha \\ \sin \alpha \end{array} \right)$$ It is obvious that this solution is periodic in $\alpha$ with period $2\pi$. We thus see that the complete solution when carried beyond the initial singularity of the FLRW spacetime continues in the $(\phi_1,\phi_2)$ plane along a line parallel to the original solution but with opposite impact parameter, and travelling in the opposite direction in parameter time. Thus the extended system is given $\phi_i = (A_i \log |t| +\sign(t) B_i)$, valid wherever $t \neq 0$. It is interesting to note here that it is only when we introduce potentials that the functional form of the gravitational action comes into play; if we had replaced the $h^2$ term with $f(h)$ in the contact Hamiltonian, $\H^{\c}$ then $\frac{\d\phi_1}{\d\phi_2}$ would be unaffected, and we’d still have the space of straight lines, and so the shape space compactification would have been the same. This is broken by interactions between the fields, as we will see in section \[ShapeSec\]. Let us now consider a set of $n$ scalar fields. Each of these will have have its dynamics given in terms of time $t$ by \_i = A\_i t + B\_i for constants $A_i$ and $B_i$. Following the same procedure as above, we expect that there will be a well defined relational motion in terms of the compactified shape space. We begin by using coordinates on the $n$-sphere: $$\label{SphericalShapeCoordinates2} \left( \begin{array}{c}\phi_1\\ \phi_2 \\...\\ \phi_n\end{array}\right) =|\tan\beta| \left( \begin{array}{c} \cos\alpha_1 \\ \sin \alpha_1 \cos \alpha_2 \\ ... \\ \sin \alpha_1 ... \sin \alpha_{n-1} \end{array} \right)$$ The geometric idea here is the same as above – considering the hyperplane in the $\phi_i$ and mapping onto an $n$-sphere by placing the sphere tangential to the plane at the north pole, and associating points on each via a line from the center of the sphere to the plane. The radial component of our motion is contained in $\tan \beta$. We know that for free fields this radius will tend to infinity as we approach the singularity at $t \rightarrow 0$, which in these terms is the equator of the $n$-sphere, $\beta=\pi/2$.Hhence we again parametrize our motion in terms of $\beta$ and examine the dynamics as we approach this point. From the equations of motion for the scalar fields we can make the relation between $\beta$ and $t$ explicit: \^2 = \_i\^2 = A\_i\^2 (t)\^2 + A\_i B\_i t + B\_i\^2 and hence to leading order $|\tan \beta|$ approaches infinity as $\log t$. From this we see that the leading order contribution to $\dot{\beta}$ is given by $1/t(\log t)^2$. For the $\alpha_i$ we can perform a similar analysis of their asymptotic structures. We first note that \_i = and we find its velocity to be = (\_i(\_[i+1]{}\_[i+1]{} + ... + \_n \_n) - \_i (\_[i+1]{}\^2 + ... + \_n\^2 ) Here we note that the bracketed term on the right hand side appears to have a term that grows as $\log(t)^2/t^2$. However this term is exactly zero, and hence the leading order contribution is in fact \_i = ( (A\_i \_[j&gt;i]{} A\_j B\_j - B\_i \_[j&gt;i]{} A\_j\^2) + o(t\^[-1]{}) ) and therefore to leading order $\dot{\alpha}$ approaches the singularity as $1/(t\log t)^2$. Hence at the singularity we find that $\frac{d\alpha_i}{d\beta}$ tends to a constant and the system can be continued beyond this point. It is a (long) exercise to show that the solutions of this system correspond to geodesic motion on the $n$-sphere. Full Shape Dynamical System {#ShapeSec} =========================== Since we know what the asymptotic behaviour will look like for most well-behaved potentials,[^1] we can set up our complete shape system. Initially it might appear that a good choice would be to pick $p=\sqrt{\dot{\phi}_1^2+\dot{\phi}_2^2}$ and $\tan{\theta}=\frac{\dot{\phi}_2}{\dot{\phi}_1}$. However, as the system asymptotes to the motion of a free field, the velocity of the field will become parallel to the field ($\theta \rightarrow \alpha$), therefore we cannot simply break the velocity into polar coordinates. Likewise asymptotically the velocity of the field will outgrow the field values ($p \gg \tan{\beta}$). It turns out that a good set of variables are = = p e\^[-||]{} \[defs\] which are asymptotically distinct from $\alpha$ and $\beta$ and thus contain solution determining data, and are Lipschitz continuous at $\beta \rightarrow \frac{\pi}{2}$. The complete dynamics of our system can now be expressed completely in these terms. Denoting a derivative with respect to $\beta$ with a prime, \[EoM\] ’ &=&\ ’ &=& ( - ) +\ ’ &=& ( - )\ &-& Thus our system is asymptotically (assuming that $V$ is well behaved) $\alpha' = \lambda, \lambda' = 0, \chi' = -\frac{\sqrt{3} \lambda^2 \chi}{4}$. The equations of motion (\[EoM\]) are Lipschitz continuous and thus we satisfy the conditions of the Picard-Lindelöf theorem. This holds even at $\beta=\pi/2$, which corresponds to the initial singularity of the cosmological system. Therefore there is a unique continuation of any given solution through $\beta=\pi/2$. As we have shown that the system of equations described in equations (\[EoM\]) is well defined at the initial singularity, it is logical to then ask if there are places at which this description breaks down. It is clear from the evolution of $\alpha$ that the equations are singular when $\lambda \cos\beta=\pm 1$. From the definition of $\lambda$ we see that this is a point at which the velocity of the scalar fields is orthogonal to their position. Since $\tan\beta$ represents the distance from the origin in the $(\phi_1,\phi_2)$ plane, this is unsurprising. At this point the motion of the fields has no radial component, and hence $\beta$ is unchanging while the other variables are, and hence derivatives of these with respect to $\beta$ will become infinite. The motion remains well defined when expressed in terms of the fields and their velocities alone, however, and thus this is simply a poor choice of representation of the system at this point. It is entirely analogous to the singularity in $\frac{\d r}{\d\theta}$ in the polar coordinate description of a straight line; there at the point of closest approach to the origin $r$ is it a minimum whilst $\theta$ changes, and thus in these variables the system appears singular. However, the equation $y=mx+c$ remains well defined at this point. The question then arises as to whether this could be the asymptotic behaviour as $\beta \rightarrow \pi/2$. The answer is that if $V$ obeys the conditions of regularity then it cannot, as the motion asymptotes to that of a free field. Recovering the free field in these circumstances amounts to setting $V$ and its derivatives to zero. Upon doing so we note two interesting features; the first is that $\chi$ becomes unimporant in dynamics, as it only enters the equations of motion for $\alpha$ and $\lambda$ through terms proportional to derivatives of the potential. Hence we can integrate the equation of motion for $\lambda$ directly in this case to obtain \_ = \[lambdafree\] wherein $\lambda_0$ is a constant. Reintroducing this into the equation of motion for $\alpha$ gives the equation for a straight line in the $(\phi_1,\phi_2)$ plane. Second, we see that this feature only relied upon there being non-zero derivatives of $V$. Had we introduced a pure cosmological constant term (corresponding to $V$ being constant) the equations of motion would have been unaffected. From equation \[lambdafree\] we see that = and so is decreasing (to zero) at the singularity. Thus when the fall-off conditions for the potential are obeyed, the system is always integrable through the singularity. Extending our system to $n$ scalar fields is a simple exercise; first we note that at any point in the evolution of the system we can choose coordinates for the scalar fields such that the position and velocity of the fields are in the plane spanned by $\phi_1$ and $\phi_2$. If we describe the positions and velocities of our fields in terms of spherical coordinates, with positions given by equation \[SphericalShapeCoordinates2\] and velocities given by $$\label{SphericalShapeCoordinates3} \left( \begin{array}{c}\phi_1\\ \phi_2 \\...\\ \phi_n\end{array}\right) =p \left( \begin{array}{c} \cos\theta_1 \\ \sin \theta_1 \cos \theta_2 \\ ... \\ \sin \theta_1 ... \sin \theta_{n-1} \end{array} \right)$$ then this amounts to setting $\alpha_2,...,\alpha_{n-1}$ and $\theta_2,...,\theta_n$ to zero locally. Thus the equations of motion for the position and velocity in the plane remain as in equations \[EoM\] locally, defining $\lambda=\frac{\sin(\alpha_1-\theta_1)}{\cos\beta}$; \[EoM2\] \_1’ &=&\ ’ &=& ( - ) +\ The presence of the potentials causes the motion to accelerate from the plane ’\_j = and hence again we see that for potentials which are well behaved (in the sense that both the potential and its derivatives fall off sufficiently) the motion becomes asymptotically planar and again describes a geodesic on the $(n-1)$-sphere. Similarly, we see that if the potentials are well behaved then the equations remain Lipschitz continuous, and hence the Picard-Lindelöf theorem guarantees that there is a unique trajectory which continues the solution through $\beta=\pi/2$. Conserved Quantities on Non-orientable Manifolds {#NonOrientableSec} ================================================ The singularity of FLRW cosmology is a point at which the spacetime geometry endows (any chosen fiducial cell embedded in) the spatial slice with zero volume. The volume form, induced by the space-time volume form pulled back to a $t=constant$ slice, is $\Vol_\Sigma = v \d x\d y\d z$. The existence of a volume form is equivalent to orientability of the manifold itself; non-orientable manifolds of dimension $d$ do not have any nowhere-vanishing d-forms. In extensions of our system beyond a point at which the volume form vanishes, it is therefore natural to ask whether this might be an indication that the manifold is not globally orientable. The volume form is directly linked to the conserved quantities of any Lagrangian system through Noether’s theorem; it is almost universally assumed that any manifold providing a basis for physics must be orientable. Here we will relax this somewhat. To provide an example of such a system, let us consider a Mobius strip formed by taking a section of $\R^2$ and placing the usual twist on the boundaries. Specifically we will take = (0,y) \~(1,1-y) and we will endow this space with a metric $\d s^2 = \d x^2+\d y^2$ everywhere except the points of identification $x=0,1$. This metric admits two diads up to a choice of overall sign; $\mathbf{e}_1 = \d x, \d y$ and $\mathbf{e}_2 = \d x, -\d y$. Hence our choices of volume form are $\Vol=\pm \d x\wedge \d y$. Let us consider the motion of a free particle on this manifold. The Lagrangian for such a particle (again, defined away from the edges) is Ł= + And we find from this that the two momenta $P_x=\dot{x}, P_y=\dot{y}$ are conserved quantities. This is, of course, the statement that free particles follow the geodesics of the manifold. In the Hamiltonian language we know that the system is determined by a Hamiltonian and symplectic structure = + = P\_x x + P\_y y Suppose we now want to extend this to what happens to a particle that crosses $x=0$, say. In such a case we can extend geodesics across this point by insisting that the path taken is a minimum of the distance as defined on all points away from $x=0$ and that the path is continuous across the join. Away from the join, the geodesics will be geodesics of the metric (straight lines in $\R^2$) and so we need only consider the point of intersection with the join, and connect the points with straight lines away from it. On doing so we find that geodesics connecting two points $K=(K_x,K_y)$ and $Q=(Q_x,Q_y)$ are given by minimizing (K,Q) = (K,(0,y\_i))+(1,(1-y\_i),Q)= + across all choices of the $y_i$, the coordinate of the intersection of the path with $x=0$. Upon doing so we find that the geodesic crossing the join between $K$ and $Q$ is equivalent to taking a second copy of the space attached at the join but flipped in the $y$ direction, and finding a geodesic on the combined space with the metric $\d s^2 = \d x^2 + \d Y^2$, where $Y=y$ for $x>0$ and $Y=1-y$ for $x<0$. This is equivalent to saying that we match across the join by swapping $\mathbf{e}_1$ for $\mathbf{e}_2$. This is shown in figure \[GeoMobius\]. One useful way to visualize this is to consider a physical Möbius strip formed by taking a strip of paper and connecting it with the usual twist. The double cover amounts to viewing the two sides of the paper, and our geodesic is a straight line along the paper which starts on one side and ends on the other. A circle with an arrow pointing clockwise on one side will appear to point anticlockwise when viewed from the other side, illustrating the choice between $\mathbf{e}_1$ and $\mathbf{e}_2$. ![The geodesic on the Mobius strip shown on the left on a single copy of the space. We see that the line is discontinuous under the identification and has a sharp corner at the join. On the right is the same line but represented in a double cover of the space where we have used the diad $\mathbf{e}_2$ on the left and $\mathbf{e}_1$ on the right, swapping the orientation between copies of the space. With this choice we recover what we would expect a geodesic to look like on the double cover with metric given in terms of $x$ and $Y$.[]{data-label="GeoMobius"}](GeoMobius.pdf "fig:"){height="100pt"} ![The geodesic on the Mobius strip shown on the left on a single copy of the space. We see that the line is discontinuous under the identification and has a sharp corner at the join. On the right is the same line but represented in a double cover of the space where we have used the diad $\mathbf{e}_2$ on the left and $\mathbf{e}_1$ on the right, swapping the orientation between copies of the space. With this choice we recover what we would expect a geodesic to look like on the double cover with metric given in terms of $x$ and $Y$.[]{data-label="GeoMobius"}](GeoMobius2.pdf "fig:"){height="100pt"} If we turn our attention back to the conserved quantities it would appear that across the join a particle following a geodesic will have preserved $P_x$ but $P_y \rightarrow -P_y$. There are several ways to interpret this. We could first consider that what is actually conserved is not in fact $\dot{y}$ but rather $\sign(\det(\mathbf{e})) \dot{y}$. In doing so we would see that we have swapped diads across the join. An equivalent way to see this is to consider that the conserved quantity corresponds to the interior product of the Hamiltonian vector field and the symplectic structure applied to the symmetry $s =\frac{\partial}{\partial y}$. Upon parallel transporting $s_y$ across the join, it is inverted $s'_y=-\frac{\partial}{\partial y}$, and hence the conserved quantity takes the value $P'_y=-P_y$. However, perhaps the most intuitive way to understand this is to consider that the Lagrangian is equivalent to defining on the double cover Ł’ = + and the conserved quantity really corresponds to $P_Y = \dot{Y}$. The issue of how chirality is treated in a relational context has been a long-standing issue in the philosophy of physics, dating back to Kant. The fundamental issue relates to objects which are counterparts, yet incongruous, with the common example being left and right hands. In the case we examine here, we find that within our system the descriptions are intrinsically indistinguishable; one a notion of handedness has been chosen, one can indeed distinguish between, clockwise and anticlockwise. However, given a single rotating object the system would not be sufficient to determine whether that object rotated clockwise or anticlockwise. For an excellent review of such issues see [@Pooley] and references therein[^2]. Let us now examine what happens in the FLRW case. Since the dynamics is captured in essence by the case of two massless scalar fields, we will take this as our example, however the results we obtain are more generic. We noted in section \[Free\] that the solution $\phi_i = A_i \log(t) + B_i$ are continued across the singularity by extending to $t<0$ through taking $\phi_i = A_i \log |t| + \sign(t)B_i$, and thus $\dot{\phi}_i = \sign(t) \frac{A_i}{t}$. Naively we might think that this inversion had violated a conservation law; the reversal of a velocity would seem to be in contradiction to the conservation of momentum. However, recall that the momentum that is conserved in an FLRW cosmology is $p_\phi=v\dot{\phi}$. Thus, since in our solution $v \propto t$ we see that this is indeed a conserved charge if we make the stipulation that $v$ is the *signed* volume of the fiducial cell. Equivalently, we should consider the connection across the singularity as being moving from the description of the spacetime geometry by the right-handed tetrad $\mathbf{e}_R$ to the left-handed triad $\mathbf{e}_L$. In this case the conservation law still holds; it is simply that in moving from one representation of our system to another it appears that we have undergone a parity inversion. Solving the Friedmann equation and interpreting the evolution as being that of a space-time geometry away from the singularity we see that the metric can be expressed in the usual FLRW terms appearing as two universes joined back-to-back at the singularity. However, in order to have the conserved quantities remain conserved across the singularity itself, we must switch tetrads across this point. This appears, choosing to represent $s=\sign(t)$: s\^2 = -t\^2 + |t|\^ (x\^2+y\^2+z\^2 ) = (t, s|t|\^ (x,y,z)) \[extended\] and hence we can uniquely extend our space-time across the singularity in this manner. At this point the question of the Hawking–Penrose singularity theorems [@Hawking] arises. In particular, how can we know that geodesics are connected across the singularity? One might expect that since the spatial metric has vanished at the singularity, any path orthogonal to the time direction will have no length, and thus we could connect any points in space here. In essence this is the infinite focussing on which the theorems are based. To answer this we must again engage with a relational description; consider the null geodesic to be the path taken by a photon on the space-time, ignoring backreaction. In coordinate terms, let us choose without loss of generality a null geodesic whose spatial component is parallel to the x axis in the region of space-time covered by $t>0$. The equation of motion is then, in coordinate terms, $x=x_\mathrm{0} + \frac{3}{2} t^\frac{2}{3}$. If we choose, as we did in the case of the Mobius strip example, to enforce continuity between points of the space-time manifold covered by different choices of tetrad, and further to conserve the momentum of the photon as measured in this way (respecting the signed volume of space), then there is a unique geodesic on the other side of the singularity which connects to this point. This is given by $x=x_o + \frac{3}{2} |t|^\frac{2}{3}$, with the tetrad given in equation (\[extended\]). We note that the conditions of the Hawking–Penrose theorem are satisfied, and hence the conclusion that there is infinite focusing of geodesics is valid. The distance between any two photons as measured using the space-time geometry will tend to zero as we approach the singularity. However, there is still a unique continuation of the paths beyond this point, which corresponds to the above trajectory as described in the geometry in which we swap tetrads. What the singularity theorems require is that we model physics in terms of a space-time defined by a metric on a globally orientable manifold. However, if we relax that assumption just on the singular surface itself, we see that there is a natural physical continuation beyond this point when described in relational terms. To be clear about this continuation, we note that the Hawking–Penrose theorems show that there is not a unique continuation through the singularity of the space-time geometry as determined by the Einstein equations. Our results are completely in agreement with this. We do not claim that the Einstein equations give such a continuation. However, the relational system of scalar fields can be continued beyond this point, and at all points away from the singularity their behaviour is consistent with that described by the Einstein equations coupled to the matter. Thus we know that on both sides of the singularity there is a consistent space-time picture. What we have then shown is that if we want to further identify trajectories of matter across the singularity such that conservation laws, specifically the conservation of momentum, hold then this in turn makes a unique identification. In this identification we can connect two FLRW cosmologies back-to-back at the singularity and continue geodesics from one into the other. This continuation is motivated by our example of the Möbius strip, and in common with that example we see that orientation is inverted in the continuation. To relate back to the issue of handedness, given a “left hand” on one side of the singularity we see that it is continued to a hand on the other side. The hands that are congruous to it on one side remain congruous on the other. Since two incongruous “hands” that propagate across the singularity can be arbitrarily chosen to be called left or right, we cannot uniquely say whether a left hand on one side of the singularity remains a left hand on the other. We are free to choose whether to call this again a left hand or a right hand as the physical system makes no distinction between the two. However, if we wish to retain the conserved quantities both in magnitude and sign across this point, we see that this choice is equivalent to taking a left hand on one side of the singularity and calling the object that it propagates to on the other a right hand. Beyond Isotropy: The Bianchi I Spacetime {#BeyondSec} ======================================== The results we have obtained in the case of the flat FLRW system can be extended to anisotropic cosmological solutions. These are of particular importance in the study of cosmological singularities as they are believed to capture the complete behaviour of an inhomogeneous system [@BKL; @Berger; @Ringstrom; @AR]. More precisely, the BKL conjecture states that in the vicinity of a singularity, the dynamics becomes local, oscillatory and vacuum dominated, with the only matter of import being massless scalar fields. Thus if we know the behaviour of homogeneous cosmologies with scalar fields, we should capture dynamics in the neighborhood of generic singularities. The case of a single free field providing the matter content was covered in [@Through], where the spacetime in consideration was Bianchi IX, though the results are general. In this section we will show that the results of this paper are entirely in line with those. Here we will present the case of a Bianchi I spacetime; the results are in fact more generic, but since the asymptotic motion of quiescent solutions is parallel to that obtained in the Bianchi I case this will be sufficient to demonstrate how the result holds. The Bianchi classification labels geometries by the commutativity classes of the three Killing vector fields that define homogeneity. These in turn contribute to the Ricci scalar, which in the 3+1 decomposition of GR acts as a potential for anisotropies [@Ellis; @Uggla; @Uggla2]. This scales as $v^{4/3}$, and hence will differ from the potentials of scalar fields, so we will need an extension of our relational ideas to describe the resulting shape systems in full generality [@Indistinguish]. Fortunately in the case where the Killing vectors all commute the Ricci scalar is exactly zero, and thus we can bypass this issue. This is the Bianchi I system described with metric: s\^2 = -t\^2 +v\^[2/3]{} (e\^[-]{} x\^2 + e\^[--]{} y\^2 + e\^[ q\_2]{} z\^2 ) and when written in these terms, the gravitational Lagrangian for the anisotropic space-time has the same form as that of an isotropic space-time with massless scalar fields: Ł= v(- + + +Ł\_m ) Therefore we can continue this system beyond the singularity in exactly the same way as we did for the case of scalar fields, identifying $q_1$ and $q_2$ with $\phi_1$ and $\phi_2$, say, and numbering the rest of our fields $\phi_3$ to $\phi_{n+2}$. In the case of a single additional massless scalar field, we see that the continuation of geodesic motion on the 3-sphere corresponds exactly to the continuation that was shown in [@Through] for the Bianchi I case. Discussion {#SecDiscussion} ========== Let us recapitulate the major results shown in this paper. We have seen that the relational motion of scalar fields in an FLRW cosmology can be described entirely in terms of the fields themselves, and without direct reference to the geometry on which their dynamics takes place. This is due to the dynamical similarity of the space of solutions. We compactified the space of fields on a sphere, as this most closely matches the way in which Misner variables were modelled in the anisotropic case. In this description the relational equations of motion are derived directly from a contact Hamiltonian and thus need never reference the space-time geometry in the first place. When the potentials for the fields are well-behaved, these equations satisfy the conditions of the Picard-Lindelöf theorem at the point corresponding to the singularity of the original space-time, and thus have a unique, deterministic extension beyond this point, and we have shown how this continuation is manifested in terms of the fields themselves. We then take these relational obesrvables as being the fundamental ontological content of our theory. As such we are able to reconstruct a space-time on the other side of the singularity, and by choosing to enforce the conservation across the singularity of quantities that are conserved everywhere else, we arrive at a unique description of the geometry. This description corresponds to having two FLRW cosmologies glued back-to-back at their respective singularities with an orientation reversal between them. In terms of tetrads, we swap a left-handed tetrad, $\mathbf{e_1}$, for a right-handed one, $\mathbf{e_2}$, in crossing. We must reiterate here that the continuation that we have described is not a direct consequence of Einstein’s equations for the geometrical degrees of freedom. The equations of motion for quantities such as the Hubble rate become non-deterministic at the singularity, and therefore Einstein’s equations provide no unique continuation by themselves. However, the equations of motion for the relational observables that constitute our empirically accessible content do indeed remain deterministic. Therefore we can reconstruct a geometrical description from this relational content. As the system exhibits both dynamical similarity and has a choice of orientation of the tetrads that determine the geometry, there is a choice in how this reconstruction is done. This choice is no different than that which is available to any observer modelling the evolution of an FLRW cosmology from observations; the value of the scale factor at any given time is a free choice (typically picked to be unity today for efficiency of description). Likewise any given metric can be equally described by left and right-handed tetrads. By exploiting this freedom to choose, at the singularity itself, how we glue together solutions we are able to find a description in which conserved quantities retain their values across the singularity. However, this choice is not in itself unique; we could have chosen to invert the conservation laws, for example, at the price of geodesics being sharply ’kinked’ at the singularity itself. The important thing to note is that this choice is a choice of the geometric framework which we use to describe the system, but does not affect the underlying relational dynamics. Any such choice would give equivalent results. The choice we have made is simply the one in which a consistent idea of momentum (for example) can be made. Physically it holds no more meaning than describing the motion of a particle in a given frame until a certain time and then switching frames. Consider describing a particle that falls past a window in terms of its height above the base of the window until it reaches this base, and thereafter in terms of the height above the ground. The description in terms of coordinates may be discontinuous between frames, but the underlying physics is unaffected. Throughout this paper we have worked with scalar fields as our matter content. The motivation for this is twofold; first they are the simplest form of matter to encode in a Lagrangian framework and thus are simple to translate into the contact geometry in which we model our system, and second massless scalar fields are the dominant form of matter on approach to the initial singularity. Further insight into the ways in which we continue solutions across the singularity may be gained by working with vector fields or pseudo-scalars. For vector matter it will be necessary to work with anisotropic solutions, as a homogeneous vector field has by necessity a preferred direction and thus would break the isotropy of the system. Thus far our attention has been largely focussed on cosmological solutions and their singularities. There is a natural parallel with black hole interiors, as spherically symmetric space-times having spatial slices $\R \times S^2$ are Kantowski-Sachs cosmological solutions. Therefore it is possible to extend work in the anisotropic case to include these solutions and investigate the singularities at the centers of black holes. Preliminary work indicates that there is a unique continuation through these points, however the complete solution is a work in progress. The existence of a classical continuation of our system beyond an initial singularity poses significant questions for quantum gravity. For long it has been assumed that the indeterminacy introduced by GR at singularities is good motivation to treat such points as places to look for the effects of quantum gravity. In the FLRW case arguments are made along the lines of dimensional analysis. The energy density of any matter present in a cosmological system must approach the Planck density, at which point it is argued that quantum gravity effects should become important. However, in the case of anisotropic cosmologies there are singularities even in the vacuum solutions. Even in the presence of matter, on approach to the singularity it is believed that the anisotropic geometric contributions to the evolution are dominant over matter contributions. Since there exists a classical continuation this motivation is lost; it may be that we have been looking for quantum gravity in the wrong places. Acknowledgements {#acknowledgements .unnumbered} ================ The author is grateful to Julian Barbour, Sean Gryb and Flavio Mercati for helpful discussion and comments. [^1]: These will turn out to be potentials $V$ where $e^{-\sqrt{\frac{3}{4}(\phi_1^2+\phi_2^2)}}V$ and its derivatives $e^{-\sqrt{\frac{3}{4}(\phi_1^2+\phi_2^2)}}\frac{\partial V}{\partial \phi_i}$ are finite. [^2]: We are grateful to Sean Gryb for making pointing out the extant literature in response to an early draft of this paper.

--- abstract: 'In this work, we evaluate the energy spectra of baryons which consist of two heavy and one light quarks in the MIT bag model. The two heavy quarks constitute a heavy scalar or axial vector diquark. Concretely, we calculate the spectra of $|q(QQ'')>_{1/2}$ and $|q(QQ'')>_{3/2}$ where $Q$ and $Q''$ stand for $b$ and/or $c$ quarks. Especially, for $|q(bc)>_{1/2}$ there can be a mixing between $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ where the subscripts 0 and 1 refer to the spin state of the diquark (bc), the mixing is not calculable in the framework of quantum mechanics (QM) as the potential model is employed, but can be evaluated by the quantum field theory (QFT). Our numerical results indicate that the mixing is sieable' --- Evaluation of Spectra of Baryons Containing Two Heavy Quarks in Bag Model Da-Heng He$^1$, Ke Qian$^1$, Yi-Bing Ding$^{2,6}$, Xue-Qian Li$^{1,5,6}$ and Peng-Nian Shen$^{4,3,5,6}$ 1\. Department of Physics, Nankai University, Tianjin 300071, China;\ 2. Graduate School of The Chinese Academy of Sciences, Beijing, 100039, China,\ 3. Institute of High Energy Physics, CAS, P.O. Box 918(4), Beijing 100039, China\ 4. Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China\ 5. Institute of Theoretical Physics, CAS, P.O. Box 2735, Beijing, 100080, China.\ 6. China Center of Advanced Science and Technology (World Laboratory), P.O.Box 8730, Beijing 100080, China Introduction ============= At present, the non-perturbative QCD which dominates the low energy physics phenomena, is not fully understood yet, a systematic and reliable way for evaluating the non-perturbative QCD effects on such as the hadron spectra and hadronic matrix elements is lacking. Fortunately, however, for the heavy flavor mesons or baryons which at least contain one b or c quarks (antiquarks) the situation becomes simpler due to an extra $SU_f(2)\otimes SU_s(2)$ symmetry[@Isgur]. The studies in this field provide us with valuable information about the QCD interaction and its low energy behavior. Among all the interesting subjects, the hadron spectra would be the first focus of attention. The spectra of the $J/\psi$ and $\Upsilon$ families have been thoroughly investigated in different theoretical approaches. Commonly, the spectra are evaluated in the potential model inspired by QCD, where the QCD Coulomb-type potential is directly derived from the one-gluon-exchange mechanism, and the confinement term originating from the non-perturbative QCD must be introduced by hand[@Rosner]. For the heavy quarkonia where only heavy quark flavors are involved, the potential model definitely sets a good theoretical framework for describing such systems where relativistic effects are small compared to the mass scale. An alternative model, the bag model can also provide a reasonable confinement for quarks. In fact, for light hadrons, especially light baryons, the bag model may be a better framework for describing their static behaviors. The MIT bag model has some advantages [@Jaffe]. First, even though it does not hold a translational invariance, the quarks inside the hadron bag obey the relativistic Dirac equation and moreover, they can be described in the Quantum Field Theory (QFT) framework, namely there exist creation and annihilation operators for the constituents of the hadron. The latter property is very important for this work, that we can calculate a mixing between $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ (the notations will be explained below) states and it is impossible in the Quantum Mechanics (QM) framework. Another interesting subject is if the diquark structure which consists of two quarks and resides in a color-anti-triplet $\bar 3$, exists in baryons. Its existence, in fact, is till in dispute. For light diquark which is composed of two light quarks, the relativistic effects are serious and the bound state should be loose. By contraries, two heavy quarks (b and c) can constitute a stable bound state of $\bar 3$, namely a diquark which serves as a source of static color field[@Falk]. As a matter of fact, the un-penetrable bag boundary which provides the confinement conditions to the constituents of the hadron, is due to the long-distance non-perturbative QCD effects, to evaluate the spectra, one needs to include the short-distance interaction between the constituents and it can be calculated in the framework of the perturbative QCD. In this work, we are going to evaluate the spectra of baryons which contains two heavy quarks (b and/or c) and a light quark and take the light-quark-heavy-diqaurk picture which obviously is reasonable for the case of concern. For evaluating the hadron spectra, the traditional method is the potential model. For baryons, the quark-diquark picture can reduce the three-body problem into a two-body problem and leads to a normal Schrödinger equation. Solving the Schrödinger equation, one can get the binding energy of quark and diquark[@Ebert; @Tong]. In recent years, remarkable progresses have been made along this direction. The authors of refs. [@Kiselev] have carefully studied the short-distance and long-distance effects, then derived a modified potential and obtained the spectroscopy of the baryons which contain two heavy quarks by using the non-relastivistic Schrödinger equation. Meantime, in Ebert et al.’s new work, the light quark is treated as a fully relativistic object and the potential is somehow different from that in their earlier work[@Ebert2]. In their works, not only the ground states of such baryons are obtained, but also the excited states are evaluated. However, the potential model has two obvious drawbacks. First, even though the diquark is heavy, the constituent quark mass of the light quark is still comparable to the linear momentum which is of order of $\Lambda_{QCD}$. Thus the reduced mass is not large and the relativistic effects are still significant. Secondly, working in the framework of QM, it is impossible to estimate the mixing of $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ where the subscripts 0 and 1 of the (bc)-diquark denote the total spin of the subsystem, i.e. the (bc)-diquark (we only consider the ground state of $l=0$). The reason is that there are no creation and annihilation operators in the traditional QM framework, so the transition $(bc)_1+q\rightarrow (bc)_0+q$, i.e. $A+q\rightarrow S+q$ where the notation A and S refer to the axial-vector and scalar diquarks respectively, is forbidden, even though the transition is calculable in QFT. On other side, the MIT bag model does not suffer from the two drawbacks. In this picture, since the diquark is heavy, it hardly moves so that can be supposed to sit at the center of the bag, whereas the light quark is freely moving in the bag and its equation of motion is the relativistic Dirac equation with a certain boundary condition[@Jaffe] and both the quark and diquark are quantized in the QFT. Thus the relativistic effects are automatically included. Secondly, one can deal with a possible conversion of the constituents in the bag in terms of QFT, namely one can let a constituent be created or annihilated, thus the transition $A+q\rightarrow S+q$ is allowed and the corresponding mixing of $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ is calculable. Usually the bag model is not very applicable to the light mesons because the spherical boundary is not a good approximation for the two-body system. Even though the quark-diquark structure is a two-body system, the aforementioned problem does not exist because the diquark is much heavier than the light quark. The picture is in analog to the solar system or an atom where only one valence electron around the heavy nucleus, and spherical boundary would be a reasonable choice. In this work, following the literature [@Jaffe], we treat the short-distance QCD interaction between the light quark and heavy diquark perturbatively. Since the interaction energy $E_{int}(R)$ is not diagonal for $|q(bc)_1>_{1/2}$ and $|q(bc)_0>_{1/2}$, we may diagonalize the matrix to obtain the eigenvalues and eigenfunctions which would be the masses of the baryons with flavor $q(bc)$ and spin 1/2. Moreover, for the other baryons $|q(bb)_1>_{1/2(3/2)}$ $|q(cc)_1>_{1/2(3/2)}$ $|q(bc)_1>_{3/2}$, the diquark must be an axial vector due to the Pauli principle[@Close]. The paper is organized as follows, after the introduction, we derive all the formulation of $E_{int}(R)$ and $M_B$ in Sec.II, then in Sec.III we present the numerical results and all concerned parameters, finally the last section is devoted to discussions.\ Formulation ============ 1\. A brief review of the MIT bag model The wavefunction of a light quark in the MIT bag obeys the Dirac equation for free fermion and a boundary condition which forbids the quark current to penetrate the bag boundary. It has a form[@Jaffe] $$q(\bf{r},t)=\frac{N(\chi)}{\sqrt{4\pi}} \left ( \begin{array}{cc} (\frac{\omega+m}{\omega})^{1/2}ij_{0}(\chi r/R)U \\-(\frac{\omega-m}{\omega})^{1/2}j_{1}(\chi r/R){\mbox{\boldmath $\sigma$}}\cdot {\bf r}U, \end{array} \right )$$ with $$N^{-2}(\chi)=R^{3}j_{0}^{2}(\chi)\frac{2\omega(\omega-1/R)+m/R}{\omega(\omega-m)},$$ where $j_l$ is a spherical Bessel function, $U$ is a two-component Pauli spinor, the eigen-energy is $$\omega(m,R)=\frac{[\chi^{2}+(mR)^{2}]^{1/2}}{R},$$ and the eigenvalue $\chi$ satisfies an equation $$\tan(\chi)=\frac{\chi}{1-mR-[\chi^{2}+(mR)^{2}]^{1/2}}.$$ In our picture, the two heavy quarks constitute a diquark which is a boson-like bound state of color $\bar 3$ and because it is heavy, it hardly moves, but sits at the center of the bag. Its wavefunction can be written as[@Ebert1] $$\begin{aligned} \label{heavy} \psi(r) &=& {N\over 4\pi}e^{{\Lambda r^2\over 2}}\;\;\;\;\;\; {\rm for\;the\;scalar\;diquark};\\ \psi_{\mu}(r) &=& {N\over 4\pi}e^{{\Lambda r^2\over 2}}\eta_{\mu}\;\;\;\;\;\; {\rm for\;the\;axial\; vector\;diquark},\end{aligned}$$ where $\eta_{\mu}$ is the polarization vector which is normalized as $\eta^2=-1$, $\Lambda$ is a parameter and will be discussed later in the text.\ 2. Formulation for the baryon spectra. In the CM frame of the baryon, the total mass of the baryon can be written as $$M_{B}=M_{D}+\omega+E_{int}(R)+{4\over 3}\pi R^{3}B-{z\over R}$$ where $E_{int}(R)$ is the interaction energy between the diquark and light quark, $B$ is the bag constant and $z$ is a constant for the zero-point energy. $\omega$ and $M_D$ are the eigen-energy of the free light quark and the mass of the heavy diquark which are given in the literature[@Jaffe; @Ebert]. In this work, following the standard procedure [@Landau; @Jaffe], we are going to calculate the interaction energy $E_{int}(R)$. Generally, the interaction hamiltonian in the bag can be expressed as $$\label{HDD} H_{D'D}=-\frac{\lambda_{1}^{a}}{2}\frac{\lambda^{a}_{2}}{2}g_{s}^{2} \int\overline{q}({\bf x})\gamma^{\mu}q({\bf x})D_{\mu\nu}\Psi^{\ast}({\bf y}) \frac{<D'|J^{\nu}|D>}{\sqrt{MM'}}\Psi({\bf y})d^{3}xd^{3}y.$$ It is noted that without mixing, i.e. $H_{D'D}$ is diagonal, $E_{int}=H_{D'D}$, however, if there is non-diagonal $H_{D'D}$, $E_{int}$ is the eigenvalues of the hamiltonian matrix. The expectation value of the Casimir operator $<0|\lambda_{1}^{a}\lambda_{2}^{a}|0>=-16/3$, the strong coupling $g_s^{2}=4\pi\alpha_{s}$, $q({\bf x})$ is the wavefunction of the free light quark, $<D'|J^{\nu}|D>$ is the effective vertex for $DD'g$ and $D_{\mu\nu}$ is the gluon propagator in the Coulomb gauge[@Lee]. The form of such a propagator in the configuration space reads $$\begin{aligned} G(\mathbf{r},\mathbf{r}_{0})=\frac{1}{4\pi}[\frac{1}{|\mathbf{r}- \mathbf{r}_{0}|}+\sum_{l=0}^{\infty}\ \frac{(l+1)(1-k)}{l+(l+1)k}\frac{(rr_{0})^{l}}{R^{2l+1}}P_{l}(cos\theta)].\end{aligned}$$ In this work, $\mathbf{r_{0}}$ is small, thus the main contribution comes from the $l=0$ component. Then, the expression can be simplified as: $$\begin{aligned} G(\mathbf{r},\mathbf{r}_{0})=\frac{1}{4\pi}[\frac{1}{|\mathbf{r}- \mathbf{r}_{0}|}+\frac{\tilde{k}}{R}].\end{aligned}$$ It is noted that the term $\frac{\tilde{k}}{R}$ is due to the mirror charge effect[@Lee], in fact, it indeed corresponds to the zero-point energy in the bag as Jaffe et al. suggested[@Jaffe]. Actually, $\tilde{k}$ is related to the vacuum property and still serves as a free parameter that cannot be determined from any underlying theory yet. We choose $\tilde{k}=0.87$ to fit the most recent lattice result about the spectrum of $(ccq)_{\frac{1}{2}^{+}}$. 3\. The D’Dg effective vertex If we only consider the ground states of the diquark, namely the two heavy quarks are in $l=0$ color-anti-triplet state, the diquark can be either a scalar (denoted as $S$) with $s=0$ or an axial vector (denoted as $A$) of $s=1$. The effective vertices can be derived by the quantum field theory under the heavy quark limit[@Guo]. The effective vertex for $SS'g$ is $$<S'|J^{\nu}|S>=\sqrt{MM'}(f_{1}v'^{\nu}+f_{2}v^{\nu})$$ and the $AA'g$ effective vertex is of the form $$\begin{aligned} <A'|J^{\nu}|A> &=& \sqrt{MM'}[f_{3}(\eta\cdot\eta'^{\ast})v'^{\nu} +f_{4}(\eta'^{\ast}\cdot\eta)v^{\nu} +f_{5}(\eta\cdot v')(\eta'^{\ast}\cdot v)v'^{\nu} \nonumber \\&& {}+f_{6}(\eta\cdot v')(\eta'^{\ast}\cdot v)v^{\nu} +f_{7}{\eta'^{\ast}}^{\nu}(\eta\cdot v')+f_{8}(\eta'^{\ast}\cdot v)\eta^{\nu}]\end{aligned}$$ where $v,\; v',\; \eta_{\mu}$ and $\eta_{\mu}'$ are the four-velocities and polarization vectors of $D$ and $D'$ respectively, $f_i'$s are the form factors at the vertices. As we did in our previous work [@Tong] where the effective potential model was employed, for convenience of calculation, we would write the polarization into a spin-operator which acts on the wavefunction of the axial-vector diquark as $$\eta_{\mu}={1\over\sqrt {2}}({\mbox{\boldmath $\beta$}}\cdot {\bf s},{\bf s}),$$ where higher order relativistic corrections proportional to ${{\bf p}^2\over M_D^2}$ are neglected and the factor $1/\sqrt 2$ is a normalization factor because $<s^2>=s(s+1)=2$. It is worth noticing that $s_i$ and $s_j$ do not commute with each other, thus one must be careful about their order in deriving formula. The effective vertex for ASg is written as $$\begin{aligned} <A'|J^{\nu}|S> &=& \sqrt{MM'}[f_{11}{\eta'^{\ast}}^{\nu}+f_{12}(\eta'^{\ast}\cdot v)v'^{\nu} \nonumber \\&& {}+f_{13}(\eta'^{\ast}\cdot v)v^{\nu} +f_{14}i\epsilon^{\nu l \rho \sigma}\eta'^{\ast}_{l}v'_{\rho}v_{\sigma}].\end{aligned}$$ Here we must stress that for convenience of calculation, we turn $\eta_{\mu}$ into the quantum spin-operator as evaluating the interacting energy for $|q(QQ')>$, but as pointed above, we cannot calculate the mixing between $|q(bc)_1>_{1/2}$ and $|q(bc)_0>_{1/2}$ in QM, but need to carry out the derivation in QFT instead. Then, we have to keep the polarization $\eta_{\mu}$ in a 4-vector form as $$\eta_{\mu}^{\pm}=\mp {1\over\sqrt 2}(0,1,\pm i,0)\;\;{\rm and}\;\; \eta^0_{\mu}=(0,0,0,1)$$. Because the diquark is very heavy and hardly moves, $|{\mbox{\boldmath $\beta$}}|\ll 1$, one can use the polarization vector in the reference frame where the diquark is at rest. In the heavy quark limit, we have $$\begin{aligned} && f_{1}=f_{2}=f_{7}=f_{8}=-f_{3}=-f_{4}=f_{14}=1 \nonumber \\&& {}f_{11}=f_{12}=f_{13}=0,\end{aligned}$$ and for very heavy diquark the above approximation holds[@Guo]. 4\. The interaction energy To calculate the interaction energy, the basic formula is eq.(\[HDD\]). For the heavy diquark, $|{\bf p}\ll M_D$, thus the relativistic corrections proportional to and higher than ${{\bf p}^2\over M_D^2}$ can be safely ignored. Based on the approximations, one can easily obtain the interaction energies. For the baryon where the diquark is a scalar of $\bar 3$, the interaction energy between the light quark and the scalar diquark is corresponding to the transition matrix element of $<q,S|H_{eff}|q,S>$, and can be expressed as $$\begin{aligned} H_{SS} &=& -\frac{\lambda_{1}^{a}}{2}\frac{\lambda_{2}^{a}}{2}\int\int\overline{q}(\mathbf{x})\gamma^{\mu}q(\mathbf{x})D_{\mu\nu}\Psi^{\ast}(\mathbf{y}) [f_{1}v'^{\nu} +f_{2}v^{\nu}]\Psi(\mathbf{y})d^{3}xd^{3}y\nonumber \\ {}&=&g^{2}_{s}N\int^{R}_{0}[j_{0}^{2}(\frac{\chi r_{x}}{R})+j_{1}^{2}(\frac{\chi r_{x}}{R})]\frac{2}{r_{x}}d^{3}x,\end{aligned}$$ where ${\bf x}$ is the spatial coordinate of the light quark, ${\bf y}$ is that of the heavy diquark, ${\bf r}={\bf x}-{\bf y}$ is the relative coordinate of the quark and diquark. For the baryon where the diquark is an axial vector, the matrix element $<q,A|H_{eff}|q,A>$ is written as formula of $H_{AA\frac{1}{2}}$: $$\begin{aligned} H_{AA\frac{1}{2}} &=& -\frac{\lambda_{1}^{a}}{2}\frac{\lambda_{2}^{a}}{2}\int\int\overline{q}(\mathbf{x})\gamma^{\mu}q(\mathbf{x})D_{\mu\nu}\Psi^{\ast}(\mathbf{y}) [f_{3}(\eta\cdot\eta'^{\ast})v'^{\nu} +f_{4}(\eta'^{\ast}\cdot\eta)v^{\nu} \nonumber \\&& {}+f_{5}(\eta\cdot v')(\eta'^{\ast}\cdot v)v'^{\nu} +f_{6}(\eta\cdot v')(\eta'^{\ast}\cdot v)v^{\nu} +f_{7}\eta'^{\ast}(\eta\cdot v')\nonumber \\&& {}+f_{8}(\eta'^{\ast}\cdot v)\eta^{\nu}]\Psi(\mathbf{y})d^{3}xd^{3}y\nonumber \\ {}&=&g^{2}_{s}N\int^{R}_{0}[j_{0}^{2}(\frac{\chi r_{x}}{R})+j_{1}^{2}(\frac{\chi r_{x}}{R})]d^{3}x\int(-\frac{1}{|\mathbf{r_{x}-\mathbf{r_{y}}}|}+\frac{k}{R})|\Psi(y)|^{2}d^{3}y \nonumber \\&& {}+g^{2}_{s}\frac{C_{\alpha\beta}}{M}\int^{R}_{0}\overline{q_{\alpha}}(\frac{\chi r_{x}}{R})\gamma^{i}q_{\beta}(\frac{\chi r_{x}}{R})d^{3}x\int(\mathbf{S}\times\mathbf{q})^{i}(-\frac{1}{|\mathbf{r_{x}}-\mathbf{r_{y}|}}+\frac{k}{R})|\Psi(\mathbf{y})|^{2}d^{3}y\end{aligned}$$ where $C_{\alpha\beta}$ refers to the spin projections of the quarks in the baryon. The transitional momentum $\mathbf{q}$ will change into the form $-i\nabla$ in configurational space acting on the relative position, and we then get: $$\begin{aligned} H_{AA\frac{1}{2}}&=&g^{2}_{s}N\int^{R}_{0}[j_{0}^{2}(\frac{\chi r_{x}}{R})+j_{1}^{2}(\frac{\chi r_{x}}{R})]d^{3}x\int(-\frac{1}{|\mathbf{r_{x}-\mathbf{r_{y}}}|}+\frac{k}{R})|\Psi(y)|^{2}d^{3}y \nonumber \\&& {}+g^{2}_{s}\frac{C_{\alpha\beta}}{M}\int^{R}_{0}\overline{q}_{\alpha}(\frac{\chi r_{x}}{R})\gamma^{i}q_{\beta}(\frac{\chi r_{x}}{R})d^{3}x\int\frac{[\mathbf{s}\times(\mathbf{r_{x}}-\mathbf{r_{y}})]^{i}}{|\mathbf{r_{x}}-\mathbf{r_{y}}|^{3}}|\Psi(\mathbf{y})|^{2}d^{3}y\end{aligned}$$ where ${\bf q}$ is the exchanged momentum between the quark and diquark, and in the configuration space of the bag, it is an operator acting only on the relative coordinate ${\bf r}$ as $-i\nabla_{\bf r}$ and $$\begin{aligned} \int e^{i\mathbf{q}\cdot\mathbf{r}}\frac{\mathbf{q}}{|\mathbf{q}|^{2}} \frac{d^{3}q}{(2\pi)^{3}}=\frac{i\mathbf{r}}{r^{3}}.\end{aligned}$$ We finally obtain an integral which must be carried out numerically if we take the form of $|\Psi(y)|^{2}$ which is treated as $\delta$ function into our consider, we will finally read: $$\begin{aligned} &H_{AA\frac{1}{2}}& = g^{2}_{s}N\int^{R}_{0}[j_{0}^{2}(\frac{\chi r_{x}}{R})+j_{1}^{2}(\frac{\chi r_{x}}{R})](-\frac{1}{r_{x}}+\frac{k}{R})d^{3}x \nonumber \\&& {}+g^{2}_{s}\frac{N'}{M}\int^{R}_{0}j_{0}(\frac{\chi r_{x}}{R})j_{1}(\frac{\chi r_{x}}{R})<s_{1}m'_{1}s_{2}m'_{2}|[\frac{(\mathbf{\sigma}\cdot\mathbf{S})}{r_{x}^{2}}\nonumber \\&& {}-\frac{(\mathbf{\sigma}\cdot\mathbf{r_{x}})(\mathbf{S}\cdot\mathbf{r_{x}})}{r_{x}^{4}}]|s_{1}m_{1}s_{2}m_{2}>d^{3}x\end{aligned}$$ For the baryon of spin 3/2, which is composed of the light quark of spin 1/2 and an axial vector diquark of spin 1, the interaction energy is $$\begin{aligned} &H_{AA\frac{3}{2}}& = g^{2}_{s}N\int^{R}_{0}[j_{0}^{2}(\frac{\chi r_{x}}{R})+j_{1}^{2}(\frac{\chi r_{x}}{R})]\frac{2}{r_{x}}d^{3}x \nonumber \\&& {}+g^{2}_{s}\frac{N'}{M}\int^{R}_{0}j_{0}(\frac{\chi r_{x}}{R})j_{1}(\frac{\chi r_{x}}{R})<\frac{1}{2},\frac{1}{2},1,1|[\frac{(\mathbf{\sigma}\cdot\mathbf{S})}{r_{x}^{2}}\nonumber \\&& {}-\frac{(\mathbf{\sigma}\cdot\mathbf{r_{x}})(\mathbf{S}\cdot\mathbf{r_{x}})}{r_{x}^{4}}]|\frac{1}{2},\frac{1}{2},1,1>d^{3}x\end{aligned}$$ For a mixing hamiltonian which originates from the transition $S+q\rightarrow A+q$, the non-diagonal interaction element is $$\begin{aligned} H_{AS}&=&\int\int\overline{q}(\mathbf{x})\gamma^{\mu}q(\mathbf{x})D_{\mu\nu}\Psi^{\ast}(\mathbf{y})(i\epsilon^{\nu l\rho\sigma}\eta'^{\ast}_{l}v'_{\rho}v_{\sigma})\Psi(\mathbf{y}) d^{3}xd^{3}y\end{aligned}$$ After a straightforward integration, we obtain $$\begin{aligned} &H_{AS}& = g^{2}_{s}\frac{N''}{M}\int^{R}_{0}j_{0}(\frac{\chi r_{x}}{R})j_{1}(\frac{\chi r_{x}}{R})<s_{1}m'_{1}|[\frac{(\mathbf{\sigma}\cdot\mathbf{S})}{r_{x}^{2}}\nonumber \\&& {}-\frac{(\mathbf{\sigma}\cdot\mathbf{r_{x}})(\mathbf{S}\cdot\mathbf{r_{x}})}{r_{x}^{4}}]|s_{1}m_{1}>d^{3}x\end{aligned}$$ It is noted that here $\eta_{\mu}$ remains as a four-vector, and it reduces into a three-vector ${\mbox{\boldmath $\eta$}}$ which is not an operator. With the mixing between $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$, one can write the real eigenstates of $(qbc)_{1/2}$ and $(qbc)'_{1/2}$ as $$\begin{aligned} |\frac{1}{2},\frac{1}{2}>=C|\frac{1}{2},\frac{1}{2},0,0>+D(-\sqrt{\frac{1}{3}} |\frac{1}{2},\frac{1}{2},1,0>+\sqrt{\frac{2}{3}}|\frac{1}{2},-\frac{1}{2},1,1>),\end{aligned}$$ where C and D are the coefficients to be determined. The interaction energy is $$\begin{aligned} &<\frac{1}{2},\frac{1}{2}|H|\frac{1}{2},\frac{1}{2}>& =C^{2}<\frac{1}{2},\frac{1}{2},0,0|H|\frac{1}{2},\frac{1}{2},0,0>+D^{2}(\frac{1}{3}<\frac{1}{2},\frac{1}{2},1,0|H|\frac{1}{2},\frac{1}{2},1,0>\nonumber \\&& {}+\frac{2}{3}<\frac{1}{2},-\frac{1}{2},1,1|H|\frac{1}{2},-\frac{1}{2},1,1>-\frac{2\sqrt{2}}{3}<\frac{1}{2},\frac{1}{2},1,0|H|\frac{1}{2},-\frac{1}{2},1,1>)\nonumber \\&&{}+CD(-\sqrt{\frac{1}{3}}<\frac{1}{2},\frac{1}{2},0,0|H|\frac{1}{2},\frac{1}{2},1,0>+\sqrt{\frac{2}{3}}<\frac{1}{2},\frac{1}{2},0,0|H|\frac{1}{2},-\frac{1}{2},1,1>)\nonumber \\&& {}+ CD(-\sqrt{\frac{1}{3}}<\frac{1}{2},\frac{1}{2},1,0|H|\frac{1}{2},\frac{1}{2},0,0>\nonumber \\&& {}+\sqrt{\frac{2}{3}}<\frac{1}{2},-\frac{1}{2},1,1|H|\frac{1}{2},\frac{1}{2},0,0>),\end{aligned}$$ which contains both diagonal and non-diagonal interaction elements. It is easy to set it into a matrix form as $$H=\left ( \begin{array}{cc} H_{SS} \ H_{SA} \\ H_{AS} \ H_{AA} \end{array} \right ),$$ and the corresponding Schrödinger equation is $$\left ( \begin{array}{cc} H_{SS} \ H_{SA} \\ H_{AS} \ H_{AA} \end{array} \right ) \left ( \begin{array}{cc} C \\ D \end{array} \right )=E \left ( \begin{array}{cc} C \\ D \end{array}\right ).$$ Diagonalizing the matrix, one can solve C and D which would determine the fraction of $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ in the eigenstates. The numerical results ======================= In this work, we have the input parameters as $ m_{u}=m_{d}=m_{q}\approx 0$, $m_{s}=0.279$ GeV, $M_{cc}=3.26\;{\rm GeV},\; M_{bb}=9.79\;{\rm GeV},\; M_{bc}=6.52\;{\rm GeV}$ [@Ebert]. $\alpha_{s}$=0.23, $B= (0.145 {\rm GeV})^{4}$. As argued above, the zero-point energy is not considered. Minimizing the expression of $M_B$ in eq.(1) with respect to the bag radius $R$, we obtain an equation and then determine the R-value. Substituting the obtained R-value into the expressions, we achieve the following table. In the table, we list our numerical results for the baryons which contain two heavy quarks, meanwhile the results obtained in other approaches are presented in the table for a clear comparison. ------------------- ----------- ------------------- --------------- --------- --------- --------- --------- --------- --------- $Notation$ $content$ $J^{p}$ $M_{B}$ $M_{B}$ $M_{B}$ $M_{B}$ $M_{B}$ $M_{B}$ $M_{B}$ (our results) \[5\] \[7\] \[8\] \[14\] \[15\] \[16\] $\Xi_{cc}$ (cc)q $\frac{1}{2}^{+}$ 3.55 3.66 3.48 3.620 3.55 3.66 3.61 $\Xi^{*}_{cc}$ (cc)q $\frac{3}{2}^{+}$ 3.59 3.81 3.61 3.727 3.64 3.74 3.68 $\Omega_{cc}$ (cc)s $\frac{1}{2}^{+}$ 3.73 3.76 3.59 3.778 3.66 3.74 3.71 $\Omega^{*}_{cc}$ (cc)s $\frac{3}{2}^{+}$ 3.77 3.89 3.73 3.872 3.73 3.82 3.76 $\Xi_{bb}$ (bb)q $\frac{1}{2}^{+}$ 10.10 10.23 10.09 10.202 - 10.34 - $\Xi^{*}_{bb}$ (bb)q $\frac{3}{2}^{+}$ 10.11 10.28 10.11 10.237 - 10.37 - $\Omega_{bb}$ (bb)s $\frac{1}{2}^{+}$ 10.28 10.32 10.21 10.359 - 10.37 - $\Omega^{*}_{bb}$ (bb)s $\frac{3}{2}^{+}$ 10.29 10.36 10.26 10.389 - 10.40 - $\Xi_{cb}$ (cb)q $\frac{1}{2}^{+}$ 6.80 6.95 6.82 6.933 - 7.04 - $\Xi'_{cb}$ (cb)q $\frac{1}{2}^{+}$ 6.87 7.00 6.85 6.963 - 6.99 - $\Xi^{*}_{cb}$ (cb)q $\frac{3}{2}^{+}$ 6.85 7.02 6.90 6.980 - 7.06 - $\Omega_{cb}$ (cb)s $\frac{1}{2}^{+}$ 6.98 7.05 6.93 7.088 - 7.09 - $\Omega'_{cb}$ (cb)s $\frac{1}{2}^{+}$ 7.05 7.09 6.97 7.116 - 7.06 - $\Omega^{*}_{cb}$ (cb)s $\frac{3}{2}^{+}$ 7.02 7.11 7.00 7.130 - 7.12 - ------------------- ----------- ------------------- --------------- --------- --------- --------- --------- --------- --------- Table 1. The baryon spectra Conclusion and Discussion ========================== In this work, we evaluate the spectra of baryons which contains two heavy quarks in the MIT bag model. It is an approach parallel to the potential model which is widely adopted for studying heavy hadrons. One can notice from the table given in the text that the numerical results in the two approaches are consistent with each other by the order of magnitude, but there are obvious distinction in numbers. The confinement in both the potential model and the bag model is artificially introduced which may reflect the non-perturbative QCD behavior in certain ways, but since none of them are derived from the first principle, one cannot expect them to be precise and it is reasonable that they result in different numbers which are related to the physics pictures and phenomenological parameters. As a matter of fact, in general the parameters are obtained by fitting data and while calculating the spectra of the lowest states of heavy quarkonia, various potential forms and sets of parameters can meet data. The MIT bag model is an alternative model and this work may be complimentary to the potential model. No matter in the potential model or the bag model, there is a zero-point energy problem, which manifest the vacuum property of QCD and is not calculable at present. The zero-point energy would determine the exact positions of each baryon and its excited states, but dies not influence the relative distances between the states. In the bag model, situation is a bit different, because $E_0\propto 1/R$, when to obtain $M_B$, we differentiate $M_B$ with respect to $R$ and the minimum determines the R-value. Since the energy of free light quark is $\sqrt {\chi^2+(mr)^2}/R$ has the same form of $E_0$, thus its contribution can be attributed to the little shift of the quark mass. The gaps between various states are not affected. In the future when data of the spectra are accumulated, one can come back to make more accurate adjustment of all the phenomenological parameters. In this work, we ignore the zero-point energy as argued above. The advantages of using the bag model to evaluate the spectra are two-folds. First the light quark obeys the relativistic Dirac equation, secondly, one can use the QFT to evaluate the possible mixing between $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ which is impossible in the regular QM framework. In our case, when we calculate the relativistic corrections to the interaction between quark and diquark, we only keep the terms up to order ${\bf p}/M$, which results in splitting of $B_{1/2}$ and $B_{3/2}$ of the same flavor combination. Because the diquark is very heavy and $M\gg |{\bf p}|$, this approximation would not bring up significant changes. As a conclusion, we evaluate the spectra of heavy baryons containing two heavy quarks in the MIT bag model. The results are qualitatively consistent with that obtained in the potential model, but numerically differ by a few percents. Moreover, we obtain a relatively large mixing between $|q(bc)_0>_{1/2}$ and $|q(bc)_1>_{1/2}$ which will be tested in the future measurements. Since there are no data available so far, we cannot fix a few parameters such as the zero-point energy, and one can definitely expect some deviations from the real values. Once the data are accumulated in the future, we can fin-tune the model and parameters. Definitely, the data will provide us with valuable information about the model and parameters. Even though the present B-factories BaBar and Belle cannot produce such baryons, TEVATRON and LHC which will begin running in 2007, may measure the spectra of such baryons, especially the long-expected Next-Linear-Collider (NLC) will offer us an idealistic place for studying such hadron spectra and our knowledge on the hadron structure can be greatly enriched. Acknowledgement: This work is partially supported by the National Natural Science Foundation of China. [1]{} N. Isgur and M. Wise, Phys.lett.[**B232**]{}(1989)113;[**B237**]{}(1990)527. C. Quigg and J. Rosner, Phys.Rep. [**56**]{} (1979) 167; W. Lucha, F. Schröberl and D. Gromes, Phys.Rep. [**200**]{} (1991) 127. T. EdGrand,R.L. Jaffe, et al. Phys.Rev.[**D12**]{}(1975)2068. A. Falk, M. Luke, M. Savage and M. Wise, Phys.Rev. [**D49**]{} (1994)555. D.Ebert et al. Z.Phys. [**C76**]{}(1997)111. S. Tong et al. Phys.Rev. [**D62**]{}(2000)054024. V.V.Kiselev, A.K.Likhoded, O.N.Pakhomova, V.A.Saleev, Phys.Rev. D66 (2002) 034030 D. Ebert, R. N. Faustov, V. O. Galkin, A. P. Martynenko Phys.Rev. [**D66**]{}(2002)014008 F. Close, [*An Introduction to Quarks and Partons*]{}, Academic Press Inc. (London) Ltd. 1979, London. D. Ebert, T. Feldmann, C. Kettner and H. Reinhardt, Z.Phys. [**C71**]{}(1996)329. V. Berestetskii, E. Lifshitz and L. Pitaevskii, [*Quantum Electrodynamics*]{}, Pergamon Press, 1982, New York. T.D. Lee, Phys.Rev. [**19**]{} (1979)1802. X.H. Guo, H.Y. Jin and X.Q. Li, Phys.Rev. [**D53**]{} (1996) 1153; Y.B. Dai,X.H. Guo and C.S. Huang, Nucl.Phys.[**B412**]{}(1994)277. J.M.Flymn, F.Mescia and A.S.B. “Tariq Spectroscopy of Doubly Charmed Baryons in Lattice QCD” R.Roncaglia, D.B.linchtenberg et al.Phys.Rev.D 52(1995)1722 J.G.K$\ddot{o}rner$, M.Kr$\ddot{a}$mer and D.Pirjol, Prog.Part.Nucl.Phys.33(1994)787

--- abstract: 'This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order HJB partial differential equation to a hierarchy of first order PDEs, followed by a numerical algorithm to solve the first few of the resulting first order PDEs. This method is applicable to stochastic systems with a relatively large number of degrees of freedom, and does not seem to suffer from the curse of dimensionality. Computer code implementation of the method using modest computational resources runs essentially in real time. We apply the method to solve a general portfolio construction problem.' author: - | **Sakda Chaiworawitkul**\ JPMorgan Chase\ New York, NY 10179\ USA - | **Patrick S. Hagan**\ Mathematics Institute\ 24-29 St Giles\ Oxford University\ Oxford, OX1 3LB\ UK - | **Andrew Lesniewski**\ Department of Mathematics\ Baruch College\ One Bernard Baruch Way\ New York, NY 10010\ USA date: | First draft: December 3, 2013\ This draft: title: | **Semiclassical approximation in stochastic optimal control\ I. Portfolio construction problem** --- \[sec:Introduction\]Introduction ================================ The stochastic Hamilton-Jacobi-Bellman (HJB) partial differential equation is the cornerstone of stochastic optimal control theory ([@FS92], [@YZ99], [@P09]). Its solution, the value function, contains the information needed to determine the optimal policy governing the underlying dynamic optimization problem. Analytic closed form solutions to the HJB equation are notoriously difficult to obtain, and they are limited to problems where the underlying state dynamics has a simple form. Typically, these solutions are only available for systems with one degree of freedom. A variety of numerical approaches to stochastic optimal control have been studied. An approach based on the Markov chain approximation is developed in [@KD01]. This approach avoids referring to the HJB equation altogether, and is, instead, based on a suitable discretization of the underlying stochastic process. Other recent approaches, such as [@FLO7], [@KLP13], and [@AK14], rely on ingenious discretization schemes of the HJB equation. These numerical methods are generally limited to systems with low numbers of degrees of freedom, as they are susceptible to the “curse of dimensionality”. In this paper, we present a methodology for effectively solving a class of stochastic HJB equations for systems with $n$ degrees of freedom, where $n$ is a moderately large number ($\lessapprox 200$). The solution methodology is based on an analytic approximation to the full HJB equation which reduces it to an infinite hierarchy of first order partial differential equations. This is accomplished by means of an asymptotic expansion analogous to the Wentzel-Kramers-Brillouin (WKB) method used in quantum mechanics, optics, quantitative finance, and other fields of applied science, see e.g. [@BO99], [@KC85]. The first in the hierarchy of equations is the classical Hamilton-Jacobi (HJ) equation which is analogous to the equation describing the motion of a particle on a Riemannian manifold[^1] subject to external forces. Its structure is somewhat less complicated than that of the full HJB equation, and its properties have been well understood. The solution to this equation is in essence the most likely trajectory for the optimal control of the stochastic system. Similar ideas, within a completely different setup have been pursued in [@T11] and [@HDM14]. The remaining equations are linear first order PDEs, with progressively more complex structure of coefficient functions. The approximate character of the solution of the HJB equation that we discuss is twofold. Firstly, we solve the Hamilton-Jacobi equation and the first of the linear PDEs in the hierarchy only. The WKB expansion is asymptotic, and the expectation is that these two equations capture the nature of the actual solution close enough. The remaining members of the hierarchy are neglected as they are believed that they contain information which does not significantly affect the shape of the solution. We refer to this approximation as the semiclassical (or eikonal) approximation in analogy with a similar approximation in physics. Interestingly, there is a class of non-trivial stochastic optimal control problems for which the semiclassical approximation produces the actual exact solutions. Two examples of such problems are discussed in the paper. Secondly, the solutions to the two leading order PDEs are constructed through numerical approximations. The key element of the numerical algorithm is a suitable symplectic method of numerical integration of Hamilton’s canonical equations, which are the characteristic equations of the HJ equation. Here, we use the powerful Störmer-Verlet (or leapfrog) method [@HLW03], [@LR04] to construct numerically the characteristics. Furthermore, we use a Newton-type search method in order to construct the numerical solution to the HJ equation out of the characteristics. This method uses a system of variational equations associated with Hamilton’s equations. This work has been motivated by our study of a stochastic extension of the continuous time version of the Markowitz mean variance portfolio optimization. The methodology developed here should, however, provide a practical method for implementation of the resulting portfolio construction. We believe, however, that the method is of broader interest and can be applied to a class of stochastic optimization problems outside of portfolio construction theory. \[sec:hjbEq\]Portfolio construction problem and the HJB equation ================================================================ We assume that the underlying source of stochasticity is a standard $p$-dimensional Wiener process $Z\oft\in\bR^p$ with independent components, $$\eE[dZ\oft dZ\oft^\tT]=\id dt.$$ Here, $\id$ denotes the $p\times p$ identity matrix. We let $(\Omega,(\sF)_{t\geq 0},\eP)$ denote the filtered probability space, which is associated with the Wiener process $Z$. We formulate the portfolio construction problem as the following stochastic control problem. We consider a controlled stochastic dynamical system whose states are described by a multi-dimensional diffusion process $(X\oft,W\oft)$, which takes values in $\cU\times\bR$, where $\cU\subset\bR^n$ is an open set. The components $X^i$, $i=1,\ldots,n$, of $X$ represent the prices of the individual assets in the portfolio, and $W$ is total value of the portfolio. We assume that $n\leq p$. The allocations of each of the assets in the portfolio are given by an $(\sF)_{t\geq 0}$-adapted process $\varphi\oft\in\bR^n$. The dynamics of $(X,W)$ is given by the system of stochastic differential equations: $$\label{eq:xDyn} \begin{split} dX\oft&=a(X\oft)dt+b(X\oft)dZ\oft,\\ X\of0&=X_0. \end{split}$$ The drift and diffusion coefficients $\cU\ni x\to a(x)\in\bR^n$ and $\cU\ni x\to b(x)\in\mathrm{Mat}_{n,p}(\bR)$, respectively, satisfy the usual Hölder and quadratic growth conditions, which guarantee the existence and uniqueness of a strong solution to this system. Note that we are not requiring the presence of a riskless asset in the portfolio: such assumption is unrealistic and unnecessary. If one wishes to consider a riskless asset, it is sufficient to take a suitable limit of the relevant components of $a$ and $b$. The process $W$ is given by $$\label{eq:yDyn1} \begin{split} dW\oft&=\varphi\oft^\tT dX\oft,\\ W\of0&=W_0. \end{split}$$ Explicitly, equation reads: $$\label{eq:yDyn} dW\oft=\varphi\oft^\tT a(X\oft)dt+\varphi\oft^\tT b(X\oft)dZ\oft.$$ We refer to the process $W$ as the investor’s wealth process. We assume that the investor has a finite time horizon $T$ and the utility function $U$. We shall assume that $U$ is a member of the HARA family of utility functions, see Appendix \[sec:UtilityFunctions\] for the their definition and summary of properties. The investor’s objective is to maximize the expected utility of his wealth at time $T$. We are thus led to the following cost functional: $$J[\varphi]=\eE\big[U(W(T))\big],$$ which represents the investor’s objective function. Let $$\label{eq:covDef} \cC\ofx=b\ofx^\tT b\ofx$$ denote the instantaneous covariance matrix of the price processes. For technical reasons, we shall make the following additional assumptions on the functions $a:\cU\to\bR^n$ and $b:\cU\to\mathrm{Mat}_{n,p}(\bR)$: - [The functions $a\ofx$ and $b\ofx$ are three times continuously differentiable for all $x\in\cU$.]{} - [The matrix $\cC\ofx$ is positive definite for all $x\in\cU$.]{} In particular, the function $x\to\cC\ofx^{-1}$ is three times continuously differentiable. Our goal thus is to find the optimal policy $\varphi^\ast$ which maximizes the expected utility if the terminal value of $W$. In other words, we are seeking the $\varphi^\ast$ such that $$\varphi^\ast=\mathop{\arg\sup}_\varphi\;\eE\big[U(W(T))\big]. \label{eq:OptimizationProblem}$$ We solve this optimization problem by invoking stochastic dynamic programming, see eg. [@FS92], [@YZ99] or [@P09]. The key element of this approach is the value function $J(t,x,w)$. It is determined by two requirements: - [it satisfies Bellman’s principle of optimality, $$J(t,X\oft,W\oft)=\sup_{\varphi}\;\eE\big[J(t+dt,X(t+dt),W(t+dt))|\sF_t\big],$$ for all $0\leq t<T$, and]{} - [it satisfies the terminal condition, $$J(T,X(T),W(T))=U(W(T)).$$]{} These conditions lead to the following nonlinear PDE for the value function, $$\dot{J}+\sup_{\varphi}\,\big\{a^\tT\nabla_x J+\varphi^\tT a\nabla_w J+\frac{1}{2}\,\tr(\cC\nabla^2_x J)+\varphi^\tT\cC\nabla^2_{xw} J+\frac{1}{2}\,\varphi^\tT\cC\varphi\nabla^2_w J\big\}=0,$$ namely the stochastic Hamilton-Jacobi-Bellman equation, subject to the terminal condition $$J(T,x,w)=U(w). \label{eq:defValueFunctionTermimal}$$ In order to solve the HJB equation, we choose $\varphi=\varphi^\ast$ which formally maximizes the expression inside the curly parentheses above. In other words, $\varphi^*$ satisfies $$(\nabla^2_w J)\cC\varphi+a\nabla_w J+\cC\nabla^2_{xw} J=0.$$ This leads to the following condition: $$\varphi^\ast=-\frac{\nabla^2_{xw} J}{\nabla^2_w J}-\frac{\nabla_w J}{\nabla ^2_w J}\,\cC^{-1} a,$$ known as the first order condition. Substituting $\varphi^*$ back to the HJB equation yields $$\label{eq:hjbRed} \dot{J}+a^\tT\nabla_x J+\frac{1}{2}\,\tr(\cC\nabla_x^2 J)-\frac{1}{2\nabla^2_w J}\,(\nabla^2_{xw}J+\cC^{-1}a\nabla_w J)^\tT\cC(\nabla^2_{xw}J+\cC^{-1} a\nabla_w J)=0.$$ We solve this equation by means of the following Ansatz: $$J(t,x,w)=\Gamma(t,x)U\ofw.$$ Using Proposition \[thm:haraProps\] we find that $\Gamma$ satisfies the following non-linear PDE: $$\label{eq:hjbGamma} \dot{\Gamma}+a^\tT\nabla_x \Gamma+\frac{1}{2}\,\tr(\cC\nabla^2\Gamma)+\frac{\kappa}{2}\,(\nabla\log\Gamma+\cC^{-1} a)^\tT \cC(\nabla\log\Gamma+\cC^{-1} a)\Gamma=0,$$ subject to the terminal condition $$\Gamma(T,x)=1.$$ The constant $\kappa$ depends only on the utility function and is given explicitly by . Since it will lead to no ambiguity, we have suppressed the subscript $x$ in the derivatives with respect to $x$. Note that the optimal control $\varphi^*$ has the following expression in terms of $\Gamma$ and $U$: $$\label{eq:optCont} \varphi^\ast=\frac{1}{A_U\ofw}\big(\nabla\log\Gamma+\cC^{-1} a\big),$$ where $A_U\ofw$ is the absolute risk aversion coefficient of the utility $U$. \[sec:wkbExp\]WKB expansion of the HJB equation =============================================== We shall write down the solution to equation in terms of an asymptotic expansion, namely the WKB expansion. The first few terms of this expansion yield an approximate solution which is sometimes referred as the semiclassical or eikonal approximation. The WKB asymptotic expansion is based on the assumption that the covariance matrix $\cC$ is “small” in some sense. To this end we scale the covariance matrix, $$\cC\to\varepsilon \cC,$$ where $\varepsilon$ is a parameter used to keep track of the order of magnitude in terms of $\cC$. At the end of the calculation, $\varepsilon$ is set back to $1$. Then, equation takes the form: $$\label{eq:hjbGammaEps} \dot{\Gamma}+a^\tT\nabla \Gamma+\frac{\varepsilon}{2}\,\tr(\cC\nabla^2\Gamma)+\frac{\varepsilon\kappa}{2}\,(\nabla\log\Gamma+\varepsilon^{-1}\cC^{-1} a)^\tT \cC(\nabla\log\Gamma+\varepsilon^{-1}\cC^{-1} a)\Gamma=0.$$ We seek a solution to the equation above in the form $$\Gamma(t,x)=\exp\big(\tfrac{1}{\varepsilon}\,S(t,x)\big),$$ where $S(t,x)$ has a finite limit as $\varepsilon\to 0$. Substituting this Ansatz into , we find that the equation for $S$ reads $$\label{eq:hjbS} \dot{S}+(\kappa+1)a^\tT\nabla S+\frac{\kappa+1}{2}\,(\nabla S)^\tT\cC\nabla S+\frac{\kappa}{2}\,a^\tT\cC^{-1} a +\frac{\varepsilon}{2}\,\tr(\cC\nabla^2 S)=0.$$ The optimal control expressed in terms of $S$ takes the following form: $$\varphi^\ast=\frac{1}{A_U}\Big(\cC^{-1}a+\frac{1}{\varepsilon}\,\nabla S\Big).$$ We assume that $S$ has an asymptotic expansion in powers of $\varepsilon$, $$S(t,x)=S^0(t,x)+S^1(t,x)\varepsilon+S^2(t,x)\varepsilon^2+\mathrm{O}(\varepsilon^3).$$ Substituting this expansion into equation yields an infinite hierarchy of equations: $$\label{eq:wkbHierarchy} \begin{split} \dot{S}^0&+\frac{\kappa+1}{2}\,(\nabla S^0)^\tT\cC\nabla S^0+(\kappa+1)a^\tT\nabla S^0+\frac{\kappa}{2}\,a^\tT\cC^{-1} a=0,\\ \dot{S}^1&+(\kappa+1)(a+\nabla S^0)^\tT\cC\nabla S^1+\frac{1}{2}\,\tr(\cC\nabla^2 S^0)=0,\\ \dot{S}^2&+(\kappa+1)(a+\nabla S^0)^\tT\cC\nabla S^2+\frac{\kappa+1}{2}\,(\nabla S^1)^\tT\cC\nabla S^1+\frac{1}{2}\,\tr(\cC\nabla^2 S^1)=0,\\ &\ldots\,,\\ \end{split}$$ where each of the $S^j$’s satisfies the terminal condition: $$\label{eq:bdCondS} S^j(T,x)=0,\text{ for } j=0,1,2,\dots .$$ The first of these equations is non-linear in $S^0$. Each of the subsequent equations is a linear PDE, with coefficients that depend on the solutions of the preceding equations. We define the variables $p$ dual to $x$ by $$\label{eq:pdef} p\triangleq\nabla S^0,$$ and refer to $p$ as the canonical momenta conjugate with $x$. We can then write the first of the equations as $$\label{eq:hjEqu} \dot{S}^0+H(x,\nabla S^0)=0,$$ where the Hamiltonian $H(x,p)$ is given by $$\label{eq:hamDef} H(x,p)=\frac{1}{2\gamma}\,p^\tT\cC\ofx p+\frac{1}{\gamma}\,p^\tT a\ofx+V(x),$$ where $$\label{eq:vDef} V\ofx=\frac{\kappa}{2}\,a\ofx^\tT\cC\ofx^{-1} a\ofx.$$ This non-linear PDE is the classical Hamilton-Jacobi equation, see e.g. [@CH53], [@E92]. Its solution gives the leading order approximation to the solution of the stochastic Hamilton-Jacobi-Bellman equation. From the physics point of view, the Hamiltonian describes the dynamics of a particle of mass $\gamma$ moving on a suitable Riemannian manifold in the potential $V\ofx$ and subject to an additional velocity dependent force[^2]. The solutions to the remaining linear equations in the hierarchy yield sub-leading “stochastic” corrections to the classical solution: $$\label{eq:wkbApprPhi} \Gamma(t,x)=\exp\big(\tfrac{1}{\varepsilon}\,S^0(t,x)+S^1(t,x)\big)\big(1+O(\varepsilon)\big).$$ This approximation is analogous to the eikonal approximation in classical optics or the semiclassical approximation in classical mechanics. \[sec:solHier\]Solving the WKB hierarchy ======================================== We shall now describe the methodology for solving the WKB hierarchy . Since each equation in the hierarchy is a first order PDE, the appropriate approach consists in applying the method of characteristics, see e.g. [@CH53] and [@E92]. We begin by solving the Hamilton-Jacobi equation . To this end, we recall that its characteristic equations are given by: $$\label{eq:charEqHJ} \begin{split} \dot{x}\ofs&=\nabla_p H(x\ofs,p\ofs),\\ \dot{p}\ofs&=-\nabla_x H(x\ofs,p\ofs),\\ \dot{z}\ofs&=p\ofs^\tT\nabla_p H(x\ofs,p\ofs)-H(x\ofs,p\ofs), \end{split}$$ where $z\ofs=S^0(s,x\ofs)$. These equations are subject to the terminal condition: $$\label{eq:termValCharEqHJ} \begin{split} x(T)&=y\,,\\ p(T)&=0\,,\\ z(T)&=0, \end{split}$$ where the terminal conditions for $p$ and $z$ are consequences of and . The first two of the characteristic equations are canonical Hamilton’s equations associated with the Hamiltonian $H$. Classic results of the theory of ordinary differential equations, see eg. [@CL55], guarantee the existence and uniqueness of the solution to the above terminal value problem, at least for $T$ sufficiently small. Furthermore, the solution depends smoothly on the terminal value $y$. In order to analyze Hamilton’s equations, assume first that $\kappa\neq-1$. They read then: $$\label{eq:hamEqs1} \begin{split} \dot{x}&=\frac{1}{\gamma}\,(\cC\ofx p+a\ofx),\\ \dot{p}&=-\nabla_x\Big(\frac{1}{2\gamma}\,p^\tT\cC\ofx p+\frac{1}{\gamma}\,p^\tT a\ofx+V\ofx\Big),\\ \end{split}$$ or, explicitly, $$\label{eq:hamEqs2} \begin{split} \dot{x}&=\frac{1}{\gamma}\,(\cC\ofx p+a\ofx),\\ \dot{p}_i&=-\frac{1}{2\gamma}\,p^\tT\,\frac{\d\cC\ofx}{\d x^i}\, p-\frac{1}{\gamma}\,p^\tT\,\frac{\d a\ofx}{\d x^i}\, -\frac{\kappa}{2}\,a\ofx^\tT\,\frac{\d\cC\ofx^{-1}}{\d x^i}\,a\ofx-\kappa a\ofx^\tT\cC\ofx^{-1}\,\frac{\d a\ofx}{\d x^i}\,,\\ \end{split}$$ for $i=1,\ldots,n$. In Section \[sec:numSec\] we shall describe an efficient algorithm to solve these equations numerically. It is now easy to write down the solution to the Hamilton-Jacobi equation. Indeed, the integral $$\label{eq:s0Expl} \begin{split} S^0(t,x\oft)&=-\int_t^T\Big(p\ofs^\tT dx\ofs-H(x\ofs,p\ofs)ds\Big)\\ &=-\int_t^T\Big(\frac{1}{2\gamma}\,p\ofs^\tT\cC(x\ofs)p\ofs-V(x\ofs)\Big)ds \end{split}$$ defines the solution to the Hamilton-Jacobi equation along the characteristic $x\ofs,p\ofs$. In order to find the solution $S^0(t,x)$, for each $x\in\cU$, we eliminate $y$ by inverting the function $y\to x\oft$. Specifically, we write the solution $x\oft$ in the form $$\label{eq:phiDef} \begin{split} x\oft&=x(t,y)\\ &=\Phi_t(y), \end{split}$$ which emphasizes the dependence of the trajectory on the terminal value. We have suppressed the terminal value for $p$ as it is always required to be zero. Then, for each $t<T$ is a diffeomorphism of $\cU$. We set $$\label{eq:solHjEq} S^0(t,x)=S^0(t,x(t,\Phi^{-1}_t(x))).$$ This is the desired solution to the Hamilton-Jacobi equation. The second equation in is an inhomogeneous linear first order partial differential equations and it can be readily solved by means of the method of characteristics. Note that, on a characteristic $(x\ofs, p\ofs)$, $$\dot{x}\ofs=\frac{1}{\gamma}\,(\cC(x\ofs) a(x\ofs)+\nabla S^0(s,x\ofs)).$$ Therefore, along $x\ofs$, the equation for $S^1$ can be written as an ordinary differential equation, $$\frac{d}{ds}\,S^1(s,x\ofs)+\frac{1}{2}\,\tr\big(\cC(x\ofs)\nabla^2 S^0(s,x\ofs))\big)=0,$$ and thus its solution reads: $$S^1(t,x\oft)=\frac{1}{2}\,\int_t^T\tr\big(\cC(x\ofs)\nabla^2 S^0(s,x\ofs)\big)ds.$$ In analogy with , we write $$\label{eq:psiDef} \begin{split} p\oft&=p(t,y)\\ &=\Psi_t(y). \end{split}$$ Then $$\begin{split} p(t,x)&\triangleq\nabla_x S^0(t,x)\\ &=\Psi_t(\Phi^{-1}_t(x)), \end{split}$$ and we can write $S^1(t,x)$ as $$\label{eq:s1Sol} S^1(t,x)=\frac{1}{2}\,\int_t^T\tr\big(\cC(x\ofs)\nabla p(s,x\ofs)\big)ds.$$ Likewise, the solution to the third equation in can be written explicitly as $$\label{eq:s2Sol} \begin{split} S^2(t,x\oft)&=\frac{1}{2\gamma}\,\int_t^T(\nabla S^1(s,x\ofs))^\tT\cC(x\ofs)\nabla S^1(s,x\ofs)ds\\ &+\frac{1}{2}\,\int_t^T \tr\big(\cC(x\ofs)\nabla^2 S^1(s,x\ofs)\big)ds, \end{split}$$ along a characteristic $x\ofs$. Note that the solution requires knowledge of $S^1$, which in turn requires knowledge of $S^0$. We can continue this process to solve for $S^n$, with the understanding that the complexity of the solution increases in $n$. Let us now consider the case of $\kappa=-1$, which corresponds to the CARA utility function. This turns out to be a Hamilton’s canonical equations read: $$\begin{split} \dot{x}\ofs&=0,\\ \dot{p}\ofs&=\nabla V(x\ofs),\\ \dot{z}\ofs&=V(x\ofs), \end{split}$$ where $V\ofx$ is defined by . Consequently, $$\begin{split} x\ofs&=y,\\ p\ofs&=-\nabla V(y)(T-s),\\ z\ofs&=-V(y)(T-s) \end{split}$$ and thus the solution to the Hamilton-Jacobi equation reads $$S^0(t,x)=-V(x)(T-t).$$ Furthermore, we find easily that $$S^1(t,x)=\frac{1}{4}\,\tr\big(\cC\ofx\nabla^2 V\ofx\big)(T-t)^2,$$ and $$S^2(t,x)=\frac{1}{24}\,\big(\tr(\cC\ofx\nabla^2)^2\big) V\ofx(T-t)^3$$ are the solutions to the second and third equations of the WKB hierarchy, respectively. \[sec:exampSec\]Generalized Merton portfolio models =================================================== In this section we illustrate the expansion method developed above with a class of portfolio models that are frequently discussed in the literature. Namely, we consider a portfolio of assets whose price dynamics are of the form: $$\label{eq:sepDyn} \begin{split} dX^i\oft&=\mu^i(X^i\oft)dt+\sigma^i(X^i\oft) dB_i\oft,\\ X^i\of0&=X^0, \end{split}$$ i.e. the drift $\mu^i(X^i)\in\bR$ and $\sigma^i(X^i)\in\bR$ are functions of $X^i$ only. The Brownian motions $B_i\oft$ above are correlated, $$\eE[dB_i\oft dB_j\oft]=\rho_{ij}dt.$$ This dynamics becomes a special case of , if we set $$B\oft=Z\oft L,$$ where $Z\oft$ is the standard $n$-dimensional Brownian motion and $L$ is the lower triangular matrix in the Cholesky decomposition, $\rho=L^\tT L$. The model specification is natural if we believe that the return and volatility of an asset are local functions of that asset’s price only, while the dependence between the assets is a function of the portfolio. Models of this type generalize dynamic portfolio models introduced and studied by Merton [@M69], [@M71]. The covariance matrix and its inverse in this model are given by $$\label{eq:locMetTens} \begin{split} \cC_{ij}\ofx&=\rho_{ij}\sigma^i(x^i)\sigma^j(x^j)\,,\\ (\cC\ofx^{-1})_{ij}&=\frac{(\rho^{-1})_{ij}}{\sigma^i(x^i)\sigma^j(x^j)}\,. \end{split}$$ Hence, $$V\ofx=\frac{\kappa}{2}\,\mu\ofx^\tT\cC\ofx^{-1}\mu\ofx,$$ and consequently, $$\frac{\d}{\d x^i}\, V\ofx=\kappa\Big(\frac{d\mu^i(x^i)}{d x^i}-\mu^i(x^i)\,\frac{d\log\sigma^i(x^i)}{d x^i}\Big)\,\sum\nolimits_j(\cC\ofx^{-1})_{ij}\mu^j(x^j).$$ Hence, Hamilton’s equations for this model read: $$\begin{split} \dot{x}^i&=\frac{1}{\gamma}\Big((\cC\ofx p)_i+\mu^i(x^i)\Big),\\ \dot{p}_i&=-\frac{1}{\gamma}\,p_i\,\Big(\frac{d\log\sigma^i(x^i)}{dx^i}(\cC\ofx p)_i +\frac{d\mu^i(x^i)}{dx^i}\Big)\\ &\quad-\kappa\Big(\frac{d\mu^i(x^i)}{d x^i}-\mu^i(x^i)\,\frac{d\log\sigma^i(x^i)}{d x^i}\Big)\,(\cC\ofx^{-1}\mu(x))_i. \end{split}$$ These equations are subject to the terminal value conditions: $$\label{eq:termValCond} \begin{split} x(T)=y,\\ p(T)=0. \end{split}$$ Let us consider two explicit examples: (i) a portfolio of lognormal assets, and (ii) a portfolio of mean reverting normal assets. A special feature of these examples is that the semiclassical approximations are, in fact, the exact solutions. As a special case of the model above, we consider a portfolio of $n$ assets each of which follows the lognormal process, i.e. $$\label{eq:lnDyn} \begin{split} \mu^i(x^i)&=\mu_i x^i,\\ \sigma^i(x^i)&=\sigma_i x^i, \end{split}$$ where $\mu_i$ and $\sigma_i$ are constant coefficients referred to as the return and lognormal volatility, respectively. This is essentially the original Merton model. Note that, in this model, $$V\ofx=\frac{\kappa}{2}\,\mu^\tT C^{-1}\mu$$ is constant. Here we have set $C_{ij}=\rho_{ij}\sigma_i\sigma_j$, for $1\leq i,j\leq n$. Hamilton’s equations read: $$\begin{split} \dot{x}^i&=\frac{1}{\gamma}\Big((\cC\ofx p)_i+\mu_i x^i\Big),\\ \dot{p}_i&=-\frac{1}{\gamma}\,p_i\,\Big(\frac{1}{x^i}\,(\cC\ofx p)_i+\mu_i\Big). \end{split}$$ Since $p(T)=0$, the second of the equations has the unique solution $$p_i\ofs=0.$$ Hence, $$x^i\ofs=y^i e^{-(\mu_i/\gamma)(T-t)}$$ is the unique solution to the first equation subject to the terminal condition $p(T)=y$. These are the characteristics of the Hamilton-Jacobi equation. This implies that $$S^0(t,x)=\frac{\kappa}{2}\,\mu^\tT C^{-1}\mu(T-t),$$ and $$S^j(t,x)=0,$$ for all $j\geq 1$. Consequently, $$\begin{split} \varphi^\ast_i&=\frac{1}{A_U(w)}\,(\cC\ofx^{-1}\mu\ofx)_i\\ &=\frac{1}{A_U(w)}\,\frac{(C^{-1}\mu)_i}{x^j}\,. \end{split}$$ The semiclassical solution is exact and it coincides with Merton’s original solution. Another tractable portfolio model arises as follows. We consider a portfolio of $n$ assets each of which follows the Ornstein-Uhlenbeck process, i.e. $\mu^i\ofx=\lambda_i(\bar{\mu}^i-x^i)$, and $\sigma^i(x^i)=\sigma_i$, where $\lambda_i$ is the speed of mean reversion of asset $i$, $\bar{\mu}^i$ is its mean reversion level, and $\sigma_i$ is its instantaneous volatility. Note that in this model $V\ofx$ is quadratic, $$V\ofx=\frac{\kappa}{2}\,(\bar{\mu}-x)^\tT\Lambda \cC^{-1}\Lambda(\bar{\mu}-x),$$ where $\Lambda\in\matn$ is the diagonal matrix with entries $\lambda_i$, $i=1,\ldots, n$. As a result, Hamilton’s equations can be solved in closed form. Indeed, we find that they form a linear system: $$\label{eq:linHam} \frac{d}{dt} \begin{pmatrix} x\\ p \end{pmatrix} =A \begin{pmatrix} x\\ p \end{pmatrix} +m,$$ where $$\begin{split} A&= \begin{pmatrix} -\gamma^{-1}\Lambda& \gamma^{-1}\cC\\ -\kappa\Lambda\cC^{-1}\Lambda& \gamma^{-1}\Lambda \end{pmatrix},\\ m&= \begin{pmatrix} \gamma^{-1}\Lambda\bar{\mu}\\ \kappa\Lambda\cC^{-1}\Lambda\bar{\mu} \end{pmatrix}. \end{split}$$ The solution to the system subject to the terminal conditions $p(T)=0$ and $x(T)=y$ reads: $$\begin{pmatrix} x\ofs\\ p\ofs \end{pmatrix} =e^{-(T-s)A}\Bigg( \begin{pmatrix} y\\ 0 \end{pmatrix} +A^{-1}m\Bigg)-A^{-1}m,$$ where the exponential denotes the matrix exponential function. These are the characteristics of the Hamilton-Jacobi equation. This representation allows us to explicitly construct the maps $\Phi_t$ and $\Psi_t$ in and , respectively. Indeed, they are linear in $y$ and, consequently, $\Phi_t^{-1}$ and $\Psi_t^{-1}$ are linear functions as well. As a consequence of , $S^0(t,x)$ is an explicitly computable quadratic function of $x$. Since in the current model $\cC$ is independent of $x$, formula implies that $S^1(t,x)$ is non-zero but independent of $x$. Inspection of the WKB hierarchy shows immediately that $$S^j(t,x)=0,$$ for all $j\geq 2$. Consequently, we obtain the following formula for the optimal control: $$\varphi^\ast=\frac{1}{A_U(w)}\,\big(\cC^{-1}\Lambda(\bar{\mu}-x)+\nabla S^0(t,x)\big),$$ with no further corrections. As in the case of the lognormal model, this semiclassical solution turns out to be exact. \[sec:numSec\]Numerical implementation of the solution ====================================================== Interesting cases for which the Hamilton-Jacobi equation admits a closed form solution are scarce. In fact, even in the case of constant covariance matrix, cannot, in general, be solved in closed form. In this section we discuss a numerical solution method for the Hamilton-Jacobi equation, and indeed (at least in principle) the entire WKB hierarchy, that is efficient and accurate for systems with a relatively large ($\lessapprox 200$) number of degrees of freedom. Let $$\begin{split} x\oft&=\Phi_t(y),\\ p\oft&=\Psi_t(y), \end{split}$$ denote the solution to Hamilton’s equations with terminal condition . Our goal is to compute $S^0(t,x)$ and $S^1(t,x)$ for all $0\leq t\leq T$ and $x\in\cU$. This amounts to an effective numerical implementation of the solutions constructed in Section \[sec:solHier\] by means of the method of characteristics. We proceed in the following steps. - [*Step 1*. For a given terminal value $y$ and each $t<T$, compute $x\oft=\Phi_t(y)$ and $p\oft=\Psi_t(y)$.]{} - [*Step 2*. Given $x\in\cU$ and $t<T$ find $y$ such that $x\oft=x$. This is equivalent to inverting the function $y\to\Phi_t\ofy$.]{} - [*Step 3*. Given $x\in\cU$ and $t<T$, compute $S^0(t,x)$.]{} - [*Step 4*. Given $x\in\cU$ and $t<T$, compute $S^1(t,x)$.]{} We shall now describe these steps in detail. *Step 1*. In order to construct the pair $(\Phi_t, \Psi_t)$ we use the Störmer-Verlet / leapfrog method of integrating Hamilton’s equations [@HLW03], [@LR04]. Other popular numerical methods, such as Euler’s method or the Runge-Kutta method, tend to perform poorly when applied to a Hamiltonian system. This can be traced to the fact that these methods do not respect the underlying symplectic structure, and, in particular, do not preserve the volume in the phase space of the system. The leapfrog method is an ingenious way of discretizing a Hamiltonian system, so that it defines a symplectic map. As a additional bonus, the Störmer-Verlet / leapfrog scheme is order $h^2$ accurate. Specifically, we discretize the time range $[t,T]$, $$\label{eq:discT} t_k=t+kh,\text{ if } k=0,1,\ldots,N,$$ where the time step $h=(T-t)/N$ is chosen suitably. We replace the continuous time Hamiltonian system by a discrete time dynamical system, and let $\hat{x}_k$ and $\hat{p}_k$ denote the approximate values of $x(t_k)$ and $p(t_k)$, respectively. We require that $\hat{x}_k$ and $\hat{p}_k$ follow the numerical scheme: $$\label{eq:genLeap} \begin{split} \hat{p}_{k-\frac{1}{2}}&=\hat{p}_{k}+\frac{h}{2}\,\nabla_x H(\hat{x}_k,\hat{p}_{k-\frac{1}{2}})\,,\\ \hat{x}_{k-1}&=\hat{x}_k-\frac{h}{2}\,\big(\nabla_p H(\hat{x}_k,\hat{p}_{k-\frac{1}{2}})+\nabla_p H(\hat{x}_{k-1},\hat{p}_{k-\frac{1}{2}})\big),\\ \hat{p}_{k-1}&=\hat{p}_{k-\frac{1}{2}}+\frac{h}{2}\,\nabla_x H(\hat{x}_{k-1},\hat{p}_{k-\frac{1}{2}}), \end{split}$$ where we have introduced half intervals values $\hat{p}_{k-\frac{1}{2}}$. The presence of these intermediate values of the momentum is the crux of the leapfrog method and it assures that the scheme is symplectic. Notice that the first and second equations in are implicit in $\hat{p}_{k-\frac{1}{2}}$ and $\hat{x}_{k-1}$, respectively. Calculating the derivatives yields $$\label{eq:LeapConcIF} \begin{split} \hat{p}_{k-\frac{1}{2}}&=\hat{p}_k+\frac{h}{2\gamma}\,\nabla a(\hat{x}_k)\hat{p}_{k-\frac{1}{2}}+\frac{h}{2\gamma}\,\hat{p}_{k-\frac{1}{2}}^\tT \nabla \cC(\hat{x}_k)\hat{p}_{k-\frac{1}{2}}+\frac{h}{2}\,\nabla V(\hat{x}_k),\\ \hat{x}_{k-1}&=\hat{x}_k-\frac{h}{2\gamma}\,\big(\cC(\hat{x}_k) +\cC(\hat{x}_{k-1})\big)\hat{p}_{k-\frac{1}{2}}-\frac{h}{2\gamma}\,\big(a(\hat{x}_k)+a(\hat{x}_{k-1})\big),\\ \hat{p}_{k-1}&=\hat{p}_{k-\frac{1}{2}}+\frac{h}{2\gamma}\,\nabla a(\hat{x}_{k-1})\hat{p}_{k-\frac{1}{2}}+\frac{h}{2\gamma}\,\hat{p}_{k-\frac{1}{2}}^\tT \nabla \cC(\hat{x}_{k-1})\hat{p}_{k-\frac{1}{2}}+\frac{h}{2}\,\nabla V(\hat{x}_{k-1}). \end{split}$$ This system is subject to the terminal condition: $$\begin{split} \hat{x}_N&=y,\\ \hat{p}_N&=0. \end{split}$$ Note that the first two relations in cannot, in general, be solved explicitly for $\hat{p}_{k-\frac{1}{2}}$ and $\hat{x}_{k-1}$, respectively, and thus they need to be solved numerically. This can be efficiently accomplished, for example by means of Newton’s method with the initial guess $\hat{p}_{k-\frac{1}{2}}=\hat{p}_k$ and $\hat{x}_{k-1}=\hat{x}_k$. Indeed, in practice, a few iterations of Newton’s method yield a very accurate solution. Solving this system yields an approximate flow map $\hat{\Phi}_t$. Throughout the reminder of this section we shall suppress the hat over $x,p$, etc. keeping in mind that all the quantities are numerical approximations to the true values. *Step 2*. In order to carry out the next step, we develop an algorithm for inverting the flow map $\Phi_t:\cU\to\cU$ defined above. From the existence theory of ordinary differential equations, $x$ and $p$ depend smoothly on the terminal value $y$. Hence, the sensitivities $\nabla_y\Phi$ and $\nabla_y\Psi$ satisfy the following system of equations: $$\label{eq:varEqs} \begin{split} \frac{d}{dt}\,\nabla_y\Phi&=\nabla^2_{px}H\,\nabla_y\Phi+\nabla^2_{pp}H\,\nabla_y\Psi,\\ \frac{d}{dt}\,\nabla_y\Psi&=-\nabla^2_{xx}H\,\nabla_y\Phi-\nabla^2_{xp}H\,\,\nabla_y\Psi, \end{split}$$ subject to the terminal condition $$\begin{split} \nabla_y\Phi(T,y)&=\id,\\ \nabla_y\Psi(T,y)&=0. \end{split}$$ Equations of this type are known as variational equations, see e.g. [@G27]. Consider now an approximation of the variational system , in which the second derivatives of $H$ are evaluated at the constant trajectory $(x\oft,p\oft)=(y,0)$. We thus obtain the following linear system with constant coefficients: $$\begin{split} \dot{F}&=Q\ofy F+R\ofy G,\\ \dot{G}&=-U\ofy F-Q\ofy G, \end{split}$$ where the matrices $Q$, $R$, and $U$ are given explicitly by $$\begin{split} Q\ofy&=(\kappa+1)\nabla a\ofy^\tT,\\ R\ofy&=(\kappa+1)\cC\ofy,\\ U\ofy&=\nabla^2 V\ofy. \end{split}$$ Note that $F_t\ofy\equiv F(t,y)$ is an approximation to $\nabla\Phi_t\ofy$, the gradient of the function $y\to\Phi_t\ofy$. This linear system can be written in a more compact form as $$\frac{d}{dt} \begin{pmatrix} F\\ G\\ \end{pmatrix} =M\ofy \begin{pmatrix} F\\ G\\ \end{pmatrix},$$ where $$M\ofy= \begin{pmatrix} Q\ofy & R\ofy\\ -U\ofy & -Q\ofy\\ \end{pmatrix},$$ subject to the terminal condition $$\begin{pmatrix} F_T\ofy\\ G_T\ofy\\ \end{pmatrix} = \begin{pmatrix} \id\\ 0\\ \end{pmatrix}.$$ This problem has a unique solution, namely $$\label{eq:linSol} \begin{pmatrix} F_t\ofy\\ G_t\ofy\\ \end{pmatrix} =e^{-(T-t)M\ofy} \begin{pmatrix} \id\\ 0\\ \end{pmatrix},$$ where, as before, the exponential denotes the matrix exponential function. This solution can readily be implemented in computer code [@GL]. Now, our next goal is to solve for $y$ the equation $$\Phi_t\ofy-x=0.$$ To this end, we use a Newton-type method. Finding the gradient $\nabla\Phi_t\ofy$ is computationally very expensive, and it may be susceptible to numerical inaccuracies. Fortunately, for convergence purposes, it is sufficient to approximate it with $F_t\ofy$, which we have just computed explicitly. In the Appendix we justify this procedure, by proving that it converges for $T$ sufficiently small. The following pseudocode implements this search algorithm: $$\begin{split} &eps\gets 10^{-13}\\ &y\gets x\\ &err\gets1.0\\ &\texttt{while}(err>eps)\\ &\qquad z\gets y-F_t\ofy^{-1}(\Phi_t\ofy-x)\\ &\qquad err\gets\|z-y\|\\ &\qquad y\gets z \end{split}$$ The norm $\|\cdot\|$ above denotes the usual Euclidean norm in $\bR^n$. *Step 3*. We are now ready to compute the value of $S^0(t,x)$. In Step 3 we have found $y=\Phi_t^{-1}(x)$. Using the algorithm explained in Step 1, we construct the discrete trajectory $(x_{t_k},p_{t_k})$. We write the integral as a sum of integrals over the segments $[t_k,t_{k+1}]$, $$S^0(t,x)=\sum_{k=0}^{N-1}\,I^0_{t_k,t_{k+1}}.$$ We denote the integrand in by $L(x\ofs,p\ofs)$, and calculate each of the subintegrals according to Simpson’s rule: $$\begin{split} I^0_{a,b}&=\int_a^b L(x(s),p(s))ds\\ &\approx\frac{1}{6}\big(L(x(a),p(a))+4L(x(m),p(m))+L(x(b),p(b))\big)(b-a), \end{split}$$ where $m=\tfrac{a+b}{2}$ is the midpoint between $a$ and $b$. *Step 4*. We break up the integral in into the sum of integrals over $[t_k,t_{k+1}]$, $$S^1(t,x)=\sum_{k=0}^{N-1}\,I^1_{t_k,t_{k+1}}.$$ Reusing the discrete trajectory $(x_{t_k},p_{t_k})$ calculated in Step 3, we compute each subintegral using Simpson’s rule: $$\begin{split} I^1_{a,b}&=\frac{1}{2}\int_a^b\tr\big(\cC(x\ofs)\nabla p(s,x\ofs)\big)ds\\ &\approx\frac{1}{6}\,\tr\big(\cC(x(a))\nabla p(a,x(a))+4\cC(x(m))\nabla p(m,x(m))+\cC(x(b))\nabla p(b,x(b))\big)(b-a). \end{split}$$ The first order partial derivatives in $\nabla p$ in the expression above are calculated as central finite differences: $$\frac{\d}{\d x^j}\, p_j(t_k,x(t_k))\approx\frac{p_{t_{k+1}}-p_{t_{k-1}}}{2(x_{t_{k+1}}-x_{t_{k-1}})}\,.$$ \[sec:UtilityFunctions\]The HARA family of utility functions ============================================================ We let $U\ofv$ denote a utility function, i.e. a twice differentiable concave function. Recall that the *absolute risk aversion* coefficient associated with $U\ofv$ is defined by $$A_U\ofv=-\frac{U^{\prime\prime}\ofv}{U^{\prime}\ofv}\;, \label{eq:ara}$$ while the *relative risk aversion* coefficient is given by $$\begin{split} R_U\ofv&=-\frac{vU^{\prime\prime}\ofv}{U^{\prime}\ofv}\\ &=vA_U\ofv\;. \end{split} \label{eq:rra}$$ In this paper we consider the following four utility functions: - [The *hyperbolic absolute risk aversion* (HARA) utility, $$U_{\rm{HARA}}\left(v;a,b,\gamma\right)=\frac{\gamma}{1-\gamma}\left(a+\frac{b}{\gamma}\,v \right)^{1-\gamma}\;. \label{eq:HARA}$$]{} - [The *constant relative risk aversion* (CRRA) utility, $$U_{\rm{CRRA}}\left(v;\gamma\right)=\frac{v^{1-\gamma}}{1-\gamma}\;. \label{eq:CRRA}$$ Note that $\gamma$ is the (constant) relative risk aversion coefficient associated with this utility function, $R_U\ofv=\gamma$.]{} - [The *constant absolute risk aversion* (CARA) utility, $$U_{\rm{CARA}}\left(v;\gamma\right)=-\frac{e^{-\gamma v}}{\gamma}\;, \label{eq:CARA}$$ Note that $\gamma$ is the (constant) absolute risk aversion coefficient associated with this utility function, $A_U\ofv=\gamma$.]{} - [The *logarithmic* (Bernoulli) utility, $$U_{\rm{LOG}}\ofv=\log\ofv.$$]{} It is well known that the HARA utility includes the CRRA, CARA, and logarithmic utility functions as limit cases. Indeed, $$\begin{split} U_{\rm{CRRA}}\left(v;\gamma\right)&=U_{\rm{HARA}}\left(v;0,\gamma^{-\gamma/\left(1-\gamma\right)},\gamma\right),\\ U_{\rm{CARA}}\left(v;\gamma\right)&=\frac{1}{\gamma}\;\lim_{c\to\infty}U_{\rm{HARA}}\left(v;1,\gamma,c\right),\\ U_{\rm{LOG}}\ofv&=\lim_{\gamma\to 1}U_{\rm{CRRA}}\left(v;\gamma\right).\\ \end{split}$$ The following proposition is used in Section \[sec:hjbEq\]. \[thm:haraProps\] Let $U$ be a twice differentiable function. The ratio $$-\frac{U'\ofv^2}{U''\ofv U\ofv}$$ is a constant if and only if $U$ is a HARA utility function. In this case, its value $\kappa$ is given by $$\kappa= \begin{cases} \left(1-\gamma\right)/\gamma,& \text{ for the HARA and CRRA utilities,}\\ -1,& \text{ for the CARA utility,}\\ 0,& \text { for the log utility.} \end{cases}\label{eq:kappaDef}$$ The proof of this proposition is a straightforward calculation and we omit it. \[sec:convSec\]Convergence of the modified Newton method ======================================================== The purpose of this Appendix is to prove that the Newton-type method described in Section \[sec:numSec\] converges, at least in the case if the time horizon $T$ is sufficiently short. Our proof uses familiar techniques of numerical analysis [@SB02] and systems of ordinary differential equations [@CL55]. We first state the following general fact. \[thm:newtMeth\][Let $\cB\subset\bR^n$ be a compact set, and let $h:\cB\to\cB$ be a twice continuously differentiable function. Assume that $h$ has a unique simple zero $y^*\in\cB$, $$\begin{split} h(y^*)&=0,\\ \nabla h(y^*)&\neq 0. \end{split}$$ Let $F:\cB\to\matn$ be a continuously differentiable function such that $F\ofy^{-1}$ exists for all $y\in\cB$, and the following two conditions are satisfied. There is a $0<\delta<1$, such that $$\label{eq:cond1} \|\id-F\ofy^{-1}\nabla h\ofy\|\leq\delta/2,$$ and $$\label{eq:cond2} \|\nabla\big(F\ofy^{-1}\big)h\ofy\|\leq\delta/2,$$ for all $y\in\cB$. Then the map $$f\ofy=y-F\ofy^{-1}h\ofy$$ is a contraction of $\cB$ into itself.]{} We verify easily that conditions and imply that $\|\nabla f\ofy\|\leq 1-\delta$, uniformly in $y\in\cB$. Hence, $f$ is a contraction. As a consequence of this proposition and the contraction principle, the sequence $$\label{eq:nrSeq} \begin{split} y_1&=f(y_0)\\ &=y_0-F(y_0)^{-1}h(y_0),\\ y_2&=f(y_1)\\ &=y_1-F(y_1)^{-1}h(y_1),\\ &\ldots, \end{split}$$ where $y_0\in\cB$ is arbitrary, converges to $y^*$. Next, we shall show that the above proposition applies to our specific situation. The sequence will furnish the modified Newton method used in Section \[sec:numSec\]. [Assume that the Hamiltonian $H(x,p)$ is three times continuously differentiable, and let $h\ofy=\Phi_t\ofy-x$ and $F\ofy=F_t\ofx$. Then, there are a $T>0$ and a compact set $\cB\ni y$ such that conditions and of Proposition \[thm:newtMeth\] are satisfied. ]{} By the general theory of ordinary differential equations (see e.g. [@CL55]), we note first that under our assumptions, there is a $T> 0$ such that $\Phi_t\ofy$ and $\Psi_t\ofy$ are unique solutions of the terminal value problem - , and they have continuous derivatives with respect to $y$. Furthermore, it is clear from that $F_t\ofy$ is nonzero for all $y$, and $$\label{eq:bound1} \max_{y\in\cB}\|F_t\ofy^{-1}\|<\infty,$$ $$\label{eq:bound2} \max_{y\in\cB}\|\nabla\big(F_t\ofy^{-1}\big)\|<\infty.$$ Now, in order to prove , it is sufficient to prove that given an $\eta>0$, there is a $T>0$ such that $$\label{eq:etaIneq} \|F\ofy-\nabla h\ofy\|\leq\eta,$$ for all $t\leq T$. Indeed, $$\begin{split} \|\id-F\ofy^{-1}\nabla h\ofy\|&=\|F\ofy^{-1}(F\ofy-\nabla h\ofy)\|\\ &\leq\max_{y\in\cB}\|F\ofy^{-1}\|\|F\ofy-\nabla h\ofy\|\\ &\leq\mathrm{const}\times\eta. \end{split}$$ In order to prove , we set $$D_t(y)= \begin{pmatrix} \nabla_y\Phi_t\ofy\\ \nabla_y\Psi_t\ofy \end{pmatrix} - \begin{pmatrix} F_t\ofy\\ G_t\ofy \end{pmatrix},$$ and $$N_t(y)= \begin{pmatrix} \nabla^2_{xp}H(\Phi_t\ofy,\Psi_t\ofy)&\nabla^2_{xp}H(\Phi_t\ofy,\Psi_t\ofy)\\ \nabla^2_{xp}H(\Phi_t\ofy,\Psi_t\ofy)&\nabla^2_{xp}H(\Phi_t\ofy,\Psi_t\ofy) \end{pmatrix}.$$ Then $D_t\ofy$ satisfies the following system of differential equations: $$\begin{split} \dot{D}_t\ofy&=N_t\ofy \begin{pmatrix} \nabla_y\Phi_t\ofy\\ \nabla_y\Psi_t\ofy \end{pmatrix} -M\ofy \begin{pmatrix} F_t\ofy\\ G_t\ofy \end{pmatrix}\\ &=M\ofy D_t\ofy+E_t\ofy, \end{split}$$ where $$E_t\ofy=(N_t\ofy-M\ofy) \begin{pmatrix} \nabla_y\Phi_t\ofy\\ \nabla_y\Psi_t\ofy \end{pmatrix},$$ subject to the terminal condition $D_T\ofy=\id$. Hence $$D_t\ofy=\int_T^t e^{(t-s)M\ofy}E_s\ofy ds.$$ As a consequence $$\begin{split} \|D_t\ofy\|&\leq\int_T^t\|e^{(t-s)M\ofy}E_s\ofy\|\,ds\\ &\leq\mathrm{const}\max_{t\leq s\leq T}\|E_s\ofy\|\,T\\ &\leq\mathrm{const}\max_{(x,p)\in\cB\times\cP}\|\nabla^3 H(x,p)\|\,T, \end{split}$$ where the constant is independent of $T$. The set $\cP$ is a bounded subset of $\bR^n$ which contains the trajectory of $p\oft$, for $0\leq t\leq T$. Since the maximum above is finite, we conclude that $\|D_t\ofy\|\leq\mathrm{const}\,T$. Condition is a consequence of and the fact that we can choose $\cB$ sufficiently small so that $\|h\ofy\|$ is less than any given number. [99]{} Aguilar, C. O., and Kener, A. J.: Numerical solutions to the Bellman equation of optimal control, *J. Optimization Theory and Applications*, **160**, 527 - 552 (2014). Bender, C. M., and Orszag, S. A.: *Advanced Mathematical Methods for Scientists and Engineers*, Springer Verlag (1999). Coddington, E. 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B., Damle, A., and Burdick, J. W.: Linear Hamilton Jacobi Bellman Equations in high dimensions, arxiv:1404.1089v1 (2014). Kevorkian, J., and Cole, J. D.: *Perturbation Methods in Applied Mathematics*, Springer (1985). Kharroubi, I., Langrene, N., and Pham, H.: A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization, preprint (2013). Kushner, H. J., and Dupuis, P. G.: *Numerical Methods for Stochastic Control Problems in Continuous Time*, Springer (2001). Leimkuhler, B., and Reich, S.: *Simulating Hamiltonian Dynamics*, Cambridge University Press (2004). Merton, R. C.: Lifetime portfolio selection under uncertainty: the continuous-time case, *Rev. Econ. Stat.*, **51**, 247-257 (1969). Merton, R. C.: Optimum consumption and portfolio rules in a continuous-time model, *J. Econ. Theory*, **3**, 373 – 413 (1971). Pham, H.: *Continuous-time Stochastic Control and Optimization with Financial Applications*, Springer (2009). Todorov, E.: Finding the most likely trajectories of optimally-controlled stochastic systems, *World Congress of the International Federation of Automatic Control*, 20728 - 4734 (2011). Stoer, J., and Bulrisch, R.: *Introduction to Numerical Analysis*, Springer (2002). Touzi, N.: *Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE*, Pringer (2012). Yong, J., and Zhou, X. Y.: *Stochastic Controls*, Springer Verlag (1999). [^1]: The language of Riemannian geometry provides a natural, albeit somewhat technical, framework for WKB expansion of the HJB equation, and we intend to discuss it in a separate paper. [^2]: Alternatively, one can interpret it as a motion on $\cU$ in a magnetic field with potential $-a\ofx$ subject to the external potential $-\frac{1}{2}\,a\ofx^\tT\cC\ofx^{-1} a\ofx$.

--- abstract: 'Beyond traditional security methods, unmanned aerial vehicles (UAVs) have become an important surveillance tool used in security domains to collect the required annotated data. However, collecting annotated data from videos taken by UAVs efficiently, and using these data to build datasets that can be used for learning payoffs or adversary behaviors in game-theoretic approaches and security applications, is an under-explored research question. This paper presents [VIOLA]{}, a novel labeling application that includes (i) a workload distribution framework to efficiently gather human labels from videos in a secured manner; (ii) a software interface with features designed for labeling videos taken by UAVs in the domain of wildlife security. We also present the evolution of [VIOLA]{} and analyze how the changes made in the development process relate to the efficiency of labeling, including when seemingly obvious improvements did not lead to increased efficiency. [VIOLA]{} enables collecting massive amounts of data with detailed information from challenging security videos such as those collected aboard UAVs for wildlife security. [VIOLA]{} will lead to the development of new approaches that integrate deep learning for real-time detection and response.' author: - | Elizabeth Bondi, Debarun Kar, Venil Noronha, Donnabell Dmello, Milind Tambe\ \ \ Fei Fang\ \ \ - | Arvind Iyer, Robert Hannaford\ \ \ bibliography: - 'Gamesec2017.bib' title: Video Labeling for Automatic Video Surveillance in Security Domains ---

--- abstract: 'Motivated by our observation of fast echo decay and a surprising coherence freeze, we have developed a pump-probe spectroscopy technique for vibrational states of ultracold $^{85}$Rb atoms in an optical lattice to gain information about the memory dynamics of the system. We use pump-probe spectroscopy to monitor the time-dependent changes of frequencies experienced by atoms and to characterize the probability distribution of these frequency trajectories. We show that the inferred distribution, unlike a naive microscopic model of the lattice, correctly predicts the main features of the observed echo decay.' author: - Samansa Maneshi - Chao Zhuang - 'Christopher R. Paul' - 'Luciano S. Cruz' - 'Aephraim M. Steinberg' title: 'Coherence freeze in an optical lattice investigated via pump-probe spectroscopy' --- Characterizing decoherence mechanisms is a crucial task for experiments aiming to control quantum systems, e.g., for quantum information processing (QIP). In this work, we demonstrate how two-dimensional (2D) pump-probe spectroscopy may be extended to provide important information on these mechanisms. As a model system, we study quantum vibrational states of ultracold atoms in an optical lattice. In addition to being a leading candidate system for QIP [@BrennenJaksch], optical lattices are proving a versatile testing ground for the development of quantum measurement and control techniques [@OMandel; @Anderlini] and a powerful tool for quantum simulations, e.g. the study of Anderson localization and the Hubbard model [@MottAnderson]. In our experiment, we study the vibrational coherence of $^{85}$Rb atoms trapped in a shallow one-dimensional standing wave. Through our 2D pump-probe technique, we obtain detailed microscopic information on the frequency drift experienced by atoms in the lattice, enabling us to predict the evolution of coherence. Since the pioneering development of the technique in NMR[@Jeener-Ernst], 2D spectroscopy has been widely used to obtain high-resolution spectra and gain information about relaxations, couplings, and many-body interactions, in realms ranging from NMR [@Ernst] to molecular spectroscopy [@Mukamel-Jonas; @Hybl; @Brixner; @MillerNature] to semiconductor quantum wells [@Cundiff; @KWStone]. Here, we show that similar powerful techniques can be applied to the quantized center-of-mass motion of trapped atoms, and more generally, offer a new tool for the characterization of systems in QIP and quantum control. ![(Color online) Two typical measurements of echo amplitude vs. time. The echo pulse and the observed echo envelope are centered at times $t_p$ and $2t_p$, respectively. After an initial decay, echo amplitude stays constant for about $1ms$ forming a plateau, before decaying to zero. The average lattice depths are $20E_R$ (circles) and $18E_R$ (squares).[]{data-label="fig1"}](Fig1.eps) We have previously measured the evolution of coherence between the lowest two vibrational states of potential wells [@Ours]. The dephasing time is about $0.3ms$ ($T^{\star}_2$). This dephasing is partly due to an inhomogeneous distribution of lattice depths as a result of the transverse Gaussian profile of the laser beams. To measure the homogeneous decoherence time ($T_2$), we perform pulse echoes, measuring the echo amplitude as a function of time [@Ours]. Figure \[fig1\] shows two typical measurements of echo amplitude carried out on different dates under slightly different conditions such as different average lattice depths and different dephasing times. The echo amplitude initially decays with a time constant of about $0.7ms$, which is much faster than the photon scattering time ($\sim 60ms$) in the lattice. It then exhibits a $1ms$-long coherence freeze followed by a final decay. Absent real decoherence on the short time scale of $1ms$, only loss of frequency memory would inhibit the appearance of echoes. This loss comes about when atoms experience time-varying frequencies. We use 2D pump-probe spectroscopy to monitor this frequency drift. Our 2D pump-probe spectroscopy is essentially a version of spectral hole-burning for vibrational states. By monitoring the changes in the hole spectrum as a function of time we gain information on the atoms’ frequency drift. Information obtained from our 2D spectra enables us to characterize the temporal decay of frequency memory and through our simulations we find that “coherence freeze" is related to the shape of this memory loss function. Similar plateaus in echo decay and a two-stage decay of echo amplitude have been observed in a Cooper-pair box [@Nakamura], for a single electron spin in a quantum dot [@Vandersypen] and for electron spins in a semiconductor [@SClark]. Those plateaus or two-stage decays have been either explained through [*[a priori]{}*]{} models or simply described phenomenologically. Here, we are introducing an experimental technique to directly probe the origin of plateaus. The periodic potential in our experiment is formed by interfering two laser beams blue-detuned by $ 25GHz$ from the D2 transition line, $F=3 \shortrightarrow F^{\prime}=4$ ($\lambda=780nm$), thus trapping atoms in the regions of low intensity, which minimizes the photon scattering rate and the transverse forces. The two laser beams intersect with parallel linear polarizations at an angle of $\theta = (49.0 \pm 0.2)^{\circ}$, resulting in a spacing of $L=(0.930 \pm 0.004) \mu m$ between the wells. Due to gravity, the full effective potential also possesses a “tilt” of $2.86 E_R$ per lattice site, where $E_R=\frac{h^2}{8mL^2}$ is the effective lattice recoil energy. The photon scattering time in our experiment is $\approx 60ms$ and the Landau-Zenner tunneling times for transitions from the lowest two levels are greater than $ 160ms$. Atoms are loaded to the lattice during a molasses cooling stage and prepared in the ground vibrational state by adiabatic filtering [@StefanQPT]. Due to the short coherence length of atoms in optical molasses ($60 nm$ at $10 \mu K$), there is no coherence between the wells. We measure populations of atoms in the ground vibrational, the first excited, and the (lossy) higher excited states $P_1$, $P_2$, and $P_{L}$, respectively, by fluorescence imaging of the atomic cloud after adiabatic filtering [@StefanQPT]. The pump and probe pulses are sinusoidal phase modulations of one of the laser beams forming the lattice. The modulation is of the form $\phi(t)=A(t)[1-cos(\omega_m t)]$, where $A(t)$ is a square envelope function with amplitude $2\pi/72$ and $\omega_m$ is a variable frequency. The duration of each pulse is $8$ cycles, i.e., $T=8 (2\pi/\omega_m)$. This phase modulation shakes the lattice back and forth periodically, coupling vibrational states of opposite parity. To first-order in modulation amplitude, the phase-modulating part of the Hamiltonian has the same form as the electric dipole Hamiltonian. The inhomogeneous spectrum of vibrational excitations is measured in an average lattice depth of $24E_R$ by applying probe pulses at different frequencies and measuring state populations. Figure \[fig2\](a) shows state populations $P_1$, $P_2$, and $P_{L}$ (black circles) as a function of probe frequency. We then measure the pump-probe spectrum for a fixed delay. A pump pulse with a specific frequency is applied, exciting atoms in wells whose vibrational transition frequency matches the frequency of the pump pulse, therefore burning a hole in the spectrum of the ground state population. After a delay, probe pulses at different frequencies are applied, coupling the ground and excited states of atoms whose frequencies match those of the probe pulses. The red squares in Fig. \[fig2\](a) show populations $\Pi_1$, $\Pi_2$, and $\Pi_{L}$ as a function of probe frequency for a delay of $2ms$ and at a pump frequency of $6.45kHz$. The central part of $\Pi_1$ shows the hole burnt into the spectrum of the ground state population. To characterize the hole, we plot the difference between the pump-probe and the probe-alone spectra, $\Delta P_1= \Pi_1-P_1$, in Fig. \[fig2\](b). We monitor the frequency drift of atoms by changing the delay between the pump and probe pulses. Figure \[fig2\](c) shows the r.m.s. width of $\Delta P_1$ as a function of delay. The r.m.s. width increases with increasing delay until it approaches the inhomogeneous width of the lattice. For pump-probe delays shorter than $2ms$, the coherence present between the lowest two states results in Ramsey fringes making it impractical to extract useful spectra from the measurements. ![(Color online) Experiment:(a) black circles: probe-alone populations, $P_1, P_2,$ and $P_{L}$. Red squares: pump-probe populations, $\Pi_1, \Pi_2, \Pi_{L}$ for a pump at $6.45kHz$ and a delay of $2ms$. The circled region shows the hole-burning signal in $P_1$ and $\Pi_1$. (b) The difference spectrum $\Delta P_1 = \Pi_1 -P_1$. (c) Growth in the r.m.s. width of a Gaussian fit to $\Delta P_1$ as a function of delay. The bar centered at $(600 \pm 25)Hz$ is the inhomogeneous width determined from the $P_1$ spectrum. The green line centered at $(417 \pm 5)Hz$ indicates the instrumental width expected due to the spectra of our pulses.[]{data-label="fig2"}](Fig2.eps) Repeating the above procedure for different pump frequencies, we obtain a 2D spectrum of $\Delta P_1$ for a fixed delay as a function of $\omega_0$, the pump frequency at time $t_0$, and $\omega_n$, the probe frequency at $t_n$. We interpret the measured 2D spectrum as the conditional probability, $P(\omega_n|\omega_0)$, of an atom having frequency $\omega_n$ at $t_n$ given that at time $t_0$ its frequency was $\omega_0$. The joint probability is $P(\omega_n, \omega_0)=P(\omega_n|\omega_0) P(\omega_0)$, where $P(\omega_0)$ is the probe-alone spectrum. Figures \[fig3\](a-b) show the experimental joint probabilities $P(\omega_2, \omega_0)$ and $P(\omega_5, \omega_0)$, for delays of $2ms$ and $5ms$, respectively. By measuring joint probabilities for three different delays, $P(\omega_2, \omega_0)$, $P(\omega_3, \omega_0)$ , and $P(\omega_5, \omega_0)$, we construct a joint distribution of three frequencies at three different times in order to characterize how frequencies change as a function of time (frequency trajectories) in the first $5ms$. Details of our analysis are provided in the next two paragraphs. ![Contour plots: measured joint probabilities (a) $P(\omega_2, \omega_0)$, (b) $P(\omega_5, \omega_0)$; convolved marginals (c) $g(\omega_2, \omega_0)$, (d) $g(\omega_5, \omega_0)$ obtained by convolving bare marginals $M(\omega_n, \omega_0)$ with the power spectra of the pump and probe pulses; bare marginals (e) $M(\omega_2, \omega_0)$, (f) $M(\omega_5, \omega_0)$.[]{data-label="fig3"}](Fig3.eps) We assume a skew-normal distribution $\mathpzc{SN}(\boldsymbol{\omega})$ [@Azzalini], which is a generalization of the normal distribution. The multivariate skew-normal distribution is defined as $\mathpzc{SN}(\boldsymbol{\omega})=2 \psi(\boldsymbol{\omega}; \pmb{\Omega}) \Psi(\boldsymbol{\alpha}^T \boldsymbol{\omega})$, where $\psi$ is the three-dimensional normal density function, and $\Psi$ is the cumulative distribution function. $\boldsymbol{\omega}=(\boldsymbol{\tilde{\omega}}- \boldsymbol{\mu})$, where $\boldsymbol{\tilde{\omega}}=(\omega_0, \omega_2, \omega_5)$ denotes frequencies at three different times. $\boldsymbol{\mu}=(\mu_0, \mu_2, \mu_5)$ and $\boldsymbol{\alpha}=(\alpha_0, \alpha_2, \alpha_5)$ are the three frequency means and shape parameters, respectively. $\pmb{\Omega}$ is the covariance matrix whose diagonal terms are variances $\{\sigma_{i}^2\}$, and whose off-diagonal terms are covariances $\{\sigma_{i}\sigma_{j} \rho_{ij}\}$, where $\{\rho_{ij}\}$ are the mutual correlation coefficients. We estimate the best values for these parameters by using a genetic algorithm, as described below. The bivariate marginals of $\mathpzc{SN}(\boldsymbol{\omega})$, denoted as $M(\omega_n, \omega_m)$, are calculated by integrating $\mathpzc{SN}$ with respect to the third frequency, e.g., $M(\omega_2, \omega_0)=\int d\omega_5 \, \, \mathpzc{SN}(\boldsymbol{\omega})$. These marginals are then convolved with the power spectra of the pump and probe pulses to obtain the three bivariate convolved marginals $g(\omega_n, \omega_m)$ for comparison to the three measured spectra, $P(\omega_n, \omega_m)$. Similarly, the univariate marginals of $\mathpzc{SN}(\boldsymbol{\omega})$ are calculated and convolved with the power spectra of the probe pulses for comparison to the three measured probe-alone spectra, $P(\omega_n)$. Then, the sum of the squared residuals for all data points is calculated. The convolved marginal $g(\omega_5, \omega_2)$ is compared to the experimental $P(\omega_3, \omega_0)$ with the assumption that the frequency changes experienced by atoms in the $(2 \shortrightarrow 5)ms$ interval are the same as in the $(0 \shortrightarrow 3)ms$ interval. From repeated runs of the genetic algorithm function in MatLab, we find $12$ sets of best fit parameters, and take their average. Figures \[fig3\](c-f) show the two convolved marginals $g(\omega_2, \omega_0)$, $g(\omega_5, \omega_0)$ and the corresponding bare marginals $M(\omega_2, \omega_0)$, $M(\omega_5, \omega_0)$ thus obtained. The distribution parameters are means $\{\bar{\mu}_0, \bar{\mu}_2, \bar{\mu}_5\}=\{5.92(4), 5.92(3), 5.96(6)\} kHz$; r.m.s. widths $\{\bar{\sigma}_0, \bar{\sigma}_2, \bar{\sigma}_5\}=\{0.77(5), 0.82(4), 0.83(7)\} kHz$; mutual correlation coefficients $\{\bar{\rho}_{02}, \bar{\rho}_{25}, \bar{\rho}_{05}\}= \{0.88(2), 0.82(3), 0.80(4)\}$; and shape parameters $\{\bar{\alpha}_0, \bar{\alpha}_2, \bar{\alpha}_5\}= \{2.6(5), 3.2(4), 4.0(2.0)\}$. The numbers in parentheses are the standard deviations calculated from the $12$ optimization runs. The means and r.m.s. widths stay approximately constant. Furthermore, the mutual correlations $\bar{\rho}_{25}$ and $\bar{\rho}_{05}$ are essentially equal and within two standard deviations of $\bar{\rho}_{02}$. The increase in the anti-diagonal width of $M(\omega_5, \omega_0)$ compared to $M(\omega_2, \omega_0)$ is a signature of the slight decrease in the mutual correlation coefficient $\bar{\rho}_{05}$ compared to $\bar{\rho}_{02}$. ![(Color online) (a,b,c) (ccw from top left): Samples of smoothed frequency trajectories selected according to their probabilities. Trajectories are constrained to originate from the frequency range $\Delta \omega_0=0.5kHz$ and to pass through a window of width $\Delta \omega_2=0.52kHz$ at $t=2ms$. $\Delta \omega_0$ is centered at $6.5kHz$ and $\Delta \omega_2$ is centered at: (a) $6.5kHz$, (b) $7.07kHz$, and (c) $5.93kHz$. The sideways curves show the final probabilities $P_f(\omega_5)$. (d) Projection of $\mathpzc{SN}(\boldsymbol{\omega})$ onto the plane defined by frequency differences $( \omega_{2} - \omega_0)$ and $(\omega_{5} - \omega_2)$, showing a negative correlation between the two time intervals.[]{data-label="fig4"}](Fig4New.eps) The bare joint distribution $\mathpzc{SN}(\boldsymbol{\omega})$ is our estimate of the probability of frequency trajectories. Since $\mathpzc{SN}$ is a function of frequencies at three different times, frequency trajectories are constructed by interpolating between frequencies at these times for all possible combinations of $\omega_0$, $\omega_2$, and $\omega_5$. Figures \[fig4\](a-c) show three groups of smoothed [@pchip] trajectories that are constrained to originate from the frequency range $\Delta \omega_0$ and to pass through a window of width $\Delta \omega_2$ at $t=2ms$. Drawn for each group are small samples of trajectories selected according to their probability. The sideways curves are the final probabilities at $t=5ms$ calculated from $P_f(\omega_5) = \int_{\Delta \omega_0} d \omega_0 \int_{\Delta \omega_2} d\omega_2 \, \mathpzc{SN}(\boldsymbol{\omega})$. The most probable trajectories (Fig. \[fig4\](a)) stay within one standard deviation (width of the univariate skew-normal marginal at $t=0$) from the center of distribution. The next most probable trajectories (Fig. \[fig4\](b)) are the ones where frequency initially increases in the first $2ms$ followed by trajectories where frequency decreases in the first $2ms$ (Fig. \[fig4\](c)). Figure \[fig4\](d) is a projection of $\mathpzc{SN}(\boldsymbol{\omega})$ onto the plane defined by frequency differences, $(\omega_2- \omega_0)$ and $(\omega_5- \omega_2)$. Each quadrant corresponds to either an increase or decrease in the two time intervals, $(0 \shortrightarrow 2)ms$ and $(2 \shortrightarrow 5)ms$. The negative correlation between the two time intervals indicates that a rising frequency in the first time interval is usually followed by a falling frequency in the second interval and vice versa. This is in sharp contrast to the positive correlation we would obtain by considering the ballistic expansion of atoms through a spatial Gaussian distribution of well depths. In that case, there are no trajectories where an initial frequency decrease is followed by an increase. Although ballistic expansion would give rise to similar time scales for echo amplitude decay it does not predict the formation of plateaus. This shows the power of 2D spectroscopy in characterizing all frequency trajectories without reference to any [*a priori*]{} model. The echo amplitude can be predicted by calculating the net accumulated phase, $\varphi( \tau\, ;\, \boldsymbol{\tilde{\omega}})$, along each trajectory before and after the echo pulse, and by weighting each trajectory according to its probability $\mathpzc{SN}(\boldsymbol{\omega})$, $$\epsilon(\tau) = \left| \int d\boldsymbol{\tilde{\omega}} \, \, \mathpzc{SN}(\boldsymbol{\omega}) \, \, e^{-i\varphi(\tau \, ;\, \boldsymbol{\tilde{\omega}})} \right| \label{echoamp}$$ where $\varphi( \tau\, ;\, \boldsymbol{\tilde{\omega}})=\int_{t_1}^{\tau =2t_1} dt'' \omega(t'') -\int_0^{t_1} dt' \omega(t') $. As Fig. \[fig5\] shows, the decay time constant of echo amplitude predicted by the bare probability distribution $\mathpzc{SN}(\boldsymbol{\omega})$ is similar to the measured value. Depending on the exact method of interpolation (smooth or linear), the details of the echo decay vary but we always see a two-stage decay. We hypothesize that the non-decaying values of mutual correlation coefficients $\bar{\rho}_{25}$ and $\bar{\rho}_{05}$ lead to the formation of the plateau. By repeating simulations with reduced values of $\bar{\rho}_{25}$ and $\bar{\rho}_{05}$, we find that for values of $(\bar{\rho}_{25} - \bar{\rho}_{05} > 0.4)$ the plateau disappears. A more accurate calculation of echo decay would require knowledge of the joint probability distribution of multiple frequencies at multiple times on a finer time scale. The complexity of the task of obtaining data and extracting a joint distribution grows rapidly as the time scale is made finer. The extracted $\mathpzc{SN}(\boldsymbol{\omega})$ is our best estimate for a trivariate joint distribution. An exact reconstruction of the distribution requires knowledge of higher-order conditional probabilities such as $P(\omega_5 | \omega_2, \omega_0)$. In principle, it would be possible to extract this conditional probability from a two-pump, one-probe measurement. Simulations, however, show that this signal might be very difficult to extract in practice [@mythesis]. ![(Color online) A selection of echo curves, $\epsilon (\tau)$, generated by different $\mathpzc{SN}(\boldsymbol{\omega})$ resulting from repeated optimization trials. The inset shows the echo decay curve generated by $\mathpzc{SN}(\boldsymbol{\omega})$ calculated from the average values of parameters.[]{data-label="fig5"}](Fig5.eps) In conclusion, we have developed a two-dimensional pump-probe spectroscopy technique for vibrational states of ultracold atoms in optical lattices to gain information about the memory dynamics of the system without any need for [*[a priori]{}*]{} physical models. Our method is a direct frequency-domain characterization of the system for different delays between the pump and probe pulses. By measuring populations of the quantum vibrational states, we infer information about the underlying correlations, which in turn enables us to predict how coherence evolves in the system. By directly extracting a best estimate trivariate joint distribution $\mathpzc{SN}(\boldsymbol{\omega})$ from the measured 2D spectra at three different delays, and constructing trajectories accordingly, we have calculated echo amplitude as a function of time. The inferred distribution correctly predicts the main features of the observed echo decay and it suggests that persistence of frequency-frequency correlations leads to the formation of plateaus. The similarity between the predicted echo decay curve and the directly measured ones demonstrates the power of 2D spectroscopy in providing information on memory dynamics, information which could not always have been foreseen. 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--- author: - 'P. E. Shtykovskiy$^{1,2 \,*}$, M. R. Gilfanov$^{2,1}$' title: '**High Mass X-ray Binaries and Recent Star Formation History of the Small Magellanic Cloud**' --- [Astronomy Letters, Vol. 33, No. 7, 2007, pp. 437-454. Translated from Pis’ma v Astronomicheskii Zhurnal, Vol. 33, No. 7, 2007, pp. 492-512.]{} We study the relation between high-mass X-ray binary (HMXB) population and recent star formation history (SFH) for the Small Magellanic Cloud (SMC). Using archival optical SMC observations, we have approximated the color-magnitude diagrams of the stellar population by model stellar populations and, in this way, reconstructed the spatially resolved SFH of the galaxy over the past 100 Myr.We analyze the errors and stability of this method for determining the recent SFH and show that uncertainties in the models of massive stars at late evolutionary stages are the main factor that limits its accuracy. By combining the SFH with the spatial distribution of HMXBs obtained from XMM-Newton observations, we have derived the dependence of the HMXB number on the time elapsed since the star formation event. The number of young systems with ages 10 Myr is shown to be smaller than the prediction based on the type-II supernova rate. The HMXB number reaches its maximum $\sim$20–50 Myr after the star formation event. This may be attributable, at least partly, to a low luminosity threshold in the population of X-ray sources studied, Lmin$\sim10^{34}$ erg/s. Be/X systems make a dominant contribution to this population, while the contribution from HMXBs with black holes is relatively small. [**Key words:**]{} high mass X-ray binaries, Small Magellanic Cloud, star formation. [$^{*}$ E-mail: pavel@hea.iki.rssi.ru]{} INTRODUCTION {#introduction .unnumbered} ============ High-mass X-ray binaries (HMXBs) are close binary systems in which the compact object (a black hole or a neutron star) accretes matter from an early-type massive star. Because of the short lifetime of the donor star, they are closely related to recent star formation and, in the simplest picture, their number should be roughly proportional to the star formation rate of the host galaxy. Indeed, Chandra observations of nearby galaxies suggest that, to the first approximation, the HMXB luminosity function follows a universal power law whose normalization is proportional to the star formation rate (SFR) of the host galaxy (Grimm et al. 2003). On the other hand, obvious considerations based on the present view of the evolution of binary systems suggest that the relation between HMXB population and star formation should be more complex than a linear one. There is also experimental evidence for this. For example, previously (Shtykovskiy and Gilfanov 2005a), we showed that the linear relation between the number of HMXBs and the SFR cannot explain their spatial distribution over the Large Magellanic Cloud (LMC), because their number does not correlate with the H$_{\alpha}$ line intensity, a well-known SFR indicator. The largest number of HMXBs is observed in the region of moderate star formation LMC 4, while they are virtually absent in the most active star-forming region in the LMC, 30 Dor. Previously (Shtykovskiy and Gilfanov 2005a), we suggested that this discrepancy could arise from the dependence of the HMXB number on the time elapsed since the star formation event. Indeed, the age of the stellar population in 30 Dor is $\approx1-2$ Myr, which is not enough for the formation of compact objects even from the most massive stars and, accordingly, for the appearance of accreting X-ray sources. At the same time, the characteristic age of the stellar population in LMC 4, $\approx10-30$ Myr, is favorable for the formation of an abundant HMXB population. Thus, on the spatial scales corresponding to individual star clusters, the linear relation between the HMXB number and the instantaneous SFR does not hold and the recent star formation history (SFH) on time scales of the order of the lifetime of the HMXB population, i.e., $\sim2-100$ Myr, should be taken into account. Obviously, the number of active HMXBs at a certain time is determined by the total contribution from systems of different ages according to the dependences of the star formation history SFR(t) and a certain function $\eta_{HMXB}(t)$ describing the dependence of the HMXB number on the time elapsed since the star formation event. The universal relation N$_{HMXB}=A\times$SFR on the scales of galaxies results from the spatial averaging of $\eta_{HMXB}(t)$ over star-forming regions of different ages. The Small Magellanic Cloud (SMC) is an ideal laboratory that allows these and other aspects of HMXB formation and evolution to be studied. Indeed, owing to its appreciable SFR and small distance (60 kpc), there are dozens of known HMXBs in it. On the other hand, the SMC proximity makes it possible to study in detail its stellar population and, in particular, to reconstruct its SFH. Another peculiarity of the SMC, namely, its low metallicity, makes it potentially possible to study the effect of the heavy-element abundance on the properties of the HMXB population. In this paper, we use XMM-Newton observations of the SMC (Shtykovskiy and Gilfanov 2005b) and archival optical observations (Zaritsky et al. 2002) to analyze the relation between the number of HMXBs and the recent SFH of the galaxy. Our goal is to derive the dependence of the HMXB number on the time elapsed since the star formation event. EVOLUTION OF THE HMXB POPULATION AFTER THE STAR FORMATION EVENT {#sec:hmxbevol} =============================================================== To describe the evolution of the HMXB population, let us introduce a function $\eta_{HMXB}(t)$ that describes the dependence of the number of observed HMXBs with luminosities above a given value on the time t elapsed since the star formation event normalized to the mass of the formed massive stars: $$\begin{aligned} \eta_{HMXB}(t)=\frac{N_{HMXB}(t)}{M(>8M_{\odot})} \label{eq:etahmxbteor1}\end{aligned}$$ where M($>$8 M$_{\odot}$) is the mass of the stars more massive than 8 M$_{\odot}$ formed in the star formation event and N$_{HMXB}(t)$ is the number of HMXBs with luminosities exceeding a certain threshold. The luminosity of $10^{34}$ erg/s that corresponds to the sensitivity achieved by XMM-Newton in the SMC observations is taken as the latter. Obviously, the function $\eta_{HMXB}(t)$ is non-zero only in a limited time interval. Indeed, the first X-ray binaries appear only after the formation of the first black holes and/or neutron stars. The lifetimes of the stars that explode as type II supernovae (SNe II) to produce a compact object lie in the interval from $\approx2-3$ Myr for the most massive stars, $\approx100$ M$_{\odot}$, to $\approx40$ Myr for stars with a mass of $\approx$8 M$_{\odot}$, the least massive stars capable of producing a compact object. In this picture, it would be natural to expect the X-ray binaries in which the compact object is a black hole to appear first and the (probably more abundant) population of accreting neutrons stars to appear next. On the other hand, the HMXB lifetime is limited by the lifetime of the companion star. Since the least massive companion stars observed during an active X-ray phase have a mass of $\approx6 M_{\odot}$, this lifetime is $\sim60$ Myr for a single star when the peculiarities of the stellar evolution in binary systems are disregarded. Given the mass transfer from the more massive star to the future donor star, this lifetime can be slightly modified. This also includes the X-ray source stage proper with characteristic time scales much shorter than those considered above, $\sim10^3-10^6$ yr, depending on the type of the companion star and the binary parameters. Obviously, the function $\eta_{HMXB}(t)$ must be closely related to the rate of SNe II $\eta_{SNII}(t)$ producing a compact object. To the first approximation, the relation may be assumed to be linear: $$\begin{aligned} \eta_{HMXB}(t)= A\cdot\eta_{SNII}(t) \label{eq:etahmxbteor0}\end{aligned}$$ The supernova rate can be easily determined from the stellar mass–lifetime relation (Schaller et al. 1992) and the initial mass function (IMF), which below is assumed to be a Salpeter one in the range 0.1–100M$_{\odot}$. Note that the IMF shape in the range of low masses is unimportant for us, since all of the relations are eventually normalized to the mass of massive stars with M$>$8 M$_{\odot}$. The normalization in Eq. (2) can be calculated using the N$_{HMXB}$–SFR calibration from Grimm et al. (2003). This relation was derived from Chandra observations of nearby galaxies and corresponds to the time integral of the function $\eta(t)$): $$\int \eta_{HMXB}(t) dt=\frac{N_{HMXB}(L_X>L_{X,min})}{SFR} \label{eq:etahmxbteornorm}$$ As the limits of integration in Eq. (3), we choose 2 and 40 Myr in accordance with the above reasoning. In what follows, we will compare the experimental dependence $\eta_{HMXB}(t)$ obtained from X-ray and optical SMC observations with predictions of the simple model specified by Eqs. (2) and (3). Clearly, Eq. (2) is based on the assumption that the X-ray phase comes immediately after the formation of a compact object, i.e., it disregards the evolution of the companion star in the binary system. A more rigorous description of the HMXB evolution requires resorting to population synthesis models (see, e.g., Popov and Prokhorov 2004; Belczynski et al. 2005), which is outside the scope of this paper. On the other hand, the experimental dependence $\eta_{HMXB}(t)$, whose derivation is the goal of this paper, can be used by the creators and users of population synthesis models to test and calibrate these models. Experimental Determination of the Function $\eta_{HMXB}(t)$ ----------------------------------------------------------- The number of HMXBs observed in a spatial region X at time t is a convolution of the function $\eta_{HMXB}(t)$ with the star formation history SFR(t,X) in this region: $$N_{HMXB}(t,X)=\int SFR(t-\tau,X)\eta_{HMXB}(\tau) d\tau. \label{eq:etahmxbobs}$$ Solving the inverse problem formulated by this equation, we can impose constraints on the dependence $\eta_{HMXB}(t)$ from observations. This requires the following: 1. Identifying the HMXB population in the galaxy. 2. Reconstructing the spatially-resolved star formation history SFR(t,X). Obviously, we need only the recent SFH from the current time to the time in the past corresponding to the maximum HMXB lifetime (i.e., $\sim 50-100$ Myr). 3. Solving the inverse problem formulated by Eq. (4) given $N_{HMXB}(X)$ and SFR(t,X) for a large set of regions X. Obviously, a galaxy with a rich HMXB population and a SFH that changes significantly from place to place is required to perform this procedure. Because of its proximity and appreciable SFR, the SMC is one of the most natural candidates for such a galaxy. This paper is structured as follows. We describe the SFH reconstruction technique and apply it to the SMC, solve the inverse problem given by Eq. (4), and find the function $\eta_{HMXB}(t)$ for HMXBs in the SMC. Next, we discuss the results obtained and summarize our conclusions. THE STAR FORMATION HISTORY IN THE SMALL MAGELLANIC CLOUD {#sec:sfh} ======================================================== To reconstruct the SFH, we will use a method based on the analysis of color-magnitude diagrams (see, e.g., Gallart et al. 2005). This method uses the fact that stars of different ages (and metallicities) occupy different positions in the color-magnitude diagram. The SFH can be determined by comparing the distributions of stars in it with predictions of stellar evolution models. Applying this method requires optical photometry at least in two bands. There are several realizations of this method; one of the most commonly used realizations was described by Dolphin (1997), Aparicio et al. (1997), and Dolphin (2002) and consists of the following steps: 1. Generating synthetic color-magnitude diagrams in the required ranges of metallicities and ages on the basis of stellar evolution models. Each diagram is the probability distribution in color-magnitude space for a coeval model stellar population. 2. Correcting the synthetic diagrams for the incompleteness and photometric errors. Allowance for the interstellar extinction and the distance to the galaxy. 3. Approximating the observed color-magnitude diagrams by a linear combination of the derived synthetic models. Estimating the uncertainties of the solution. Because of their proximity, the Magellanic Clouds are attractive objects for star formation studies. It is not surprising that a number of papers are devoted to the SFH in them (see, e.g., Holtzman et al. 1999; Dolphin 2000). In particular, note the paper by Harris and Zaritsky (2004), who reconstructed the spatially resolved SFH of the SMC. However, in all of these studies, the star formation was considered in a wide range of ages, with the emphasis being inevitably on time scales of $\sim$ Gyr. In contrast, we are interested in the SFH for the youngest stellar population. As will be shown below, its reconstruction has several peculiarities that have escaped attention previously. Therefore, we adapted the SFH reconstruction method to meet the requirements of our problem by concentrating on the time interval 0–100 Myr. Synthetic Color-Magnitude Diagrams {#sec:synthcmd} ---------------------------------- The first step in generating synthetic color-magnitude diagrams is to choose the model isochrones that define the region occupied by a coeval stellar population. In what follows, we use the isochrones from Girardi et al. (2002) (the so-called “Padova isochrones”) covering wide ranges of ages (log t = 6.60–10.25), metallicities (Z = 0.0001–0.03), and masses (0.15–70 M$_{\odot}$). All model calculations are performed for the color-magnitude diagrams in (U–B, B) and (B–V , V) spaces. The theoretical isochrones relate the mass of a star of a certain age to its position in the diagram. Therefore, the probability of filling some region in it can be easily determined from the corresponding mass interval M$_i$–M$_{i+1}$ and the IMF, which below is assumed to be a Salpeter one: $$p(M_{i},M_{i+1})=\frac{M_{i+1}^{-\Gamma}-M_{i}^{-\Gamma}}{M^{-\Gamma}_{max}-M^{-\Gamma}_{min}}, \label{eq:occprob}$$ where $\Gamma=1.35$, $M$ is the initial mass of the star, $M_{min}$=0.1 $M_{\odot}$, and M$_{max}$=100 $M_{\odot}$. Note that the IMF deviations from the Salpeter one in the range of low masses affect only the normalization of the derived SFH rather than its shape. This is because we analyze the color-magnitude diagrams only for a relatively massive stellar population. The SFH sensitivity to the IMF deviations from the Salpeter one in the range of high masses is discussed in the Section “Checking the SFH Reconstruction Procedure.” Equation (5) allows us to calculate the probabilities of filling various regions in the color-magnitude diagram that are needed to fit the observations by a model. This is convenient to do using model photometry generated by the Monte Carlo method. The total number of model stars must be large enough to minimize the contribution from Poisson noise. In our case, the number of stars is $>10^5$ per isochrone (which corresponds to 10$^8$ stars in the mass range 0.1–100 M$_{\odot}$). However, before generating model photometry, we must make several more steps, including the choice of an age range, an age step, ametallicity range, a binary fraction and isochrone interpolation. These steps are considered below. First, we found that the isochrones need to be interpolated. Indeed, the magnitude difference at adjacent points can be 0.5. Therefore, we perform a linear interpolation of the magnitudes in such a way that the magnitude step does not exceed 0.01. In choosing an age interval and its binning, we will keep in mind that we are interested only in the recent star formation. This allows us to exclude the old population from our analysis and, thus, to avoid problems related to the incompleteness of the optical catalog at faint magnitudes (see the Subsection “Binning of Color-Magnitude Diagrams”). Below, we reconstruct the SFH in the time interval log t = 6.6–8.0. We also include the isochrones in the time interval log t = 8.0–8.6 in our model to avoid the distortion of the solution at log t$\leq$8.0 due to the older population. Initially, the time step in the isochrones is $\Delta\log(t)=0.05$. The simple tests show that this resolution is excessive in terms of the photometry used (see the Subsection “Optical Photometry”). Therefore, we combine the isochrones into groups, each with 3–4 isochrones, thereby obtaining the time step $\Delta\log(t)=0.2$. The binary fraction is also important in generating model stellar populations, since the binary stars in the color-magnitude diagram will appear as single stars with distorted photometry. As the binary fraction, we use the standard value of $f_{binary}=0.5$. Following Harris and Zaritsky (2004), we will assume that the mass of the companion star is taken from an independent Salpeter IMF. The influence of these assumptions on the derived SFH is discussed in the Section “Checking the SFH Reconstruction Procedure.” ### Metallicity. {#sec:metallicity} The heavy-element abundance is an important parameter in the evolution of a star. The positions of stars with different metallicities in the color-magnitude diagram will differ almost at all evolutionary stages. For example, since an increase in metallicity is accompanied by an increase in opacity, it causes the main sequence to be displaced toward the less bright and cooler stars. However, metallicity plays the most important role at the final stages of stellar evolution. For instance, the position of a star in the color-magnitude diagram for (super)giants depends critically on the heavy-element abundance. This can give rise, for example, to partial degeneracy between age and metallicity for red supergiants. Therefore, choosing the isochrones with the proper metallicity (or metallicity range) is very important for reconstructing the SFH. The metal abundance in the Magellanic Clouds is known to be low. For example, the metallicity of the interstellar medium in the SMC is 0.6 dex lower than that of the local medium in our Galaxy (Russell and Dopita 1992). Note also that the SMC metallicity has gradually increased with time due to continuous star formation. Therefore, a self-consistent description of the SFH should take into account the spread in metallicity. Several attempts have been made to describe quantitatively the heavy-element enrichment history of the SMC. The typical metallicities lie in the range from $[Fe/H]\approx-1.25$ (Z$\approx0.001$) for the old population to $[Fe/H]\approx-0.5$ (Z$\approx0.006$) for the young population (see, e.g., Pagel and Tautvaisiene 1998). Since we are interested in the recent star formation in the SMC, a component relatively rich in heavy elements is expected to dominate among the stellar population used in the calculations. However, a spread in metallicity exists even for the young population (see, e.g., Harris and Zaritsky 2004; Maeder et al. 1999; and references therein). Therefore, to choose the metallicity suitable for the spatial regions used to reconstruct the SFH, we visually compare the observed color-magnitude diagrams with the model isochrones. The effect of the heavy-element abundance is most pronounced for the red supergiant branch. Note that the isochrones in the region of red supergiants in the range $Z\sim 0.004-0.008$ under consideration do not intersect, i.e., there is no degeneracy between metallicity and age. We found that the locations of the red supergiant branches in most regions are satisfactorily described by the Z = 0.004 isochrones. However, in one region, the (B–V, V) diagram is described better by Z = 0.008, while Z = 0.004 is more suitable for the (U–B, B) diagram. Below, we use Z = 0.004 everywhere, except for this region where Z = 0.008 is used. We also analyze the dependence of our results on the chosen metallicity (see below). ### Interstellar extinction and distance. {#sec:synthcmd1} The derived synthetic diagrams should also be corrected for the interstellar extinction and the SMC distance. As the distance modulus for the SMC, we take m–M = 18.9 (Westerlund 1997), corresponding to a distance of D$\approx$60 kpc. Zaritsky et al. (2002) showed that the extinction for the stellar population in the SMC changes from region to region and differs for hot and cool stars and obtained the distributions of extinction for different regions (http://ngala.as.arizona.edu/dennis/smcext.html). We correct the synthetic photometry using these distributions just as was done by Harris and Zaritsky (2004). For the young stars ($logt < 7.0$), we take the distribution of extinction corresponding to hot stars. For the older population, the distribution of extinction for hot stars is used only for the population fraction $f=1-0.5\cdot(\log(t)-7)$, while the fraction 1–f of stars exhibit extinction corresponding to cool stars. Finally, all of the stars older than 1 Gyr have the distribution of extinction corresponding to cool stars. ### Photometric errors and completeness. {#sec:photoerror} The synthetic color-magnitude diagrams should take into account the photometric errors and the incompleteness of the optical catalog at low fluxes. The most important source of errors is the telescope’s limited resolution. Clearly, the resolution-related photometry distortions depend on the spatial density of stars and will be at a maximum where this density is high. Artificial star tests – reconstructing the photometry of model stars placed on real images through the standard procedures used in compiling a real catalog – are a standard method of solving this problem. The subsequent comparison of the reconstructed photometry with the model one allows the distortions produced by this factor to be estimated as a function of the spatial density of stars. Obviously, this requires input optical data. In addition, there are factors whose contribution is much more difficult to estimate quantitatively (e.g., the systematic uncertainties in the calibration). Analysis of the star catalog used shows that these are actually present (see the Subsection “Optical Photometry”). If the photometric errors are moderately large, then the problem of photometric errors can be solved by choosing a special binning of the color-magnitude diagram, more specifically, using a grid with wider color and magnitude intervals than the characteristic photometry distortions. Since we are interested only in the recent SFH, we can also exclude faint stars for which the problem of photometric errors is more serious. Excluding faint stars also solves the problem with the incompleteness of the catalog. On the other hand, it is clear that we cannot make the cells in the color-magnitude diagram too large, because this can give rise to additional degeneracy in the solution. Since there were no input optical data for the SMC at our disposal, we chose the second path – optimizing the binning of the color-magnitude diagram (for more detail, see the Subsection “Binning of the Color-Magnitude Diagram”). SFH reconstruction. {#sec:sfhfit} ------------------- Using the synthetic photometry obtained, we can approximate the observed distribution of stars in the color-magnitude diagram ($n_i$) by linear combinations of model stellar populations ($A_{i,j}$ ): $$n_i=\sum_j A_{i,j}\times x_j, \label{eq:cmdapprox}$$ where i is the cell number in the diagram and j is the time interval number. The amplitudes $x_j$ minimizing the discrepancy $\Vert Ax-n\Vert$ are the sought-for SFH. Since the problem in question is ill-conditioned, we use an iterative Lucy-Richardson method (Lucy 1974) for its solution. Using the initial approximation to the solution, this procedure calculates a vector that approaches the maximum likelihood solution with increasing number of iterations. The solution after iteration i is regularized in the sense that the method retains non-negativity of the initial solution and that it is smoother than themaximum likelihood solution. An important feature of the method is the choice of a stopping criterion (Lucy 1994) – the number of iterations giving an optimum solution. Obviously, the stopping criterion is determined by the character of the problem. For example, at low noise in the input data, the maximum likelihood solution is close to the true one, while in the reverse situation with large errors, fitting the data with a high accuracy is equivalent to attempting to describe the noise. Below, we define a stopping criterion suitable for our problem by reconstructing the SFH of a model stellar population and studying the behavior of the likelihood function L depending on the number of iterations (see below): $$L=\sum_i (\mu-N_i\cdot \ln\mu). \label{eq:statistics}$$ {width="100.00000%"} ### Optical photometry. {#sec:optcat} As the stellar population photometry necessary to reconstruct the SFH, we used the Magellanic Clouds Photometric Survey (MCPS) catalog for the SMC (Zaritsky et al. 2002). To reconstruct the SFH, we use the (U–B, B) and (B–V , V) diagrams. The catalog also presents I-band photometry, i.e., the additional (V–I, I) diagram could be used. However, we found that the I magnitude is often absent (I = 0) for bright stars, with the most significant loss of photometry being observed among the red supergiants. Since the latter play an important role in reconstructing the recent SFH, we decided to exclude the (V–I, I) diagram from our analysis. Note also the problem with the U photometry of the catalog. As described in Zaritsky et al. (2004), Zaritsky et al. (2002) corrected the U–B color using the photometry from Massey (2002) calibrated (in the initial version) from faint dwarfs. In addition, Zaritsky et al. (2002) replaced part of the photometry for bright stars with the photometry from Massey (2002). As a result, the U–B color for blue supergiants may be unreliable. This is clearly seen in the (U–B, B) diagram as the displacement of the blue supergiant sequence by $\sim0^m.3$ relative to the model (see, however, the “Section Checking the SFH Reconstruction Procedure” for a discussion of the reliability of stellar evolution models for supergiants). Our binning of the color-magnitude diagram into large cells allows the effect of this kind of uncertainties to be minimized. Therefore, we expect this problem to be not critical in our procedure. Another problem described by Harris and Zaritsky (2004), more specifically, the need for displacing the B-V color by $0^m.1-0^m.2$ in some regions, will not affect strongly our results for the same reason. For test purposes, we also used the OGLE catalog (Udalski et al. 1998) containing B, V, I photometry, but covering only part of the SMC. {width="100.00000%"} ### Contribution from foreground stars. {#sec:foreground} Obviously, the catalog of stars toward the SMC also contains Galactic stars that can introduce distortions into the color-magnitude diagrams. To estimate their contribution, we constructed the color-magnitude diagrams for 10 outermost MCPS fields (each with an area of 12$\arcmin\times$12$\arcmin$) where the contribution from SMC stars is at a minimum (Fig. 1). When comparing the densities of stars in Figs. 1 and 2, we should take into account the fact that the total area of the fields used to construct the diagram for Galactic foreground stars shown in Fig. 1 is approximately twice the area of the sky used to construct the diagram in Fig. 2 (equal to the area of the XMM-Newton field of view). Obviously, the contribution from Galactic foreground stars is negligible in most of the color-magnitude diagram. The only region where Galactic stars can introduce noticeable distortions is the blue supergiant branch in the (U–B, B) diagram in the range of colors near U–B = 0. However, as we see from Fig. 1, they are easily identified in the (B–V, V) diagram, since they are separated from both blue and red supergiants in it. This forms the basis for our foreground star rejection algorithm. We determined the region in the (B–V, V) diagram that, on the one hand, includes most of the Galactic foreground stars superimposed on the SMC blue supergiant branch in the (U–B, B) diagram and, on the other hand, the contribution from SMC stars to it is negligible. This region is highlighted by the dashed line in Figs. 1 and 2. All of the stars lying in this region (they are marked by triangles in both diagrams in Fig. 1) are then excluded from the analysis in both (B–V, V) and (U–B, B) diagrams. ### Binning of the color-magnitude diagrams. {#sec:cmdgrid} To compare the model color-magnitude diagrams with the observations, we must specify their binning. There are two approaches to this problem – uniform and more complex grids. Whereas the former is more objective, the latter makes it possible to avoid problems related to the photometric errors and uncertainties in the stellar evolution. In any case, the choice of a grid must take into account the existence of extended structures corresponding to long stages of stellar evolution in the diagram. The main sequence and blue and red supergiants are most important in determining the recent SFH. Since the core hydrogen burning is the longest phase of stellar evolution, the number of stars on the main sequence is at its maximum and the latter plays a major role in determining the SFH. In principle, the SFH can be reconstructed based only on the main sequence, without invoking other stages of stellar evolution (see, e.g., Dohm-Palmer et al. 1997). However, this method places heavy demands on the photometric accuracy, because the blue supergiants are close to the upper part of the main sequence (Figs. 2 and 3). {width="100.00000%"} The supergiant branches are important regions in the color-magnitude diagram and complement the main sequence when reconstructing the SFH. However, whereas the evolution of a main-sequence star has been studied well, the evolution of supergiants is more uncertain. This is because the supergiants are very sensitive to such aspects of the model as mass loss, convection, etc. Uncertainties in the latter can strongly affect the manifestation of supergiants in the color-magnitude diagram. The best known outstanding problem here is the blue-to-red supergiant ratio (B/R). As was shown by Langer and Maeder (1995), there are no stellar evolution models that are capable of explaining self-consistently the dependence of B/R on metallicity in a wide range of the latter (see also Gallart et al. 2005). The region that we use to compare the observations with the model consists of two strips: one covers the main sequence and the blue supergiant branch and the other covers the region of red supergiants (see Fig. 3). The width of each strip is taken to be much larger than the scatter in photometry, which also allows the effect of uncertainties in the stellar evolution models to be reduced (for more detail, see the Section “Checking the SFH Reconstruction Procedure”). At the same time, it is small enough for the contribution from Galactic foreground stars to be at a minimum (Fig. 1). Since each strip has only one color interval, using this grid is equivalent to simultaneously fitting two luminosity functions. The scatter in magnitude is less important than the scatter in color, because all of the features in the color-magnitude diagram are elongated along the magnitude axis. Its effect is equivalent to convolving the SFH with the function defined by the distribution of photometric errors. We take dm = 0.25 as the width of the magnitude interval. The magnitude threshold for the main sequence was chosen to be V$_{lim}$=18.25 and B$_{lim}$=18.25. This allowed us to avoid problems with the incompleteness of the catalog and with the photometry distortion through the superposition of stars. Indeed, for the MCPS catalog, the completeness is large for $\sim20^m$ stars and the magnitude errors (including the star superposition effect) for bright stars are smaller than the chosen width of the color and magnitude intervals (Zaritsky et al. 2002). We used a higher threshold for red supergiants to avoid the contribution from the old low-metallicity stellar population. ### Uncertainty of the solution. {#sec:sfhfit3} To estimate the statistical uncertainty in the reconstructed SFH, we analyzed the stability of our solution to Poisson noise in the number of stars by the bootstrap method. We calculated the expected number of stars in each cell in the color-magnitude diagram from our solution. Next, we drew their realization by assuming a Poisson distribution for the number of stars, which was then used as input data in the SFH reconstruction code. This procedure was repeated many times and the rms scatter of the solutions obtained was taken as the error. Checking the SFH Reconstruction Procedure. {#sec:sfhval} ------------------------------------------ To check the SFH reconstruction procedure, we performed a number of tests. First of all, to check the general functioning of the algorithm and its implementation, we reconstructed the SFHs for various model stellar populations. Subsequently, we investigated the adequacy of the stellar evolution models and the accuracy with which the observed color-magnitude diagrams are approximated. Finally, we analyzed the stability of the solution to photometric errors and its sensitivity to various model parameters, such as the metallicity, the binary fraction, and the IMF slope. For the first test, we chose a model stellar population whose SFH consists of several bursts alternating with periods of quiescence. The number of stars in the model population was close to that observed in the SMC within the XMM-Newton field of view. This test allows us to check the SFH reconstruction procedure, the degeneracy between adjacent time intervals, and to analyze the dependence of the solution on the stopping criterion in the Lucy-Richardson method. The results are presented in Fig. 4, which shows the behavior of the likelihood function depending on the number of iterations and the model and reconstructed SFHs. We compare two solutions – one long before the saturation of the likelihood function (200 iterations) and the other close to its saturation (1000 iterations). Obviously, the latter corresponds much better to the model. The model SFH used in the first test is implausibly complicated. As a more realistic example, we chose the actual SFH obtained for one of the SMC fields and used the model stellar population corresponding to it as input data in our code. As we see from Fig. 4, the model SFH is smoother in this case. As in the previous case, the best solution is achieved close to the saturation of the likelihood function. Based on the results of these and other tests, we concluded that the best solution is achieved near the saturation of the likelihood function. In other words, the problem has such a character (the number of stars etc.) that the solution obtained requires no (or almost no) significant regularization. To verify the adequacy of the model isochrones, we analyzed how well the model describes the color-magnitude diagram for the actual stellar population. Figure 5 presents the model and observed luminosity functions for themain sequence with blue supergiants and for red supergiants summed over all of the SMC fields used in this paper. These luminosity functions correspond to the two regions shown in Fig. 2. We see that the model agrees well with the observations for faint magnitudes, but in the region of bright stars ($B,V\la 13.5$) the model prediction for the main sequence and blue supergiants exceeds appreciably the observations. As our tests showed, this excess is related to blue supergiants – the two clearly seen features in the model luminosity function at V$\approx$13 and $\approx11.5$, which are much less pronounced in the data, are unequivocally identified with them. This is clear from an examination of the color-magnitude diagram for the model population in the (U–B, B) diagram in Fig. 3. The problem with the excess for the brightest stars can be partly removed if the metallicity is assumed to be Z = 0.008 for all fields. In this case, however, the locations of the supergiant branches in the color-magnitude diagram will be in poorer agreement with the data. On the other hand, the luminosity function for red supergiants is described well by the model. Obviously, the discrepancy between the data and the model results from uncertainties in modeling the supergiants, whichmanifest themselves as the problem of the blue-to-red supergiant ratio mentioned above. To estimate how strongly this affects the reconstructed SFH, we analyzed the sensitivity of the solution to the choice of stellar evolution models and metallicity, more specifically, we reconstructed the SFH using the Padova isochrones with Z = 0.004 and Z = 0.008 and the Geneva isochrones (Charbonnel et al. 1993) with Z = 0.004. The latter use the same convection criterion as the Padova ones. However, as was pointed out by Langer and Maeder (1995), they give different predictions for the occurrence frequency of supergiants. A visual comparison of the two model populations showed that the locations of the supergiant branches in the color-magnitude diagram predicted by the Geneva isochrones differ significantly from those predicted by the Padova isochrones. As we see from Fig. 6a, the solutions obtained with these two models also differ from one another. Since the observed diagrams are described by the Padova isochrones much better, the solution obtained with the latter is probably more realistic and below we take it as the main one. A similar situation is also observed for the solutions obtained with the same (Padova) isochrones, but with different metallicities – they are statistically incompatible with one another, although the differences are appreciably smaller than than in the case of different stellar evolution models (Fig. 6b). The general tendency in the behavior of the solution is retained, because the main constraints on the SFH are imposed by the distribution of stars along the main sequence, on which the stellar evolution is modeled much better than on the supergiant branches. As a result of the model inadequacy, the solution also slightly depends on the choice of the region under consideration in the color-magnitude diagram. This is illustrated by Fig. 6b, which shows the solution on a grid with a more stringent magnitude threshold for supergiants. As we see from Fig. 6, the solutions differ, but less than in the previous cases. {width="80.00000%"} As has already been noted above, we could reconstruct the SFH using only main-sequence stars, thereby avoiding the supergiant-related problems. However, the accuracy of the photometry available at our disposal is insufficient for the latter to be reliably separated from the main-sequence stars. Thus, imperfectness of the models for massive stars on which the present stellar evolution models are based limits the reconstruction accuracy of the recent star formation history. Only the general behavior of the SFH has a reasonable accuracy, while the individual features in it should be interpreted with caution. To minimize the effect of such uncertainties, below we coarsen the grid in time by combining two time bins into one. As a result, four of them remain in the interval log t = 6.6–8.0 instead of eight. As we see from Fig. 7, although this does not solve all of the problems considered above, it allows the uncertainties in the solution related to them to be reduced appreciably. As will be clear in the subsequent analysis, a higher time resolution is not required for the problem under consideration, because the accuracy of determining the sough-for function $\eta_{HMXB}(t)$ is limited by the Poisson noise associated with the relatively small number of HMXBs in the SMC. It is interesting to compare the SFH that we obtained with that from Harris and Zaritsky (2004), whose method differs significantly from ours. It should be kept in mind that Harris and Zaritsky (2004) investigated the SFH in a wide range of ages and did not concentrate on the features related to the reconstruction of recent star formation. The two SFHs are shown in Fig. 7a. We see that they are in satisfactory agreement at t$\ga20$ Myr and differ on shorter time scales. The largest discrepancy is observed in the second time bin corresponding to $\log(t)\approx 7.0-7.3$. For a quantitative comparison of the accuracies of the two SFHs, let us consider the number of red supergiants formed in this time bin predicted by these two dependences. The stars formed in the bin $\log(t)\approx 7.0-7.3$ that have become red supergiants by now had initial masses in the range $\approx12-22M_\odot$. Their current positions in the color-magnitude diagram are roughly limited by the intervals of magnitudes V = 12.0–13.5 and colors B–V=1.4–1.8. The SFH obtained in this paper predicts 57 stars in these magnitude and color intervals, while according to Harris and Zaritsky (2004), their number must be 15. The numbers of stars are shown for the set of all fields for which the SFH was obtained in Fig. 7. As would be expected, the predictions differ by almost a factor of 4. We emphasize that both solutions are based on the same stellar evolution models and identical assumptions about the IMF, the binary fraction, and the stellar mass distribution in binary systems. These numbers should be compared with the observed number of red supergiants, 50. Obviously, the solution obtained in this paper describes better the population of massive young stars. Since the mass of the stars formed in this time bin is low, this difference affects weakly the end result, as we demonstrate below. To verify that the solution is only weakly sensitive to photometric errors, we performed two tests. In the first test, we introduced noise into the actual photometry by shifting the magnitudes by random values distributed uniformly in the interval from -0.2 to +0.2. Subsequently, we reconstructed the SFH using the original and distorted photometries. As is clear from Fig. 8, the solution depends weakly even on such large errors. As the second test, we compared the SFHs for the actual stellar population obtained using two different catalogs, OGLE and MCPS. Since the OGLE catalog provides photometry only in the B, V, I bands, we use only the (B-V, V) diagram. Obviously, this procedure is equivalent to reconstructing the SFH for one stellar population with the errors taken from different distributions. The derived SFHs are in good agreement with one another (Fig. 8). {width="80.00000%"} In constructing the synthetic color2013magnitude diagrams, we assumed a Salpeter IMF. Since we use only the upper part of the color-magnitude diagram, only the behavior of the mass function for massive stars is important to us. Although the stellar mass distribution in (massive) star clusters is known to follow the Salpeter mass function up to the highest masses, the mass function of the field stars may be steeper (Massey 2003). To check how strongly the solution depends on the presumed IMF slope, we reconstructed the SFH in one of the SMC fields by assuming a steeper slope, $\Gamma=1.7$. The derived SFH does not differ greatly from the solution obtained with the standard value of $\Gamma=1.35$, except that its normalization is a factor of 2 higher (see Fig. 9). Formally, this difference stems from the fact that we assume the same IMF slope in the entire mass range $0.1-100M_\odot$. The end result of our calculations will be virtually unchanged, since it was normalized to the total mass of the massive stars $M>8~M_{\odot}$, while the difference of the normalization is attributable to the low-mass stars. In generating the synthetic diagrams, another significant assumption is made with regard to the fraction of binary systems and their distribution in mass ratio. The binary fraction can exceed f$_{binary}$=0.5 adopted here as the standard one, while the distribution in binary component mass ratio is nearly flat (see, e.g., Kobulnicky et al. 2006). To analyze the dependence of the solution on these assumptions, we reconstructed the SFH for one of the fields by assuming that the distribution in component mass ratio is flat and that all stars are in binaries (f$_{binary}$=1). In both cases, the solution is found to be close to that obtained with standard parameters (see Fig. 9; the solution with a different binary fraction is not provided, since it is almost identical to the standard one). However, the conversion coefficient from the number of stars to the stellar mass depends on these parameters. Therefore, the normalization of the derived SFH may differ. Thus, for example, the SFH normalization for f$_{binary}$=1 increases by a factor of $\approx1.3$. Results: The SFH in the SMC {#sec:sfhresults} --------------------------- Based on the results of previous sections, we reconstructed the SFH in the SMC. This was done separately for each of the regions observed with XMM-Newton and used in Shtykovskiy and Gilfanov (2005b) to search for HMXBs. The pointing at CF Tuc, which is displaced considerably from the SMC center and contains no HMXBs, constitutes an exception. The combined SFH for these regions is shown in Fig. 10. EVOLUTION OF THE HMXB POPULATION AFTER THE STAR FORMATION EVENT {#sec:etahmxbevol} =============================================================== Having the spatially resolved star formation history SFR(t,X), let us turn to the solution of Eq. (4). To construct the function $N_{HMXB}(t,X)$, we used our catalog of HMXBs in the SMC (Shtykovskiy and Gilfanov 2005b), from which we selected HMXBs brighter than $10^{34}$ erg/s. This threshold corresponds to the detection of $\approx$75% of the sources (see Shtykovskiy and Gilfanov 2005b). To take into account the incompleteness of the catalog, we will divide our solution $\eta_{HMXB}(t)$ by 0.75. The spatial variable X in Eq. (4) is basically the index numbering the XMM-Newton fields of view – Eq. (4) is written for each pointing.After discretization, we obtained a system of eight linear equation for four unknowns. The number of unknowns is determined by the number of time bins in the interval log t = 6.6–8.0. As above, we used the iterative Lucy–Richardson method to obtain a regularized solution. To find the stopping criterion, we solved the problem based on the model function $\eta(t)$. Using the SFHs SFR(t,X) and the function $\eta(t)$ based on the SN II rate, we calculated the expected numbers of HMXBs in eight spatial regions in the SMC and their Poisson realizations. The total number of model sources is close to the actual number of HMXBs in the SMC. The reconstructed dependence $\eta(t)$ is shown in Fig. 11 for two stopping criteria; one is close to the plateau in the likelihood function (100 iterations) and the other is long before it (20 iterations). As we see from the figure, the latter is in better agreement with the model and, in addition, has smaller errors. This means that the problem is ill-posed and its solution requires regularization. RESULTS AND DISCUSSION {#sec:etahmxbevolfin} ====================== The derived dependence of the HMXB number on the time elapsed since the star formation event, $\eta_{HMXB}(t)$, is shown in Fig. 12a. The uncertainty of the solution was calculated in the same way as above in the Subsection “Uncertainty of the Solution”. To take into account the incompleteness of the catalog of HMXBs in the SMC, we multiplied the normalization of the solution obtained by the factor 1.3. The theoretical curve in Fig. 12a corresponds to the model based on the SN II rate and normalized using the N$_{HMXB}$–SFR calibration (Grimm et al. 2003), as described in the Section “Evolution of the HMXB Population after the Star Formation Event”. The solutions shown in Fig. 12a were obtained both for the SFH determined in this paper and for the SFH from Harris and Zaritsky (2004). We see that the solutions are compatible, despite a certain difference between the two SFHs in the lower time bins (Fig. 7) – as was mentioned above, the accuracy of the solution is limited by the Poisson noise due to the relatively small number of HMXBs in the SMC. As is clear from Fig. 12a, the HMXB formation efficiency does not exceed the prediction based on the mean N$_{HMXB}$–SFR relation for the local Universe. The abundance of HMXBs in the SMC is the result of a specific form of the recent SFH in this galaxy, namely, its high rate $\sim50$ Myr ago. The specific form of $\eta_{HMXB}(t)$ differs significantly from the behavior of the SN II rate: the HMXB number reaches its maximum 20–50 Myr after the star formation event, i.e., on time scales of the order of or longer than the explosion time of the last supernova with the formation of a neutron star. Note also the paucity of the youngest systems compared to the model predictions. Obviously, most of the young systems correspond to HMXBs with black holes, since they are the first to be formed after the star formation event. This shortage is not unexpected from an observational point of view, since most of the HMXBs in the SMC are known to be pulsars with Be companions. However, it is of great interest from the standpoint of the theory of formation and evolution of binary systems. Obviously, this behavior is related to the evolution of a companion star whose lifetime can reach $\sim 60$ Myr for a single $6M_{\odot}$ star (when the evolution effects in the binary system are disregarded). Another important factor is the evolution of the neutron star spin period (Illarionov and Sunyaev 1975). Population synthesis models are an adequate tool for studying these effects. As an example, Fig. 12a shows the time dependence of the number of Be/X systems with neutron stars derived by Popov et al. (1998) based on calculations using the “Scenario Machine”. The systems of other classes (e.g., neutron stars with supergiants) are much less numerous, given the luminosity threshold of 10$^{33}$ erg/s chosen by the authors. Therefore, we provide no curves for them. To be able to compare the absolute number of Xray sources with the results of our observations, we renormalized the theoretical dependence to the number of systems brighter than $10^{34}$ erg/s. For this purpose, we used the luminosity function for HMXBs in the SMC obtained previously (Shtykovskiy and Gilfanov 2005b). Note that its slope in the range of low luminosities is slightly smaller than the standard value of 0.6 (Grimm et al. 2003). As we see from Fig. 12a, there is good agreement with the observations both in the shape of the dependence and in its normalization in the time interval 5–20 Myr in which the models by Popov et al. (1998) are valid. Note that Popov et al. (1998) performed their calculations by assuming a solar heavy-element abundance, while the details of the population of X-ray sources depend on metallicity (Dray 2006). Obviously, our experimental dependence can be used to test and “calibrate” the population synthesis models and to clarify the various aspects of the evolution of binary systems. Figure 12b shows the age distribution of HMXBs in the SMC, which is the product of the reconstructed dependence $\eta_{HMXB}(t)$ by the mass of the stars formed in the corresponding time bins. As is clear from Fig. 12b, the HMXB population in the SMC is rather old, $\tau\approx20-50$ Myr. Dray (2006) also reached a similar conclusion by analyzing the observed distributions of HMXB periods and luminosities and by comparing them with the results of population synthesis models. She also suggested the existence of a relatively recent intense star formation event in the SMC. When the results shown in Fig. 12 are interpreted, it should be kept in mind that we used X-ray sources with luminosities L$_X\geq10^{34}$erg/s, i.e., faint sources dominate in our sample, to reconstruct the time dependence of the HMXB number. It would be interesting to look at the behavior of the function $\eta_{HMXB}(t)$ for bright sources, e.g., L$_X\ga10^{37}$erg/s. Indeed, although the luminosity of a specific binary depends on the size of its orbit, one may expect its mean X-ray luminosity to rise with increasing mass of the companion star. Bright X-ray binaries will then be, on average, younger than faint ones due to the shorter lifetime of more massive stars. This conclusion is also supported by the observational fact that brighter sources in star-forming galaxies are, on average, closer to young star clusters (see, e.g., Kaaret al. 2004). Therefore, the time dependence of the number of bright sources will differ from that shown in Fig. 12. However, such a study cannot be performed for the SMC because of its insufficiently high SFR and, accordingly, small number of bright sources. Note also that the N$_{HMXB}$–SFR and L$_X$–SFR relations from Grimm et al. (2003) are based on Chandra observations of bright HMXBs in other galaxies. Therefore, one might expect these relations to break down for a lower luminosity threshold. This effect will be unimportant for the L$_X$–SFR relation, since the total X-ray luminosity of the HMXB population is determined mainly by bright sources in view of the shape of their luminosity function. However, the total number of sources is determined by the more numerous faint sources. Therefore, one might expect noticeable deviations from a linear relation in the N$_{HMXB}$–SFR relation when the luminosity threshold is lowered. CONCLUSIONS {#sec:discussion} =========== We considered the relation between the HMXB population and the SFH of the host galaxy. The number of HMXBs can be represented as a convolution (Eq. (4)) of the star formation history SFR(t) with the function $\eta_{HMXB}(t)$ describing the dependence of the HMXB number on the time elapsed since the star formation event. Thus, the evolution of the HMXB population after the star formation event can be reconstructed by analyzing the distribution of HMXBs in stellar complexes with different SFHs. Using archival optical observations, we reconstructed the spatially resolved SFH in the SMC over the past $\sim$100 Myr (Fig. 10). For this purpose, the observed color-magnitude diagrams of the stellar population were approximated by linear combinations of model isochrones. We analyzed the stability and errors of this method for reconstructing the recent SFH and showed that its accuracy is limited by the uncertainties in the currently available models for the evolution of massive stars. However, the systematic error introduced by this factor may be ignored, since the main source of uncertainty in the solution is the Poisson noise due to the relatively small number of HMXBs in the part of the SMC investigated by XMM-Newton. Using the derived SFHs and the spatial distribution of HMXBs in the SMC from Shtykovskiy and Gilfanov (2005b), we reconstructed the function $\eta_{HMXB}(t)$ that describes the dependence of the HMXB number on the time elapsed since the star formation event (Fig. 12). We compared the derived dependence with the behavior of the SN II rate. The HMXB number reaches its maximum $\sim$20–50 Myr after the star formation event, which is comparable to or exceeds the lifetime of a $8M_\odot$ star. This is much later than the maximum of the SN II rate. In addition, note the shortage of the youngest systems. Observationally, this manifests itself in the absence (or an extremely small number) of HMXBs with black holes in the SMC. This behavior is related to the evolution of the companion star and the neutron star spin period and is consistent with the population synthesis model calculations (Popov et al. 1998). When these results are interpreted, it should be kept in mind that the function $\eta_{HMXB}(t)$ depends on the luminosity threshold used to select the X-ray sources. In our analysis, we used a sample with a low luminosity threshold, L$_{min}\sim 10^{34}$ erg/s. In such a sample, low-luminosity sources, mostly Be/X systems, mainly contribute to the number of sources, while the relative contribution from systems with black holes and/or O/B supergiants, which must constitute the majority of sources in the lower time bin in Fig. 12, is small. Therefore, the time dependence of the number of bright sources (e.g., $>10^{37}$ erg/s) will differ from that shown in Fig. 12. The HMXB formation efficiency in the SMC does not exceed the prediction of the N$_{HMXB}$–SFR calibration (Grimm et al. 2003). 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--- abstract: 'We present measurements of the large-scale cosmic-ray anisotropies in right ascension, using data collected by the surface detector array of the Pierre Auger Observatory over more than 14 years. We determine the equatorial dipole component, $\vec{d}_\perp$, through a Fourier analysis in right ascension that includes weights for each event so as to account for the main detector-induced systematic effects. For the energies at which the trigger efficiency of the array is small, the “East-West” method is employed. Besides using the data from the array with detectors separated by 1500 m, we also include data from the smaller but denser sub-array of detectors with 750 m separation, which allows us to extend the analysis down to $\sim 0.03$ EeV. The most significant equatorial dipole amplitude obtained is that in the cumulative bin above 8 EeV, $d_\perp=6.0^{+1.0}_{-0.9}$%, which is inconsistent with isotropy at the 6$\sigma$ level. In the bins below 8 EeV, we obtain 99% CL upper-bounds on $d_\perp$ at the level of 1 to 3 percent. At energies below 1 EeV, even though the amplitudes are not significant, the phases determined in most of the bins are not far from the right ascension of the Galactic center, at $\alpha_{\rm GC}=-94^\circ$, suggesting a predominantly Galactic origin for anisotropies at these energies. The reconstructed dipole phases in the energy bins above 4 EeV point instead to right ascensions that are almost opposite to the Galactic center one, indicative of an extragalactic cosmic ray origin.' author: - 'A. Aab, P. Abreu, M. Aglietta, I.F.M. Albuquerque, J.M. Albury, I. Allekotte, A. Almela, J. Alvarez Castillo, J. Alvarez-Muñiz, G.A. Anastasi, L. Anchordoqui, B. Andrada, S. Andringa, C. Aramo, P.R. Araújo Ferreira, H. Asorey, P. Assis, G. Avila, A.M. Badescu, A. Bakalova, A. Balaceanu, F. Barbato, R.J. Barreira Luz, K.H. Becker, J.A. Bellido, C. Berat, M.E. Bertaina, X. Bertou, P.L. Biermann, T. Bister, J. Biteau, A. Blanco, J. Blazek, C. Bleve, M. Boháčová, D. Boncioli, C. Bonifazi, L. Bonneau Arbeletche, N. Borodai, A.M. Botti, J. Brack, T. Bretz, F.L. Briechle, P. Buchholz, A. Bueno, S. Buitink, M. Buscemi, K.S. Caballero-Mora, L. Caccianiga, L. Calcagni, A. Cancio, F. Canfora, I. Caracas, J.M. Carceller, R. Caruso, A. Castellina, F. Catalani, G. Cataldi, L. Cazon, M. Cerda, J.A. Chinellato, K. Choi, J. Chudoba, L. Chytka, R.W. Clay, A.C. Cobos Cerutti, R. Colalillo, A. Coleman, M.R. Coluccia, R. Conceição, A. Condorelli, G. Consolati, F. Contreras, F. Convenga, C.E. Covault, S. Dasso, K. Daumiller, B.R. Dawson, J.A. Day, R.M. de Almeida, J. de Jesús, S.J. de Jong, G. De Mauro, J.R.T. de Mello Neto, I. De Mitri, J. de Oliveira, D. de Oliveira Franco, V. de Souza, J. Debatin, M. del Río, O. Deligny, N. Dhital, A. Di Matteo, M.L. Díaz Castro, C. Dobrigkeit, J.C. D’Olivo, Q. Dorosti, R.C. dos Anjos, M.T. Dova, J. Ebr, R. Engel, I. Epicoco, M. Erdmann, C.O. Escobar, A. Etchegoyen, H. Falcke, J. Farmer, G. Farrar, A.C. Fauth, N. Fazzini, F. Feldbusch, F. Fenu, B. Fick, J.M. Figueira, A. Filipčič, M.M. Freire, T. Fujii, A. Fuster, C. Galea, C. Galelli, B. García, A.L. Garcia Vegas, H. Gemmeke, F. Gesualdi, A. Gherghel-Lascu, P.L. Ghia, U. Giaccari, M. Giammarchi, M. Giller, J. Glombitza, F. Gobbi, G. Golup, M. Gómez Berisso, P.F. Gómez Vitale, J.P. Gongora, N. González, I. Goos, D. Góra, A. Gorgi, M. Gottowik, T.D. Grubb, F. Guarino, G.P. Guedes, E. Guido, S. Hahn, R. Halliday, M.R. Hampel, P. Hansen, D. Harari, V.M. Harvey, A. Haungs, T. Hebbeker, D. Heck, G.C. Hill, C. Hojvat, J.R. Hörandel, P. Horvath, M. Hrabovský, T. Huege, J. Hulsman, A. Insolia, P.G. Isar, J.A. Johnsen, J. Jurysek, A. Kääpä, K.H. Kampert, B. Keilhauer, J. Kemp, H.O. Klages, M. Kleifges, J. Kleinfeller, M. Köpke, G. Kukec Mezek, A. Kuotb Awad, B.L. Lago, D. LaHurd, R.G. Lang, M.A. Leigui de Oliveira, V. Lenok, A. Letessier-Selvon, I. Lhenry-Yvon, D. Lo Presti, L. Lopes, R. López, A. López Casado, R. Lorek, Q. Luce, A. Lucero, A. Machado Payeras, M. Malacari, G. Mancarella, D. Mandat, B.C. Manning, J. Manshanden, P. Mantsch, A.G. Mariazzi, I.C. Mariş, G. Marsella, D. Martello, H. Martinez, O. Martínez Bravo, M. Mastrodicasa, H.J. Mathes, J. Matthews, G. Matthiae, E. Mayotte, P.O. Mazur, G. Medina-Tanco, D. Melo, A. Menshikov, K.-D. Merenda, S. Michal, M.I. Micheletti, L. Miramonti, D. Mockler, S. Mollerach, F. Montanet, C. Morello, G. Morlino, M. Mostafá, A.L. Müller, M.A. Muller, S. Müller, R. Mussa, M. Muzio, W.M. Namasaka, L. Nellen, M. Niculescu-Oglinzanu, M. Niechciol, D. Nitz, D. Nosek, V. Novotny, L. Nožka, A Nucita, L.A. Núñez, M. Palatka, J. Pallotta, M.P. Panetta, P. Papenbreer, G. Parente, A. Parra, M. Pech, F. Pedreira, J. Pȩkala, R. Pelayo, J. Peña-Rodriguez, L.A.S. Pereira, J. Perez Armand, M. Perlin, L. Perrone, C. Peters, S. Petrera, T. Pierog, M. Pimenta, V. Pirronello, M. Platino, B. Pont, M. Pothast, P. Privitera, M. Prouza, A. Puyleart, S. Querchfeld, J. Rautenberg, D. Ravignani, M. Reininghaus, J. Ridky, F. Riehn, M. Risse, P. Ristori, V. Rizi, W. Rodrigues de Carvalho, J. Rodriguez Rojo, M.J. Roncoroni, M. Roth, E. Roulet, A.C. Rovero, P. Ruehl, S.J. Saffi, A. Saftoiu, F. Salamida, H. Salazar, G. Salina, J.D. Sanabria Gomez, F. Sánchez, E.M. Santos, E. Santos, F. Sarazin, R. Sarmento, C. Sarmiento-Cano, R. Sato, P. Savina, C. Schäfer, V. Scherini, H. Schieler, M. Schimassek, M. Schimp, F. Schlüter, D. Schmidt, O. Scholten, P. Schovánek, F.G. Schröder, S. Schröder, S.J. Sciutto, M. Scornavacche, R.C. Shellard, G. Sigl, G. Silli, O. Sima, R. Šmída, P. Sommers, J.F. Soriano, J. Souchard, R. Squartini, M. Stadelmaier, D. Stanca, S. Stanič, J. Stasielak, P. Stassi, A. Streich, M. Suárez-Durán, T. Sudholz, T. Suomijärvi, A.D. Supanitsky, J. Šupík, Z. Szadkowski, A. Taboada, O.A. Taborda, A. Tapia, C. Timmermans, P. Tobiska, C.J. Todero Peixoto, B. Tomé, G. Torralba Elipe, A. Travaini, P. Travnicek, C. Trimarelli, M. Trini, M. Tueros, R. Ulrich, M. Unger, M. Urban, L. Vaclavek, J.F. Valdés Galicia, I. Valiño, L. Valore, A. van Vliet, E. Varela, B. Vargas Cárdenas, A. Vásquez-Ramírez, D. Veberič, C. Ventura, I.D. Vergara Quispe, V. Verzi, J. Vicha, L. Villaseñor, J. Vink, S. Vorobiov, H. Wahlberg, A.A. Watson, M. Weber, A. Weindl, L. Wiencke, H. Wilczyński, T. Winchen, M. Wirtz, D. Wittkowski, B. Wundheiler, A. Yushkov, E. Zas, D. Zavrtanik, M. Zavrtanik, L. Zehrer, A. Zepeda, M. Ziolkowski, F. Zuccarello' title: | Cosmic-ray anisotropies in right ascension\ measured by the Pierre Auger Observatory --- Introduction ============ The distribution of cosmic-ray (CR) arrival directions is expected to provide essential clues to understanding the CR origin. Being charged particles, they are significantly deflected by the magnetic fields present in our galaxy [@hav15] and, for those arriving from outside it, also by the extragalactic magnetic fields [@fere]. Since the deflections get smaller for increasing rigidities, it is only at the highest energies that one may hope to observe localized flux excesses associated with individual CR sources. On the other hand, as the energies lower and the deflections become large, the propagation eventually becomes diffusive and it is likely that only large-scale patterns, such as a dipolar flux modulation, may be detectable. However, the small amplitudes of these anisotropies make their observation quite challenging. Due to the Earth’s rotation, cosmic-ray observatories running for long periods of time have an almost uniform exposure in right ascension. This enables them to achieve a high sensitivity to the modulation of the flux in this angular coordinate. In particular, for a dipolar cosmic-ray flux the first-harmonic modulation in right ascension provides a direct measurement of the projection of the dipole in the equatorial plane, $\vec{d}_\perp$. The possible sources of systematic uncertainties that could affect these measurements, such as those from remaining non-uniformities of the exposure or those related to the effects of atmospheric variations, can often be accounted for. Even when this is not possible, as can happen when the trigger efficiency of the array is small, methods that are insensitive to these systematic effects can be adopted to reconstruct $\vec{d}_\perp$, although they have a somewhat reduced sensitivity to the modulations. On the other hand, at low energies the number of events detected is large, what tends to enhance the statistical sensitivity of the measurements. The projection of the dipole along the Earth rotation axis $d_z$ can, in principle, be reconstructed from the study of the azimuthal modulation of the CR fluxes. This requires accounting in detail for the effects of the geomagnetic field on the air showers, which can affect the reconstruction of the CR energies in an azimuthally dependent way. Also, the presence of a tilt of the array can induce a spurious contribution to $d_z$. When the trigger efficiency of the array is small, these effects may lead to systematic uncertainties that cannot be totally corrected for, particularly given the azimuthal dependence of the trigger efficiency arising from the actual geometry of the surface detector array of the Pierre Auger Observatory. Due to these limitations, we will here restrict our analysis to the determination of $\vec{d}_\perp$ through the study of the distribution in right ascension of the events recorded in different energy bins. We note that the determination of $d_z$ for energies $E\geq4$ EeV, for which that detector has full efficiency for zenith angles up to 80$^\circ$, was discussed in detail in @lsa2015 [@science; @uhedip]. At $E\geq8$ EeV, a significant first-harmonic modulation in right ascension, corresponding to an amplitude $d_\perp \sim 6$%, has been detected by the Pierre Auger Observatory [@science]. The reconstructed direction of the three-dimensional dipole suggests a predominant extragalactic origin of the CR anisotropies at energies above 4 EeV, and the dipolar amplitudes obtained in different bins show a growing trend with increasing energies [@science; @uhedip]. The phase in right ascension of the dipolar modulation of the flux determined above 8 EeV is $\alpha_d\simeq 100^\circ$. This is nearly opposite to the phases measured at PeV energies by IceCube and IceTop [@ic12; @ic16], which lie not far from the Galactic center direction which is at $\alpha_{\rm GC} = -94^\circ$. Also the KASCADE-Grande measurements, involving CR energies from few PeV up to few tens of PeV, lead to phases lying close to the right ascension of the Galactic center, even though the measured amplitudes are not statistically significant [@KG].[^1] All this is in agreement with the expectation that for energies above that of the knee of the CR spectrum, which corresponds to the steepening taking place at $\sim 4$ PeV, the outward diffusive escape of the CRs produced in the Galaxy should give rise to a dipolar flux component having its maximum not far from the Galactic center direction. Also at energies above few EeV, where the propagation would become more rectilinear, a continuous distribution of Galactic sources should give rise to a dipolar component not far from the GC direction [@uhedip]. Departures from these behaviors could however result if the CR source distribution is not symmetric with respect to the Galactic center (such as in the presence of a powerful nearby CR source), in the presence of drift motions caused by the regular Galactic magnetic field components [@ptus], or when the contribution from the extragalactic component becomes sizable. Note that the expected direction of a dipole of extragalactic origin will depend on the (unknown) distribution of the CR sources and on the effects of the deflections caused by the Galactic magnetic field, as was discussed in detail in @uhedip. The change from a Galactic CR origin towards a predominantly extragalactic origin is expected to take place somewhere above the knee. More precise measurements of the large-scale anisotropies, filling the gap between the IceCube/IceTop or KASCADE-Grande measurements and the dipole determined by the Pierre Auger Observatory above 8 EeV, should provide information about this transition. In fact, although at energies below 8 EeV the reported dipolar amplitudes are not significant, indications that a change in the phase of the anisotropies in right ascension takes place around few EeV are apparent in the Pierre Auger Observatory measurements [@App11; @LSA2012; @LSA2013; @ICRC13; @ICRC15]. One has to keep in mind in this discussion that the energy at which the total anisotropy becomes of predominantly extragalactic origin may be different from the energy at which the CR flux becomes of predominantly extragalactic origin, since the intrinsic anisotropies of each component are likely different. We present here an update of the measurements of the large-scale anisotropies that are sensitive to the equatorial component of a dipole, for the whole energy range from $\sim 0.03$ EeV up to $\geq32$ EeV, covering more than three decades of energy. The results above 4 EeV are an update of those presented in @uhedip, including two more years of data, corresponding to an increase in the exposure by 20%. At lower energies, we provide a major update of the latest published results [@ICRC15], with 50% more exposure for the SD1500 array and twice as much for the SD750 array. At energies below 2 EeV, possible systematic effects related to the reduced trigger efficiency could be significant. To study the modulation in right ascension in this regime we have then to resort to the “East-West” method, which has larger associated uncertainties but is not affected by most of the systematic effects [@na89; @ew]. At energies below 0.25 EeV, it proves convenient to use the data from the sub-array of detectors with 750 m spacing which, although being much smaller, can detect a larger number of events at these energies. The Observatory and the dataset =============================== The Pierre Auger Observatory [@NIM2015], located near the city of Malargüe in western Argentina (at latitude $35.2^\circ$ South), is the largest existing CR observatory. Its surface detector array (SD) consists of water-Cherenkov detectors having each one 12 tonnes of ultra-pure water viewed by three 9 inch phototubes. The main array, SD1500, consists of detectors distributed on a triangular grid with separations of 1,500 m that span an area of 3,000 km$^2$. A smaller sub-array, SD750, covers an area of 23 km$^2$ with detectors separated by 750 m, making it sensitive also to smaller CR energies. These arrays sample the secondary particles of the air showers reaching ground level. In addition, the fluorescence detector (FD) consists of 27 telescopes that overlook the SD array. The FD can determine the longitudinal development of the air showers by observing the UV light emitted by atmospheric nitrogen molecules excited by the passage of the charged particles of the shower. This fluorescence light can be detected during clear moonless nights, with a corresponding duty cycle of about 15% [@NIM2015]. The SD arrays have instead a continuous operation, detecting events with a duty cycle close to 100%. They also have a more uniform (and simpler to evaluate) exposure. This is why the studies of the large-scale anisotropies that we perform here are based on the much larger number of events recorded by the surface arrays. For the SD1500 array, the dataset considered in this work includes events with energies above 0.25 EeV that were detected from 2004 January 1 up to 2018 August 31. For energies below 4 EeV, it includes events with zenith angles up to $60^\circ$, allowing coverage of 71% of the sky, and the quality trigger applied requires that all the six detectors surrounding the one with the largest signal be active at the time the event is detected. For energies above 4 EeV, more inclined events can be reliably reconstructed [@inclined] and hence the zenith-angle range is extended up to $80^\circ$, allowing coverage of 85% of the sky. Moreover, given that at these energies the number of detectors triggered by each shower is large (4 or more detectors for more than 99% of the events), we also include in this case events passing a relaxed trigger condition, allowing that one of the six detectors that are neighbors to the one with the largest signal be missing or not functioning, provided that the reconstructed shower core be contained inside a triangle of nearby active detectors [@science]. The integrated exposure of the array for $\theta \le 60^\circ$ and using the strict trigger selection is 60,700 km$^2$sryr, while that for $\theta \le 80^\circ$ and relaxing the trigger is 92,500km$^2$sryr. The CR arrival directions are reconstructed from the timing of the signals in the different triggered stations, and the angular resolution is better than 1.6$^\circ$ [@NIM2015], so that it has negligible impact on the reconstruction of the dipole. The energies of the events with $\theta\leq 60^\circ$ are assigned in terms of the reconstructed signals at a reference distance from the shower core of 1000 m. They are corrected for atmospheric effects, accounting for the pressure and air density variations following @jinst17, as well as for geomagnetic effects, following @geo. The inclined events, whose signals are dominantly produced by the muonic component of the showers, have a negligible dependence on atmospheric variations, while geomagnetic effects are already taken into account in their reconstruction [@inclined]. Their energies are assigned in terms of the estimated muon number at ground level. The SD1500 array has full trigger efficiency for $E\geq 2.5$ EeV if one considers events with $\theta\leq 60^\circ$, and for $E\geq 4$ EeV for events with $\theta\leq 80^\circ$. The energies of the CRs are calibrated using the hybrid events measured simultaneously by the SD and FD detectors, in the regimes of full trigger efficiency. For lower energies, in which case we consider events with $\theta\leq 60^\circ$, the energy assignment is performed using the extrapolation of the corresponding calibration curve. The energy resolution for events with $\theta\leq 60^\circ$ is about 7% above 10 EeV, and degrades for lower energies, reaching about 20% at 1 EeV, while the systematic uncertainty in the energy scale is 14% (see @spectrum for details). The more inclined events have an energy resolution of 19%, with a similar systematic uncertainty [@inclined]. For energies below 0.25 EeV, and down to $\sim 0.03$ EeV (below which the trigger efficiency is tiny), we use the events from the denser and smaller SD750 array, since the accumulated statistics is larger. The dataset comprises events with zenith angles up to $55^\circ$ detected from 2012 January 1 up to 2018 August 31. The trigger applied requires that all six detectors around the one with the largest signal be functioning and the associated exposure is 234 km$^2$sryr. The energies are assigned in terms of the reconstructed signals at a reference distance from the shower core of 450 m. They are corrected for atmospheric effects following @jinst17. The SD750 array has full trigger efficiency for $E\geq 0.3$ EeV if one considers events with $\theta\leq 55^\circ$ [@NIM2015]. The energies are calibrated with hybrid events observed in the regime of full trigger efficiency and below that threshold the energy assignment is performed on the basis of the extrapolation of the corresponding calibration curve. At 0.3 EeV the energy resolution is about 18% [@coleman]. The analysis method =================== The weighted first-harmonic analysis in the right ascension angle $\alpha$, often referred to as Rayleigh analysis, provides the Fourier coefficients as $$\label{eq:fcoef} a=\frac{2}{\mathcal{N}}\sum_{i=1}^N w_i\cos \alpha_i ,~~~~~~ b=\frac{2}{\mathcal{N}}\sum_{i=1}^N w_i\sin \alpha_i,$$ where the sums run over all $N$ detected events. The weights $w_i$, which are of order unity, account for the effects of the non-uniformities in the exposure as a function of time, with the normalization factor being ${\mathcal{N}}\equiv \sum_i w_i$. The amplitude and phase of the first-harmonic modulation are given by $r=\sqrt{a^2+b^2}$ and $\varphi=\arctan(b/a)$. The probability to obtain an amplitude larger than the one measured as a result of a fluctuation from an isotropic distribution is $P(\ge r)=\exp(-\mathcal{N}r^2/4)$. To obtain the weights, we permanently monitor the number of active unitary detector cells, corresponding to the number of active detectors that are surrounded by an hexagon of working detectors or, when considering the relaxed trigger condition above 4 EeV, we also account for detector configurations with only five active detectors around the central one. We obtain from this the exposure of the Auger Observatory in bins of right ascension of the zenith of the array, $\alpha^0$. This angle is given by $\alpha^0(t_i)\equiv 2\pi t_i/T_s ~ (\mathrm{mod} ~ 2\pi)$, with the origin of time being taken such that $\alpha^0(0)=0$. The sidereal-time period, $T_s\simeq 23.934$ h, corresponds to one extra cycle per year with respect to the solar frequency. The fraction of the total exposure that is associated to a given $\alpha^0$ bin, taken to be of 1.25$^\circ$ width (5 minutes), is proportional to the total number of unitary cells in that bin, $N_{\mathrm{cell}}(\alpha^0)$. The weights $w_i$ account for the relative variations of $N_{\mathrm{cell}}$ as a function of $\alpha^0$, i.e. $$w_i =\left(\frac{N_{\mathrm{cell}}(\alpha^0(t_i))}{\langle N_{\mathrm{cell}}\rangle}\right)^{-1},$$ with $\langle N_{\mathrm{cell}}\rangle=1/(2\pi)\int_0^{2\pi} \mathrm{d}\alpha^0~ N_{\mathrm{cell}}(\alpha^0)$. Including these weights in the Fourier coefficients eliminates the spurious contribution to the amplitudes associated to the non-uniform exposure in right ascension. We note that if one were to consider periods of only a few months, the resulting modulation of $N_{\mathrm{cell}}(\alpha^0)$ could amount to an effect of a few percent on the modulation in right ascension of the event rates. However, after considering several years, the modulations that appear on shorter time scales tend to get averaged out, with the surviving effects being now typically at the level of about $\pm 0.5$%. The effects of the tilt of the SD array [@LSA2012], which is inclined on average by $\sim 0.2^\circ$ towards $\phi\simeq-30^\circ$ (i.e. towards the South-East), can also be accounted for by adding an extra factor in the weights [@uhedip]. However, this is actually only relevant when performing the Fourier analysis in the azimuth variable $\phi$, something we will not perform here. When the triggering of the array is not fully efficient, there are additional systematic effects related to the interplay between the atmospheric effects in the air-shower development and the energy-dependent trigger efficiency. In particular, changes in the air density modify the Molière radius determining the lateral spread of the electromagnetic component of the showers. The fall-off of the signal at ground level is preferentially harder under hot weather conditions and steeper under cold ones. The detection efficiency of the SD is thus expected to follow these variations to some extent, being on average larger when the weather is hot than when it is cold. As a consequence, one could expect that, at energies below full trigger efficiency, a spurious modulation could appear at the solar frequency. Moreover, we have found that the amplitude of the modulation of the rates at the antisidereal frequency, which is that corresponding to one cycle less per year than the solar frequency, suggests that spurious unaccounted effects become relevant below 2 EeV. In particular, the Fourier amplitude corresponding to the antisidereal time period $T_{\rm as}=24.066$ h in the bin \[1, 2\] EeV is $r=0.005$. This has a probability of arising as a fluctuation of less than 0.1%. A non-negligible antisidereal amplitude could for instance appear in the presence of daily and seasonal systematic effects which are not totally accounted for. Since in this case comparable spurious amplitudes could be expected in the sidereal and antisidereal sidebands [@fast], we only use the Rayleigh method described before in the bins above 2 EeV. We have checked that in the bins above 2 EeV the amplitudes at both the solar and antisidereal frequencies are consistent with being just due to fluctuations, so that there are no signs indicating that surviving systematic effects could be present at the sidereal frequency at these energies (see Table \[tab:sol-asid\] in the Appendix).[^2] Alternatively, one can use for the energies below 2 EeV the differential [East-West]{} (EW) method [@ew], which is based on the difference between the counting rates of the events measured from the East sector and those from the West sector. Since the exposure is the same for events coming from the East and for those coming from the West[^3], and also the spurious modulations due to the atmospheric effects are the same in both sectors, the relative difference between both rates, $(E-W)/(E+W)$, is not sensitive to these experimental and atmospheric systematic effects. This allows one to reconstruct in a clean way the modulation of the rate itself, without the need to apply any correction but at the expense of a reduced sensitivity to the amplitude of the CR flux modulations. In this approach [@ew], the first-harmonic amplitude and phase are calculated using a slightly modified Fourier analysis that accounts for the subtraction of the Western sector from the Eastern one. The Fourier coefficients are defined as $$a_{\rm EW}=\frac{2}{N}\sum_{i=1}^N \cos(\alpha^0(t_i)-\xi_i),~~~~~~b_{\rm EW}=\frac{2}{N}\sum_{i=1}^N \sin(\alpha^0(t_i)-\xi_i),$$ where $\xi_i=0$ for events coming from the East and $\xi_i=\pi$ for those coming from the West, so as to easily implement the subtraction of data from the two hemispheres. In the case in which the dominant contribution to the flux modulation is purely dipolar, the amplitude $r_{\rm EW}=\sqrt{a_{\rm EW}^2+b_{\rm EW}^2}$ and phase $\varphi_{\rm EW}=\arctan(b_{\rm EW}/a_{\rm EW})$ obtained with this method are related to the ones from the Rayleigh formalism through $r=\frac{\pi\langle\cos\delta\rangle}{2\langle\sin\theta\rangle}r_{\rm EW}$ and $\varphi=\varphi_{\rm EW}+\pi/2$, where $\langle\cos\delta\rangle$ is the average of the cosine of the declination of the events and similarly $\langle\sin\theta\rangle$ is the average of the sine of their zenith angles [@ew]. The probability to obtain an amplitude larger than the one measured as a result of a fluctuation from an isotropic distribution is $P(\ge r_{\rm EW})=\exp(-Nr_{\rm EW}^2/4)$. The amplitude of the equatorial dipole component is related to the amplitude of the first-harmonic modulation through $d_\perp\simeq r/\langle\cos\delta\rangle$, and its phase $\alpha_d$ coincides with the first-harmonic phase $\varphi$. Right ascension modulation from 0.03 EV up to $E\geq 32$ EV =========================================================== In Table \[tab:dper\], we report the results for the reconstructed equatorial dipole in different energy bins, covering the range from $\sim 0.03$ EeV up to $E\geq 32$ EeV. The energies defining the boundaries of the bins are $2^n$EeV, with $n=-5,-4,...,4,5$. As mentioned previously, the results are obtained from the study of the right ascension modulation using different methods and datasets. We use the weighted Rayleigh analysis in the energy bins above 2 EeV, for which the systematic effects associated with the non-saturated detector efficiency and to the effects related to atmospheric variations are well under control. When this is not the case, we report the results of the East-West method which, although having larger uncertainties, is quite insensitive to most sources of systematic effects in the right ascension distribution. For energies above 0.25 EeV, we report the results obtained with the data from the SD1500 array, while for lower energies we use the dataset from the SD750 array since, having a lower threshold, it leads to a larger number of events despite the reduced size of the array. In that case, given that the SD750 array is not fully efficient below 0.3 EeV, we just use the East-West method. ![Reconstructed equatorial-dipole amplitude (left) and phase (right). The upper limits at 99% CL are shown for all the energy bins in which the measured amplitude has a chance probability greater than 1%. The gray bands indicate the amplitude and phase for the energy bin $E\geq 8$ EeV. Results from other experiments are shown for comparison [@ic12; @ic16; @KG].[]{data-label="fig:dper"}](dper_0818_all_ul-eps-converted-to.pdf "fig:") ![Reconstructed equatorial-dipole amplitude (left) and phase (right). The upper limits at 99% CL are shown for all the energy bins in which the measured amplitude has a chance probability greater than 1%. The gray bands indicate the amplitude and phase for the energy bin $E\geq 8$ EeV. Results from other experiments are shown for comparison [@ic12; @ic16; @KG].[]{data-label="fig:dper"}](dper_pha_0818_all-eps-converted-to.pdf "fig:") For each energy bin, we report in Table \[tab:dper\] the number of events $N$, the amplitude $d_\perp$, the uncertainty $\sigma_{x,y}$ of the components $d_x$ or $d_y$, the right ascension phase of the dipolar modulation $\alpha_d$, the chance probability $P( \ge d_\perp)$ and, when the measured amplitude has a probability larger than 1%, we also report the 99% CL upper limit on the amplitude of the equatorial dipole $d_\perp^{\rm\small UL}$. The upper limits on the first-harmonic amplitude at a given confidence level CL (${\rm CL}=0.99$ for 99% CL) are derived from the distribution for a dipolar anisotropy of unknown amplitude, marginalized over the dipole phase, requiring that $$\int_{0}^{r^{\rm\small UL}}\mathrm{d}r\,\frac{r}{\sigma^2}\exp\left[-\frac{r^2+s^2}{2\sigma^2}\right]I_0\left(\frac{rs}{\sigma^2}\right) = {\rm CL},$$ with $I_0(x)$ the zero-order modified Bessel function, $s$ the measured amplitude and the dispersion being $\sigma=\sqrt{2/\mathcal{N}}$ for the Rayleigh analysis while $\sigma=(\pi\langle\cos\delta\rangle/2\langle\sin\theta\rangle)\sqrt{2/N}$ for the East-West method. These bounds on the first-harmonic amplitude are then converted into the corresponding upper limit for the amplitude of the equatorial dipole using that $d_\perp^{\rm\small UL}=r^{\rm\small UL}/\langle\cos\delta\rangle$. For the uncertainties in the phase, we use the two-dimensional distribution marginalized instead over the dipole amplitude $r$ [@linsley]. In Table \[tab:dperew\] in the Appendix we also report the results obtained above 2 EeV with the East-West method, which are consistent with those obtained with the Fourier analysis in Table \[tab:dper\] but have larger uncertainties. Fig. \[fig:dper\] shows the equatorial dipole amplitude (left panel) and phase (right panel) that were determined in all the energy bins considered, as reported in Table \[tab:dper\]. Also shown are the results obtained by the IceCube, IceTop and KASCADE-Grande experiments in the 1–30 PeV range [@ic12; @ic16; @KG]. We also show the 99% CL upper limit $d_\perp^{\rm UL}$ in the cases in which the measured amplitude has more than 1% probability to be a fluctuation from an isotropic distribution. The results for the integral bin with $E\geq8$ EeV, that was considered in @science, is shown as a gray band. A trend of increasing amplitudes for increasing energies is observed, with values going from $d_\perp\simeq 0.1$% at PeV energies, to $\sim 1$% at EeV energies and reaching $\sim 10$% at 30 EeV. Regarding the phases, a transition between values lying close to the right ascension of the Galactic center, $\alpha_d\simeq \alpha_{\rm GC}$, towards values in a nearly opposite direction, $\alpha_d\simeq 100^\circ$, is observed to take place around a few EeV. ![Components of the dipole in the equatorial plane for different energy bins above 0.25 EeV (left panel) and below 1 EeV (right panel). The horizontal axis corresponds to the component along the direction $\alpha=0$ while the vertical axis to that along $\alpha=90^\circ$. The radius of each circle corresponds to the 1$\sigma$ uncertainty in $d_x$ and $d_y$. The Galactic center direction is also indicated. The measurements from IceCube (IC) and IceTop (IT) at PeV energies are also indicated in the right panel [@ic12; @ic16].[]{data-label="fig:circles"}](dperpxy1_c.pdf "fig:") ![Components of the dipole in the equatorial plane for different energy bins above 0.25 EeV (left panel) and below 1 EeV (right panel). The horizontal axis corresponds to the component along the direction $\alpha=0$ while the vertical axis to that along $\alpha=90^\circ$. The radius of each circle corresponds to the 1$\sigma$ uncertainty in $d_x$ and $d_y$. The Galactic center direction is also indicated. The measurements from IceCube (IC) and IceTop (IT) at PeV energies are also indicated in the right panel [@ic12; @ic16].[]{data-label="fig:circles"}](dperpxy2_c.pdf "fig:") The overall behavior of the amplitudes and phases in the $d_x$–$d_y$ plane is depicted in Fig. \[fig:circles\]. The left panel includes the energies above 0.25 EeV while the right panel those below 1 EeV. In these plots, the right ascension $\alpha_d$ is the polar angle, measured anti-clockwise from the x-axis (so that $d_x=d_\perp \cos\alpha_d$ and $d_y=d_\perp\sin\alpha_d$). The circles shown have a radius equal to the 1$\sigma$ uncertainties $\sigma_{x,y}$ in the dipole components $d_{x,y}$ (reported in the Table 1), effectively including $\sim 39$% of the two-dimensional confidence region. One can appreciate in this plot how the amplitudes decrease for decreasing energies, and how the phases change as a function of the energy, pointing almost in the opposite direction of the Galactic center above 4 EeV and not far from it below 1 EeV. The values of the anisotropy parameters obtained above are based, by construction, on the event content in the energy intervals under scrutiny. The finite resolution on the energies induces bin-to-bin migration of events. Due to the steepness of the energy spectrum, the migration happens especially from lower to higher energy bins. This influences the energy dependence of the recovered parameters. However, given that the size of the energy bins chosen here is much larger than the resolution, the migration of events remains small enough to avoid significant distortions for the recovered values above full efficiency. For instance, given the energy resolution of the SD1500 array [@spectrum] and assuming a dipole amplitude scaling as $E^{0.8}$, as was found in @uhedip to approximately hold above 4 EeV, the impact of the migrations remains below an order of magnitude smaller than the statistical uncertainties associated to the recovered parameters. In the energy range below full efficiency, additional systematic effects enter into play on the energy estimate. We note that forward-folding simulations of the response function effects into an injected anisotropy show that the recovered parameters are not impacted by more than their current statistical uncertainties. A complete unfolding of these effects is left for future studies. It requires an accurate knowledge of the response function of the SD arrays down to low energies, which is not available at the moment. Discussion and conclusions ========================== We have updated the searches for anisotropies on large angular scales using the cosmic rays detected by the Pierre Auger Observatory. The analysis covered more than three orders of magnitude in energy, including events with $E\geq 0.03$ EeV and hence encompassing the expected transition between Galactic and extragalactic origins of the cosmic rays. This was achieved by studying the first-harmonic modulation in right ascension of the CR fluxes determined with the SD1500 and the SD750 surface detector arrays. This allowed us to determine the equatorial component of a dipolar modulation, $\vec{d}_\perp$, or eventually to set strict upper-bounds on it. For the inclusive bin above 8 EeV, the first-harmonic modulation in right ascension leads to an equatorial dipole amplitude $d_\perp= 0.060^{+0.010}_{-0.009}$, which has a probability to arise by chance from an isotropic distribution of $1.4\times 10^{-9}$, corresponding to a two-sided Gaussian significance of 6$\sigma$. The phase of the maximum of this modulation is at $\alpha_d= 98^\circ\pm 9^\circ$, indicating an extragalactic origin for these CRs. When splitting the bin above 8 EeV, as originally done in @uhedip, one finds indications of an increasing amplitude with increasing energies, and the direction of the dipole suggests that it has an extragalactic origin in all the three bins considered. A growing dipole amplitude for increasing energies could for instance be associated with the larger relative contribution to the flux that arises at high energies from nearby sources, that are more anisotropically distributed than the integrated flux from the distant ones. A suppression of the more isotropic contribution from distant sources is expected to result from the strong attenuation of the CR flux that should take place at the highest energies as a consequence of their interactions with the background radiation [@gzk1; @gzk2]. At energies below 8 EeV, none of the amplitudes are significant, and we set 99% CL upper bounds on $d_\perp$ at the level of 1 to 3%. The phases measured in most of the bins below 1 EeV are not far from the direction towards the Galactic center. All this suggests that the origin of these dipolar anisotropies changes from a predominantly Galactic one to an extragalactic one somewhere in the range between 1 EeV and few EeV. The small size of the dipolar amplitudes in this energy range, combined with the indications that the composition is relatively light [@compo], disfavor a predominant flux component of Galactic origin at $E>1$ EeV [@LSA2013]. Models of Galactic CRs relying on a mixed mass composition, with rigidity dependent spectra, have been proposed to explain the knee (at $\sim 4$ PeV) and second-knee (at $\sim 0.1$ EeV) features in the spectrum [@candia]. The predicted anisotropies depend on the details of the Galactic magnetic field model considered and, below 0.5 EeV, they are consistent with the upper bounds we obtained. An extrapolation of these models, considering that there is no cutoff in the Galactic component, would predict dipolar anisotropies at the several percent level beyond the EeV, in tension with the upper bounds in this range. The conflict is even stronger for Galactic models [@kusenko] having a light CR composition that extends up to the ankle energy (at $\sim 5$ EeV). The presence of a more isotropic extragalactic component making a significant contribution already at EeV energies could dilute the anisotropy of Galactic origin, so as to be consistent with the bounds obtained. Note that even if the extragalactic component were completely isotropic in some reference frame, the motion of the Earth with respect to that system could give rise to a dipolar anisotropy through the Compton-Getting effect [@cg]. For instance, for a CR distribution that is isotropic in the CMB rest frame, the resulting Compton-Getting dipole amplitude would be about 0.6% [@cg2]. This amplitude depends on the relative velocity and on the CR spectral slope, but not directly on the particle charge. The deflections of the extragalactic CRs caused by the Galactic magnetic field are expected to further reduce this amplitude, and also to generate higher harmonics, in a rigidity dependent way, so that the exact predictions are model dependent. The Compton-Getting extragalactic contribution to the dipolar anisotropy is hence below the upper limits obtained. More data, as well as analyses exploiting the discrimination between the different cosmic-ray mass components that will become feasible with the upgrade of the Pierre Auger Observatory currently being implemented [@augerprime], will be crucial to understand in depth the origin of the cosmic rays at these energies and to learn how their anisotropies are produced. Acknowledgments {#acknowledgments .unnumbered} =============== The successful installation, commissioning, and operation of the Pierre Auger Observatory would not have been possible without the strong commitment and effort from the technical and administrative staff in Malargüe. We are very grateful to the following agencies and organizations for financial support: Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in gratitude for their continuing cooperation over land access; Australia – the Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP); Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo Research Foundation (FAPESP) Grants No. 2010/07359-6 and No. 1999/05404-3; Ministério da Ciência, Tecnologia, Inovações e Comunicações (MCTIC); Czech Republic – Grant No. MSMT CR LTT18004, LO1305, LM2015038 and CZ.02.1.01/0.0/0.0/16013/0001402; France – Centre de Calcul IN2P3/CNRS; Centre National de la Recherche Scientifique (CNRS); Conseil Régional Ile-de-France; Département Physique Nucléaire et Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de l’Univers (SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX ANR-10-LABX-63 within the Investissements d’Avenir Programme Grant No. ANR-11-IDEX-0004-02; Germany – Bundesministerium für Bildung und Forschung (BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-Württemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen; Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg; Italy – Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE); México – Consejo Nacional de Ciencia y Tecnología (CONACYT) No. 167733; Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The Netherlands – Ministry of Education, Culture and Science; Netherlands Organisation for Scientific Research (NWO); Dutch national e-infrastructure with the support of SURF Cooperative; Poland -Ministry of Science and Higher Education, grant No. DIR/WK/2018/11; National Science Centre, Grants No. 2013/08/M/ST9/00322, No. 2016/23/B/ST9/01635 and No. HARMONIA 5–2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal – Portuguese national funds and FEDER funds within Programa Operacional Factores de Competitividade through Fundação para a Ciência e a Tecnologia (COMPETE); Romania – Romanian Ministry of Research and Innovation CNCS/CCCDI-UESFISCDI, projects PN-III-P1-1.2-PCCDI-2017-0839/19PCCDI/2018 and PN18090102 within PNCDI III; Slovenia – Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033, N1-0111; Spain – Ministerio de Economía, Industria y Competitividad (FPA2017-85114-P and FPA2017-85197-P), Xunta de Galicia (ED431C 2017/07), Junta de Andalucía (SOMM17/6104/UGR), Feder Funds, RENATA Red Nacional Temática de Astropartículas (FPA2015-68783-REDT) and María de Maeztu Unit of Excellence (MDM-2016-0692); USA – Department of Energy, Contracts No. DE-AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-SC0011689; National Science Foundation, Grant No. 0450696; The Grainger Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin American Network; and UNESCO. In Table \[tab:sol-asid\] we report the amplitudes and probabilities obtained with the SD1500 array at the solar and antisidereal frequencies, in all bins above 2 EeV for which the Rayleigh analysis was applied at the sidereal frequency. One can see that all these amplitudes are consistent with being fluctuations, showing then no signs of remaining systematic effects. We also report in Table \[tab:dperew\] the equatorial dipole amplitudes and phases obtained with the East-West method above 2 EeV, and compare them with the results for the same datasets that were obtained with the Rayleigh method (reported in Table \[tab:dper\]). The inferred equatorial dipole amplitudes turn out to be consistent, although the statistical uncertainty obtained with the East-West method is larger by a factor $\pi\langle \cos\delta\rangle/2\langle \sin\theta \rangle$ [@ew]. Given that above full trigger efficiency one has that $\langle\sin\theta\rangle\simeq 0.58$ when considering $\theta<60^\circ$, as we do for $E<4$ EeV, or $\langle\sin\theta\rangle\simeq 0.65$ when considering $\theta<80^\circ$, as we do for $E\geq 4$ EeV, and that $\langle\cos\delta\rangle\simeq 0.78$ in both zenith ranges, the statistical uncertainties obtained in the East-West analysis are larger by a factor of about 2.1 than those obtained with the Rayleigh analysis for $\theta<60^\circ$, or by a factor of about 1.9 for $\theta<80^\circ$, as can be seen in Table \[tab:dperew\]. -------------- --------- --------------------- ------------- --------------------- ------------- $ E$ \[EeV\] $N$ $r$ \[%\] $P(\geq r)$ $r$ \[%\] $P(\geq r)$ 2 - 4 283,074 $0.6^{+0.3}_{-0.2}$ 0.07 $0.5^{+0.3}_{-0.2}$ 0.20 4 - 8 88,325 $0.8^{+0.5}_{-0.3}$ 0.24 $0.5^{+0.5}_{-0.2}$ 0.59 8 - 16 27,271 $0.6^{+1.1}_{-0.2}$ 0.79 $0.5^{+1.1}_{-0.1}$ 0.83 16 - 32 7,664 $1.1^{+2.0}_{-0.3}$ 0.79 $3.1^{+1.9}_{-1.1}$ 0.16 $\geq 32$ 1,993 $1.5^{+4.4}_{-0.1}$ 0.90 $1.3^{+4.6}_{-0.0}$ 0.92 $\geq 8$ 36,928 $0.3^{+1.1}_{-0.0}$ 0.93 $1.0^{+0.8}_{-0.4}$ 0.39 -------------- --------- --------------------- ------------- --------------------- ------------- : Fourier amplitudes at the solar and antisidereal frequencies, and the probabilities to get larger values from statistical fluctuations of an isotropic distribution, for the different energy bins above 2 EeV.[]{data-label="tab:sol-asid"} [ c c | c c c c | c c c]{} & & &\ $E$ \[EeV\] & $N$ & $d_\perp$ \[%\] & $\sigma_{x,y}$ \[%\] & $\alpha_d [^\circ]$ & $P(\ge d_\perp)$ & $d_\perp$ \[%\] &$\sigma_{x,y}$ \[%\] & $\alpha_d [^\circ]$\ ------------------------------------------------------------------------ 2 - 4 & 283,074 & $0.2^{+0.9}_{-0.2}$ & 0.72 & $-16 \pm 167$ & 0.94 & $0.5^{+0.4}_{-0.2}$ & 0.34 & $-11 \pm 55$\ 4 - 8 & 88,325 & $1.7^{+1.3}_{-0.7}$ &1.1 & $41 \pm 38$ & 0.33 & $1.0^{+0.7}_{-0.4}$ & 0.61 & $69 \pm 46$\ 8 - 16 & 27,271 & $6.4^{+2.3}_{-1.7}$ &2.1 & $147 \pm 18$ & $8.3\times 10^{-3}$ & $5.6^{+1.2}_{-1.0}$ &1.1 & $ 97\pm 12$\ 16 - 32 & 7,664 & $9.3^{+4.5}_{-3.0}$ & 3.9& $67 \pm 24$ & $5.8\times 10^{-2}$ & $7.5^{+2.3}_{-1.8}$ & 2.1 &$ 80\pm 17$\ $\geq 32$ & 1,993 & $25^{+9}_{-6}$ & 7.6 & $151 \pm 17$ & $4.1\times 10^{-3}$ & $13^{+5}_{-3}$ & 4.1 & $152 \pm 19$\ ------------------------------------------------------------------------ $\geq 8$ & 36,928 & $6.6^{+2.0}_{-1.5}$ & 1.8& $132 \pm 15$ & $8.6\times 10^{-4}$ & $6.0^{+1.0}_{-0.9}$ & 0.94 & $98 \pm 9$\ Ahlers, M. 2019, , 886, L18 Al Samarai, I. for the Pierre Auger Collaboration 2016, [PoS ICRC2015]{}, 372 Bonino, R. et al. 2011, , 738, 67 Candia, J., Mollerach, S. and Roulet, E. 2003, , 05, 003 Calvez, A., Kusenko, A. and Nagataki, S. 2010, , 105, 091101 Castellina, A. for the Pierre Auger Collaboration 2019, [EPJ Web Conf.]{}, 210, 06002 Coleman, A. for the Pierre Auger Collaboration 2019, [PoS ICRC2019]{}, 225 Compton, A.H. and Getting, I.A. 1935, PhRv, 47, 817 Farley, F.J.M. and Storey, J.R. 1954, [Proc. Phys. Soc.]{} A, 67, 996 Feretti, L. et al. 2012, , 20, 54 Greisen, K. 1966, , 16, 748 Haverkorn, M. 2015, [Astrophys. Space Sci. Library]{}, 407, 483 IceCube Collaboration 2012, , 746, 33 IceCube Collaboration 2016, , 826, 220 Kachelriess, M. and Serpico, P.D. 2006, [Phys. Lett.]{} B, 640, 225 KASCADE-Grande Collaboration 2019, , 870, 91 Linsley, J. 1975, , 34, 1530 Nagashima, K. et al. 1989, [Il Nuovo Cimento]{} C, 12, 695 Ptuskin, V.S. et al. 1993, , 268, 726 Sidelnik, I. for the Pierre Auger Collaboration 2013, [Proceeding of the 33rd ICRC]{}, arXiv:1307.5059 The Pierre Auger Collaboration 2011a, [Astropart. Phys.]{}, 34, 627 The Pierre Auger Collaboration 2011b, , 11, 022 The Pierre Auger Collaboration 2012, , 203, 34 The Pierre Auger Collaboration 2013, , 762, L13 The Pierre Auger Collaboration 2014a, , 90, 122006 The Pierre Auger Collaboration 2014b, , 08, 019 The Pierre Auger Collaboration 2015a, , 802, 111 The Pierre Auger Collaboration 2015b, [NIM]{} A, 798, 172 The Pierre Auger Collaboration 2017a, [Science]{}, 357, 1266 The Pierre Auger Collaboration 2017b, [JINST]{}, 12, P02006 The Pierre Auger Collaboration  2018, , 868, 4 Verzi, V. for the Pierre Auger Collaboration 2019, [PoS ICRC2019]{}, 450 Zatsepin, G.T. and Kuzmin, V.A. 1966, [JETP Lett.]{}, 4, 78 [^1]: Hints of anisotropies on smaller angular scales were also found recently in a reanalysis of KASCADE-Grande data [@ahlers]. [^2]: Given that, for events with zenith angles smaller than 60$^\circ$, the trigger efficiency is larger than $\sim 95$% above 2 EeV, the efficiency related systematic effects are negligible above this threshold. [^3]: A possible tilt of the array in the East-West direction, giving just a constant term in the East-West rate difference, does not affect the determination of the first-harmonic modulation.

--- abstract: 'We consider normal-form games with ${{n}}$ players and two strategies for each player where the payoffs are Bernoulli random variables. We define the associated to a strategy profile as the sum of the payoffs of all players divided by ${{n}}$. We assume that payoff vectors corresponding to different profiles are i.i.d., and the payoffs within the same profile are conditionally independent given some underlying random parameter. Under these conditions we examine the asymptotic behavior of the that correspond to the optimum, to the best and to the worst . We perform a detailed analysis of some particular cases showing that these random quantities converge, as ${{n}}\to\infty$, to some function of the models’ parameters. Moreover, we show that these functions exhibit some interesting phase-transition phenomena.' address: - '$^{\dagger}$ Dipartimento di Economia e Finanza, LUISS, Viale Romania 32, 00197 Roma, Italy.' - '$^{\#}$ Dipartimento di Economia e Finanza, LUISS, Viale Romania 32, 00197 Roma, Italy.' author: - 'Matteo Quattropani$^{\dagger}$' - 'Marco Scarsini$^{\#}$' bibliography: - 'bibrandomPoA.bib' title: Efficiency of equilibria in random binary games --- Introduction {#se:intro} ============ The concept of is central in game theory. [@Nash:PNAS1950; @Nash:AM1951] proved that every finite game admits . In general, may fail to exist. Given that the concept of is epistemically more clearly understood than the one of , it is important to understand how rare it is to have games without . One way to address the problem is to consider games in normal form whose payoffs are random. In a random game the number of is also a random variable, whose distribution is interesting to study. It is known that this distribution depends on the assumptions made on the distribution of the random payoffs. The simplest case that has been considered in the literature deals with i.i.d. payoffs having a continuous distribution function. This implies that ties happen with probability zero. Even in this simple case, although it is easy to compute the expected number of , the characterization of their exact distribution is non-trivial. Asymptotic results exist as either the number of players or the number of strategies for each player diverge. In both cases the number of converges to a Poisson distribution with parameter $1$. Generalizations of the simple case can be achieved either by removing the assumptions that all payoffs are independent or by allowing for discontinuities in their distribution functions, or both. In both cases the number of diverges and some holds. To the best of our knowledge, the literature on this topic has focused on the distribution of the number of but not on their , i.e., the sum of the payoffs of each player. The issue of efficiency of equilibria and its measure has received an attention for more than a century and, at the end of the last millennium, has led to the definition of the as a pessimistic measure of inefficiency [@KouPap:STACS1999; @Pap:ACMSTC2001], followed by the as its optimistic counterpart [@SchSti:P14SIAM2003; @AnsDasKleTarWexRou:SIAMJC2008]. The is the ratio of the optimum over the of the worst equilibrium. The is the ratio of the optimum over the of the best equilibrium. It is interesting to study how these three quantities behave in a random game. Our contribution ---------------- We consider a model with ${{n}}$ players and two strategies for each player. Payoffs are assumed to be random. To be more precise the payoff vectors corresponding to each strategy profile are assumed to be i.i.d. and payoffs within the same strategy profile ${\boldsymbol{{{s}}}}$ to be conditionally i.i.d. Bernoulli random variables, given a parameter $\Phi({\boldsymbol{{{s}}}})$ distributed according to the probability law ${\pi}$ on $[0,1]$. A model with a similar dependence structure was considered in @RinSca:GEB2000, but there the payoffs have a Gaussian distribution. We will study the asymptotic behavior of the in this game as ${{n}}\to\infty$. In particular, we focus our analysis on the optimal , on the of the best, the worst, and the *typical* . As a preliminary step, we will consider the asymptotic behavior of the random number of . We consider three relevant cases for the measure ${\pi}$. First we look at the case where the support of ${\pi}$ is the whole interval $[0,1]$ and we show that the asymptotic behavior of the number of does not depend on ${\pi}$. Moreover we show that in this case the asymptotic behavior of the of the optimum, and the best equilibrium coincide and have maximal , i.e., equal to 1. On the other hand, we show that efficiency of the worst depends on ${\pi}$ only through its mean. The same analysis is performed for the case in which ${\pi}$ is the Dirac mass at $p\in(0,1)$, which corresponds to i.i.d. payoffs. Finally we deal with a model where the dependence within the profile depends on a single parameter $q$ and perform the same asymptotic analysis as a function of $p$ and $q$. For each of these models we analyze the behavior of the best and worst equilibria as a function of the relevant parameters, showing some interesting irregularities. The techniques we use in this paper are standard in the probabilistic literature, and amount mostly to first and second moment analysis, large deviations and calculus. Nonetheless, a refined analysis of a perturbation of the large deviation rate of binomial random variables is required to provide precise asymptotic results on the phase-transition mentioned in the abstract. Related literature ------------------ The distribution of the number of in games with random payoffs has been studied for a number of years. Many papers assume the random payoffs to be i.i.d. from a continuous distribution. Under this hypothesis, several papers studied the asymptotic behavior of random games, as the number of strategies grows. For instance, [@Gol:AMM1957] showed that in zero-sum two-person games the probability of having a goes to zero. He also briefly dealt with the case of payoffs with a Bernoulli distribution. [@GolGolNew:JRNBSB1968] studied general two-person games and showed that the probability of having at least one converges to $1-\operatorname{e}^{-1}$. [@Dre:JCT1970] generalized this result to the case of an arbitrary finite number of players. Other papers have looked at the asymptotic distribution of the number of , again when the number of strategies diverges. [@Pow:IJGT1990] showed that, when the number of strategies of at least two players goes to infinity, the distribution of the number of converges to a $\operatorname{\mathsf{{Poisson}}}(1)$. She then compared the case of continuous and discontinuous distributions. [@Sta:GEB1995] derived an exact formula for the distribution of in random games and obtained the result in [@Pow:IJGT1990] as a corollary. [@Sta:MOR1996] dealt with the case of two-person symmetric games and obtained Poisson convergence for the number of both symmetric and asymmetric . In all the above models, the expected number of is in fact 1. Under different hypotheses, this expected number diverges. For instance, [@Sta:MSS1997; @Sta:EL1999] showed that this is the case for games with vector payoffs and for games of common interest, respectively. [@RinSca:GEB2000] weakened the hypothesis of i.i.d. payoffs; that is, they assumed that payoff vectors corresponding to different strategy profiles are i.i.d., but they allowed some dependence within the same payoff vector. In this setting, they proved asymptotic results when either the number of players or the number of strategies diverges. More precisely, if each payoff vector has a multinormal exchangeable distribution with correlation coefficient $\rho$, then, if $\rho$ is positive, the number of diverges and a central limit theorem holds. [@Rai:P7YSM2003] used Chen-Stein method to bound the distance between the distribution of the normalized number of and a normal distribution. His result is very general, since it does not assume continuity of the payoff distributions. [@Tak:GEB2008] considered the distribution of the number of in a random game with two players, conditionally on the game having nondecreasing best-response functions. This assumption greatly increases the expected number of . [@DasDimMos:AAP2011] extended the framework of games with random payoffs to graphical games. Strategy profiles are vertices of a graph and players’ strategies are binary, like in our model. Moreover, their payoff depends only on their strategy and the strategies of their neighbors. The authors studied how the structure of the graph affects existence of and they examined both deterministic and random graphs. [@AmiColSca:arXiv2019] showed that in games with ${{n}}$ players and two actions for each player, the key quantity that determines the behavior of the number of is the probability that two different payoffs assume the same value. They then studied the behavior of best-response dynamics in random games. The issue of solution concepts in games with random payoffs has been explored by various authors in different directions. For instance, [@Coh:PNAS1998] studied the probability that Nash equilibria (both pure and mixed) in a finite random game maximize the sum of the players’ payoffs. This bears some relation with what we do in this paper. The fact that selfish behavior of agents produces inefficiencies goes back at least to [@Pig:Macmillan1920] and has been studied in various fashions in the economic literature. Measuring inefficiency of equilibria in games has attracted the interest of the algorithmic-game-theory community around the change of the millennium. Efficiency of equilibria is typically measured using either the or the . The , i.e., the ratio of the optimum over the of the worst equilibrium, was introduced by [@KouPap:STACS1999] and given this name by [@Pap:PACM2001]. The , i.e., the ratio of the optimum over the of the best equilibrium, was introduced by [@SchSti:P14SIAM2003] and given this name by [@AnsDasKleTarWexRou:SIAMJC2008]. The reader is referred for instance to [@RouTar:inCUPress2007] for the basic concepts related to inefficiency of equilibria. ### Connections with random Constraint Satisfaction Problems (CSP) and partially-oriented percolation A CSP amounts to find an initialization for a set of $n$ variables taking value in a finite alphabet, say $\{{{\texttt}{0}},{{\texttt}{1}}\}$, subject to a certain number of constraints. Example of problems in this class are classical in the computer science literature, e.g. SAT, graph coloring, independent set, etc. See, among others, [@coja9random; @mezard2009information]. Clearly, a binary game can be phrased as a CSP by considering pure Nash equilibria as the solution concept. Random CSP have attracted at lot of attention in the physics community, where a number of deep conjectures on the behavior of the solution set have been developed, and only part of them have been recently rigorously proved by mathematicians (see, e.g., [@AchPer:2SAT2004; @abbemonta:2sat2010; @DSS:kSAT2015]). Given a law on the space of instances of a CSP, the first problem lies in the analysis of the size of the solution set, which is a random subset of $\{0,1\}^n$. In [@AmiColSca:arXiv2019] the authors noticed that a random binary game can be phrased as a marked partially oriented percolation on the hypercube. Strategy profiles represent vertices of the hypercube, each vertex has an array mark, which corresponds to the utilities of the players under the corresponding strategy profile. We place an oriented arc between two profiles if and only if they are neighbors in the hypercube and the mark in the differing coordinate is strictly larger in the arrival vertex. In this framework, the set of Nash equilibria coincide with the set of vertices having out-degree equal to zero, i.e., sinks. In the physicists’ language, in [@AmiColSca:arXiv2019] the authors computed the *quenched free-energy* of the model, see \[eq:coll-scar\]. In this work we consider a closely related CSP, in which we enlarge the set of constraints: a “solution” is a pure Nash equilibrium with a certain social utility. In the percolation representation of the problem, we aim at controlling the number of sinks with a given sum of the entries in the mark. We will see how this additional constraint affects the free-energy and, in general, we refine the analysis of the solution set in [@AmiColSca:arXiv2019] under the binary-payoff assumption. We stress that, by our analysis, in the case of binary random games a “vanilla” second-moment argument (see [@AchPer:2SAT2004]) is sufficient to control the quenched free-energy of the random CSP. Organization of the paper {#suse:organization} ------------------------- The rest of the paper is organized as follows. \[se:general\] is devoted to a second moment analysis under no assumption on the distribution $F$. In section \[se:continuous,se:indep,se:dependent-payoffs\] we present a precise analysis of the efficiency of equilibria for three different specific choices for $F$. Finally, \[se:proofs\] is devoted to proofs. General model {#se:general} ============= We consider a game with ${{n}}$ players. We use the symbol ${\bracks{{{n}}}}$ for the set of players. Each player can choose one action in $\braces{0,1}$. Then the set ${{\Sigma}}$ of strategy profiles is the Cartesian product $\times_{{{i}}\in{\bracks{{{n}}}}}\braces{0,1}$. As usual, the symbol $\oplus$ will denote the binary XOR operator, defined as $$\label{eq:XOR} 1 \oplus 0=0 \oplus 1 = 1,\quad 0\oplus 0 = 1 \oplus 1 = 0.$$ Therefore, $\oplus$-adding $1$ changes one action into the other. Moreover, for every ${\boldsymbol{{{s}}}}=({{s}}_1,\dots,{{s}}_n)\in{{\Sigma}}$ we let the symbol ${\boldsymbol{{{s}}}}_{-{{i}}}$ denote the strategy profile in which the action of player ${{i}}$ is unspecified, so that ${\boldsymbol{{{s}}}}=({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}})$, for all ${\boldsymbol{{{s}}}}\in{{\Sigma}}$ and ${{i}}\in[n]$. Let $\operatorname{\mathsf{{NE}}}$ denote the set of Nash equilibria, i.e., $$\operatorname{\mathsf{{NE}}}:=\left\{{\boldsymbol{{{s}}}}\in{{\Sigma}}\:|\: u_{{i}}({\boldsymbol{{{s}}}})\ge u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1),\quad\forall i\in[n] \right\}.$$ For ${{i}}\in[{{n}}]$, ${{u}}_{{{i}}} \colon {{\Sigma}}\to {\mathbb{R}}$ denotes player ${{i}}$’s payoff function. We further assume that the payoffs are *binary*, in the sense that $$\label{eq:binary-payoff} {{u}}_{{i}}({\boldsymbol{{{s}}}})\in\braces{0,1},\quad\forall{{i}}\in{\bracks{{{n}}}},\:\forall {\boldsymbol{{{s}}}}\in{{\Sigma}}.$$ We will refer to such games with the name of *binary games*. We will be interested in the behavior of the following quantities: $$\begin{aligned} &\label{eq:SU} \text{\acfi{SU}\acused{SU} } & \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}}) &\coloneqq \sum_{{{i}}\in[{{n}}]}{{u}}_{{{i}}}({\boldsymbol{{{s}}}}),\\ &\label{eq:ASU} \text{\acfi{ASU}\acused{ASU} } & \operatorname{\mathsf{ {ASU}}}({\boldsymbol{{{s}}}}) &\coloneqq \tfrac{1}{n}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}}). \end{aligned}$$ In particular, we will focus on the extremes of the social utility, in the sense that we consider the following objects $$\begin{aligned} &\text{social utility of the \acfi{SO}\acused{SO} } & \operatorname{\mathsf{{SO}}}({\boldsymbol{{{s}}}}) &\coloneqq \max_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}}),\\ &\text{social utility of the \acfi{BEq}\acused{BEq} } & \operatorname{\mathsf{{Beq}}}({\boldsymbol{{{s}}}}) &\coloneqq \max_{{\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}}),\\ &\text{social utility of the \acfi{WEq}\acused{WEq} } & \operatorname{\mathsf{{Weq}}}({\boldsymbol{{{s}}}}) &\coloneqq \min_{{\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}}). \end{aligned}$$ In what follows, we will consider binary games with random payoffs. More precisely, for every choice of ${{n}}\in{\mathbb{N}}$ we will consider a probability measure on the set of binary games with ${{n}}$ players as follows. Consider a random *potential function*, $\Phi:{{\Sigma}}\to [0,1]$, such that $$\big(\Phi({\boldsymbol{{{s}}}})\big)_{{\boldsymbol{{{s}}}}\in{{\Sigma}}},\quad \text{i.i.d.}\quad \Phi({\boldsymbol{{{s}}}})\sim {\pi},$$ for some probability measure ${\pi}$ with $\operatorname{supp}({\pi})\subseteq[0,1]$. Notice that considering the common-interests game with payoffs $$u_{{i}}({\boldsymbol{{{s}}}})=\Phi({\boldsymbol{{{s}}}}),\qquad\forall {{i}}\in[{{n}}],\:\forall{\boldsymbol{{{s}}}}\in{{\Sigma}}$$ we have a potential game. In our model, instead, we consider a *discrete perturbation* of the potential structure, in the sense that we use the potential $\Phi$ just to model dependences between payoffs of different players under the same profile. More precisely, given the value of the potential at a given profile ${\boldsymbol{{{s}}}}$, i.e. $\Phi({\boldsymbol{{{s}}}})$, the utility of the players are $n$ i.i.d. Bernoulli random variables of parameter $\Phi({\boldsymbol{{{s}}}})$. Moreover, we assume independence of payoffs vectors under different profiles. We will call $p={\mathsf{{E}}}[\Phi({\boldsymbol{{{s}}}})]$. Notice that, marginally, $$\label{eq:probab-X-R} {\mathsf{{P}}}(u_{{i}}({\boldsymbol{{{s}}}})=1)= p.$$ \[eq:probab-X-R\] implies that the marginal distribution of the payoffs does not depend on the specific choice of ${\pi}$, but only on its expectation. In the following section we will present precise results concerning three specific but significant examples: - **Fully supported potential:** ${\pi}$ is fully supported in the whole interval $[0,1]$. - **Dirac potential:** ${\pi}$ is the Dirac mass at $p$. Notice that in this case the sequence $(u_{{i}}({\boldsymbol{{{s}}}}))_{{{i}}\in[n],{\boldsymbol{{{s}}}}\in{{\Sigma}}}$ is i.i.d.. For this reason we will refer to this model as the *independent case*. - **Dichotomous potential:** For some $q\in[0,1]$, ${\pi}$ is the convex combination of two Dirac masses, i.e., for every $I\subseteq[0,1]$, $${\pi}(I)=(1-p)\delta_{(1-q)p}(I)+p\delta_{q+(1-q)p}(I).$$ Notice that if $q=0$ we are back to the independent case, while if $q=1$ we have a.s. a common-interests game. We stress that an interpolation of the techniques used in what follows are in principle sufficient to study the general model with arbitrary distribution of the potential. In fact, in this first section we will investigate the first and the second moment of the set of solutions, i.e., the set of equilibria, without any assumption on the measure ${\pi}$. As we will see, the expected number of equilibria grows exponentially with the number of players, regardless of the specific form of ${\pi}$. Moreover, the independence of the payoffs across different profiles is sufficient to ensure that the random number of equilibria is well approximated by its expectation. \[pr:mean\] For any probability measure ${\pi}$ with mean $p$ we have $$\label{eq:expect} {\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|]=\int_0^1 \(2-2p(1-x) \)^nd{\pi}(x),$$ and $$\liminf_{n\to\infty}\frac{1}{n}\log{\mathsf{{E}}}\[|\operatorname{\mathsf{{NE}}}| \]\ge\log \frac{3}{2}.$$ \[pr:var\] For any probability measure ${\pi}$ we have $$\label{eq:var} \lim_{n\to\infty}\frac{{\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|^2]}{{\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|]^2}=1.$$ The next corollary follows immediately by Chebyshev’s inequality and \[pr:mean,pr:var\]. For any probability measure ${\pi}$, if exists some $c\in[\log(3/2),\log(2)]$ such that $$\lim_{n\to\infty}\frac{1}{n}\log{\mathsf{{E}}}\[|\operatorname{\mathsf{{NE}}}| \]=c,$$ then $$\label{eq:logNE} \frac{1}{n}\log|\operatorname{\mathsf{{NE}}}|\overset{{\mathsf{{P}}}}{\longrightarrow} c.$$ For the independent model \[eq:expect\] reads $${\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|]=(1+\alpha(p))^n,$$ where $$\alpha(p):={{p}}^{2}+(1-{{p}})^2\ge \frac{1}{2}.$$ In fact, the independent model is a particular instance of the more general one introduced in [@AmiColSca:arXiv2019], where the authors show that $$\label{eq:coll-scar} \tfrac{1}{{{n}}}\log|\operatorname{\mathsf{{NE}}}|\overset{{\mathsf{{P}}}}{\longrightarrow}\log\parens*{1+\alpha({{p}})}.$$ In fact, the analogue of \[eq:coll-scar\] can be proved for other models, as it is stated by \[eq:logNE\]. The same phenomenon occurs for the set of equilibria with a certain social utility, as soon as the expected size of this set grows exponentially in $n$. More precisely, if we call $$\label{eq:def-WZ} W_k=\left\{{\boldsymbol{{{s}}}}\in {{\Sigma}}\:|\:\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \right\},\qquad Z_k=\left\{{\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\:|\:\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \right\}\subset W_k,$$ the following proposition holds. \[pr:var2\] Let $Q=|Z_k|$ or $Q=|W_k|$ for some $k\in\{0,\dots, n\}$. Then, for any probability measure ${\pi}$ for which $$\lim_{n\to\infty}\frac{1}{n}\log{\mathsf{{E}}}\[Q \]=c>1,$$ we have $$\label{eq:var2} \lim_{n\to\infty}\frac{{\mathsf{{E}}}[Q^2]}{{\mathsf{{E}}}[Q]^2}=1,$$ and, consequently, $$\frac{1}{n}\log Q\overset{{\mathsf{{P}}}}{\longrightarrow} c.$$ Fully supported potential {#se:continuous} ========================= In this section we focus on the case in which ${\pi}$ is fully supported in $[0,1]$. We will show that, under this assumption, the number of equilibria grows at the maximal possible rate. \[th:num-ne-dep\] If ${\pi}$ is fully supported in $[0,1]$, then $$\lim_{n\to\infty} \frac{1}{n}\log |\operatorname{\mathsf{{NE}}}|\overset{{\mathsf{{P}}}}{\longrightarrow} \log 2.$$ Moreover, if $\widehat{\operatorname{\mathsf{{NE}}}}_\varepsilon$ is the set of equilibria having average social utility greater than $1-\varepsilon$, then for all $\varepsilon>0$, $$\lim_{n\to\infty} \frac{|\widehat{\operatorname{\mathsf{{NE}}}}_\varepsilon|}{ |\operatorname{\mathsf{{NE}}}|}\overset{{\mathsf{{P}}}}{\longrightarrow}1.$$ Notice that when $u_{{i}}({\boldsymbol{{{s}}}})=1$ for all ${{i}}$ then the profile ${\boldsymbol{{{s}}}}$ is automatically a pure equilibrium. On the other hand, if the social utility of ${\boldsymbol{{{s}}}}$ is $x n$ for some $x\in(0,1)$, then the probability that ${\boldsymbol{{{s}}}}$ is an equilibrium is exponentially small, with a rate depending only on $x$ and $p$, more precisely, $$\label{eq:cost} {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\mid \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=xn \)=\[(1-p)^{(1-x)}\]^n.$$ The rationale underlying \[th:num-ne-dep\] is that—given that ${\pi}$ is fully supported—for all $\varepsilon>0$ there exists a fraction $\delta$ of strategy profiles with average social utility larger than $1-\varepsilon$. For those profiles, the probability of being an equilibrium has a small exponential cost. In other words, the proof of \[th:num-ne-dep\] shows that in this framework best equilibria are optimal and have average social utility equal to $1$. Moreover, most of the equilibria share this property. On the other hand, for the behavior of the worst equilibria, we need to analyze the exponential rate in \[eq:cost\]. The following \[th:efficiency-dependence\] shows that, if $p<\frac{1}{2}$, then arbitrary bad equilibria exist. On the other hand, if $p$ is sufficiently large, the worst equilibria have a typical average social utility, which depends only on $p$. \[th:efficiency-dependence\] If ${\pi}$ is fully supported in $[0,1]$ , then $$\label{eq:conv-smallp-corr} \frac{1}{n}(\operatorname{\mathsf{{SO}}},\operatorname{\mathsf{{Beq}}},\operatorname{\mathsf{{Weq}}})\overset{{\mathsf{{P}}}}{\longrightarrow}\(1,1,h(p) \),$$ where $h:(0,1)\to[0,1]$ is the non-decreasing continuous function defined as $$\label{eq:def-h} h(p):=\begin{cases}0&\text{if }p\le\frac{1}{2},\\ \frac{\log\(2(1-p)\)}{\log(1-p)}&\text{if }p>\frac{1}{2}. \end{cases}$$ ![Plot of the function $h(p)$ defined in \[eq:def-h\].](ploth){width="7cm"} Independent payoffs {#se:indep} =================== In this section we analyze the independent model which, as mentioned above, is a particular instance of the model in [@AmiColSca:arXiv2019]. In this framework, the study of the behavior of the random variable $\operatorname{\mathsf{{SO}}}$ is somehow classical in the probabilistic literature. In fact, the latter can be thought of as the maximum of $2^n$ independent random variables with law Bin$(n,p)$. Therefore, the analysis of $\operatorname{\mathsf{{SO}}}$ relies on the study of the large deviation rate of a sequence of Binomial trials and has been performed in details, e.g., in [@durrett:maxima79]. Clearly, when one focuses on $\operatorname{\mathsf{{Beq}}}$ ($\operatorname{\mathsf{{Weq}}}$) the analysis is more complicated, due to the fact that dependencies arise when restricting the maximization (minimization) to the random domain $\operatorname{\mathsf{{NE}}}$. In this context, the behavior of $\operatorname{\mathsf{{Beq}}}$ and $\operatorname{\mathsf{{Weq}}}$ can be determined by a precise analysis of the interplay of two different factors: the exponential cost needed to have a large average social utility (i.e., equal to some $x>p$) and the exponential cost of being an equilibrium given an average social utility equal to $x$. Such a competition realizes in the phenomena described in the forthcoming \[th:convergence-p,pr:phase-transition\]. \[th:convergence-p\] There exists three explicit functions $$\label{eq:x-x-x} x_{\operatorname{\mathsf{opt}}},x_{\operatorname{\mathsf{beq}}},x_{\operatorname{\mathsf{weq}}}:[0,1]\to[0,1],$$ such that $$\frac{1}{{{n}}}\big(\operatorname{\mathsf{{SO}}},\operatorname{\mathsf{{Beq}}},\operatorname{\mathsf{{Weq}}}\big)\overset{{\mathsf{{P}}}}{\longrightarrow}\big(x_{\operatorname{\mathsf{opt}}}(p),x_{\operatorname{\mathsf{beq}}}(p),x_{\operatorname{\mathsf{weq}}}(p) \big).$$ The limit quantities for $\operatorname{\mathsf{{SO}}}$, $\operatorname{\mathsf{{Beq}}}$ and $\operatorname{\mathsf{{Weq}}}$—seen as a functions of ${{p}}$—display an interesting behavior. In particular, there exists some threshold for the value of ${{p}}$ before/after which the functions stay constant. \[pr:phase-transition\] The functions $x_{\operatorname{\mathsf{opt}}}$ and $x_{\operatorname{\mathsf{beq}}}$ are both increasing on the interval $(0,1/2)$ and are identically equal to $1$ on the interval $[1/2,1]$. The function $x_{\operatorname{\mathsf{weq}}}$ is identically $0$ on the interval $(0,1-\sqrt{2}/2)$ and is increasing on the interval $[1-\sqrt{2}/2,1]$. We stress that in this model efficiency can always be “nearly achieved” at equilibrium, in the sense that the ratio $x_{\operatorname{\mathsf{opt}}}/x_{\operatorname{\mathsf{beq}}}$ is near 1 for all the value of $p\in(0,1)$, see \[fig:x-x-x\]. Notice that by choosing $p=1/2$ we are considering the uniform measure on the space of binary games with $n$ players. In other words, properties that hold with high probability in the model with potential distribution $\pi(\cdot)=\delta_{1/2}(\cdot)$ are shared by a fraction of binary games that approaches $1$ as $n$ grows to infinity. Therefore, if $p=\frac{1}{2}$, we can rephrase \[pr:phase-transition\] as a counting problem and obtain the following result. \[corollary\] For all $\varepsilon>0$ consider the set ${\ensuremath{\mathcal G}}_n$ of all the binary games with $n$ players and the subset $\widetilde{{\ensuremath{\mathcal G}}}_{n,\varepsilon}$ of binary games having $\operatorname{\mathsf{{SO}}},\operatorname{\mathsf{{Beq}}}\in[1-\varepsilon,1]$ and $\operatorname{\mathsf{{Weq}}}\in[x_{\operatorname{\mathsf{weq}}}(1/2)-\varepsilon,x_{\operatorname{\mathsf{weq}}}(1/2) +\varepsilon ]$. Then, for all $\varepsilon>0$, $$\lim_{n\to\infty}\frac{|\widetilde{{\ensuremath{\mathcal G}}}_{n,\varepsilon}|}{|{\ensuremath{\mathcal G}}_n|}=1.$$ Roughly, \[corollary\] states that asymptotically almost every binary game has $$\operatorname{\mathsf{{PoS}}}\approx 1\qquad\text{and}\qquad \operatorname{\mathsf{{PoA}}}\approx4.4034.$$ ![Numerical approximation of the functions defined in \[eq:x-x-x\]. In blue: $x_{\operatorname{\mathsf{opt}}}(p)$. In orange: $x_{\operatorname{\mathsf{beq}}}(p)$. In green: $x_{\operatorname{\mathsf{weq}}}(p)$.[]{data-label="fig:x-x-x"}](bestworst.pdf){width="7cm"} Underlying dichotomous potential {#se:dependent-payoffs} ================================ We now consider the dichotomous potential case, which can be equivalently defined as follows. For every ${\boldsymbol{{{s}}}}\in{{\Sigma}}$ consider an auxiliary sequence of random variables $\(X({\boldsymbol{{{s}}}})\)_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}$ i.i.d. $\text{Bern}(p)$, a sequence $\(R_{{i}}({\boldsymbol{{{s}}}}) \)_{{{i}}\in[n],{\boldsymbol{{{s}}}}\in{{\Sigma}}}$ of i.i.d. $\text{Bern}(q)$ and, finally, a sequence $\(Y_{{i}}({\boldsymbol{{{s}}}}) \)_{{{i}}\in[n],{\boldsymbol{{{s}}}}\in{{\Sigma}}}$ of i.i.d. $\text{Bern}(p)$. Moreover, we assume all these sequences to be independent. We then define the game as follows: $$u_{{i}}({\boldsymbol{{{s}}}})=\begin{cases} X({\boldsymbol{{{s}}}})&\text{if }R_{{i}}({\boldsymbol{{{s}}}})=1\\ Y_{{i}}({\boldsymbol{{{s}}}})&\text{if }R_{{i}}({\boldsymbol{{{s}}}})=0\\ \end{cases},\qquad\forall{\boldsymbol{{{s}}}}\in{{\Sigma}},\:\forall {{i}}\in[n].$$ Roughly, with probability $q$ the payoff of player ${{i}}$ under ${\boldsymbol{{{s}}}}$ copies the random potential $X({\boldsymbol{{{s}}}})$, while, with the remaining probability, it is an independent $\operatorname{\mathsf{{Bern}}}(p)$ random variable. Our next theorem shows that the exponential size of $|\operatorname{\mathsf{{NE}}}|$ increases monotonically with the correlation parameter $q$. Moreover, asymptotically the efficiency of almost every equilibrium corresponds to the value that optimizes the competition of the exponential costs mentioned in \[se:indep\]. We will give an explicit expression for this value and we will show that it is increasing in $q$ for every fixed $p\in(0,1)$. Therefore, the presence of correlation not only increases the number of Nash equilibria, but also their typical efficiency. \[th:num-ne-q\] For all $(p,q)\in[0,1]\times[0,1]$, we have $$\label{eq:thm3-1} \frac{1}{{{n}}}\log|\operatorname{\mathsf{{NE}}}|\overset{{\mathsf{{P}}}}{\longrightarrow}\log\big( 1+\alpha(p)+2qp(1-{{p}})\big).$$ Moreover, given any $\varepsilon>0$ and defining $$\label{eq:thm3-2} \widehat{\operatorname{\mathsf{{NE}}}}_{\varepsilon}=\left\{{\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\::\:\big|\tfrac{1}{{{n}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})-x^+(p,q) \big| < \varepsilon\right\},$$ where $$x^+(p,q)=\frac{q+p(1-q)}{1-p(1-q)+{{p}}^{2}(1-q)},$$ we have $$\label{eq:efficiency-q} \frac{ |\widehat{\operatorname{\mathsf{{NE}}}}_{\varepsilon}|}{ |\operatorname{\mathsf{{NE}}}|}\overset{{\mathsf{{P}}}}{\longrightarrow}1.$$ Since $x^+(1/2,0)=2/3$, an immediate consequence of \[th:num-ne-q\] is that, in the same spirit of \[corollary\], a uniformly sampled equilibrium in a uniformly random binary game has an average social utility of $2/3$. ![The figures represent numerical approximations of $x_{\operatorname{\mathsf{opt}}}(p,q)$ (blue), $x_{\operatorname{\mathsf{beq}}}(p,q)$ (orange) and $x_{\operatorname{\mathsf{weq}}}(p,q)$ (green) as function of $q$, when $p=0.2,\:0.4,\:0.6$ and $0.75$, respectively. In third picture, the vertical segment around $x\approx 0.7$ is the discontinuous jump mentioned in \[pr:phase-transition-q\].[]{data-label="fig:x-x-x-q"}](p02 "fig:"){width="6cm"}![The figures represent numerical approximations of $x_{\operatorname{\mathsf{opt}}}(p,q)$ (blue), $x_{\operatorname{\mathsf{beq}}}(p,q)$ (orange) and $x_{\operatorname{\mathsf{weq}}}(p,q)$ (green) as function of $q$, when $p=0.2,\:0.4,\:0.6$ and $0.75$, respectively. In third picture, the vertical segment around $x\approx 0.7$ is the discontinuous jump mentioned in \[pr:phase-transition-q\].[]{data-label="fig:x-x-x-q"}](p04 "fig:"){width="6cm"}\ ![The figures represent numerical approximations of $x_{\operatorname{\mathsf{opt}}}(p,q)$ (blue), $x_{\operatorname{\mathsf{beq}}}(p,q)$ (orange) and $x_{\operatorname{\mathsf{weq}}}(p,q)$ (green) as function of $q$, when $p=0.2,\:0.4,\:0.6$ and $0.75$, respectively. In third picture, the vertical segment around $x\approx 0.7$ is the discontinuous jump mentioned in \[pr:phase-transition-q\].[]{data-label="fig:x-x-x-q"}](p06 "fig:"){width="6cm"}![The figures represent numerical approximations of $x_{\operatorname{\mathsf{opt}}}(p,q)$ (blue), $x_{\operatorname{\mathsf{beq}}}(p,q)$ (orange) and $x_{\operatorname{\mathsf{weq}}}(p,q)$ (green) as function of $q$, when $p=0.2,\:0.4,\:0.6$ and $0.75$, respectively. In third picture, the vertical segment around $x\approx 0.7$ is the discontinuous jump mentioned in \[pr:phase-transition-q\].[]{data-label="fig:x-x-x-q"}](p08 "fig:"){width="6cm"} We will now establish the analogue of \[th:convergence-p\] for the general model with $q\ge0$. Notice that the limit functions in this case depend on the interplay of the two parameters $p$ and $q$. \[th:convergence-pq\] There exists three functions $$x_{\operatorname{\mathsf{opt}}},x_{\operatorname{\mathsf{beq}}},x_{\operatorname{\mathsf{weq}}}:(0,1)\times[0,1]\to[0,1],$$ such that $$\label{eq:conv} \frac{1}{{{n}}}\big(\operatorname{\mathsf{{SO}}},\operatorname{\mathsf{{Beq}}},\operatorname{\mathsf{{Weq}}}\big)\overset{{\mathsf{{P}}}}{\longrightarrow}\big(x_{\operatorname{\mathsf{opt}}}(p,q),x_{\operatorname{\mathsf{beq}}}(p,q),x_{\operatorname{\mathsf{weq}}}(p,q) \big).$$ Given \[eq:conv\], it is natural to analyze the behavior of the limit quantities as functions of the two parameters ${{p}}$ and $q$, in the same vein of \[pr:phase-transition\]. In this case, we will fix the parameter $p\in(0,1)$ and vary the correlation parameter $q\in[0,1]$. We now show that these functions exhibit different kinds of irregularity depending on the choice of ${{p}}$. \[pr:phase-transition-q\] For all $(p,q)\in(0,1)\times[0,1]$ the function $ x_{\operatorname{\mathsf{weq}}}(p,q)$ has the following properties 1. If $p\in\big[0,1-\tfrac{\sqrt{2}}{2}\big]$, then $x_{\operatorname{\mathsf{weq}}}(p,q)=0$ for every $q\in[0,1]$. 2. If $p\in\big[1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big]$, then $x_{\operatorname{\mathsf{weq}}}(p,\cdot)$ is continuous in $[0,1]$. Moreover, calling $$\rho(p):=\frac{4p-2 {{p}}^{2}-1}{2 (1-{{p}}) p},$$ $ x_{\operatorname{\mathsf{weq}}}(p,\cdot)\in C_1\big((0,\rho(p))\big)$ with $$\frac{d}{d q} x_{\operatorname{\mathsf{weq}}}(p,q)<0,$$ and $ x_{\operatorname{\mathsf{weq}}}(q,p)=0$ for every $q>\rho(p)$. 3. If $p>\frac{1}{2}$, $ x_{\operatorname{\mathsf{weq}}}(p,q)>0$ for every $q\in[0,1]$. 4. There exist some critical $p_c\in(1/2,1)$ ($p_c\approx 0.732$) such that - If $p\in (p_c,1)$, then $x_{\operatorname{\mathsf{weq}}}(p,q)$ is continuous for $q\in[0,1]$. - If $p\in (1/2,p_c)$, then $x_{\operatorname{\mathsf{weq}}}(p,q)$ is continuous for $q\in[0,1]\setminus\{q^*(p) \}$, where $$q^*(p):=\frac{1-2p+2{{p}}^{2}}{2{{p}}^{2}}.$$ Moreover, $$\lim_{q\uparrow q^*(p) } x_{\operatorname{\mathsf{weq}}}(p,q)<\lim_{q\downarrow q^*(p)} x_{\operatorname{\mathsf{weq}}}(p,q).$$ As we remarked above, in the independent model the efficiency is approximatively achieved at equilibrium, in the sense that, for all $p\in(0,1)$, $x_{\operatorname{\mathsf{opt}}}$ and $x_{\operatorname{\mathsf{beq}}}$ are not far apart. The following proposition shows that the same is true in the model with dependent payoffs. \[pr:opt-beq-p\] For all $p,q\in(0,1)$ the functions $x_{\operatorname{\mathsf{opt}}}(p,q)$ and $x_{\operatorname{\mathsf{beq}}}(p,q)$ satisfy, $$x_{\operatorname{\mathsf{opt}}}(p,q)=x_{\operatorname{\mathsf{opt}}}(q+(1-q)p,0),\qquad x_{\operatorname{\mathsf{beq}}}(p,q)\ge x_{\operatorname{\mathsf{beq}}}(q+(1-q)p,0).$$ Proofs {#se:proofs} ====== In this section we present the main proofs of the results of \[se:general,se:continuous,se:indep,se:dependent-payoffs\]. In particular, in \[se:proof-general\] we deal with the moments results in \[se:general\]. In \[se:proof-convergence\] we prove the convergences in \[th:convergence-p,th:convergence-pq\]. In \[se:proof-convergence\] we prove the phase transition outlined in \[pr:phase-transition,pr:phase-transition-q\]. Finally, in \[se:proof-continuous\] we will prove the results in \[se:continuous\]. We will adopt the following notation. For every ${{n}}\in{\mathbb{N}}$, $\parens*{\Omega^{({{n}})},{\mathsf{{P}}}_{{\pi}}^{({{n}})}}$ denotes the probability space introduced above, where ${\pi}$ is the probability law of the potential and ${{n}}$ is number of players. Since we are interested in the asymptotic scenario in which the number of players grows to infinity, we will usually drop the dependence on ${{n}}$ in the notation. Moreover, when the choice of ${\pi}$ is clear by the context, we will also drop the dependence on $F$. We say that a sequence of real random variables $(X_{{{n}}})_{{{n}}\in{\mathbb{N}}}$ in the product probability space $\times_{{{n}}\in{\mathbb{N}}}\big(\Omega^{({{n}})},{\mathsf{{P}}}_{{\pi}}^{({{n}})}\big)$ *converges to* $\ell\in{\mathbb{R}}$ in probability (denoted by $X_{{{n}}}\overset{{\mathsf{{P}}}}{\longrightarrow}\ell$), if $$\forall\varepsilon>0,\quad\lim_{{{n}}\to\infty}{\mathsf{{P}}}_{{\pi}}^{(n)}\big(\big|X_{{{n}}}-\ell\big|<\varepsilon \big)=1.$$ Moments estimates {#se:proof-general} ----------------- We start the subsection by proving \[pr:mean\], namely the expected size of the set of equilibria. We start by computing the probability that a given profile ${\boldsymbol{{{s}}}}$ is an equilibrium by conditioning on the value of the potential at ${\boldsymbol{{{s}}}}$, namely $$\begin{aligned} {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\)=&\int_0^1 {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\mid \Phi({\boldsymbol{{{s}}}})=x \) dF(x)\\ =&\int_0^1 \[1-p(1-x) \]^n dF(x),\\ \end{aligned}$$ where we used the fact that $$\begin{aligned} {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\mid \Phi({\boldsymbol{{{s}}}})=x \)=&{\mathsf{{P}}}\(\not\exists {{i}}\in[n]\text{ s.t. } u_{{i}}({\boldsymbol{{{s}}}})=0,\:u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}} , {{s}}_{{i}}\oplus 1)=1 \)\\ =&\prod_{{{i}}\in[n]}\big(1-{\mathsf{{P}}}\( u_{{i}}({\boldsymbol{{{s}}}})=0,\:u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1)=1\)\big)\\ =&\[1-{\mathsf{{P}}}\( u_1({\boldsymbol{{{s}}}})=0,\:u_1({\boldsymbol{{{s}}}}_{-1},{{s}}_1\oplus 1)=1\) \]^n\\ =& \[1-p(1-x) \]^n. \end{aligned}$$ Therefore, by the linearity of the expectation, $${\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|]=2^n{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\),$$ from which the first part of the thesis follows. Notice that, regardless of the specific choice of $p$ the expected number of equilibria grows exponentially in $n$. In fact, by the fact that ${\pi}$ has mean $p$, there exists some $\varepsilon>0$ such that ${\mathsf{{P}}}(\Phi({\boldsymbol{{{s}}}})\ge p)>\varepsilon$. Then $$\begin{aligned} {\mathsf{{E}}}\[|\operatorname{\mathsf{{NE}}}| \]=&\int_{0}^{1}\(2-2p(1-x) \)^nd{\pi}(x)\\ \ge& \int_p^1\(2-2p(1-x) \)^nd{\pi}(x)\\ \ge& \varepsilon\(2-2p(1-p) \)^n\\ \ge&\varepsilon\(\frac{3}{2} \)^n.\end{aligned}$$ Hence, $$\liminf_{n\to\infty }\frac{1}{n}\log {\mathsf{{E}}}\[|\operatorname{\mathsf{{NE}}}| \]\ge \log \frac{3}{2}.$$ We now aim at computing the second moment of the quantities $|Z_k|$, $|W_k|$ and $|\operatorname{\mathsf{{NE}}}|$. We start the proof with the analysis of the second moment of $|W_k|$, which is easier to compute. Indeed, for every distinct ${\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in{{\Sigma}}$, thanks to the independence of the payoffs vector across profile, we have $${\mathsf{{P}}}({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in W_k)={\mathsf{{P}}}({\boldsymbol{{{s}}}}\in W_k )^2.$$ Therefore, $$\begin{aligned} \label{eq:bound2m} {\mathsf{{E}}}[|W_k|]^2\le {\mathsf{{E}}}[|W_k|^2]=&\sum_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}\sum_{{{\boldsymbol{{{s}}}}'}\in{{\Sigma}}}{\mathsf{{P}}}({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in W_k)\\ =&\sum_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in W_k)+\sum_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}\sum_{{{\boldsymbol{{{s}}}}'}\neq{\boldsymbol{{{s}}}}}{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in W_k)^2\\ \le& {\mathsf{{E}}}[|W_k|]+{\mathsf{{E}}}[|W_k|]^2. \end{aligned}$$ In particular, if there exists some subset of values $k\in[n]$ and some $\varepsilon>0$ for which $$\label{eq:k-largeavg} \liminf_{n\to\infty}\frac{1}{n}\log{\mathsf{{E}}}[|W_k|]\ge 1+\varepsilon,$$ then \[eq:bound2m,eq:k-largeavg\] ensure that for those $k$’s the following asymptotic estimate holds $$\frac{{\mathsf{{E}}}[|W_k|^2]}{{\mathsf{{E}}}[|W_k|]^2}\ge 1-\frac{1}{(1+\varepsilon)^n}.$$ The above argument fails if the $W_k$ is replaced by its subset $Z_k$. This is due to the fact that $${\mathsf{{P}}}({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k)\neq{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2.$$ Since the proof is identical for both $|Z_k|$ and $|\operatorname{\mathsf{{NE}}}|$, we will prove the lemma using $|Z_k|$. The proof for $|\operatorname{\mathsf{{NE}}}|$ is similar. We claim that for every ${\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}$ differing in at least three strategies, the events $\{{\boldsymbol{{{s}}}}\in Z_k\}$ and $\{{{\boldsymbol{{{s}}}}'}\in Z_k\}$ are independent. We notice that the event $\{{\boldsymbol{{{s}}}}\in Z_k\}$ is measurable with respect to the $\sigma$-field $$\label{eq:sigma1} \sigma\big(\{u_{{i}}({\boldsymbol{{{s}}}})\}, \{u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1) \}\::\:{{i}}\in[n]).$$ We remark that in the independent case, i.e., ${\pi}(\cdot)=\delta_p(\cdot)$, if ${{\boldsymbol{{{s}}}}'}$ differs from ${\boldsymbol{{{s}}}}$ in at least two strategies we have that the events $\{{\boldsymbol{{{s}}}}\in Z_k \}$ and $\{{{\boldsymbol{{{s}}}}'}\in Z_k \}$ are measurable with respect to independent $\sigma$-fields, hence they are independent. On the other hand, in the general case, the events $\{{\boldsymbol{{{s}}}}\in Z_k \}$, $\{{{\boldsymbol{{{s}}}}'}\in Z_k \}$ are still measurable with respect to the $\sigma$-fields of the type in \[eq:sigma1\], nonetheless, if ${\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}$ differ in a only one or two strategies, such $\sigma$-fields are not independent. Notice that, in particular, $\{{\boldsymbol{{{s}}}}\in Z_k \}$ is measurable with respect to the $\sigma$-field generated by the complete information about the payoffs of all the players in the neighboring profiles, i.e., $$\sigma\big(\{u_{{i}}({\boldsymbol{{{s}}}}) \}, \{\boldsymbol{u}(({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1)) \}\::\:i\in[n] \big).$$ Clearly, if ${\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}$ differs in more than two strategies, then they are measurable with respect to independent $\sigma$-fields, hence are independent. We now want to upper bound the probability of the event $\{{\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k\}$ when the two profiles ${\boldsymbol{{{s}}}}$ and ${{\boldsymbol{{{s}}}}'}$ differs in only one or two strategies. We start by analyzing the case in which there exists a unique ${{i}}\in[n]$ such that $${{\boldsymbol{{{s}}}}'}=\big({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1\big).$$ Then, $$\begin{aligned} \label{eq:cond-dist-1} {\mathsf{{P}}}\({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k \)=&{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \){\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\:|\:{\boldsymbol{{{s}}}}\in Z_k \)\\ \nonumber=&{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \){\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\:|\:u_i({{\boldsymbol{{{s}}}}'})\le u_i({\boldsymbol{{{s}}}}) \)\\ \nonumber=&{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \)\frac{{\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\cap u_i({{\boldsymbol{{{s}}}}'})\le u_i({\boldsymbol{{{s}}}}) \)}{{\mathsf{{P}}}\( u_i({{\boldsymbol{{{s}}}}'})\le u_i({\boldsymbol{{{s}}}}) \)}\\ \nonumber=&\frac{1}{1-p(1-{{p}})}{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \){\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\cap u_i({{\boldsymbol{{{s}}}}'})= u_i({\boldsymbol{{{s}}}}) \). \end{aligned}$$ Notice that the probability of the intersection must satisfy, uniformly in $k$, $${\mathsf{{P}}}^{({{n}})}\({\boldsymbol{{{s}}}}\in Z_k\cap u_i({{\boldsymbol{{{s}}}}'})= u_i({\boldsymbol{{{s}}}}) \)=(1-{{p}})^2{\mathsf{{P}}}^{(n-1)}\({\boldsymbol{{{s}}}}\in Z_k \)+{\mathbf{1}}_{k>0}\cdot {{p}}^{2}\cdot {\mathsf{{P}}}^{(n-1)}\({\boldsymbol{{{s}}}}\in Z_{k-1} \)=\Theta\({\mathsf{{P}}}^{({{n}})}\({\boldsymbol{{{s}}}}\in Z_k\) \).$$ Hence, we can conclude that for all distinct ${\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in{{\Sigma}}$ differing in exactly one strategy, uniformly in $k$, $${\mathsf{{P}}}\({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k \)=\Theta\( {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \)^2 \).$$ Let us now consider the case in which ${\boldsymbol{{{s}}}}$ and ${{\boldsymbol{{{s}}}}'}$ are at distance $2$. In other words, assume that there are two distinct players ${{i}}$ and ${{j}}$, such that $${{\boldsymbol{{{s}}}}'}=\big({\boldsymbol{{{s}}}}_{-{{i}}{{j}}},{{s}}_{{{i}}}\oplus1,{{s}}_{{{j}}}\oplus1\big).$$ Consider also the intermediate strategies $${\boldsymbol{{{s}}}}'':=\big({\boldsymbol{{{s}}}}_{-{{j}}},{{s}}_{{j}}\oplus 1 \big)=\big({{\boldsymbol{{{s}}}}'}_{-{{i}}},{{{{s}}}'}_{{i}}\oplus 1 \big),\qquad {\boldsymbol{{{s}}}}''':=\big({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1 \big)=\big({{\boldsymbol{{{s}}}}'}_{-{{j}}},{{{{s}}}'}_{{j}}\oplus 1 \big).$$ Arguing as in \[eq:cond-dist-1\] we get $$\begin{aligned} \label{eq:cond-dist-2} {\mathsf{{P}}}\({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k \)=&{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \){\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\mid {\boldsymbol{{{s}}}}\in Z_k \)\\ \nonumber=&{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \){\mathsf{{P}}}\({{\boldsymbol{{{s}}}}'}\in Z_k\mid u_{{i}}({\boldsymbol{{{s}}}})\ge u_{{i}}({\boldsymbol{{{s}}}}''')\cap u_{{j}}({\boldsymbol{{{s}}}})\ge u_{{j}}({\boldsymbol{{{s}}}}'') \)\\ =&\Theta\big({\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2 \big). \end{aligned}$$ We can now compute the second moment. Denote by $\mathcal{N}_\ell({\boldsymbol{{{s}}}})$ the set of strategy profiles differing from ${\boldsymbol{{{s}}}}$ in exactly $\ell$ coordinates. By the asymptotic estimates in \[eq:cond-dist-1,eq:cond-dist-2\] we can conclude that there exists some constant $C=C(p)>0$, such that $$\begin{aligned} \label{eq:estimate-2m}{\mathsf{{E}}}[|Z_k|^2]=&\sum_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}\sum_{{{\boldsymbol{{{s}}}}'}\in{{\Sigma}}}{\mathsf{{P}}}\({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k \)\\ \nonumber=&\sum_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}\[{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)+\sum_{{{\boldsymbol{{{s}}}}'}\in{\ensuremath{\mathcal N}}_1({\boldsymbol{{{s}}}})}{\mathsf{{P}}}({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k)+\sum_{{{\boldsymbol{{{s}}}}'}\in{\ensuremath{\mathcal N}}_2({\boldsymbol{{{s}}}})}{\mathsf{{P}}}({\boldsymbol{{{s}}}},{{\boldsymbol{{{s}}}}'}\in Z_k)+\sum_{{{\boldsymbol{{{s}}}}'}\in{\ensuremath{\mathcal N}}_{\ge 3}({\boldsymbol{{{s}}}})}{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2 \]\\ \nonumber\le& {\mathsf{{E}}}|Z_k|+C\cdot 2^n\cdot\(n+n(n-1)\){\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2 +2^n\cdot\(2^n-n-n(n-1)-1 \){\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2\\ \nonumber=&{\mathsf{{E}}}|Z_k|+(1+o(1))2^{2n}{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2={\mathsf{{E}}}|Z_k|+(1+o(1)){\mathsf{{E}}}[|Z_k|]^2. \end{aligned}$$ Therefore, if $\liminf_{n\to\infty}\frac{1}{n}\log{\mathsf{{E}}}|Z_k|>1+\varepsilon$ for some $\varepsilon>0$, $$\label{eq:ratio-exp-small} \frac{{\mathsf{{E}}}[|Z_k|^2]}{{\mathsf{{E}}}[|Z_k|]^2}\le 1+\frac{C\cdot 2^{{{n}}}\cdot n^2 {\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2}{2^{2n}\cdot {\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)^2}+\frac{1}{{\mathsf{{E}}}|Z_k|}\le 1+\frac{1}{(1+\varepsilon)^n} \qedhere$$ Proof of the convergence {#se:proof-convergence} ------------------------ Notice that \[th:convergence-p\] is a particular case of \[th:convergence-pq\], with ${{q}}=0$. In this section we will use a unified approach, showing directly \[th:convergence-pq\] and obtaining \[th:convergence-p\] by taking ${{q}}=0$. The first lemma deals with the expected size of the sets in \[eq:def-WZ\]. We remind to the reader that the entropy of a Bernoulli$(x)$ is defined as $H:(0,1)\to(0,\infty)$, where $$H(x):=-x\log(x)-(1-x)\log(1-x).$$ The following definitions are needed. \[def:fg\] Consider the following bounded analytic functions - The function $f_{W}:(0,1)^2\to[0,2]$ is defined as $$f_{W}(p,x):=2p^x(1-{{p}})^{1-x}e^{H(x)}.$$ - The function $f_{Z}:(0,1)^2\to[0,2]$ is defined as $$f_{Z}(p,x):=2p^x(1-{{p}})^{2(1-x)}e^{H(x)}.$$ - The function $g_{W}:(0,1)^3\to[0,2]$ is defined as $$g_{W}(p,q,x):=\max \left\{f_{W}(q+(1-q)p,x),f_{W}((1-q)p,x)\right\}.$$ - The function $g_{Z}^+:(0,1)^3\to[0,2]$ is defined as $$g_{Z}^+(p,q,x)=f_{W}(q+(1-q)p,x)(1-{{p}})^{1-x}.$$ - The function $g_{Z}^-:(0,1)^3\to[0,2]$ is defined as $$g_{Z}^-(p,q,x)=f_{W}((1-q)p,x)(1-{{p}})^{1-x}.$$ - The function $g_{Z}:(0,1)^3\to[0,2]$ is defined as $$g_{Z}(p,q,x):=(1-{{p}})^{1-x}g_{W}(p,q,x)=\max\left\{g_{Z}^-(p,q,x),g_{Z}^+(p,q,x) \right\}.$$ Notice that, for all $(p,x)\in(0,1)^2$ $$g_{W}(p,0,x)=f_{W}(p,x),\qquad g_{Z}^-(p,0,x)=g_{Z}^+(p,0,x)=g_{Z}(p,0,x)=f_{Z}(p,x).$$ Moreover, for all $p\in(0,1)$, the functions defined in \[def:fg\] admit the limits $x\uparrow1$ and $x\downarrow0$, see \[fact:f,fact:g\]. Therefore, for all $p\in(0,1)$ we can extend the functions in \[def:fg\], as functions of the second variable, to continuous functions in $[0,1]$. \[se:other-proofs\] The forthcoming \[fact:f,fact:g\] establish some easy facts about the behavior of the functions defined in \[def:fg\], which can be checked by direct computation. \[fact:f\] The functions $f_{W}$ and $f_{Z}$ have the following properties: 1. \[fact1\] For every $p\in(0,1)$ $$\frac{\partial}{\partial x}\log f_{W}(p,x)=\log(\eta(p,x)),$$ where $$\label{eq:def-eta} \eta(p,x):=\frac{p}{1-p}\cdot\frac{1-x}{x}.$$ Hence, fixed any $p\in(0,1)$ $$\frac{\partial}{\partial x} f_{W}(p,x)=0\iff x=p.$$ Moreover, $$f_{W}(p,p)=2.$$ 2. \[fact2\] For every $x\in(0,1)$ $$\frac{\partial}{\partial p}\log f_{W}(p,x)=\tau(p,x),$$ where $$\label{eq:def-tau} \tau(p,x):=\frac{x-p}{p(1-{{p}})}.$$ Hence, fixed any $x\in(0,1)$ $$\frac{\partial}{\partial p}f_{W}(p,x)=0\quad\iff\quad p=x.$$ 3. \[fact3\] For every $p\in(0,1)$ $$\lim_{x\uparrow 1}f_{W}(p,x)=2p,\qquad \lim_{x\downarrow 0}f_{W}(p,x)=2-2p.$$ 4. \[fact4\] For every $p,x\in(0,1)$ $$f_{W}(p,x)> f_{Z}(p,x).$$ 5. \[fact5\] For every $p\in(0,1)$ $$\frac{\partial}{\partial x}\log f_{Z}(p,x)=\log\(\beta(p,x)\)$$ where $$\label{eq:def-beta} \beta(p,x):=\frac{p}{(1-{{p}})^2}\cdot\frac{1-x}{x}$$ hence, fixed any $p\in(0,1)$ $$\frac{\partial}{\partial x}f_{Z}(p,x)=0\quad\iff\quad x=\widehat{x}(p):=\frac{p}{1-p+{{p}}^{2}},$$ moreover $$f_{Z}\(p,\widehat x(p)\)=1+{{p}}^{2}+(1-{{p}}^{2})=:1+\alpha(p).$$ 6. \[fact6\] For every $x\in(0,1)$ $$\frac{\partial}{\partial p}\log f_{Z}(p,x)=\upsilon(p,x),$$ where $$\label{eq:def-upsilon} \upsilon(p,x):=\frac{x-p(2-x)}{p(1-{{p}})}.$$ Hence, fixed any $x\in(0,1)$ $$\frac{\partial}{\partial p}f_{Z}(p,x)=0\quad\iff\quad p=\frac{x}{2-x}.$$ 7. \[fact7\] For every $p\in(0,1)$ $$\lim_{x\downarrow 0}f_{Z}(p,x)=2(1-{{p}})^2,\qquad \lim_{x\uparrow 1}f_{Z}(p,x)=2p.$$ \[fact:g\] The functions $g^+_{Z}(p,q,x)$ and $g^-_{Z}(p,q,x)$, defined in \[def:fg\], have the following properties: 1. \[g1\] For every $p,q\in(0,1)$ $$\begin{aligned} \frac{\partial}{\partial x}\log g_{Z}^+(p,q,x)=&\log(\beta_+(p,q,x)),\qquad \frac{\partial}{\partial x}\log g_{Z}^-(p,q,x)=\log(\beta_-(p,q,x)) \end{aligned}$$ where $$\label{eq:def-beta+-} \beta_+(p,q,x):=\frac{q+(1-q)p}{(1-{{p}})^2(1-q)}\cdot \frac{1-x}{x},\qquad\beta_-(p,q,x):=\frac{(1-q)p}{(1 - p) (1 - p (1 - q))}\cdot \frac{1-x}{x}.$$ 2. \[g2\] For all $p,q\in(0,1)$ there exists two points $$\label{eq:def-x+-} x^+(p,q):=\frac{q+p(1-q)}{1-p(1-q)+{{p}}^{2}(1-q)},\qquad x^-(p,q):=\frac{p(1-q)}{1-p+{{p}}^{2}(1-q)}$$ such that $$\frac{\partial}{\partial x}g_{Z}^\pm(x)\begin{cases} >0&\text{if }x<x^\pm(p,q),\\ =0&\text{if }x=x^\pm(p,q),\\ <0&\text{if }x>x^\pm(p,q).\\ \end{cases}$$ 3. \[g3\] For every $p,q,x\in(0,1)$ $$\frac{\partial}{\partial q}\log g^+_{Z}(p,q,x)=\frac{q+p (q-1)+x}{(q-1) (p (q-1)+q)},$$ $$\frac{\partial}{\partial q}\log g^-_{Z}(p,q,x)=\frac{p (q-1)+x}{(q-1) (p (q-1)+1)}.$$ Moreover, $$\frac{\partial}{\partial q} g^\pm_{Z}(p,q,x)\begin{cases} >0&\text{if }x<\upsilon^\pm(p,x),\\ =0&\text{if }x=\upsilon^\pm(p,x),\\ <0&\text{if }x>\upsilon^\pm(p,x),\\ \end{cases}$$ where $$\upsilon^-(p,x):=\frac{p-x}{p},\qquad \upsilon^+(p,x):=\frac{x-p}{1-p}.$$ 4. \[g4\] For every $p,q\in(0,1)$ $$g_{Z}^+(p,q,x)\begin{cases} <g_{Z}^-(p,q,x)&\text{if }x<\gamma(p,q),\\ >g_{Z}^-(p,q,x)&\text{if }x=\gamma(p,q),\\ >g_{Z}^-(p,q,x)&\text{if }x>\gamma(p,q),\\ \end{cases}$$ where, $$\label{eq:def-gamma} \gamma(p,q):=\frac{\log\theta_1(p,q) }{\log\theta_2(p,q)},$$ $$\label{eq:def-theta12} \theta_1(p,q):=\frac{(1-{{p}})(1-q)}{1-p(1-q)},\qquad\theta_2(p,q):=\theta_1(p,q)\frac{p(1-q)}{q-p(1-q)}.$$ 5. \[g5\] For every $p,q\in(0,1)$ $$\lim_{x\downarrow0}g^-_{Z}(p,q,x)=2 (1 -p) (1 - p (1 - q)),\qquad \lim_{x\downarrow0}g^+_{Z}(p,q,x)=2 (1 -p)^2 (1 - q),$$ and $$\lim_{x\uparrow1}g^-_{Z}(p,q,x)=2 p(1-q),\qquad \lim_{x\uparrow1}g^+_{Z}(p,q,x)=2\big(q+p(1-q)\big).$$ Notice that if $q=0$ we are back to the independent case, indeed $g_{Z}^\pm(p,0,x)=f_{Z}(p,x)$ for every $p,x\in(0,1)$ and $$\beta_\pm(p,0,x)=\frac{p}{(1-{{p}})^2}\cdot \frac{1-x}{x} ,\qquad x^{\pm}(p,0)= \frac{p}{1-p+{{p}}^{2}}\qquad g_{Z}^\pm(p,0,x^\pm(p,0))=1+\alpha(p).$$ \[le:mean-q\] For all $(p,q)\in(0,1)\times[0,1)$, $\varepsilon>0$, $k\in[\varepsilon n,(1-\varepsilon)n]$ we have $$\frac{1}{{{n}}}\log {\mathsf{{E}}}|W_{k}|\sim \log g_{W}\big(p,q,\tfrac{k}{n}\big),\qquad \frac{1}{{{n}}}\log {\mathsf{{E}}}|Z_{k}|\sim \log g_{Z}\big(p,q,\tfrac{k}{n}\big).$$ By independence of payoffs under different strategy profile, we have $$\begin{aligned} {\mathsf{{E}}}[|W_k|]=&2^n{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in W_k)\\ =&2^n\cdot \left[p\cdot {\mathsf{{P}}}\left(\text{Bin}(n,q+(1-q)p)=k \right)+(1-{{p}})\cdot {\mathsf{{P}}}\left(\text{Bin}(n,(1-q)p)=k \right) \right]\\ =&p\cdot 2^n\cdot \left(\binom{{{n}}}{k}(q+(1-q)p)^k(1-q-(1-q)p)^{n-k} \right)+\\ &+(1-{{p}})\cdot 2^n\cdot \left(\binom{{{n}}}{k}((1-q)p)^k(1-(1-q)p)^{n-k} \right), \end{aligned}$$ where, to get the second equality, we conditioned on the outcome of $X({\boldsymbol{{{s}}}})$, defined at the beginning of \[se:dependent-payoffs\]. Fix $\varepsilon>0$, and pick some $k\in\[\varepsilon n,(1+\varepsilon)n\]$. Let $x=\frac{k}{{{n}}}$, and notice that using the asymptotic approximation $$\binom{n}{xn}=e^{(1+o(1))H(x)n},$$ we can estimate $$\begin{aligned} \label{eq:est-mean-W} {\mathsf{{E}}}[|W_k|]=& p\cdot\left[(1+o(1))2 e^{H(x)}(q+(1-q)p)^x\left(1-q-(1-q)p\right)^{1-x} \right]^n+\\ \nonumber&+(1-{{p}})\cdot\left[(1+o(1))2 e^{H(x)}((1-q)p)^x\left(1-(1-q)p\right)^{1-x} \right]^n\\ \nonumber=&p\cdot[(1+o(1))f_{W}(q+(1-q)p,x)]^n+(1-{{p}})\cdot [(1+o(1)) f_{W}((1-q)p,x)]^n. \end{aligned}$$ Notice that the convex coefficients ${{p}}$ and $1-p$ are absorbed in the $(1+o(1))$ error within the squared brackets. Hence, if $k\in[\varepsilon n,(1-\varepsilon)n]$ for any $\varepsilon>0$, $$\begin{aligned} {\mathsf{{E}}}[|W_k|]=&[(1+o(1))\max\{f_{W}(q+(1-q)p,\tfrac{k}{n}),f_{W}((1-q)p)\}]^n. \end{aligned}$$ By taking the logarithm and normalizing, $$\frac{1}{{{n}}}\log{\mathsf{{E}}}|W_k|\sim\max\{\log f_{W}(q+(1-q)p,\tfrac{k}{n}),\log f_{W}((1-q)p,\tfrac{k}{n})\}.$$ We now compute the expected size of $Z_k$. By conditioning on the social utility of the profile ${\boldsymbol{{{s}}}}\in{{\Sigma}}$ $$\begin{aligned} {\mathsf{{E}}}[|Z_k|]=&2^n{\mathsf{{P}}}({\boldsymbol{{{s}}}}\in Z_k)\\ =&2^n {\mathsf{{P}}}({\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\:|\:{\boldsymbol{{{s}}}}\in W_k){\mathsf{{P}}}({\boldsymbol{{{s}}}}\in W_k). \end{aligned}$$ Notice that, conditioning on any $\omega\in\{{\boldsymbol{{{s}}}}\in W_k \}$, the probability that ${\boldsymbol{{{s}}}}$ is an equilibrium is exactly the probability of the event $$\label{eq:cond-nash} \left\{u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1)=0,\:\forall {{i}}\text{ s.t. }u_{{i}}({\boldsymbol{{{s}}}})=0 \right\}.$$ By the independence across ${{i}}$’s in the events in \[eq:cond-nash\], we conclude that $${\mathsf{{P}}}\(\bigcap_{{{i}}\colon u_{{i}}({\boldsymbol{{{s}}}})=0}u_{{i}}({\boldsymbol{{{s}}}}_{-{{i}}},{{s}}_{{i}}\oplus 1)=0 \:\big\vert\: {\boldsymbol{{{s}}}}\in W_k \)=(1-{{p}})^{n-k}.$$ Hence, using the same argument as in \[eq:est-mean-W\], $$\begin{aligned} {\mathsf{{E}}}[|Z_k|]=&2^n\cdot(1-{{p}})^{n-k}\cdot \left[p\cdot {\mathsf{{P}}}\left(\text{Bin}(n,q+(1-q)p)=k \right)+(1-{{p}})\cdot {\mathsf{{P}}}\left(\text{Bin}(n,(1-q)p)=k \right) \right]\\ =&\[(1+o(1))\max\{g^-_{Z}(p,q,\tfrac{k}{n}),g^+_{Z}(p,q,\tfrac{k}{n}) \}\]^n. \end{aligned}$$ Therefore, if $k\in[\varepsilon n,(1-\varepsilon)n]$ for any $\varepsilon>0$, $$\frac{1}{{{n}}}\log{\mathsf{{E}}}\[|Z_{k}|\]\sim\max\{\log g^-_{Z}(p,q,\tfrac{k}{n}),\log g^+_{Z}(p,q,\tfrac{k}{n}) \}. \qedhere$$ We can use the result in \[le:mean-q\] to show \[th:num-ne-q\], which controls the number and typical efficiency of equilibria. In order to compute the expectation of $|\operatorname{\mathsf{{NE}}}|$ it is sufficient to notice that $$\begin{aligned} {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\)=&p\:{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\mid X({\boldsymbol{{{s}}}})=1 \)+(1-p)\:{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\mid X({\boldsymbol{{{s}}}})=0 \)\\ =&p\:\(1-p(1-q)(1-p) \)^n+(1-p)\:(1-p(q+(1-q)(1-p))^n. \end{aligned}$$ Therefore $$\begin{aligned} \label{eq:mean} {\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}| ]=2^n{\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}\)=&p\:\[2\( 1-p(1-q)(1-p) \) \]^n(1+o(1))\\ =&p\:\[1+\alpha(p)+2qp(1-p) \]^n\\ =&\Omega\(\(3/2 \)^n \) \end{aligned}$$ Therefore, by \[pr:var\], $|\operatorname{\mathsf{{NE}}}|/{\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}|]\overset{{\mathsf{{P}}}}{\longrightarrow} 1, $ and by \[eq:mean\] we get \[eq:thm3-1\]. At this point, in order to show the validity of \[eq:efficiency-q\] it is sufficient to notice that for every sufficiently small $\varepsilon>0$ $$1<\frac{1}{n}\log\({\mathsf{{E}}}[|\operatorname{\mathsf{{NE}}}\setminus\widehat{\operatorname{\mathsf{{NE}}}}_\varepsilon |]\)<\log\(1+\alpha(p)+2qp(1-p) \)$$ and again by \[pr:var\], $|\widehat{\operatorname{\mathsf{{NE}}}}_\varepsilon|/{\mathsf{{E}}}[|\widehat{\operatorname{\mathsf{{NE}}}}_\varepsilon|]\overset{{\mathsf{{P}}}}{\longrightarrow} 1 $. Notice that if there exists some $\delta>0$ such that $g_{W}(p,q,k/n)\ge 1+\delta$ then the expectation ${\mathsf{{E}}}|W_{k}|=\omega(1)$, while if $g_{W}(p,q,k/n)\le1-\delta$ then the expectation ${\mathsf{{E}}}|W_{k}|=o(1)$. The main idea of the proof of \[th:convergence-pq\] goes as follows. If for some triple $(p,q,x)$ we have $g_{Z}(p,q,x)<1$, then by Markov’s inequality we can infer that asymptotically there are no Nash equilibria with average social utility $x+o(1)$. On the other hand, if $g_{Z}(p,q,x)>1$, the set of equilibria with average social utility $x+o(1)$ is not empty; this will be proved by the control on the second moments in \[pr:var2\]. We start by using \[le:mean-q\] to control the probability that the set $Z_k$ is empty. As remarked above, for every $k$ that is not too close to $0$ or $n$, the expectation of $|Z_k|$ is exponential of rate $g_{NE}(p,q,k/n)\in[0,2]$. Hence, in what follows we will be interested in studying the solution of the following equations $$g_{W}(p,q,x)=1\quad\text{and}\quad g_{Z}(p,q,x)=1.$$ ![In Blue: $f_{Z}(p,x)$ as function of $x$. In Orange: $f_{W}(p,x)$ as function of $x$. The value of ${{p}}$ in the four pictures is, respectively, $p=0.25,1-\sqrt{2}/2,0.4$. The green line is at height $1$. []{data-label="fig:rates"}](f1f2025 "fig:"){width="4.5cm"} ![In Blue: $f_{Z}(p,x)$ as function of $x$. In Orange: $f_{W}(p,x)$ as function of $x$. The value of ${{p}}$ in the four pictures is, respectively, $p=0.25,1-\sqrt{2}/2,0.4$. The green line is at height $1$. []{data-label="fig:rates"}](f1f2pc "fig:"){width="4.5cm"} ![In Blue: $f_{Z}(p,x)$ as function of $x$. In Orange: $f_{W}(p,x)$ as function of $x$. The value of ${{p}}$ in the four pictures is, respectively, $p=0.25,1-\sqrt{2}/2,0.4$. The green line is at height $1$. []{data-label="fig:rates"}](f1f204 "fig:"){width="4.5cm"} See \[fig:rates,fig:example\] for a representation of the functions $g_{W}(p,q,x)$ and $g_{Z}(p,q,x)$ as a function of the variable $x$ for some fixed values of ${{p}}$ and ${{q}}$. For instance, when ${{q}}=0$, for every given $p\in(0,1)$, the smallest $x$ for which $f_{Z}(p,x)\ge 1$ is our proxy for the of the worst equilibrium, while the largest $x$ for which $f_{Z}(p,x)\ge 1$ is our proxy for the of the best equilibrium. Similarly, the optimum can be obtained by looking at the largest $x$ for which $f_{W}(p,x)\ge 1$. We formalize this intuition in the following proposition, which is the technical version of the more readable \[th:convergence-pq\]. ![Plot of the functions $g_{W}(0.5,0.75,x)$ (in orange) and $g_{Z}(0.5,0.75,x)$ (in blue). The fact that the orange curve lies above the line at height one, means that the expectation of the number of strategy with $k$ diverges exponentially fast, for all $k\in[0,n]$. On the other hand, the regions where the blue curve lies below the green line denotes values of which will not appear within the strategies in $\operatorname{\mathsf{{NE}}}$.[]{data-label="fig:example"}](example-p05-q075){width="7cm"} \[pr:2mm\] Fix $p,q\in(0,1)\times[0,1)$ and any $\varepsilon,\delta>0$. 1. \[it:pr:2mm-1\] Call $$\begin{aligned} \label{eq:N+} N_{\varepsilon,\delta}^+:=&\{k\in[\delta n,(1-\delta)n]\::\: g_{W}\big(p,q,\tfrac{k}{{{n}}}\big)\ge 1+\varepsilon \},\\ \label{eq:N-} N_{\varepsilon,\delta}^-:=&\{k\in[\delta n,(1-\delta)n]\::\: g_{W}\big(p,q,\tfrac{k}{{{n}}}\big)\le 1-\varepsilon \},\end{aligned}$$ we have $$\begin{aligned} \label{eq:empty1} \lim_{{{n}}\to\infty}{\mathsf{{P}}}\big(\cup_{ k\in N_{\varepsilon,\delta}^-}W_{k}=\varnothing\big)=1,\qquad \lim_{{{n}}\to\infty}{\mathsf{{P}}}\big(\forall k\in N^{+}_{\varepsilon,\delta},\: W_k\neq\varnothing \big)=1.\end{aligned}$$ 2. \[it:pr:2mm-2\] Called $$\begin{aligned} \label{eq:M+} M_{\varepsilon,\delta}^+:=&\{k\in[\delta n,(1-\delta)n]\::\: g_{Z}\big(p,q,\tfrac{k}{{{n}}}\big)\ge 1+\varepsilon \},\\ \label{eq:M-} M_{\varepsilon,\delta}^-:=&\{k\in[ \delta n,(1-\delta)n]\::\: g_{Z}\big(p,q,\tfrac{k}{{{n}}}\big)\le 1-\varepsilon \},\end{aligned}$$ we have, $$\begin{aligned} \label{eq:empty2} \lim_{{{n}}\to\infty}{\mathsf{{P}}}\big(\cup_{k\in M_{\varepsilon,\delta}^-}Z_{k}=\varnothing\big)=1,\qquad \lim_{{{n}}\to\infty}{\mathsf{{P}}}\big(\forall k\in M^{+}_{\varepsilon,\delta},\: Z_k\neq\varnothing \big)=1.\end{aligned}$$ We prove \[it:pr:2mm-1\]; the proof of \[it:pr:2mm-2\] is similar. Fix a pair $p,q\in(0,1)\times[0,1]$ and $\varepsilon,\delta>0$. Pick some $k\in N_{\varepsilon,\delta}^-$. We have $${\mathsf{{P}}}\(|W_{k}|\ge 1 \)\le {\mathsf{{E}}}[|W_k|],$$ hence, for every sufficiently large $n$ $$\frac{1}{n}\log{\mathsf{{P}}}\(|W_{k}|\ge 1 \)\le 1-\varepsilon,$$ where the first is Markov inequality and the second estimate follows from \[le:mean-q\] and holds uniformly in $k\in N_{\varepsilon,\delta}^-$. By applying the union bound we get the first limit in \[eq:empty1\]. On the other hand, by \[pr:var2\] and the second moment method, [see @AloSpe:Wiley2016 Ch. 4] , $${\mathsf{{P}}}\(W_k\neq\varnothing \)\ge \frac{{\mathsf{{E}}}[|W_k|]^2}{{\mathsf{{E}}}[|W_k|^2]}\ge 1-\frac{1}{(1+\varepsilon)^n}.$$ We can therefore deduce the second limit in \[eq:empty1\] by applying a union bound over $k\in N_{\varepsilon,\delta}^+$. The results in \[th:convergence-p,th:convergence-pq\] now follow as a corollary of \[pr:2mm\]. The claim follows directly by \[eq:empty1,eq:empty2\]. We show the result for the $\operatorname{\mathsf{{Weq}}}$, since the proofs for $\operatorname{\mathsf{{Beq}}}$ and $\operatorname{\mathsf{{SO}}}$ are identical. Fix $\varepsilon,\delta>0$. Notice that, by \[eq:M+\], $$\begin{aligned} \tfrac{1}{{{n}}}\min_{{\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=\tfrac{1}{{{n}}}\min\{k\in[n]\::\: Z_k\neq\varnothing \}\le \tfrac{1}{{{n}}}\min\{k\in M_{\varepsilon,\delta}^+ \}=:\tfrac{1}{{{n}}}\overline{k}_{\varepsilon,\delta}\end{aligned}$$ On the other hand, by \[eq:empty2\], we have $$\tfrac{1}{{{n}}}\min\{k\in[n]\::\: Z_k\neq\varnothing \}\ge \tfrac{1}{{{n}}}\max\{k\in M_{\varepsilon,\delta}^- \:|\: k\le \overline{k}_{\varepsilon,\delta}\}=:\tfrac{1}{{{n}}}\underline{k}_{\varepsilon,\delta},$$ where we define $\underline{k}_{\varepsilon,\delta}=0$ if the set in its definition is empty. By continuity of $g_{Z}(p,q,\cdot)$ and the fact that $g_{Z}(p,q,\cdot)$ and its derivative are bounded around 0 and 1 (see \[fact:g\]), we have that for all $\eta>0$, we can find ${{n}}_0,\varepsilon$ and $\delta$ such that, for all $n>n_0$, $$\bigg|\tfrac{1}{{{n}}}\overline{k}_{\varepsilon,\delta}-\tfrac{1}{{{n}}}\underline{k}_{\varepsilon,\delta}\bigg|<\eta,$$ and $$\big|\tfrac{1}{{{n}}}\overline{k}_{\varepsilon,\delta}-\inf\big\{x\in(0,1)\::\:g_{Z}(p,q,x)\ge 1 \big\} \big|<\eta.$$ Hence, $$\label{eq:first-char-xweq} \lim_{{{n}}\to\infty}{\mathsf{{P}}}\(\big| \tfrac{1}{{{n}}}\min_{{\boldsymbol{{{s}}}}\in\operatorname{\mathsf{{NE}}}}\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})-\inf\big\{x\in(0,1)\:: \:g_{Z}(p,q,x)\ge 1\big\}\big|\le\eta \)=1,$$ which concludes the proof. As a byproduct of the above proof we get the following characterization of the limit functions defined in \[th:convergence-pq\]. By \[eq:first-char-xweq\], for all $p,q\in(0,1)\times[0,1]$, the function $x_{\operatorname{\mathsf{weq}}}$ defined in \[th:convergence-pq\] admits the following implicit representation $$\begin{aligned} \label{eq:weq}x_{\operatorname{\mathsf{weq}}}(p,q)=&\inf\{x\in(0,1)\:: \:g_{Z}(p,q,x)\ge 1\}.\end{aligned}$$ Similarly, the functions $x_{\operatorname{\mathsf{opt}}}$ and $x_{\operatorname{\mathsf{beq}}}$ admit the representations $$\begin{aligned} \label{eq:xopt}x_{\operatorname{\mathsf{opt}}}(p,q)=&\sup\{x\in(0,1)\:: \:g_{W}(p,q,x)\ge 1\},\\ \label{eq:beq}x_{\operatorname{\mathsf{beq}}}(p,q)=&\sup\{x\in(0,1)\:: \:g_{Z}(p,q,x)\ge 1\}.\end{aligned}$$ The phase-transition in \[pr:phase-transition\] is then a consequence of the analysis of the functions in \[eq:weq,eq:xopt\], \[eq:beq\]. We start by analyzing the function $x_{\operatorname{\mathsf{opt}}}:(0,1/2)\to[0,1]$. By \[fact:f\]\[fact3\], we have $$\lim_{x\uparrow 1}\log f_{W}(p,x)<0.$$ This inequality, together with the continuity of $f_{W}(p,x)$, justifies the characterization of $x_{\operatorname{\mathsf{opt}}}(p)$ as the largest solution in $x\in(0,1)$ of the equation $$\label{eq:implicit-def-xopt} \log f_{W}(p,x)=0.$$ In order to use implicit function theorem, we need to check that, for every fixed $p\in(0,1/2)$, the partial derivative with respect to $x$ of the function $f_{W}(p,x)$ does not vanish at the largest solution of \[eq:implicit-def-xopt\]. Notice that $x_{\operatorname{\mathsf{opt}}}(p)>p$ since $f_{W}(p,p)=2$. Moreover, thanks to \[fact:f\]\[fact1\] for every $p\in(0,1/2)$ and $x\in(p,1)$ we have $$\eta(p,x)<1\qquad\Longrightarrow\qquad \frac{\partial}{\partial x}\log f_{W}(p,x)<0,$$ where $\eta$ is defined as in \[eq:def-eta\]. Relying again on the rough estimate $x_{\operatorname{\mathsf{opt}}}(p)>p$, together with \[fact:f\]\[fact2\], we get $$\tau(p,x)>0\qquad\Longrightarrow\qquad\frac{\partial}{\partial p}\log f_{W}(p,x_{\operatorname{\mathsf{opt}}}(p))>0,$$ where $\tau$ is defined as in \[eq:def-tau\]. Therefore, by the implicit function theorem, the implicit function $x_{\operatorname{\mathsf{opt}}}(p)$ defined by \[eq:implicit-def-xopt\] admits, for all $p\in(0,1/2)$, derivative of the form $$\frac{d}{dp}x_{\operatorname{\mathsf{opt}}}(p)\big\rvert_{p_0}=-\frac{\frac{\partial}{\partial p}\log f_{W}(p,x)\big\rvert_{p_0,x_{\operatorname{\mathsf{opt}}}(p_0)}}{\frac{\partial}{\partial x}\log f_{W}(p,x)\big\rvert_{p_0,x_{\operatorname{\mathsf{opt}}}(p_0)}}>0.$$ We proceed similarly for $x_{\operatorname{\mathsf{beq}}}(p)$ in the same interval $(0,1/2)$. By \[fact:f\]\[fact7\] $$\lim_{x\uparrow 1}\log f_{W}(p,x)<0.$$ Hence, by continuity of $f_{Z}(p,x)$, the function $x_{\operatorname{\mathsf{beq}}}(p)$ is defined implicitly as the largest solution in $(0,1)$ of the equation $$\label{eq:implicit-def-xbeq-weq} \log f_{Z}(p,x)=0.$$ Notice that, thanks to \[fact:f\]\[fact5\], $$\label{eq:partial-x-fne} \frac{\partial}{\partial x}\log f_{Z}(p,x)<0,\qquad\forall x>\widehat{x}(p):=\frac{p}{1-p+{{p}}^{2}}.$$ Since $$\label{eq:rough2} \log f_{Z}(p,\widehat{x}(p))=\log(1+\alpha(p))>0,$$ it holds that $x_{\operatorname{\mathsf{beq}}}(p)> \widehat{x}(p)$. Moreover, by \[fact:f\]\[fact6\], we have $$\label{eq:partial-p-fne} \frac{\partial}{\partial p}\log f_{Z}(p,x)>0\qquad\forall x>\frac{2p}{1+p}.$$ Since $$\label{eq:rough1} f_{Z}\(p,\frac{2p}{1+p}\)=(1+p) (2-2 p)^{\frac{2}{p+1}-1} >1,$$ we have $$x_{\operatorname{\mathsf{beq}}}>\frac{2p}{1+p}.$$ In conclusion, by the implicit function theorem, the function $x_{\operatorname{\mathsf{beq}}}$ is $C_1(0,1/2)$ and $$\frac{d}{dp}x_{\operatorname{\mathsf{beq}}}(p)\big\rvert_{p_0}>0.$$ The behavior of the function $x_{\operatorname{\mathsf{opt}}}$ and $x_{\operatorname{\mathsf{beq}}}$ in the interval $(1/2,1)$ follows by the continuity of $f_{W}$ and $f_{Z}$ together with \[fact:f\]\[fact4\] and \[fact:f\]\[fact7\]. We are left to show the regularity properties of the function $x_{\operatorname{\mathsf{weq}}}$. If we restrict to $p\in(1-\sqrt{2}/2,1)$, we can characterize $x_{\operatorname{\mathsf{weq}}}$ as the smallest solution in $x\in(0,1)$ of the equation in \[eq:implicit-def-xbeq-weq\]. Notice that by \[fact:f\]\[fact3\], \[eq:rough1\], and the continuity of $f_{Z}(p,\cdot)$ we deduce that $$x_{\operatorname{\mathsf{weq}}}( p)<\frac{2 p}{1 + p}.$$ For the same reason, given \[eq:rough2\], we have $x_{\operatorname{\mathsf{weq}}}( p)<\frac{p}{1-p+{{p}}^{2}}$. By the converse of the inequalities in \[eq:partial-x-fne,eq:partial-p-fne\], we obtain $$\begin{aligned} \frac{\partial}{\partial p} \log f_{Z}(p,x)<0&\iff x<\frac{2 p}{1 + p}\\ \frac{\partial}{\partial x}f_{Z}(p,x)>0&\iff x<\frac{p}{1-p+{{p}}^{2}}, \end{aligned}$$ So, we can apply the implicit function theorem and conclude that the function $x_{\operatorname{\mathsf{weq}}}(p)$ admits a derivative $$\frac{d}{dp}x_{\operatorname{\mathsf{weq}}}(p)\big\rvert_{p_0} >0\qquad\forall p_0\in(1-1/\sqrt{2},1 ).$$ To conclude, we need to show that $x_{\operatorname{\mathsf{weq}}}$ is constantly zero in the interval $\big(0,1-\tfrac{\sqrt{2}}{2}\big)$. This is just a simple consequence of \[fact:f\]\[fact7\]. In fact, $$\lim_{x\uparrow 0}\log f_{W}(p,x)>0\qquad\iff\qquad p<1-\frac{\sqrt{2}}{2}. \qedhere$$ We are now going to prove the phase-transition phenomenon described in \[pr:phase-transition-q\]. We recall to the reader that here we assume $p$ to be fixed while the moving parameter is the correlation $q$. By steps: (i) \[xtilde1\] By \[fact:g\]\[g5\] we have $$\lim_{x\downarrow 0}g_{Z}^-(p,q,x)=2(1-{{p}})(1-p(1-q))>1\iff q>\rho(p).$$ Notice that $$\rho(p)>0\iff p>1-\frac{1}{\sqrt{2}}.$$ The result follows immediately (ii) We note that $\rho(p) \colon (1-1/\sqrt{2},1/2)\to(0,1)$ is a bijection. Moreover, by the implicit function theorem, for all $q_0\in(0,\rho(p))$, $$\begin{aligned} \frac{d}{dq}\tilde x_{\operatorname{\mathsf{weq}}}(p,q)\big\rvert_{q_0}=&-\frac{\frac{\partial }{\partial q}\log g^-_{Z}(p,q,x)\big\rvert_{q_0,\tilde x_{\operatorname{\mathsf{weq}}}(p,q_0)}}{\frac{\partial }{\partial x}\log g^-_{Z}(p,q,x)\big\rvert_{q_0,\tilde x_{\operatorname{\mathsf{weq}}}(p,q_0)}}\\ \label{eq:numdem}=&-\frac{p (1-q)-x}{(1-q) (1-p (1-q)) \log \left(\beta^-(p,q,x)\right)}\bigg\rvert_{q_0,\tilde x_{\operatorname{\mathsf{weq}}}(p,q_0)},\end{aligned}$$ where $\beta^{-}$ is defined as in \[eq:def-beta+-\]. We aim at showing that the signs of numerator and denominator in \[eq:numdem\] coincide. Note that we have $$\label{eq:st1} \log\beta^-(p,q, x_{\operatorname{\mathsf{weq}}}(p,q))>0,\qquad\forall p\in\big(1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big),\:\forall q>\rho(p).$$ In fact, the sign of $\log\beta^-(p,q,\cdot)$ is the sign of the partial derivative of $g_{Z}^-(p,q,x)$ with respect to $x$, which is positive at $x_{\operatorname{\mathsf{weq}}}(p,q)$. On the other hand, by definition, $$\label{eq:st2} (1-q)(1-p(1-q))>0.$$ Hence, we are left with showing that $$\label{eq:st3} p(1-q)- x_{\operatorname{\mathsf{weq}}}(p,q)>0,\qquad\forall p\in\big(1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big),\:\forall q>\rho(p).$$ Since for $p\in(1-\sqrt{2}/{2},1/2)$ the quantity $ x_{\operatorname{\mathsf{weq}}}(p,q)$ is the smallest solution of the equation $$\log g^-_{Z}(p,q,x)=0,$$ it is sufficient to check that $$\label{eq:st3bis} \log g^-_{Z}(p,q,p(1-q))>0,\qquad\forall p\in\big(1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big),\:\forall q>\rho(p).$$ We can rewrite \[eq:st3bis\] as $$\label{eq:st3tris} \log(2)+(1-p(1-q))\log(1-{{p}})>0,\qquad\forall p\in\big(1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big),\:\forall q>\rho(p).$$ Notice that the following inequality is stronger than \[eq:st3tris\]: $$\label{eq:st4} \log(2)+\log(1-{{p}})>0,\qquad\forall p\in\big(1-\tfrac{\sqrt{2}}{2},\tfrac{1}{2} \big),$$ and the latter is trivially true. (iii) By the same argument used to prove \[xtilde1\], we have $$\lim_{x\downarrow 0}g_{Z}^-(p,q,x)<1 \iff q<\rho(p).$$ To conclude, notice that $\rho(p)>1$ if $p\in(1/2,1)$. (iv) For every $p\in(1/2,1)$ call $q^*(p)$ the unique solution of the equation in $q$ $$\label{eq:max1} \log g_{Z}^-(p,q,x^-(p,q))=\log \left(2 \left({{p}}^{2} (1-q)+(1-{{p}})\right)\right)= 0.$$ For all $p\in(1/2,1)$ the function $g_{Z}^-(p,\cdot,x^{-}(p,\cdot))$ is decreasing in $q\in(0,1)$, and the unique solution of the equation in \[eq:max1\] is given by $$q^*(p)=\frac{1-2p+2{{p}}^{2}}{2{{p}}^{2}},\quad\forall p\in(1/2,1).$$ The existence of a unique solution to \[eq:max1\] implies that, for all $p\in(1/2)$, there exist a unique value of $q$ for which the maximum attained by the curve $g^{-}_{Z}(p,q,\cdot)$ is exactly $1$. Having in mind the plots in \[fig:2nd-pt,fig:1st-pt\], we are interested in understanding whether $$x^-(p,q^*(p)) \lessgtr \inf\{x\in(0,1)\::\: g_{Z}^+(p,q^*(p),x)\ge 1 \}.$$ In order to do so, we aim at analyzing the map $$\label{eq:fun-max-pq-plus} p\mapsto g_{Z}^+\big(p,q^*(p),x^-(p,q^*(p))\big),$$ namely, the value assumed by the function $g_{Z}^+$ at the point in which the function $g_{Z}^-$ attains its maximum height, i.e., 1. We start by claiming that the function in \[eq:fun-max-pq-plus\] is increasing, hence there exists a unique solution in $(1/2,1)$ of the equation in ${{p}}$ $$\label{eq:def-imp-pc} g^+_{Z}\big(p,q^{*}(p),x^-(p,q^*(p))\big)=1.$$ We define $p_c$ the solution of \[eq:def-imp-pc\]; numerically, $p_c\approx 0.731642$. As suggested by the plots in \[fig:2nd-pt,fig:1st-pt\], we expect two different behaviors of the function $x_{\operatorname{\mathsf{weq}}}(p,\cdot)$ when $p\in(1/2,p_c)$ (see \[fig:2nd-pt\]), and when $p\in(p_c,1)$ (see \[fig:1st-pt\]). ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.6<p_c$ and $q=0.5,0.68,0.74,0.77$, respectively. Recall that $g_{Z}(p,q,x)=g_{Z}^-(p,q,x)$ on the left of the dashed line $x=\gamma(p,q)$, while $g_{Z}(p,q,x)=g_{Z}^+(p,q,x)$ on the right. Similarly $g_{W}(p,q,x)=f_{W}((1-q)p,x)$ on the left of the dashed line, while $g_{W}(p,q,x)=f_{W}(q+(1-q)p,x)$ on the right.[]{data-label="fig:2nd-pt"}](newfuns-p06-q05 "fig:"){width="5cm"} ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.6<p_c$ and $q=0.5,0.68,0.74,0.77$, respectively. Recall that $g_{Z}(p,q,x)=g_{Z}^-(p,q,x)$ on the left of the dashed line $x=\gamma(p,q)$, while $g_{Z}(p,q,x)=g_{Z}^+(p,q,x)$ on the right. Similarly $g_{W}(p,q,x)=f_{W}((1-q)p,x)$ on the left of the dashed line, while $g_{W}(p,q,x)=f_{W}(q+(1-q)p,x)$ on the right.[]{data-label="fig:2nd-pt"}](newfuns-p06-q068 "fig:"){width="5cm"}\ ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.6<p_c$ and $q=0.5,0.68,0.74,0.77$, respectively. Recall that $g_{Z}(p,q,x)=g_{Z}^-(p,q,x)$ on the left of the dashed line $x=\gamma(p,q)$, while $g_{Z}(p,q,x)=g_{Z}^+(p,q,x)$ on the right. Similarly $g_{W}(p,q,x)=f_{W}((1-q)p,x)$ on the left of the dashed line, while $g_{W}(p,q,x)=f_{W}(q+(1-q)p,x)$ on the right.[]{data-label="fig:2nd-pt"}](newfuns-p06-q074 "fig:"){width="5cm"} ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.6<p_c$ and $q=0.5,0.68,0.74,0.77$, respectively. Recall that $g_{Z}(p,q,x)=g_{Z}^-(p,q,x)$ on the left of the dashed line $x=\gamma(p,q)$, while $g_{Z}(p,q,x)=g_{Z}^+(p,q,x)$ on the right. Similarly $g_{W}(p,q,x)=f_{W}((1-q)p,x)$ on the left of the dashed line, while $g_{W}(p,q,x)=f_{W}(q+(1-q)p,x)$ on the right.[]{data-label="fig:2nd-pt"}](newfuns-p06-q077 "fig:"){width="5cm"} ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.8>p_c$ and $q=0.2,0.45,0.52,0.55$, respectively. The dashed line lies at $x=\gamma(p,q)$.[]{data-label="fig:1st-pt"}](newfuns-p08-q02 "fig:"){width="5cm"} ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.8>p_c$ and $q=0.2,0.45,0.52,0.55$, respectively. The dashed line lies at $x=\gamma(p,q)$.[]{data-label="fig:1st-pt"}](newfuns-p08-q045 "fig:"){width="5cm"}\ ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.8>p_c$ and $q=0.2,0.45,0.52,0.55$, respectively. The dashed line lies at $x=\gamma(p,q)$.[]{data-label="fig:1st-pt"}](newfuns-p08-q052 "fig:"){width="5cm"} ![Plot of $g_{Z}(p,q,x)$ (blue) and $g_{W}(p,q,x)$ (orange) when $p=0.8>p_c$ and $q=0.2,0.45,0.52,0.55$, respectively. The dashed line lies at $x=\gamma(p,q)$.[]{data-label="fig:1st-pt"}](newfuns-p08-q055 "fig:"){width="5cm"} In order to show the monotonicity of the map in \[eq:fun-max-pq-plus\] it is sufficient to proceed by explicit computation. In fact, $$\log g_{Z}^+\big(p,q^*(p),x^-(p,q^*(p))\big)=\log\(3 - \frac{1}{p} - 2 p\) + \(2 - \frac{1}{p}\) \log\(\frac{ 1-(3 - 4 p) p}{(2 p-1)^2 (1 - p)}\),$$ take the derivative $$\frac{d}{dp} \log g_{Z}^+\big(p,q^*(p),x^-(p,q^*(p))\big)=\frac{-8p^3+10{{p}}^{2}-8p+3}{p(2p-1)(4{{p}}^{2}-3p+1)}+\frac{1}{{{p}}^{2}}\log\(\frac{4{{p}}^{2}-3p+1}{(1-{{p}})(2p-1)^2} \),$$ and notice that, for all $p\in(1/2,1)$, $$\frac{d}{dp} \log g_{Z}^+\big(p,q^*(p),x^-(p,q^*(p))\big)>0.$$ Therefore, there exists a unique $p_c\in(1/2,1)$ for which $$g^-_{Z}\(p_c,q^{*}(p_c),x^-(p_c,q^*(p_c))\)=g^+_{Z}\(p_c,q^{*}(p_c),x^-(p_c,q^*(p_c))\)=1.$$ Moreover, thanks to \[fact:g\]\[g4\], the value of $p_c$ can be further characterized as the unique $p\in(1/2,1)$ such that $$x^-(p_c,q^*(p_c))=\gamma(p_c,q^*(p_c)),$$ where $\gamma$ is defined as in \[eq:def-gamma\]. In conclusion, if $p\in(1/2,p_c)$ the function $x_{\operatorname{\mathsf{weq}}}(p,\cdot)$ is discontinuous at $q^*(p)$. On the other hand, if $p\in(p_c,1)$, the map $$\label{eq:final} q\mapsto\inf\left\{x\in(0,1)\::\:\max\big( \log g_{Z}^-(p,q,x),\log g_{Z}^+(p,q,x) \big)= 0 \right\},$$ is continuous in $[0,1)$. In fact, by \[eq:max1\], $g_{Z}^-(p,q,x^{-}(p,q))$ is decreasing in $q$, hence, if $q\in(q^*(p),1)$, $x_{\operatorname{\mathsf{weq}}}(p,q)=\inf\left\{x\in(0,1)\::\:\log g_{Z}^+(p,q,x)= 0 \right\}$, which is clearly continuous. Conversely, if $q\in(0,q^*(p))$, then the function $ g_{Z}^-(p,q,\cdot)$ is increasing in $(0,\gamma(p,q))$, and the function $g_{Z}^+(p,q,\cdot)$ is increasing in a neighborhood of $\gamma(p,q)$. Hence, \[eq:final\] can be written as $$\label{eq:Scenario} q\mapsto \begin{cases} \inf\left\{x\in(0,1)\::\: \log g_{Z}^-(p,q,x)= 0 \right\}&\text{if }q<q^{\Box}({{p}}),\\ \inf\left\{x\in(0,1)\::\: \log g_{Z}^+(p,q,x)= 0 \right\}&\text{if }q>q^{\Box}({{p}}),\\ \end{cases}$$ where $q^{\Box}({{p}})$ solves $$g_Z^-(p,q^{\Box}({{p}}),\gamma(p,q^{\Box}({{p}})))=1.$$ By definition of $\gamma$, $$g_Z^+(p,q^{\Box}({{p}}),\gamma(p,q^{\Box}({{p}})))=1.$$ hence, there cannot be any discontinuity when passing from the first to the second branch of \[eq:Scenario\]. By \[eq:xopt\] and \[fact:f\], $$\begin{aligned} x_{\operatorname{\mathsf{opt}}}(p,q)=&\sup\big\{x\in(0,1)\::\:\max\(f_W((1-q)p,0),f_{W}(q+(1-q)p,x) \)\ge 1 \big\}\\ =&\sup\big\{x\in(0,1)\::\:f_{W}(q+(1-q)p,x)\ge 1 \big\}\\ =&x_{\operatorname{\mathsf{opt}}}(q+(1-q)p,0). \end{aligned}$$ On the other hand, by \[eq:beq\], $$\begin{aligned} x_{\operatorname{\mathsf{beq}}}(p,q)=&\sup\big\{x\in(0,1)\::\:(1-{{p}})^{1-x}\max\(f_W((1-q)p,x),f_{W}(q+(1-q)p,x) \)\ge 1 \big\}\\ =&\sup\big\{x\in(0,1)\::\:(1-{{p}})^{1-x}f_{W}(q+(1-q)p,x)\ge 1 \big\}\\ \ge&\sup\big\{x\in(0,1)\::\:(1-{{p}})^{1-x}f_{W}(p,x)\ge 1 \big\}\\ =&x_{\operatorname{\mathsf{beq}}}(q+(1-q)p,0). \end{aligned}$$ where the inequality above follow from \[fact:f\]\[fact2\], which implies that $\forall (q,x)\in(0,1)^2$ $$(1-{{p}})^{1-x}f_W(q+(1-q)p,x)\ge(1-{{p}})^{1-x}f_W(p,x). \qedhere$$ Proof for the fully supported potential {#se:proof-continuous} --------------------------------------- In this subsection we focus on the case in which ${\pi}$ is fully supported in $[0,1]$. More precisely, we will assume that for every $I\subset[0,1]$ with $\text{Leb}(I)=\varepsilon$ there exists some $\delta=\delta(\varepsilon)$ such that $$\label{hp:cont} \pi(I)\ge \delta.$$ We start by showing a concentration result for the number of profiles with a large potential. By \[hp:cont\], for any fixed $\varepsilon>0$ there exists some $\delta=\delta(\varepsilon)>0$ such that $$\begin{aligned} {\mathsf{{P}}}\big(\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon \big)=\delta. \end{aligned}$$ It follows that the expected number of profiles with potential in the interval $ [1-\varepsilon,1]$ is $$\begin{aligned} {\mathsf{{E}}}\[\left|\left\{{\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon \right\}\right| \]=\delta 2^n. \end{aligned}$$ Moreover, by the Chernoff bound, for any constant $\gamma\in(0,1)$ $${\mathsf{{P}}}\(\operatorname{\mathsf{{Binomial}}}(2^n,\delta)< (1-\gamma)\delta 2^n \)\le \exp\(-\Theta(2^n) \).$$ Hence, considering the family of events $${\ensuremath{\mathcal E}}_{\varepsilon,c}:=\left\{ \left|\left\{{\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon] \right\}\right|>c2^n\right\},$$ there exists some sufficiently small constant $c=c(\varepsilon)>0$ for which $$\lim_{n\to\infty}{\mathsf{{P}}}\({\ensuremath{\mathcal E}}_{\varepsilon,c} \)=1.$$ Notice now that if $\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon$ then the probability that $\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=n$ can be lower bounded by $${\mathsf{{P}}}\(\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})={{n}}\mid \Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon \)\ge (1-\varepsilon)^n.$$ Therefore, $$\begin{aligned} {\mathsf{{E}}}\[|Z_{{n}}| \]\ge& {\mathsf{{E}}}\[|Z_{{n}}|\mid \left|\left\{{\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon] \right\}\right|>c2^n \]{\mathsf{{P}}}\( \left|\left\{{\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\ge 1-\varepsilon] \right\}\right|>c2^n\)\\ \ge& c(2(1-\varepsilon))^n \end{aligned}$$ By \[pr:var2\], $$\frac{|Z_n|}{{\mathsf{{E}}}\[|Z_{{n}}| \]}\overset{{\mathsf{{P}}}}{\longrightarrow}1,$$ from which, by taking $\varepsilon\to0$, follows $$\frac{1}{n}\log\(|Z_n|\)\overset{{\mathsf{{P}}}}{\longrightarrow}2.$$ Fix some ${\boldsymbol{{{s}}}}\in{{\Sigma}}$ and notice that $${\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in Z_k \)={\mathsf{{P}}}\(\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \){\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\mid \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \).$$ It is worth noting that the quantity ${\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\mid \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \)$ depends only on $p$, regardless of the specific form of ${\pi}$. In fact, $$\begin{aligned} {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\mid \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})=k \)=&{\mathsf{{P}}}\(u_{{i}}({\boldsymbol{{{s}}}})=0 \)^{n-k}\\ =&\(\int_0^1 (1-x) d{\pi}(x)\)^{n-k}\\ \label{eq3} =&\(1-p \)^{n-k}. \end{aligned}$$ We notice further that, if $\Phi({\boldsymbol{{{s}}}})=x$ then, by the law of large numbers, for all $\varepsilon>0$ $$\begin{aligned} \label{eq1} \lim_{n\to\infty}{\mathsf{{P}}}\(\frac{\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})}{n}\in[x-\varepsilon,x+\varepsilon] \:\big\rvert\:\Phi({\boldsymbol{{{s}}}})\in[x-\varepsilon/2,x+\varepsilon/2] \)=1.\end{aligned}$$ Being ${\pi}$ fully supported we have $$\begin{aligned} {\mathsf{{P}}}\(\Phi({\boldsymbol{{{s}}}})\in[x-\varepsilon/2,x+\varepsilon/2]\)=\Omega(1),\end{aligned}$$ and therefore $$\begin{aligned} {\mathsf{{E}}}\[|\left\{ {\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\in[x-\varepsilon/2,x+\varepsilon/2]\right\}|\]=\Theta(2^n),\end{aligned}$$ so that by the independence of the sequence $(\Phi({\boldsymbol{{{s}}}}))_{{\boldsymbol{{{s}}}}\in{{\Sigma}}}$ we have $$\begin{aligned} \label{eq2} \frac{|\left\{ {\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\in[x-\varepsilon/2,x+\varepsilon/2]\right\}|}{{\mathsf{{E}}}\[|\left\{ {\boldsymbol{{{s}}}}\::\:\Phi({\boldsymbol{{{s}}}})\in[x-\varepsilon/2,x+\varepsilon/2]\right\}|\]}\overset{{\mathsf{{P}}}}{\longrightarrow}1.\end{aligned}$$ Therefore, by \[eq1,eq2\] we conclude that for any arbitrarily small interval $[x-\varepsilon,x+\varepsilon]\in[0,1]$ we have $\Theta(2^n)$ profiles with such an . Moreover, by \[eq3\], a profile with $x\pm \varepsilon$ is a $\operatorname{\mathsf{{NE}}}$ with probability $$\begin{aligned} (1-p)^{(1-x+\varepsilon)n}\le {\mathsf{{P}}}\({\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}}\mid \operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})\in [(x-\varepsilon)n,(x+\varepsilon)n] \)\le (1-p)^{(1-x-\varepsilon)n}.\end{aligned}$$ Hence, $$\begin{aligned} \label{eq:eps} [2(1-p)^{(1-x+\varepsilon)}]^{n}\le{\mathsf{{E}}}\[\left|\left\{{\boldsymbol{{{s}}}}\::\: {\boldsymbol{{{s}}}}\in \operatorname{\mathsf{{NE}}},\:\operatorname{\mathsf{ {SU}}}({\boldsymbol{{{s}}}})/n\in[x-\varepsilon,x+\varepsilon] \right\} \right| \]\le[2(1-p)^{(1-x-\varepsilon)}]^{n}.\end{aligned}$$ The theorem follows by letting $\varepsilon\to 0$ in \[eq:eps\], by using \[pr:var2\] and by noting that $$2(1-p)^{(1-x)}=1\qquad\iff\qquad p\ge\frac{1}{2}\text{ and }x=h(p).$$ Acknowledgments {#acknowledgments .unnumbered} --------------- Both authors are members of GNAMPA-INdAM and of COST Action GAMENET. This work was partially supported by the GNAMPA-INdAM Project 2020 “Random walks on random games” and PRIN 2017 project ALGADIMAR.

--- abstract: 'Effective electrostatic interactions between colloidal particles, coated with polyelectrolyte brushes and suspended in an electrolyte solvent, are described via linear response theory. The inner cores of the macroions are modeled as hard spheres, the outer brushes as spherical shells of continuously distributed charge, the microions (counterions and salt ions) as point charges, and the solvent as a dielectric continuum. The multi-component mixture of macroions and microions is formally mapped onto an equivalent one-component suspension by integrating out from the partition function the microion degrees of freedom. Applying second-order perturbation theory and a random phase approximation, analytical expressions are derived for the effective pair interaction and a one-body volume energy, which is a natural by-product of the one-component reduction. The combination of an inner core and an outer shell, respectively impenetrable and penetrable to microions, allows the interactions between macroions to be tuned by varying the core diameter and brush thickness. In the limiting cases of vanishing core diameter and vanishing shell thickness, the interactions reduce to those derived previously for star polyelectrolytes and charged colloids, respectively.' author: - 'H. Wang' - 'A. R. Denton' title: 'Effective Electrostatic Interactions in Suspensions of Polyelectrolyte Brush-Coated Colloids' --- Introduction ============ Polyelectrolytes [@PE1; @PE2] are ionizable polymers that dissolve in a polar solvent, such as water, through dissociation of counterions. Solutions of polyelectrolytes are complex mixtures of macroions and microions (counterions and salt ions) in which direct electrostatic interactions between macroions are screened by surrounding microions. Polyelectrolyte chains, grafted or adsorbed by one end to a surface at high concentration, form a dense brush that can significantly modify interactions between surfaces in solution. When attached to colloidal particles, [*e.g.*]{}, latex particles in paints or casein micelles in milk [@Tuinier02], polyelectrolyte brushes can stabilize colloidal suspensions by inhibiting flocculation [@Evans; @Hunter]. Biological polyelectrolytes (biopolymers), such as proteins in cell membranes, can modify intercellular and cell-surface interactions. Conformations and density profiles of polyelectrolyte (PE) brushes have been studied by a variety of experimental, theoretical, and simulation methods, including dynamic light scattering [@Guo-Ballauff01], small-angle neutron scattering [@Mir95; @Guenoun98; @Groenewegen00], transmission electron microscopy [@Groenewegen00], neutron reflectometry [@Tran99], surface adsorption [@Hariharan-Russel98], atomic force microscopy [@Mei-Ballauff03], self-consistent field theory [@Miklavic88; @Misra89; @Misra96; @Zhulina-Borisov97; @Gurovitch-Sens99; @Borisov01; @Klein-Wolterink-Borisov03], scaling theory [@Pincus91; @Schiessel-Pincus98; @Borisov01; @Klein-Wolterink-Borisov03], Poisson-Boltzmann theory [@Miklavic90], Monte Carlo simulation [@Miklavic90], and molecular dynamics simulation [@Seidel00; @Likos02]. Comparatively few studies have focused on electrostatic interactions between PE brush-coated surfaces. Interactions between neutral surfaces – both planar and curved (spherical) – with grafted PE brushes have been modeled using scaling theory [@Pincus91], while interactions between charged surfaces coated with oppositely-charged PEs have been investigated for planar [@Miklavic90] and spherical (colloidal) surfaces [@Podgornik95] via Monte Carlo simulation and a variety of theoretical methods. While microscopic models that include chain and microion degrees of freedom provide the most realistic description of PE brushes, simulation of such explicit models for more than one or two brushes can be computationally demanding. The purpose of the present paper is to develop an alternative, coarse-grained theoretical approach, based on the concept of effective interactions, which may prove useful for predicting thermodynamic and other bulk properties of suspensions of PE brush-coated colloids. Modeling each brush as a spherical shell of continuously distributed charge, we adapt linear response theory, previously developed for charged colloids [@Silbert91; @Denton99; @Denton00] and PEs [@Denton03], to derive effective electrostatic interactions. The theory is based on mapping the multi-component mixture onto an equivalent one-component system of “pseudo-macroions" by integrating out from the partition function the degrees of freedom of the microions. Within the theory, microions play three physically important roles: reducing (renormalizing) the bare charge on a macroion; screening direct Coulomb interactions between macroions; and generating a one-body volume energy. The volume energy – a natural by-product of the one-component reduction – contributes to the total free energy and can significantly influence thermodynamic behavior of deionized suspensions. Outlining the remainder of the paper, Sec. \[Model\] defines the model suspension of PE brush-coated colloids; Sec. \[Theory\] reviews the linear response theory; Secs. \[Analytical Results\] and \[Numerical Results\] present analytical and numerical results for counterion density profiles, effective pair interactions, and volume energies in bulk suspensions; and finally, Sec. \[Conclusions\] summarizes and concludes. Model {#Model} ===== The system of interest is modeled as a suspension of $N_m$ spherical, core-shell macroions of charge $-Ze$ (valence $Z$), core radius $a$, and PE brush shell thickness $l$ (outer radius $R=a+l$), and $N_c$ point counterions of charge $ze$ in an electrolyte solvent in volume $V$ at temperature $T$ (see Fig. \[PEbrush\]). The core is assumed to be neutral, the macroion charge coming entirely from the PE shell. Assuming a symmetric electrolyte and equal salt and counterion valences, the electrolyte contains $N_s$ point salt ions of charge $ze$ and $N_s$ of charge $-ze$. The microions thus number $N_+=N_c+N_s$ positive and $N_-=N_s$ negative, for a total of $N_{\mu}=N_c+2N_s$. Global charge neutrality in a bulk suspension constrains macroion and counterion numbers via $ZN_m=zN_c$. Number densities of macroions, counterions, and salt ions are denoted by $n_m$, $n_c$, and $n_s$, respectively. Within the primitive model of ionic liquids [@HM], the solvent is treated as a dielectric continuum of dielectric constant $\epsilon$, which acts only to reduce the strength of Coulomb interactions between ions. In PE solutions, the counterions can be classified into four regions: (1) those within narrow tubes enclosing the PE chains, of radius comparable to the Bjerrum length, $\lambda_B=e^2/(\epsilon k_{\rm B}T)$; (2) those outside of the tubes but still closely associated with the chains; (3) those not closely associated with the chains, but still inside of the PE shells; and (4) those entirely outside of the macroions. Counterions in regions (1)-(3) can be regarded as trapped by the macroions, while those in region (4) are free to move throughout the suspension. Within region (1), the counterions may be either condensed and immobilized on a chain or more loosely bound and free to move along a chain. These chain-localized (condensed or mobile) counterions tend to distribute uniformly along, and partially neutralize, the chains. In our model, counterions in regions (1) and (2) act to renormalize the bare macroion valence. The parameter $Z$ thus should be physically interpreted as an [*effective*]{} macroion valence, generally much lower than the bare valence (number of ionizable monomers). From the Manning counterion condensation criterion [@PE1], according to which the linear charge density of a PE chain saturates at $\sim e/\lambda_B$, we can expect the bare charge in an aqueous solution to be renormalized down by at least an order of magnitude. The local number density profiles of charged monomers in the PE brushes, $\rho_{\rm mon}(r)$, and of counterions, $\rho_c(r)$, are modeled here as continuous, spherically symmetric distributions. Charge discreteness can be reasonably neglected if we ignore structure on length scales shorter than the minimum separation between charges. Spherical symmetry of charge distributions can be assumed if intra-macroion chain-chain interactions, which favor isotropic distribution of chains, dominate over inter-macroion interactions, which favor anisotropy. The density profile of charged monomers depends on the conformations of chains in the PE shells. Electrostatic repulsion between charged monomers tends to radially stretch and stiffen PE chains. Indeed, neutron scattering experiments [@Guenoun98] on diblock (neutral-charged) copolymer micelles, as well as simulations [@Likos02], provide strong evidence that the arms of spherical PE brushes can exhibit rodlike behavior. Here we assume the ideal case of fully stretched chains of equal length – a porcupine conformation [@Pincus91] – and model the charged monomer number density profile by $$\rho_{\rm mon}(r)~=~~\left\{ \begin{array} {l@{\quad\quad}l} ~~~0, & r>R \\ {\displaystyle \frac{Z}{4\pi lr^2}}, & a < r \leq R\\ ~~~0, & r\leq a, \end{array} \right .\label{monomerdensity}$$ where $r$ is the radial distance from the macroion’s center. The model thus neglects configurational entropy of the PE chains, although it does include the entropy of the microions. Theory {#Theory} ====== For the model suspension defined above, our goal is to predict distributions of microions inside and outside of the PE brushes and effective interactions between macroions. Adapting the general response theory approach previously applied to charged colloids [@Silbert91; @Denton99; @Denton00] and PE solutions [@Denton03], we reduce the multi-component mixture to an equivalent one-component system governed by effective interactions, and approximate the effective one-component Hamiltonian via perturbation theory. To simplify notation, we initially ignore salt ions. The Hamiltonian then decomposes, quite generally, into three terms: $$H=H_m(\{{\bf R}\})+H_c(\{{\bf r}\})+H_{mc}(\{{\bf R}\},\{{\bf r}\}), \label{H}$$ where $\{{\bf R}\}$ and $\{{\bf r}\}$ denote collective coordinates of macroions and counterions, respectively. The first term in Eq. (\[H\]), $$H_m=H_{\rm hc}+\frac{1}{2}\sum_{i\neq j=1}^{N_m}v_{mm}(|{\bf R}_i-{\bf R}_j|), \label{Hp}$$ is the macroion Hamiltonian, which includes a hard-core contribution $H_{\rm hc}$ (kinetic energy and hard-core interactions), and an electrostatic contribution due to the bare Coulomb pair interaction potential $$v_{mm}(r)=\frac{Z^2e^2}{\epsilon r} \label{vmm}$$ at center-center separation $r$. The second term in Eq. (\[H\]), $$H_c=K_c+\frac{1}{2}\sum_{i\neq j=1}^{N_c}v_{cc}(|{\bf r}_i-{\bf r}_j|), \label{Hc}$$ is the Hamiltonian of the counterions with kinetic energy $K_c$ interacting via the Coulomb pair potential $v_{cc}(r)=z^2e^2/\epsilon r$. The third term in Eq. (\[H\]), $$H_{mc}=\sum_{i=1}^{N_m}\sum_{j=1}^{N_c}v_{mc}(|{\bf R}_i-{\bf r}_j|), \label{Hmc1}$$ is the macroion-counterion interaction, which also may be expressed in the form $$H_{mc}~=~\int{\rm d}{\bf R}\,\rho_m({\bf R})\int{\rm d}{\bf r} \,\rho_c({\bf r})v_{mc}(|{\bf R}-{\bf r}|), \label{Hmc2}$$ where $\rho_m({\bf R})=\sum_{i=1}^{N_m} \delta({\bf R}-{\bf R}_i)$ and $\rho_c({\bf r})=\sum_{j=1}^{N_c} \delta({\bf r}-{\bf r}_j)$ are the macroion and counterion number density operators, respectively, and $v_{mc}(|{\bf R}-{\bf r}|)$ is the macroion-counterion interaction potential (to be specified in Sec. \[Analytical Results\]). The mixture of macroions and counterions can be formally reduced to an equivalent one-component system by integrating out the counterion coordinates. Denoting traces over counterion and macroion coordinates by ${\left\langle}~{\right\rangle}_c$ and ${\left\langle}~{\right\rangle}_m$, respectively, the canonical partition function can be expressed as $${\cal Z}~=~{\left\langle}{\left\langle}\exp(-\beta H){\right\rangle}_c{\right\rangle}_m ~=~{\left\langle}\exp(-\beta H_{\rm eff}){\right\rangle}_m, \label{part}$$ where $\beta=1/k_BT$, $H_{\rm eff}=H_m+F_c$ is the effective one-component Hamiltonian, and $$F_c~=~-k_BT\ln{\left\langle}\exp\Bigl[-\beta(H_c+H_{mc})\Bigr]{\right\rangle}_c \label{Fc1}$$ is the free energy of a nonuniform gas of counterions in the presence of the macroions. Now regarding the macroions as an “external" potential for the counterions, we invoke perturbation theory [@Silbert91; @Denton99; @Denton00; @HM] and write $$F_c~=~F_0~+~\int_0^1{\rm d}\lambda\,{\left\langle}H_{mc}{\right\rangle}_{\lambda}, \label{Fc2}$$ where $F_0=-k_BT\ln{\left\langle}\exp(-\beta H_c){\right\rangle}_c$ is the reference free energy of the unperturbed counterions, the $\lambda$-integral charges the macroions ([*i.e.*]{}, the PE brushes) from neutral to fully charged, $H_{mc}$ represents the perturbing potential of the macroions acting on the counterions, and ${\left\langle}H_{mc}{\right\rangle}_{\lambda}$ is the mean value of this potential in a suspension of macroions charged to a fraction $\lambda$ of their full charge. Two formal manipulations prove convenient. First, we convert the free energy of the unperturbed counterions to that of a classical one-component plasma (OCP) by adding and subtracting, on the right side of Eq. (\[Fc2\]), the energy of a uniform compensating negative background [@note1], $E_b=-N_cn_c\hat v_{cc}(0)/2$. Here $n_c=N_c/[V(1-\eta_{\rm hc})]$ is the average density of counterions in the free volume – [*i.e.*]{}, the total volume reduced by the volume fraction $\eta_{\rm hc}=(4\pi/3)n_ma^3$ of the macroion hard cores – and $\hat v_{cc}(0)$ is the $k \to 0$ limit of the Fourier transform of $v_{cc}(r)$. Equation (\[Fc2\]) then becomes $$F_c~=~F_{\rm OCP}~+~\int_0^1{\rm d}\lambda\,{\left\langle}H_{mc}{\right\rangle}_{\lambda} -E_b, \label{Fc3}$$ where $F_{\rm OCP}=F_0+E_b$ is the free energy of a homogeneous OCP excluded from the colloidal hard cores. Second, we express $H_{mc}$ in terms of Fourier components: $${\left\langle}H_{mc}{\right\rangle}_{\lambda}~=~\frac{1}{V}\sum_{{\bf k}\neq 0} \hat v_{mc}(k) \hat\rho_m({\bf k}) {\left\langle}\hat\rho_c(-{\bf k}){\right\rangle}_{\lambda} + \frac{1}{V}\lim_{k\to 0}\left[\hat v_{mc}(k) \hat\rho_m({\bf k}) {\left\langle}\hat\rho_c(-{\bf k}){\right\rangle}_{\lambda}\right], \label{Hmck}$$ where $\hat v_{mc}(k)$ is the Fourier transform of the macroion-counterion interaction and where $\hat\rho_m({\bf k})=\sum_{j=1}^{N_m}\exp(-i{\bf k}\cdot{\bf R}_j)$ and $\hat\rho_c({\bf k})=\sum_{j=1}^{N_c}\exp(-i{\bf k}\cdot{\bf r}_j)$ are Fourier components of the macroion and counterion number density operators. The $k=0$ term is singled out because the number of counterions, $N_c=\hat\rho_c(0)$, does not respond to the macroion charge, but rather is fixed by the constraint of global charge neutrality. Further progress requires approximations for the counterion free energy. Applying second-order perturbation (linear response) theory, the counterions are assumed to respond linearly to the macroion external potential: $$\rho_c({\bf r})~=~\int{\rm d}{\bf r}'\,\chi({\bf r}-{\bf r}') \int{\rm d}{\bf r}''\,\rho_m({\bf r}'')v_{mc}({\bf r}'-{\bf r}'') \label{rhocr}$$ or $$\hat\rho_c({\bf k})~=~\chi(k)\hat v_{mc}(k)\hat\rho_m({\bf k}), \qquad k\neq 0, \label{rhock}$$ where $\chi(k)$ is the linear response function of the OCP. Combining Eqs. (\[Fc3\])-(\[rhock\]), the effective Hamiltonian can be expressed in the form of the Hamiltonian of a one-component pairwise-interacting system: $$H_{\rm eff}~=~H_{\rm hc}~+~\frac{1}{2}\sum_{i\neq j=1}^{N_m} v_{\rm eff}(|{\bf R}_i-{\bf R}_j|)~+~E_0, \label{Heff}$$ where $v_{\rm eff}(r)=v_{mm}(r)+v_{\rm ind}(r)$ is an effective electrosatic macroion-macroion pair interaction that augments the bare macroion interaction $v_{mm}(r)$ by a counterion-induced interaction $$\hat v_{\rm ind}(k)~=~\chi(k)\left[\hat v_{mc}(k)\right]^2. \label{vindk}$$ The final term in Eq. (\[Heff\]) is the volume energy, $$E_0~=~F_{\rm OCP}~+~\frac{N_m}{2}\lim_{r\to 0} v_{\rm ind}(r)~+~N_m\lim_{k\to 0}\left[-\frac{1}{2}n_m\hat v_{\rm ind}(k)+n_c\hat v_{mc}(k)+\frac{Z}{2z}n_c\hat v_{cc}(k)\right], \label{E0}$$ which emerges naturally from the one-component reduction. Although independent of the macroion coordinates, the volume energy depends on the average macroion density and thus has the potential to significantly influence thermodynamics. Evidently, the effective interactions depend on the macroion structure through the specific form of the macroion-counterion interaction $v_{mc}$ in Eqs. (\[vindk\]) and (\[E0\]). The OCP linear response function, proportional to the corresponding static structure factor $S(k)$, may be obtained from liquid-state theory [@HM]. In practice, the OCP is weakly correlated, with coupling parameter $\Gamma=\lambda_B/a_c \ll 1$, where $a_c=(3/4\pi n_c)^{1/3}$ is the counterion sphere radius. For example, for hard-sphere macroions of radius $a=50$ nm, valence $Z=500$, and volume fraction $\eta_{\rm hc}=0.01$, in water at room temperature ($\lambda_B=0.714$ nm), we find $\Gamma\simeq 0.02$. As in previous work on charged colloids [@Denton99; @Denton00] and polyelectrolytes [@Denton03], we adopt the random phase approximation (RPA), which is valid for weakly-coupled plasmas. The RPA equates the OCP two-particle direct correlation function to its exact asymptotic limit: $c^{(2)}(r)=-\beta v_{cc}(r)$. Using the Ornstein-Zernike relation, $S(k)=1/[1-n_c\hat c^{(2)}(k)]$, the linear response function then takes the form $$\chi(k)~=~-\beta n_c S(k)~=~-\frac{\beta n_c}{(1+\kappa^2/k^2)}, \label{chi}$$ where $\kappa=\sqrt{4\pi n_cz^2\lambda_B}$ is the Debye screening constant (inverse screening length). Note that the screening constant, which involves the density of counterions in the free volume, naturally incorporates the excluded volume of the macroion cores. With $\chi(k)$ specified, the counterion density can be calculated from the macroion-counterion interaction and Eq. (\[rhock\]) for a given macroion distribution (see Sec. \[Analytical Results\]). Finally, salt ions can be easily incorporated by introducing additional response functions [@Denton00]. The pair interaction and volume energy are then modified only through a redefinition of the Debye screening constant: $\kappa=\sqrt{4\pi (n_c+2n_s)z^2\lambda_B}$, where $n_s=N_s/[V(1-\eta_{\rm hc})]$ is the average number density of salt ion pairs in the free volume. Generalization of response theory to incorporate leading-order nonlinear microion response entails three-body effective interactions, as well as corrections to the effective pair potential and volume energy [@Denton04]. Nonlinear effects are generally significant, however, only in concentrated, deionized suspensions of highly charged macroions [@Denton04] and are here ignored. It has been shown that response theory, combined with the RPA, is formally equivalent to Poisson-Boltzmann theory [@Denton04]. Both approaches rely on mean-field approximations that neglect microion fluctuations and predict microion distributions of the same general form, aside from a distinction in the screening constant, which response theory corrects for excluded volume of the macroion cores. Advantages of response theory over Poisson-Boltzmann theory are its predictions of (1) the entire effective Hamiltonian, including the one-body volume energy, which is essential for a complete description of phase behavior [@Denton99; @Denton00; @Silbert91; @vRH; @Graf; @vRDH; @Warren], and (2) a more accurate expression for the Debye screening constant that incorporates the macroion excluded-volume correction. Analytical Results {#Analytical Results} ================== For our porcupine model of a spherical PE brush with $1/r^2$ monomer density profile, Gauss’s law gives the electric field around a macroion as $$E(r)~=~\left\{ \begin{array} {l@{\quad\quad}l} -\frac{\displaystyle Ze}{\displaystyle \epsilon}\frac{\displaystyle 1}{\displaystyle r^2}, & r>R \\ -\frac{\displaystyle Ze}{\displaystyle \epsilon}\frac{\displaystyle r-a}{\displaystyle lr^2}, & a < r \leq R\\ ~~~~ 0 , & r\leq a. \end{array} \right. \label{Estar}$$ Integration over $r$ yields the electrostatic potential energy between a brush and a counterion: $$v_{mc}(r)~=~\left\{ \begin{array} {l@{\quad\quad}l} ~~~~~-\frac{\displaystyle Zze^2}{\displaystyle \epsilon r}, & r>R \\ -\frac{\displaystyle Zze^2}{\displaystyle \epsilon l}\left[1-\frac{\displaystyle a}{\displaystyle r}-\ln\left(\frac{\displaystyle r}{\displaystyle R}\right)\right], & a < r \leq R\\ -\frac{\displaystyle Zze^2}{\displaystyle \epsilon l}\left[\alpha-\ln{\left(\frac{\displaystyle a}{\displaystyle R}\right)}\right], & r \leq a, \end{array} \right. \label{vmcrbrush}$$ where $\alpha$ is an arbitrary constant, which arises because $v_{mc}(r)$ is not uniquely defined inside the hard core. Following van Roij and Hansen [@vRDH], we choose $\alpha$ below by requiring that the counterion density vanish inside the hard core. Fourier transforming Eq. (\[vmcrbrush\]) yields $$\hat v_{mc}(k)~=~ -\frac{4\pi Zze^2}{\epsilon k^3 l}{\rm G}(ka,kR;\alpha), \label{vmckbrush}$$ where the function ${\rm G}(ka,kR;\alpha)$ is defined as $${\rm G}(x_{1},x_{2};\alpha)~=~{\rm sinc}(x_{2})-{\rm sinc}(x_{1})-\alpha[x_{1}\cos(x_{1})-\sin(x_{1})], \label{functionG}$$ with ${\rm sinc}(x)\equiv\int_0^x{\rm d}u\,\sin(u)/u$. We can now calculate, in the dilute limit, the counterion number density profile around a single macroion, taking $\hat\rho_m({\bf k})=1$. From Eqs. (\[rhock\]), (\[chi\]), and (\[vmckbrush\]), the Fourier component of the density profile is $$\hat\rho_c(k)~=~\frac{Z}{z}\frac{\kappa^2}{kl(k^2+\kappa^2)} {\rm G}(ka,kR;\alpha), \label{rhockbrush}$$ which in real space takes the form $$\rho_c(r)~=~\frac{Z}{z}\frac{\kappa}{4\pi lr}~\left\{ \begin{array} {l@{\quad\quad}l} {\rm S}(\kappa a,\kappa R;\alpha)~e^{-\kappa r}, & r>R \\ {\rm S}(\kappa a,\kappa r;\alpha)~e^{-\kappa r} +{\rm Ec}(-\kappa r,-\kappa R){\rm sinh}(\kappa r), & a < r \leq R\\ \left[{\rm Ec}(-\kappa a,-\kappa R)+\alpha(1+\kappa a)e^{-\kappa a}\right]{\rm sinh}(\kappa r), & r \leq a. \end{array} \right. \label{rhocrbrushfull}$$ For simplicity, we have introduced two functions, ${\rm S}(x_{1},x_{2};\alpha)$ and ${\rm Ec}(x_{1},x_{2})$, which are defined, respectively, as $${\rm S}(x_{1},x_{2};\alpha)={\rm shi}(x_{2})-{\rm shi}(x_{1})-\alpha[x_{1}\cosh(x_{1})-\sinh(x_{1})] \label{functionS}$$ and $${\rm Ec}(x_{1},x_{2})={\rm Ei}(x_{2})-{\rm Ei}(x_{1}), \label{findAlan}$$ where ${\rm shi}(x)\equiv\int_0^x{\rm d}u\,\sinh(u)/u$ denotes the hyperbolic sine integral function and ${\rm Ei}(x)\equiv\int_{-\infty}^{x}{\rm d}u\,e^u/u$ is the exponential integral function. Now setting the counterion density to zero within the hard core \[[*i.e.*]{}, $\rho_c(r)=0, r\leq a$, in Eq. (\[rhocrbrushfull\])\] fixes the constant $\alpha$: $$\alpha~=~-\frac{e^{\kappa a}}{1+\kappa a}{\rm Ec}(-\kappa a, -\kappa R). \label{constant}$$ Integrating Eq. (\[rhocrbrushfull\]) over the spherical shell volume of a PE brush yields the fraction of counterions inside a brush: $$f_{\rm in}~=~1-\frac{1+\kappa R}{\kappa l}~e^{\displaystyle -\kappa R}~{\rm S}(\kappa a,\kappa R;\alpha). \label{ncinside-brush}$$ From this expression, it is clear that the counterion distribution, within the model, is determined entirely by two independent dimensionless parameters, $\kappa a$ and $\kappa l$, [*i.e.*]{}, the ratios of the macroion core radius and brush thickness, respectively, to the Debye screening length. From Eqs. (\[vindk\]) and (\[vmckbrush\]), the induced electrostatic pair interaction is given by $$\hat v_{\rm ind}(k) ~=~-\frac{4\pi Z^2e^2}{\epsilon}~\frac{\kappa^2}{l^2 k^4(k^2+\kappa^2)}~ [{\rm G}(ka,kR;\alpha)]^2, \label{vindk-brush}$$ whose Fourier transform is $$v_{\rm ind}(r)~=~-\frac{2Z^2e^2\kappa^2}{\pi\epsilon l^2 r} ~\int_0^{\infty}{\rm d}k\,\frac{\sin(kr)}{k^3(k^2+\kappa^2)} ~[{\rm G}(ka,kR;\alpha)]^2. \label{vindr-brush}$$ For nonoverlapping brushes, Eq. (\[vindr-brush\]) can be reduced to the analytical form $$v_{\rm ind}(r)~=~-\frac{Z^2e^2}{\epsilon r}~+~\frac{Z^2e^2}{\epsilon}~\left[\frac{{\rm S}(\kappa a,\kappa R;\alpha)}{\kappa l }\right]^2~\frac{e^{-\kappa r}}{r}, \qquad r>2R. \label{vindr>2R}$$ After adding to Eq. (\[vindr&gt;2R\]) the bare Coulomb potential between the spherical macroions \[Eq. (\[vmm\])\], the residual effective pair interaction is $$v_{\rm eff}(r)~=~\frac{Z^2e^2}{\epsilon}~\left[\frac{{\rm S}(\kappa a,\kappa R;\alpha)}{\kappa l }\right]^2~\frac{e^{-\kappa r}}{r}, \qquad r>2R. \label{veffr>2R}$$ Thus, within the coarse-grained PE brush model and at the level of linear response theory, nonoverlapping PE brushes are predicted to interact via an effective Yukawa pair potential of the same screened-Coulomb form as the long-range limit of the DLVO potential [@DLVO] for charged colloids. This result is consistent with previous linear response results for charged hard spheres [@Denton99; @Denton00], which interact via the DLVO effective pair potential $$v_{\rm eff}(r)~=~\frac{Z^2e^2}{\epsilon}~\left(\frac{e^{\kappa R}} {1+\kappa R}\right)^2~\frac{e^{-\kappa r}}{r}, \qquad r>2R, \label{vDLVO}$$ and for PE stars [@Denton03], which interact via $$v_{\rm eff}(r)~=~\frac{Z^2e^2}{\epsilon}~\left[\frac{{\rm shi} (\kappa R)}{\kappa R}\right]^2~\frac{e^{-\kappa r}}{r}, \qquad r>2R. \label{vstar}$$ Note that the screening constant, $\kappa$, in the pair potential depends on the total density of microions – inside and outside of the brushes – since all microions respond to the macroion charge. We do not consider here overlapping brushes, in which case steric interactions between chains also should be included [@Likos02]. Finally, the volume energy is obtained from Eqs. (\[E0\]), (\[vmckbrush\]), (\[vindk-brush\]), and (\[vindr-brush\]), as $$\begin{aligned} E_0~&=&~F_{\rm OCP}~-~N_m\frac{Z^2e^2\kappa^2}{\pi\epsilon l^{2}} \int_0^{\infty}{\rm d}k\,\frac{[{\rm G}(ka,kR;\alpha)]^2} {k^2(k^2+\kappa^2)}~-~(N_+-N_-)\frac{k_BT}{2}. \label{E0-star}\end{aligned}$$ Assuming weakly-coupled microion plasmas, the OCP free energy is well approximated by its ideal-gas limit: $$F_{\rm OCP}~=~N_+[\ln(n_+\Lambda_+^3)-1]~+~N_-[\ln(n_-\Lambda_-^3)-1], \label{FOCP}$$ where $\Lambda_{\pm}$ are the thermal de Broglie wavelengths of the positive/negative microions. The physical interpretation of the volume energy is straightforward. The first term on the right side of Eq. (\[E0-star\]) represents the entropy of free microions, the second term the electrostatic energy of microion-macroion interactions, and the third term accounts for the background substraction. If the macroion valence $Z$ is allowed to vary with concentration ([*e.g.*]{}, through counterion condensation), then $E_0$ should be supplemented by the macroion self energy. We emphasize that, because of its dependence on the average macroion concentration, the volume energy has the potential to influence thermodynamic phase behavior. As a check of the present results, it can be shown that in the two limiting cases of vanishing PE shell thickness ($l\to 0$, with Z fixed) and, independently, vanishing hard core diameter ($a\to 0$) all analytical results reduce to those given in refs. [@Denton99; @Denton00] and [@Denton03], respectively. Numerical Results and Discussion {#Numerical Results} ================================ To illustrate applications of the theory developed above, we present numerical results for the case of monovalent counterions ($z=1$) in aqueous suspensions at room temperature ($\lambda_B=0.714$ nm). Figure \[rhoc3in1\] shows the predicted counterion profiles around three different types of macroion, all of the same outer radius $R=50$ nm, valence $Z=500$, and reduced number density $n_mR^3=0.01$, for a salt-free suspension. The chosen valence is within the upper limit suggested by charge renormalization theory [@Alexander84] for this size of macroion: $Z<{\cal O}(10)R/\lambda_B$. For a star macroion, the counterion density diverges logarithmically towards the center [@Denton03], while for brush-coated and bare hard-sphere macroions the counterion densities remain finite. Figure \[ncbrush\] displays the corresponding internal counterion fraction, [*i.e.*]{}, fractional counterion penetration, as a function of $\kappa R$. For fixed ratio of hard-core radius to outer radius, $a/R$, the internal counterion fraction increases monotonically with $\kappa R$, reflecting increasing permeability of the macroions to counterions with decreasing screening length ([*e.g.*]{}, increasing salt concentration). On the other hand, when $\kappa R$ is fixed, the counterion penetration decreases upon thinning of the PE brush (increasing $a/R$). In the limit of vanishing brush thickness ($l/R\to 0$, $a/R\to 1$), Eq. (\[ncinside-brush\]) reduces to $$f_{\rm in}(l \rightarrow 0)=\frac{\kappa a}{\kappa a + 1} \kappa l+O(l^2). \label{fin_L_0}$$ Counterions are predicted to penetrate PE brush-coated macroions less efficiently than stars. Penetration of macroions by counterions can strongly influence screening of bare Coulomb interactions. Thus, effective pair interactions between brush-coated macroions depend sensitively on the thickness of the PE brush. To illustrate, Fig. \[veff3in1\] shows the effective pair potential for the same three macroion types as in Figs. \[rhoc3in1\] and \[ncbrush\] and for two salt concentrations, $c_s=0$ M and $c_s=100$ $\mu$M, corresponding to different Debye screening constants $\kappa$. For identical system parameters, the strength of the Yukawa pair interaction for nonoverlapping brush macroions is intermediate between that for hard-sphere and star macroions. Figure \[amp3in1\] compares the dependence of the macroion-size-dependent amplitude of $v_{\rm eff}(r)$, $r>2R$, on Debye screening constant for the three macroion types. The amplitude increases with $\kappa R$ for fixed ratio of hard-core to outer radius, while for fixed $\kappa R$ the amplitude increases from the star limit to the hard-sphere limit as the PE brush thins to infinitesimal thickness ($a/R\to 1$). Conclusions {#Conclusions} =========== Summarizing, polyelectrolyte-coated colloids provide a valuable conceptual bridge between charged colloids and polyelectrolytes. In this paper, linear response theory is applied to bulk suspensions of spherical colloidal particles coated with PE brushes. Assuming stiff, radially stretched PE chains, we model each brush as a spherically symmetric shell of continuously distributed charge, the charge density varying with radial distance $r$ as $1/r^2$. By formally integrating out the microion degrees of freedom, the Hamiltonian of the macroion-microion mixture is mapped onto the effective Hamiltonian of an equivalent one-component system. Predictions of the theory include microion density profiles, effective electrostatic interactions between pairs of (nonoverlapping) macroions, and a state-dependent one-body volume energy, which contributes to the total free energy. The theory presented here may provide a practical guide for choosing system parameters to achieve desired interactions. The main conclusions of this study are: (1) Trapping of counterions inside a spherical PE brush is highly sensitive to variations in the core radius, brush thickness, and Debye screening length of the solution. For fixed ratio of core to outer radius, the fraction of trapped counterions increases monotonically with increasing outer radius or decreasing screening length. For fixed ratio of outer radius to screening length, the fraction of trapped counterions decreases monotonically from a maximum in the limit of vanishing core radius (PE star macroion) to zero in the limit of vanishing shell thickness (hard-sphere macroion). (2) Within the linear response approximation, the effective pair interaction between nonoverlapping macroions has a Yukawa (screened-Coulomb) form. (3) By varying core radius and brush thickness, effective interactions between PE brush-coated macroions can be tuned – in both amplitude and range – between interactions for hard-sphere and star macroions. For fixed ratio of core radius to outer radius, the amplitude of the pair interaction increases monotonically with increasing outer radius or decreasing screening length, while for fixed ratio of outer radius to screening length, the amplitude increases monotonically from the star-limit to the hard-sphere limit. The range of the pair interaction, governed by the Debye screening length, depends on the hard-core volume fraction and so can be varied by adjusting the core radius. The range of validity of the coarse-grained model and linearized theory studied here, and the accuracy of the predicted Yukawa form of effective pair interaction, including amplitude and range, could be directly tested by future simulations of more explicit models of PE-grafted colloids. 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T. Hellweg, C. D. Dewhurst, W. Eimer, and K. Kratz, [*Langmuir*]{} [**20**]{}, 4330 (2004). ![\[PEbrush\] (a) Polyelectrolyte (PE) brush-coated colloidal sphere and, (b) model considered here, in which the PE monomer charge distribution is assumed continuous and varying as $1/r^2$, $a<r<a+l$.](figbrush1.ps){width="0.6\columnwidth"} ![\[rhoc3in1\] Counterion number density profiles of three types of spherical macroion of outer radius $R=50$ nm, valence $Z=500$, and reduced number density $n_mR^3=0.01$ in water at room temperature ($\lambda_B=0.714~{\rm nm}$): PE brush-coated macroion \[solid curve from Eq. (\[rhocrbrushfull\])\], PE star \[dashed curve from Eq. (20) of ref. [@Denton03]\], and charged hard sphere \[dot-dashed curve from Eq. (32) of ref. [@Denton99]\]. For the brush-coated macroion, the hard-core radius is $a=25$ nm and the PE shell thickness is $l=25$ nm.](figbrush2.ps){width="0.6\columnwidth"} ![\[ncbrush\] Fraction of counterions \[from Eq. (\[ncinside-brush\])\] trapped inside PE brush as a function of the dimensionless parameter $\kappa R$ (ratio of outer radius to Debye screening length) for several values of $a/R$ (ratio of core radius to outer radius).](figbrush3.ps){width="0.6\columnwidth"} ![\[veff3in1\] Effective electrostatic interactions between pairs of nonoverlapping macroions of outer radius $R=50$ nm, valence $Z=500$, and reduced number density $n_mR^3=0.01$ in room temperature water ($\lambda_B=0.714~{\rm nm}$) at salt concentrations (a) $c_s=0$ mol/l ($\kappa R\simeq 0.95$) and (b) $c_s=100$ $\mu$mol/l ($\kappa R\simeq 1.9$): PE brush-coated spherical macroions \[solid curves from Eq. (\[veffr&gt;2R\])\]; PE stars \[dashed curves from Eq. (\[vstar\])\]; and charged hard spheres \[dot-dashed curves from Eq. (\[vDLVO\])\].](figbrush4a.ps "fig:"){width="0.6\columnwidth"} ![\[veff3in1\] Effective electrostatic interactions between pairs of nonoverlapping macroions of outer radius $R=50$ nm, valence $Z=500$, and reduced number density $n_mR^3=0.01$ in room temperature water ($\lambda_B=0.714~{\rm nm}$) at salt concentrations (a) $c_s=0$ mol/l ($\kappa R\simeq 0.95$) and (b) $c_s=100$ $\mu$mol/l ($\kappa R\simeq 1.9$): PE brush-coated spherical macroions \[solid curves from Eq. (\[veffr&gt;2R\])\]; PE stars \[dashed curves from Eq. (\[vstar\])\]; and charged hard spheres \[dot-dashed curves from Eq. (\[vDLVO\])\].](figbrush4b.ps "fig:"){width="0.6\columnwidth"} ![\[amp3in1\] Amplitude of Yukawa effective electrostatic interactions between pairs of nonoverlapping brushes, stars and charged hard spheres vs. Debye screening constant, normalized to unity at $\kappa R=0$ \[from Eqs. (\[veffr&gt;2R\])-(\[vstar\])\].](figbrush5.ps){width="0.6\columnwidth"}

--- abstract: 'The paper is devoted to the relationship between the continuous Markovian description of Lévy flights developed previously (*Lubashevsky et al., Phys. Rev. E **79** (2009) 011110, **80** (2009) 031148; Eur. Phys. J. B **78** (2010) 207, **82** (2011) 189*) and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of continuous time random walks. To simplify understanding the key points of the technique to be created, our consideration is confined to the one-dimensional model for continuous random motion of a particle with inertia. Its dynamics governed by stochastic self-acceleration is described as motion on the phase plane $\{x,v\}$ comprising the position $x$ and velocity $v=dx/dt$ of the given particle. A notion of random walks inside a certain neighborhood $\mathcal{L}$ of the line $v=0$ (the $x$-axis) and outside it is developed. It enables us to represent a continuous trajectory of particle motion on the plane $\{x,v\}$ as a collection of the corresponding discrete steps. Each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the “boundary” of $\mathcal{L}$. As demonstrated, the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude, which endows the derived random walks along the $x$-axis with the characteristic properties of Lévy flights. Using the developed classification of random trajectories a certain parameter-free core stochastic process is constructed. Its peculiarity is that all the characteristics of Lévy flights similar to the exponent of the Lévy scaling law are no more than the parameters of the corresponding transformation from the particle velocity $v$ to the related variable of the core process. In this way the previously found validity of the continuous Markovian model for all the regimes of Lévy flights is explained. Based on the obtained results an efficient “single-peak” approximation is constructed. In particular, it enables us to calculate the basic characteristics of Lévy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations.' address: 'University of Aizu, Ikki-machi, Aizu-Wakamatsu, Fukushima 965-8560, Japan' author: - Ihor Lubashevsky title: | Equivalent continuous and discrete realizations of Lévy flights:\ Model of one-dimensional motion of inertial particle --- Lévy flights ,nonlinear Markovian processes ,random motion trajectories ,extreme fluctuations ,power-law heavy tails ,time scaling law ,continuous time random walks Introduction ============ During the last two decades there has been a great deal of research into Lévy type stochastic processes in various systems (for a review see, e.g., Ref. [@CTRW]). According to the accepted classification of the Lévy type transport phenomena [@CTRW], Lévy flights are Markovian random walks characterized by the divergence of the second moment of walker displacement $x(t)$, i.e., $\left<[x(t)]^2\right> \to \infty$ for any time scale $t$. It is caused by a power-law asymptotics of the distribution function $\mathcal{P}(x,t)$. For example, in the one-dimensional case this distribution function exhibits the asymptotic behavior $\mathcal{P}(x,t)\sim [\overline{x}(t)]^\alpha/x^{1+\alpha}$ for $x\gg\overline{x}(t)$, where $\overline{x}(t)$ is the characteristic length of the walker displacements during the time interval $t$ and the exponent $\alpha$ belongs to the interval $0<\alpha<2$. The time dependence of the quantity $\overline{x}(t)$ obeys the scaling law $\overline{x}(t)\propto t^{1/\alpha}$. Lévy flights are met, for instance, in the motion of tracer particles in turbulent flows [@Swinney], the diffusion of particles in random media [@Bouchaud], human travel behavior and spreading of epidemics [@Brockmann], or economic time series in finance [@Stanley]. As far as the developed techniques of modeling such stochastic processes are concerned, worthy of mention are, in particular, the Langevin equation with Lévy noise (see, e.g., Ref. [@Weron]) and the corresponding Fokker-Planck equations [@Schertzer1; @Schertzer2; @CiteNew1; @CN100], the description of anomalous diffusion with power-law distributions of spatial and temporal steps [@Fogedby1; @Sokolov], Lévy flights in heterogeneous media [@Fogedby2; @Honkonen; @BrockmannGeisel; @citeNNN3; @citeNNN4] and in external fields [@BrockmannSokolov; @Fogedby3], constructing the Fokker-Planck equation for Lévy type processes in nonhomogeneous media [@CiteNew2; @CiteNew3; @CiteNew4], first passage time analysis and escaping problem for Lévy flights [@fptp1; @fptp1Chech; @fptp2; @fptp3; @fptp4; @fptp5; @fptp6; @CiteNew5; @CiteNew6; @citeNNN1; @citeNNN2]. One of the widely used approaches to coping with Lévy flights, especially in complex environment, is the so-called continuous time random walks (CTRW) [@CTRW1; @CTRW2]. It models, in particular, a general class of Lévy type stochastic processes described by the fractional Fokker-Planck equation [@CTRW3]. Its pivot point is the representation of a stochastic process at hand as a collection of random jumps (steps) $\{\delta\mathbf{x}, \delta t\}$ of a wandering particle in space and time as well. In the frameworks of the coupled CTRW the particle is assumed to move uniformly along a straight line connecting the initial and terminal points of one step. In this case the discrete collection of steps is converted into a continuous trajectory and the velocity $\mathbf{v}=\delta\mathbf{x}/\delta t$ of motion within one step is introduced. As a result, the given stochastic process is described by the probabilistic properties, e.g., of the collection of random variables $\{\mathbf{v},\delta t \}$. A more detailed description of particle motion lies beyond the CTRW model. Unfortunately, for Lévy flights fine details of the particle motion within one step can be important especially in heterogeneous media or systems with boundaries because of the divergence of the moment $\left<[\delta\mathbf{x}(\delta t)]^2\right>$. Broadly speaking, it is due to a Lévy particle being able to jump over a long distance during a short time. The fact that Lévy flights can exhibit nontrivial properties on scales of one step was demonstrated in Ref. [@CiteNew5] studied the first passage time problem for Lévy flights based on the leapover statistics. Previously a new approach to tackling this problem was proposed [@we1; @we2; @we3; @we33]. It is based on the following nonlinear stochastic differential equation with white noise $\xi(t)$ $$\label{int:1} \tau\frac{dv}{dt} = - \lambda v + \sqrt{\tau\big(v_a^2+v^2\big)} \xi(t)$$ governing random motion of a particle wandering, e.g., in the one-dimensional space $\mathbb{R}_x$. Here $v=dx/dt$ is the particle velocity, the time scale $\tau$ characterizes the delay in variations of the particle velocity which is caused by the particle inertia, $\lambda$ is a dimensionless friction coefficient. The parameter $v_a$ quantifies the relative contribution of the additive component $\xi_a(t)$ of the Langevin force with respect to the multiplicative one $v \xi_m(t)$ which are combined within one term $$\label{int:2} v_a\xi_a(t)+ v\xi_m(t)\Rightarrow\sqrt{\big(v_a^2+v^2\big)} \xi(t)\,.$$ Here we have not specified the type of the stochastic differential equation because in the given case all the types are interrelated via the renormalization of the friction coefficient $\lambda$. It should be noted that models similar to Eq.  within replacement can be classified as the generalized Cauchy stochastic process [@Konno] and has been employed to study stochastic behavior of various nonequilibrium systems, in particular, lasers [@22], on-off intermittency [@26], economic activity [@27], passive scalar field advected by fluid [@28], etc. Model  generates continuous Markovian trajectories obeying the Lévy statistics on time scales $t\gg\tau$ [@we1; @we2; @we3]. Using a special singular perturbation technique [@we2] it was rigorously proved for the superdiffusive regime matching $1<\alpha<2$ [@we1; @we2] and also verified numerically for the quasiballistic ($\alpha = 1$) and superballistic ($0<\alpha<1$) regimes [@we3]. Moreover, the main expressions obtained for the distribution function $\mathcal{P}(x,t)$ and the scaling law $\overline{x}(t)$ within the interval $1< \alpha < 2$ turn out to hold also for the whole region $0<\alpha <2$ [@we3]. After its generalization [@we33] model  generates truncated Lévy flights as well. The given approach can be regarded as a continuous Markovian realization of Lévy flights. Indeed, it is possible to choose the system parameters in such a way that, on one hand, the “microscopic” time scale $\tau$ be equal to an arbitrary small value given beforehand and, on the other hand, the system behavior remain unchanged on scales $t\gg\tau$ [@we1; @we2]. The goal of this paper is to elucidate the fundamental features of this approach, to explain the found validity of model  for describing Lévy flights of all the regimes, and to construct a certain generalization of the continuous time random walks that admits a rather fine representation of the particle motion within one step. The latter feature will demonstrate us a way to overcome the basic drawback of the CTRW model, it appeals to different mechanisms in modeling the particle motion within individual steps and the relative orientation of neighboring steps. Continuous Markov model of Lévy flights {#1DModel} ======================================= Model ----- Following [@we1; @we3; @we33] let us consider random walks $\{x(t)\}$ of an inertial particle wandering in the one-dimensional space $\mathbb{R}_x$ whose velocity $v=dx/dt$ is governed by the following stochastic differential equation written in the dimensionless form $$\label{sec1:eq1} \frac{dv}{dt}=-\alpha v k(v)+\sqrt{2}g(v)\circ\xi(t)\,.$$ Here the constant $\alpha$ is a system parameter meeting the inequality $$\label{sec1:alpha} 0<\alpha<2\,,$$ the positive function $k(v)>0$ allows for nonlinear friction effects, $\xi(t)$ is the white Gaussian noise with the correlation function $$\label{sec1:wngf} \left\langle \xi(t)\xi(t')\right\rangle =\delta(t-t')\,,$$ and whose intensity $g(v)>0$ depending on the magnitude the particle velocity describes the cumulative effect of the additive and multiplicative components of the Langevin random force as noted in Introduction. The coefficient $\sqrt2$ has been introduced for the sake of convenience. The units of the spatial and temporal scales have been chosen in such a manner that the equalities $$\label{sec1:kg_v0} k(0) = 1\quad \text{and}\quad g(0) = 1$$ hold. Equation  is written in the Stratonovich form, which is indicated with the multiplication symbol $\circ$ in the product of the white noise $\xi(t)$ and its intensity $g(v)$. To avoid possible misunderstanding we note that this equation has been written in the Hänggi-Klimontovich form in Refs. [@we1; @we3; @we33]. In the kinetic theory of gases quantities relative to $k(v)$ and $g(v)$ and describing regular and random effects caused by the scattering of atoms or molecules are usually called “kinetic” coefficients (see, e.g., Ref. [@LLKT]). For this reason the functions $k(v)$ and $g(v)$ together with other quantities to be derived from them will be also referred to as the kinetic coefficients. In the present paper the main attention is focused on the special case when the kinetic coefficients $k(v)$ and $g(v)$ are of the form $$\label{sec1:kg0} k_0(v) = 1\quad \text{and}\quad g_0(v) = \sqrt{1+v^2}$$ and the random walks $\{x(t)\}$ generated by model  can be classified as Lévy flights for large time scales, i.e., for $t\gg1$ in the chosen units [@we1; @we2; @we3]. However, where appropriate, the general form of these kinetic coefficients will be used to demonstrate a certain universality of the results to be obtained and the feasibility of their generalization, for example, to the truncated Lévy flights [@we33]. Nevertheless, in further mathematical manipulations leading to particular results the following assumption about the behavior of the kinetic coefficients $$\label{sec1:kgaa} k(v)\approx 1\,,\quad g(v)\approx v \quad\text{for}\quad 1\lesssim v \lesssim v_c\,, \quad\text{and}\quad \frac{vk(v)}{g^2(v)} > B \quad\text{for}\quad v \gtrsim v_c$$ will be accepted beforehand. Here $B>0$ is some positive constant and $v_c \gg 1$ is a certain critical velocity characterizing the region where the generated random walks deviate from Lévy flights in properties. Naturally, case  obeys these conditions in the limit $v_c\to\infty$. To make it easier to compare the results to obtained in the following sections with the characteristic properties of the velocity fluctuations governed by Eq. , here let us find the stationary distribution function $P^\text{st}(v)$ of the particle velocity $v$. The Fokker-Planck equation describing the dynamics of the velocity distribution $P(v,t)$ and matching Eq.  is written as (see, e.g., Ref. [@MyBook]) $$\label{sec1:FPv} \frac{\partial P}{\partial t} = \frac{\partial}{\partial v}\left\{ g(v) \frac{\partial[g(v) P]}{\partial v} + \alpha vk(v)P \right\}.$$ Its stationary solution $P^\text{st}(v)$ meets the equality $$g(v) \frac{\partial[g(v) P^\text{st}]}{\partial v} + \alpha vk(v)P^\text{st} = 0$$ whence it follows that $$\begin{aligned} \label{sec1:Pst} P^\text{st}(v) & = \frac{C_v}{g(v)}\exp\left[-\alpha \int\limits_0^v \frac{uk(u)}{g^2(u)}du\right]\,,\\ \intertext{where the constant} \label{sec1:PstC} C_v &= \left\{2 \int\limits_{0}^{\infty}\frac{dv}{g(v)}\exp\left[-\alpha \int\limits_0^v \frac{uk(u)}{g^2(u)}du\right]\right\}^{-1}\end{aligned}$$ is specified by the normalization of the distribution function $P(v,t)$ to unity. Additive noise representation ----------------------------- To elucidate the general mechanism responsible for anomalous properties of the stochastic process at hand let us pass from the velocity $v$ to a new variable $\eta = \eta(v)$ introduced via the equation $$\label{sec2:u} \frac{d\eta}{dv} = \frac{1}{g(v)}\quad\text{subject to the condition}\quad \eta(0)= 0 \,.$$ Equation  is of the Stratonovich form, thus, we may use the standard rules of change of variables in operating with it. Therefore relation  between the variables $v$ and $\eta$ enables us to reduce Eq.  to the following one $$\label{sec1:eq2} \frac{d\eta}{dt}=-\alpha\phi(\eta)+\sqrt{2}\xi(t)\,,$$ where the function $\phi(\eta)$ is specified by the expression $$\label{sec1:phi} \phi(\eta) = \left. \frac{vk(v)}{g(v)}\right|_{v=v(\eta)}\,.$$ The function $\phi(\eta)$ together with the parameter $\alpha$ determine the rate of the particle regular drift in the $\eta$-space, $\mathbb{R}_\eta$. In particular, in case  we have $$\label{sec1:voneta0} v = \sinh (\eta)\qquad\text{and}\qquad \phi_0(\eta) = \tanh(\eta)\,.$$ It should be noted that the multiplication symbol $\circ$ has been omitted in Eq.  because the noise $\xi(t)$ enters it additively and, thus, all the types of this stochastic differential equation have the same form (for details see, e.g., Ref. [@MyBook]). Let us also introduce into consideration the potential $$\label{sec1:Phi} \Phi(\eta) = \int\limits_0^\eta\phi(\zeta)d\zeta \equiv \int\limits_0^{v(\eta)} \frac{u k(u)}{g^2(u)}du$$ which will be used below; in case  it can be written as $$\label{sec1:Phi0} \Phi_0(\eta) = \ln\left[\cosh(\eta)\right]\,.$$ Equation  together with expression  enable us to find the time pattern $\{\eta(t)\}$ for a given realization of the noise $\{\xi(t)\}_{-\infty}^{+\infty}$. Leaving expression  aside for a moment and keeping in mind solely the governing equations  and we may state that the constructed stochastic process $\{\eta(t)\}$ and the random variations $\{v(t)\}$ of the particle velocity are related to each other via the white noise $\xi(t)$ and are of the same level of generality. Only relation  treated as the definition of the function $\eta = \eta(v)$ causes us to regard the time variations $\{\eta(t)\}$ as a stochastic process derived from $\{v(t)\}$. However, expression  may be read in the “opposite way” as the definition of the function $v = v(\eta)$. In this case the process $\{\eta(t)\}$ plays the role of the noise source endowing the particle motion with stochastic properties and the particle velocity $v=v(\eta)$ becomes a derivative random variable. The particle displacement $x(t)$ in the space $\mathbb{R}_x$ during the time interval $t$ $$\label{sec1:Phi0x} x(t) = \int\limits_0^t v\left[\eta(t')\right]dt'$$ is also a derivative stochastic process of $\{\eta(t)\}$. Integral  can be treated in the Riemann sense for a given pattern $\{\eta(t)\}$ because the correlation function $\left<\eta(t)\eta(t')\right>$ of the stochastic process $\eta(t)$ is smooth, in particular, has no singularity at $t=t'$. It should be noted that the situation would be much more complex if we dealt with nonlinear integrals of white noise because the correlation function of white noise proportional to $\delta(t-t')$ comes up at the boundary of integration regions [@MyBook]. As in the previous section let us find the stationary distribution function $p^\text{st}(\eta)$ of the variable $\eta$ which will be used below. Going in the same, way we appeal to the Fokker-Planck equation $$\label{FPeta} \frac{\partial p}{\partial t} = \frac{\partial }{\partial \eta}\left[\frac{\partial p}{\partial \eta} + \alpha \frac{d\Phi(\eta)}{d\eta}p\right]\,,$$ matching Eq.  and governing the evolution of the distribution function $p(\eta,t)$. In the limit $t\to\infty$ its solution gives us the desired result $$\label{petast} p^\text{st}(\eta) = \left[2\int\limits^{\infty}_{0}e^{-\alpha\Phi(\eta')} \,d\eta'\right]^{-1} e^{-\alpha\Phi(\eta)}\,.$$ It should be noted that Exps.  and , as it must, coincide with each other within the cofactor $d\eta/dv$ (see Exp. ) coming from the transformation of the elementary volume in the transition $\mathbb{R}_v\to \mathbb{R}_\eta$. Trajectory classification {#sec:TC} ------------------------- ![The typical form of the time pattern $\{v(t)\}$ generated by model  in the frameworks of ansatz . The two frames depict the same time pattern for different time scales. Based on the results presented in Ref. [@we1], the value $\alpha = 1.6$ was used.[]{data-label="F1"}](fig1.pdf){width="0.65\columnwidth"} As shown in Refs. [@we1; @we2], it is extreme fluctuations in the particle velocity, i.e., large amplitude peaks in the time pattern $\{v(t)\}$ that are responsible for the random walks at hand exhibiting the properties of Lévy flights. Figure \[F1\] illustrates the multiscale structure of the pattern $\{v(t)\}$. It turns out that for any time interval $t\gg1$ only a few peaks with the largest amplitudes inside it contribute mainly to the particle displacement [@we1]. The amplitudes of such velocity fluctuations are distributed according to the power-law and their characteristic magnitude grows with the observation time interval $t$ also according to the power-low. The former feature gives rise to the Lévy distribution of the particle displacement, the latter one is responsible for the Lévy time scaling [@we2]. These peculiarities make it attractive to employ the following classification of particle trajectories. The general idea is to partition any random trajectory $\{x(t)\}$ in such a manner that its fragments $\{x(t)\}_i^{i+1}$ or, more strictly, their “projections” $\{v(t)\}_i^{i+1}$ onto the velocity space $\mathbb{R}_v$ can be treated as random walks *inside* or *outside* a certain neighborhood $\mathcal{L}$ of the origin $v=0$. In this case it would be possible to regard the fragments of random walks outside the region $\mathcal{L}$ as individual peaks of the time pattern $\{v(t)\}$. Naturally, instead of the space $\mathbb{R}_v$ the space $\mathbb{R}_\eta$ of the variable $\eta$ can be equivalently used, so in what follows the corresponding neighborhood in the space $\mathbb{R}_\eta$ will be designated with the same symbol $\mathcal{L}$ to simplify notations. Besides, to operate with the desired partition efficiently it has to be countable. A direct implementation of this idea, however, faces a serious obstacle. The matter is that a random trajectory is not a smooth curve. Therefore, although for a wandering particle it is possible to calculate the probability of getting the boundary of $\mathcal{L}$ for the first time, the question about crossing this boundary for the second time is meaningless. The particle will cross the boundary immediately after the first one and ordering such intersection points in the countable manner seems to be just impossible. The trajectory classification to be constructed below enables us to overcome this problem and, thus, to implement the desired partition. Previously the key elements of this classification were used in describing grain boundary diffusion [@Keigan1] and subdiffusion along a comb structure [@Keigan2]. Figure \[F2\] demonstrates the required classification of the particle random walks $\{\eta(t)\}$ in the diagram form. Let us describe it step by step. () First of all, just to simplify subsequent mathematical manipulations we confine our consideration to the upper half-space $\mathbb{R}^+_\eta = \{\eta\geq 0\}$ assuming its boundary, $\eta = 0$, to be reflecting. The symmetry of the kinetic coefficient $\phi(\eta)$, namely, $\phi(-\eta) = \phi(\eta)$, allows it. Then two values $\eta_l$ and $\eta_u$ such that $$\label{sec1:etaul} 0 < \eta_l < \eta_u$$ have to be introduced. The choice of their specific magnitudes will be reasoned below. Here, leaping ahead, we note that in the frameworks of ansatz  the value $\eta_l$ should not be a too large number enabling us, nevertheless, to approximate the kinetic coefficient $\phi_0(\eta)=\tanh(\eta)$ by the sign-function for $|\eta|>\eta_l$, $$\label{sec1:sign} \phi_0(\eta)\approx \operatorname{sign}(\eta) = \begin{cases} 1 &\text{if $\eta>0$}\,,\\ 0 &\text{if $\eta = 0$\,,}\\ -1 &\text{if $\eta<0$}\,. \end{cases}$$ For example, for $\eta_l = 2.0$ the estimates of $\tanh(\eta_l)\approx 0.96$ and $v_l = \sinh(\eta_l)\approx 3.6$ hold by virtue of . The value $\eta_u$ just should not exceed $\eta_l$ substantially. So in some sense $$\label{sec1:etaul0} 1 \lesssim \eta_l \qquad\text{and}\qquad \eta_u -\eta_l \lesssim 1$$ is the optimal choice. Certainly, the final results describing the particle displacement in the real space $\mathbb{R}_x$ do not depend on the two values. () Again to simplify mathematical constructions without loss of generality let us consider trajectories $\{\eta(t')\}_{t'=0}^{t'=t}$ originating from the point $\eta_u$, i.e., set $\eta(t')|_{t'=0}=\eta_u$. Their terminal point $\eta = \eta(t')|_{t'=t}$ can take an arbitrary value. () Now we are able to specify the desired partition. Each trajectory $\{\eta(t')\}_{t'=0}^{t'=t}$ is represented as a certain collection of fragments shown in Fig. \[F2\], which is written symbolically as $$\label{revis:1} \mathbb{G}_t^{(n)}(\eta) = \mathbb{F}_{01}^\text{out} \otimes \mathbb{F}_{12}^\text{in} \otimes \mathbb{F}_{23}^\text{out} \otimes \mathbb{F}_{34}^\text{in}\otimes\ldots\otimes \mathbb{G}_n^\text{out/in}(\eta)\,.$$ If there is only one fragment in the collection $\mathbb{G}_t^{(n)}(\eta)$, namely, the last fragment, then the number $n$ of the internal fragments is set equal to zero, $n=0$. The meaning of these fragments is as follows. - The first fragment $\mathbb{F}^\text{out}_{01}$ represents the motion of the particle outside the interval $\mathcal{L}_l=[0,\eta_l)$ until it gets the point $\eta_l$ for the first time at a time moment $t_1>0$. Further such particle motion will be referred to as random walks *outside* the neighborhood $\mathcal{L}$ of the origin $\eta=0$. - The next fragment $\mathbb{F}^\text{in}_{12}$ matches the particle wandering inside the interval $\mathcal{L}_u=[0,\eta_u)$ until it gets the point $\eta_u$ for the first time at a certain moment $t_2>t_1$. Particle motion of this type will be referred to as random walks *inside* the neighborhood $\mathcal{L}$. - The following two fragments $\mathbb{F}^\text{out}_{23}$ and $\mathbb{F}^\text{in}_{34}$ are similar to the first and second ones, $\mathbb{F}^\text{out}_{01}$ and $\mathbb{F}^\text{in}_{12}$, respectively. The time moments corresponding to their terminal points are designated as $t_3$ and $t_4$. A sequence of such fragments alternating one another makes up the internal part of the given trajectory. - Sequence  ends with a final fragment which can be of two configurations, $\mathbb{G}^\text{out}_{n}(\eta)$ and $\mathbb{G}^\text{in}_n(\eta)$. - The former one, $\mathbb{G}^\text{out}_{n}(\eta)$, is related to the particle motion starting from the point $\eta_u$ at time $t_n$ and getting the point $\eta$ at time $t$ without touching the boundary point $\eta=\eta_l$ of the interval $\mathcal{L}_l$. This configuration exists only for $\eta > \eta_l$ and is characterized by an even number of the intermediate points, $n = 2k$, of the trajectory partition. - The latter one, $\mathbb{G}^\text{in}_n(\eta)$, is similar to the former configuration within the exchange of the start and boundary points; now $\eta_l$ is the start point whereas $\eta_u$ is the boundary point of the interval $\mathcal{L}_u$. For the configuration $\mathbb{G}^\text{in}_n(\eta)$ to exist the terminal point $\eta$ has to meet the inequality $\eta < \eta_u$. The corresponding number of the partition points is odd, $n = 2k+1$. It should be noted that when the terminal point $\eta$ of the given trajectory belongs to the interval $\eta_l < \eta<\eta_u$ both the configurations exist. - In addition, for $\eta >\eta_l$ there is a special configuration $\mathbb{G}_t^\text{(0)}(\eta)$ of the trajectories that start from the point $\eta_u$ at the initial time $t=0$ and get the point $\eta$ at time $t$ without touching the boundary point of $\mathcal{L}_l$, i.e., $\eta = \eta_l$. It represents sequence  with no internal fragments, $n=0$. This configuration is actually equivalent to $\mathbb{G}^\text{out}_n(\eta)$ with $t_n=0$. () The introduced fragments of particle trajectories may be characterized by other classification parameters, denoted symbolically as $\boldsymbol{\Theta}^{\text{out/in}}_{i,i+1}$, in addition to the initial and terminal time moments of their realization, $t_i$ and $t_{i+1}$. Figure \[F2\] shows such parameters for the terminal fragments of $\mathbb{G}^{(n)}_t(\eta)$ too. The collection $\{\boldsymbol{\Theta}^{\text{out/in}}_{i,i+1}\}$ is determined by the specific details we want to know about the time pattern $\{\eta(t')\}_{t'=0}^{t'=t}$. In the present paper all the random walks in $\mathbb{R}_v$ will be separated into different groups considered individually according to the largest amplitude $\theta$ attained by the variable $\eta$ during the corresponding time interval $[0,t]$. As will be clear below, this classification can implemented imposing the addition requirement on each fragment of random walks outside the region $\mathcal{L}$ that bounds variations of the variable $\eta$ inside it. Namely, we assume the variable $\eta$ not to exceed $\theta$, $$\label{sec1:theta} \eta_l < \eta(t) < \theta \qquad \text{for}\qquad t_i < t < t_{i+1}\quad\text{and}\quad\forall i\,.$$ In this case each set $\boldsymbol{\Theta}^\text{out}_{i,i+1} = \{\theta\}$ contains only one parameter taking the same value for all of them. For the terminal fragment $\mathbb{G}_n^\text{out}(\eta)$ a similar condition holds. For the random walks inside the region $\mathcal{L}$ there are no additional classification parameters, so the sets $\boldsymbol{\Theta}^\text{in}_{i,i+1}$ are empty. $\blacksquare$ The constructed partition enables us to develop a more sophisticated description of the pattern $\{\eta(t)\}$, in particular, consider its configurations where the amplitudes of its $m$ largest peaks take given values $\theta_1,\theta_2,\ldots, \theta_m$. For sure, such analysis will allow us to penetrate much deeper into the properties of Lévy flights, which, however, is worthy of individual investigation. The remaining part of the paper will be devoted to the statistical properties of random walks described in terms of the constructed partition. Before this, however, let us discuss the relationship between the given classification of the particle motion and the well-known model of continuous time random walks (CTRW). Constructed partition as a generalized CTRW model ------------------------------------------------- The CTRW model imitates random motion of particles by assigning to each jump of a wandering particle a jump length $x$ and a waiting time $t$ elapsing between two successive jumps which are distributed with a certain joint probability density $\psi(x,t)$. When it is possible to write this probability density as the product of the individual probability densities of jump length, $\psi_x(x)$, and waiting time, $\psi_t(t)$, i.e., $\psi(x,t)= \psi_x(x)\psi_t(t)$, the two quantities can be regarded as independent random variables and the model is called decoupled CTRW. Another widely met version of this model, coupled CTRW, writes the probability density $\psi(x,t)$ as the product of two functions $\psi(x,t)=\psi_x(x)\psi_v(x/t)$ or $\psi(x,t)=\psi_t(t)\psi_v(x/t)$. In the given case the jump parameters $x$ and $t$ are no longer independent random variables; the independent variables are the jump length $x$ (or the waiting time $t$) and the mean particle velocity $v= x/t$. The continuous implementation of the coupled CTRW is based on the assumption that within one jump the particle moves along the straight line connecting its initial and terminal points with the fixed velocity $v=x/t$. The constructed partition can be regarded as a certain generalization of CTRW that allows one to consider the detailed structure of elementary steps. Indeed, a pair of succeeding fragments of random walks inside, $\mathbb{F}^\text{in}_{i,i+1}$, and outside, $\mathbb{F}^\text{out}_{i+1,i+2}$, the region $\mathcal{L}$ form an elementary step of the equivalent discrete random walks whose jump length and waiting time are random variables. If the random walks inside the region $\mathcal{L}$ do not contribute substantially to the particle displacement $x$ we may speak about a stochastic process similar to coupled CTRW. The main difference between the two processes is due to the fact that the particle under consideration does not move uniformly within the fragment $\mathbb{F}^\text{out}_{i+1,i+2}$. So the knowledge of the corresponding mean velocity is not enough to describe the real particle motion. The random walks inside the region $\mathcal{L}$ can be essential for the particle motion in space when, for example, a wandering particle spends the main time inside the region $\mathcal{L}$. In this case the corresponding stochastic process may be categorized as coupled-decoupled CTRW. In the case under consideration, as will be demonstrated below, the random walks inside the region $\mathcal{L}$ are not responsible for the Lévy type behavior of the particle motion. However, the constructed partition actually does not require the normal behavior of random walks near the origin $v=0$, so it could be also used for modeling stochastic processes with kinetic coefficients exhibiting singularities at $v=0$. Under such conditions the random walks inside the region $\mathcal{L}$ are also able to cause anomalous properties of wandering particles, giving rise to the power-law distribution of the waiting time. Green function -------------- Continuing the constructions of Sec. \[sec:TC\] the present section analyzes the statistical properties of the random trajectories $\{\eta(t')\}_{t'=0}^{t'=t}$ meeting condition  that are imposed upon all the fragments of the random walks outside the region $\mathcal{L}$. It will be done based on the calculation of the Green function $\mathcal{G}(\eta,t)$, i.e., the probability density of finding the particle at the point $\eta$ at time $t$ provided initially, $t=0$, it is located at the point $\eta_u$ and does not cross the boundary $\eta=\theta$ within the time interval $(0,t)$. The developed classification of random walks illustrated by the diagrams in Fig. \[F2\] enables us to write $$\label{sec2:G0} \mathcal{G}(\eta,t)= \Theta_\text{H}(\eta -\eta_l) \mathcal{G}^\vartriangle(\eta,t) + \Theta_\text{H}(\eta_u -\eta) \mathcal{G}^\triangledown(\eta,t)\,,$$ where we have introduced the functions \[sec2:Gab\] $$\begin{aligned} \label{sec2:Ga} \mathcal{G}^\vartriangle(\eta,t) & = \sum_{k=0}^\infty \int\limits_0^t dt' \mathcal{G}^\text{out}(\eta,t-t') \mathfrak{P}_k(t') \,, \\ \label{sec2:Gb} \mathcal{G}^\triangledown(\eta,t) &= \sum_{k=0}^\infty \int\limits_0^t dt' \int\limits_0^{t'} dt'' \mathcal{G}^\text{in}(\eta,t-t') \mathcal{F}^\text{out}(t'-t'') \mathfrak{P}_k(t'') \,,\end{aligned}$$ and the Heaviside step function $$\label{sec2:Heav} \Theta_\text{H}(\eta) = \begin{cases} 1, & \text{if $\eta > 0$},\\ 0, & \text{if $\eta < 0$} \end{cases}$$ to combine the cases $0\leq \eta < \eta_l$, $\eta_l < \eta < \eta_u$, and $\eta_u < \eta < \theta$ in one formula. Here $\mathcal{G}^\text{out}(\eta,t)$ is the probability density of finding the particle at the point $\eta$ at time $t$ provided it starts from the point $\eta_u$ remains inside the region $\eta_l < \eta<\theta$ within the time interval $0<t'<t$. The function $\mathcal{G}^\text{in}(\eta,t)$ is actually the same probability density within the replacement of the start point $\eta_u\to\eta_l$ and now the localization region is $0\leq \eta < \eta_u$. The function $\mathfrak{P}_k(t)$ specifies the probabilistic weight of the pattern $\mathbb{P}(t|k)$ shown in Fig. \[F3\] which comprises $k$ fragments of random walks outside the region $\mathcal{L}$ and $k$ fragments of random walks inside it. The former fragments will be referred also as to peaks of the pattern $\mathbb{P}(t|k)$. The probabilistic weight of the pattern $\mathbb{P}(t|k)$ is given by the following formula for $k>0$ \[sec2:G2\] $$\begin{aligned} \label{sec2:G2a} \mathfrak{P}_k(t) &= \idotsint\limits_{0<t_1<t_2 \cdots < t_{2k-1} < t} dt_1dt_2\ldots dt_{2k-1} \prod_{i=0}^{k-1} \mathcal{F}^\text{in}(t_{2i+2}-t_{2i+1}) \mathcal{F}^\text{out}(t_{2i+1}-t_{2i}) \\ \intertext{with $t_0=0$ and for $k = 0$, by definition,} \label{sec2:G2b} \mathfrak{P}_0(t) &= \delta(t)\,. \end{aligned}$$ In Exps.  and the functions $\mathcal{F}^\text{out}(t)$ and $\mathcal{F}^\text{in}(t)$ with the corresponding values of the argument $t$ determine the probabilistic weights of the fragments $\mathbb{F}^\text{out}_{i,i+1}$ and $\mathbb{F}^\text{in}_{i,i+1}$, respectively. The meaning of the former function is the probability density of reaching the point $\eta_l$ for the first time at the moment $t$ provided the particle is located initially at the point $\eta_u$. The meaning of the latter function is the same within the replacement $\eta_l \leftrightarrow \eta_u$. To simplify the notations, the quantities $\eta_u$, $\eta_l$, and $\theta$ have been omitted from the argument list of the corresponding functions noted above. Integrals and are of the convolution type, therefore, it is convenient to pass from the time variable $t$ to the corresponding variable $s$ using the Laplace transform $$\label{sec2:Lapl} W(s) := \mathcal{W}^L(s) =\int_0^\infty dt e^{-st}\mathcal{W}(t)\,,$$ where $\mathcal{W}(t)$ is one of the functions entering Exps.  and . In these terms the given integrals are reduced to algebraic expressions, namely, $$\label{sec2:G2L} \mathfrak{P}^L_k(s) = \left[{F}^\text{in}(s) {F}^\text{out}(s)\right]^k\,.$$ and, thus, \[sec2:GabL\] $$\begin{aligned} \label{sec2:GaL} {G}^\vartriangle(\eta,s) & = \sum_{k=0}^\infty \left[{F}^\text{in}(s) {F}^\text{out}(s)\right]^k {G}^\text{out}(\eta,s) = \frac{{G}^\text{out}(\eta,s)}{1-{F}^\text{in}(s) {F}^\text{out}(s)}\,, \\ \label{sec2:GbL} {G}^\triangledown(\eta,s) &= \sum_{k=0}^\infty \left[{F}^\text{in}(s) {F}^\text{out}(s)\right]^k {F}^\text{out}(s){G}^\text{in}(\eta,s) = \frac{{F}^\text{out}(s){G}^\text{in}(\eta,s)}{1-{F}^\text{in}(s) {F}^\text{out}(s)} \end{aligned}$$ because $|{F}^\text{in}(s) {F}^\text{out}(s)|<1$. As demonstrated in \[App:inL\] the functions $F^\text{in}(s),\, F^\text{out}(s)\approx 1$ for $s\ll 1$ provided, in addition, the inequality $\theta \gg 1$ holds. Therefore dealing with time scales $t\gg 1$ only the dependence of the denominator on the argument $s$ should be taken into account in evaluating Exps. . The other terms may be calculated within the limit $s\to 0$. In this way, using formula  derived in \[AppF\], the Green function components $G^{\vartriangle,\triangledown}(\eta,s)$ determined by Exps.  can be represented as $$\label{sec2:New1} G^{\vartriangle,\triangledown}(\eta,s) = \left[ s - \left.\frac{dp^\text{st}(\eta')}{d\eta'}\right|_{\eta'=\theta}\right]^{-1} \frac{\alpha G^\text{out,\,in}(\eta,s)\big|_{s\to 0}}{\left[e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}\right]\int_0^{\infty}d\eta' e^{-\alpha\Phi(\eta')}}$$ in the leading order. Then appealing to Exp.  the desired Green function or, speaking more strictly, its Laplace transform $G(\eta,s)$ specified by Exp.  is written as $$\label{sec2:New2} G(\eta,s) = \left[ s - \left.\frac{dp^\text{st}(\eta')}{d\eta'}\right|_{\eta'=\theta}\right]^{-1} p^\text{st}(\eta)\,.$$ It should be reminded that here $p^\text{st}(\eta)$ is the stationary distribution function of the random variable $\eta$ after merging the half-spaces $\{\eta >0\}$ and $\{\eta <0\}$. Expression  for the Green function together with the constructed probabilistic weight $\mathfrak{P}_k(t)$ of the peak pattern $\mathbb{P}(t|k)$, Exps.  and , are the main results of the given section. They enable us to consider the probabilistic properties of individual peaks in the pattern $\mathbb{P}(t|k)$ instead of random trajectories as whole entities, which is the subject of the next section. Peak pattern statistics ======================= As follows from the governing equation  fluctuations of the particle velocity $v$ are correlated on time scales about unity, $t\sim 1$. Therefore, in the analysis of Lévy flights based on model  within the limit $t\gg 1$ the information about the particle velocity $\eta$ (or $v$ in the initial units) is redundant. So to rid our constructions of such details let us integrate Exp.  over all the possible values of the velocity $\eta$, i.e. over the region $0<\eta<\theta$. Since in this context the influence of the upper boundary $\theta \gg 1$ is ignorable, the integration region may be extended over the whole interval $0<\eta<\infty$. Then denoting the result of this action on the left-hand side of Exp.  by $$\label{sec22:0} \mathfrak{G}_L(s) = \int\limits_0^\infty G(\eta,s)\,d\eta$$ we get $$\label{sec22:1} \mathfrak{G}_L(s) = \left[ s - \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right]^{-1} = \tau \sum_{k= 0}^\infty \left[F^\text{in}(s)F^\text{out}(s)\right]^k.$$ In deriving the second equality of Exp.  formula  for the probabilistic weight of the composed unit $\mathbb{F}_{i,i+1}^\text{out}\otimes\mathbb{F}_{i+1,i+2}^\text{in}$ of the peak pattern $\mathbb{P}(t\mid k)$ has been used. Namely, first, it has enabled us to write $$\label{sec22:2} F^\text{in}(s)F^\text{out}(s) = 1 - \tau \left[ s - \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right]\,,$$ where the introduced time scale $$\label{sec22:3} \tau = \frac1{\alpha}\left[{e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}}\right] \int\limits_{0}^\infty d\eta'\, e^{-\alpha\Phi(\eta')}$$ can be interpreted as the mean duration of this unit. Finally, the equality $$\sum_{k=0}^\infty \left[F^\text{in}(s)F^\text{out}(s) \right]^k = \frac1{1-F^\text{in}(s)F^\text{out}(s)}$$ has led us to Exp. . Since the limit of large time scales, $t\gg 1$, is under consideration, the inequalities $s\ll 1$ and $\theta \gg 1$ are assumed to hold beforehand. In this case, by virtue of Exp. , $$1-F^\text{in}(s)F^\text{out}(s)\ll 1$$ and in sum  the terms with $k\gg 1$ contribute mainly to its value. Thereby we may confine our consideration to peak patterns $\mathbb{P}(t\mid k)$ that are composed of many elementary units. It allows us to convert the discrete sum $\sum^\infty_{k=0}(\ldots)$ into a continuous integral as follows $$\label{sec22:4} \sum_{k=0}^\infty (\ldots)\Rightarrow \int\limits_0^\infty dk(\ldots)$$ and make use of the approximation $$\label{sec22:5} \left[F^\text{in}(s)F^\text{out}(s)\right]^k = \exp\left\{-k \tau \left[ s - \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right]\right\}\,.$$ In these terms formula  reads $$\label{sec22:6} \mathfrak{G}_L(s) = \tau\int\limits_0^\infty dk \exp\left\{-k \tau \left[ s - \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right]\right\} {} = \tau\int\limits_0^\infty dt\int\limits_0^\infty dk\, e^{-st} \delta(t-k\tau) \exp\left\{k \tau \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right\}.$$ The last expression enables us to represent the original function $\mathfrak{G}(t)$ of the Laplace transform $\mathfrak{G}_L(s)$ as $$\label{sec22:7} \mathfrak{G}(t) = \tau\int\limits_0^\infty dk\, \delta(t-k\tau) \exp\left\{k \tau \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right\} =\exp\left\{t \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right\}.$$ In other words, within the given limit it is possible to consider that only the pattern $\mathbb{P}(t\mid k)$ with $k=t/\tau$ (or more strictly $k=[t/\tau]$) contributes to the function $\mathfrak{G}(t)$. The latter statement is the pivot point in our constructions that admits another interpretation of the function $\mathfrak{G}(t)$ and its efficient application in the theory of Lévy flights. Appealing to approximation  let us rewrite Exp.  in the form $$\label{sec22:8} \mathfrak{G}(t)= \left[F^\text{in}(0)F^\text{out}(0)\right]^{k_t}\qquad\text{with}\qquad k_t=\frac{t}{\tau}\,.$$ In what follows we will not distinguish between the ratio $t/\tau$ and the derived integer $k_t=[t/\tau]$ because for $k\gg 1$ their difference is of minor importance and will keep in mind this integer where appropriate in using the ratio $t/\tau$. Therefore formula  can be read as \[sec22:12\] $$\begin{aligned} \label{sec22:9} \mathfrak{G}(t) & = \int\limits_0^\infty \mathfrak{P}_{k_t}(t')\,dt'\qquad\text{for $t\gg\tau\sim 1$} \\ \intertext{or, by virtue of \eqref{sec2:G2},} \label{sec22:12a} \mathfrak{G}(t) &= \idotsint\limits_{0<t_1<t_2 \cdots < t_{2k} < \infty} dt_1dt_2\ldots dt_{2k} \prod_{i=0}^{k_t-1} \mathcal{F}^\text{in}(t_{2i+2}-t_{2i+1}) \mathcal{F}^\text{out}(t_{2i+1}-t_{2i})\,.\end{aligned}$$ Expressions  are one of the main technical results obtained in the present work. In particular, they allow us to state the following. \[PropAdd:1\] The probability $\mathfrak{G}(t)$ of finding the particle whose “velocity” $\eta$ is localized inside the region $|\eta| <\theta$ during the time interval $(0,t)$ for $\theta\,,t\gg 1$ is equal to the probability $\mathfrak{P}_{k_t}(t')$ of the pattern $\mathbb{P}(t'|k_t)$ with $k_t = t/\tau$ peaks after the averaging over its duration $t'$. It should be noted that the given statement has been directly justified assuming the initial particle velocity to be equal to $\eta_0 = \eta_u$. However, as can be demonstrated, it holds for any $\eta_0\ll\theta$. Also it is worthwhile to underline the fact that within this description the physical time $t$ is replaced by the number $k_t$ of peaks forming the pattern $\mathbb{P}(t'|k_t)$, whereas its actual duration $t'$ does not matter. It becomes possible due to a certain self-averaging effect. In particular, according to Exp. , the duration of the units $\{\mathbb{F}_{i,i+1}^\text{out}\otimes\mathbb{F}_{i+1,i+2}^\text{in}\}$ should be distributed with the exponential law, at least, on scales $t\gg \tau$. As a result the duration $t'$ of the pattern $\mathbb{P}(t'|k_t)$ changes near its mean value $k_t\tau=t$ with relatively small amplitude. Besides, to simplify further explanations, when appropriate the probability $\mathfrak{G}(t)$ will be also referred to as the probability of the pattern $\mathbb{P}(k_t)$, where the argument $t'$, its duration, is omitted to underline that the required averaging over $t'$ has been performed. Expression  admits another interpretation of the developed random walk classification. \[Prop1\] The probability $\mathfrak{G}(t)$ is equal to the probability of the particle returning to the initial point $\eta = \eta_u$ for $k_t=t/\tau$ times sometime after. The return events are understood in the sense determined by the sequence of alternate jumps between the two boundaries $\{\eta_l,\eta_u\}$ of the layer $\mathcal{L}$. The following proposition will play a significant role in further constructions. \[Prop2\] The probability $\mathfrak{G}(t)$ of the peak pattern $\mathbb{P}(k_t)$ is specified by the expression $$\label{Prop.eq.1} \mathfrak{G}(t) = \exp\left\{t \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right\}$$ and the probability of its one unit, i.e., the probability of one return to the initial point $\eta = \eta_u$ sometime after is evaluated by the expression $$\label{Prop.eq.2} \mathfrak{g}(t) = 1+\tau\left. \frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\,.$$ Formula  stems directly from Exp. , whereas formula  is a consequence of Exps.  and , where in the latter one the parameter $s$ is set equal to zero, $s=0$. Core stochastic process ======================= As noted above spatial displacement of the wandering particle is manly caused by extreme fluctuations in its velocity. Therefore the direct contribution of the random walks inside the layer $\mathcal{L}$ to the particle displacement is ignorable. We have to take into account only the fact that during a certain time the wandering particle or, more strictly, its velocity $\eta$ (or $v$ in the original units) is located inside this layer. Thus, if two given stochastic processes differ from each other in their properties *only inside* the layer $\mathcal{L}$ but are characterized by the same probabilistic weight $\mathcal{F}^\text{in}(t)$ of the fragments $\{\mathbb{F}^\text{in}_{i,i+1}\}$, then they can be regarded as equivalent. In the case under consideration, i.e., for $t\gg 1$ it is sufficient to impose the latter requirement on the Laplace transform $F^\text{in}(s)$ of this weight within the linear approximation in $s\ll 1$. The purpose of the present section is to construct a certain stochastic process that, on one hand, is equivalent in this sense to the process governed by model . On the other hand, its description should admit an efficient scaling of the corresponding governing equation, which will enable us to single out the basic mechanism responsible for the found basic features of particle motion. The other characteristics of particle motion can be explained by appealing to the corresponding coefficients of this scaling. The desired equivalent process will be called the core stochastic process for these reasons. Construction ------------ ![Illustration of constructing an effective stochastic process $\{\eta'(t)\}$ with the V-type potential that is equivalent to the initial process $\{\eta(t)\}$ from the standpoint of the particle displacement in the space $\mathbb{R}_x$.[]{data-label="F5"}](fig5.pdf){width="0.6\columnwidth"} The key points in constructing the core stochastic process are illustrated in Fig. \[F5\]. The construction is based on the transformation $$\begin{aligned} \label{sec3:1} \eta & = \eta'+ \operatorname{sign}(\eta') \Delta_\alpha \\ \intertext{and the replacement of the potential $\Phi_0(\eta)$ by its linear extrapolation to the region $\eta\lesssim 1$} \label{sec3:2} \Phi_0(\eta) & \rightarrow |\eta'| + \Delta_\alpha - \ln2\,.\end{aligned}$$ The constant $\Delta_\alpha$ has to be chosen such that in both the cases the function $F^\text{in}(s)$ take the same form within the linear approximation in $s$. Appealing to \[AppFIN\], namely, formula  we see that this requirement is reduced practically to the equality $$\label{sec3:3} \int\limits_0^\infty e^{-\alpha\Phi_0(\eta)} d\eta = \int\limits_{\Delta_\alpha}^\infty e^{-\alpha(\eta-\ln2)}d\eta$$ because the potential $\Phi_0(\eta)$ differs considerably from its linear interpolation  only in the region $\eta \lesssim 1$. The direct calculation of the integrals entering equality  for function  yields the desired value $$\label{sec3:4} \Delta_\alpha = \frac1{\alpha}\ln\left[ \frac{4\Gamma(\alpha)}{\alpha\Gamma^2(\alpha/2)} \right]\,,$$ where $\Gamma(\ldots)$ is the gamma function. Figure \[F6\] shows the value $\Delta_\alpha$ as a function of the parameter $\alpha\in(0,2)$. It should be noted that previously the value of $\Delta_\alpha$ has been implicitly assumed to be less than $\eta_l$, which is justified by Fig. \[F6\]. Indeed, on one side, the boundary $\eta_l$ is initially assumed to belong to the region wherein the function $\Phi_0(\eta)$ admits the linear approximation, i.e., the estimate $\eta_l\gtrsim1$ is assumed to hold beforehand. On the other side, the maximal value of $\Delta_\alpha$ is about 0.35 according to Fig. \[F6\]. ![The magnitude of the parameter $\Delta_\alpha$ vs the possible values of the parameter $\alpha$.[]{data-label="F6"}](fig6.pdf){width="0.45\columnwidth"} In what follows we will confine our consideration to the special case  matching the ideal Lévy flights. It enables us to convert from the stochastic process $\{\eta(t)\}$ to the effective stochastic process $\{\eta'(t)\}$ using transformation  and to reduce Eq.  to the stochastic equation $$\label{sec3:5} \frac{d\eta'}{dt}=-\alpha\operatorname{sign}(\eta')+\sqrt{2}\xi(t)$$ governing the analyzed Lévy flights in the equivalent way. Then, via the next transformation $\eta'\to \mathfrak{u}$ specified by the expressions $$\label{sec3:6} \eta' = \frac1{\alpha}\,\mathfrak{u}\,,\qquad t = \frac1{\alpha^2}\,\mathfrak{t}$$ involving also the transformation of the time scales $t\to \mathfrak{t}$ equation  is rewritten as follows $$\label{sec3:7} \frac{d\mathfrak{u}}{d\mathfrak{t}}=-\operatorname{sign}(\mathfrak{u})+\sqrt{2}\xi(\mathfrak{t})\,.$$ It is a parameter-free stochastic differential equation with additive white noise such that $$\label{sec3:8} \left< \xi(\mathfrak{t})\right> = 0\,, \qquad \left< \xi(\mathfrak{t}) \xi(\mathfrak{t}')\right> = \delta(\mathfrak{t}-\mathfrak{t}')\,.$$ In other words, we have constructed the stochastic process $\{\mathfrak{u}(\mathfrak{t})\}$ governed by Eq.  that can be treated as the desired *core process*. It is of the same form for all the types of one-dimensional Lévy flights, at least, Lévy flights described by models similar to Eq.  inside the region where the cut-off effects are not significant. The basic parameters characterizing the generated Lévy flights such as the exponent of the Lévy scaling law depend on the system parameters, in particular, the coefficient $\alpha$ via the transformation from $\mathfrak{u}(\mathfrak{t})$ to $v(t)=dx/dt$. Combining together Exps. , , and we can write this transformation as follows $$\label{sec3:9} \frac{dx}{dt} = \sinh\left\{ \frac{\mathfrak{u}(\mathfrak{t})}{\alpha} + \Delta_\alpha \operatorname{sign}\left[\mathfrak{u}(\mathfrak{t})\right] \right\}$$ which is completed by the second proportionality of Exps.  relating $t$ to $\mathfrak{t}$. Summarizing aforesaid we get the following statement. \[Prop3\] The Lévy flights $\{x(t)\}$ governed by model  with the kinetic coefficients  are equivalently described by the parameterless stochastic process $\{\mathfrak{u}(\mathfrak{t})\}$ obeying the stochastic differential equation  with additive white noise. The variable $\mathfrak{u}$ and the initial variable $x$ are related via Exp.  and the second proportionality of Exps. . Propositions \[PropAdd:1\]–\[Prop3\] are the basic results of the present paper. In what follows we will make use of them to demonstrate the fact that model  does describe the Lévy flights of superballistic, quasiballistic, and superdiffusive regimes matching $0<\alpha < 1$, $\alpha = 1$, and $1<\alpha <2$, respectively. It should be reminded that this correspondence was strictly proved only for the superdiffusive Lévy flights [@we2] and for the other regimes it was demonstrated numerically [@we3]. A more sophisticated analysis of such Lévy flights based on the constructed representation is worthy of an individual publication. Asymptotic properties of the core stochastic process ---------------------------------------------------- Let us analyze the probabilistic properties of the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ for the core stochastic process $\mathfrak{u}(\mathfrak{t})$ in the limit $\theta\gg1$. To do this, the asymptotic behavior of distribution functions describing the statistics of time patterns related to $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ as $\theta\to\infty$ will be considered in detail. Dealing with the core stochastic process we may set the first boundary $\mathfrak{u}_l$ of the layer $\mathcal{L}_\mathfrak{u}$ equal to zero, $\mathfrak{u}_l=0$, without loss of generality. The stationary distribution function $\mathfrak{p}^\text{st}(\mathfrak{u})$ of the random variable $\mathfrak{u}$ is determined by the expression $$\label{sec3:10} \mathfrak{p}^\text{st}(\mathfrak{u}) = e^{-\mathfrak{u}}$$ after merging the half-spaces $\{\mathfrak{u}<0\}$ and $\{\mathfrak{u}>0\}$, which stems directly from Exps.  and after setting $\phi(\eta) = 1$ and $\alpha=1$ in these formulas. The probability $\mathfrak{G_u}(\mathfrak{t},\theta)$ is determined by the corresponding path integral over all the trajectories $\{\mathfrak{u}(\mathfrak{t}')\}_{0}^{\mathfrak{t}}$ meeting the inequality $0<\mathfrak{u}(\mathfrak{t}')< \theta$ for $0<\mathfrak{t}'<\mathfrak{t}$; now the value $\theta$ is noted directly in the list of the arguments of the function $\mathfrak{G_u}(\mathfrak{t},\theta)$. Therefore the function $$\label{sec3:11} \mathfrak{F}(\mathfrak{t},\theta) := \frac{d\mathfrak{G}(\mathfrak{t},\theta)}{d\theta} = \mathfrak{t}e^{-\theta}\cdot \exp\left\{-\mathfrak{t}e^{-\theta}\right\}$$ is the probability density that the random variable $\mathfrak{u}$ attains the maximal value equal to $\theta$, i.e., $$\label{sec3:12} \theta = \max_{0<\mathfrak{t}'<\mathfrak{t}}\mathfrak{u}(\mathfrak{t}')$$ inside the interval $(0,\mathfrak{t})$. The latter equality in Exp.  stems directly from Proposition \[Prop2\] and formula . According to Proposition \[PropAdd:1\] the function $\mathfrak{G_u}(\mathfrak{t},\theta)$ admits the interpretation in terms of the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$, where the number of peaks or, what is the same, the number of basic units $k_\mathfrak{t} := k_t =\mathfrak{t}/(\alpha^2\tau)$ is regarded as a fixed parameter of the random walk classification. In contrast, the specific time moments $\{\mathfrak{t}_i\}$ of the pattern partition including the terminal point $\mathfrak{t}_n = \mathfrak{t}$ (Fig. \[F3\]) are treated as internal classification parameters over which the averaging must be performed. Under these conditions the function $\mathfrak{G_u}(\mathfrak{t},\theta)$ is just the the probability of the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ generated by the random variable $\mathfrak{u}$ and containing exactly $k_\mathfrak{t}$ peaks. Therefore the function $\mathfrak{F}(\mathfrak{t},\theta)$ introduced via Exp.  gives the probability density that the maximum attained by the variable $\mathfrak{u}$ within at least one of the $k_\mathfrak{t}$ peaks is equal to $\theta$. Naturally, the maxima attained by this variable inside the other peaks must be less than or equal to $\theta$. Going in a similar way the probability density $\mathfrak{f}(\theta)$ of the extreme value $\theta$ for one peak is written as $$\label{sec3:13} \mathfrak{f}(\theta) := \frac{d\mathfrak{g}(\theta)}{d\theta} =\tau_\mathfrak{u} e^{-\theta}\,.$$ Here the value $\theta$ has the meaning $$\label{sec3:14} \theta = \max_{\mathfrak{t}'\in\text{a given peak}}\mathfrak{u}(\mathfrak{t}')$$ and, as before, $\tau_\mathfrak{u}$ is the mean duration of individual peaks. Function  is plotted in Fig. \[F7\] and its asymptotic behavior as $\theta\to \infty$ enables us to introduce a single-peak approximation. ### Single-peak approximation ![Probability density $\mathfrak{F}(\theta,\mathfrak{t})$ of the extreme value $\theta$ being attained by the random variable $\mathfrak{u}$ within the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ (curve 1) and its form in the frameworks of the single-peak approximation (curve 2). In plotting these data the time $\mathfrak{t} =50$ was used.[]{data-label="F7"}](fig7.pdf){width="0.5\columnwidth"} In the region $\theta \gtrsim \theta_\mathfrak{t}$, where $\mathfrak{t}>0$ and $$\label{sec3:15} \theta_\mathfrak{t} = \ln(\mathfrak{t})$$ so that $\mathfrak{t}e^{-\theta_\mathfrak{t}} = 1$, the asymptotic behavior of the extreme value probability $\mathfrak{F}(\theta,\mathfrak{t})$ is specified by the expression $$\label{sec3:15.1} \mathfrak{F}(\theta,\mathfrak{t})\approx \mathfrak{t}e^{-\theta}$$ by virtue of . Formula  can be reproduced in the following way. Let us assume, at first, that the maximal value $\theta$ is attained inside a certain peak $i$. Then inside the other $(k_\mathfrak{t}-1)$ peaks the random variable $\mathfrak{u}$ has to belong to the interval $\mathfrak{u}\in(0,\theta)$. The probability of the first event is given by the function $\mathfrak{f}(\theta)$. The probability of the second event can be written as $$ \mathfrak{g}(\theta)^{k_\mathfrak{t}-1}\approx \left[1 -\mathfrak{f}(\theta)\right]^{k_\mathfrak{t}}\,.$$ In deriving this formula Exps. , , and have been used as well as the inequality $k_\mathfrak{t} = \mathfrak{t}/\tau_\mathfrak{u}\gg 1$ has been taken into account. The latter inequality enables us to confine out consideration to the patten configurations for which $\mathfrak{f}(\theta)\ll 1$ and, thus, to ignore the difference between $k_\mathfrak{t}$ and $(k_\mathfrak{t}-1)$. Therefore, the probability of the extreme value $\theta$ being attained inside peak $i$ is $$ \mathfrak{f}(\theta)\left[1 -\mathfrak{f}(\theta)\right]^{k_\mathfrak{t}}\approx \mathfrak{f}(\theta)\exp\left\{-k_\mathfrak{t} \mathfrak{f}(\theta)\right\}\,.$$ The choice of peak $i$ is arbitrary. So to find the probability of the extreme value $\theta$ being attained somewhere inside the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ we have to multiply the last expression by the number $k_\mathfrak{t} = \mathfrak{t}/\tau_\mathfrak{u}$ of peaks, which gives rise to the equality $$\label{sec3:18} \mathfrak{F}(\theta,\mathfrak{t}) = k_\mathfrak{t} \mathfrak{f}(\theta) \exp\left\{-k_\mathfrak{t} \mathfrak{f}(\theta)\right\}$$ coinciding with Exp.  by virtue of . In the case $\theta \gtrsim \theta_\mathfrak{t}$ the inequality $k_\mathfrak{t}\mathfrak{f}(\theta)\ll 1$ holds and the exponential multiplier in Exp.  can be ignored, leading to Exp. . In other words, the extreme value $\theta$ is attained actually within one peak whereas the effect of the other peaks on its probabilistic properties is ignorable. Summarizing these arguments we draw the conclusion below. \[Prop4\] In the region $\theta \gtrsim \theta_\mathfrak{t}$ the probability density $\mathfrak{F}(\theta,\mathfrak{t})$ of the variable $\mathfrak{u}$ attaining the extremal value $\theta$ within the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ exhibits the asymptotic behavior that can be represented in the following way. The value $\theta$ is attained exactly within one peak. Variations of the random variable $\mathfrak{u}$ inside the other peaks may be assumed to be small in comparison with $\theta$ and the condition $\mathfrak{u}\leq \theta$ does not really affect their probabilistic properties. Naturally, when a given value of $\theta$ falls outside the region $\theta \gtrsim \theta_\mathfrak{t}$, i.e., $\theta \lesssim \theta_\mathfrak{t}$ variations of the random variable $\mathfrak{u}$ in many peaks become comparable with $\theta$. As a result, the fact that the condition $\mathfrak{u}<\theta$ is imposed on all the peaks of $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ is essential and the single-peak approximation does not hold. In particular, for $\theta \lesssim \theta_\mathfrak{t}$ the probability density $\mathfrak{F}(\theta,\mathfrak{t})$ has to deviate substantially from its asymptotics  as illustrated in Fig. \[F7\]. Now let us apply these constructions to the analysis of the particle displacement in the space $\mathbb{R}_x$. Asymptotic properties and scaling law of spatial particle displacement ====================================================================== The purpose of this section is to demonstrate the fact that for a particle whose random motion is governed by model  the asymptotic behavior of the distribution function $\mathcal{P}(x,t)$ of its spatial displacement $x$ during the time interval $t$ is determined by the asymptotic properties of the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$. By virtue of Proposition \[Prop4\] it can be reformulated as follows. \[Prop5\] Large fluctuations in the spatial displacement $x$ of the wandering particle during the time interval $t\gg1$ are related mainly to its motion inside the single peak of the pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$ whose amplitude is maximal in comparison with the other peaks and attains extremely large values. Proposition \[Prop5\] is actually the implementation of the single-peak approximation applied to describing the particle motion in the space $\mathbb{R}_x$. Appealing to the notion of the core stochastic process $\{\mathfrak{u(t)}\}$ we also can formulate the next statements. \[Prop7\] The region $\mathbb{L}_x(t)=\{x : |x|\gg \overline{x}(t)\}$ of large fluctuations in the particle spatial displacement $x$ directly matches the region $\theta\gtrsim\theta_\mathfrak{t}$ of the extreme values $\theta$ attained by the random variable $\mathfrak{u(t)}$ inside the peak pattern $\mathbb{P}_\mathfrak{u}(k_\mathfrak{t})$. \[Prop8\] The asymptotic behavior of the distribution function $\mathcal{P}(x,t)$ in the region $\mathbb{L}_x(t)$ is practically specified by the asymptotic behavior of the probability density $\mathfrak{F}(\theta,\mathfrak{t})$ inside the region $\theta\gtrsim\theta_\mathfrak{t}$. \[Prop9\] The lower boundary $\overline{x}(t)$ of the region $\mathbb{L}_x(t)$ quantifying the characteristic displacement of the wandering particle during the time interval $t$ can be evaluated using the single-peak approximation and setting $\theta$ equal to the lower boundary of the $\theta$-region wherein the single-peak approximation holds, $\theta\sim \theta_\mathfrak{t}$. Below in this section, at first, we will accept Propositions \[Prop5\]–\[Prop9\] as hypotheses and using them construct the distribution function $\mathcal{P}(x,t)$. Then the comparison of the results to be obtained below with the previous rigorous results [@we2; @we3] justifies these Propositions. Integrating expression  over the time interval $(0,t)$ we get the relationship between the particle spatial displacement $x(t)$ during the given time interval and the time pattern $\{\mathfrak{u(t')}\}_{\mathfrak{t'}=0}^{\mathfrak{t'=t}}$ of the core stochastic process $$\label{new:1} x(t) = \frac1{\alpha^2}\int\limits_{0}^{\mathfrak{t}}\sinh\left\{ \frac{\mathfrak{u}(\mathfrak{t'})}{\alpha} + \Delta_\alpha \operatorname{sign}\left[\mathfrak{u}(\mathfrak{t'})\right] \right\} d\mathfrak{t'}\,.$$ Here the time scale transformation  has been taken into account. According to Proposition \[Prop5\] the time integration in Exp.  can be reduced to the integration over the largest peak $\mathbb{P}_\theta$ of the pattern $\{\mathfrak{u(t')}\}_{\mathfrak{t'}=0}^{\mathfrak{t'=t}}$ provided extreme fluctuations of the random variable $x(t)$ are under consideration. In addition, Proposition \[Prop7\] claims that the random variable $\mathfrak{u}$ inside the peak $\mathbb{P}_\theta$ attains values much larger than unity. The latter holds, at least, within a certain neighborhood of the maximum $\theta$ attained by the variable $\mathfrak{u}$ inside the peak $\mathbb{P}_\theta$. In this case formula  can be rewritten as $$\label{sec4:1} x(t) =\frac{e^{\Delta_\alpha}}{2\alpha^2} \int\limits_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})}{\alpha}\right\}d\mathfrak{t}\,.$$ In obtaining this expression we implicitly have assumed without loss of generality that the variable $\mathfrak{u(t)}$ takes positive values inside the peak $\mathbb{P}_\theta$. The spatial displacement $x$ of the wandering particle and its velocity maximum $$\label{sec4:2} \vartheta = \frac{e^{\Delta_\alpha}}2 \exp\left\{\frac{\mathfrak{\theta}}{\alpha}\right\}$$ attained inside the peak $\mathbb{P}_\theta$ are partly independent variables. Indeed, for example, their ratio $$\label{sec4:2a} \frac{x}{\vartheta} = \frac1{\alpha^2} \int_{\mathfrak{t}\in \mathbb{P}_\theta}\exp\left\{\frac{\mathfrak{u}(\mathfrak{t})-\theta}{\alpha}\right\}d\mathfrak{t}$$ depends on the details of the pattern $\mathfrak{u}(\mathfrak{t})$ in the vicinity of its maximum $\theta$. Nevertheless, these details seem not to be too essential; they determine mainly some cofactors of order unity, see also Ref. [@we1]. To justify the latter statements, first, Figure \[F8\] depicts two trajectories $\mathfrak{u}(\mathfrak{t})$ implementing the peak $\mathbb{P}_\theta$. The difference in their forms explains the partial independence of the variable $x$ and $\vartheta$. Second, Figure \[F9\] demonstrates the statistical properties of such trajectories. The shown patterns were obtained in the following way. A collection of random trajectories similar to ones shown in Fig. \[F8\] were generated based on Eq.  with the discretization time step of 0.01. All the trajectories started from the point $\mathfrak{u}_u = 1$ and terminated when crossing the boundary $\mathfrak{u}_l = 0$ for the first time. Only the trajectories that passed through the layer $(\theta,\theta+1)$ with $\theta = 10$ without touching the upper boundary $\theta+1 = 11$ were taken into account. Then for each trajectory the time moment $\mathfrak{t}_\text{max}$ of attaining the corresponding maximum $\mathfrak{u}_\text{max}$ was fixed and the trajectory as a whole was shifted along the time axis that the point $\mathfrak{t}_\text{max}$ be located at the time origin $\mathfrak{t}=0$. In this way all the trajectories were rearranged that their maxima be located at the same point on the time axis. The total number of the trajectories constructed in this way was equal to $10^5$. Then the plane $\{\mathfrak{t,u}\}$ was partitioned into cells of $0.1\times 0.1$ size and the discretization points of individual trajectories fell into each cell were counted. Finally their numbers were renormalized to the obtained maximum. The left window in Fig. \[F9\] exhibits the obtained distribution of these values called the distribution pattern of $\mathfrak{u(t)}$. Actually this pattern visualizes the regular trend in the dynamics of the variable $\mathfrak{u(t)}$ near the extreme point $\theta$ and its scattering around it. As should be expected, the regular trend of $\mathfrak{u(t)}$ matches the optimal trajectory $$\label{new:2} \mathfrak{u}_\text{opt}(\mathfrak{t}) = \theta - |\mathfrak{t-t}_\text{max}|$$ of the system motion towards the maximum $\theta$ and away from it. The symmetry of this pattern is worthy of being noted because only the left branch of the optimal trajectory  matching the motion towards the maximum ($\mathfrak{t< t}_\text{max}$) is related to extreme fluctuations in the time dynamics of $\mathfrak{u(t)}$. The right one is no more than a “free” motion of particle under the regular drift. The right window depicts actually the same pattern in units of the particle elementary displacement, see Exp. , ![Example of random trajectories $\{\mathfrak{u}(\mathfrak{t})\}$ implementing the peak $\mathbb{P}_\theta$. Initially, $\mathfrak{t}=0$, both of them start from the point $\mathfrak{u}_u$ and terminate when crossing the lower boundary $\mathfrak{u}_l = 0$ for the first time. This figure also illustrates the technique of constructing the distribution function of the maximal value $\theta$ attained by *continuous* random walks $\{\mathfrak{u}(\mathfrak{t})\}$ via counting the trajectories passing through the layer $(\theta,\theta+d\theta)$ without touching its upper boundary $u = \theta+d\theta$. The division of the result by the thickness $d\theta$ of the analyzed layer yields the probability *density* $\mathfrak{f}(\theta)$.[]{data-label="F8"}](fig8.pdf){width="0.5\columnwidth"} ![Spatial patterns visualizing the regular trend in the dynamics of the random variable $\mathfrak{u(t)}$ in the vicinity of the attained maximal value $\theta$ and the scattering of $\mathfrak{u(t)}$ around the regular trend. The left window depicts this pattern on the plane $\{\mathfrak{t,u}\}$, the right window maps this pattern on the plane $\{\mathfrak{t},\delta x\}$. Here $\delta x$ is the elementary displacement of the wandering particle along the axis $x$ during the time interval $d\mathfrak{t}$ or, what is actually the same after the corresponding normalization, the particle velocity $v$ normalized to the velocity $\vartheta$ corresponding to $\mathfrak{u}=\theta$. In obtaining these data Eq.  and Exp.  with $\alpha = 1$ were used; the extreme value was set equal to $\theta = 10$. The details of constructing the given patterns are described in the text.[]{data-label="F9"}](fig9.jpg){width="0.9\columnwidth"} $$\delta x \propto \exp \left\{\frac{\mathfrak{u}(\mathfrak{t})}{\alpha}\right\} d\mathfrak{t}$$ during the time step $d\mathfrak{t}$. It plots this pattern normalized to the particle velocity $\vartheta$ attained at $\mathfrak{u} = \theta$. As seen in this figure, fluctuations in the variable $x$ should be comparable with its mean value or less than it. In constructing the given patterns the value of $\alpha=1$ was used. Therefore, in spite of the partial independence of the random variables $x$ and $\vartheta$ solely the statistical properties of the variable $\vartheta$ are responsible for the Lévy characteristics of the generated random walks. For the sake of simplicity in the present analysis we confine our consideration to the regular model of the time variations $\mathfrak{u(t)}$ near the extremum point, i.e., the ansatz $$\label{sec4:3} \mathfrak{u}(\mathfrak{t}) = \mathfrak{u}_\text{opt}(\mathfrak{t}) = \theta - |\mathfrak{t-t}_\text{max}|$$ will be used in calculating integral . An approach enabling us to go beyond this approximation will be published somewhere else. Substituting into we get $$\label{sec4:4} x =\frac{e^{\Delta_\alpha}}{\alpha} \exp\left\{\frac{\theta}{\alpha}\right\}\,.$$ Then using formula  and Exp.  for the extreme value probability density $\mathfrak{F}(\theta,\mathfrak{t})$ we obtain the expression $$\label{sec4:5} \mathcal{P}(x,t) = \frac{4\alpha\Gamma(1+\alpha)}{\alpha^\alpha \Gamma^2(\alpha/2)}\cdot \frac{t}{x^{1+\alpha}}$$ giving us the asymptotics of the distribution function $\mathcal{P}(x,t)$ of the particle spatial displacement $x$ during the time interval $t$ when $x\gg \overline{x}(t)$ and the expression $$\label{sec4:6} \overline{x}(t)= \left[\frac{4\Gamma(1+\alpha)}{\alpha^\alpha \Gamma^2(\alpha/2)}\cdot t\,\right]^{1/\alpha}$$ evaluating the characteristic spatial distance $\overline{x}(t)$ passed by the particle during the time interval $t$. Expression  has been derived via the relationship between the probability functions $$\mathcal{P}(x,t) = \mathfrak{F}(\theta,\mathfrak{t}) \left[\frac{dx}{d\theta}\right]^{-1}$$ and Exp.  was obtained setting $\theta = \theta_\mathfrak{t}$ in Exp. . The rigorous formula for the asymptotic behavior of the function $\mathcal{P}(x,t)$ was obtained in Ref. [@we2] using a singular perturbation technique for $1<\alpha<2$ and then verified numerically also for $0<\alpha \leq 1$ [@we3]. Following Ref. [@we3] we rewrite it as $$\label{sec4:7} \mathcal{P}^\text{rig}(x,t) = \frac{4\alpha}{2^\alpha \Gamma^2(\alpha/2)}\cdot \frac{t}{x^{1+\alpha}}\,.$$ Whence we see that the rigorous expression and the expression obtained using ansatz  coincide with each other within the factor $$\label{cofactor} \Omega(\alpha)= \left(\frac{2}{\alpha}\right)^\alpha \Gamma(1+\alpha)\in(1,2.05)$$ for $\alpha\in(0,2)$. The fact that the obtained coefficient $\Omega(\alpha)$ is really about unity for all the values of the parameter $\alpha$ under consideration justifies Propositions \[Prop5\]–\[Prop9\]. Conclusion and closing remarks ============================== The work has been devoted to the relationship between the continuous Markovian model for Lévy flights developed previously [@we1; @we2; @we3; @we33] and their equivalent representation in terms of discrete steps of a wandering particle. The present analysis has been confined to the one-dimensional model of continuous random motion of a particle with inertia. Its dynamics is studied in terms of random motion on the phase plane $\{x,v\}$ comprising the position $x$ and velocity $v=dx/dt$ of the given particle. Time variations in the particle velocity are considered to be governed by a stochastic differential equation whose regular term describes “viscous” friction with, maybe, a nonlinear friction coefficient $k(v)$. Its stochastic term containing white Gaussian noise, the random Langevin force, allows for the stochastic self-acceleration phenomenon which can be of different nature. The stochastic self-acceleration is taken into account via the noise intensity $g(v)$ growing with the particle velocity $v$. Spacial attention is payed to the ideal case where the friction coefficient $k$ is constant and the noise intensity $g(v)\propto v$ becomes proportional to the particle velocity $v$ when the latter exceeds some threshold $v_a$. It is the case where the generated random walks exhibit the main properties of Lévy flights. Namely, first, the distribution function $\mathcal{P}(x,t)$ of the particle displacement $x$ during the time interval $t$ possesses the power-law asymptotics. Second, the characteristic length $\overline{x}(t)$ of particle displacement during the time interval $t$ scales with $t$ also according to the power-law. The characteristic feature of the considered stochastic process is the fact that such nonlinear dependence of the noise intensity on the particle velocity gives rise to a multiscale time pattern $\{v(t)\}$ and the spatial particle displacement is mainly caused by the velocity extreme fluctuations, i.e., large peaks of the given pattern [@we1]. In particular, if we consider the velocity pattern $\{v(t)\}$ of duration $t$ then the particle displacement $x$ within the corresponding time interval can be evaluated as $x\sim \vartheta_t \tau$, where $\vartheta_t$ is the amplitude of the largest peak available in the given pattern and $\tau$ is a “microscopic” time scale characterizing the velocity correlations. As a result, the statistical properties of the velocity fluctuations cause the Lévy time scaling of the characteristic length $\overline{x}(t)$ of particle displacement and endow the distribution function $\mathcal{P}(x,t)$ with the appropriate power-law asymptotics. This feature has made it attractive to represent a trajectory of the wandering particle or, speaking more strictly, the time pattern $\{v(t)\}$ as a sequence of peaks of duration about $\tau$ and to consider each peak as a certain implementation of one discrete step of the particle motion. Unfortunately such an approach cannot be constructed directly because the time pattern $\{v(t)\}$ as a random trajectory is not a smooth curve. To overcome this obstacle a complex neighborhood $\mathcal{L}$ of the line $v= 0$ on the phase plane $\{x,v\}$ has been introduced. It contains two boundaries $|v| = v_u$ and $|v| = v_l$, where the choice of the parameters $v_u$, $v_l$ meeting the inequality $v_l < u_u$ is determined by the simplicity of mathematical constructions. The presence of the two boundaries has enabled us to introduce the notion of random walks outside and inside the neighborhood $\mathcal{L}$. The notion of random walks outside $\mathcal{L}$ describes a fragment of the particle motion in the region $|v|> v_l$ without touching the lower boundary $|v|=v_l$ until the particle gets it for the first time. Random walks inside the neighborhood $\mathcal{L}$ correspond to a fragment of the particle motion inside the region $|v|<v_u$ without touching the upper boundary $|v| = v_u$ again until the particle gets it for the first time. For the analyzed phenomena the initial particle velocity does not matter provided it is not too large. Therefore the initial particle velocity was set equal to $v_u$ without loss of generality. In this way any trajectory of the particle motion is represented as a sequence of alternate fragments of random walks inside and outside the neighborhood $\mathcal{L}$. A complex unit (basic unit) made of two succeeding fragments of the particle random motion inside and outside $\mathcal{L}$ may be treated as a continuous implementation of one step of the equivalent discrete random walks. The individual duration and the resulting length of these basic units are partly correlated random variables. It enables us to regard the constructed representation of random trajectories as a certain generalization of continuous time random walks (CTRW). The main difference between the CTRW model and the model developed here is the fact that the particle is not assumed to move uniformly along the straight line connecting the terminal points of one step. For the analyzed model  it has been demonstrated that the particle motion inside the neighborhood $\mathcal{L}$ practically contributes only to the duration of the basic units. The particle motion outside $\mathcal{L}$ determines the spatial displacement as well as contributes to the basic unit duration too. In the given model the kinetic coefficients, i.e., the friction coefficient $k(v)$ and the noise intensity $g(v)$ exhibit no singularities at $v=0$. Therefore the distribution of the basic unit duration has no anomalous properties and, thereby, there is a linear relationship $$k = \gamma t$$ between the running time $t$ and the number of basic units $k$, naturally, for $t\gg \tau$ and, so, for $k\gg 1$. The proportionality coefficient $\gamma$ has been obtained as a certain function of the model parameters. As a result, in describing statistical properties of such random walks the system may be characterized by the number $k$ of basic units imposing no requirements on the duration of the pattern $\{v(t)\}$ as a whole. It should be pointed out that the developed classification of random trajectories holds within rather general assumptions about Markovian stochastic processes. So it can be generalized to models where the kinetic coefficients exhibit essential singularities in the region of small velocities. However, in this case there is no direct proportionality between $t$ and $k$, moreover, the integer $k$ has to be treated as a certain random variable; a similar situation is met in modeling grain boundary diffusion as a stochastic processes of the subdiffusion type [@Keigan1; @Keigan2]. Using the constructed trajectory classification the analyzed stochastic process has been reduced to a certain universal stochastic process $\{\mathfrak{u(t)}\}$ for which the corresponding governing equation contains no parameters. Moreover, the white Gaussian noise enters this equation in the additive manner and the regular drift is a piece-wise constant function, namely, the $\operatorname{sign}(\mathfrak{u})$-function. It has been called the core stochastic process. All the basic parameters characterizing the generated random walks, e.g., the exponent of the Lévy scaling law, are specified by the model parameters via the corresponding coefficients in the transformation from the variable $\mathfrak{u}$ of the core stochastic process $\{\mathfrak{u(t)}\}$ to the original particle velocity $v$. Namely, it is the transformation $\mathfrak{u}\to v$ as well as the linear transformation of the core process time to the “physical” time, $\mathfrak{t}\to t$. This, in particular, explains us why the main results rigorously obtained for the superdiffusive regime of Lévy flights hold also for the other possible regimes as demonstrated numerically [@we3]. In order to elucidate the basic properties of the random walks at hand the developed technique has been applied to the core stochastic process to construct the peak pattern $\mathbb{P}_\mathfrak{u}(k):=\{\mathfrak{u(t)}\}$ subjected to the condition $|\mathfrak{u(t)}|<\theta$ for all the moments of time, where $\theta\gg1$ is a certain given value. Based on the probabilistic properties of the core stochastic process it has enabled us also to introduce the probability density $\mathfrak{F(\theta,t)}$ that the maximum attained by the variable $|\mathfrak{u(t)}|$ inside the pattern $\mathbb{P}_\mathfrak{u}(k)$ is equal to $\theta$. The probability density $\mathfrak{F(\theta,t)}$ has been studied in detail in the present paper. In particular, first, it has been found out that the region of the asymptotic behavior of $\mathfrak{F(\theta,t)}$ is specified by the inequality $\theta \gtrsim \theta_\mathfrak{t}$, where $\theta_\mathfrak{t} = \ln(\mathfrak{t})$. Second, the so-called single-peak approximation has been justified. It states that if the value $\theta \gtrsim \theta_\mathfrak{t}$ is attained within a given peak of the pattern $\mathbb{P}_\mathfrak{u}(k)$ then the variations of the variable $\mathfrak{u}$ inside the other peaks may be assumed to be small in comparison with $\theta$ and the condition $|\mathfrak{u}|\leq \theta$ does not really affect their statistics. Then, as been demonstrated, the asymptotic properties of the generated random walks can be formulated in terms of the core stochastic process as follows. - On time scales $t\gg\tau$ large fluctuations in the spatial displacement $x$ of the wandering particle are implemented mainly via its motion within the single peak of the pattern $\mathbb{P}_\mathfrak{u}(k)$ whose amplitude is maximal in comparison with the other peaks and attains extremely large values. - The region $|x|\gg \overline{x}(t)$ of large fluctuations in the particle spatial displacement $x$ matches directly the region $\theta\gtrsim\theta_\mathfrak{t}$ of the extreme values $\theta$ attained by the variable $\mathfrak{u(t)}$ inside the pattern $\mathbb{P}_\mathfrak{u}(k)$. - The asymptotic behavior of the distribution function $\mathcal{P}(x,t)$ in the region $|x|\gg \overline{x}(t)$ is practically specified by the asymptotic behavior of the probability density $\mathfrak{F}(\theta,\mathfrak{t})$ inside the region $\theta\gtrsim\theta_\mathfrak{t}$. - The lower boundary $\overline{x}(t)$ of the asymptotic behavior of $\mathcal{P}(x,t)$ determines the characteristic displacement of the wandering particle during the time interval $t$. So it can be evaluated using the single-peak approximation and setting the value $\theta$ equal $\theta_\mathfrak{t}$, i.e., $\theta\sim \theta_\mathfrak{t}$. It should be reminded that $\theta_\mathfrak{t}$ is also the lower boundary of the $\theta$-region wherein the single-peak approximation holds. In addition it is worthwhile to note that, first, in studying the extreme characteristics of particle motion outside the neighborhood $\mathcal{L}$ the shape of the velocity pattern $\{v(t)\}$ can be approximated using the most probable trajectory $\{\mathfrak{u_\text{opt}(t)}\}$. In particular using these results the asymptotics of the distribution function $\mathcal{P}(x,t)$ has be constructed and demonstrated to coincide with the rigorous results within a cofactor about unity. It opens a gate to modeling such processes in heterogeneous media constructing the most optimal trajectories of particle motion in nonuniform environment. Second, the first item above can be regarded as a certain implementation of the single-peak approximation. It is based on the use of the condition $|\mathfrak{u(t)}|<\theta$ in constructing the pattern $\mathbb{P}_\mathfrak{u}(k)$. However, the developed classification of random trajectories admits a more sophisticated analysis of anomalous stochastic processes. To do this it is necessary to consider a more complex system of restrictions $|\mathfrak{u(t)}|_i<\theta_i$ with the independent external boundaries $\{\theta_i\}$ for different peaks. In this case it would be possible to speak about several peaks of different large amplitude and to go beyond the single-peak approximation. Acknowledgments {#acknowledgments .unnumbered} =============== The work was supported in part by the JSPS “Grants-in-Aid for Scientific Research” Program, Grant 245404100001, as well as the Competitive Research Funding of the University of Aizu, Project P-25, FY2012. Probabilistic properties of random walks inside and outside the layer $\mathcal{L}$ {#App:inL} =================================================================================== It should be noted beforehand that in the present Appendix no approximation of the potential $\Phi(\eta)$ introduced by expression  will be used, only the general properties of the kinetic coefficients $k(v)$ and $g(v)$ are taken into account. It enables us to make use of the results to be obtained here in further generalizations, e.g., to allow for the cutoff effects. Terminal fragments: the limit case $s\to 0$ and $\theta\to\infty$ {#AppG} ----------------------------------------------------------------- The limit $\theta\to\infty$ describes the situation when the upper boundary $\eta=\theta$ of the analyzed region $[0,\theta)\ni\eta$ is placed rather far away from the origin $\eta=0$ and its effect on the random particle motion is ignorable. In this case, from the general point of view, the terminal fragments shown in Fig. \[F2\] are no more than random walks starting from at a given point $\eta_0$ and reaching another point $\eta$ in a time $t$ without touching a certain boundary $\eta=\zeta$. Their probabilistic properties are described, in particular, by the probability density $\mathcal{G}(\eta,t|\eta_0,\zeta)$ of finding the random walker at the point $\eta$ in the time $t$. The Laplace transform $G(\eta,s|\eta_0,\zeta)$ of this function obeys the following forward Fokker-Planck equation matching the Langevin equation  (see, e.g., Ref. [@Gardiner]) $$\label{AppG:fFP} s G = \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial \eta} + \alpha\frac{d\Phi(\eta)}{d\eta} G\right] + \delta(\eta-\eta_0)\,.$$ For random walks inside the layer $\mathcal{L}_\zeta=[0,\zeta)$ with the initial point $\eta_0 < \zeta$ Eq.  should be subjected to the boundary conditions \[AppG:0U\] $$\begin{aligned} \label{AppG:0Ua} \left[\frac{\partial G}{\partial \eta}+\alpha\frac{d\Phi}{d\eta} G\right]_{\eta = 0} & = 0\,, & \left. G \right|_{\eta = \zeta} & = 0\,. \\ \intertext{For random walks outside the layer $\mathcal{L}_\zeta$ with $\eta_0>\zeta$ the corresponding boundary conditions are} \label{AppG:0Ub} \left[\frac{\partial G}{\partial \eta}+\alpha\frac{d\Phi}{d\eta}G\right]_{\eta \to \infty} & \to 0\,, & \left. G \right|_{\eta = \zeta}& = 0\,.\end{aligned}$$ Since the effect of time in the analyzed phenomena is mainly caused by the properties of the peak pattern $\mathbb{P}(t|k)$ and the time scales $t\gg1$ are of the primary interest, dealing with the terminal fragments we may confine our consideration to the limit $s\to0$. In this case Eq.  is reduced to the following $$\label{AppG:fFP0} \frac{\partial}{\partial \eta}\left[\frac{\partial G}{\partial \eta} + \alpha\frac{d\Phi(\eta)}{d\eta} G\right] = - \delta(\eta-\eta_0)$$ and after simple mathematical manipulations using the method of variation of constants for solving difference equations we get the desired expression for random walks inside the layer $\mathcal{L}_\zeta$ \[AppG:GL\] $$\begin{aligned} \label{AppG:GLin} G^\text{in}(\eta|\eta_0,\zeta):= G(\eta,s|\eta_0,\zeta)\big|_{\substack{\eta_0<\zeta\\ s\to0}} & = e^{-\alpha\Phi(\eta)}\int\limits_{\eta_0}^\zeta e^{\alpha\Phi(\eta')} \Theta_\text{H}(\eta'-\eta)\,d\eta' \\ \intertext{and for random walks outside the layer $\mathcal{L}_\zeta$} \label{AppG:GLout} G^\text{out}(\eta|\eta_0,\zeta) := G(\eta,s|\eta_0,\zeta)\big|_{\substack{\eta_0>\zeta\\ s\to0}} & = e^{-\alpha\Phi(\eta)}\int\limits^{\eta_0}_\zeta e^{\alpha\Phi(\eta')} \Theta_\text{H}(\eta-\eta')\,d\eta'\,,\end{aligned}$$ where $\Theta_\text{H}(\ldots)$ is the Heaviside step function determined by Exp. . The first terminal fragment $\mathbb{G}^\text{in}_n(\eta)$ (Fig. \[F2\]) matches the analyzed random walks starting at the point $\eta_l$ and reaching the point $\eta$ in the time $\Delta t = t-t_n$ without touching the boundary $\eta=\eta_u > \eta_l$. So the probabilistic weight $\mathcal{G}^\text{in}(\eta,\Delta t)$ of this fragment is specified by the expression $$\mathcal{G}^\text{in}(\eta,\Delta t) = \mathcal{G}(\eta,\Delta t|\eta_0,\zeta)\big|_{\substack{\eta_0=\eta_l\\\zeta=\eta_u}}$$ and its Laplace transform ${G}^\text{in}(\eta,s)$ in the limit $s\to0$ takes the form \[AppG:GLFinal\] $$\label{AppG:GLinFinal} G^\text{in}(\eta,s)\big|_{s\to0} = e^{-\alpha\Phi(\eta)}\int\limits_{\eta_l}^{\eta_u} e^{\alpha\Phi(\eta')} \Theta(\eta'-\eta)\,d\eta'$$ by virtue of . The second terminal fragment $\mathbb{G}^\text{out}_n(\eta)$ (Fig. \[F2\]) is also represented by the given random walks starting at the point $\eta_u$ and reaching the point $\eta$ in the time $\Delta t = t-t_n$ without touching the boundary $\eta_l$. Thereby the Laplace transform ${G}^\text{out}(\eta,s)$ of its probabilistic weight $\mathcal{G}^\text{out}(\eta,\Delta t)$ is specified by Exp. , namely, $$\label{AppG:GLoutFinal} G^\text{out}(\eta,s)\big|_{s\to0} = e^{-\alpha\Phi(\eta)}\int\limits_{\eta_l}^{\eta_u} e^{\alpha\Phi(\eta')} \Theta(\eta-\eta')\,d\eta'\,.$$ It should be noted that in both of Exps.  the quantity $\eta$ can take any arbitrary value $\eta\in (0,\infty)$ rather than a value from the corresponding interval only. Indeed, if the taken value falls outside this interval the relevant expression will give out the probability density equal to zero. The latter feature enables us to represent the construction reproducing actually the right-hand side of Exp.  as \[AppG:GGFinal\] $$\begin{gathered} \label{AppG:GGFinal1} \Theta_\text{H}(\eta -\eta_l) G^\text{out}(\eta,s)\big|_{s\to0} + \Theta_\text{H}(\eta_u -\eta) G^\text{in}(\eta,s)\big|_{s\to0} = \\ G^\text{out}(\eta,s)\big|_{s\to0} + G^\text{in}(\eta,s)\big|_{s\to0} = e^{-\alpha\Phi(\eta)}\int\limits_{\eta_l}^{\eta_u} e^{\alpha\Phi(\eta')} d\eta'\,.\end{gathered}$$ In deriving this expression the identity $\Theta(\eta-\eta')+ \Theta(\eta'-\eta) \equiv 1$ has been taken into account. In the cause under consideration it is assumed that the potential $\Phi(\eta)$ can be approximated by a linear function of $\eta$ in the region $\eta\sim 1$, namely, $\Phi(\eta)\approx C + \eta$, where $C$ is some constant. Under such conditions the previous expression is reduced to $$\label{AppG:GGFinal2} \Theta_\text{H}(\eta -\eta_l) G^\text{out}(\eta,s)\big|_{s\to0} + \Theta_\text{H}(\eta_u -\eta) G^\text{in}(\eta,s)\big|_{s\to0} = \frac1{\alpha} e^{-\alpha\Phi(\eta)}\left[e^{\alpha\Phi(\eta_u)} - e^{\alpha\Phi(\eta_l)} \right]\,.$$ In particular, these expressions demonstrate that in the limit $s\to 0$, i.e., when $t\to\infty$ the cumulative contribution of the two terminal fragments to the probability density of finding the random walker at the point $\eta$ is equal to the steady state distribution of the random variable $\eta$ within a certain factor of proportionality. Fragments of the peak pattern $\mathbb{P}(t\mid k)$ {#AppF} --------------------------------------------------- Generally the individual fragments of the peak pattern represented in Fig. \[F3\] can be regarded as random walks of a particle that initially ($t=0$) is located at a point $\eta$ and gets a boundary $\zeta$ for the first time at a moment $t$. If the initial point $\eta$ is located outside the interval $\mathcal{L}_\zeta := [0,\zeta)$, i.e., $\eta >\zeta$, then the addition condition is imposed on the particle motion; it is not allowed for it to touch or cross the distant boundary $\theta\gg1$. The probabilistic properties of these fragments are described by the probability density $\mathcal{F}(\zeta,t|\eta,\theta)$ that the particle starting from the point $\eta$ at $t=0$ gets the boundary $\zeta$ at the moment $t$ for the first time and, in addition when applicable, during its motion never crosses the distant boundary $\theta$. The Laplace transform ${F}(\zeta,s|\eta,\theta)$ of this function determined by Exp.  obeys the following backward Fokker-Planck equation matching the Langevin equation  (see, e.g., Ref. [@Gardiner]) $$\label{AppF:bFP} s F = \frac{\partial^2 F}{\partial \eta^2} - \alpha\frac{d\Phi(\eta)}{d\eta} \frac{\partial F}{\partial \eta}\,.$$ For the random walks inside the layer $\mathcal{L}_\zeta$, i.e. when $0\leq\eta<\zeta$ Eq.  should be subjected to the boundary conditions \[AppB:BC\] $$\begin{aligned} \label{AppF:BCin} F\big|_{\eta = \zeta} & = 1\,, & \left.\frac{\partial F}{\partial \eta}\right|_{\eta = 0} & = 0\,,\\ \intertext{and for the random walks outside the layer $\mathcal{L}_\zeta$, i.e. when $\zeta<\eta<\theta$ the relevant boundary conditions are} \label{AppF:BCOut} F\big|_{\eta = \zeta} & = 1\,, & F\big|_{\eta = \theta}& = 0\,.\end{aligned}$$ Since the details of solving Eq.  for the random walks inside and outside the layer $\mathcal{L}_\zeta$ are different we will analyze the two cases individually assuming the inequality $s\ll1$ to hold beforehand, which matches large time scales $t\gg1$. ### Random walks inside the layer $\mathcal{L}_\zeta$ {#AppFIN} Because values of $\eta_l$ and $\eta_u$ about unity are of the prime interest, see the corresponding discussion in Sec. \[sec:TC\], here we may consider the thickness of the layer $\mathcal{L}_\zeta$ to be also about unity, $\zeta\sim 1$. In this case at the zero-th approximation in $s$ the solution of Eq.  subject to the boundary conditions  is equal to unity. So at the first approximation in $s$ Eq.  can be rewritten as $$\label{AppF:bFPin} s = \frac{\partial^2 F}{\partial \eta^2} - \alpha\frac{d\Phi(\eta)}{d\eta} \frac{\partial F}{\partial \eta}\,.$$ Using the method of variation of constants Eq.  under conditions  is integrated directly, yielding the desired expression $$\label{AppF:FGin} F^\text{in}(\zeta,s|\eta) = 1- s \int\limits_\eta^{\zeta}d\eta'\, e^{\alpha\Phi(\eta')} \int\limits_0^{\eta'}d\eta'' e^{-\alpha\Phi(\eta'')}\,.$$ Here the parameter $\theta$ has been omitted from the list of arguments and the superscript ‘in’ has been added to denote the analyzed region explicitly. Expression  immediately enables us to write the Laplace transform of the individual probabilistic weights $\mathcal{F}^\text{in}(\Delta t)$ of the fragments $\mathbb{F}^\text{in}_{i,i+1}:= \mathbb{F}^\text{in}(\Delta t)$ with $\Delta t = t_{i+1}-t_i$ (Fig. \[F2\]) in the form $$\label{AppF:Fin} F^\text{in}(s) = 1- s \int\limits_{\eta_l}^{\eta_u}d\eta'\, e^{\alpha\Phi(\eta')} \int\limits_0^{\eta'}d\eta'' e^{-\alpha\Phi(\eta'')}$$ within the approximation of large time scales $t\gg1$ or, equivalently, for $s\ll1$. ### Random walks outside the layer $\mathcal{L}_\zeta$ Since the analyzed phenomena are governed by large fluctuations in the particle velocity, the upper boundary of the region under consideration $\eta\in(\zeta,\theta)$ is presumed to be a rather distant point, $\theta\gg1$, in addition to the assumption $s\ll1$. Then let us seek the desired solution of Eq.  subject to the boundary conditions  in the form $$\label{AppF:FF} F(\eta) = A_s F_s(\eta)+ A_0 F_0(\eta)\,,$$ where $A_s$, $A_0$ are some constants and the functions $F_s(\eta)$, $F_0(\eta)$ are specified via the expression $$\label{AppF:Fk} F_{s,0}(\eta) = \exp\left\{ \int\limits_{\zeta}^\eta k_{s,0}(\eta')\,d\eta' \right\}\,.$$ The boundary conditions  imposed on solution  enable us to calculate the coefficients $A_s$, $A_0$, and, then, to rewrite solution  in the form $$\label{AppF:FFfin} F(\eta) = \frac{F_0(\theta)F_s(\eta) - F_s(\theta)F_0(\eta)}{F_0(\theta) - F_s(\theta)} = F_s(\eta)- \frac{F_s(\theta)}{F_0(\theta)-F_s(\theta)}\cdot\left[F_0(\eta)-F_s(\eta)\right]\,.$$ In deriving Exps.  the identity $F_0(\zeta)=F_s(\zeta)=1$ stemming directly from definition  has been taken into account. The substitution of into shows the function $k(\eta):=k_{s,0}(\eta)$ to obey the Riccati equation $$\label{AppF:Ric} s = k^2 +\frac{dk}{d\eta} -\alpha\frac{d\Phi(\eta)}{d\eta} k\,.$$ In the region $\eta\gtrsim 1$ (when also $\zeta\gtrsim1$) the potential $\Phi(\eta)$ introduced via Exp.  is approximately a linear function of its argument, namely, $\Phi(\eta)\approx\eta + C$, where $C$ is a certain constant, by virtue of the accepted assumption . In this case $d\Phi(\eta)/d\eta \approx 1$ and the Riccati equation  has two solutions $$\begin{aligned} \label{AppF:ks:0} k_s(\eta)& \approx -\frac{s}{\alpha}\,, \\ \label{AppF:k0:0} k_0(\eta)& \approx \alpha\,,\end{aligned}$$ written in the approximation of leading terms in $s\ll1$. Since we can choose any two *specific independent* solutions of the Riccati equation  let us impose on the desired solutions $k=k_{s,0}(\eta)$ the requirement that in the region $\eta\gtrsim1$ both of them tend to the constant values , , respectively. In accordance with the results to be obtained the solution $k_s(\eta)$ can be treated as a small quantity of order $s$ due to the monotonous growth of the potential $\Phi(\eta)$ with its argument $\eta$. It enables us to omit the quadratic term from the Riccati equation  and, then, using the method of variation of constants write the desired solution as $$\label{AppF:ks:1} k_s(\eta) = -s e^{\alpha\Phi(\eta)}\int\limits_\eta^\infty e^{-\alpha\Phi(\eta')}\,d\eta'\,.$$ Whence it follows, in particular, that the function $F_s(\eta)$ is decreasing one and its asymptotic behavior as $s\to 0$ is specified by the expression $$\label{AppF:FonS} F_s(\eta) = 1 - s\int\limits_\zeta^\eta d\eta' e^{\alpha\Phi(\eta')}\int\limits_{\eta'}^\infty d\eta'' e^{-\alpha\Phi(\eta'')}\,.$$ For the analyzed phenomena we need to know only the characteristics of small deviations of the function $F_s(\eta)$ from unity for $s\ll 1$, which actually has enabled us to construct the function $k_s(\eta)$ within the accuracy of the leading term in $s$. In constructing the second solution $k_0(\eta)$ we may set the left-hand side of equation  equal to zero and, then, rewrite it as follows $$\label{AppF:k0:1} k = \alpha\frac{d\Phi(\eta)}{d\eta} -\frac{d\ln(k)}{d\eta} \,.$$ In the case under consideration the deviation of the function $\Phi(\eta)$ from the linear dependence can occur only on scales $\eta\gg1$. So we may confine ourselves to the first iteration in , yielding us the desired approximation $$\label{AppF:k0:2} k_0(\eta) = \alpha\frac{d\Phi(\eta)}{d\eta} -\frac{d}{d\eta}\ln\left[\frac{d\Phi(\eta)}{d\eta}\right]\,.$$ Expression  and definition  immediately enable us to specify the function $F_0(\eta)$, namely, $$\label{AppF:F0} F_0(\eta) = \left[\frac{d\Phi(\eta)}{d\eta}\right]^{-1} e^{\alpha\left[\Phi(\eta)-\Phi(\zeta)\right]}\,.$$ In obtaining this formula we have set $d\Phi(\eta)/d\eta=1$ at $\eta=\zeta\gtrsim 1$. In particular, this formula demonstrates us that the function $F_0(\eta)$ exhibits the exponential growth as the variable $\theta$ runs to large values. So the contribution of the finite magnitude of the parameter $s\ll 1$ to the function $F_0(\eta)$ becomes essential only when the argument $\eta$ attains such large values that the function $F_0(s)$ even in the leading approximation in $s$, i.e., for $s=0$ has already got exponentially large values. This region is of minor interest in the frameworks of the present analysis, which has enabled us to construct the solution $k_0(\eta)$ setting $s=0$. For the same reasons we may confine ourselves to approximation  in evaluating the first term on the right-hand side of Exp.  and in the second term set $F_s(\eta) = F_s(\theta) = 1$. It is due to the fact that for the most interesting magnitudes of the parameter $\theta$ the effects caused by the finite magnitude of the parameter $s$ are ignorable. In this way using formulas  and we gets the approximation $$\label{AppF:Fst} F(\eta) = 1 - s \int\limits_\zeta^\eta d\eta' e^{\alpha\Phi(\eta')}\int\limits_{\eta'}^\infty d\eta''\, e^{-\alpha\Phi(\eta'')} - \left.\frac{d\Phi(\eta)}{d\eta}\right|_{\eta=\theta}\left[\frac{e^{\alpha\Phi(\eta)}-e^{\alpha\Phi(\zeta)}}{e^{\alpha\Phi(\theta)}}\right]\,.$$ Setting here $\eta=\eta_u$ and $\zeta = \eta_l$ we immediately get the desired expression for the Laplace transform of the individual probabilistic weights $\mathcal{F}^\text{out}(\Delta t)$ of the fragments $\mathbb{F}^\text{out}_{i,i+1}:= \mathbb{F}^\text{out}(\Delta t)$ with $\Delta t = t_{i+1}- t_i$ (Fig. \[F2\]) in the form $$\label{AppF:Fout} F^\text{out}(s) = 1 - s \int\limits_{\eta_l}^{\eta_u} d\eta' e^{\alpha\Phi(\eta')}\int\limits_{\eta'}^\infty d\eta''\, e^{-\alpha\Phi(\eta'')} - \left.\frac{d\Phi(\eta)}{d\eta}\right|_{\eta=\theta}\left[\frac{e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}}{e^{\alpha\Phi(\theta)}}\right]\,.$$ It should be reminded that Exp.  as well as assumes the estimate $d\Phi(\eta)/d\eta=1$ to hold for $\eta,\eta_l,\eta_u,\zeta\gtrsim 1$. ### Composed element of the peak pattern $\mathbb{P}(t\mid k)$ The construction of the peak pattern $\mathbb{P}(t\mid k)$, see Fig. \[F3\] and Exp. , is based on the repetition of the same unit, a composed element made of random walks outside the region $\mathcal{L}$ followed by random walks inside it. The probabilistic weight $\mathcal{F}(t)$ of this unit is determined by its Laplace transform equal to the product $$\label{AppF:Unit1} F(s) = F^\text{in}(s)\cdot F^\text{out}(s)\,.$$ Whence taking into account formulas  and we immediately get the expression \[AppF:Unit2\] $$\begin{gathered} \label{AppF:Unit2a} F(s) = 1 - s \int\limits_{\eta_l}^{\eta_u} d\eta' e^{\alpha\Phi(\eta')}\int\limits_{0}^\infty d\eta''\, e^{-\alpha\Phi(\eta'')} - \left.\frac{d\Phi(\eta)}{d\eta}\right|_{\eta=\theta}\left[\frac{e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}}{e^{\alpha\Phi(\theta)}}\right] \\ {}=1 - \frac1{\alpha}\left[{e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}}\right] \left[{s} \int\limits_{0}^\infty d\eta''\, e^{-\alpha\Phi(\eta'')} + {\alpha}\left.\frac{d\Phi(\eta)}{d\eta}\right|_{\eta=\theta}{e^{-\alpha\Phi(\theta)}}\right]\,.\end{gathered}$$ Using formula  for the stationary distribution function $p^\text{st}(\eta)$ of the random variable $\eta$ and actually merging the half-spaces $\{\eta>0\}$ and $\{\eta <0\}$ the obtained expression is reduced to the following $$\label{AppF:Unit2b} F(s) =1 - \frac1{\alpha}\left[{e^{\alpha\Phi(\eta_u)}-e^{\alpha\Phi(\eta_l)}}\right] \left[{s} - \left.\frac{dp^\text{st}(\eta)}{d\eta}\right|_{\eta=\theta}\right]\int\limits_{0}^\infty d\eta'\, e^{-\alpha\Phi(\eta')}\,.$$ It gives us the desired probabilistic weight in the frameworks of the analyzed case, i.e., for large time scales, $t\gg 1$, or equivalently $s\ll 1$, the distant upper boundary of the region of random walks, $\theta \gg 1$, and the accepted assumption about the behavior of the potential $\Phi(\eta)$ in the region $\eta\sim 1$, namely, $\Phi(\eta) \approx C + \eta$, where $C$ is a constant. 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--- abstract: 'A virtual graphical construction is made to show the difference between neutrino and anti-neutrino oscillations in the presence of CP violation with CPT conservation.' author: - | R. G. Moorhouse$^b$\ $^b$University of Glasgow,Glasgow G12 8QQ, U.K.\ title: Note on a Pattern from CP Violation in Neutrino Oscillations --- Introduction ============ There is interest in the possibility that CPT violation may occur and then show in neutrino oscillation experiments[@BL],[@CDK], [@MINOS].. However this may be, CP violation is long established and it is of importance to seek it in neutrino oscillation results. If one takes a conservative point of view that nature conserves CPT, and there are 3 generations of neutrinos, then the consequences of CP violation in neutrino oscillation become more definite. In particular if CP were conserved $\nu$ transition probabilities would be the same as $\bar{\nu}$ transition probabilities while the occurence therein of CP violation makes these different [@Kayser]. This differencc is dependent on the neutrino oscillation parameter $L/E$, $L$ being the distance of travel from creation to detection and $E$ the energy of the initial neutrino: and it is also sensitive to the value of the small ratio of neutrino mass squared differences. There results a complicated 2-variable dependence in addition to the linear dependence on the leptonic Jarlskog parameter, $J_{lep}$[@HSW]. Pattern for the difference between $\nu$ and $\bar\nu$ oscillations =================================================================== The input to the formula for neutrino transition probabilities is largely from the mixing matrix elements $U_{\alpha i}$ where $\alpha$ is one of the 3 flavour indices and $i$ one of the 3 mass eigenstate indices. From these 9 elements plaquettes [@BD] can be constructed, these being phase invariant products of 2 $U$ elements multiplied by products of 2 $U^{\star}$ elements which occur in transition probabilities $\nu_{\alpha} \to \nu_{\beta}$ of neutrino beams. Here $\alpha,\beta$ are flavour indices of beam neutrinos. The construction is as follows. Greek letters denoting flavour indices $(e,\mu,\tau)$ and Roman letters mass eigenstate indices there are 9 plaquettes,labelled $\Pi_{\alpha i}$: $$\Pi_{\alpha i} \equiv U_{\beta j}U^{\star}_{\beta k}U_{\gamma k}U^{\star}_{\gamma j} \label{119}$$ where $\alpha,\beta,\gamma$ are non-equal and in cyclic order and $i,j,k$ are also non-equal and in cyclic order (The pattern discussed in this paper applies for an inverted hierarchy as well as for the normal hierarchy; that is there is no necessary association between a particular $\alpha$ and a particular $i$.) . Making use of well-known formalism [@Kayser] the beam transition probability for $\nu_{\alpha} \to \nu_{\beta}$,$\alpha \neq \beta$ can be written as $$\begin{aligned} P(\nu_{\alpha} \to \nu_{\beta})= -4\sum_{i=1}^3 \Re(\Pi_{{\gamma}i})\sin^2((m_{{\nu}k}^2-m_{{\nu}j}^2) L/4E)\\ +2\sum_{i=1}^3 \Im(\Pi_{{\gamma}i})\sin((m_{{\nu}k}^2-m_{{\nu}j}^2) L/2E)\label{120}\end{aligned}$$ where L is the length travelled by neutrino energy E from creation to annihilation at detection. The survival probability, $P(\nu_{\alpha} \to \nu_{\alpha})$, (given in [@Kayser]) can be calculated from the transition probabilities above. So the $3 \times 3$ plaquette matrix $\Pi$ and the neutrino mass eigenstate values squared differences carry all the information on transition and survival probabilities of a given beam. The last term (\[120\]) being only non-zero when CP is not conserved. Indeed all the nine $\Im \Pi_{{\alpha} i}$ are equal and equal to [@HDS] the leptonic Jarlskog invariant; $$\Im \Pi_{{\alpha} i}=J_{lep}$$. So J, as usual, is signalling CP violation. With CPT invariance the transition probability for anti-neutrinos $P(\bar{\nu}_{\alpha} \to \bar{\nu}_{\beta};\Pi )= P(\nu_{\alpha} \to \nu_{\beta};\Pi^\star)$ [@Kayser]. Thus the contribution of CP violation in anti-neutrino transitions is of the same magnitude but opposite sign to that in neutrino transitions, giving rise to a, in principle measurable, difference in the overall probability since the CP conserving contributions are the same.. The part of the probability (\[120\]) arising from CP violation is 2$J\xi$ where $$\xi= \sum_{i=1}^3 \sin((m_{{\nu}k}^2-m_{{\nu}j}^2)L/2E) \label{121}$$ This sum of sine functions (the sum of whose arguments is zero) may readily be transformed to $$\xi= 4\sin(x_d)\sin(y_d)\sin(x_d+y_d) \label{122}$$ $$\begin{aligned} (x_d,y_d)=(d_1L/4E,d_2L/4E) \\ d_1=(m_{{\nu}2}^2-m_{{\nu}1}^2), d_2=(m_{{\nu}3}^2-m_{{\nu}2}^2)\label{123}\end{aligned}$$ . Now consider the function $$\Xi(x,y) \equiv 4\sin(x)\sin(y)\sin(x+y) \label{126}$$ where in $\Xi (x,y)$ the arguments $x,y$ are freely varying and not restricted as in $\xi$. This function $\Xi$ has multiple maxima,minima with values $+3\sqrt3/2,-3\sqrt3 /2$ at arguments say $x_m$ and $y_m$ which are integer multiples of $\pi/3$ (but obviously not spanning all such integer multiples). Given mass squared differences then $\xi(L/E)$ is a function varying only with $L/E$ and the above maxima and minima cannot generally be attained. However one can distinguish regions of $L/E$ where relatively high values of $\xi$ are attained. These are, naturally, given by values of $x_d,y_d$ near to $x_m,y_m$ points of $\Xi$. These latter points can be located in the $(x,y)$ plane through the necessary condition that there the first derivatives of $\Xi$ with respect to both $x$ and $y$ should vanish. A simple geometrical picture can be given as follows. On the $x$ and $y$ positive quartile of the plane construct a square grid with neighbouring grid lines a distance $\pi/3$ apart resulting in a pattern of squares of side $\pi/3$. All the maximum and minimum points of $\Xi$ are at intersection points of the grid lines and are given by $$\begin{aligned} (x_m,y_m)=(1+3l,1+3k)\pi/3 \label{127}\\ (x_m,y_m)=(2+3l,2+3k)\pi/3 \label{128}\end{aligned}$$ where $l$ and $k$ are any non-negative integers. The points $(\ref{127})$ have $\Xi=3\sqrt3/2$ and the points $(\ref{128})$ have $\Xi=-3\sqrt3/2$. It is near these special points in the $x,y$ plane that $\xi(L/E)$ (eqn. \[121\]) has numerically large values. Note that for seeking observation of CP violation using the difference between $\nu$ and $\bar{\nu}$ transitions it does not matter whether $\xi$ is positive or negative so both maximum and minimum points of $\Xi$ are equally potentially important. As $L/E$ varies the points $(x_d,y_d)$ (\[123\]) trace a straight line in the $(x,y)$ plane starting at $(0,0)$ and ascending as $(L/E)$ increases. This line of $\xi$ makes a small angle arctan$(d_1/d_2)$ with the y-axis and pases through the archipelago of special points given by (\[127\],\[128\]). Points $(x_d,y_d)$ on the line of $\xi$ which are close to the $\Xi$ special points (\[127\],\[128\]) give numerically large values of $\xi$ and the associated values of $L/E$ signify neutrinos whose transtions contain a relatively large CP violating part. To give an idea of how much of the plane has a value of $\Xi$ near maximum or minimum then the value of $\Xi$ near the special points should be evaluated. Let $\delta$ be the distance between a near point and the special point which it is near to. Then near a maximum or minimum $\Xi=\pm 3\sqrt3/2(1- \Delta)$ respectively. where $\Delta \leq 2\delta^2$ . . Thus within an area limited by $\delta = .2$ (noting that a grid square has sides length $\pi/3$) the value of $\Xi$ is nearly equal to that at the special grid point. So the structure of $\Xi$ is such that for certain intervals (not large) of $L/E$ the contribution of $J\xi$ to CP violation in neutrino transitions (measured by the difference between $\nu$ and $\bar{\nu}$ transitions) is much bigger than an average over larger intervals. Such an interval of $L/E$ is when the line of $L/E$ passes near a peak or trough of $\Xi$. The example that follows is only illustrative, though by happenstance rather striking. There are, obviously, uncertaities in the prescription due to considerable relative uncertainties in the value (though certainly small) of $d_1/d_2$ and also in the different plaquette values found in different theories. Take $$d_1=8.0\times 10^{-5} eV^2, d_2=2.5\times 10^{-3} eV^2,$$ so that $d_1/d_2=.032, d_2/d_1=31.25$. Consider the grid line $y=62\pi/3$ (given by $k=20$ eqn.\[128\]). On this grid line $\Xi$ has a minimum value $-3\sqrt3/2$ at $x=2\pi/3$ and the line of $\xi$ crosses the grid line at $x=(2\pi/3- .016)$ which means that $\xi$ has almost attained the minimum value, $-3\sqrt3/2$,and that $L/4E = 62\pi/3d_2 = 0.248 \times 10^5\pi/3$. The values of the plaquettes defined from the MNS mixing matrix elements, $U_{\alpha i}$,depend on the phases of these elements. A theory can give the moduli and phases of these elements and the consequent plaquette have the virtue of being obviously invariant under phase redefinitions of the eigenstates $\nu_{\alpha}$ and of the eigenstates $\nu_i$. In the absence of experimental data on the phases of the $U_{\alpha i}$, but the existence of some considerable experimental guidance on the moduli it seems not unreasonable to use, as a specimen for present purposes, one of the theories which gives a matrix of moduli squared resembling that of the tri-bimaximal mixing hypothesis [@HPS]. The particular theory used [@M] incorporates the values of the mass squared differences given and used above and has the matrix of MNS modulus squared elements:. $$\left[\begin{array}{ccc} .638 & .344 & .017\\ .260 & .331 & .409\\ .102 & .325 & .573 \end{array}\right], \label{132}$$ bearing a distinct resemblance to the postulated ’ideal’ structure of this matrix in tri-bimaximal mixing [@HPS]: (The theoretical model [@M] produces the MNS matrix using the normal hierarchy.) As previously noted $\Im \Pi_{{\alpha} i}=J_{lep}$ (for all 9 elements of $\Pi$) and the contribution of CP violation to the transition probabilities is given by $$P_{CP}(\nu_{\alpha} \to \nu_{\beta})=2\xi(L/E)J_{lep} \label{129}$$ for all $\alpha$ not equal to $\beta$. For this particular model $J_{lep}=.01744$ and at the nearly minimum point discussed above $$\begin{aligned} P(\bar{\nu}_\mu \to \bar{\nu}_e)=P(\nu_e \to \nu_\mu) = .3470\label{130}\\ P(\nu_\mu \to \nu_e)=P(\bar{\nu}_e \to \bar{\nu}_\mu) = .5276\label{131}\end{aligned}$$ the difference of 0.1806. It should be emphasized that this large difference depends not only on the nearness to a special point, which is a concept independent of the any particular model of the MNS matrix but also on the particular model having a maybe atypically large value of $J_{lep}$. Naturally this type of experimental comparison may yield some knowledge of $J_{lep}$. The numerical value of $L/E$ given above in units $ev^{-2}$ may, on inserting the appropriate dimensionful value of $\hbar c$, be related to the experimental conditions through $$L/4Eev^{-2}=1.266L(km)/E(Gev)=0.248 \times 10^5\pi/3$$. Values of $L/E$ of this order may be appropriate for atmospheric muon neutrinos created on the opposite side of the earth to the detector. However the large value of $L/E$ highlights the probable accelerator experiment difficulties of getting near to some of the special grid points of $\Xi$; this sharply depends on the precise value of the gradient of the line of $\xi$. It is clear that there is at present no reliable prediction of detailed exprimental results; most importantly because the precise slope of the line of $\xi$ in the $(x,y)$ plane is uncertain, being the ratio of neutrino mass differences. Rather, any value of the construction is that experiment may give information on its physically significant parameters. Information may of course be obtained by computer evaluation of the transition probabilities (\[120\]) for very many multiple parameter choices, diligent attention enabling the construction of cognitive or computer maps. The author thanks David Sutherland and Colin Froggatt for comments on this note.. [8]{} G.Barenboim and J.D.Lykken, arXiv: 0908.2993\[hep-ph\] G.Barenboim, L Borissov and J.D.Lykken, arXiv: 0212.116\[hep-ph\] D.Choudhury,A. Datta and A.Kundu arXiv:1007.2923\[hep-ph\]. P. Adamson et al.arXiv:1007.2791\[hep-ph\]. B.Kayser ’Neutrino mass, mixing and flavor’ in 2008 Review of Particle Physics: C. Amaler et al.,Physics Letters B667, 1(2008). P. F. Harrison, W. G. Scott and T. J. Weiler, arXiv: 0908.2993\[hep-ph\] J. D. Bjorken and I. Dunietz,Phys. Rev. D36, 2109 (1987). P F Harrison, S Dallison and W G Scott, arXiv:0904.3071\[hep-ph\] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys.Lett. B530, 167 (2002). R. G. Moorhouse, Phys. Rev. D77,053008 (2008)

--- abstract: 'We performed terahertz magneto-optical spectroscopy of FeSe thin film to elucidate the charge carrier dynamics. The measured diagonal (longitudinal) and off-diagonal (Hall) conductivity spectra are well reproduced by two-carrier Drude model, from which the carrier densities, scattering times and effective masses of electron and hole carriers are determined in a wide range of temperature. The hole density decreases below the structural transition temperature while electron density increases, which is attributed to the band structure modification in the electronic nematic phase. The scattering time of the hole carrier becomes substantially longer than that of the electron at lower temperature, which accounts for the increase of the positive dc Hall coefficient at low temperature.' author: - Naotaka Yoshikawa - Masayuki Takayama - Naoki Shikama - Tomoya Ishikawa - Fuyuki Nabeshima - Atsutaka Maeda - Ryo Shimano bibliography: - 'refs\_FeSeFaraday.bib' title: | Charge carrier dynamics of FeSe thin film investigated by\ terahertz magneto-optical spectroscopy --- =1 Since the discovery of iron-based superconductors (FeSCs), tremendous research efforts have been devoted to reveal the pairing mechanism of superconductivity. The elucidation of interplay between the nematic order, antiferromagnetic spin order, and superconductivity emergent in FeSCs has been believed to provide a clue to understand the emergent superconductivity. Among FeSCs, FeSe provides a unique playground to study the role of nematicity, because it lacks the long-range magnetic order in the nematic phase that appears below the tetragonal-orthorhombic structural transition temperature $T_{\mathrm{s}}\simeq \SI{90}{\kelvin}$, as evidenced by a significant electronic anisotropy from transport and nuclear magnetic resonance (NMR) spectral properties[@McQueen:2009hs; @Baek:2014gs; @Bohmer:2015fk]. While the superconducting transition temperature $T_{\mathrm{c}}$ of bulk FeSe is $\sim\SI{9}{K}$ at ambient pressure[@Hsu:2008ep], it shows a remarkable tunability. $T_{\mathrm{c}}$ increases to as high as under hydrostatic pressure[@Medvedev:2009ex; @Imai:2009hw; @Mizuguchi:2008bn; @Sun:2016dh], and single-layer FeSe grown on SrTiO$_3$ shows $T_{\mathrm{c}}$ up to [@Ge:2014hc; @He:2013cn; @Tan:2013jb]. Electron doping by ionic-gating in FeSe thin flakes enhances the superconductivity toward [@Lei:2016gl; @Shiogai:2015fw; @Hanzawa:2017fa; @Kouno:2018fp]. Intercalation also enhances [$T_{\mathrm{c}}$]{} by a similar doping effect in addition to an effect of separating the layers[@BurrardLucas:2012fm]. One important key to understand the [$T_{\mathrm{c}}$]{} increase of FeSe is considered to be a change of the Fermi surface topology. The high tunability of the electronic structure of FeSe achieved by various ways is related to its extremely small effective Fermi energy, which has been demonstrated in FeSe[@Kasahara:2014gt] as well as FeSe$_{1-x}$Te$_x$[@Lubashevsky:2012br; @Okazaki:2014im]. FeSe is a semimetal with the Fermi surface consisting of hole pockets around the Brillouin zone center $\mathit{\Gamma}$ point and electron pockets around the zone corner $M$ point. The low-energy electronic structure around the Fermi level has been experimentally revealed by angle-resolved photoemission spectroscopy (ARPES)[@Shimojima:2014kc; @Zhang:2015fx; @Watson:2015kn; @Nakayama:2014eo; @Fanfarillo:2016kz]. ARPES studies have also shown a significant modification of the band structure below $T_{\mathrm{s}}$ which is attributed to the development of an electronic nematicity. For the understanding of unconventional superconductivity in FeSe, it is also indispensable to investigate the charge carrier dynamics in normal and nematic phases as well as in superconducting phase. The Hall resistivity measured by magneto-transport shows an unusual temperature dependence with the sign change owing to the nearly compensated electron and hole carriers[@Kasahara:2014gt; @Watson:2015hx; @Sun:2016by; @Nabeshima:2018fi]. In bulk FeSe, the presence of a small number of highly mobile electron-like carrier at the nematic phase was also identified by the Hall resistivity[@Watson:2015hx] and mobility spectrum analysis[@Huynh:2014ch], which could be attributed to the Dirac-like dispersion near the $M$ point[@Tan:2016cd]. However, the complexity of the multi-band Fermi surfaces of FeSe makes it difficult to grasp the properties of charge carriers only by dc transport measurements. This is because the characterization of the carriers by dc transport measurements needs to assume some models such as a compensated two-band model, where the compensated electron is assumed to have same carrier density as that of the hole ($n_e=n_h$)[@Watson:2015hx; @Nabeshima:2018fi; @Huynh:2014ch; @Ovchenkov:2017fp]. Although three band model can also be used by including the nonlinear term when dc transport is measured up to high magnetic field, the characterization is not complete because the mobility, which is determined by dc transport in addition to carrier densities, is a function of effective mass and scattering time. For more detailed characterization of charge carriers, quantum oscillations are a well-established technique[@Watson:2015kn; @Watson:2015hx; @Terashima:2014ft; @Audouard:2015hp]. However, quantum oscillations are able to be observed typically in bulk crystals grown by the vapor transport techniques. Thus, the observation of quantum oscillations of FeSe has been limited in bulk single crystals and at very low temperature. Therefore, the properties of charge carriers of FeSe in a wide range of temperature in particular across the structural phase transition temperature have remained to be clarified. In this study, we investigate the charge dynamics in a thin-film FeSe by terahertz (THz) magneto-optical spectroscopy. The obtained diagonal (longitudinal) and off-diagonal (Hall) conductivity spectra are well described by two-carrier Drude model, from which the carrier densities, scattering times and effective masses of electron and hole carriers were independently determined. The temperature dependence of THz magneto-optical spectra revealed the significant change of the carrier densities below $T_{\mathrm{s}}$, which is plausibly attributed to the band structure modification in the nematic phase. The scattering time of the hole carrier substantially increases at lower temperature, which explains the peculiar temperature dependence of the dc Hall coefficient in FeSe thin films. ![(a) Temperature dependence of dc resistivity $\rho$ (red line) and $d\rho/dT$ (blue line) of the FeSe thin film. A kink anomaly at $T_{\mathrm{s}}$ is indicated by black arrow. Inset shows an enlarged view of the resistivity curve around [$T_{\mathrm{c}}$]{} $\sim\SI{3}{K}$. (b) Schematic of our THz magneto-spectroscopy. (c) Faraday rotation spectrum and (d) ellipticity spectrum induced by FeSe film at with the magnetic field of .](Fig1.pdf){width="\columnwidth"} A FeSe thin film with the thickness of 46 nm was fabricated on LaAlO$_3$ (LAO) substrate by pulsed-laser deposition method[@Imai:2010ez; @Imai:2010jl]. The temperature dependence of dc resistivity shows the superconducting transition at [$T_{\mathrm{c}}$]{} $\sim\SI{3}{K}$ defined by the zero resistivity (Fig. 1(a)). A kink anomaly in $d\rho/dT$ curve indicates the structural transition at $T_{\mathrm{s}}\sim \SI{80}{K}$. Figure 1(b) shows the schematic of our THz magneto-spectroscopy based on THz time-domain spectroscopy (THz-TDS)[@Ikebe:2008cq; @Ikebe:2009it; @Shimano:2013ez]. The output of a mode-locked Ti:sapphire laser with the pulse duration of 110 fs, center wavelength of 800 nm, and repetition rate of 76 MHz was focused onto a p-type $(111)$ InAs surface to generate THz pulses. Linearly polarized THz incident pulses were focused on the sample placed in a split-type superconducting magnet which can produce the magnetic field up to in Faraday configuration, that is, the magnetic field is parallel to the wavevector of the THz wave. The THz-wave was detected by electro-optical sampling with a $(110)$ ZnTe crystal. By measuring the waveform of the parallel polarization component defined as $E_x(t)$ and perpendicular polarization component $E_y (t)$ of the transmitted THz pulses by using wire-grid polarizers, the Faraday rotation angle $\theta$ and ellipticity $\eta$ induced by the FeSe film in the magnetic field can be obtained. Here, the approximated expression $E_y (\omega)/E_x (\omega)\sim \theta(\omega)+i\eta(\omega)$ for small Faraday rotation angle was used. Figures 1(c) and 1(d) show the THz Faraday rotation angle and ellipticity, respectively, under $B = \SI{7}{T}$ at the sample temperature $T = \SI{7}{K}$. The error bars indicate the standard deviations determined by the multiple measurements, confirming that the obtained signal $\theta(\omega)$ and $\eta(\omega)$ well exceed the noise level. Figures 2(a) and 2(b) show the real- and imaginary-part of the longitudinal optical conductivity spectrum $\sigma_{xx} (\omega)$ of FeSe at , respectively, given by the normal transmission type THz-TDS without magnetic field. The combination of the obtained $\sigma_{xx} (\omega)$, $\theta(\omega)$ and $\eta(\omega)$ provides the optical Hall conductivity spectrum $\sigma_{xy} (\omega)$ through the following equation for a thin film under small rotation angle approximations: $$\begin{aligned} \theta+i\eta\sim\frac{\sigma_{xy}(\omega)d}{\left( 1+n_{\mathrm{sub}}(\omega)-\frac{i\omega dn_{\mathrm{sub}}(\omega)}{c}\right)c\varepsilon_0 + \sigma_{xx}(\omega)d}\end{aligned}$$ where $n_{\mathrm{sub}}$ is the refractive index of the substrate, $d$ the thickness of FeSe film, $c$ speed of light, and $\varepsilon_0$ permittivity of vacuum. Figures 2(c) and 2(d) show the real- and imaginary-part of the optical Hall conductivity spectrum with $B = \SI{7}{T}$, respectively. The data set of the complex longitudinal and Hall conductivity spectra was well fitted by the Drude model with an electron carrier and a hole carrier: $$\begin{aligned} \sigma_{xx}(\omega,B=0)&=\frac{n_e q_e^2 \tau_e}{m^*_e}\frac{1}{1-i\omega \tau_e}+\frac{n_h q_h^2 \tau_h}{m^*_h}\frac{1}{1-i\omega \tau_h}\\ \sigma_{xy}(\omega,B)&=\frac{n_e q_e^2 \tau_e}{m^*_e}\frac{\omega_{c,e}\tau_e}{(1-i\omega \tau_e)^2-\omega_{c,e}^2}\notag\\ &\quad+\frac{n_h q_h^2 \tau_h}{m^*_h}\frac{\omega_{c,h}\tau_h}{(1-i\omega \tau_h)^2-\omega_{c,h}^2}\end{aligned}$$ {width="\columnwidth"} where $q_e=-e$ ($q_h=+e$), $n_e$ ($n_h$), $\tau_e$ ($\tau_h$), $m_e^*$ ($m_h^*$), $\omega_{c,e}=q_e B/m_e^*$ ($\omega_{c,h}=q_h B/m_h^*$) represent the charge of carrier, carrier density, scattering time, effective mass, cyclotron frequency of the electrons (holes), respectively. Notably, the optical Hall conductivity is sensitive to the carrier type (electron-like or hole-like) while the longitudinal conductivity is formally independent of the sign of the charge carriers. The sign change around in the imaginary-part of $\sigma_{xy} (\omega)$ indicates that two types of carriers exist with different sign of charge, which can be attributed to those in the electron pocket and hole pocket in FeSe. We can determine the complete set of the parameters describing charge carrier dynamics, $n_e=\SI{5.4E19}{cm^{-3}}$, $n_h=\SI{1.7E20}{cm^{-3}}$, $m_e=0.86m_0$, $m_h=2.90m_0$, $\tau_e=\SI{0.11}{ps}$, $\tau_h=\SI{0.45}{ps}$. We also checked the magnetic field dependence of the complex optical Hall conductivity. The good agreement between the experiments and the two-carrier Drude model is also confirmed in the magnetic field dependence of the diagonal and off-diagonal conductivity spectra. The effective mass of the hole evaluated by quantum oscillation and ARPES studies are typically around $4m_0$, which is consistent with our result[@Watson:2015kn; @Terashima:2014ft; @Audouard:2015hp; @Phan:2017ew]. The effective mass of the electron, on the other hand, differs among various reports. The present THz magneto-spectroscopy shows the similar effective mass with that deduced by the band curvature measured by ARPES[@Phan:2017ew; @Watson:2016dy]. The carrier densities $n_e$ and $n_h$ are also reasonable compared with the various transport measurements[@Watson:2015hx; @Nabeshima:2018fi; @Huynh:2014ch]. The ARPES measurements showed that the Fermi surface around the zone center consists of the outer and inner hole pockets above the structural transition temperature, while the inner hole pocket moves completely below the Fermi level in the nematic phase[@Watson:2015kn]. Thus, the carrier density and effective mass of the hole at evaluated by the THz magneto-spectroscopy are most likely attributed to that of the outer hole pocket. On the other hand, the Fermi surface around $M$ point consists of at least two electron pockets in the nematic phase. Here the electrons in the outer electron pocket are considered to contribute dominantly to the transport and charge dynamics, so the obtained parameters of the electron are probably those of the outer electron pocket. {width="\columnwidth"} Next, we investigated the temperature dependence of conductivity spectra. Figures 3(a) and 3(b) show the complex longitudinal optical conductivity spectra from to . The conductivity spectrum shows a broadening at higher temperature indicating the shorter scattering time of the carriers. The optical Hall conductivity spectra in Figs. 3(c) and 3(d) shows more notable change where it goes to almost zero at . We performed the two-carrier Drude fitting as done in Fig. 2 for various temperature and extracted the parameters describing the observed complex conductivity spectra. Here we fixed effective masses of the electron and hole to those obtained for . From the fitting, we elucidate that zero Hall conductivity in the THz frequency at is a consequence of almost completely compensated electron and hole carriers at this temperature. Figure 4 summarizes the temperature dependence of the extracted charge carrier parameters. The scattering time of the hole is shortened at higher temperature while that of the electron weakly depends on the temperature as shown in Fig. 4(a). By considering that the impurity scattering dominates at $T=0$ limit, it is found that the impurity scattering rate of the electron is larger than that of the hole. This could be derived from the about three times lighter effective mass of the electron than the hole. Since the ARPES measurements revealed that the Fermi wave numbers $k_F$ of the electron and the hole pockets are similar to each other[@Shimojima:2014kc; @Zhang:2015fx; @Watson:2015kn; @Phan:2017ew; @Watson:2016dy], the Fermi velocity $v_{\mathrm{F}}=k_{\mathrm{F}}/m^*$ of the electron is larger than that of the hole. The carrier with the larger Fermi velocity is expected to have the larger scattering rate with the impurities with certain density. Concomitantly, the hole density substantially reduces at lower temperature below $T_{\mathrm{s}}$ while that of electrons increases. These behaviors seem to correlate with the band structure change between tetragonal and orthorhombic phase accompanied by electronic nematicity. As for the hole pockets around $\mathit{\gamma}$ point, the ARPES revealed that the inner hole band moves below the Fermi energy in the nematic phase[@Watson:2015kn], leading to the smaller hole carrier density. The $d_{xy}$ band which forms the outer electron-pocket at the $M$ point is also pushed down with decreasing temperature, resulting in the increase of electron density at lower temperature below $T_{\mathrm{s}}$. A decrease of carrier densities below the structural transition temperature has been observed in bulk FeSe by the dc magneto-transport measurement[@Watson:2015hx]. The strong anisotropy of the scattering rate induced by the enhancement of spin fluctuations below $T_{\mathrm{s}}$ was suggested to explain the drop of the effective electron and hole densities without a change of Fermi surface volume. However, our observation of the simultaneous increase of the hole density and decrease of the electron density cannot be explained only by the strongly anisotropic carrier scattering. The present THz magneto-optical spectroscopy unambiguously reveals the temperature dependence of the electron and hole densities, which suggests the Fermi surface modification in the nematic phase. The reduction of carrier density below $T_{\mathrm{s}}$ has also been observed by conventional far-infrared optical reflectivity measurement[@Nakajima:2017cw]. By using the scattering time and effective mass, we further evaluate the mobility $\mu=e\tau/m^*$ for each carrier as shown in Fig. 4(c). The mobility of the hole increases at lower temperature as a result of the increase of the scattering time. The temperature dependence of the dc Hall coefficient described as $R_{\mathrm{H}}\approx \sigma_{xy}(\omega=0)/(\sigma_{xx}(\omega=0)^2 B)$ is plotted (red circles) in Fig. 4(d). It shows an excellent agreement with the Hall coefficient obtained by dc magneto-transport measurement (grey solid curve), indicating that the dc transport and the THz response are described by the common physical origin of the charge carrier dynamics. Accordingly, the peculiar temperature dependence of the Hall coefficient that increases monotonically toward the low temperature is dominantly attributed to the increase of hole scattering time. {width="\columnwidth"} In summary, we performed THz magneto-optical spectroscopy to investigate the charge carrier dynamics in FeSe. The obtained diagonal and off-diagonal conductivity spectra are well described by two-carrier Drude model, from which the carrier densities, scattering times and effective masses of electron and hole carriers are determined in a wide temperature range. We found the significant temperature dependence of the carrier densities of electrons and holes below $T_{\mathrm{s}}$ , which is most likely attributed to the band structure modification at the structural phase transition. The scattering time of the hole carrier becomes substantially longer than that of the electron at lower temperature, which results in the positive dc Hall coefficient at low temperature observed in the present FeSe thin film. We demonstrated that THz magneto-optical spectroscopy is a powerful tool for investigating the charge carrier dynamics of FeSe. The application of the technique to FeSe system under pressure, with ionic gating, and thin film or single-layer on various substrates would pave a unique pathway to access the charge carrier properties, thereby providing a deep insight into their correlation with the nematicity and the superconductivity. This work was supported by JSPS KAKENHI (Grants No. 18H05324, No. 18H05846, and 18H04212).

--- abstract: 'Recent results from the NA48/2 and NA62 kaon decay-in-flight experiments at CERN are presented. A precision measurement of the helicity-suppressed ratio $R_K$ of the $K^\pm\to e^\pm\nu$ and $K^\pm\to\mu^\pm\nu$ decay rates has been performed using the full dedicated data set collected by the NA62 experiment ($R_K$ phase); the result is in agreement with the Standard Model expectation. New measurements of the $K^\pm\to\pi^\pm\gamma\gamma$ decay at the NA48/2 and NA62 experiments provide further tests of the Chiral Perturbation Theory. A planned measurement of the branching ratio of the ultra-rare $K^+\to\pi^+\nu\bar\nu$ decay at 10% precision is expected to represent a powerful test of the Standard Model.' author: - 'Evgueni Goudzovski   for the NA48/2 and NA62 collaborations' title: 'Kaon experiments at CERN: recent results and prospects' --- INTRODUCTION {#introduction .unnumbered} ============ In 2003–04, the NA48/2 experiment has collected at the CERN SPS the world largest sample of charged kaon decays, with the main goal of searching for direct CP violation in the $K^\pm\to3\pi$ decays [@ba07]. In 2007–08, the NA62 experiment ($R_K$ phase) collected a large minimum bias data sample with the same detector but modified data taking conditions, with the main goal of measuring the ratio of the rates of the $K^\pm\to\ell^\pm\nu$ decays ($\ell=e,\mu$). The large statistics accumulated by both experiments has allowed the studies of a range of rare $K^\pm$ decay modes. The main stage of the NA62 experiment, expected to start physics data taking in 2014, aims at measuring the $K^+\to\pi^+\nu\bar\nu$ decay rate. The recent results and prospects of these experiments are discussed here. BEAM AND DETECTOR IN 2003–08 ============================ The beam line has been designed to deliver simultaneous narrow momentum band $K^+$ and $K^-$ beams derived from the primary 400 GeV/$c$ protons extracted from the CERN SPS. Central beam momenta of 60 GeV/$c$ and 74 GeV/$c$ have been used. The beam kaons decayed in a fiducial decay volume contained in a 114 m long cylindrical vacuum tank. A detailed description of the detector used in 2003–08 is available in [@fa07]. The momenta of charged decay products are measured in a magnetic spectrometer, housed in a tank filled with helium placed after the decay volume. The spectrometer comprises four drift chambers (DCHs), two upstream and two downstream of a dipole magnet which gives a horizontal transverse momentum kick of $120~\mathrm{MeV}/c$ or $265~\mathrm{MeV}/c$ to singly-charged particles. Each DCH is composed of eight planes of sense wires. A plastic scintillator hodoscope (HOD) producing fast trigger signals and providing precise time measurements of charged particles is placed after the spectrometer. A 127 cm thick liquid krypton (LKr) electromagnetic calorimeter located further downstream is used for lepton identification and as a photon veto detector. Its 13248 readout cells have a transverse size of approximately 2$\times$2 cm$^2$ each, without longitudinal segmentation. LEPTON UNIVERSALITY TEST WITH 2007–08 DATA ========================================== Decays of pseudoscalar mesons to light leptons ($P^\pm\to\ell^\pm\nu$, denoted $P_{\ell 2}$ below) are suppressed in the Standard Model (SM) by helicity considerations. Ratios of leptonic decay rates of the same meson can be computed very precisely: in particular, the SM prediction for the ratio $R_K=\Gamma(K_{e2})/\Gamma(K_{\mu 2})$ is [@ci07] $$\label{Rdef} R_K^\mathrm{SM} = \left(\frac{m_e}{m_\mu}\right)^2 \left(\frac{m_K^2-m_e^2}{m_K^2-m_\mu^2}\right)^2 (1 + \delta R_{\mathrm{QED}})=(2.477 \pm 0.001)\times 10^{-5},$$ where $\delta R_{\mathrm{QED}}=(-3.79\pm0.04)\%$ is an electromagnetic correction. Within extensions of the SM involving two Higgs doublets, $R_K$ is sensitive to lepton flavour violating effects induced by loop processes with the charged Higgs boson ($H^\pm$) exchange [@ma06]. A recent study [@gi12] has concluded that $R_K$ can be enhanced by ${\cal O}(1\%)$ within the Minimal Supersymmetric Standard Model. However, the potential new physics effects are constrained by other observables such as $B_s\to\mu^+\mu^-$ and $B_u\to\tau\nu$ decay rates [@fo12]. On the other hand, $R_K$ is sensitive to the neutrino mixing parameters within SM extensions involving a fourth generation of quarks and leptons [@la10]. The analysis strategy is based on counting the numbers of reconstructed $K_{e2}$ and $K_{\mu 2}$ candidates collected concurrently. Therefore the analysis does not rely on the absolute beam flux measurement, and several systematic effects cancel at first order. The study is performed independently for 40 data samples (10 bins of reconstructed lepton momentum and 4 samples with different data taking conditions) by computing the ratio $R_K$ as $$R_K = \frac{1}{D}\cdot \frac{N(K_{e2})-N_{\rm B}(K_{e2})}{N(K_{\mu2}) - N_{\rm B}(K_{\mu2})}\cdot \frac{A(K_{\mu2})}{A(K_{e2})} \cdot \frac{f_\mu\times\epsilon(K_{\mu2})} {f_e\times\epsilon(K_{e2})}\cdot\frac{1}{f_\mathrm{LKr}}, \label{eq:rkcomp}$$ where $N(K_{\ell 2})$ are the numbers of selected $K_{\ell 2}$ candidates $(\ell=e,\mu)$, $N_{\rm B}(K_{\ell 2})$ are the numbers of background events, $A(K_{\mu 2})/A(K_{e2})$ is the geometric acceptance correction, $f_\ell$ are the efficiencies of $e$/$\mu$ identification, $\epsilon(K_{\ell 2})$ are the trigger efficiencies, $f_\mathrm{LKr}$ is the global efficiency of the LKr calorimeter readout (which provides the information used for electron identification), and $D$ is the downscaling factor of the $K_{\mu2}$ trigger. The data sample is characterized by high values of $f_\ell$ and $\epsilon(K_{\ell 2})$ well above 99%. A Monte Carlo (MC) simulation is used to evaluate the acceptance correction and the geometric part of the acceptances for most background processes entering the computation of $N_B(K_{\ell 2})$. Particle identification, trigger and readout efficiencies and the beam halo background are measured directly from control data samples. Two selection criteria are used to distinguish $K_{e2}$ and $K_{\mu2}$ decays. Kinematic identification is based on the reconstructed squared missing mass assuming the track to be a electron or a muon: $M_{\mathrm{miss}}^2(\ell) = (P_K - P_\ell)^2$, where $P_K$ and $P_\ell$ ($\ell = e,\mu$) are the kaon and lepton 4-momenta (Fig. \[fig:mm2\]). A selection condition $M_1^2<M_{\mathrm{miss}}^2(\ell)<M_2^2$ is applied; $M_{1,2}^2$ vary across the lepton momentum bins depending on resolution. Lepton type identification is based on the ratio $E/p$ of energy deposit in the LKr calorimeter to track momentum measured by the spectrometer. Particles with $(E/p)_{\rm min}<E/p<1.1$ ($E/p<0.85$) are identified as electrons (muons), where $(E/p)_{\rm min}$ is 0.90 or 0.95, depending on momentum. The numbers of selected $K_{e2}$ and $K_{\mu 2}$ candidates are 145,958 and $4.2817\times 10^7$ (the latter pre-scaled at trigger level). The background contamination in the $K_{e2}$ sample has been estimated by MC simulations and, where possible, direct measurements to be $(10.95\pm0.27)\%$. The largest background contribution is the $K_{\mu2}$ decay with a mis-identified muon via the ‘catastrophic’ bremsstrahlung process in the LKr. To reduce the uncertainty due to background subtraction, the muon mis-identification probability $P_{\mu e}$ has been measured as a function of momentum using dedicated data samples. The contributions to the systematic uncertainty of the result include the uncertainties on the backgrounds, helium purity in the spectrometer tank (which influences the detection efficiency via bremsstrahlung and scattering), beam simulation, spectrometer alignment, particle identification and trigger efficiency. The result of the measurement, combined over the 40 independent samples taking into account correlations between the systematic errors, is $$R_K = (2.488\pm 0.007_{\mathrm{stat.}}\pm 0.007_{\mathrm{syst.}})\times 10^{-5} = (2.488\pm0.010)\times 10^{-5}.$$ The stability of $R_K$ measurements in lepton momentum bins and for the separate data samples is shown in Fig. \[fig:rkfit\]. The result is consistent with the Standard Model expectation, and the achieved precision dominates the world average. $K^\pm\to\pi^\pm\gamma\gamma$ DECAY MEASUREMENTS WITH 2004 AND 2007 DATA ======================================================================== Measurements of radiative non-leptonic kaon decays provide crucial tests for the ability of the Chiral Perturbation Theory (ChPT) to explain weak low energy processes. In the ChPT framework, the $K^\pm\to\pi^\pm\gamma\gamma$ decay receives two non-interfering contributions at lowest non-trivial order ${\cal O}(p^4)$: the pion and kaon [*loop amplitude*]{} depending on an unknown ${\cal O}(1)$ constant $\hat{c}$ representing the total contribution of the counterterms, and the [*pole amplitude*]{} [@ec88]. Higher order unitarity corrections from $K\to3\pi$ decays, including the main ${\cal O}(p^6)$ contribution as well as those beyond ${\cal O}(p^6)$ due to using the phenomenological values of $K\to3\pi$ amplitudes, have been found to modify the decay spectrum significantly; in particular, they lead to non-zero differential decay rate at zero diphoton invariant mass [@da96]. The total decay rate is predicted to be ${\rm BR}(K^\pm\to\pi^\pm\gamma\gamma) \sim 10^{-6}$, with the pole amplitude contributing 5% or less [@da96; @ge05]. The ChPT predictions for the decay spectra for several values of $\hat{c}$ are presented in Fig. \[fig:br-vs-c\]: the diphoton mass spectra exhibit a characteristic cusp at twice the pion mass due to the dominant pion loop amplitude. Experimentally, the only published $K^\pm\to\pi^\pm\gamma\gamma$ observation is that of 31 $K^+$ decay candidates in the kinematic region $100~{\rm MeV}/c<p_\pi^*<180~{\rm MeV}/c$ ($p_\pi^*$ is the $\pi^+$ momentum in the $K^+$ frame) by the BNL E787 experiment [@ki97]. New measurements of this decay have been performed using minimum bias data sets collected during a 3-day special NA48/2 run in 2004 with 60 GeV/$c$ $K^\pm$ beams, and a 3-month NA62 run in 2007 with 74 GeV/$c$ $K^\pm$ beams. The latter set has been collected with a set of downscaled trigger conditions with an effective downscaling factor of about 20. The effective kaon fluxes collected in 2004 and 2007 are similar, but the background conditions and resolution on kinematic variables differ significantly. Signal events are selected in the region $z=(m_{\gamma\gamma}/m_K)^2>0.2$ to reject the $K^\pm\to\pi^\pm\pi^0$ background peaking at $z=0.075$. The $\pi^\pm\gamma\gamma$ mass spectra, with the MC simulation expectations of the signal and background contributions, are displayed in Fig. \[fig:pigg-m\]: 147 (175) decays candidates are observed in the 2004 (2007) data set, with backgrounds contaminations of 12% (7%) from $K^\pm\to\pi^\pm\pi^0(\pi^0)(\gamma)$ decays with merged photon clusters in the LKr calorimeter. \[fig:pigg-z\] The data spectra of the $z$ kinematic variable, together with the signal and background expectations, are displayed in Fig. \[fig:pigg-z\]: they clearly exhibit the cusp at two-pion threshold as predicted by the ChPT. The values of the $\hat{c}$ parameter in the framework of the ChPT ${\cal O}(p^4)$ and ${\cal O}(p^6)$ parameterizations according to [@da96] have been measured by the performing likelihood fits to the data. The preliminary results of the fits are presented in Table \[tab:chat\]: they are in agreement with the earlier BNL E787 ones. The uncertainties are dominated by the statistical ones; the systematic errors and mainly due to uncertainties of the background estimates. The ${\cal O}(p^6)$ parametrization involves a number of external inputs. In this analysis, they have been fixed as follows: the polynomial contribution terms are $\eta_1=2.06$, $\eta_2=0.24$ and $\eta_3=-0.26$ as suggested in [@da96], while the $K^\pm\to3\pi^\pm$ amplitude parameters come from a fit to the experimental data [@bi03]. Along with the separate 2004 and 2007 results, the combined results are presented in Table \[tab:chat\]. The combination takes into account the large positive correlation of the systematic uncertainties of the two measurements. 2004 data 2007 data Combined -------------------------------- -------------------------------------------- -------------------------------------------- -------------------------------------------- $\hat{c}$, ${\cal O}(p^4)$ fit $1.36\pm0.33_{\rm stat}\pm0.07_{\rm syst}$ $1.71\pm0.29_{\rm stat}\pm0.06_{\rm syst}$ $1.56\pm0.22_{\rm stat}\pm0.07_{\rm syst}$ $\hat{c}$, ${\cal O}(p^6)$ fit $1.67\pm0.39_{\rm stat}\pm0.09_{\rm syst}$ $2.21\pm0.31_{\rm stat}\pm0.08_{\rm syst}$ $2.00\pm0.24_{\rm stat}\pm0.09_{\rm syst}$ BR, ${\cal O}(p^6)$ fit $(0.94\pm0.08)\times10^{-6}$ $(1.06\pm0.07)\times10^{-6}$ $(1.01\pm0.06)\times 10^{-6}$ : The preliminary results of the fits to the $K^\pm\to\pi^\pm\gamma\gamma$ diphoton mass spectra to the ChPT parameterizations [@da96]. The quoted BR values correspond to the full kinematic range.[]{data-label="tab:chat"} THE ULTRA-RARE DECAY $K^+\to\pi^+\nu\bar\nu$ ============================================ Among the flavour changing neutral current $K$ and $B$ decays, the $K\to\pi\nu\bar\nu$ decays play a key role in the search for new physics through the underlying mechanisms of flavour mixing. These decays are strongly suppressed in the SM (the highest CKM suppression), and are dominated by top-quark loop contributions. The SM branching ratios have been computed to an exceptionally high precision with respect to other loop-induced meson decays: ${\rm BR}(K^+\to\pi^+\nu\bar\nu)=8.22(75)\times 10^{-11}$ and ${\rm BR}(K_L\to\pi^0\nu\bar\nu)=2.57(37)\times 10^{-11}$; the uncertainties are dominated by parametric ones, and the irreducible theoretical uncertainties are at a $\sim 1\%$ level [@br11]. The extreme theoretical cleanness of these decays remains also in certain new physics scenarios. Experimentally, the $K^+\to\pi^+\nu\bar\nu$ decay has been observed by the BNL E787/E949 experiments, and the measured branching ratio is $\left(1.73^{+1.15}_{-1.05}\right)\times 10^{-10}$ [@ar09]. The achieved precision is inferior to that of the SM expectation. The main goal of the NA62 experiment at CERN is the measurement of the $K^+\to\pi^+\nu\bar\nu$ decay rate at the 10% precision level, which would constitute a significant test of the SM. The experiment is expected to collect about 100 signal events in two years of data taking, keeping the systematic uncertainties and backgrounds low. Assuming a 10% signal acceptance and the SM decay rate, the kaon flux should correspond to at least $10^{13}$ $K^+$ decays in the fiducial volume. In order to achieve a small systematic uncertainty, a rejection factor for generic kaon decays of the order of $10^{12}$ is required, and the background suppression factors need to be measured directly from the data. In order to achieve the required kaon intensity, signal acceptance and background suppression, most of the NA48/NA62 apparatus used until 2008 is being replaced with new detectors. The CERN SPS extraction line used by the NA48 experiment is capable of delivering beam intensity sufficient for the NA62. Consequently the new setup is housed at the CERN North Area High Intensity Facility where the NA48 was located. The decay in flight technique will be used; optimisation of the signal acceptance drives the choice of a 75 GeV/$c$ charged kaon beam with 1% momentum bite. The experimental setup is conceptually similar to the one used for NA48: a $\sim 100$ m long beam line to form the appropriate secondary beam, a $\sim 80$ m long evacuated decay volume, and a series of downstream detectors measuring the secondary particles from the $K^+$ decays in the fiducial decay volume. The signal signature is one track in the final state matched to one $K^+$ track in the beam. The integrated rate upstream is about 800 MHz (only 6% of the beam particles are kaons, the others being mostly $\pi^+$ and protons). The rate seen by the detector downstream is about 10 MHz, mainly due to $K^+$ decays. Timing and spatial information are required to match the upstream and downstream track. Backgrounds come from kaon decays with a single reconstructed track in the final state, including accidentally matched upstream and downstream tracks. The background suppression profits from the high kaon beam momentum. A variety of techniques will be employed in combination in order to reach the required level of background rejection. They can be schematically divided into kinematic rejection, precise timing, highly efficient photon and muon veto systems, and precise particle identification systems to distinguish $\pi^+$, $K^+$ and positrons. The above requirements drove the design and the construction of the subdetector systems. The main NA62 subdetectors are: a differential Cherenkov counter (CEDAR) on the beam line to identify the $K^+$ in the beam; a silicon pixel beam tracker; guard-ring counters surrounding the beam tracker to veto catastrophic interactions of particles; a downstream spectrometer composed of 4 straw chambers operating in vacuum; a RICH detector to distinguish pions and muons; a scintillator hodoscope; a muon veto detector. The photon veto detectors will include a series of annular lead glass calorimeters surrounding the decay and detector volume, the NA48 LKr calorimeter, and two small angle calorimeters to provide hermetic coverage for photons emitted at close to zero angle to the beam. The design of the experimental apparatus and the R&D of the new subdetectors have been completed. The experiment is under construction, and the first technical run is scheduled for October–December 2012. CONCLUSIONS {#conclusions .unnumbered} =========== The NA48/2 and NA62 experiments at CERN have recently accomplished several precision measurements of rare $K^\pm$ decays. The $R_K$ phase of the NA62 experiment provided the most precise measurement of the ratio of leptonic $K^\pm$ decay rates $R_K=(2.488\pm0.010)\times 10^{-5}$. This result is consistent with the SM expectation, and constrains multi-Higgs and fourth generation new physics scenarios. NA48/2 and NA62 experiments have performed new measurements of the $K^\pm\to\pi^\pm\gamma\gamma$ decay, significantly improving the precision of ChPT tests with this channel. The ultra-rare $K^+\to\pi^+\nu\bar\nu$ decay represents a unique environment to search for new physics. The NA62 experiment, aiming to collect ${\cal O}(100)$ events of this decay, is being constructed and is preparing for a technical run in 2012. [99]{} R. Batley [*et al.*]{}, Eur. Phys. J [**C52**]{} (2007) 875. V. Fanti [*et al.*]{}, Nucl. Instrum. 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--- abstract: 'We extend to a generalized pseudoeffect algebra (GPEA) the notion of the exocenter of a generalized effect algebra (GEA) and show that elements of the exocenter are in one-to-one correspondence with direct decompositions of the GPEA; thus the exocenter is a generalization of the center of a pseudoeffect algebra (PEA). The exocenter forms a boolean algebra and the central elements of the GPEA correspond to elements of a sublattice of the exocenter which forms a generalized boolean algebra. We extend to GPEAs the notion of central orthocompleteness, prove that the exocenter of a centrally orthocomplete GPEA (COGPEA) is a complete boolean algebra and show that the sublattice corresponding to the center is a complete boolean subalgebra. We also show that in a COGPEA, every element admits an exocentral cover and that the family of all exocentral covers, the so-called exocentral cover system, has the properties of a hull system on a generalized effect algebra. We extend the notion of type determining (TD) sets, originally introduced for effect algebras and then extended to GEAs and PEAs, to GPEAs, and prove a type-decomposition theorem, analogous to the type decomposition of von Neumann algebras.' address: 'Department of Mathematics an Statistics, Univ. of Massachusetts, Amherst, MA, USA; Štefánikova 49, 814 73 Bratislava, Slovakia' author: - 'David J. Foulis, Sylvia Pulmannová and Elena Vinceková' title: The exocenter and type decomposition of a generalized pseudoeffect algebra --- Introduction {#sc:Intro} ============ Our purpose in this article is to define and study extensions to generalized pseudoeffect algebras of the notions of the center, central orthocompleteness, central cover, type determining sets and type decompositions for an effect algebra, resp. for a pseudoeffect algebra (see [@FPType; @COEA; @ExoCen; @CenGEA; @TDPA; @GFP]). Effect algebras (EAs) [@FandB] were originally introduced as a basis for the representation of quantum measurements [@BLM], especially those that involve fuzziness or unsharpness. Special kinds of effect algebras include orthoalgebras, MV-algebras, Heyting MV-algebras, orthomodular posets, orthomodular lattices, and boolean algebras. An account of the axiomatic approach to quantum mechanics employing EAs can be found in [@DvPuTrends]. Several authors have studied or employed algebraic structures that, roughly speaking, are EAs “without a largest element." These studies go back to M.H. Stone’s work [@Stone] on generalized boolean algebras; later M.F. Janowitz [@Jan] extended Stone’s work to generalized orthomodular lattices. More recent developments along these lines include [@FandB; @HedPu; @KR; @KCh; @MI; @PV07; @Zdenka99; @W]. The notion of a (possibly) non-commutative effect algebra, called a pseudoeffect algebra, was introduced and studied in [@DV1; @DV2; @D]. Whereas a prototypic example of an effect algebra is the order interval from $0$ to a positive element in a partially ordered abelian group, an analogous interval in a partially ordered non-commutative group is a prototype of a pseudoeffect algebra. Pseudoeffect algebras “without a largest element", called generalized pseudoeffect algebras, also have been studied in the literature [@DvVepo; @DvVegen; @PVext; @XieLi]. The classic decomposition of a von Neumann algebra as a direct sum of subalgebras of types I, II and III [@MvN], which plays an important role in the theory of von Neumann algebras, is reflected by a direct sum decomposition of the complete orthomodular lattice (OML) of its projections. The type-decomposition for a von Neumann algebra is dependent on the von Neumann-Murray dimension theory, and likewise the early type-decomposition theorems for OMLs were based on the dimension theories of L. Loomis [@L] and of S. Maeda [@M]. Decompositions of complete OMLs into direct summands with various special properties were obtained in [@CChM; @K; @R] without explicitly employing lattice dimension theory. More recent and considerably more general results on type-decompositions based on dimension theory can be found in [@GW]. Dimension theory for effect algebras was developed in [@HandD]. As a continuation of the aforementioned work, the theory of so called type determining sets was introduced and applied, first to obtain direct decompositions for centrally orthocomplete effect algebras [@FPType; @COEA], and later for centrally orthocomplete pseudoeffect algebras [@TDPA]. While direct decompositions of effect algebras and pseudoeffect algebras are completely described by their central elements [@D; @GFP], for the generalized structures without a top element, we need to replace the center by the so called exocenter, which is composed of special endomorphisms, resp. ideals [@ExoCen; @Je00]. The present paper is organized as follows. In Section \[sc:GPEAs\], we introduce basic definitions and facts concerning generalized pseudoeffect algebras (GPEAs). In Section \[sc:ExoCenter\] we introduce the notion of the exocenter of a GPEA and study its properties. Section \[sc:CenterGPEA\] is devoted to central elements in a GPEA and relations between the center and the exocenter. The notion of central orthocompleteness is extended to GPEAs in Section \[sc:CO\] where it is shown that the center of a centrally orthocomplete GPEA (COGPEA) is a complete boolean algebra. In Section \[sc:ExoCenCover\] we introduce the exocentral cover, which extends the notion of a central cover for an EA. In Section \[sc:TDsets\], we develop the theory of type determining sets for GPEAs and show some examples. Finally, in Section \[sc:TypeDecomp\], we develop the theory of type decompositions of COGPEAs into direct summands of various types. We note that COGPEAs are, up to now, the most general algebraic structures for which the theory of type determining sets has been applied to obtain direct decompositions. Generalized pseudoeffect algebras {#sc:GPEAs} ================================= We abbreviate ‘if and only if’ as ‘iff’ and the notation $:=$ means ‘equals by definition’. \[def:gpea\] A *generalized pseudoeffect algebra* (GPEA) is a partial algebraic structure $(E,\oplus,0)$, where $\oplus$ is a partial binary operation on $E$ called the *orthosummation*, $0$ is a constant in $E$ called the *zero element*, and the following conditions hold for all $a,b,c\in E$: 1. 1. (*associativity*) $(a\oplus b)$ and $(a\oplus b)\oplus c$ exist iff $b\oplus c$ and $a\oplus (b\oplus c)$ exist and in this case $(a\oplus b)\oplus c=a\oplus (b\oplus c)$. 2. (*conjugacy*) If $a\oplus b$ exists, then there are elements $d,e\in E$ such that $a\oplus b=d\oplus a=b\oplus e$. 3. (*cancellation*) If $a\oplus b=a\oplus c$, or $b \oplus a=c\oplus a$, then $b=c$. 4. (*positivity*) If $a\oplus b=0$, then $a=b=0$. 5. (*zero element*) $a\oplus 0$ and $0\oplus a$ always exist and are both equal to $a$. As a consequence of (GPEA3), the elements $d$ and $e$ in (GPEA2) are uniquely determined by $a$ and $b$. Following the usual convention, we often refer to a GPEA $(E,\oplus,0)$ simply as $E$. If $E$ and $F$ are GPEAs, then a mapping $\phi\colon E\to F$ is a *GPEA-morphism* iff, for all $a,b\in E$, if $a\oplus b$ exists in $E$, then $\phi(a)\oplus\phi(b)$ exists in $F$ and $\phi(a\oplus b) =\phi(a)\oplus \phi(b)$. If $\phi\colon E\to F$ is a bijective GPEA-morphism and $\phi\sp{-1}\colon F\to E$ is also a GPEA-morphism, then $\phi$ is a *GPEA-isomorphism*. In what follows, $(E,\oplus,0)$ is a generalized pseudoeffect algebra. In general, lower case Latin letters $a,b,c,...,x,y,z$, with or without subscripts, will denote elements of $E$. If we write an equation involving an orthosum, e.g. $x\oplus y=z$, we tacitly assume its existence. \[df:leqetc\] The relation $\leq$ is defined on the GPEA $E$ by $$a\leq b \mbox{ iff } a\oplus x=b \mbox{ for some } x\in E$$ or equivalently (in view of (GPEA2)), by $$a\leq b \mbox{ iff } y\oplus a=b \mbox{ for some } y\in E.$$ If $a\leq b$, then by (GPEA3) the elements $x$ and $y$ such that $a\oplus x=y\oplus a=b$ are uniquely determined by $a$ and $b$, and we define the (left and right) differences $$a/b:=x \mbox{ and } b\backslash a:=y.$$ In the event that $a\leq b$ and $a/b$ coincides with $b\backslash a$, we also define $$b\ominus a:=a/b=b\backslash a.$$ We say that elements $p$ and $q$ in $E$ are *orthogonal*, in symbols $p\perp q$, iff $p\oplus q$ and $q\oplus p$ both exist and are equal. The GPEA $E$ is *commutative* iff $p\perp q$ holds whenever $p\oplus q$ is defined. Evidently, if either $a/b$ or $b\backslash a$ exists, then both exist and $a\leq b$; conversely, if $a\leq b$, then both $a/b$ and $b\backslash a$ exist and $b=a\oplus(a/b)=(b\backslash a)\oplus a$. Also, if $b\ominus a$ exists, then $a,b\ominus a\leq b$, $a\perp(b\ominus a)$ and $a\oplus(b\ominus a)=(b\ominus a)\oplus a =b$. We note that a commutative GPEA is the same thing as a *generalized effect algebra* [@Zdenka99]. The GPEA $E$ is partially ordered by $\leq$ and $0$ is the smallest element in $E$. The cancellation laws in (GPA3) are easily extended to $\leq$ as follows: $$\mbox{If }a\oplus b\leq a\oplus c,\mbox{or if }b\oplus a\leq c\oplus a, \mbox{ then }b\leq c.$$ An existing supremum (resp. infimum) in the partially ordered set (poset) $E$ of elements $a$ and $b$ is denoted by $a\vee b$ (resp. by $a\wedge b$). We say that $a$ and $b$ are *disjoint* iff $a\wedge b=0$. We note that a GPEA-morphism preserves inequalities and corresponding left and right differences. An important example of a GPEA ([@DvVepo], Example 2.3) is a subset of the positive cone in a partially ordered group (po-group). Let $(G,+,0, \leq)$ be a po-group with $G^+:=\{g\in G: 0\leq g\}$. Let $G^0$ be a nonempty subset of $G^+$ such that for all $a,b\in G^0$, if $b\leq a$ then $-a+b,\ b-a\in G^0$. Then $(G^0,\oplus,0)$, where $\oplus$ is the group addition restricted to those pairs of elements whose sum is again in $G^0$, is a GPEA whose partial order coincides with the group partial order restricted to $G^0$. \[le:SlashProps\] Let $a,b,c,d\in E$ with $a\leq b$. Then[:]{} 1. $b\backslash a,\ a/b\leq b$ and $(b\backslash a)/b=b\backslash(a/b)=a$. 2. $d\leq a/b\Leftrightarrow a\oplus d\leq b\Leftrightarrow d\leq b$ and $a\leq b\backslash d$. 3. If $b\oplus d$ exists, then $a/(b\oplus d)=(a/b)\oplus d$, $a\oplus d$ exists, and $a\oplus d\leq b\oplus d$. Also, if $d\oplus b$ exists, then $(d\oplus b)\backslash a=d\oplus(b\backslash a)$, $d\oplus a$ exists, and $d\oplus a\leq b\oplus a$. 4. If $a\leq b\leq c$, then $a/c=a/b\oplus b/c$ and $c\backslash a=c \backslash b\oplus b\backslash a$. \(i) As $b=b\backslash a\oplus a$, we get $(b\backslash a)/b=a$, and $b=a \oplus a/b$ implies $b\backslash (a/b)=a$. \(ii) If $d\leq a/b$, then $\exists x\in E$ with $d\oplus x=a/b$, so $(a\oplus d) \oplus x=a\oplus(d\oplus x)=b$, and therefore $a\oplus d\leq b$. If $a\oplus d \leq b$, then $\exists y\in E$, $y\oplus(a\oplus d)=(y\oplus a)\oplus d=b$, whence $d\leq b$, $y\oplus a=b\backslash d$, and $a\leq b\backslash d$. Thus $d\leq a/b\Rightarrow a\oplus d\leq b\Rightarrow d\leq b\text{\ and\ }a\leq b \backslash d$. Proofs of the converse implications are straightforward. \(ii) Assume that $b\oplus d$ exists. Then $a\leq b\leq b\oplus d$ and $a \oplus a/(b\oplus d)=b\oplus d=(a\oplus a/b)\oplus d=a\oplus((a/b)\oplus d)$, whence $a/(b\oplus d)=(a/b)\oplus d$ by cancellation. Also, as $a\leq b$, we have $b\backslash a\oplus a=b$, whence $b\oplus d=(b\backslash a\oplus a) \oplus d=b\backslash a\oplus(a\oplus d)$, whence $a\oplus d$ exists and $a\oplus d\leq b\oplus d$. The remaining assertion is proved analogously. \(iv) As $a\leq b\leq c$, we have $a\oplus(a/b\oplus b/c)=(a\oplus a/b) \oplus b/c=b\oplus b/c=c=a\oplus a/c$, whence $a/b\oplus b/c=a/c$ by cancellation. The second equality is proved similarly. \[le:oplusdist\] Let $e\in E$, and let $(f\sb{i})\sb{i\in I}$ be a family of elements of $E$ such that the supremum $f:=\bigvee\sb{i\in I}f\sb{i}$ exists in $E$. Suppose that $e\oplus f$ [(]{}resp. $f\oplus e$[)]{} exists. Then $e\oplus f\sb{i}$ [(]{}resp. $f\sb{i}\oplus e$[)]{} exists for all $i\in I$, the supremum $\bigvee\sb{i\in I}(e\oplus f\sb{i})$ [(]{}resp. the supremum $\bigvee\sb {i\in I}(f\sb{i}\oplus e)$[)]{} exists in $E$, and $e\oplus f=\bigvee\sb{i \in I}(e\oplus f\sb{i})$ [(]{}resp. $f\oplus e=\bigvee\sb{i\in I}(f\sb{i} \oplus e)$[)]{}. We prove the lemma under the hypothesis that $e\oplus f$ exists. The proof under the alternative hypothesis is similar. For each $i\in I$, we have $f\sb{i}\leq f$, and therefore $e\oplus f\sb{i}$ exists and $e\oplus f \sb{i}\leq e\oplus f$ (Lemma \[le:SlashProps\] (iii)). Suppose that $e\oplus f\sb{i}\leq b\in E$ for all $i\in I$, i.e., there exists $x\sb{i}$ with $b=(e\oplus f\sb{i})\oplus x\sb{i}=e\oplus (f\sb{i}\oplus x\sb{i})$. Then $e\leq b$ and $f\sb{i}\leq f\sb{i}\oplus x\sb{i}=e/b$ for all $i\in I$, whence $f\leq e/b$, and it follows from Lemma \[le:SlashProps\] (ii) that $e \oplus f\leq b$, proving that $e\oplus f=\bigvee\sb{i\in I}(e\oplus f\sb{i})$. By (GPEA1), we may omit parentheses in expressions such as $a\oplus b\oplus c$. By recursion, the partial operation $\oplus$ can be extended to finite sequences $e_1,e_2,\ldots,e_n$ as follows: The orthosum $e_1\oplus\cdots \oplus e_n$ exists iff the elements $f:=e_1\oplus e_2\oplus\cdots\oplus e_{n-1}$ and $f\oplus e_n$ both exist, and then $e_1\oplus\cdots\oplus e_n :=f\oplus e_n$. In general, the orthosum may depend on the order of its orthosummands. In a similar way, by recursion, we also define *orthogonality* and the corresponding *orthosum* for a finite sequence of elements in $E$, and it turns out that the orthosum does not depend on the order of the orthosummands. Therefore, in the obvious way, we define orthogonality and the corresponding orthosum for finite families in $E$. (We understand that the empty family in $E$ is orthogonal and that its orthosum is $0$.) The notion of orthogonality and the orthosum for arbitrary families is defined as follows: A family $(e_i)_{i\in I}$ in $E$ is said to be *orthogonal* iff every finite subfamily $(e_i)_{i\in F}$ ($I\supseteq F$ is finite) is orthogonal in $E$. The family $(e_i)_{i\in I}$ is *orthosummable* with *orthosum* $\oplus_{i\in I} e_i$ iff it is orthogonal and the supremum $\bigvee_{F\subseteq I}(\oplus_{i\in F} e_i)$ over all finite subsets $F$ of $I$ exists in $E$, in which case $\oplus_{i\in I}e_i:=\bigvee_{F\subseteq I} (\oplus_{i\in F} e_i)$. \[le:veeopluswedge\] Let $e,f\in E$. If $e\perp f$ and $e\vee f$ exists in $E$, then $e\wedge f$ exists in $E$, $(e\vee f)\perp(e\wedge f)$, and $e\oplus f=(e\vee f) \oplus(e\wedge f)$. As $e\perp f$, we have $e\oplus f=f\oplus e$. Evidently $e,f\leq e\oplus f$, so $e\leq e\vee f\leq e\oplus f$, and by Lemma \[le:SlashProps\] (iv), $e/(e\vee f)\oplus(e\vee f)/(e\oplus f)=e/(e\oplus f)=f$, whence $(e\vee f) /(e\oplus f)\leq f$. Likewise, $(e\vee f)/(e\oplus f)\leq e$. Suppose that $d\leq e,f$. By Lemma \[le:SlashProps\] (iii), $f\leq f\oplus(e\backslash d) =(f\oplus e)\backslash d=(e\oplus f)\backslash d$. Likewise, $e\leq(e\oplus f) \backslash d$, and we have $e\vee f\leq(e\oplus f)\backslash d$; hence by Lemma \[le:SlashProps\] (ii), $d\leq (e\vee f)/(e\oplus f)$. This proves that $(e\vee f)/(e\oplus f)=e\wedge f$, from which we obtain $e\oplus f= (e\vee f)\oplus (e\wedge f)$. Similarly, by considering $(e\oplus f) \backslash(e\vee f)$, which is again under $e$ and $f$, and arguing that $(e\oplus f)\backslash(e\vee f)=e\wedge f$, we find that $e\oplus f= (e\wedge f)\oplus (e\vee f)$. \[def:pea\] A *pseudoeffect algebra* (PEA) is a partial algebraic structure $(E,\oplus,0,1)$, where $\oplus$ is a partial operation and $0$ and $1$ are constants, and the following hold: 1. 1. $a\oplus b$ and $(a\oplus b)\oplus c$ exist iff $b\oplus c$ and $a\oplus (b\oplus c)$ exist, and in this case $(a\oplus b)\oplus c=a\oplus (b\oplus c)$. 2. There is exactly one $d\in E$ and exactly one $e\in E$ such that $a \oplus d=e\oplus a=1$. 3. If $a\oplus b$ exists, there are elements $d,e\in E$ such that $a\oplus b=d\oplus a=b\oplus e$. 4. If $1\oplus a$ or $a\oplus 1$ exists, then $a=0$ The partial ordering for a PEA is defined in the same way as the partial ordering for a GPEA. It is easy to see, that a PEA is the same thing as a GPEA with a greatest element. We claim the following statement from ([@DvVepo], Proposition 2.7): \[pr:interval\] Let $(E,\oplus,0)$ be a GPEA and let $u\in E$. Then $(E[0,u],\oplus\sb{u},0,u)$ is a PEA, where $E[0,u]:=\{ a\in E:a\leq u\}$ and where $a\oplus\sb{u}b$ is defined for $a,b\in E[0,u]$ iff $a\oplus b$ exists in $E$ and $a\oplus b\leq u$, in which case $a\oplus\sb{u}b:=a\oplus b$. \[df:Ideal\] An *ideal* of the GPEA $E$ is a nonempty subset $I\subseteq E$ such that: 1. If $a\in I$, $b\in E$, and $b\leq a$, then $b\in I$. 2. If $a,b\in I$ and $a\oplus b$ exists, then $a\oplus b\in I$. If $I$ is an ideal in $E$, then $I$ is said to be *normal* iff, 1. whenever $a,x,y\in E$ and $a\oplus x=y\oplus a$, then $x\in I\Leftrightarrow y\in I$. \[df:CentralIdeal\] We say, that an ideal $S$ in the GPEA $E$ is *central*, or equivalently, that it is a *direct summand* of $E$, iff there is an ideal $S'$ in $E$ such that 1. $a\in S, b\in S'\Rightarrow a\perp b$, and 2. every $a\in E$ can be uniquely written a an orthosum $a=a_1\oplus a_2$ with “coordinates" $a_1\in S$ and $a_2\in S'$. We write $E=S\oplus S'$ iff (1) and (2) hold. If $E=S\oplus S'$, then $S'$ is also a central ideal (direct summand) in $E$, $S'$ is uniquely determined by $S$ (cf. the proof of [@CenGEA Lemma 4.3]), and all GPEA calculations on $E$ can be conducted “coordinatewise" in the obvious sense. If $E=S\oplus S'$, we refer to $S$ and $S'$ as *complementary direct summands* of $E$. \[pr:DirSumNormal\] Any central ideal [(]{}direct summand[)]{} of a GPEA $E$ is normal. Let $S$ be a central ideal of $E$ with $S'$ as its complementary direct summand, and assume that $a,x,y\in E$ with $a\oplus x=y\oplus a$. We can write $a$ uniquely as $a=a\sb{1}\oplus a\sb{2}$ with $a\sb{1}\in S$ and $a\sb{2}\in S'$. Then $a\sb{1}\oplus a\sb{2}\oplus x=y\oplus a\sb{1} \oplus a\sb{2}$. Suppose that $x\in S$. Then, as $a\sb{2}\in S'$, we have $x\perp a\sb{2}$, so $a\sb{2}\oplus x=x\oplus a\sb{2}$, whence $a\sb{1}\oplus x\oplus a\sb{2}=y\oplus a\sb{1}\oplus a\sb{2}$, and by cancellation $a\sb{1}\oplus x=y\oplus a\sb{1}$. Therefore, $y\leq a\sb{1} \oplus x\in S$, and it follows that $y\in S$. By a similar argument, if $y\in S$, then $x\in S$. The notion that $E$ is a direct sum $E=S\oplus S'$ of two central ideals is extended to finitely many direct summands $E=S\sb{1}\oplus S\sb{2} \oplus\cdots\oplus S\sb{n}$ in the obvious way, each $S\sb{i}$, $i= 1,2,...,n$, being a central ideal (direct summand) in $E$ with complementary direct summand $(S\sb{i})'=S\sb{1}\oplus\cdots S\sb{i-1}\oplus S\sb{i+1} \cdots\oplus S\sb{n}$. The exocenter of a GPEA {#sc:ExoCenter} ======================= \[df:ExoCen\] The exocenter of the GPEA $E$, denoted by ${{\Gamma\!\sb{\rm ex}}}(E)$, is the set of all mappings $\pi: E\rightarrow E$ such that for all $e,f\in E$ the following hold: 1. 1. $\pi: E\rightarrow E$ is a PGEA-endomorphism of $E$, that is: if $e\oplus f$ exists, then $\pi e\oplus \pi f$ exists and $\pi(e\oplus f)=\pi e \oplus\pi f$. 2. $\pi$ is idempotent (i.e., $\pi (\pi e)=\pi e$). 3. $\pi$ is decreasing (i.e., $\pi e\leq e$). 4. $\pi$ satisfies the following orthogonality condition: if $\pi e=e$ and $\pi f=0$, then $e\perp f$ (i.e., $e\oplus f=f\oplus e$). If $\pi\in {{\Gamma\!\sb{\rm ex}}}(E)$ and $e\in E$, then as $\pi e\leq e$ by (EXC3), we can (and do) define $\pi\,'e:=(\pi e)/e$ for all $e\in E$. \[le:piprime\] If $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and $e\in E$, then $\pi\,'e=(\pi e)/e=e\backslash(\pi e) =e\ominus\pi e$ and $\pi e\perp\pi\,' e\text{\ with\ }\pi e\oplus\pi\,' e =\pi\,'e\oplus\pi e=e.$ Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and $e\in E$. As $\pi e\leq e$, both $\pi\,'e=(\pi e) /e$ and $e\backslash(\pi e)$ are defined, and with $x:=(\pi e)/e$ and $y:=e\backslash(\pi e)$, we have $\pi e\oplus x=e=y\oplus\pi e$. We apply the mapping $\pi$ and obtain $\pi e\oplus\pi x=\pi e=\pi y\oplus \pi e$; hence $\pi x=\pi y=0$ and by (EXC4), $\pi e\oplus x=x\oplus\pi e =e$ and also $\pi e\oplus y=y\oplus\pi e=e$. Therefore by cancellation, $\pi\,'e=(\pi e)/e=x=y=e\backslash(\pi e)=e\ominus\pi e$, and $\pi e \perp\pi\,'e$ with $\pi e\oplus\pi\,'e=\pi\,'e\oplus\pi e=e.$ \[th:EXCprop\] If $\pi\in {{\Gamma\!\sb{\rm ex}}}(E)$, then for all $e,f\in E$ the following hold[:]{} 1. $\pi(\pi\,'e)=\pi\,'(\pi e)=0$. 2. $\pi\,'\in {{\Gamma\!\sb{\rm ex}}}(E)$ and $(\pi\,')'=\pi$. 3. If $e\leq \pi f$, then $e=\pi e$. 4. If $e\leq f$, then $\pi e=e\wedge\pi f$. 5. $\pi(E):=\{ \pi e: e\in E\} =\{ e\in E: e=\pi e\}$ is an ideal in $E$. 6. $\pi(E)$ is sup/inf-closed in $E$ [(]{}i.e., $\pi(E)$ is closed under the formation of existing suprema and infima in $E$ of nonempty families in $\pi(E)$[)]{}. 7. If $e\in \pi (E)$ and $f\in \pi\,'(E)$, then $e\perp f, e\oplus f= e\vee f$ and $e\wedge f=0$. 8. For each element $e\in E$ there are uniquely determined elements $e_1\in \pi (E), e_2\in \pi\,'(E)$ such that $e=e\sb{1}\oplus e\sb{2}$; in fact, $e\sb{1}=\pi e$ and $e\sb{2}=\pi\,'e$. 9. If $e=e_1\oplus e_2, f=f_1\oplus f_2$, where $e_1,f_1\in \pi (E)$, $e_2,f_2\in \pi\,'(E)$, then $e\oplus f$ exists iff both $e_1\oplus f_1$ and $e_2\oplus f_2$ exist. 10. $\pi\,'(E)=\{ f\in E: f\wedge e=0,\,\forall e\in \pi (E)\}$. \(i) $\pi(\pi\,'e)=\pi(e\backslash\pi e)=\pi e\backslash\pi\pi e=\pi e\backslash \pi e=0$ and $\pi\,'(\pi e)=\pi e\backslash\pi\pi e=0$ too. \(ii) By Lemma \[le:SlashProps\] (i), $(\pi\,')' e=e\backslash\pi\,'e=e \backslash(\pi e/e)=\pi e$. To prove that $\pi\,'$ is a GPEA-endomorphism of $E$, suppose that $e\oplus f$ exists. Then by (EXC1) $\pi\,'(e\oplus f) =(e\oplus f)\backslash(\pi e\oplus \pi f)$, whence $\pi\,'(e\oplus f) \oplus\pi e\oplus\pi f=e\oplus f$ and so by Lemma \[le:SlashProps\] (iii), $\pi\,'(e\oplus f)\oplus\pi e=(e\oplus f)\backslash\pi f=e\oplus(f\backslash \pi f)=e\oplus\pi\,'f$. As $\pi\pi e =\pi e$ and by (i), $\pi\pi\,'(e\oplus f) =0$, we have $e\perp\pi\,'f$ by (EXC4), whence $\pi e\oplus\pi\,'(e\oplus f)= \pi\,'(e\oplus f)\oplus\pi e=e\oplus\pi\,'f$, i.e., $\pi\,'(e\oplus f)= \pi e/(e\oplus\pi\,'f)$, and a second application of Lemma \[le:SlashProps\] (iii) yields $\pi\,'(e\oplus f)=(\pi e/e)\oplus\pi\,'f=\pi\,'e\oplus\pi\,'f$. Thus, $\pi\,'$ satisfies (EXC1). Moreover, by (i), $\pi\,'(\pi\,'e)=\pi\,' (e\backslash\pi e)=\pi\,'e\backslash\pi\,'\pi e=\pi\,'e$, whence $\pi\,'$ satisfies (EXC2). Obviously, (EXC3) holds for $\pi\,'$. Finally to prove that $\pi\,'$ satisfies (EXC4), suppose that $\pi\,'e=e$ and $\pi\,'f =0$. Then $\pi e=0$ because $\pi\,'e=e\backslash\pi e=e$ and $\pi f=f$ because $\pi\,'f=f\backslash\pi f=0$. Therefore, since $\pi$ satisfies (EXC4), we have $e\perp f$, and $\pi\,'$ also satisfies (EXC4). Therefore, $\pi\,'\in{{\Gamma\!\sb{\rm ex}}}(E)$. \(iii) If $e\leq \pi f$, then $e\backslash\pi e=\pi\,'e\leq \pi\,'\pi f=0$, whence $e=\pi e$. \(iv) Suppose that $e\leq f$. Then $\pi e\leq e$ and $\pi e\leq \pi f$. Suppose that $d\leq e,\pi f$. Since $d\leq\pi f$, (iii) implies that $d=\pi d\leq \pi e$, so $\pi e=e\wedge\pi f$. \(v) If $e=\pi e$, then $e\in\pi(E)$. Vice versa, if $e\in\pi(E)$, then $e= \pi f$ for some $f\in E$, so $\pi e=\pi\pi f=\pi f=e$, and we have $\pi(E)= \{e\in E: e=\pi e\}$. \(vi) Assume that $(e_i)_{i\in I}\subseteq\pi(E)$ and $e=\bigvee_{i\in I} e_i$ exists in $E$. As $e_i\leq e$, we have $e_i=\pi e_i\leq\pi e$ for all $i\in I$, whence $e\leq\pi e$. But also $\pi e\leq e$ and thus $\pi e=e\in\pi(E)$. Since $\pi(E)$ is an ideal, it is automatically closed under the formation of existing infima in $E$ of nonempty families in $\pi(E)$. \(vii) Let $e\in\pi(E)$ and $f\in\pi\,'(E)$. Then $e=\pi e$, and $\pi f=\pi \pi\,'f=0$, whence by (EXC4) $e\perp f$. Clearly, $e,f\leq e\oplus f$. If now $e,f\leq d\in E$, then $e=\pi e\leq\pi d$ and $f=\pi\,'f\leq\pi\,'d$, thus $e\oplus f\leq\pi d\oplus\pi\,'d=d$, whence $e\oplus f=e\vee f$. Finally, by Lemma \[le:veeopluswedge\], $e\wedge f=0$. \(viii) Obviously, $e=\pi e\oplus\pi\,'e$, $\pi e\in\pi(E)$ and $\pi\,'e\in \pi\,'(E)$. Suppose $e=e_1\oplus e_2$ with $e_1\in\pi(E)$, $e_2\in\pi\,'(E)$. Then $e_1=\pi e_1$, $e_2=\pi\,'e_2$, $\pi e=\pi e_1\oplus\pi e_2=e_1$, and $\pi\,'e=\pi\,'e_1\oplus\pi\,'e_2=e_2$. \(ix) Suppose $e=e_1\oplus e_2$ and $f=f_1\oplus f_2$, with $e_1,f_1\in\pi(E)$, $e_2,f_2\in\pi\,'(E)$. If $e_1\oplus f_1$ and $e_2\oplus f_2$ both exist, then $e_1 \oplus f_1\in\pi(E)$ and $e_2\oplus f_2\in\pi\,'(E)$ by (v). Then by (vii) $(e_1 \oplus f_1)\perp(e_2\oplus f_2)$, so $(e_1\oplus f_1)\oplus(e_2\oplus f_2)$ exists and equals $e_1\oplus(f_1\oplus e_2)\oplus f_2=e_1\oplus(e_2\oplus f_1) \oplus f_2=(e_1\oplus e_2)\oplus(f_1\oplus f_2)=e\oplus f$. If, on the other hand, $e\oplus f$ exists, then $e_1\oplus e_2\oplus f_1\oplus f_2$ exists and equals $e_1 \oplus f_1\oplus e_2\oplus f_2$, which implies that $e_1\oplus f_1$ and $e_2\oplus f_2$ both exist. \(x) Assume that $f\wedge e=0$ for all $e\in\pi(E)$. As $f=f_1\oplus f_2$ with $f_1\in \pi(E)$, $f_2\in\pi\,'(E)$, we have $f_1=f\wedge f_1=0$, whence $f=f_2\in\pi\,'(E)$. The converse follows from (vii). \[le:circ\] Let $\xi,\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Then[:]{} - $\xi\circ\pi=\pi\circ\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$. - $\xi=\xi\circ\pi\,\Leftrightarrow\,\xi e\leq\pi e,\,\forall\, e\in E\, \Leftrightarrow\,\xi (E)\subseteq\pi (E)$. \(i) Since $\xi(\pi e)\leq\pi e$, part (iii) of Theorem \[th:EXCprop\] yields $\xi(\pi e)=\pi(\xi(\pi e))$. Also, since $\pi e\leq e$ and both, $\pi$ and $\xi$ are order-preserving mappings, it follows that $\xi(\pi e)=\pi(\xi(\pi e)) \leq\pi(\xi e)$. By symmetry $\pi(\xi e)\leq\xi(\pi e)$, which gives $\xi\circ \pi=\pi\circ\xi$. Obviously, $\xi\circ\pi$ is a GPEA-endomorphism. Furthermore, $(\xi\circ\pi) \circ(\xi\circ\pi)=\xi\circ\pi\circ\pi\circ\xi=\xi\circ\pi\circ\xi=\xi\circ \xi\circ\pi=\xi\circ\pi$, whence $\xi\circ\pi$ is idempotent. Moreover, $(\xi\circ\pi)e=\xi(\pi e)\leq\pi e\leq e$, so (EXC3) holds. Finally, suppose that $e,f\in E$ with $e=\xi(\pi e)$ and $\xi(\pi f)=0$. Then $e=\pi(\xi e)$, so $e=\xi e=\pi e$. We put $d:=\pi f$, so that $\xi d=0$, $d\leq f$, $d=\pi d= \pi f$, and $\pi\,'f=(\pi f)/f=f\backslash\pi f=d/f=f\backslash d$. Therefore, $\pi\,'f\oplus d=d\oplus\pi\,'f=f$. As $e=\xi e$ and $\xi d=0$, (EXC4) implies that $e\perp d$, i.e., $e\oplus d=d\oplus e$. Also, $\pi(e\oplus d)=\pi e \oplus\pi d=e\oplus d$, and it follows from $\pi(\pi\,'f)=0$ and (EXC4) that $(e\oplus d)\perp\pi\,'f$. Consequently, $$e\oplus f=e\oplus d\oplus\pi\,'f=\pi\,'f\oplus e\oplus d=\pi\,'f\oplus d \oplus e=f\oplus e,$$ proving that $\xi\circ\pi$ satisfies (EXC4). \(ii) If $\xi=\xi\circ\pi$, then $\xi e=\xi(\pi e)\leq\pi e$ for all $e\in E$. Conversely, if $\xi e\leq\pi e$ for all $e\in E$, then $\xi e=\xi(\xi e)\leq \xi(\pi e)$. Also, as $\xi(\pi e)\leq\xi e$ always holds, $\xi e=\xi(\pi e)$ for all $e\in E$, which means that $\xi=\xi\circ\pi$. Now if $\xi e\leq\pi e$ for all $e\in E$, then if $e\in\xi(E)$, we get $e=\xi e\leq\pi e$, whence $\pi e=e\in\pi(E)$. Conversely, if $\xi (E)\subseteq\pi(E)$, then every $\xi e \in\pi(E)$, thus by (i), $\xi e=\pi(\xi e)=\xi(\pi e)\leq\pi e$. \[th:boolalg\] Let $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and let $e\in E$. Then ${{\Gamma\!\sb{\rm ex}}}(E)$ is partially ordered by $\xi\leq\pi\Leftrightarrow\,\xi=\xi\circ\pi\,\Leftrightarrow\,\xi e\leq\pi e, \,\forall e\in E\,\Leftrightarrow\,\xi (E)\subseteq\pi (E)$, with $0$ [(]{}the zero mapping[)]{} as the smallest element and $1$ [(]{}the identity mapping[)]{} as the largest element. Moreover, ${{\Gamma\!\sb{\rm ex}}}(E)$ is a boolean algebra with $\pi\mapsto\pi\,'$ as the boolean complementation, with $\pi\wedge\xi= \pi\circ\xi=\xi\circ\pi$, and with $\pi\vee\xi=(\pi\,'\circ\xi\,')'$. Let $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$. By Lemma \[le:circ\], $\leq$ is a partial order on ${{\Gamma\!\sb{\rm ex}}}(E)$ and $0\leq\pi\leq 1$ holds for every $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Clearly, $\pi \circ\xi$ is the infimum $\pi\wedge\xi$ of $\pi$ and $\xi$ in ${{\Gamma\!\sb{\rm ex}}}(E)$. We also have $\pi\wedge\xi=0$ iff $\pi(\xi e)=0$ for every $e\in E$, which is equivalent to $\pi(e\backslash\xi e)=\pi e,\,\forall\, e\in E$. But this means that $\pi(\xi\,'e)=\pi e,\,\forall\, e\in E$, that is $\pi\circ\xi\,'=\pi$, which holds iff $\pi\leq\xi\,'$. So by [@GrGLT Theorem 4, p. 49], ${{\Gamma\!\sb{\rm ex}}}(E)$ is a boolean algebra, $\pi\,'$ is the complement of $\pi$ in ${{\Gamma\!\sb{\rm ex}}}(E)$, and $\pi \vee\xi=(\pi\,'\circ\xi\,')'$. \[le:DisjointPiXi\] Let $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$ with $\pi\wedge\xi=0$ and let $e,f\in E$. Then[:]{} - If $e\in\pi(E), f\in\xi(E)$, then $e\perp f$ and $e\oplus f\in (\pi\vee\xi)(E)$, $e\oplus f=e\vee f$ and $e\wedge f=0$. - $\pi e\perp\xi e, (\pi\vee\xi)e=\pi e\vee\xi e=\pi e\oplus\xi e$ and $\pi e\wedge\xi e=0$. \(i) By the hypotheses $e=\pi e$ and $f=\xi f$. As $\pi f=\pi(\xi f)=0$ (by Theorem \[th:boolalg\]), we get $\pi\,'f=f\backslash\pi f=f$. Therefore, $f\in\pi\,'(E)$, and by Theorem \[th:EXCprop\] (vii), $e\perp f$, $e\oplus f =e\vee f$ and $e\wedge f=0$. Also $e=\pi e\leq(\pi\vee\xi)e\leq e$, whence $(\pi\vee\xi)e=e$. Likewise, $(\pi\vee\xi)f=f$, whence $e\oplus f=(\pi\vee \xi)(e\oplus f)\in(\pi\vee\xi)(E)$. \(ii) We need only replace $e$ by $\pi e$ and $f$ by $\xi e$ in (i) to obtain $\pi e\perp\xi e$, $\pi e\oplus\xi e=\pi e\vee\xi e$ and $\pi e\wedge\xi e=0$. As $\pi\wedge\xi=0$ in the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$, we have $\pi\leq\xi\,'$, whence $\pi e=(\pi\wedge\xi\,')e=(\pi\circ\xi\,')e=\pi(\xi\,'e)$. Thus, combining the equalities $\xi e\oplus \xi\,'e=e$ and $\pi e\oplus(\pi\,' \circ\xi\,')e=\pi(\xi\,'e)\oplus\pi\,'(\xi\,'e)=\xi\,'e$, we obtain $\xi e \oplus\pi e\oplus(\pi\,'\circ\xi\,')e=e$. Therefore, as $(\pi\,'\circ \xi\,')'e\oplus(\pi\,'\circ\xi\,')e=e$, we infer by cancellation that $(\pi\vee\xi)e=(\pi\,'\circ\xi\,')'e=\pi e\oplus\xi e=\pi e\vee\xi e$. \[th:FinitePwiseDisjointPi\] Let $\pi_1,\pi_2,\ldots ,\pi_n$ be pairwise disjoint elements of the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$ and let $e\in E, e_i\in\pi_i(E)$ for $i=1,2,\ldots ,n$. Then[:]{} - $(e_i)_{i=1,2,\ldots ,n}$ is an orthogonal sequence in $E$ and $\oplus_{i=1}^n e_i=\bigvee_{i=1}^n e_i$. - $(\pi_ie)_{i=1}^n$ is an orthogonal sequence in $E$ and $(\pi_1\vee \pi_2\vee\dots\vee\pi_n)e=\oplus_{i=1}^n\pi_i e=\bigvee_{i=1}^n\pi_i e$. For $n=1$ the assertions hold trivially, and the results for $n=2$ are consequences of Lemma \[le:DisjointPiXi\]. The results for an arbitrary $n\in\mathbb{N}$ then follow from a straightforward induction argument. \[th:finitepointwisesup/inf\] Let $\pi_1,\pi_2,\ldots ,\pi_n\in{{\Gamma\!\sb{\rm ex}}}(E)$, $e\in E$. Then[:]{} - $(\pi_1\wedge\pi_2\wedge\dots\wedge\pi_n)e=\pi_1 e\wedge\pi_2 e \wedge\dots\wedge\pi_n e$. - $(\pi_1\vee\pi_2\vee\dots\vee\pi_n)e=\pi_1 e\vee\pi_2 e\vee\dots \vee\pi_n e$. We will prove the assertions for $n=2$ and the general cases will then follow by induction. \(i) Obviously, $(\pi\wedge\xi)e\leq\pi e,\xi e$. Suppose now that $f\leq \pi e,\xi e$. Then $f=\pi f=\xi f$ by Theorem \[th:EXCprop\] (iii) and therefore $f=(\pi\circ\xi)f\leq(\pi\circ\xi)\xi e=(\pi\circ\xi\circ\xi)e= (\pi\circ\xi)e=(\pi\wedge\xi)e$. \(ii) Working in the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$, we can write $\pi\vee\xi$ as a pairwise disjoint supremum: $$\pi\vee\xi=(\pi\wedge\xi)\vee(\pi\wedge\xi\,')\vee(\pi\,'\wedge\xi).$$ Then we use Theorem \[th:FinitePwiseDisjointPi\] to get $$(\pi\vee\xi)e=(\pi\wedge\xi)e\vee(\pi\wedge\xi\,')e\vee(\pi\,'\wedge\xi)e$$ where $\pi e=(\pi\wedge\xi)e\vee(\pi\wedge\xi\,')e$ and $\xi e=(\pi\wedge\xi)e \vee(\pi\,'\wedge\xi)e$. Therefore $(\pi\vee\xi)e=\pi e\vee\xi e$. As is easily confirmed, a cartesian product of GPEAs, with the obvious pointwise operations and relations, is again a GPEA. \[th:finitecartesianprod\] Let $\pi_1,\pi_2,\ldots ,\pi_n$ be pairwise disjoint elements of ${{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi_1\vee\pi_2\vee\ldots\vee\pi_n=1$ and let $X$ be the cartesian product of $\pi_i(E)$ for $i=1,2\ldots ,n$. Then for $(e_1,e_2,\ldots ,e_n)\in X$, the sequence $(e_i)_{i=1}^n$ is orthogonal in $E$ and $\oplus_{i=1}^n e_i= \bigvee_{i=1}^n e_i$. Moreover, $\Phi: X\rightarrow E$ defined by $\Phi (e_1,e_2,\ldots ,e_n):=e_1\oplus e_2\oplus\ldots\oplus e_n$, is a GPEA-isomorphism and for every $e\in E$, $\Phi^{-1}e=(\pi_1 e,\pi_2 e,\ldots, \pi_n e)\in X$. The first part has already been proved in Theorem \[th:FinitePwiseDisjointPi\]. To prove that $\Phi$ is a GPEA-morphism, let $(e_1,e_2,\ldots ,e_n), (f_1,f_2,\ldots ,f_n)\in X$ and let $e_i \oplus f_i$ exist for all $i=1,2,\ldots ,n$. Then $(e_1\oplus f_1,e_2 \oplus f_2,\ldots ,e_n\oplus f_n)\in X$ and so $(e_i\oplus f_i)_{i=1}^n$ is an orthogonal sequence. Using Theorem \[th:EXCprop\] (ix) and induction, we get $\oplus_{i=1}^n(e_i\oplus f_i)=\oplus_{i=1}^{n-1}(e_i \oplus f_i)\oplus e_n\oplus f_n=(\oplus_{i=1}^{n-1} e_i)\oplus(\oplus_{i=1} ^{n-1} f_i)\oplus e_n\oplus f_n$. But, since $f_i$ for $i=1,2\ldots ,n-1$ are all orthogonal to $e_n$, we have $(\oplus_{i=1}^{n-1} e_i)\oplus(\oplus_{i=1} ^{n-1} f_i)\oplus e_n\oplus f_n=(\oplus_{i=1}^{n-1} e_i)\oplus e_n\oplus (\oplus_{i=1}^{n-1} f_i)\oplus f_n=(\oplus_{i=1}^n e_i)\oplus (\oplus_{i=1} ^n f_i)$, whence $\Phi: X\rightarrow E$ is a GPEA-morphism. Define $\Psi: E\to X$ by $\Psi(e):=(\pi_1 e,\pi_2 e,\ldots ,\pi_n e)$ for all $e \in E$. Then $\Psi$ is also a GPEA-morphism and by Theorem \[th:FinitePwiseDisjointPi\] (ii), $\Phi\circ\Psi$ is the identity on $E$. Now consider $\pi_i e_j$ for $i,j=1,2,\ldots ,n$. We have $\pi_i e_j=\pi_i (\pi_j e_j)=(\pi_i\wedge\pi_j)e_j$. Thus $\pi_i e_j=0$ for $i\not=j$ and $\pi_i e_j=e_j$ for $i=j$ and so $\Psi\circ\Phi$ is the identity on $X$. Consequently $\Psi=\Phi^{-1}$ and $\Phi$ is a GPEA-isomorphism. According to the previous theorem, we may consider $E$ as a direct sum $E=\pi_1(E)\oplus\pi_2(E)\oplus\dots\oplus\pi_n(E)$ whenever $\pi_i$ are pairwise disjoint elements of ${{\Gamma\!\sb{\rm ex}}}(E)$ and $\bigvee_{i=1}^n\pi_i=1$. In particular, $E=\pi(E)\oplus\pi\,'(E)$ for every $\pi$ in the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$. \[th:CentId=piE\] If $S\subseteq E$, then the following statements are equivalent: - $S$ is a central ideal [(]{}direct summand[)]{} of $E$. - There exists $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $S=\pi(E)$. Assume that $E=S\oplus S'$. We define for $e\in E$: $\pi e:=s$ where $e= s\oplus t$, $s\in S$, $t\in S'$. Then $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Indeed, (EXC1) and (EXC2) hold trivially and since $s\leq s\oplus t$, (EXC3) also holds. If $e,f\in E$ are such that $\pi e=e$ and $\pi f=0$, then $e\in S$ and $f\in S'$, thus $e\perp f$ and so (EXC4) holds too. If, on the other hand, $\pi\in {{\Gamma\!\sb{\rm ex}}}(E)$ and $S=\pi(E)$, then $E=S\oplus\pi\,'(E)$ and so $S$ is a central ideal. \[co:piEnormal\] If $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$, then $\pi(E)$ is a normal ideal in $E$. By Theorem \[th:CentId=piE\], $\pi(E)$ is a central ideal in $E$, and by Proposition \[pr:DirSumNormal\], every central ideal in $E$ is normal. \[co:CIposet\] Let us partially order the set $C$ of all central ideals [(]{}direct summands[)]{} of $E$ by inclusion. Then there is an order isomorphism between ${{\Gamma\!\sb{\rm ex}}}(E)$ and $C$ given by: $\pi\leftrightarrow S$ iff $\pi(E)=S$. Moreover, if $\pi(E)=S$, then $\pi\,'(E)$ is the direct summand $S'$ of $E$ that is complementary to $S$. \[th:PtwisePi\] Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and let $(e_i)_{i\in I}$ be a family of elements in $E$. Then[:]{} - If $\bigvee_{i\in I} e_i$ exists in $E$, then so does $\bigvee_ {i\in I} \pi e_i$ and $\pi(\bigvee_{i\in I} e_i)=\bigvee_{i\in I} \pi e_i$. - If $I\not=\emptyset$ and $\bigwedge_{i\in I} e_i$ exists in $E$, then so does $\bigwedge_{i\in I} \pi e_i$ and $\pi(\bigwedge_{i\in I} e_i)= \bigwedge_{i\in I} \pi e_i$. - If $(e_i)_{i\in I}$ is orthosummable, then so is $(\pi e_i) _{i\in I}$ and $\pi(\oplus_{i\in I} e_i)=\oplus_{i\in I}\pi e_i$. \(i) Put $e:=\bigvee_{i\in I} e_i$. As $e_i\leq e$, we also have $\pi e_i \leq\pi e$ for all $i\in I$. Now suppose that $\pi e_i\leq f$ for all $i\in I$. Then $\forall i\in I$: $\pi e_i=\pi (\pi e_i)\leq\pi f$. But we also have $\pi\,' e_i\leq\pi\,' e$ for all $i\in I$. So by (vii) and (viii) in Theorem \[th:EXCprop\], $e_i=\pi e_i\oplus\pi\,' e_i=\pi e_i\vee\pi\,' e_i\leq \pi f\vee\pi\,' e=\pi f\oplus\pi\,' e$ for all $i\in I$. Thus $e\leq\pi f \oplus\pi\,' e$ so $\pi e\leq\pi f\oplus\pi (\pi\,' e)=\pi f\leq f$. Hence $\pi e=\bigvee_{i\in I} \pi e_i$. \(ii) Put $e:=\bigwedge_{i\in I} e_i$. As $e\leq e_i$, we have $\pi e \leq\pi e_i$ for all $i\in I$. Suppose $f\in E$ with $f\leq\pi e_i$ for all $i\in I$. As $I\not=\emptyset$, Theorem \[th:EXCprop\] (iii) implies that $f=\pi f$. Because $\pi e_i\leq e_i$, we have $f\leq e_i$ for all $i\in I$. Therefore $f\leq e$ and $\pi f=f\leq \pi e$. \(iii) For any finite subset $F$ of $I$, as $\pi$ is a GPEA-endomorphism, $\pi(\oplus_{i\in F}e_i)=\oplus_{i\in F}\pi e_i$. As $\oplus_{i\in I} \pi e_i=\bigvee_{F}\oplus_{i\in F}\pi e_i=\bigvee_{F}\pi(\oplus_{i\in F} e_i)=\pi\bigvee_{F} (\oplus_{i\in F} e_i)=\pi(\bigvee_{i\in I} e_i)$, the desired result follows from (i). The center of a GPEA {#sc:CenterGPEA} ==================== \[df:Center\] An element $c\in E$ is *central* iff for every $a, b\in E$, the following hold: - There exist $a_1, a_2\in E$ such that $a_1\leq c$, $a_2\oplus c$ exists and $a=a_1\oplus a_2$. - If $a\leq c$ and if $b\oplus c$ exists, then $a\perp b$. - If $a,b\leq c$ and $a\oplus b$ exists, then $a\oplus b\leq c$. - If $a\oplus c$, $b\oplus c$ and $a\oplus b$ exist, then $a \oplus b\oplus c$ exists. We denote the set of all central elements of the GPEA $E$ by $\Gamma(E)$. \[le:CentProp\] Let $a,x,y\in E$ and let $c\in\Gamma(E)$. Then[:]{} 1. The elements $a_1$ and $a_2$ in [(]{}C1[)]{} of Definition \[df:Center\] are unique and $a_1\perp a_2$. 2. $\forall\, a\in E$, $a\oplus c$ exists iff $a\perp c$ iff $c\oplus a$ exists. 3. If $x\oplus y$ exists in $E$ and at least one of the elements $x$, $y$ is central, then $x\perp y$. \(i) Suppose that $a\sb{1},a\sb{2},b\sb{1},b\sb{2}\in E$ with $a=a\sb{1}\oplus a\sb{2}=b\sb{1}\oplus b\sb{2}$, where $a_1,b_1 \leq c$ and both $a_2\oplus c$ and $b_2\oplus c$ exist. Then by (C2), we have $a_1\perp a_2$ and $b_1\perp b_2$, whence $a=a_1 \oplus a_2=a_2\oplus a_1=b_1\oplus b_2=b_2\oplus b_1$. As $a\sb{1}\leq c$, there exists $d\in E$ such that $a\sb{1}\oplus d=c$ and we have $a\sb{2}\oplus c=a\sb{2}\oplus a\sb{1}\oplus d=b\sb{2}\oplus b \sb{1}\oplus d$. Since $b\sb{1},d\leq c$, (C3) implies that $b \sb{1}\oplus d\leq c$, whence $a\sb{2}\oplus c\leq b\sb{2}\oplus c$, and it follows by cancellation that $a\sb{2}\leq b\sb{2}$. By symmetry, $b\sb{2}\leq a\sb{2}$, so $a\sb{2}=b\sb{2}$, and therefore $a\sb{1} =b\sb{1}$ by cancellation. \(ii) If $a\oplus c$ exists, then as $c\leq c$, we have $a\perp c$ by (C2). As $a\perp c$, then $c\oplus a$ exists. Finally, suppose that $c\oplus a$ exists. Then by (C1), there exist $d\sb{1},d\sb{2}\in E$ with $c\oplus a=d\sb{1}\oplus d\sb{2}$, where $d\sb{1}\leq c$ and $d\sb{2}\oplus c$ exists. As $c\leq c$ and $d\sb{2}\oplus c$ exists, (C2) implies that $c\perp d\sb{2}$. Also, by part (i), $d\sb{1}\perp d \sb{2}$, and since $d\sb{1}\leq c$, we have $c\oplus a=d\sb{1}\oplus d \sb{2}=d\sb{2}\oplus d\sb{1}\leq d\sb{2}\oplus c=c\oplus d\sb{2}$, whence $a\leq d\sb{2}$ by cancellation. Thus, $a\leq d\sb{2}$ and $d\sb{2}\perp c$, so $a\oplus c$ exists by Lemma \[le:SlashProps\] (iii). Part (iii) follows immediately from (ii). \[th:centr\] If $c\in E$, then the following are equivalent: - $c$ is central, i.e., $c\in\Gamma(E)$. - $E[0,c]$ is a central ideal [(]{}direct summand[)]{} of $E$. - $E$ decomposes as a direct sum $E=E[0,c]\oplus\{f\in E: f\perp c\}$. \(i) $\Rightarrow$ (ii): If $c$ is central, then by (C3), $E[0,c]$ is an ideal. We prove that it is moreover a central ideal; that is, there exists another ideal, namely $E[0,c]':=\{e\in E: e\perp c\}$, such that $E=E[0,c]\oplus E[0,c]'$. By Definition \[df:Center\] and Lemma \[le:CentProp\] (ii), for every $e\in E$ there exist $e_1,e_2\in E$ such that $e=e_1\oplus e_2$, where $e_1\in E[0,c]$ and $e_2\in E[0,c]'$. It will be sufficient to show that $E[0,c]'$ is an ideal in $E$. If $d\leq e$ and $e\in E[0,c]'$, then by Lemma \[le:SlashProps\] (iii), $d\oplus c$ exists; whence, as $c\in\Gamma(E)$, we have $d\in E[0,c]'$. Finally, suppose that $e,f\in E[0,c]'$ and $e\oplus f$ exists. Then by (C4), $e \oplus f\oplus c$ exists, and again, as $c\in\Gamma(E)$, it follows that $e\oplus f\in E[0,c]'$. \(ii) $\Rightarrow$ (iii): If $E[0,c]$ is a central ideal in $E$, then there is an ideal $E[0,c]'$ such that $E=E[0,c]\oplus E[0,c]'$. Evidently, if $f \in E[0,c]'$, then $f\perp c$. Conversely, if $f\in E$ with $f\perp c$, then $f=s\oplus t$, where $s\leq c$ and $t\in E[0,c]'$. As $s\leq f$ and $f \perp c$, we get $s\perp c$ and since $E[0,c]$ is an ideal, $s\oplus c \leq c$, which entails $s=0$. Thus $f=t\in E[0,c]'$ and $E[0,c]'=\{f\in E: f\perp c\}$. \(iii) $\Rightarrow$ (i): Let $E=E[0,c]\oplus\{f\in E: f\perp c\}$. We prove (C1)–(C4). (C1) follows directly from the fact, that every $e\in E$ can be written as $e=e_1\oplus e_2$, where $e_1\in E[0,c]$ and $e_2\in\{f\in E: f\perp c\}$. To prove (C2), suppose that $a\leq c$ and $b\oplus c$ exists. Then, we can write $b\oplus c=e\sb{1}\oplus f\sb{1}$ where $e\sb{1}\in E[0,c]$, $f\sb{1}\in\{f\in E: f\perp c\}$, and $e\sb{1}\perp f\sb{1}$. Therefore, $b\oplus c=f\sb{1}\oplus e\sb{1}\leq f\sb{1}\oplus c$, so $b\leq f\sb{1}$ by cancellation; hence, since $\{f\in E: f\perp c\}$ is an ideal, it follows that $b\in\{f\in E: f\perp c\}$. Now we have $a\in E[0,c]$ and $b\in\{f\in E:f\perp c\}$, whence $a\perp b$, proving (C2). Because $E[0,c]$ is an ideal, (C3) follows immediately. For (C4), suppose $a\oplus c$, $b\oplus c$ and $a\oplus b$ all exist. As a consequence of (C2) and the fact that $c\leq c$, we have $a\perp c$ and $b\perp c$, i.e., $a,b\in\{f\in E:f\perp c\}$. Again, since $\{f\in E:f\perp c\}$ is an ideal, we infer that $a\oplus b\perp c$, so $a\oplus b\oplus c$ exists, proving (C4). \[df:pisbc\] If $c\in \Gamma(E)$, then by Theorems \[th:centr\] and \[th:CentId=piE\], there exists uniquely determined mapping in ${{\Gamma\!\sb{\rm ex}}}(E)$, henceforth denoted by $\pi_c$, such that $\pi_c(E)=E[0,c]$. \[co:pic\] Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Then the following statements are equivalent: 1. There exists a largest element $c\in\pi(E)$. 2. $\pi(E)=E[0,c]$. 3. $c\in\Gamma(E)$, $\pi=\pi_c$, and $\pi\,'(E)=\{f\in E:f\perp c\}$. Since $\pi(E)$ is an ideal in $E$, $\pi(E)=E[0,c]$ iff $c$ is the largest element in $\pi(E)$. The rest follows by Theorem \[th:CentId=piE\], Theorem \[th:centr\], and Definition \[df:pisbc\]. If $c\in\Gamma(E)$ and $d\in E$ with $c\leq d$, then there exists $x:=d\backslash c\in E$ with $x\oplus c=d$, and since $c\in\Gamma(E)$, it follows from Lemma \[le:CentProp\] (iii) that $x\perp c$, whence $c\oplus x=d$ also holds, i.e., $x=c/d$. Consequently, $d\ominus c= d\backslash c=c/d$ exists (Definition \[df:leqetc\]). In particular, $d\ominus c$ is defined for $c,d\in\Gamma(E)$ iff $c\leq d$, and if $c\leq d$, then by part (x) of the next theorem, $d\ominus c\in\Gamma(E)$ and we have $d=c\oplus(d\ominus c)=(d\ominus c)\oplus c$. We omit the proofs of the following two theorems as they can be obtained by easy modifications of the proofs of [@ExoCen Lemma 4.5, Theorem 4.6]. \[th:ceprop\] Let $c,d\in\Gamma(E)$, $e\in E$. Then[:]{} 1. $\pi_c e=e\wedge c$. 2. $\pi_c d=\pi_d c=c\wedge d$. 3. $e\wedge c=0\,\Leftrightarrow\, e\in(\pi_c)'(E)\,\Leftrightarrow\, e\perp c$. 4. $c\wedge d\in\Gamma(E)$ and $\pi_{c\wedge d}=\pi_c\wedge\pi_d$. 5. $c\wedge d=0\,\Leftrightarrow\,\pi_c\wedge\pi_d=0\,\Leftrightarrow\, c\perp d$. 6. If $c\perp d$, then $c\oplus d=c\vee d\in\Gamma(E)$ and $\pi_{c \oplus d}=\pi_{c\vee d}=\pi_c\vee\pi_d$. 7. $\pi_c$ is the smallest $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi c=c$. 8. If $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and $h\in E$, then $h\in\Gamma(E)$ iff $\pi e =e\wedge h$ for all $e\in E$, and in this case, $\pi=\pi_h$. 9. $c\leq d\,\Leftrightarrow\,\pi_c\leq\pi_d$. 10. If $c\leq d$, then $d\ominus c$ exists, $d\ominus c\in\Gamma(E)$ and $\pi_{d\ominus c}=\pi_d\wedge(\pi_c)'$. 11. $c\vee d$ exists in $E$, $c\vee d\in\Gamma(E)$ and $\pi_{c\vee d} =\pi_c\vee\pi_d$. \[th:centgea\] [(i)]{} $\{\pi_c: c\in\Gamma(E)\}$ is a sublattice of the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$, and as such, it is a generalized boolean algebra. [(ii)]{} $\Gamma(E)$ is a commutative lattice-ordered sub-GPEA (hence sub-GEA) of $E$. [(iii)]{} The mapping $c\mapsto\pi_c$ from $\Gamma(E)$ onto $\{\pi_c: c\in \Gamma(E)\}$ is a lattice isomorphism. [(iv)]{} $\Gamma(E)$ is a generalized boolean algebra, i.e., a distributive and relatively complemented lattice with smallest element $0$. [(v)]{} $E$ is a PEA iff $\{\pi_c: c\in\Gamma(E)\}={{\Gamma\!\sb{\rm ex}}}(E)$. If $\phi$ is a mapping defined on $E$ and $S\subseteq E$, then $\phi|\sb{S}$ denotes the restriction of $\phi$ to $S$. The proofs of parts (i)–(iv) of the next theorem are easy modifications of the proofs of [@CenGEA Theorem 4.13, (i)–(iv)]; part (v) follows as in the proof of ; and with the aid of part (v), part (vi) follows as in the proof of [@CenGEA Theorem 4.13 (v)]. \[th:mis\] Let $\xi, \pi \in {{\Gamma\!\sb{\rm ex}}}(E)$. Then[:]{} 1. $\xi|\sb{\pi(E)}\in {{\Gamma\!\sb{\rm ex}}}(\pi(E))$. 2. If $\tau \in {{\Gamma\!\sb{\rm ex}}}(\pi(E))$, then $\tau\circ\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. 3. $\xi\mapsto\xi|\sb{\pi(E)}$ is a surjective boolean homomorphism of ${{\Gamma\!\sb{\rm ex}}}(E)$ onto ${{\Gamma\!\sb{\rm ex}}}(\pi(E))$. 4. If $p\in \pi(E)$, then $\pi(E)[0,p]$ and $E[0,p]$ coincide both as sets and as pseudoeffect algebras. 5. If $p\in E$, then $\pi(E[0,p])=E[0,\pi p]=\pi(E)[0,\pi p]$. 6. $\Gamma(\pi(E))=\Gamma(E)\cap\pi(E)$. \[le:nova\] If $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ and $k\in E$, then $\pi|\sb{E[0,k]}\in{{\Gamma\!\sb{\rm ex}}}(E[0,k])$. We prove that $\pi|\sb{E[0,k]}$ satisfies (EXC1)–(EXC4) for the PEA $E[0,k]$. Let $a,b\in E[0,k]$. We have $\pi|\sb{E[0,k]}a=\pi a\leq a \leq k$, so $\pi|\sb{E[0,k]}\colon E[0,k]\to E[0,k]$. To prove (EXC1), suppose that $a\oplus\sb{k}b=a\oplus b\leq k$. Then $\pi|\sb{E[0,k]} (a\oplus\sb{k}b)=\pi(a\oplus b)=\pi(a)\oplus\pi(b)\leq a\oplus b\leq k$, so $\pi|\sb{E[0,k]}$ is a GPEA-endomorphism of $E[0,k]$. Conditions (EXC2) and (EXC3) hold trivially. To prove (EXC4), suppose that $\pi| \sb{E[0,k]}a=\pi a=a$ and $\pi|\sb{E[0,k]}b=\pi b=0$. Then $a\perp b$, so $a\oplus b=b\oplus a$. Also $\pi\,'b=b$, and by Lemma \[le:DisjointPiXi\] (i) with $\xi:=\pi\,'$, $a\oplus b=b\oplus a= a\vee b\leq k$. Therefore, $a\oplus\sb{k}b=a\oplus b=b\oplus a=b \oplus\sb{k}a$, i.e., $a$ is orthogonal to $b$ in $E[0,k]$, proving (EXC4). Central orthocompleteness {#sc:CO} ========================= \[df:GammaexOrghogonal\] We say that elements $e,f\in E$ are *${{\Gamma\!\sb{\rm ex}}}$-orthogonal* iff there are $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi\wedge\xi=0$, $\pi e=e$ and $\xi f=f$. More generally, an arbitrary family $(e_i)_{i\in I}$ in $E$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff there is a pairwise disjoint family $(\pi_i) _{i\in I}$ in ${{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi_i e_i=e_i$ for all $i\in I$. As is easily seen, elements $e,f\in E$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff there is a direct sum decomposition $E=S\oplus S'$ such that $e\in S$ and $f\in S'$. \[le:co\] [(i)]{} A finite family $(e_i)_{i=1}^n$ in $E$ is pairwise ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff it is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal and then it is orthogonal with $\oplus_{i=1}^n e_i=\bigvee_{i=1}^n e_i$. [(ii)]{} If an arbitrary family $(e_i)_{i\in I}\in E$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, then it is orthogonal and it is orthosummable iff its supremum exists in $E$, in which case $\oplus_{i\in I} e_i=\bigvee_{i\in I} e_i$. \(i) Clearly, a subfamily of a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. It is also clear from the definition, that every ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family is pairwise ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. We prove both the converse and orthogonality by induction on $n$. For $n=1$ the assertion obviously holds. Suppose now the statement holds for $(n-1)$ elements, $n>1$ and assume that $(e_i)_{i=1}^n$ is a pairwise ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family. Then by the induction hypotheses, $(e_i)_{i=1}^{n-1}$ is orthogonal, $\oplus_{i=1}^{n-1} e_i=\bigvee_{i=1} ^{n-1}e_i$, and there exist pairwise disjoint mappings $\xi_i\in{{\Gamma\!\sb{\rm ex}}}(E)$ with $\xi_i e_i=e_i$ for $i=1,2,\ldots ,n-1$. Moreover, $e_i$ and $e_n$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal for $i=1,2,\ldots,n-1$; hence there exist $\alpha_i, \beta_i\in{{\Gamma\!\sb{\rm ex}}}(E)$ with $\alpha_i\wedge\beta_i=0$, $\alpha_i e_i=e_i$, and $\beta_i e_n=e_n$. For $i=1,2,\ldots ,n-1$, put $\pi_i:=\xi_i\wedge\alpha_i$ and put $\pi_n:=\bigwedge_{i=1}^{n-1}\beta_i$. Then $\pi_i\in{{\Gamma\!\sb{\rm ex}}}(E)$ are pairwise disjoint and $\pi_i e_i=e_i$ for $i=1,2,\ldots ,n$, so the family $(e_i)_{i=1}^n$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. We now put $\pi:=\bigvee_{i=1}^{n-1} \pi_i$ to get $\pi\wedge\pi_n=0$, $\pi(\oplus_{i=1}^{n-1} e_i)=\oplus_ {i=1}^{n-1}\pi e_i=\oplus_{i=1}^{n-1} e_i$, and $\pi_n e_n=e_n$; hence by Lemma \[le:DisjointPiXi\] (i), $(\oplus_{i=1}^{n-1} e_i)\perp e_n$ and $\oplus_{i=1}^n e_i=(\oplus_{i=1}^{n-1} e_i)\oplus e_n=(\vee_{i=1}^{n-1} e_i)\vee e_n=\vee_{i=1}^n e_i$. \(ii) If $(e_i)_{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, then every finite subfamily is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal and by (i), $\oplus_{i\in F}e_i=\bigvee_{i\in F}e_i$, where $F$ is any finite subset of $I$. Therefore $\bigvee_{i\in I}e_i= \bigvee_F(\bigvee_{i\in F}e_i)=\bigvee_F(\oplus_{i\in F} e_i)=\oplus_ {i\in I}e_i$. \[le:coce\] [(i)]{} $c,d\in\Gamma (E)$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff $\pi_c\wedge\pi_d =0$ iff $c\perp d$ iff $c\wedge d=0$. [(ii)]{} A family of central elements is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff it is orthogonal iff it is pairwise orthogonal iff it is pairwise disjoint. \(i) If $\pi_c\wedge\pi_d=0$, then $c$ and $d$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal by definition. If $c,d$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, then there exist $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi c=c$, $\xi d=d$ and $\pi\wedge\xi=0$. But $\pi_c\leq\pi$ and $\pi_d\leq\xi$ by Theorem \[th:centgea\] (vii), thus $\pi_c$ and $\pi_d$ are disjoint too. The remaining equivalences follow from Theorem \[th:ceprop\] (v). \(ii) If the family $(c_i)_{i\in I}$ of central elements in $E$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, then by Lemma \[le:co\] (ii) it is orthogonal. If it is orthogonal, then by the definition of orthogonality it is pairwise orthogonal. If it is pairwise orthogonal, then by Theorem \[th:ceprop\] (v) it is pairwise disjoint. Finally, suppose that $(c_i)_{i\in I}$ is pairwise disjoint. Then by Theorem \[th:ceprop\] (v) again, $(\pi\sb{c\sb{i}}) \sb{i\in I}$ is a pairwise disjoint family in ${{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi\sb{c\sb{i}}c\sb{i}=c\sb{i}$ for all $i\in I$, so $(c_i)_{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. \[de:COGPEA\] The generalized pseudo-effect algebra $E$ is *centrally orthocomplete* (COGPEA) iff it satisfies the following conditions: - Every ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family in $E$ is orthosummable, i.e. (Lemma \[le:co\] (ii)), it has a supremum) in $E$. - If $e\in E$ is such that $e\oplus e_i$ (resp. $e_i\oplus e)$ exists for every element of a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family $(e_i)_{i\in I} \subset E$, then $e\oplus(\oplus_{i\in I} e_i)$ (resp. $(\oplus_{i \in I}e_i)\oplus e$) exists in $E$. \[th:COGPEAp’wisedisj\] Let $E$ be a COGPEA and $(\pi_i)_{i\in I}$ a pairwise disjoint family in ${{\Gamma\!\sb{\rm ex}}}(E)$. Let $(e_i)_{i\in I}$, $(f_i)_{i\in I}$ be families of elements in $E$ such that $e_i\oplus f_i$ exists for all $i\in I$ and $e_i,f_i\in \pi_i(E)$. Then[:]{} 1. $(e_i)_{i\in I}$, $(f_i)_{i\in I}$, and $(e_i\oplus f_i)_{i\in I}$ are ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, hence orthosummable. 2. $\oplus_{i\in I} e_i=\bigvee_{i\in I} e_i$, $\oplus_{i\in I} f_i= \bigvee_{i\in I} f_i$ and $\oplus_{i\in I}(e_i \oplus f_i)=\bigvee_{i\in I} (e_i\oplus f_i)$. 3. $(\oplus_{i\in I} e_i)\oplus(\oplus_{i\in I} f_i)$ exists. 4. $(\oplus_{i\in I} e_i)\oplus(\oplus_{i\in I} f_i)=\oplus_{i\in I} (e_i\oplus f_i)=\bigvee_{i\in I}(e_i\oplus f_i)$. Since $e_i$, $f_i$ belong to $\pi_i(E)$ for every $i\in I$, so does $e_i \oplus f_i$. Thus (i) follows directly from (CO1) and the definition of ${{\Gamma\!\sb{\rm ex}}}$-orthogonality, and (ii) is implied by Lemma \[le:co\] (ii). \(iii) Put $e:=\oplus_{i\in I} e_i=\bigvee_{i\in I} e_i$ and $f:= \oplus_{i\in I}f_i=\bigvee_{i\in I} f_i$. By hypotheses $e_i\oplus f_i$ exists for every $i\in I$, and for $i\not=j$, $e_i\oplus f_j$ also exists by Lemma \[le:DisjointPiXi\] (i). Applying (CO2) we find that $e_i\oplus f$ exists for all $i\in I$, and applying (CO2) once more we conclude that $e\oplus f$ exists too. \(iv) As $e\oplus f$ exists, so does $e\sb{i}\oplus f$ for every $i\in I$, and therefore by Lemma \[le:oplusdist\], $e\sb{i}\oplus f=\bigvee\sb{j\in I} (e\sb{i}\oplus f\sb{j})$. Therefore a second application of Lemma \[le:oplusdist\] yields $$\label{eq:p'wisedisj01} e\oplus f=(\bigvee\sb{i\in I}e_i)\oplus f=\bigvee\sb{i\in I}(e\sb{i} \oplus f)=\bigvee\sb{i\in I}\bigvee\sb{j\in I}(e\sb{i}\oplus f\sb{j}) =\bigvee\sb{i,j\in I}(e\sb{i}\oplus f\sb{j}).$$ Also, by Lemma \[le:DisjointPiXi\] (i), for all $i,j\in I$, $$\label{eq:p'wisedisj02} i\not=j\Rightarrow e\sb{i}\oplus f\sb{j}=e\sb{i}\vee f\sb{j} \leq(e\sb{i}\oplus f\sb{i})\vee(e\sb{j}\oplus f\sb{j})\leq \bigvee\sb{i\in I}(e\sb{i}\oplus f\sb{i}).$$ Combining (\[eq:p’wisedisj01\]) and (\[eq:p’wisedisj02\]), and using (ii) above, we conclude that $e\oplus f=\bigvee\sb{i\in I}(e\sb{i}\oplus f \sb{i})=\oplus\sb{i\in I}(e\sb{i}\oplus f\sb{i})$. \[th:DisjSup\] If $E$ is a COGPEA and $(\pi_i)_{i\in I}$ is a pairwise disjoint family of elements in ${{\Gamma\!\sb{\rm ex}}}(E)$, then the supremum $\bigvee_{i\in I} \pi_i$ exists in the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$ and for every $e\in E$,  $(\bigvee \sb{i\in I}\pi\sb{i})e=\bigvee\sb{i\in I}\pi\sb{i}e=\oplus\sb{i\in I} \pi\sb{i}e$. Let $e,f\in E$ and $i,j\in I$. The family $(\pi_i)_{i\in I}$ is pairwise disjoint and $\pi_i(\pi_i e)=\pi_i e$ for every $i\in I$, whence $(\pi_i e) _{i\in I}$ is a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family in $E$. Thus by (CO1) $(\pi_i e) _{i\in I}$ is orthosummable with $\oplus_{i\in I}\pi_i e=\bigvee_{i\in I} \pi_i e$ (Lemma \[le:co\] (ii)). We define $\pi\colon E\to E$ by $\pi e :=\bigvee_{i\in I}\pi_i e=\oplus_{i\in I}\pi_i e$. It will be sufficient to prove that $\pi$ is in ${{\Gamma\!\sb{\rm ex}}}(E)$ and that it is the supremum of $(\pi_i)_{i\in I}$ in ${{\Gamma\!\sb{\rm ex}}}(E)$. Suppose $e\oplus f$ exists, so that $\pi\sb{i}(e\oplus f)=\pi\sb{i}e \oplus\pi\sb{i}f$ for all $i\in I$. In Theorem \[th:COGPEAp’wisedisj\], put $e\sb{i}:=\pi\sb{i}e$ and $f\sb{i}:=\pi\sb{i}f$ for all $i\in I$ to infer that $\pi e\oplus \pi f$ exists and $$\pi e\oplus \pi f=(\oplus\sb{i}\pi\sb{i}e)\oplus(\oplus\sb{i}\pi\sb{i}f) =\oplus\sb{i}(\pi\sb{i}e\oplus\pi\sb{i}f)=\oplus\sb{i}(\pi\sb{i}(e\oplus f)) =\pi(e\oplus f),$$ which proves that $\pi$ satisfies (EXC1). We also have $\pi_i(\pi e)=\pi_i \bigvee_{j\in I}\pi_j e=\bigvee_{j\in I}\pi_i\pi_j e=\pi_i e$ by Theorem \[th:PtwisePi\] (i), whence $\pi(\pi e)=\bigvee_{i\in I}\pi_i(\pi e)= \bigvee_{i\in I}\pi_i e=\pi e$, proving (EXC2). Moreover, as $\pi_i e \leq e$ for all $i\in I$, it follows that $\pi e=\bigvee_{i\in I}\pi_i e \leq e$ and therefore (EXC3) holds. To prove (EXC4), suppose that $\pi e =e$ and $\pi f=0$. Then $\bigvee_{i\in I}\pi_i f=0$, so $\pi_i f=0$ for all $i\in I$. As $\pi_i(\pi_i e)=\pi_i e$, (EXC4) implies that $\pi_i e \perp f$ for every $i\in I$. But then, by (CO2), $e=\pi e\perp f$, and (EXC4) holds for $\pi$ too. Evidently, $\pi_i e\leq\pi e$ for every $e\in E$, whence $\pi_i\leq\pi$ for all $i\in I$. Also, if $\pi_i\leq\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$ for all $i\in I$, then $\pi_i e\leq\xi e$, so $\pi e=\bigvee_{i\in I}\pi_i e\leq\xi e$ for all $e\in E$ and thus $\pi\leq\xi$. So $\pi=\bigvee_{i\in I}\pi_i$. Since a boolean algebra is complete iff every pairwise disjoint subset has a supremum, Theorem \[th:DisjSup\] has the following corollary. \[co:completeboo\] The exocenter ${{\Gamma\!\sb{\rm ex}}}(E)$ of a COGPEA $E$ is a complete boolean algebra. We may now extend Theorem \[th:DisjSup\] in the same way as in [@ExoCen Theorem 6.9] for an arbitrary family $(\pi_i)_{i\in I}$ in the complete boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$. \[th:arbp’wisesup\] Suppose that $E$ is a COGPEA, let $(\pi_i)_{i\in I}$ be a family in ${{\Gamma\!\sb{\rm ex}}}(E)$, and let $e\in E$. Then[:]{} [(i)]{} $\bigvee_{i\in I}\pi_i e$ exists in $E$ and $(\bigvee_{i\in I}\pi_i)e=\bigvee_{i\in I}\pi_i e$. [(ii)]{} If $I\not=\emptyset$, then $\bigwedge_{i\in I}\pi_i e$ exists in $E$ and $(\bigwedge_{i\in I}\pi_i)e=\bigwedge_{i\in I}\pi_i e$. The proof of the next theorem, which extends Theorem \[th:finitecartesianprod\] to arbitrary direct sums, is analogous to the proof of [@ExoCen Theorem 6.10]. \[th:arbCartProd\] Suppose that $E$ is a COGPEA, let $(\pi_i)_{i\in I}$ be a pairwise disjoint family in the complete boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$ with $\pi:=\bigvee_{i\in I}\pi_i$, and consider the cartesian product $X:=$[$\times$]{}$_{i\in I}\pi_i(E)$. Then each element in $X$ is a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal [(]{}hence orthosummable[)]{} family $(e_i)_{i\in I}$ and $\oplus_{i\in I} e_i=\bigvee_{i\in I} e_i$. Define the mapping $\Phi:X\rightarrow\pi(E)$ by $\Phi((e_i)_{i\in I}):=\oplus_{i\in I} e_i$. Then $\Phi$ is a GPEA-isomorphism of $X$ onto $\pi(E)$ and if $e\in\pi(E)$, then $\Phi^{-1} e=(\pi_i e)_{i\in I}\in X$. \[cor:cop\] Let $E$ be a COGPEA, let $(p_i)_{i\in I}$ be a nonempty ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family in $E$ with $p:=\bigvee_{i\in I}p_i$, let $(\pi_i)_{i\in I}$ be a corresponding family of pairwise disjoint mappings in ${{\Gamma\!\sb{\rm ex}}}(E)$ such that $p_i=\pi_i p_i$ for all $i\in I$, and let $X$ be the cartesian product $X:=$[$\times$]{}$ _{i\in I} E[0,p_i]$. Then[: (i)]{} If $(e_i)_{i\in I}\in X$, then $e_i=\pi_i e _i$ for all $i\in I$, so $(e_i)_{i\in I}$ is a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, hence orthosummable family in $E$. [(ii)]{} If $(e_i)_{i\in I}\in X$ with $e:= \oplus_{i\in I} e_i$, then $\pi_i e=e_i$ for all $i\in I$. In particular, $\pi_i p=p_i$ for all $i\in I$. [(iii)]{} If $e\in E[0,p]$, then $\pi_i e =e\wedge p_i$ for all $i\in I$, $(\pi_i e)_{i\in I}\in X$ and $\bigvee_ {i\in I}\pi_i e=e$. [(iv)]{} The mapping $\Phi: X\rightarrow E[0,p]$ defined by $\Phi((e_i)_{i\in I}):=\oplus_{i\in I} e_i=\bigvee_{i\in I} e_i$ is a PEA-isomorphism of $X$ onto $E[0,p]$ and $\Phi^{-1} (e)=(\pi_i e)_i\in I\in X$ for all $e\in E[0,p]$. The following theorem can also be proved using the same arguments as in the proof of [@ExoCen Theorem 6.11] \[th:COGPEAcenter\] Suppose that $E$ is a COGPEA and $(c_i)_{i\in I}$ is a family of elements in the center $\Gamma(E)$ of $E$. Then[:]{} [(i)]{} If $I\not=\emptyset$, then $c:=\bigwedge_{i\in I} c_i$ exists in $E$, $c\in\Gamma(E)$, $\pi_c=\bigwedge _{i\in I}\pi_{c_i}$ and $c$ is the infimum of $(c_i)_{i\in I}$ as calculated in $\Gamma(E)$. [(ii)]{} If $(c_i)_{i\in I}$ is bounded above in $E$, then $d:=\bigvee_{i\in I} c_i$ exists in $E$, $d\in\Gamma(E)$, $\pi_d=\bigvee _{i\in I}\pi_{c_i}$ and $d$ is the supremum of $(c_i)_{i\in I}$ as calculated in $\Gamma(E)$. The next theorem extends the results obtained for centrally orthocomplete GEAs in [@CenGEA Lemma 7.5, Theorem 7.6]. Here we give a simplified proof. \[th:largestandboo\] Let $E$ be a COGPEA. Then[:]{} [(i)]{} There exists a largest element $u\in\Gamma(E)$ and $\Gamma(E)\subseteq \pi_u(E)=E[0,u]$. [(ii)]{} The center $\Gamma(E)$ is a complete boolean algebra. \(i) We apply Zorn’s lemma to obtain a maximal pairwise disjoint family of nonzero elements $(c_i)_{i\in I}\subseteq\Gamma(E)$. (Note that $(c_i)_{i\in I}$ could be the empty family.) By Lemma \[le:coce\], $(c_i)_{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal, and since $E$ is a COGPEA, $u:=\bigvee_{i\in I}c_i=\oplus_{i\in I}c_i$ exists in $E$. Thus the family $(c_i)_{i\in I}$ is bounded above by $u$ in $E$, and we infer from Theorem \[th:COGPEAcenter\] (ii) that $u\in\Gamma(E)$. Let $c\in \Gamma(E)$. Working in the generalized boolean algebra $\Gamma(E)$ (Theorem \[th:centgea\] (iv)), we have $c=(c\wedge u) \vee d$, where $d:=c\ominus(c\wedge u)\in\Gamma(E)$. As $d\wedge u=0$ and $c\sb{i}\leq u$, it follows that $d\wedge c\sb{i}=0$ for all $i\in I$, whence $d=0$ by the maximality of $(c_i)_{i\in I}$, and it follows that $c=c\wedge u\leq u$. Consequently, $\pi\sb{c}\leq\pi\sb{u}$, and therefore $c\in E[0,c]=\pi_c(E)\subseteq \pi_u(E)=E[0,u]$. \(ii) Since the generalized boolean algebra $\Gamma(E)$ has a unit (largest element), it is a boolean algebra, and it is complete by Theorem \[th:COGPEAcenter\]. \[th:centerless\] Let $u$ be the unit [(]{}largest element[)]{} in the complete boolean algebra $\Gamma(E)$ of the COGPEA $E$. Then[:]{} 1. The PEA $E[0,u]=\pi_u(E)$ is a direct summand of $E$ and the complementary direct summand is $(\pi_u)'(E)=\{f\in E:f\perp u\}= \{e\ominus(u\wedge e):e\in E\}$. 2. The center of $E[0,u]$ is $\Gamma(E)$, the complementary direct summand $(\pi_u)'(E)$ is centerless [(]{}i.e., its center is $\{ 0\}$[)]{}, and no nonzero direct summand of $(\pi_u)'(E)$ is a PEA. 3. If $E=H\oplus K$ where the direct summand $H$ is a PEA and $K$ is centerless, then $H=E[0,u]$ and $K=\{ f\in E:f\perp u\}$. As $u\in\Gamma(E)$, we have $\pi\sb{u}\in{{\Gamma\!\sb{\rm ex}}}(E)$ as per Definition \[df:pisbc\], by Theorem \[th:centr\], the PEA $E[0,u]= \pi\sb{u}(E)$ is a direct summand of $E$, and its complementary direct summand is $(\pi_u)'(E)=\{f\in E:f\perp u\}$. If $e\in E$, then by Theorem \[th:ceprop\] (i), $\pi\sb{u}e=u\wedge e$, whence $(\pi \sb{u})'e=\pi\sb{u}e/e=e\backslash\pi\sb{u}e=e\ominus\pi\sb{u}e =e\ominus(u\wedge e)$, and it follows that $(\pi_u)'(E)=\{e\ominus (u\wedge e):e\in E\}$. \(ii) As a consequence of Theorem \[th:largestandboo\] (i), we have $\Gamma(E)\subseteq \pi_u(E)=E[0,u]$. Therefore, by Theorem \[th:mis\] (vi), $\Gamma(E[0,u])=\Gamma(\pi_u(E))=\Gamma(E)\cap\pi_u(E)=\Gamma(E)$. Also by Theorem \[th:mis\] (vi), $\Gamma((\pi_u)'(E))=\Gamma(E)\cap (\pi_u)'(E)\subseteq \pi_u(E)\cap(\pi_u)'(E)=\{0\}$. \(iii) Assume the hypotheses of (iii). By Theorem \[th:CentId=piE\], there exists $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ with $\pi(E)=H$, so $K=\pi\,'(E)$. Since $H$ is a PEA, there is a largest element $c\in H=\pi(E)$; hence by Corollary \[co:pic\], $H=\pi(E)=E[0,c]$, $c\in\Gamma(E)$, $\pi= \pi\sb{c}$, and $K=(\pi\sb{c})'(E)=\{f\in E:f\perp c\}$. Also, since $u$ is the largest element in $\Gamma(E)$, we have $c\leq u$, whence $u\ominus c\in\Gamma(E)$ by Theorem \[th:ceprop\] (x). Furthermore, $(u\ominus c)\perp c$, therefore $u\ominus c\in K$, and by Theorem \[th:mis\] (vi) we have $u\ominus c\in\Gamma(E)\cap K=\Gamma(E) \cap(\pi_c)'(E)=\Gamma((\pi_c)'(E))=\Gamma(K)$. Consequently, as $K$ is centerless, $u\ominus c=0$, so $c=u$, $H=E[0,u]$, and $K= \{f\in E:f\perp u\}$. The exocentral cover {#sc:ExoCenCover} ==================== \[df:ExoCenCover\] If $e\in E$, and if there is the smallest mapping in the set $\{\pi\in{{\Gamma\!\sb{\rm ex}}}(E): \pi e=e\}$, we will refer to it as *exocentral cover* of $e$ and denote it by $\gamma_e$. If every element of $E$ has an exocentral cover, we say that the family $(\gamma_e)_{e\in E}$ is the *exocentral cover system* for $E$, and in this case, we also denote the set of all mappings in the exocentral cover system by $\Theta\sb{\gamma}:=\{\gamma\sb{e}:e\in E\}$. (We note that it is quite possible to have $\gamma\sb{e}=\gamma\sb{f}$ with $e\not=f$.) If $E$ is a COGPEA, then the exocentral cover $\gamma\sb{e}$ exists for every $e\in E$ and $\gamma_e=\bigwedge\{\pi\in{{\Gamma\!\sb{\rm ex}}}(E):\pi e=e\}\in{{\Gamma\!\sb{\rm ex}}}(E)$. Let $e\in E$ and put $\gamma:=\bigwedge\{\pi:\pi\in{{\Gamma\!\sb{\rm ex}}}(E),\pi e=e\}$. As the identity mapping $1$ is in the set $\{\pi\in{{\Gamma\!\sb{\rm ex}}}(E):\pi e=e\}$, it is nonempty, and by Theorem \[th:arbp’wisesup\] (ii), $$\gamma e=(\bigwedge\{\pi:\pi\in{{\Gamma\!\sb{\rm ex}}}(E),\pi e=e\})e=\bigwedge\{\pi e:\pi\in {{\Gamma\!\sb{\rm ex}}}(E),\pi e=e\}=e.$$ Therefore, $\gamma$ is the smallest mapping in the set $\{\pi\in{{\Gamma\!\sb{\rm ex}}}(E): \pi e=e\}$, so $\gamma\sb{e}=\gamma$. \[th:ExCovProp\] Let $E$ be a COGPEA and $e,f\in E$. Then[:]{} [(i)]{} $\gamma\sb{0} =0$. [(ii)]{} $\gamma\sb{e}e=e$. [(iii)]{} $e\leq f\Rightarrow \gamma_e \leq\gamma_f$. [(iv)]{} If $e\oplus f$ exists, then $\gamma_{e\oplus f} =\gamma_e\vee\gamma_f$. [(v)]{} $\gamma_{\gamma_e f}=\gamma_e\circ \gamma_f=\gamma_e\wedge\gamma_f$. [(vi)]{} $\gamma_{(\gamma_e)'f}= (\gamma_e)'\circ\gamma_f=(\gamma_e)'\wedge\gamma_f$. [(vii)]{} $\gamma\sb{e}\wedge\gamma\sb{f}\in\Theta\sb{\gamma}$. [(viii)]{} $(\gamma_e)'\wedge\gamma_f\in\Theta\sb{\gamma}$. Parts (i) and (ii) are obvious from Definition \[df:ExoCenCover\]. \(iii) If $e\leq f=\gamma_f f$, then by Theorem \[th:EXCprop\] (iii), $\gamma_f e=e$. But since $\gamma_e$ is the smallest mapping in ${{\Gamma\!\sb{\rm ex}}}(E)$ that fixes $e$, it follows that $\gamma_e\leq\gamma_f$. \(iv) Suppose that $e\oplus f$ exists. We have $(\gamma_e\vee\gamma_f)e =\gamma_e e\vee\gamma_f e=e\vee\gamma_f e=e$ because $\gamma_f e\leq e$. Similarly $(\gamma_e\vee\gamma_f)f=f$. Thus $(\gamma_e\vee\gamma_f) (e\oplus f)=(\gamma_e\vee\gamma_f)e\oplus(\gamma_e\vee\gamma_f)f=e\oplus f$, and so $\gamma_{e\oplus f}\leq\gamma_e\vee\gamma_f$. On the other hand, $e,f\leq e\oplus f$, so by (iii), $\gamma_e,\gamma_f\leq\gamma_{e\oplus f}$ and thus $\gamma_e\vee\gamma_f\leq\gamma_{e\oplus f}$. \(v) Since $\gamma_e\in{{\Gamma\!\sb{\rm ex}}}(E)$, $\gamma_e(\gamma_e f)=\gamma_e f$ and $\gamma_f(\gamma_e f)=\gamma_e(\gamma_f f)=\gamma_e f$. Therefore $\gamma_{\gamma_e f}\leq\gamma_e\wedge\gamma_f=\gamma_e\circ\gamma_f$. To prove the reverse inequality, consider $f=\gamma_e f\oplus(\gamma_e)' f$ and (iv) to obtain $\gamma_f=\gamma_{\gamma_e f}\vee\gamma_{(\gamma_e)' f}$. Also $(\gamma_e)'((\gamma_e)'f)=(\gamma_e)'f$, and as $\gamma \sb{(\gamma\sb{e})'f}$ is the smallest mapping in ${{\Gamma\!\sb{\rm ex}}}(E)$ that fixes $(\gamma\sb{e})'f$, we have $\gamma_{(\gamma_e)'f}\leq (\gamma_e)'$. But then $\gamma_e\wedge\gamma_{(\gamma_e)'f}=0$ and thus $\gamma_e\circ \gamma_f=\gamma_e\wedge\gamma_f=(\gamma_e\wedge\gamma_{\gamma_e f}) \vee(\gamma_e\wedge\gamma_{(\gamma_e)' f})=\gamma_e\wedge\gamma_{\gamma_e f} \leq\gamma_{\gamma_e f}$. \(vi) By (v), $(\gamma_e)'\wedge\gamma_{\gamma_e f}=(\gamma_e)'\wedge \gamma_e\wedge\gamma_f=0$. Also, as in the proof of (v), we have $\gamma_f=\gamma_{(\gamma_e)'f}\vee\gamma_{\gamma_e f}$ and $\gamma \sb{(\gamma\sb{e})'f}\leq(\gamma\sb{e})'$. Therefore, $(\gamma_e)' \wedge\gamma_f=[(\gamma_e)'\wedge\gamma_{(\gamma_e)'f}]\vee[(\gamma_e)' \wedge\gamma_{\gamma_e f}]=\gamma_{(\gamma_e)'f}\vee 0=\gamma_ {(\gamma_e)'f}$. Parts (vii) and (viii) follow immediately from parts (v) and (vi). \[co:ThetasbgammaGBA\] With the partial order inherited from ${{\Gamma\!\sb{\rm ex}}}(E)$, $\Theta\sb{\gamma}= \{\gamma\sb{e}:e\in E\}$ is a generalized boolean algebra. By [@HDandTD Theorem 3.2] with $B:={{\Gamma\!\sb{\rm ex}}}(E)$ and $L:=\Theta \sb{\gamma}$, it will be sufficient to prove that, for all $e,f \in E$, (i) $\Theta\sb{\gamma}\not=\emptyset$, (ii) $e,f\in E\Rightarrow (\gamma\sb{e})'\wedge\gamma\sb{f}\in\Theta\sb{\gamma}$, and (iii) $\gamma\sb{e}\wedge\gamma\sb{f}=0\Rightarrow\gamma\sb{e} \vee\gamma\sb{f}\in\Theta\sb{\gamma}$. Condition (i) is obvious and (ii) follows from Theorem \[th:ExCovProp\] (viii). To prove (iii), suppose that $\gamma\sb{e}\wedge\gamma\sb{f}=0$. Then, as $e=\gamma\sb{e}e$ and $f=\gamma\sb{f}f$, Lemma \[le:DisjointPiXi\] (i) implies that $e\perp f$; hence by Theorem \[th:ExCovProp\] (iv), $\gamma\sb{e}\vee\gamma\sb{f}=\gamma\sb{e\oplus f}\in\Theta\sb{\gamma}$, proving (iii). The following definition, originally formulated for a generalized effect algebra (GEA) [@ExoCen Definition 7.1] as a generalization of the notion of a hull mapping on an effect algebra [@HandD Definition 3.1], extends to the GPEA $E$ the notion of a so-called hull system. A family $(\eta\sb{e})\sb{e\in E}$ is a *hull system* for $E$ iff (1) $\eta_0=0$, (2) $e\in E\,\Rightarrow\,\eta_e e=e$, and (3) $e,f\in E\,\Rightarrow\,\eta_{\eta_e f}=\eta_e\circ\eta_f$. If $(\eta\sb{e})\sb{e\in E}$ is a hull system for $E$, then an element $e\in E$ is *$\eta$-invariant* iff $\eta\sb{e}f=e\wedge f$ for all $f\in E$. \[th:gammahullsys\] If $E$ is a COGPEA, then $(\gamma\sb{e})\sb{e\in E}$ is a hull system for $E$, the center $\Gamma(E)$ is precisely the set of $\gamma$-invariant elements in $E$, and for $c\in\Gamma(E)$, $\gamma\sb{c}=\pi\sb{c}$. That $(\gamma\sb{e})\sb{e\in E}$ is a hull system for $E$ follows from parts (i), (ii), and (v) of Theorem \[th:ExCovProp\], and the remainder of the theorem follows from parts (i), (vii), and (viii) of Theorem \[th:ceprop\]. \[th:disjointgammasbei\] Let $E$ be a COGPEA and $(e_i)_{i\in I}\subseteq E$. Then the family $(e_i)_{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal iff $\gamma_{e_i}\wedge\gamma_ {e_j}=0$ for all $i,j\in I$, $i\not =j$. If $(\gamma\sb{e\sb{i}})\sb{i\in I}$ is pairwise disjoint, then since $\gamma\sb{e\sb{i}}e\sb{i}=e\sb{i}$, it follows that $(e\sb{i})\sb{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. Conversely, suppose that $(e\sb{i})\sb{i\in I}$ is ${{\Gamma\!\sb{\rm ex}}}$-orthogonal. Then there exists a pairwise disjoint family $(\pi_i)_ {i\in I}\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $\pi_i e_i=e_i$ for all $i\in I$. But then $\gamma_{e_i}\leq\pi_i$ for all $i\in I$, and therefore the family $(\gamma_{e_i})_{i\in I}$ is also pairwise disjoint. In view of Theorem \[th:disjointgammasbei\], a ${{\Gamma\!\sb{\rm ex}}}$-orthogonal family of elements of the COGPEA $E$ will also be called *$\gamma$-orthogonal*. Type determining sets {#sc:TDsets} ===================== \[df:fourclosures\] Let $E$ be a COGPEA and $Q,K\subseteq E$. Then we consider four closure operators on the set of all subsets $Q$ of $E$: - $[Q]_{\gamma}$ is the set of all orthosums (suprema) of $\gamma$-orthogonal families in $Q$, with the understanding that $[\emptyset]_{\gamma}=\{0\}$. - $Q^{\gamma}:=\{\gamma_e q: e\in E, q\in Q\}$. - $Q^{\downarrow}:=\bigcup_{q\in Q}E[0,q]$. - $Q'':=(Q')'$, where $Q':=\{e\in E: q\wedge e=0$ for all $q\in Q\}$. We say that - $K$ is *type-determining* (TD) set iff $K=[K]_{\gamma}= K^{\gamma}$. - $K$ is *strongly type-determining* (STD) set iff $K=[K]_ {\gamma}=K^{\downarrow}$. We note that $Q\subseteq Q''$, $P\subseteq Q\Rightarrow Q'\subseteq P'$, and $Q'=Q'''$. \[th:QK\] Let $E$ be a COGPEA and let $Q,K\subseteq E$. Then[:]{} [(i)]{} If $q\in [Q]_{\gamma}$, then there is a $\gamma$-orthogonal family $(q_i) _{i\in I}$ in $Q$ such that $q=\oplus_{i\in I}q_i=\bigvee_{i\in I}q_i$; moreover, if $e\leq q$, then $(e\wedge q_i)_{i\in I}$ is a $\gamma$-orthogonal family in $Q^{\downarrow}$ and $e=$$\oplus_{i\in I}(e\wedge q_i)=\bigvee_{i\in I} (e\wedge q_i)$. [(ii)]{} $[K^{\gamma}]_{\gamma}$ is the smallest TD subset of $E$ containing $K$. [(iii)]{} $[K^{\downarrow}]_{\gamma}$ is the smallest STD subset of $E$ containing $K$. [(iv)]{} $K'=(K')^{\downarrow}= (K^{\downarrow})'$ is STD. [(v)]{} $K'=([K^{\gamma}]_{\gamma})'= ([K^{\downarrow}]_{\gamma})'$. \(i) By the definition of $[Q]_{\gamma}$, there exists a family $(q_i) _{i\in I}$ in $Q$ such that $(\gamma_{q_i})_{i\in I}$ is a pairwise disjoint family in ${{\Gamma\!\sb{\rm ex}}}(E)$ and $q=\oplus_{i\in I}q_i=\bigvee_{i\in I} q_i$. By Theorem \[th:PtwisePi\] (i), for each $i\in I$, $\gamma_ {q_i}q=\gamma_{q_i}(\bigvee_{j\in I}q_j)=\bigvee_{j\in I}\gamma_{q_i}q_j =\bigvee_{j\in I}\gamma_{q_i}(\gamma_{q_j}q_j)=q_i$. Therefore, as $e\leq q$, we can apply Theorem \[th:EXCprop\] (iv) to obtain $\gamma_{q_i} e=e\wedge\gamma\sb{q\sb{i}}q=e\wedge q_i\in Q\sp{\downarrow}$. By Theorem \[th:ExCovProp\] (iii), $\gamma_{e\wedge q_i}\leq\gamma_{q_i}$, so the family $(e\wedge q_i)_{i\in I}$ is $\gamma$-orthogonal. Let us define $\pi:=\bigvee_{i\in I}\gamma_{q_i}$ in the complete boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$. Then by Theorem \[th:DisjSup\], $\pi q=\bigvee_{i\in I}\gamma_{q_i}q= \bigvee_{i\in I}q_i=q$, hence, as $e\leq q\in\pi(E)$, it follows by Theorems \[th:EXCprop\] (iii) and \[th:DisjSup\] that $e=\pi e=\bigvee_{i\in I} \gamma_{q_i}e=\bigvee_{i\in I}(e\wedge q_i)$. \(ii) From the definition it is clear that $[K^{\gamma}]_{\gamma}$ is contained in every TD set containing $K$. It is also easily seen that $K\subseteq[K^{\gamma}]_{\gamma}$ and $[[K^{\gamma}]_{\gamma}]_{\gamma} \subseteq [K^{\gamma}]_{\gamma}$. To prove that $([K^{\gamma}]_{\gamma})^ {\gamma}\subseteq [K^{\gamma}]_{\gamma}$, let $e\in ([K^{\gamma}]_ {\gamma})^{\gamma}$. Then there exists $h\in E$ and $q\in [K^{\gamma}]_ {\gamma}$ with $e=\gamma_h q\leq q$. By (i) with $Q:=K^{\gamma}$, we find that there exists a $\gamma$-orthogonal family $(q_i)_{i\in I}$ in $K^ {\gamma}$ such that $q=\bigvee_{i\in I} q_i$ and $e=\bigvee_{i\in I} (e\wedge q_i)$. Thus, as $q_i\leq q$ for all $i\in I$, Theorem \[th:EXCprop\] (iv) implies that $\gamma\sb{h}q\sb{i}=q\sb{i}\wedge \gamma\sb{h}q=q\sb{i}\wedge e$ for all $i\in I$. Also, as $q_i\in K^{\gamma}$ for every $i\in I$, there exist $h_i\in E$ and $k_i\in K$ such that $q_i=\gamma_{h_i}k_i$, and we have $e\wedge q\sb{i}=\gamma_h q_i =\gamma_h\gamma_{h_i}k_i=(\gamma_h\wedge\gamma_{h_i})k_i=\gamma_ {\gamma_h h_i}k_i\in K\sp{\gamma}$. Therefore the elements of the $\gamma$-orthogonal family $(e\wedge q_i)_{i\in I}$ all belong to $K^{\gamma}$ and so $e\in [K^{\gamma}]_{\gamma}$. We omit the proof of (iii) as it is similar to the proof of (ii). \(iv) Evidently, $K'=(K')^{\downarrow}=(K^{\downarrow})'$. It remains to prove that $[K']\sb{\gamma}\subseteq K'$. Let $q\in [K']_{\gamma}$, $k\in K$, and $e\in E$ with $e\leq q,k$. By (i) with $Q:=K'$, there are $\gamma$-orthogonal families $(q_i)_{i\in I}\subseteq K'$ and $(e \wedge q_i)_{i\in I}$ such that $q=\bigvee_{i\in I} q_i$ and $e= \bigvee_{i\in I}(e\wedge q_i)$. Since $e\leq k$ and $k\wedge q_i=0$, it follows that $e\wedge q_i=0$ for all $i\in I$, so $e=0$. Thus $q \wedge k=0$, whence $q\in K'$. \(v) We have $K\subseteq K''$ and as $K''=(K')'$, it is STD by (iv), hence it is TD. But then by (ii), $[K^{\gamma}]_{\gamma}\subseteq K''$, therefore $K'\subseteq ([K^{\gamma}]_{\gamma})'$. We also get $([K^{\gamma}]_{\gamma})'\subseteq K'$ because $K\subseteq [K^{\gamma}] _{\gamma}$. Similarly, $K\subseteq [K^{\downarrow}]_{\gamma}$, whence $([K^{\downarrow}]_{\gamma})'\subseteq K'$ and by (iv) and (iii), $[K^{\downarrow}]_{\gamma}\subseteq K''$; hence $K'=K'''\subseteq ([K^{\downarrow}]_{\gamma})'$. If $A$ [(]{}which may be empty[)]{} is the set of all atoms in $E$, then the STD set $A'$ is the set of all elements in $E$ that dominate no atom in $E$, and the STD set $A''$ is the set of all elements $p\in E$ such that either $p=0$ or the PEA $E[0,p]$ is atomic. \[centrTD\] The set $\Gamma(E)$ of central elements of a COGPEA $E$ is a TD subset of $E$. Obviously $\Gamma(E)\subseteq [\Gamma(E)]_{\gamma}$ and by theorem \[th:largestandboo\] (ii), $[\Gamma(E)]_{\gamma}\subseteq\Gamma(E)$. To prove that $\Gamma(E)^{\gamma}\subseteq\Gamma(E)$, let $c_1\in \Gamma(E)^{\gamma}$, so that $c_1:=\gamma_e c$ for some $e\in E$ and $c\in\Gamma(E)$. We claim that $c_1$ is the greatest element of $\gamma_{c_1}(E)$; hence by Corollary \[co:pic\], it is a central element of $E$. Indeed, if $f\in\gamma\sb{c\sb{1}}(E)$, then $f=\gamma_ {c_1} f=\gamma_{\gamma_e c} f=\gamma_e (\gamma_c f)=\gamma_e(c\wedge f) \leq\gamma_e c=c_1$ by Theorem \[th:ExCovProp\] (v) and Theorem \[th:ceprop\] (i). \[df:TypeClass\] A nonempty class $\mathcal{K}$ of PEAs is called a *type class* iff the following conditions are satisfied: (1) $\mathcal{K}$ is closed under the passage to direct summands. (2) $\mathcal{K}$ is closed under the formation of arbitrary nonempty direct products. (3) If $E_1$ and $E_2$ are isomorphic PEAs and $E_1$ is in $\mathcal{K}$, then $E_2\in\mathcal{K}$. If, in addition to (2) and (3), $\mathcal{K}$ satisfies (1$'$) $H\in \mathcal{K}, h\in H\,\Rightarrow\, H[0,h]\in\mathcal{K}$, then $\mathcal{K}$ is called a *strong type class*. \[th:TypeClass\] Let $\mathcal{K}$ be a type class of PEAs and define $K:=\{k\in E: E[0,k]\in\mathcal{K}\}$. Then $K$ is a TD subset of $E$, and if $\mathcal{K}$ is a strong type class, then $K$ is STD. Suppose $k\in K$ and $e\in E$. Then $E[0,k]\in\mathcal{K}$, $\gamma_e\in{{\Gamma\!\sb{\rm ex}}}(E)$, and by Lemma \[le:nova\], $\gamma_e| _{E[0,k]}\in{{\Gamma\!\sb{\rm ex}}}(E[0,k])$. Thus by Theorem \[th:mis\] (v) and Definition \[df:TypeClass\] (1), $E[0,\gamma_e k]=\gamma_e (E[0,k])=\gamma_e|_{E[0,k]}(E[0,k])\in\mathcal{K}$, so $K^{\gamma} \subseteq K$. If $\mathcal{K}$ is a strong type class, it is clear, that $K^{\downarrow}\subseteq K$. Finally, suppose that $k\in[K]_ {\gamma}$. Then there exists a $\gamma$-orthogonal family $(k_i)_ {i\in I}$ in $K$ such that $k=\bigvee_{i\in I} k_i$. Thus by Definition \[df:TypeClass\] (2), $X:=$[$\times$]{}$_{i\in I} E[0,k_i]\in\mathcal{K}$ and by Corollary \[cor:cop\], $X$ is PEA-isomorphic to $E[0,k]$, whence by Definition \[df:TypeClass\] (3), $E[0,k]\in\mathcal{K}$, and therefore $k\in K$. The class $\mathcal K$ of all EAs is a strong type class of PEAs; hence by Theorem \[th:TypeClass\], the set $K$ of all elements $k\in E$ such that $E[0,k]$ is an EA is an STD subset of $E$. From now on we will assume that $K$ is a TD subset of the COGPEA $E$. ${\widetilde K}:=K\cap\Gamma(E)$. \[th:kstar\] There exists $k^*\in K$ such that $\gamma\sb{k^*}$ is the largest mapping in$\{\gamma_k:k\in K\}=\{\gamma_e:e\in E, e \leq k^*\}=\Theta\sb{\gamma}[0,\gamma\sb{k^*}]$, which is a sublattice of ${{\Gamma\!\sb{\rm ex}}}(E)$, and as such, it is a boolean algebra. Moreover, ${\widetilde K}$ is a TD subset of $E$, there exists ${\widetilde k}\in{\widetilde K}$ such that $\gamma\sb{\widetilde k}$ is the largest mapping in $\{\gamma_k:k\in{\widetilde K}\}=\{\gamma_e:e \in E,e\leq{\widetilde k}\}=\Theta\sb{\gamma}[0,\gamma\sb{\widetilde k}]$, which is a sublattice of ${{\Gamma\!\sb{\rm ex}}}(E)$, and as such, it is a boolean algebra. Let us take a maximal $\gamma$-orthogonal family $(k_i)_{i\in I}\subseteq K$ and set $k^*:=\bigvee_{i\in I} k_i$. Then $k^*\in K$, because $K$ is TD subset of $E$. Let $k\in K$. As $\gamma\sb{k}k=k$ and $(\gamma\sb{k^*})' k\leq k$, we have $(\gamma\sb{k}\wedge(\gamma\sb{k^*})')k=\gamma\sb{k}k \wedge(\gamma\sb{k^*})'k=k\wedge(\gamma\sb{k^*})'k=(\gamma\sb{k^*})'k$. Also, by Theorem \[th:ExCovProp\] (vi), $\gamma_k\wedge(\gamma_{k^*})' =\gamma\sb{d}$, where $d:=(\gamma_{k^*})'k$, and since $K\sp{\gamma} \subset K$, it follows that ${\widehat k}:=(\gamma\sb{k^*})'k=\gamma\sb{d}k \in K$ with $\gamma\sb{k^*}{\widehat k}=\gamma\sb{k^*}((\gamma\sb{k^*})'k)=0$. Therefore, by Theorem \[th:ExCovProp\] (v), $\gamma\sb{\widehat k} \wedge\gamma\sb{k\sp{\ast}}=\gamma\sb{\gamma\sb{k\sp{\ast}}{\widehat k}}=0$, and since $k\sb{i}\leq k\sp{\ast}$, it follows that $\gamma\sb{\widehat k} \wedge\gamma\sb{k\sb{i}}=0$ for all $i\in I$. Consequently, $(\gamma \sb{k^*})'k={\widehat k}=0$ by the maximality of $(k_i)_{i\in I}$, therefore $k=\gamma\sb{k^*}k$, whence $\gamma\sb{k}\leq\gamma\sb{k^*}$. Suppose $k\in K$ and put $e:=\gamma\sb{k}k^*$. Then $e\leq k^*$ with $\gamma\sb{k}=\gamma\sb{k}\wedge\gamma\sb{k^*}=\gamma\sb{\gamma\sb{k}k^*} =\gamma\sb{e}$, whence $\{\gamma\sb{k}:k\in K\}\subseteq\{\gamma\sb{e} :e\in E, e\leq k^*\}$. If $e\in E$ and $e\leq k^*$, then $\gamma\sb{e} \leq\gamma\sb{k^*}$, so $\{\gamma\sb{e}:e\in E, e\leq k^*\}\subseteq \{\gamma\sb{e}:e\in E, \gamma\sb{e}\leq\gamma\sb{k^*}\}=\Theta\sb{\gamma} [0,\gamma\sb{k^*}]$. Finally, suppose $e\in E$ with $\gamma\sb{e}\leq \gamma\sb{k^*}$, and put $k:=\gamma\sb{e}k^*$. Since $K$ is TD, we have $k\in K$; moreover, $\gamma\sb{e}=\gamma\sb{e}\wedge\gamma\sb{k^*}= \gamma\sb{k}$, so $\Theta\sb{\gamma}[0,\gamma\sb{k^*}]\subseteq \{\gamma\sb{k}:k\in K\}$. By Corollary \[co:ThetasbgammaGBA\], $\Theta\sb{\gamma}$ is a generalized boolean algebra; hence the interval $\Theta\sb{\gamma} [0,\gamma\sb{k^*}]=\{\pi\in\Theta\sb{\gamma}:0\leq\pi\leq\gamma \sb{k^*}\}$ is a boolean algebra with unit $\gamma\sb{k^*}$. That ${\widetilde K}$ is a TD subset, follows from Theorem \[centrTD\] and the fact that ${\widetilde K}=K\cap\Gamma(E)$. Thus we obtain the second part of the theorem by applying the first part to ${\widetilde K}$. Since $\gamma_{k^*}\in{{\Gamma\!\sb{\rm ex}}}(E)$ is the largest element in $\{\gamma_k: k \in K\}$, it is uniquely determined by the TD set $K$. Likewise, ${\widetilde k}$ is uniquely determined by ${\widetilde K}=K\cap\Gamma(E)$, hence it also is uniquely determined by $K$, and we may formulate the following definition. \[df:gammasbK\] With the notation of Theorem \[th:kstar\], (1) $\gamma_K:=\gamma_{k^*}$ and (2) $\gamma_{\widetilde K}:=\gamma_{\widetilde k}$. \[co:gammasbK\] $\Theta\sb{\gamma}[0,\gamma\sb{K}]$ is a boolean algebra and we have[:]{} 1. $\gamma_{\widetilde K}\leq\gamma_K\in\Theta\sb{\gamma}[0,\gamma\sb{K}] \subseteq{{\Gamma\!\sb{\rm ex}}}(E)$. 2. $\gamma_K=\bigvee_{k\in K}\gamma_k$. 3. $\gamma_K$ is the smallest mapping $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ such that $K\subseteq\pi(E)$. 4. $\gamma_{\widetilde K}=\bigvee_{k\in{\widetilde K}} \gamma_k\in{{\Gamma\!\sb{\rm ex}}}(E)$. 5. $\gamma_{\widetilde K}$ is the smallest mapping $\pi\in \Theta_{\gamma}$ such that ${\widetilde K}\subseteq\pi(E)$. \(i) This is clear by Theorem \[th:kstar\], because $\{\gamma_k :k\in \tilde{K}\}\subseteq\{\gamma_k :k\in K\}$. \(ii) By Theorem \[th:kstar\], $\gamma_K$ is the largest mapping in $\{\gamma\sb{k}:k\in K\}$, from which (ii) follows immediately. \(iii) First we show that $K\subseteq\gamma_K(E)$. Indeed, if $k\in K$, then $\gamma_k\leq\gamma_K$, so $k=\gamma_k k\leq\gamma_K k\leq k$, and therefore $k=\gamma_K k\in\gamma_K(E)$. Suppose $K\subseteq\pi(E)$ for some $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Then, $k^*\in K\subseteq\pi(E)$, so $k^*= \pi k^*$. But $\gamma_{k^*}$ is the smallest mapping in ${{\Gamma\!\sb{\rm ex}}}(E)$ with the latter property, whence $\gamma_{k^*}\leq\pi$. Proofs of (iv) and (v) are similar to (ii) and (iii) with $\tilde{K}$ instead of $K$. \[df:Type\] Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Then[:]{} 1. $\pi$ is *type-$K$* iff there exists $k\in{\widetilde K}$ such that $\pi=\gamma_k$. 2. $\pi$ is *locally type-$K$* iff there exists $k\in K$ such that $\pi=\gamma_k$. 3. $\pi$ is *purely non-$K$* iff $\pi\wedge\gamma_K=0$, i.e., iff $\pi\leq(\gamma\sb{K})'$. 4. $\pi$ is *properly non-$K$* iff $\pi\wedge\gamma_ {\widetilde K}=0$, i.e., iff $\pi\leq(\gamma\sb{\widetilde K})'$. \[rm:Type\] Directly from Definition \[df:Type\] and Corollary \[co:gammasbK\], we have the following for all $\pi,\xi\in{{\Gamma\!\sb{\rm ex}}}(E)$: 1. If $\pi$ is type-$K$, then $\pi$ is locally type-$K$. 2. If $\pi$ is purely non-$K$, then $\pi$ is properly non-$K$. 3. If $\pi$ is both type-$K$ and properly non-$K$, then $\pi=0$. 4. If $\pi$ is both locally type-$K$ and purely non-$K$, then $\pi=0$. 5. If $\xi\in\Theta_{\gamma}$ and $\pi$ is type-$K$ or locally type-$K$ then so is $\pi\wedge\xi$. 6. If $\pi$ is purely non-$K$ or properly non-$K$, then so is $\pi \wedge\xi$. 7. If both $\pi$ and $\xi$ are type $K$, locally type $K$, purely non-$K$, or properly non-$K$, then so is $\pi\vee\xi$. \[th:Type\] Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. Then[:]{} 1. $\pi$ is type-$K$ iff $\pi\in\Theta_{\gamma}$ and $\pi\leq\gamma_{\widetilde K}$. 2. If $K$ is STD and $\pi$ is type-$K$, then $\pi(E)\subseteq K$. 3. $\pi$ is locally type-$K$ iff $\pi\in\Theta_{\gamma}$ and $\pi \leq\gamma_K$. 4. If $\pi$ is purely non-$K$, then $K\cap\pi(E)=\{0\}$ 5. if $\pi$ is properly non-$K$, then ${\widetilde K}\cap\pi(E) =\{0\}$. \(i) By Theorem \[th:kstar\] and Definition \[df:gammasbK\], $\{\gamma \sb{k}:k\in{\widetilde K}\}=\Theta\sb{\gamma}[0,\gamma\sb{\widetilde K}] =\{\gamma\sb{e}:e\in E, \gamma\sb{e}\leq\gamma\sb{\widetilde K}\}$, from which (i) follows immediately. \(ii) If $\pi$ is type-$K$, then $\pi=\gamma_k$ for some $k\in K\cap\Gamma(E)$, whence by Theorem \[th:gammahullsys\], $\pi=\gamma\sb{k}=\pi\sb{k}$, and therefore, since $K$ is STD, $\pi(E)=E[0,k]\subseteq K$. \(iv) Suppose that $\pi$ is purely non-$K$, i.e., $\pi\wedge\gamma\sb{K}=0$. Thus if $k\in K$, then $\gamma\sb{k}\leq\gamma\sb{K}$, whence $\pi\wedge \gamma\sb{k}=0$. Therefore, if $k\in K\cap\pi(E)$, then $k=k\wedge k= \pi k\wedge\gamma\sb{k}k=(\pi\wedge\gamma\sb{k})k=0$. The proofs of (iii) and (v) are analogous to those of (i) and (iv). \[df:faithful\] An element $f\in E$ is *faithful* iff $\gamma\sb{f}=1$. As is easily seen, if $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$, then an element $f\in\pi(E)$ is faithful in the GPEA $\pi(E)$ iff $\gamma\sb{f}=\pi$. \[ksharp\] Let $\pi\in\Theta_{\gamma}$ and put $k^{\sharp}:=\pi k^*$, where $k^*\in K$ is the element in Theorem \[th:kstar\]. Then $k^{\sharp} \in K\cap\pi(E)$ and the following conditions are mutually equivalent[:]{} 1. $\pi$ is locally type-$K$. 2. $k^{\sharp}$ is faithful in the direct summand $\pi(E)$ of $E$ [(]{}i.e., $\gamma_{k^{\sharp}}=\pi$[)]{}. 3. If $\xi\in\Theta_{\gamma}$ with $\xi\wedge\pi\not =0$, then $k^{\sharp}$ has a nonzero component $0\not=\xi k^{\sharp}$ in the direct summand $\xi(\pi(E))$ of the GPEA $\pi(E)$, and $\xi k^{\sharp}\in K$. As $\pi\in\Theta\sb{\gamma}$, there exists $d\in E$ with $\pi=\gamma_d$. Since $K$ is TD and $k^*\in K$, we have $k\sp{\sharp}=\pi k^*=\gamma\sb{d}k^*\in K$. Also, $k\sp{\sharp}=\pi k^*\in\pi(E)$, whence $k^{\sharp}\in K\cap\pi(E)$. \(i) $\Rightarrow$ (ii): If $\pi=\gamma_d$ is locally type-$K$, then $\gamma_d \leq\gamma_K=\gamma_{k^*}$ so $\gamma_{k^{\sharp}}=\gamma_{\gamma_d k^*}= \gamma_d\wedge\gamma_{k^*}=\gamma\sb{d}=\pi$. \(ii) $\Rightarrow$ (iii): Assume (ii) and the hypotheses of (iii). Then $\xi k^{\sharp}=\xi\pi k^*\in\xi(\pi(E))$, $\gamma\sb{k\sp{\sharp}}=\pi$, there exists $e\in E$ with $\xi=\gamma\sb{e}$, and $0\not=\xi\wedge\pi= \gamma\sb{e}\wedge\gamma\sb{k\sp{\sharp}}=\gamma\sb{\gamma\sb{e}k\sp{\#}} =\gamma\sb{\xi k\sp{\#}}$, so $\xi k\sp{\#}\not=0$. Also, since $K$ is TD and $k\sp{\#}\in K$, we have $\xi k\sp{\#}=\gamma\sb{e}k\sp{\#}\in K$. \(iii) $\Rightarrow$ (i): Assume (iii). We have $\pi=\gamma\sb{d}$, and since $k\sp{\#}\in K$, we also have $\gamma\sb{k\sp{\#}}\leq\gamma\sb{K}$; hence, by Theorem \[th:Type\] (iii), it will be sufficient to show that $\gamma\sb{d}\leq\gamma\sb{k\sp{\#}}$. Aiming for a contradiction, we assume that $\gamma\sb{d}\not\leq\gamma\sb{k\sp{\#}}$, i.e., by Theorem \[th:ExCovProp\] (vi), $\xi:=\gamma\sb{e}=(\gamma_{k^{\sharp}})'\wedge \gamma_d\not=0$, where $e:=\gamma\sb{k^{\#}}d$. Then $\xi\leq\gamma\sb{d} =\pi$, so $\xi\wedge\pi=\xi\not=0$. But $\xi\leq(\gamma_{k^{\sharp}})'$ implies $\xi k^{\sharp}=0$, contradicting (iii). If $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$ is locally type-$K$ and $\xi\in\Theta_{\gamma}$ with $\xi\wedge\pi\not =0$, then the direct summand $\xi(\pi(E))$ of $\pi(E)$ contains a nonzero element of $K$. The nonzero element $\xi k^{\sharp}\in K$ in Theorem \[ksharp\] belongs to $\xi(\pi(E))$ . \[le:type\] [(i)]{} There exists a unique mapping $\pi\in\Theta_{\gamma}$, namely $\pi=\gamma_K$, such that $\pi$ is locally type-$K$ and $\pi\,'$ is purely non-$K$. [(ii)]{} There exists a unique mapping $\xi\in\Theta_{\gamma}$, namely $\xi=\gamma_{\widetilde K}$, such that $\xi$ is type-$K$ and $\xi\,'$ is properly non-$K$. By Theorem \[th:Type\] (iii), $\pi$ is locally type-$K$ iff $\pi \leq\gamma_K$ and by Definition \[df:Type\] (3), $\pi\,'$ is purely non-$K$ iff $\pi\,'\wedge\gamma_K=0$, i.e., iff $\gamma_K\leq\pi$, from which (i) follows. Similarly, (ii) follows from Theorem \[th:Type\] (i) and Definition \[df:Type\] (4). Type-decomposition of COGPEA {#sc:TypeDecomp} ============================ *We maintain our standing hypothesis that $K$ is a TD subset of the COGPEA $E$.* According to Lemma \[le:type\], we have two bipartite direct decompositions $E=\pi(E)\oplus\pi\,'(E)$ and $E=\xi(E)\oplus\xi\,'(E)$, corresponding to $\pi=\gamma_K$ and $\xi=\gamma_{\widetilde K}$. Thus we may decompose $E$ into four direct summands: $$E=(\pi\wedge\xi)(E)\oplus(\pi\wedge\xi\,')(E)\oplus(\pi\,'\wedge\xi)(E) \oplus(\pi\,'\wedge\xi\,')(E)$$ one of which, namely $(\pi\,'\wedge\xi)(E)$ is necessarily $\{0\}$, because by Corollary \[co:gammasbK\] (i), $\xi\leq\pi$. Therefore we have the following *fundamental direct decomposition theorem for a COGPEA $E$ with a TD set $K\subseteq E$*. \[th:decompos\] There exist unique pairwise disjoint mappings $\pi_1, \pi_2, \pi_3\in{{\Gamma\!\sb{\rm ex}}}(E)$, namely $\pi_1=\gamma_{\widetilde K}, \pi_2=\gamma_K\wedge(\gamma_ {\widetilde K})'$, and $\pi_3=(\gamma_K)'$, such that: 1. $\pi_1\vee\pi_2\vee\pi_3=1$ so that $E=\pi_1(E)\oplus\pi_2(E)\oplus \pi_3(E)$, and 2. $\pi_1$ is type-$K$, $\pi_2$ is locally type-$K$ but properly non-$K$, and $\pi_3$ is purely non-$K$. For the existence part of the theorem, put $\pi_1=\gamma_{\widetilde K}, \pi_2=\gamma_K\wedge(\gamma_{\widetilde K})'$, and $\pi_3=(\gamma_K)'$. Obviously, $\pi\sb{1}$, $\pi\sb{2}$, and $\pi\sb{3}$ are pairwise disjoint, and since $\gamma\sb{\widetilde K}\leq\gamma\sb{K}$, it is clear that $\pi\sb{1}\vee\pi\sb{2}\vee\pi\sb{3}=1$. Evidently, $\pi \sb{1}\in\Theta\sb{\gamma}$, and by Theorem \[th:ExCovProp\] (viii), $\pi\sb{2}\in\Theta\sb{\gamma}$. Thus, by Theorem \[th:Type\] (i), $\pi\sb{1}$ is type $K$, and by Theorem \[th:Type\] (iii) $\pi\sb{2}$ is locally type $K$. Also, by parts (3) and (4) of Definition \[df:Type\], $\pi\sb{3}$ is purely non-$K$ and $\pi\sb{2}$ is properly non-$K$. To prove uniqueness, suppose that $\pi_{1a},\pi_{2a},\pi_{3a}$ are pairwise disjoint mappings in the boolean algebra ${{\Gamma\!\sb{\rm ex}}}(E)$ satisfying (i) and (ii). Then $\pi_{1a}\leq\gamma_{\tilde{K}}$ by Theorem \[th:Type\] (i), $\pi_{2a}\leq\gamma_K\wedge(\gamma_{\tilde{K}})'$ by Theorem \[th:Type\] (iii) and Definition \[df:Type\] (4), and $\pi_{3a}\leq(\gamma_K)'$ by Definition \[df:Type\] (3). Thus after an elementary boolean computation, we finally get $\pi_1=\pi_{1a}$, $\pi_2=\pi_{2a}$ and $\pi_3=\pi_{3a}$. In what follows we will obtain a decomposition of the COGPEA $E$ into types I, II and III analogous to the type decomposition of a von Neumann algebra. We shall be dealing with two TD subsets $K$ and $F$ of $E$ such that $K\subseteq F$. For the case in which $E$ is the projection lattice of a von Neumann algebra, one takes $K$ to be the set of abelian elements and $F$ to be the set of finite elements in $E$. Thus, in what follows, assume that $K$ and $F$ are TD subsets of the COGPEA $E$ such that $K\subseteq F$. By Theorem \[th:decompos\], we decompose $E$ as $E=\pi_1(E)\oplus \pi_2(e)\oplus\pi_3(E)$ and also as $E=\xi_1(E)\oplus \xi_2(E)\oplus\xi_3(E)$ where $$\begin{aligned} &\pi_1=\gamma_{\widetilde K},\ \pi_2=\gamma_K\wedge(\gamma_{\widetilde K})',\ \pi_3=(\gamma_K)',\\ &\xi_1=\gamma_{\widetilde F},\ \xi_2=\gamma_F\wedge(\gamma_{\widetilde F})',\ \xi_3=(\gamma_F)'.\end{aligned}$$ As $K\subseteq F$, it is clear that $\gamma_K\leq\gamma_F$, $\gamma_ {\widetilde K}\leq\gamma_{\widetilde F}$, $(\gamma_F)'\leq(\gamma_K)'$, and $(\gamma_{\widetilde F})'\leq(\gamma_{\widetilde K})'$. Applying Theorem \[th:decompos\], we obtain a direct sum decomposition $$E=\tau_{11}(E)\oplus\tau_{21}(E)\oplus\tau_{22}(E)\oplus\tau_{31}(E) \oplus\tau_{32}(E)\oplus\tau_{33}(E),$$ where $\tau_{ij}=\pi_i\wedge \xi_j$, for $i,j=1,2,3$. Evidently, $\tau_{11} =\pi_1$, $\tau_{33}=\xi_3$ and $\tau_{12}=\tau_{13}=\tau_{23}=0$. \[de:I,II,III\] ([@HandD Definition 6.3], [@HDandTD Definition 13.3]) Let $\pi\in{{\Gamma\!\sb{\rm ex}}}(E)$. For the TD sets $K$ and $F$ with $K\subseteq F$: - $\pi$ is *type-I* iff it is locally type-$K$, i.e., iff $\pi \in \Theta_{\gamma}$ and $\pi \leq\gamma_K$. - $\pi$ is *type-II* iff it is locally type-$F$, but purely non-$K$, i.e., iff $\pi\in \Theta_{\gamma}$ and $\pi \leq \gamma_F \wedge(\gamma_K)'$. - $\pi$ is *type-III* if it is purely non-$F$, i.e., iff $\pi \leq (\gamma_F)'$. - $\pi$ is *type-I$_F$* (respectively, *type-II$_F$*) iff it is type-I (respectively, type-II) and also type-$F$, i.e., iff $\pi\in\Theta_{\gamma}$ and $\pi\leq\gamma_K\wedge\gamma_{\tilde{F}}$ (respectively, $\pi\in\Theta_{\gamma}$ and $\pi\leq\gamma_F\wedge (\gamma_K)'\wedge\gamma_{\tilde{F}}$). - $\pi$ is *type-I$_{\neg F}$* (respectively, *type-II$_ {\neg F}$*) iff it is type-I (respectively, type-II) and also properly non-$F$, i.e., iff $\pi\in\Theta_{\gamma}$ and $\pi\leq\gamma_K\wedge (\gamma_{\tilde{F}})'$ (respectively, iff $\pi \in \Theta_{\gamma}$ and $\pi\leq \gamma_F\wedge (\gamma_K)'\wedge (\gamma_{\tilde{F}})'$). If $\pi$ is type-I, type-II, etc. we also say that the direct summand $\pi(E)$ is type-I, type-II, etc. The following theorem is the I/II/III - decomposition theorem for COGPEAs. \[th:I-II-III\] Let $E$ be COGPEA and let $K$ and $F$ be TD sets in $E$ with $K\subseteq F$. Then there are pairwise disjoint mappings $\pi_I, \pi_{II}, \pi_{III}\in {{\Gamma\!\sb{\rm ex}}}(E)$ of types I, II and III, respectively, such that $E$ decomposes as a direct sum $$E=\pi_I(E)\oplus \pi_{II}(E)\oplus \pi_{III}(E).$$ Such a direct sum decomposition is unique and $$\pi_I=\gamma_K,\ \pi_{II}=\gamma_F\wedge(\gamma_K)',\ \pi_{III}=(\gamma_F)'.$$ Moreover, there are further decompositions $$\pi_I(E)=\pi_{I_F}(E)\oplus \pi_{I_{\neg F}}(E), \ \pi_{II}(E)= \pi_{II_F}(E)\oplus \pi_{II_{\neg F}}(E),$$ where $\pi_{I_F}, \pi_{I_{\neg F}}, \pi_{II_F},\pi_{II_{\neg F}}$ are of types $I_F, I_{\neg F}, II_F, II_{\neg F}$, respectively. 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--- abstract: 'A novel multiscale method for non M-matrices using Multiscale Restricted Smoothed Basis (MsRSB) functions is presented. The original MsRSB method is enhanced with a filtering strategy enforcing M-matrix properties to enable the robust application of MsRSB as a preconditioner. Through applications to porous media flow and linear elastic geomechanics, the method is proven to be effective for scalar and vector problems with multipoint finite volume (FV) and finite element (FE) discretization schemes, respectively. Realistic complex (un)structured two- and three-dimensional test cases are considered to illustrate the method’s performance.' address: - 'Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA' - SINTEF Digital - 'Norwegian University of Science and Technology, Department of Mathematical Sciences' - 'Atmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, Livermore, CA, U.S.A.' author: - Sebastian BM Bosma - Sergey Klevtsov - 'Olav M[ø]{}yner' - Nicola Castelletto bibliography: - 'main\_arXiv.bib' title: 'Enhanced multiscale restriction-smoothed basis (MsRSB) preconditioning with applications to porous media flow and geomechanics' --- Multiscale methods ,MsRSB ,Multipoint flux approximation ,Finite element method ,Preconditioning ,Geomechanics Introduction ============ Large-scale numerical simulations are often required to understand and predict real world dynamics. In many applications, the use of high-resolution grids is required to characterize the heterogeneity of the material properties and the geometric complexity of the domains. Such simulations impose severe computational challenges and motivate the need for efficient solution schemes. Attractive multilevel strategies to achieve this are multiscale methods [@EfeHou09]. In this paper, we propose a generalization of the multiscale restriction-smoothed basis method (MsRSB) recently put forward in [@MsRSB_Moyner2016], and investigate its use as an effective preconditioner for multipoint flux approximation finite volume (FV) and finite element (FE) discretizations of second-order elliptic problems. Specifically we focus on applications to porous media flow and linear elastic geomechanics. The original idea underlying multiscale discretization methods for heterogeneous second-order elliptic problems can be traced back four decades [@StrFed79; @BabOsb83]. In essence, these methods aim at constructing accurate coarse-scale problems that preserve information of fine scale heterogeneity and can be solved at low computational cost. This is accomplished by numerically computing multiscale basis functions, which are local solutions of the original problem, that are used to both: (i) construct the coarse-scale problem, and (ii) interpolate the coarse-scale solution back to the fine-scale. Various methods to obtain these basis functions have been developed, for example generalized finite-element (GFE) methods [@BabCalOsb94], multiscale finite-element (MsFE) methods [@HouWu97], numerical-subgrid upscaling [@Arb02], multiscale mixed finite-element (MsMFE) methods [@CheHou03], multiscale finite-volume (MsFV) methods [@MSFV_Jenny2003], multiscale mortar mixed finite-element (MsMMFE) methods [@Arb_etal07], multilevel multiscale mimetic (M^3^) methods [@LipMouSvy08], multiscale mixed/mimetic finite-element (MsMFEM) [@AarKrogLie08] and generalized multiscale finite element (GMsFE) [@EfeGalHou13] methods, to name a few. In the geoscience community, multiscale methods have been extensively applied both as single-pass [@MSFV_Jenny2003] and iterative schemes [@NorBjo08; @Haj_etal08] to resolve some of the limitations of existing upscaling methods. They have established a solid framework for simulating complex subsurface flow processes, e.g. [@JuaTch08; @Nor09; @Hel_etal10; @ZhoTch12; @WanHajTch14; @Koz_etal16; @Cus_etal15; @ChuEfeLee15; @ParEdw15; @TenKobHaj16; @Lie_etal16; @LieMoyNat17; @Cus_etal18]. Multiscale methods for linear elastic problems have focused primarily on the derivation of accurate coarse space basis functions which are robust with respect to material property heterogeneities and enable scalable performance [@BucIliAnd13; @BucIliAnd14; @Spi_etal14; @MultiscaleFEM_Castelleto2017; @ChuLee19]. Applications to the poroelasticity equations include [@ZhaFuWu09; @BroVas16a; @BroVas16b; @DanGanWhe18; @Akk_etal18; @SokBasHaj19; @Cas_etal19]. The MsRSB method was proposed in the context of FV simulation for fluid flow in highly heterogeneous porous media [@MsRSB_Moyner2016]. Based on a two-grid approach, the MsRSB method constructs multiscale basis functions through restricted smoothing on the fine-scale matrix. In more detail, the basis functions, which are consistent with the local differential operators, are constructed with a cheap relaxation scheme, i.e. a weighted Jacobi iteration, similar to approaches used in smoothed aggregation multigrid methods [@VanManBre96; @VanManBre01; @Bre_etal05]. An important advantage of MsRSB is that smoothing by relaxation provides a great deal of flexibility in handling unstructured grids, an essential requirement, for example, in applications involving complex geological structures. MsRSB has been widely proven and implemented in open source and commercial simulators using a linear two-point flux approximation (TPFA) [@Lie_etal16]. Because of the two-point structure, the linear TPFA scheme is monotone [@Dro14], i.e. it preserves the positivity of the differential solution [@BerPle94], and leads to an M-matrix with a small stencil. This is the reason why linear TPFA is the scheme of choice in most engineering software. Unfortunately, the consistency of TPFA is not guaranteed for arbitrary grids and anisotropic permeability distributions, potentially leading to inaccurate results [@MRST]. Therefore, other FV methods such as multipoint flux approximation (MPFA) and/or nonlinear schemes [@Dro14; @TerMalTch17] must be considered to achieve consistent fluxes. To date, MsRSB has not been combined with MPFA or other consistent discretizations. Hence, in this paper, we focus on enhancing MsRSB to enable the solution of second-order elliptic problems using discretization methods that do not result in an M-matrix. Based on the MPFA-O method [@MPFA_Aavatsmark], we show that the MsRSB basis construction as presented in [@MsRSB_Moyner2016] can fail due to divergent iterations for an anisotropic diffusion problem. We propose a variant of the original MsRSB approach that restores the desired behavior by enforcing M-matrix properties based on a filtering strategy. We develop the new method focusing on FV discretizations for porous media single-phase flow, and extend its use to vector elliptic problems by targeting FE-based simulation of linear elastic geomechanics. The paper is structured as follows. First, the original multiscale restriction-smoothed basis method is briefly reviewed in \[sec:MsRSB\]. Second, MsRSB for an MPFA flow discretization is analyzed and the novel approach is proposed in Section \[sec:MPFA\]. Next, the proposed method is extended to geomechanics in Section \[sec:geomechanics\]. Challenging two- and three-dimensional experiments are presented to demonstrate properties, robustness and scalability of the method throughout Section \[sec:MPFA\] and \[sec:geomechanics\], including comparisons to existing methods and published results. Finally the report is concluded and future work specified. The Multiscale Restriction-Smoothed Basis method (MsRSB) {#sec:MsRSB} ======================================================== We propose a two-level preconditioning framework based on MsRSB for accelerating iterative Krylov methods to solve linear systems of the form: $${A} {\mathbf{u}} = {\mathbf{f}}, \label{eq:linsys_general}$$ where the coefficient matrix ${A} \in \mathbb{R}^{n \times n}$ arises from a finite volume (FV) or finite element (FE) discretization of a scalar or vector second-order elliptic problem. Furthermore, ${\mathbf{u}} = \{ u_i \}_{i=1}^{n} \in \mathbb{R}^{n}$ is the solution vector containing the unknown degrees of freedom, and ${\mathbf{f}} = \{ f_j \}_{j=1}^{n} \in \mathbb{R}^{n}$ is the discrete forcing term. In this work we develop the method and illustrate its performance focusing on two simple but representative models routinely employed in practical simulation of subsurface processes: (i) the incompressible single-phase flow equation, and (ii) the linear elastostatic equations. For the flow problem we will concentrate on FV fine-scale discretizations while for the elastostatics problem we will consider the FE method. A review of governing equations and the derivation of the matrix form in are provided for both models in \[app:models\_strong\_form\] and \[app:model\_problems\_discretization\], respectively. The essence of any multiscale formulation and solution algorithm lies in the construction of a representative coarse-scale problem capable of capturing fine-scale features of a high-resolution model. The connection among scales is accounted for by computing basis functions, i.e. localized fine-scale solutions, which are used to construct a coarse-scale (upscaled) problem and reconstruct a fine-scale (downscaled) solution from the coarse solution. The reconstruction stage can be represented through the *prolongation operator* ${P}$, a sparse linear operator that stores the basis function associated to each coarse degree of freedom in the corresponding column such that $$\begin{aligned} {P} : \mathbb{R}^{n_c} \rightarrow \mathbb{R}^{n_f}, {\mathbf{u}}_c \mapsto {\mathbf{u}}_f = {P}{\mathbf{u}}_c. \label{eq:prolongation}\end{aligned}$$ Here and in the following, subscripts $f$ and $c$ indicate quantities associated with the fine and the coarse problem. In particular, $n_c$ and $n_f$ denote the number of coarse- and fine-scale degrees of freedom, respectively. Assuming ${P}$, to be specified hereafter, is available, the definition of the coarse problem proceeds as follows. First the fine-scale solution in is replaced with the approximation ${\mathbf{u}} = {\mathbf{u}}_f \approx {P} {\mathbf{u}}_c$. Second the resulting residual vector, namely ${\mathbf{r}} = ({\mathbf{f}} - {A} {P} {\mathbf{u}}_c)$, is orthogonalized against $n_c$ vectors in $\mathbb{R}^{n_c}$ that form the rows of the operator ${R}$. Hence, ${\mathbf{u}}_c$ is the solution to the linear system with $n_c$ equations $$\begin{aligned} {A}_c {\mathbf{u}}_c &= {\mathbf{f}}_c & &\text{with} & {A}_c &= {R} {A} {P}, & {\mathbf{f}}_c &= {R} {\mathbf{f}}. \label{eq:lin_system_coarse}\end{aligned}$$ We refer to ${R}$ as the *restriction operator*, a linear operator mapping vectors from the fine- to the coarse-scale $$\begin{aligned} {R} : \mathbb{R}^{n_f} \rightarrow \mathbb{R}^{n_c}, {\mathbf{u}}_f \mapsto {\mathbf{u}}_c = {R}{\mathbf{u}}_f. \label{eq:restriction}\end{aligned}$$ Different options may be considered for ${R}$. If the fine-scale matrix ${A}$ is symmetric positive definite (SPD), a Galerkin orthogonalization, i.e. ${R} = {P}^T$, is typically the strategy of choice since it provides a coarse scale operator ${A}_c$ that is still SPD. Alternatively, a Petrov-Galerkin approach is often used. For example, in the MSFV method [@MSFV_Jenny2003], which is designed for diffusion problems, ${R}$ is constructed such that discrete mass conservation also holds for the coarse-scale problem. MsRSB for TPFA finite volume schemes {#sec:FvTpfaMsRSB} ------------------------------------ A crucial component of multiscale methods is the efficient and accurate construction of the prolongation operator, that is the computation of the basis functions. Originally proposed for the cell-centered finite volume solution to the diffusion equation for flow through porous media [@MsRSB_Moyner2016], the MsRSB method computes the basis functions iteratively with restricted smoothing. To describe this process, some terminology is first defined. A primary coarse grid is defined as a partitioning of the fine grid. In each coarse cell, a coarse node is chosen as the representative fine cell for that coarse cell. Furthermore, support boundary cells are defined as the cells connecting neighboring coarse nodes. Support edge cells are defined as the cells connecting a coarse node to its neighbors. Note that edge cells for one coarse node will be boundary cells for other coarse nodes. Fig. \[fig:OlavGrid\_new\] shows an example of the coarse grid structure for a flow problem. \[htbp\] [0.3]{} ![\[fig:OlavGrid\_new\] Coarse grid features for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);); and edge ((0,0) rectangle (1ex, 1ex);) cells; and (c) basis function.](./OlavCoarseGrid_new_a "fig:"){width="\linewidth"} [0.3]{} ![\[fig:OlavGrid\_new\] Coarse grid features for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);); and edge ((0,0) rectangle (1ex, 1ex);) cells; and (c) basis function.](./OlavCoarseGrid_new_b "fig:"){width="\linewidth"} [0.3]{} ![\[fig:OlavGrid\_new\] Coarse grid features for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);); and edge ((0,0) rectangle (1ex, 1ex);) cells; and (c) basis function.](./OlavCoarseGrid_new_c "fig:"){width="\linewidth"} The initial guess for the basis function associated to the $j$th coarse node consists of the characteristic function of the primary $j$th coarse cell, hence it is equal to 1 for fine scale cells belonging to the $j$th coarse cell and 0 elsewhere. Let ${L}$ and ${U}$ denote the strictly lower-triangular and upper-triangular part of the matrix [${G}$]{} which we want to compute basis functions for. We assume that [${G}$]{} has the same dimensions as the fine-scale discretization matrix ${A}$, but otherwise leave the relationship ambiguous for the time being. Let $\hat{{D}}$ be the diagonal matrix such that the entries in each row of matrix $\hat{{\ensuremath{{G}}}} = ({L} + \hat{{D}} + {U})$ sum to zero, i.e. $$\begin{aligned} [\hat{{D}}]_{ii} = - \sum_{j=1,j \ne i}^{n} [{\ensuremath{{G}}}]_{ij}, \qquad \forall i \in \{1, 2 \ldots, n \}. \label{eq:Dhat_zero_rowsum}\end{aligned}$$ Each MsRSB basis function is then iteratively smoothed by applying the following relaxation scheme $$\begin{aligned} [{P}]_{*j}^{k+1} = [{P}]_{*j}^k - \omega \hat{{D}}^{-1} \hat{{\ensuremath{{G}}}} [{P}]_{*j}^k, \label{eq:smoother}\end{aligned}$$ where $[{P}]_{*j}$ denotes the $j$th column of ${P}$, i.e. the basis function corresponding to coarse node $j$, and $k$ is the iteration count. Furthermore, $\omega$ is a relaxation factor which in this work is set to 2/3, i.e. the value warranting the optimal smoothing factor of the weighted Jacobi iteration for the homogeneous Poisson’s equation [@Saa03]. If the smoothing update extends the basis function outside of its support region, the update is adjusted to enforce that the basis support is enclosed by the boundary cells. This is done by adding overflowing basis function values to the neighboring basis functions. We refer the reader to [@MsRSB_Moyner2016] for additional details. The original MsRSB basis function construction process is also incorporated in algorithm \[alg:EnhancedMsRSB\], for a given coarse mesh and support regions. Importantly the described smoothing process theoretically guarantees the conservation of the initial partition of unity at each iteration. However, the authors note that round-off errors in a numerical implementation can cause a violation of this condition. As such, global basis function rescaling is required to guarantee the partition of unity. These are inexpensive operations which can be done every few iterations ($n_{it}$). MsRSB for non-M matrices: multipoint FV schemes {#sec:MPFA} =============================================== In order to demonstrate the challenges of applying a multiscale method to discretized systems which do not result in M-matrices, we will consider a specific fine-scale discretization of the incompressible single-phase pressure equation $$- \text{div}({\boldsymbol{\Lambda}} \cdot \text{grad} \; p) = q, \label{eq:pressure}$$ where $p$ is the scalar pressure field, $q$ a source term distributed in the domain and ${\boldsymbol{\Lambda}}$ a positive-definite tensor describing the diffusion properties of the medium. Generally, ${\boldsymbol{\Lambda}}$ is principally characterized by the permeability tensor ${\boldsymbol{\kappa}}$ and fluid viscosity $\mu$. A detailed description of the governing equations is provided in \[app:model\_flow\]. The standard approach for discretizing is a two-point flux approximation (TPFA) scheme which is only consistent when the principal axes of the permeability tensor are aligned with the grid [@EdwRog98]—i.e., for so-called ${\boldsymbol{\kappa}}$-orthogonal grids. One possible choice to solve pressure on rough grids with non-diagonal permeability tensors is the MPFA-O method [@MPFA_Aavatsmark], which may not result in M-matrices for grids of interest. In the MPFA-O method, transmissibilities are computed by enforcing continuity of fluxes over each half-face for local reconstructions of linear flow. These fluxes rely on all the cell pressure unknowns surrounding a grid node and guarantee by construction that the set of equations, related to the half faces connected to a grid node, result in a solvable system. As it is not the focus of this paper, the authors refer to the cited paper for more details on the well-established method. Furthermore, we note that the implementation of MsRSB for MPFA was done using the Matlab Reservoir Simulation Toolbox [@MRST]. In the following, we address two points: i) we will consider scalar systems discretized with MPFA-O as a proxy for the inherent difficulties in applying multiscale methods to non-M-matrices and ii) demonstrate a practical approach for implementing MsRSB or similar methods to MPFA-type discretizations for flow. While the MsRSB method is well studied for TPFA-type problems, it has not, to the authors’ knowledge, been combined with consistent discretizations in the literature. Extension to non M-matrices {#sec:fix_non_M-matrix} --------------------------- To avoid issues related to non M-matrices, a method is devised to alter, with minimal intrusion, the original fine-scale matrix ${A}$ such that it satisfies M-matrix properties. This approximation of the linear system is justified because we aim to find an approximate solution using multiscale methods, where the basis functions should account for the M-matrix like part of the system matrix. This is conceptually similar to e.g. using incompressible or steady-state basis functions for flow when the problem under consideration is nonlinear due to compressibility [@MsRSB_compressible_Moyner2016; @Compressible_MS_Zhou]. Moreover, this work’s primary objective is to employ the proposed method as a preconditioner for GMRES and other iterative Krylov solvers, where an inexpensive local solver will target any local errors in the approximation. In the following, we assume that the discretization matrix has by convention a positive diagonal and primarily negative off-diagonal entries. To enforce M-matrix properties, it is sufficient to filter out all positive-diagonal entries and construct a modified system matrix $\tilde{{A}}$ $$\begin{aligned} [\tilde{{A}}]_{ij} = min([{A}]_{ij},0), \qquad \forall (i,j) \in \{1, 2, \ldots, n \} \times \{1, 2, \ldots, n \}, \label{eq:negoffdiag}\end{aligned}$$ from which we can easily compute the basis functions by setting ${\ensuremath{{G}}}{} = \tilde{{A}}$. As the resulting matrix $\hat{{\ensuremath{{G}}}{}}$ from eq. now has zero row sum with only non-positive off-diagonal entries, the enhanced MsRSB method guarantees the robust generation of partition-of-unity basis functions with entries in $[0, 1]$. Furthermore, note that these changes do not affect a problem described by an M-matrix and as such the method can be implemented generally. To verify that the method has the desired effect, a simple test case is devised. Starting from an equidistant 2-D Cartesian grid, all grid nodes are perturbed randomly in both $x$- and $y$-direction. Additionally the grid is stretched by a factor 10 in the $y$-direction. The resulting test case primal grid is shown in Fig. \[fig:CoarseGridMPFA\]. The fine grid has 9x9 cells where we coarsen by a ratio of 3 in each direction. Furthermore all flow properties are homogeneous. Note that MPFA-O for a ${\boldsymbol{\kappa}}$-orthogonal Cartesian grid is equivalent to the TPFA method. Therefore the test case is designed to vary from an orthogonal grid substantially. \[htbp\] [0.22]{} ![\[fig:MPFAbfs\] Basis function for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);), edge ((0,0) rectangle (1ex, 1ex);) and internal cells ((0,0) rectangle (1ex, 1ex);); (c) MsRSB basis function using the original fine-scale system; and (d) MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-12}$, where $e_{it}$ is the smoother update on the internal cells (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 11 iterations is plotted in (c).](./MPFA_9x9_a){width="\linewidth"} [0.22]{} ![\[fig:MPFAbfs\] Basis function for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);), edge ((0,0) rectangle (1ex, 1ex);) and internal cells ((0,0) rectangle (1ex, 1ex);); (c) MsRSB basis function using the original fine-scale system; and (d) MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-12}$, where $e_{it}$ is the smoother update on the internal cells (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 11 iterations is plotted in (c).](./MPFA_9x9_b){width="\linewidth"} [0.22]{} ![\[fig:MPFAbfs\] Basis function for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);), edge ((0,0) rectangle (1ex, 1ex);) and internal cells ((0,0) rectangle (1ex, 1ex);); (c) MsRSB basis function using the original fine-scale system; and (d) MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-12}$, where $e_{it}$ is the smoother update on the internal cells (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 11 iterations is plotted in (c).](./MPFA_9x9_c){width="\linewidth"} [0.22]{} ![\[fig:MPFAbfs\] Basis function for the internal block of a regular 3$\times$3 partition: (a) primal grid and coarse nodes ((0,0) circle (.5ex);); (b) support boundary ((0,0) rectangle (1ex, 1ex);), edge ((0,0) rectangle (1ex, 1ex);) and internal cells ((0,0) rectangle (1ex, 1ex);); (c) MsRSB basis function using the original fine-scale system; and (d) MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-12}$, where $e_{it}$ is the smoother update on the internal cells (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 11 iterations is plotted in (c).](./MPFA_9x9_d){width="\linewidth"} The basis functions corresponding to the central coarse node are plotted in Figure \[fig:MPFAbfs\]. Figure \[fig:MPFAbforig\] displays the basis function obtained after 11 iterations when using the original linear system. It is evident from the plot that the prolongator is diverging. Figure \[fig:MPFAbfalter\] presents the converged basis function when using the altered linear system. In this case, the desired monotone basis functions are obtained. Note that due to the cancellation of certain connections in the matrix, the basis function does not spread over the full domain. This effect is especially present in the corners of the dual-region. As will be shown in numerical examples in Section \[sec:geomechanics\], although this has no impact on the final solution, it is favorable to avoid small coarsening ratios to ensure good performance. The proposed multiscale method is summarized in algorithm \[alg:EnhancedMsRSB\]. Furthermore, the multiscale preconditioning strategy employed in the numerical results is described in algorithms \[alg:setupPreconditioner\]-\[alg:applyPreconditionerTwoLevel\]. $\hat{{A}} = \min({A}, 0)$ $\hat{{A}} \gets \hat{{A}} - \texttt{diag}(\texttt{rowsum}(\hat{{A}}))$ $\hat{{D}} = \texttt{diag}(\hat{{A}})$ k = 0, $e_{it} = \infty$ $\delta{P} = - \omega \hat{{D}}^{-1} \hat{{A}} {P}$ Modify $\delta{P}$ to avoid stencil growth outside of support region ${P} \gets {P} + \delta{P} $ ${[P]}_{i*} \gets {[P]}_{i*} / \sum_j({[P]}_{ij}) $ $e_{it} = \max\limits_{i,j} (|[\delta {P}]_{ij}|), \qquad i \not \in$ support edges $k \leftarrow k+1$ **return** ${P}$ Initialize ${P}$ ${P}$ = ${{R}} \gets$ Construct restriction operator ${{R}} = {{P}}^T$ ${A}_c = {R} {A} {P}$ ${M}_c^{-1} \approx A_c^{-1}$ ${\mathbf{v_c}} = {R} {\mathbf{v}}$ ${\mathbf{w}}_c = M_c^{-1}{\mathbf{v}}_c$ ${\mathbf{w}} = {P}{\mathbf{w}}_c$ ${\mathbf{w}}$ ${\mathbf{z}} \gets {\mathbf{z}} + M_{\text{pre.}}^{-1} ({\mathbf{v}} - {A} {\mathbf{z}})$ ${\mathbf{w}}$ = ${\mathbf{z}} \gets {\mathbf{z}} + {\mathbf{w}}$ ${\mathbf{z}} \gets {\mathbf{z}} + M_{\text{post.}}^{-1} ({\mathbf{v}} - {A} {\mathbf{z}})$ **return** ${\mathbf{z}}$ MsRSB for an MPFA-O discretization: 2D test case ------------------------------------------------ The example problem is shown in Figure \[fig:mpfa\_problem\] and consists of a 100 by 100 structured grid discretizing a rectangular domain of 20 by 150 meters. We impose a unit pressure drop from $x=0$m to $x=20$m. Interior nodes are perturbed by a factor $0.2 \Psi \Delta x$ where $\Psi \in [-\frac{1}{2}, \frac{1}{2}]$ is a uniformly random variable with expected value 0. The tensor ${\boldsymbol{\Lambda}}$ has a diagonal value of $\lambda_{xx} = \lambda_{yy} = 100$ md$\cdot$cP^-1^ with off-diagonal elements $\lambda_{xy} = \lambda_{yx} = 25$ md$\cdot$cP^-1^. The combined effect of the non-orthogonal grid and the strength of the off-diagonal permeability means that directly applying the MsRSB iterative process to the system matrix results in rapid divergence of the basis functions. With the proposed regularization, however, MsRSB can be employed and satisfactory convergence rates are obtained for both Richardson and GMRES-accelerated iterations. We partition the domain into 400 coarse blocks, each comprised of a 5 by 5 segment of fine cells and solve the MPFA system to a tolerance of $10^{-8}$. The results are displayed in Figure \[fig:mpfa\_results\], where Symmetric Gauss-Seidel (SGS) or ILU(0) is used as the second stage of the preconditioner. We observe a clear improvement to convergence rates indicating that the basis functions successfully capture the local features of the system and resolve low-frequency errors. As a non-accelerated stand-alone solver, i.e. Richardson, MsRSB+ILU(0) converges in 30 iterations while ILU(0) fails to converge in 150 iterations. The set-up with the cheaper but less effective smoother SGS fails to converge in both cases although the multiscale stage again leads to a significantly higher convergence rate. The performance difference between the single- and two-level solvers is reduced when GMRES is used to accelerate the solution process, nonetheless the multiscale solvers still only require half as many iterations as smoothers alone. \[htbp\] [0.45]{} ![Grid (a) and reference solution (b) for the 2D example with a MPFA-discretized pressure equation. Note that the dimensions of the grid have been scaled for plotting visibility.[]{data-label="fig:mpfa_problem"}](./mpfa_2d_reference_a "fig:"){width="\linewidth"} [0.45]{} ![Grid (a) and reference solution (b) for the 2D example with a MPFA-discretized pressure equation. Note that the dimensions of the grid have been scaled for plotting visibility.[]{data-label="fig:mpfa_problem"}](./mpfa_2d_reference_b "fig:"){width="\linewidth"} \[htbp\] [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_RICHARDSON\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_RICHARDSON\_multiscale\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_RICHARDSON\_smoother\_ILU0x1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_RICHARDSON\_multiscale\_ILU0x1.csv]{}; ; [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_GMRES\_right\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_GMRES\_right\_multiscale\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_GMRES\_right\_smoother\_ILU0x1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./2d\_mpfa\_GMRES\_right\_multiscale\_ILU0x1.csv]{}; ; MsRSB for an MPFA-O discretization: 3D Field Test Case ------------------------------------------------------ Next a somewhat more realistic conceptual problem is considered. The test case consists of a 50 by 50 structured grid, with 30 layers in the vertical direction. The physical domain has a horizontal extent of 1000 by 1000 meters, with a vertical thickness of 100 meters. Similar to the 2D case, the nodes of the grid are perturbed to create a rough grid. Additionally, the top surface has a varying topography. The model has five different regions of a log-normally distributed diffusion tensor, with mean values of 700, 1000, 300, 800 and 100 md$\cdot$cP^-1^, respectively as shown in Figure \[fig:mpfa\_field\_perm\]. We consider a logically structured coarse mesh with block sizes of 5 by 5 by 5 fine cells, resulting in a total of 600 coarse blocks to partition the fine-grid with 75,000 cells. We apply a simple boundary condition resulting in flow from $x = 0$ to $x=1000$ meters. In engineering applications, flow would typically be driven by wells, but our goal here is to produce flow over the entire domain to verify our implementation. The reference solution is plotted in Figure \[fig:mpfa\_field\_pressure\]. The convergence rates of MsRSB+ILU(0), ILU(0), MsRSB+SGS and SGS are compared in Figure \[fig:mpfa\_field\_results\] with and without Krylov acceleration. We observe that the benefits of the multiscale stage are more significant than in the 2D example. This is likely due to the denser stencil in 3D which leads to a fundamentally more difficult linear system, even for an equivalent number of degrees of freedom. We point out that the multiscale solver with SGS smoothing exhibits robust convergence with GMRES acceleration while the smoother-only setup stagnates. We also observe that the performance of the multiscale solver with ILU(0) is excellent, using 21 and 11 iterations without and with GMRES, respectively. \[htbp\] [0.45]{} ![The permeability, grid (a) and the reference solution (b) for the 3D MPFA example.[]{data-label="fig:mpfa_field_setup"}](./mpfa_field_perm.png "fig:"){width="\linewidth"} [0.45]{} ![The permeability, grid (a) and the reference solution (b) for the 3D MPFA example.[]{data-label="fig:mpfa_field_setup"}](./mpfa_field_sol.png "fig:"){width="\linewidth"} \[htbp\] [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_RICHARDSON\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_RICHARDSON\_multiscale\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_RICHARDSON\_smoother\_ILU0x1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_RICHARDSON\_multiscale\_ILU0x1.csv]{}; ; [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_GMRES\_right\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_GMRES\_right\_multiscale\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_GMRES\_right\_smoother\_ILU0x1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./3d\_mpfa\_GMRES\_right\_multiscale\_ILU0x1.csv]{}; ; MsRSB for non-M matrices: FE simulation of linear elastic geomechanics {#sec:geomechanics} ====================================================================== In this section, we investigate the extension of the enhanced MsRSB preconditioner to linear systems arising from geomechanical problems assuming linear elastic behavior. By means of this application, we extend the method’s applicability to problems with vector unknowns. The linear elastostatics governing equations and their FE discretization using a classical displacement formulation [@Hug00] are reviewed in \[app:model\_elastostatics\] and \[app:model\_elasticity\_FE\], respectively. To compute the nodal displacement discrete solution, the FE method requires the solution of linear systems characterized by a fine scale stiffness matrix $A$ that is SPD but not an M-matrix. Therefore, in the construction of the prolongation operator, a naïve implementation of MsRSB would exhibit the same convergence issues as discussed for the pressure equation in Section \[sec:fix\_non\_M-matrix\]. To apply the enhanced MsRSB preconditioner, we observe that if the displacement degrees of freedom are ordered based on each coordinate direction, the stiffness matrix $A$ possesses a block structure, namely: $$\begin{aligned} A &= \begin{bmatrix} A_{xx} & A_{xy} & A_{xz} \\ A_{yx} & A_{yy} & A_{yz} \\ A_{zx} & A_{zy} & A_{zz} \end{bmatrix}, \label{eq:blk_stiff}\end{aligned}$$ which reflects the full coupling between $x$, $y$, and $z$ components of displacements. For preconditioning purposes, the complete matrix is often replaced with a sparser block diagonal approximation $$\begin{aligned} A^{\textsc{(sdc)}} &= \begin{bmatrix} A_{xx} & & \\ & A_{yy} & \\ & & A_{zz} \end{bmatrix}, \label{eq:blk_stiff_SDC}\end{aligned}$$ namely the *separate displacement component* (SDC) approximation proposed in [@AxeGus78]. The motivation behind the SDC approximation is that, using Korn’s inequality, one can show that ${A}^\text{(SDC)}$ is spectrally equivalent to ${A}$ [@Bla94; @GusLin98]. Note that this approximation breaks down in the incompressible elasticity limit, i.e. Poisson ratio $\nu \to 0.5$. As each diagonal block in corresponds to the finite element discretization of an anisotropic diffusion operator for the corresponding displacement component—see Remark \[rem:SDC\]—, enhanced MsRSB can be readily applied to to obtain a preconditioner for the fine-scale matrix. In FE-based elasticity simulation, two or three MsRSB basis functions are associated with displacement degrees of freedom located at each coarse node in a two- or three-dimensional space, respectively. Computing such basis functions using $A^{\textsc{(sdc)}}$ implies that the fine-scale displacement field in each coordinate direction is expressed as a linear combination of coarse nodal displacement in the corresponding direction only. This represents a major difference with a classic MSFE approach [@MultiscaleFEM_Castelleto2017] that accounts instead for additional coupling, at the price of a much denser prolongation operator, by representing the displacement solution in each direction employing basis functions in all coordinated directions. Nevertheless, we emphasize that the coarse operator, $A_c$, computed using MsRSB basis functions will still capture the full coupling influences between $x$, $y$, and $z$ displacement components. This extension to the enhanced MsRSB method is tested on a simple homogeneous problem defined in terms of dimensionless quantities with unit Lamé parameters, i.e. setting $E$=1 and $\nu$=0.25. The fine grid is chosen to be $12\times12$ Cartesian. The coarse grid has 5 coarse nodes in each direction. Finally, the grid is rescaled by a factor of $20$ in the $y$-dimension. The high aspect ratio is chosen to induce strong non-M matrix properties of the resulting linear system. Figure \[fig:FE\_demo\_a\] depicts the coarse and fine grid of the test case on the $x$-$y$ plane. To assess the robustness of the enhanced MsRSB method including the extension to vector physics, a basis function obtained using a naïve implementation of MsRSB is compared to the same basis function obtained with the enhanced method. Here, a naïve implementation of MsRSB is a straightforward application of the original method for M matrices to the linear system. Figure \[fig:bfSimpleTestGeomechAdapted\] depicts the obtained basis functions for a given coarse node. It is obvious from the plots that the naïve implementation leads to diverging basis functions whereas the enhanced method recovers the expected bi-linear interpolators of a Cartesian homogeneous problem. The results reemphasize the findings of section \[sec:fix\_non\_M-matrix\] and illustrate the performance of the enhanced MsRSB method for vector physics. \[htbp\] [0.22]{} ![\[fig:bfSimpleTestGeomechAdapted\] Basis function relative to the displacement degree of freedom in the $x$-direction associated to the innermost coarse node of a regular 4$\times$4 partition: (a) primal grid; (b) coarse ((0,0) circle (.5ex);), support boundary ((0,0) circle (.5ex);), edge ((0,0) circle (.5ex);) and internal ((0,0) circle (.5ex);) nodes; (c) MsRSB basis function using the original fine-scale system; and (d) enhanced MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-3}$, where $e_{it}$ is the smoother update on the internal nodes (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 20 iterations is plotted in (c).](./FE_12x12_a){width="\linewidth"} [0.22]{} ![\[fig:bfSimpleTestGeomechAdapted\] Basis function relative to the displacement degree of freedom in the $x$-direction associated to the innermost coarse node of a regular 4$\times$4 partition: (a) primal grid; (b) coarse ((0,0) circle (.5ex);), support boundary ((0,0) circle (.5ex);), edge ((0,0) circle (.5ex);) and internal ((0,0) circle (.5ex);) nodes; (c) MsRSB basis function using the original fine-scale system; and (d) enhanced MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-3}$, where $e_{it}$ is the smoother update on the internal nodes (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 20 iterations is plotted in (c).](./FE_12x12_b){width="\linewidth"} [0.22]{} ![\[fig:bfSimpleTestGeomechAdapted\] Basis function relative to the displacement degree of freedom in the $x$-direction associated to the innermost coarse node of a regular 4$\times$4 partition: (a) primal grid; (b) coarse ((0,0) circle (.5ex);), support boundary ((0,0) circle (.5ex);), edge ((0,0) circle (.5ex);) and internal ((0,0) circle (.5ex);) nodes; (c) MsRSB basis function using the original fine-scale system; and (d) enhanced MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-3}$, where $e_{it}$ is the smoother update on the internal nodes (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 20 iterations is plotted in (c).](./FE_12x12_c){width="\linewidth"} [0.22]{} ![\[fig:bfSimpleTestGeomechAdapted\] Basis function relative to the displacement degree of freedom in the $x$-direction associated to the innermost coarse node of a regular 4$\times$4 partition: (a) primal grid; (b) coarse ((0,0) circle (.5ex);), support boundary ((0,0) circle (.5ex);), edge ((0,0) circle (.5ex);) and internal ((0,0) circle (.5ex);) nodes; (c) MsRSB basis function using the original fine-scale system; and (d) enhanced MsRSB basis function using the filtered fine-scale system. The convergence criteria is $||e_{it}||_{\infty} < 10^{-3}$, where $e_{it}$ is the smoother update on the internal nodes (see algorithm \[alg:EnhancedMsRSB\]). However as the case with the original linear system diverges, the basis function obtained after 20 iterations is plotted in (c).](./FE_12x12_d){width="\linewidth"} MsRSB for Geomechanics: A 2D Heterogeneous Test Case ---------------------------------------------------- ![Structured 2D heterogeneous case [@MultiscaleFEM_Castelleto2017]: Young’s modulus distribution and computational mesh setup.[]{data-label="fig:2D_structured_young"}](./2D_structured_young){width="\textwidth"} To compare the enhanced MsRSB method to an existing multiscale preconditioner for geomechanics [@MultiscaleFEM_Castelleto2017], we investigate a 2D test case consisting of an elastic isotropic domain with a heterogeneous (layered) distribution of Young’s modulus, shown in Figure \[fig:2D\_structured\_young\]. Four mesh families (`cart`, `skew`, `trig` and `rand`) are considered, each with a different type of geometric distortion applied to the coarse elements. In each case a $224\times224$ fine-scale mesh and $7\times7$ coarse mesh are defined on the domain. The reader is referred to section 4.1 of [@MultiscaleFEM_Castelleto2017] for a more detailed description of the test case setup. Two Krylov solver setups are tested. The first variant is Preconditioned Conjugate Gradient (PCG), with a symmetric preconditioning operator constructed by pre– and post–smoothing the multiscale (MsRSB) operator with no-fill Incomplete Cholesky factorization (IC(0)). The second version employs Biconjugate Gradient Stabilized (BiCGStab) with a two–stage preconditioning scheme which consists of the multiscale operator as the global step and no–fill incomplete LU factorization (ILU(0)) as the local smoother. This setup is identical to the one used in [@MultiscaleFEM_Castelleto2017], except that MSFE is replaced by enhanced MsRSB to construct the multiscale operator. Table \[tab:2D\_structured\_CG\_iter\] summarizes the iteration counts using PCG for different mesh families and levels of contrast in material properties. With exception of the case with extremely large coarse elements ($7\times7$, which results in each coarse element consisting of $32\times32$ fine elements), the preconditioner offers robust performance over a variety of mesh families and types of boundary conditions. In a direct comparison against an identical setup using MSFE, from [@MultiscaleFEM_Castelleto2017], Table \[tab:2D\_structured\_BICGSTAB\_iter\] shows similar results obtained with BiCGSTab and a single ILU(0) smoothing step. Here, the proposed MsRSB method achieves similar performance compared to MSFE, with only a modest (about 20–25% on average) increase in the number of iterations for most cases, with a notable exception of Cartesian laterally constrained case, where MSFE basis functions provide an exact interpolation. Note that the BiCGStab solver applies the preconditioner twice at every iteration, which corresponds to two applications of both multiscale operator and smoother, whereas PCG employs a single application of the preconditioner, which involves two smoothing steps (pre– and post–smoothing) and only one multiscale step. Nevertheless, comparable performance is observed in terms of iteration count, suggesting PCG with symmetric preconditioning should lead to a more efficient option due to its lower cost per iteration. [cccccccccccccccc]{} $\beta$ & & & & & &\ & & `cart` & `skew` & `trig` & `rand` & & `cart` & `skew` & `trig` & `rand` & & `cart` & `skew` & `trig` & `rand`\ 1 & $56\times56$ & 5 & 7 & 7 & 6 & & 5 & 7 & 7 & 6 & & 6 & 7 & 7 & 6\ & $28\times28$ & 9 & 12 & 13 & 10 & & 9 & 13 & 13 & 11 & & 12 & 15 & 15 & 13\ & $14\times14$ & 16 & 21 & 22 & 20 & & 18 & 24 & 23 & 22 & & 24 & 27 & 27 & 23\ & $7 \times7 $ & 37 & 40 & 46 & 40 & & 41 & 45 & 48 & 45 & & 50 & 43 & 47 & 43\ 2 & $56\times56$ & 5 & 6 & 7 & 6 & & 7 & 7 & 8 & 7 & & 7 & 7 & 8 & 7\ & $28\times28$ & 8 & 11 & 12 & 10 & & 9 & 12 & 12 & 10 & & 13 & 15 & 17 & 14\ & $14\times14$ & 17 & 22 & 25 & 21 & & 19 & 24 & 27 & 24 & & 32 & 32 & 32 & 30\ & $7 \times7 $ & 47 & 44 & 52 & 48 & & 58 & 53 & 55 & 55 & & 67 & 50 & 54 & 52\ 3 & $56\times56$ & 5 & 7 & 10 & 8 & & 9 & 9 & 11 & 9 & & 10 & 10 & 11 & 10\ & $28\times28$ & 8 & 12 & 14 & 11 & & 11 & 13 & 15 & 13 & & 16 & 17 & 19 & 16\ & $14\times14$ & 21 & 28 & 42 & 30 & & 23 & 31 & 44 & 33 & & 39 & 40 & 52 & 40\ & $7 \times7 $ & 50 & 54 & 66 & 60 & & 72 & 64 & 85 & 71 & & 94 & 63 & 75 & 64\ [cccccccccccccccc]{} $\beta$ & & & & & &\ & & `cart` & `skew` & `trig` & `rand` & & `cart` & `skew` & `trig` & `rand` & & `cart` & `skew` & `trig` & `rand`\ 1 & $56\times56$ & 5 & 7 & 6 & 5 & & 4 & 7 & 6 & 5 & & 6 & 7 & 7 & 5\ & $28\times28$ & 8 & 10 & 11 & 8 & & 8 & 12 & 11 & 10 & & 13 & 16 & 17 & 14\ & $14\times14$ & 16 & 20 & 19 & 16 & & 18 & 21 & 22 & 19 & & 26 & 29 & 32 & 26\ & $7 \times7 $ & 37 & 33 & 40 & 40 & & 41 & 38 & 43 & 43 & & 40 & 47 & 44 & 42\ 2 & $56\times56$ & 4 & 6 & 6 & 5 & & 5 & 6 & 6 & 6 & & 5 & 6 & 6.5 & 5\ & $28\times28$ & 7 & 10 & 12 & 8 & & 9 & 10 & 12 & 9 & & 12 & 19 & 20 & 17\ & $14\times14$ & 17 & 26 & 26 & 18 & & 18 & 23 & 25 & 22 & & 37 & 39 & 46 & 37\ & $7 \times7 $ & 46 & 50 & 54 & 49 & & 52 & 58 & 53 & 53 & & 51 & 59 & 55 & 51\ 3 & $56\times56$ & 4 & 6 & 8 & 6 & & 6 & 8 & 10 & 9 & & 6 & 7 & 8 & 7\ & $28\times28$ & 10 & 11 & 13 & 10 & & 11 & 12 & 17 & 10 & & 17 & 19 & 19 & 18\ & $14\times14$ & 22 & 30 & 39 & 30 & & 21 & 32 & 40 & 34 & & 46 & 49 & 47 & 50\ & $7 \times7 $ & 49 & 61 & 73 & 63 & & 63 & 71 & 86 & 71 & & 62 & 67 & 71 & 58\ MsRSB for Geomechanics: A Geological 2D Cross-Section Test Case --------------------------------------------------------------- A second geomechanics test case is designed to represent a 2D cross-section of an elastic subsurface porous media domain, characterized by distinct geological layers and faults (shown in Figure \[fig:cross\_section\_sketch\]). A vertical distribution of Young’s modulus is prescribed, based on a correlation for uniaxial compressibility developed in [@Bau_etal02] and recently used in [@MultiscaleFEM_Castelleto2017]. Specifically, vertical compressibility is computed as $$\begin{aligned} c_M = 0.01241|\sigma'_y|^{-1.1342} \label{eq:compressibility2Dtestcase}\end{aligned}$$ where $$\begin{aligned} \sigma'_y = \sigma_y + p = -0.12218|z|^{1.0766} + 0.1|z| \label{eq:effectiveStress2Dtestcase}\end{aligned}$$ is the vertical effective stress, consisting of vertical total stress $\sigma_y$ \[bar\] and hydrostatic pressure $p$ (both in units of \[$\text{bar}^{-1}$\]). Young’s modulus is expressed as $$\begin{aligned} E = \frac{(1-2\nu)(1+\nu)}{(1-\nu)c_M} \label{eq:Youngs2Dtestcase}\end{aligned}$$ with a Poisson ratio $\nu$ set to 0.3 everywhere. In addition, a constant value is added to Young’s modulus in each layer, specifically the mean Young’s modulus in that layer multiplied by a layer–dependent coefficient. This is done to emulate discontinuities in material properties between layers as often encountered in real subsurface systems. The resulting distribution spans 3 orders of magnitude over the domain and is depicted in Figure \[fig:cross\_section\_young\_2D\]. The domain is gridded with an unstructured triangular mesh that conforms to the layers and faults (see Figure \[fig:cross\_section\_mesh\]) with the faults themselves considered inactive (no slip between fault surfaces). Seven different resolutions of the mesh are considered, ranging between 11,879 and 745,900 elements — see Table \[tab:cross\_section\_dims\] for detailed information on mesh resolution and corresponding problem sizes. The domain is subject to roller boundary conditions on three sides, while the ground surface is traction-free. The deformation process is driven by a constant pressure drawdown of $\Delta p = 20$ bar prescribed in a small reservoir zone inside the domain (shown in Figure \[fig:cross\_section\_mesh\]), that acts as an external distributed force. To evaluate the algorithmic scalability of the MsRSB method for mechanics, the problem is solved with a Krylov solver (namely PCG) to a relative tolerance of $10^{-8}$. Three symmetric two-stage preconditioning operators are constructed using different choices of pre- and post-smoother: 2 sweeps of $l_1$-Jacobi, symmetric Gauss-Seidel, and no-fill incomplete Cholesky factorization. The coarse grid for the multiscale solver is generated by agglomerating fine-scale cells based on face connectivity using METIS [@metis] graph partitioning software. For each mesh resolution, the ratio of fine–to–coarse elements and nodes is kept approximately the same, which results in the size of the coarse problem growing with mesh resolution. As a comparative baseline, a smoothing-only preconditioner (i.e. not involving the global multiscale step) is also applied to the problem for each choice of smoother. Krylov iteration counts are recorded to evaluate performance. Table \[tab:cross\_section\_conv\] summarizes the findings. The multiscale solver convergence remains well bounded for all mesh resolutions and only exhibits very mild mesh dependence, while the baseline approach does not scale well, in some cases failing to achieve convergence within 1000 CG iterations. This example demonstrates a clear benefit of using a multiscale approach for subsurface mechanical problems compared to relying on incomplete factorizations only. It also emphasizes the method’s excellent algorithmic scalability and mesh independence on two-dimensional unstructured grids. [0.8]{} ![Geological 2D cross–section test case. []{data-label="fig:cross_section"}](./CrossSectionSketch_a.pdf "fig:"){width="\textwidth"} [0.8]{} ![Geological 2D cross–section test case. []{data-label="fig:cross_section"}](./CrossSectionSketch_b.pdf "fig:"){width="\textwidth"} [0.8]{} [0.8]{} [rrrrrrrrrrcc]{} & & & & & &\ & & \# cell & \# node & \# dof & & \# cell & \# node & \# dof & & cell & dof\ 0 & & 11,879 & 6,119 & 12,238 & & 12 & 26 & 52 & & 989.9 & 235.4\ 1 & & 23,390 & 11,947 & 23,894 & & 25 & 51 & 102 & & 935.6 & 234.3\ 2 & & 46,932 & 23,817 & 47,634 & & 50 & 102 & 204 & & 938.6 & 233.5\ 3 & & 93,129 & 47,085 & 94,170 & & 100 & 202 & 404 & & 931.3 & 233.1\ 4 & & 186,940 & 94,165 & 188,330 & & 200 & 408 & 816 & & 934.7 & 230.8\ 5 & & 372,360 & 187,210 & 374,420 & & 400 & 803 & 1,606 & & 930.9 & 233.1\ 6 & & 745,900 & 374,350 & 748,690 & & 800 & 1,603 & 3,206 & & 932.4 & 233.5\ [cccccccccc]{} & & & & & &\ & & MsRSB & no MS & & MsRSB & no MS & & MsRSB & no MS\ 0 & & **117** & 259 & & **66** & 145 & & **47** & 97\ 1 & & **113** & 369 & & **63** & 204 & & **43** & 137\ 2 & & **122** & 517 & & **68** & 290 & & **47** & 194\ 3 & & **126** & 727 & & **71** & 407 & & **48** & 268\ 4 & & **129** & — & & **72** & 570 & & **49** & 385\ 5 & & **133** & — & & **75** & 816 & & **51** & 544\ 6 & & **134** & — & & **75** & — & & **50** & 769\ MsRSB for Geomechanics: A 3D Test Case -------------------------------------- A third test is performed on a 3D poromechanical domain representing a $16\times16\times4$ km subsurface formation. The vertical distribution of Young’s modulus prescribed by Eq. \[eq:compressibility2Dtestcase\]–\[eq:Youngs2Dtestcase\] is applied without discontinuities, and Poisson ratio is again set to 0.3. Also similar to the previous case, boundary conditions are imposed to be rollers (zero normal displacement) on all sides except for the traction-free top surface. Forcing is prescribed through pressure drawdown of $\Delta p_1 = 15$ bar and $\Delta p_2 = 22$ bar, respectively, in two reservoirs located around the center of the domain highlighted in Figure \[fig:3D\_structured\_mesh\]. The domain is initially gridded with a $70\times70\times70$ structured Cartesian grid, labeled as `cart`. It is then coarsened with a factor of 10 in each dimension, resulting in a $7\times7\times7$ coarse-scale grid. In addition, nodes of the grid are shifted, resulting in a skewed grid, shown in Figure \[fig:3D\_structured\_mesh\], labeled `skew`. For both grids, the multiscale preconditioner was constructed using the same choices of pre- and post-smoothers as in the previous example. i.e. $l_1$-Jacobi, symmetric Gauss-Seidel and no-fill incomplete Cholesky (the latter not being used with the skewed mesh due to numerical breakdowns in factorization which are not related to the multiscale method). A sequence of progressively finer coarse grids is used, containing between 5 and 14 coarse cells in each dimension. Table \[tab:3D\_structured\_results\] reports the observed iteration counts using CG as the chosen Krylov method and Figure \[fig:3D\_structured\_results\] compares convergence histories obtained using both multiscale (with a $10\times10\times10$ coarse element size) and smoother-only preconditioned CG solvers. [0.45]{} ![3D skewed mesh test case.[]{data-label="fig:3D_structured_young"}](./reservoir3d_mesh.pdf "fig:"){width="\textwidth"} [0.45]{} ![3D skewed mesh test case.[]{data-label="fig:3D_structured_young"}](./reservoir3d_young "fig:"){width="\textwidth"} The results are in line with the lower dimensional test cases. Acceptable iteration counts and good scalability is observed. Moreover, the test case indicates that the developed enhanced MsRSB method generates a robust multiscale preconditioner for 3D geomechanics or other vectorial physics yielding non-M matrices. [ccccccccccc]{} & & & & & & &\ & & & `cart` & `skew` & & `cart` & `skew` & & `cart` & `skew`\ $5\times5\times5$ & $14\times14\times14$ & & 257 & 269 & & 93 & 96 & & 24 & —\ $7\times7\times7$ & $10\times10\times10$ & & 213 & 229 & & 77 & 82 & & 21 & —\ $10\times10\times10$ & $7\times7\times7$ & & 178 & 203 & & 67 & 73 & & 17 & —\ $14\times14\times14$ & $5\times5\times5$ & & 149 & 169 & & 58 & 63 & & 13 & —\ \[htbp\] [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_smoother\_JACx2.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_multiscale\_JACx2.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_multiscale\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_smoother\_IC0x1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_cart\_7x7x7\_CG\_multiscale\_IC0x1.csv]{}; ; [0.48]{} table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_skew\_7x7x7\_CG\_smoother\_JACx2.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_skew\_7x7x7\_CG\_multiscale\_JACx2.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_skew\_7x7x7\_CG\_smoother\_SGSx1.csv]{}; ; table\[x index=0, y index=1, col sep=comma\] [./reservoir3dGrid\_skew\_7x7x7\_CG\_multiscale\_SGSx1.csv]{}; ; Conclusion ========== A novel multiscale method for non M-matrices using Multiscale Restricted Smoothed Basis(MsRSB) functions is presented. The essence of the enhanced MsRSB method consists of enforcing M-matrix properties on the fine-scale linear system based on a filtering technique. Using the resulting approximate linear system, basis functions can robustly be constructed using the original iterative MsRSB strategy. The method is demonstrated for the single phase flow problem and the linear elastic geomechanics problem. Enhanced MsRSB is validated to be effective through various test cases including heterogeneous, unstructured 2-dimensional and 3-dimensional problems . Moreover, when it is applied as a preconditioner, the results show similar iteration counts compared to existing multiscale methods while enabling increased flexibility, easier implementation and sparser systems. Studies on CPU times and computational efficiency are subject of further work. Finally, as shown for the porous media problem, the proposed method allows for the application of multiscale methods to multipoint stencils. Noting that multiscale operators are themselves inherently multipoint, the adapted MsRSB enables multilevel multiscale. Such a development has not yet been achieved in literature and is the topic of current research. Acknowledgements {#acknowledgements .unnumbered} ================ SB is supported by a named Stanford Graduate Fellowship in Science and Engineering (SGF). Funding for SK and NC was provided by Total S.A. through the FC-MAELSTROM Project. OM is funded by VISTA, which is a basic research program funded by Equinor and conducted in close collaboration with The Norwegian Academy of Science and Letters. Portions of this work were performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07-NA27344. The authors also thank Dr. Igor Shovkun and Dr. Jacques Franc for their feedback and Prof. Hamdi Tchelepi for many helpful discussions. Model problems: Governing equations {#app:models_strong_form} =================================== Let $\Omega \subset \mathbb{R}^{n_{sd}}$ and $\Gamma$ denote a domain occupied by a heterogeneous porous medium and its boundary, respectively, with ${\boldsymbol{n}}_{\Gamma}$ the unit outward normal vector to $\Gamma$, ${\boldsymbol{x}}$ the position vector in $\mathbb{R}^{n_{sd}}$, and $n_{sd}$ (= 2 or 3) the spatial dimension of the problem. Incompressible single-phase flow {#app:model_flow} -------------------------------- Let $p$ denote the pore pressure. For the application of the boundary conditions, let $\Gamma$ be decomposed as $\Gamma = \overline{\Gamma^D \cup \Gamma^N}$, where $\Gamma^D \cap \Gamma^N = \emptyset$. The strong form of the incompressible single-phase flow boundary value problem (BVP) may be formally stated as follows: given $q : \Omega \rightarrow \mathbb{R}$, $g_D : \Gamma^D \rightarrow \mathbb{R}$, and $g_N : \Gamma^N \rightarrow \mathbb{R}$, find $p : \overline{\Omega} \rightarrow \mathbb{R}$ such that $$\begin{aligned} \text{div} \; {\boldsymbol{w}}(p) &= q, && \text{in} \; \Omega && \mbox{(pressure equation)}, \label{eq:massBalanceS} \\ p &= g_{D}, && \text{on} \; \Gamma^D && \mbox{(prescribed boundary pressure)}, \label{eq:massBalanceS_DIR}\\ {\boldsymbol{w}}(p) \cdot {\boldsymbol{n}}_{\Gamma} &= g_N, && \text{on} \; \Gamma^N && \mbox{(prescribed boundary flux)}, \label{eq:massBalanceS_NEU}\\ {\boldsymbol{w}}(p) &= - {\boldsymbol{\Lambda}} \cdot \text{grad} \; p, && \text{in} \; \Omega && \mbox{(Darcy's law)}. \label{eq:Darcy}\end{aligned}$$ \[eq:model\_flow\_BVP\] Here, ${\boldsymbol{w}}(p)$ is the Darcy velocity, ${\boldsymbol{\Lambda}} = ({\boldsymbol{\kappa}} / \mu)$ is the the rank-two tensor characterizing the diffusion properties of the medium, with ${\boldsymbol{\kappa}}$ the intrinsic symmetric positive definite permeability tensor and $\mu$ the fluid viscosity, which is assumed constant, and $q$ denotes a volumetric source term. Linear elastostatics {#app:model_elastostatics} -------------------- Let ${\boldsymbol{d}} = \{ d_{\ell} \}_{\ell = 1}^{n_{sd}}$ be the displacement-vector, where $d_{\ell} = ( {\boldsymbol{e}}_{\ell} \cdot {\boldsymbol{d}} )$ are the displacement components with respect to the Euclidean basis $\{ {\boldsymbol{e}} \}_{\ell = 1}^{n_{sd}} $ in $\mathbb{R}^{n_{sd}}$. Let us consider $n_{sd}$ non-overlapping partitions of the domain boundary into two segments associated with Dirichlet, $\Gamma_{\ell}^D$, and Neumann boundary conditions, $\Gamma_{\ell}^N$, respectively, such that $\Gamma = \overline{\Gamma_{\ell}^D \cup \Gamma_{\ell}^N}$, with $\Gamma_{\ell}^D \cap \Gamma_{\ell}^N = \emptyset$, $\ell \in \{ 1, \ldots, n_{sd} \} $. The strong form of the linear elastostatic BVP reads as: given ${\boldsymbol{b}} : \Omega \rightarrow \mathbb{R}^3$, $g_{D,\ell} : \Gamma_{\ell}^D \rightarrow \mathbb{R}$, and $g_{N,\ell} : \Gamma_{\ell}^N \rightarrow \mathbb{R}$, find ${\boldsymbol{d}} : \overline{\Omega} \rightarrow \mathbb{R}^3$ such that $$\begin{aligned} - \text{div} \; {\boldsymbol{\sigma}}({\boldsymbol{d}}) &= {\boldsymbol{b}}, && \text{in} \; \Omega && \mbox{(equilibrium equations)}, \label{momentumBalanceS} \\ d_{\ell} &= g_{D,\ell}, && \text{on} \; \Gamma_{\ell}^D && \mbox{(prescribed boundary displacements)}, \label{momentumBalanceS_DIR}\\ {\boldsymbol{\sigma}}({\boldsymbol{d}}) : \left( {\boldsymbol{e}}_{\ell} \otimes {\boldsymbol{n}}_{\Gamma} \right) &= g_{N,\ell}, && \text{on} \; \Gamma_{\ell}^N && \mbox{(prescribed boundary tractions)}, \label{momentumBalanceS_NEU}\\ {\boldsymbol{\sigma}}({\boldsymbol{d}}) &= {\mathbb{C}} : \text{sym} (\text{grad} \; {\boldsymbol{d}}), && \text{in} \; \Omega && \mbox{(generalized Hooke's law)}, \label{stressStrain} \end{aligned}$$ \[eq:elasticity\_global\] with $\ell \in \{ 1, \ldots, n_{sd} \}$.. Here, ${\boldsymbol{\sigma}}({\boldsymbol{d}})$ is the rank-2 stress-tensor, respectively, ${\mathbb{C}}_{dr}$ is the rank-4 elasticity tensor, and ${\boldsymbol{b}}$ is a body force. In this work we will focus on isotropic linear elastic materials, hence only two independent elastic coefficients are required for the definition of ${\mathbb{C}}$, namely $$\begin{aligned} {\mathbb{C}} &= \lambda ({\boldsymbol{I}} \otimes {\boldsymbol{I}}) + 2G {\mathbb{I}}, \label{eq:elasticity_tensor}\end{aligned}$$ where ${\boldsymbol{I}}$ and ${\mathbb{I}}$ are the second-order and fourth-order identity tensor, respectively, and $\lambda = \frac{E \nu}{(1+\nu)(1-2\nu)}$ and $G = \frac{E}{2(1+\nu)}$ are the Lamé parameters of the material, with $E$ the Young modulus and $\nu$ the Poisson ratio. Note that the subscripts $x$, $y$, and $z$ are also used to denote a quantity associated with the spatial dimension $\ell$ equal to 1, 2, and 3, respectively. Model problems: Discrete formulation {#app:model_problems_discretization} ==================================== Incompressible single-phase flow: Finite volume formulation {#app:model_flow_FV} ----------------------------------------------------------- Given a partition $\mathcal{T}^h$ of the domain $\Omega$ consisting of non-overlapping conforming cells, a finite volume discretization of consists of writing the pressure equation for each cell (control-volume) in $\mathcal{T}^h$ in integral form [@EymGalHer00]. Let $\mathcal{F}^h$ be the set of interfaces in $\mathcal{T}^h$, namely edges ($n_{sd} = 2$), or faces ($n_{sd} = 3$). Let ${V}^h$ be the space of piecewise constant cell-wise fuctions associated with $\mathcal{T}^h$. We consider a discrete approximation for the pressure field such that $p \approx u^h \in {V}^h$. Let $g_D^h$ denote the piecewise constant interface-wise interpolant of $g_D$ having support in $\mathcal{F}^{h,D} \subset \mathcal{F}^h$, namely the set of interfaces belonging to $\Gamma^D$. Similarly, $\mathcal{F}^{h,N} \subset \mathcal{F}^h$ is the set of boundary interfaces lying on $\Gamma^N$. Finally, let $\hat{w}^{\gamma}$ denote a conservative numerical flux approximating the volumetric flux through an interface $\gamma \in \mathcal{F}^h$, namely $\hat{w}^{\gamma} \approx \int_{\gamma} {\boldsymbol{w}} \cdot {\boldsymbol{n}}_{\gamma} \, \mathrm{d}\Gamma$, with ${\boldsymbol{n}}_{\gamma}$ a unit normal vector defining a unique global orientation for $\gamma$. Based on a suitable functional dependence on $u^h$ and $g_D^h$, for linear flux approximation schemes $\hat{w}^{\gamma}(u^h,g_D^h)$ can be split as a sum of two terms; $$\begin{aligned} \hat{w}^{\gamma}(u^h,g_D^h) = \mathring{w}^{\gamma}(u^h) + \bar{w}^{\gamma}(g_D^h), \label{eq:num_flux_split}\end{aligned}$$ to highlight the contribution to the flux related to $u^h$ and $g_D^h$, respectively. Clearly, $\bar{w}^{\gamma}(g_D^h)$ is nonzero only in the presence of non homogeneous pressure boundary conditions. The first term to the right-hand side in is expressed as a linear combination of pressure values from selected cells—e.g., the two cells sharing $\gamma$ in the TPFA method, or the cells sharing at least a vertex with $\gamma$ in the MPFA-O method [@EdwRog98; @MPFA_Aavatsmark]—using constant transmissibility coefficients. A similar linear expression is utilized for $\bar{w}^{\gamma}(g_D^h)$. Nonlinear flux approximation schemes are not considered in this work. For a review and details on recent developments on finite volume discretizations for anisotropic diffusion problems in heterogeneous media, we refer the reader to [@Dro14; @TerMalTch17] and references therein. The finite volume discretization provides an approximation $u^h$ to the weak pressure solution of by solving a set of discrete balance equations that are equivalent to the following mesh-dependent variational problem: find $u^h \in {V}^h$ such that $$\begin{aligned} a^h( v, u) = F^h(v) \qquad \forall v^h \in {V}^h, \label{eq:model_flow_weak_FV}\end{aligned}$$ where the discrete bilinear form $a^h : {V}^h \times {V}^h \rightarrow \mathbb{R}$ and the discrete linear form $F: {V}^h \rightarrow \mathbb{R}$ are defined as $$\begin{aligned} &a^h(v^h,u^h) = - \sum_{\gamma \in \mathcal{F}^h \setminus \mathcal{F}^{h,N}} \llbracket v^h \rrbracket_{\gamma} \mathring{w}^{\gamma}(u^h), \label{eq:model_flow_bilin_FV}\\ &F^h(v^h) = \int_{\Omega} v^h q \, \mathrm{d}\Omega + \sum_{\gamma \in \mathcal{F}^{h,N}} \llbracket v^h \rrbracket_{\gamma} \int_{\gamma} \bar{q} \, \mathrm{d}\Gamma + \sum_{\gamma \in \mathcal{F}^h \setminus \mathcal{F}^{h,N}} \llbracket v^h \rrbracket_{\gamma} \bar{w}^{\gamma}( g_D^h). \label{eq:model_flow_functional_FV} \end{aligned}$$ Here, the symbol $\llbracket \cdot \rrbracket_{\gamma}$ denotes the jump of a quantity across an interface $\gamma\in \mathcal{F}^h$. For internal interfaces, $\llbracket v^h \rrbracket_{\gamma} = ( {v^h}_{\left| \tau_L \right.} - {v^h}_{\left| \tau_K \right.} )$, with ${v^h}_{\left| \tau_L \right.}$ and ${v^h}_{\left| \tau_K \right.}$ the restriction of $v^h$ on cells $\tau_K$ and $\tau_L$ sharing $\gamma$, with ${\boldsymbol{n}}_{\gamma}$ pointing from $\tau_K$ to $\tau_L$. For domain boundary interfaces, the jump expression simplifies to $\llbracket v^h \rrbracket_{\gamma} = - {v^h}_{\left| \tau_K \right.}$. To obtain the matrix form of the FV discrete problem, we introduce the basis $\{ \chi_i \}_{i \in \mathcal{N}^h}$ for ${V}^h$, with $\chi_i$ the characteristic function of the $i$th cell $\tau_i$ in $\mathcal{T}^h$ such that $\chi_i({\boldsymbol{x}}) = 1$, if ${\boldsymbol{x}} \in \tau_i$, $\chi_i({\boldsymbol{x}}) = 0$, if ${\boldsymbol{x}} \notin \tau_i$, and $\mathcal{N}^h = \{1, \ldots, n_{\tau} \}$ with $n_{\tau}$ the total number of cells. Hence, the approximate pressure field is expressed as $p({\boldsymbol{x}}) \approx u^h({\boldsymbol{x}}) = \sum_{i \in \mathcal{N}^h} u_i \chi_i({\boldsymbol{x}})$, with $u_i$ the unknown cell pressure values. Requiring that $u^h$ satisfy for each function of the basis itself yields the system of discrete balance equations for the unknown coefficients vector ${\mathbf{u}} = \{ u_i \}$ $$\begin{aligned} {A}_p {\mathbf{u}} = {\mathbf{f}}_p, \label{eq:model_flow_linsys_FV}\end{aligned}$$ with the system matrix ${A}_p$ and the right-hand side ${\mathbf{f}}_p$ such that $[{A}_p]_{ij} = a( \chi_i, \chi_j)$ with $\{ i, j \} \in \mathcal{N}^h \times \mathcal{N}^h$ and $\{{\mathbf{f}}_p\}_i = F( \chi_i)$, with $i \in \mathcal{N}^h$. The properties of matrix $A_p$ depend on the flux approximation scheme chosen for $\hat{w}^{\gamma}$. For example, a TPFA scheme produces a symmetric positive definite matrix whereas MPFA methods typically lead to a non-symmetric $A_p$ [@Dro14]. Linear elastostatics: Galerkin finite element formulation {#app:model_elasticity_FE} --------------------------------------------------------- The weak form of the linear elastostatics BVP is derived based on the classical displacement formulation [@Hug00] by eliminating ${\boldsymbol{\sigma}}({\boldsymbol{d}})$ in using . Under appropriate regularity assumptions, admits a unique solution ${\boldsymbol{d}}$ that can be obtained by solving an equivalent variational problem [@Hug00]. Let ${\boldsymbol{V}} = \{ {\boldsymbol{v}} \in [H^1(\Omega)]^{n_{sd}} : v_{\ell} = ( {\boldsymbol{e}}_{\ell} \cdot {\boldsymbol{v}} ) \in {V}_{\ell} \}$ denote the space of test functions, where ${V}_{\ell} = \{ v \in H^1(\Omega): v | _{\Gamma^D_{\ell}} = 0 \}$, $\ell \in \{1, \ldots, n_{sd} \}$. Let us consider an extension of the Dirichlet boundary datum $\tilde{{\boldsymbol{g}}}_D \in [H^1(\Omega)]^{n_{sd}}$ such that $( {\boldsymbol{e}}_{\ell} \cdot \tilde{{\boldsymbol{g}}}_D) =g_{D,\ell}$ on $\Gamma^D_{\ell}$, $\ell \in \{1, \ldots, n_{sd} \}$. By expressing the displacement as ${\boldsymbol{d}} = \tilde{{\boldsymbol{g}}}_D + {\boldsymbol{u}}$, the weak form of reads as: find ${\boldsymbol{u}} \in {\boldsymbol{V}}$ such that $$\begin{aligned} a( {\boldsymbol{v}}, {\boldsymbol{u}}) = F({\boldsymbol{v}}) \qquad \forall {\boldsymbol{v}} \in {\boldsymbol{V}}, \label{eq:model_elastostatics_weak}\end{aligned}$$ where the bilinear form $a : {\boldsymbol{V}} \times {\boldsymbol{V}} \rightarrow \mathbb{R}$ and the linear form $F: {\boldsymbol{V}} \rightarrow \mathbb{R}$ are defined as $$\begin{aligned} &a({\boldsymbol{v}}, {\boldsymbol{u}}) = \int_{\Omega} \text{sym} (\text{grad} \; {\boldsymbol{v}}) : {\mathbb{C}}_{dr} : \text{sym} (\text{grad} \; {\boldsymbol{u}}) \, \mathrm{d} \Omega, \label{eq:model_elastostatics_bilin_FE}\\ &F({\boldsymbol{v}}) = \int_{\Omega} {\boldsymbol{v}} \cdot {\boldsymbol{b}} \, \mathrm{d}\Omega + \sum_{\ell = 1}^{n_{sd}} \int_{\Gamma^N_{\ell}} v_\ell g_{N,\ell} \, \mathrm{d}\Gamma - \int_{\Omega} \text{sym} (\text{grad} \; {\boldsymbol{v}}) : {\mathbb{C}}_{dr} : \text{sym} (\text{grad} \; \tilde{{\boldsymbol{g}}}_D) \, \mathrm{d}\Omega \label{eq:model_elastostatics_functional_FE}\\.\end{aligned}$$ Let ${\boldsymbol{X}}^h$ be the finite element space of piecewise polynomial vector functions that are continuous in $\overline{\Omega}$ associated with a conforming triangulation $\mathcal{T}^h$ of $\Omega$. Let $\{ {\boldsymbol{\eta}}_i \}_{i \in \mathcal{N}^h}$ be the standard (vector) nodal basis for ${\boldsymbol{X}}^h$, with $\mathcal{N}^h = \{1, \ldots, n_{sd}n_n \}$ and $n_n$ the number of node points in $\mathcal{T}^h$. We define the finite dimensional counterpart of ${\boldsymbol{V}}$ as ${\boldsymbol{V}}^h = {\boldsymbol{X}}^h \cap {\boldsymbol{V}}$ and denote its basis as $\{ {\boldsymbol{\eta}}_i \}_{i \in \mathcal{N}^h_u}$, with $\mathcal{N}^h_u \subset \mathcal{N}^h$ the set of indexes of basis functions of ${\boldsymbol{X}}^h$ vanishing on $\Gamma^D_{\ell}$, $\ell \in \{1, \ldots, n_{sd} \}$. The discrete approximation to the displacement field can then be expressed as $$\begin{aligned} {\boldsymbol{d}}({\boldsymbol{x}}) \approx \tilde{{\boldsymbol{g}}}_D^h({\boldsymbol{x}}) + {\boldsymbol{u}}^h({\boldsymbol{x}}) = \sum_{\ell = 1}^{n_{sd}} \sum_{j \in \mathcal{N}^h \setminus \mathcal{N}^h_u} g_{D,\ell}(\phi_{\ell j}) \phi_{\ell j}({\boldsymbol{x}}) {\boldsymbol{e}}_{\ell} + \sum_{j \in \mathcal{N}^h_u} u_j {\boldsymbol{\eta}}_j({\boldsymbol{x}}), \label{eq:approx_u_FE} \end{aligned}$$ where $\tilde{{\boldsymbol{g}}}_D^h \in {\boldsymbol{X}}^h$ is the trivial discrete extension of the Dirichlet boundary datum such that ${\boldsymbol{e}}_{\ell} \cdot \tilde{{\boldsymbol{g}}}_D^h$ on $\Gamma^D_{\ell}$ is equal to the finite element interpolant of $g_{D,\ell}$, $\ell \in \{1, \ldots, n_{sd} \}$, $\phi_{\ell j} = ({\boldsymbol{e}}_{\ell} \cdot {\boldsymbol{\eta}}_j({\boldsymbol{x}}))$ , and ${\boldsymbol{u}}^h \in {\boldsymbol{V}}^h$ is an approximate solution to the corresponding homogeneous Dirichlet problem with $u_j$ the unknown nodal displacement degrees of freedom. Substituting $\tilde{{\boldsymbol{g}}}_D$ by $\tilde{{\boldsymbol{g}}}_D^h$ in and requiring that ${\boldsymbol{u}}^h$ satisfy for each basis function of ${\boldsymbol{V}}^h$ yields the matrix form of the variational problem, namely the system of equations for the unknown coefficients vector ${\mathbf{u}} = \{ u_j \}$ $$\begin{aligned} {A}_d {\mathbf{u}} = {\mathbf{f}}_d, \label{eq:model_elastostatics_linsys}\end{aligned}$$ with ${A}_d$ and ${\mathbf{f}}_d$ the symmetric positive definite (SPD) stiffness matrix and force vector, respectively, such that $[{A}_d]_{ij} = a( {\boldsymbol{\eta}}_i, {\boldsymbol{\eta}}_j)$ with $\{ i, j \} \in \mathcal{N}^h_u \times \mathcal{N}^h_u$ and $\{{\mathbf{f}}_d\}_i = F( {\boldsymbol{\eta}}_i)$, with $i \in \mathcal{N}^h_u$. \[rem:FEM\_Dir\_BC\] For efficiency reasons, the linear system is typically assembled ignoring the essential (Dirichlet) boundary conditions—i.e., $\{ {\boldsymbol{\eta}}_i, {\boldsymbol{\eta}}_j \}$ is the range over the bases for ${\boldsymbol{X}}^h$. This is also the strategy adopted in our implementation. The Dirichlet conditions are introduced by using a so-called *symmetric diagonalization* approach [@For_etal12]. In this approach, rows and columns of the stiffness matrix associated with displacement degrees of freedom where such conditions apply are modified while preserving symmetry, with the right-hand-side updated accordingly. \[rem:SDC\] If displacement degrees of freedom are ordered based on each coordinate direction, $A_d$ possesses a $n_{sd} \times n_{sd}$ block structure, namely: $$\begin{aligned} A_d &= \begin{bmatrix} A_{11} & \ldots & A_{1n_{sd}}\\ \vdots & \ddots & \vdots\\ A_{n_{sd}1} & \ldots & A_{n_{sd}n_{sd}} \end{bmatrix}, \label{eq:blk_stiff_gen}\end{aligned}$$ which reflects the full coupling between $\ell$-components of displacements, $\ell = \{1, \ldots, n_{sd} \}$. Each diagonal block $A_{\ell\ell}$ in corresponds to the finite element discretization of the anisotropic diffusion operator $-\text{div} ({\boldsymbol{\Lambda}} \cdot \text{grad} \; d_{\ell})$, with anisotropic diffusion tensor ${\boldsymbol{\Lambda}} = G {\boldsymbol{I}} + (G+\lambda)\ {\boldsymbol{e}}_{\ell} \otimes {\boldsymbol{e}}_{\ell} $, $\ell = \{1, \ldots, n_{sd} \}$. References {#references .unnumbered} ==========

--- abstract: | Zero-shot video classification for fine-grained activity recognition has largely been explored using methods similar to its image-based counterpart, namely by defining image-derived attributes that serve to discriminate among classes. However, such methods do not capture the fundamental dynamics of activities and are thus limited to cases where static image content alone suffices to classify an activity. For example, reversible actions such as entering and exiting a car are often indistinguishable. In this work, we present a framework for straightforward modeling of activities as a state machine of dynamic attributes. We show that encoding the temporal structure of attributes greatly increases our modeling power, allowing us to capture action direction, for example. Further, we can extend this to activity detection using dynamic programming, providing, to our knowledge, the first example of zero-shot joint segmentation and classification of complex action sequences in a larger video. We evaluate our method on the Olympic Sports dataset where our model establishes a new state of the art for standard zero-shot-learning (ZSL) evaluation as well as outperforming all other models in the inductive category for general (GZSL) zero-shot evaluation. Additionally, we are the first to demonstrate zero-shot decoding of complex action sequences on a widely used surgical dataset. Lastly, we show that that we can even eliminate the need to train attribute detectors by using off-the-shelf object detectors to recognize activities in challenging surveillance videos. author: - | Jonathan D. Jones $\textsuperscript{*}$ Tae Soo Kim$\textsuperscript{*}$ Michael Peven[^1] Jin Bai\ Zihao Xiao Yi Zhang Weichao Qiu Alan Yuille Gregory D. Hager\ Johns Hopkins University\ 3400 N. Charles Street, Baltimore, MD, USA\ [{jdjones,tkim60,mpeven,jbai12,zxiao10,yzhan286,wqiu7,ayuille1,hager}@jhu.edu]{} bibliography: - 'egbib.bib' title: 'Zero-shot Recognition of Complex Action Sequences' --- Introduction ============ When learning activity recognition models using deep neural networks, most approaches assume a fully supervised problem setting where 1) all categories of query actions are known *a priori*, 2) example instances from such categories are made available during training and 3) the pre-defined closed set of labels are supported by a large and relatively balanced set of examples. Taken together, this has led to an emphasis on ever more advanced regression-style approaches, whereby a neural network model is trained and scored on held-out examples from the same label set in an end-to-end bottom-up fashion. However, many real-world applications do not fit this model because they are naturally “open-set” problems where new labels may be defined at test time, and/or are fine-grained and compositional so that a combinatorial number of possible activity labels may exist, and/or may be data poor so that sufficient labeled training data may not exist for the desired use case. For example, in video surveillance, the goal is often to detect specific unusual activities in a zero-shot manner, *e.g.* “locate instances where a light brown package is being placed under a car by a man wearing a gray parka.” To successfully answer such a structured query, the ability of a zero-shot system to compose together detectable actor-object relational attributes in an on-demand fashion is highly desired. {width="1.0\linewidth"} In this paper, we present a framework for zero-shot recognition of complex action sequences that models an activity as a sequence of dynamic action signatures. In our framework, an action signature is a particular configuration of visually detectable entities, such as attributes, objects and relations, that describe a temporally local segment of a video. A fundamental observation in our work is that such configurations are often *dynamic*, rather than static—*i.e.* an action’s attributes change over time in a characteristic manner. For example, the act of *a person entering a vehicle* as shown in Figure \[fig:money\] can be defined as “a person near a vehicle moving into a vehicle”. This can be described as the attribute sequences `a person exists` followed by `a person does not exist` and `a vehicle exists`. In the remainder of this paper, we show that dynamic action signatures provide a powerful semantic label embedding for zero-shot activity classification and establish a new state-of-the-art zero-shot classification benchmark on a standard zero-shot-learning (ZSL) dataset, Olympic Sports [@olympic-sports-2010]. We also use our methodology to impose constraints *on the predicted action sequences themselves*, leading to the first zero-shot segmentation results on complex action sequences in a challenging surgical dataset [@jigsaws-2014], and establish, for the first time, a zero-shot baseline result that is competitive with end-to-end trained methods. Finally, in section \[sec:diva\] we eliminate any kind of supervised training on the dataset from which unseen (test) cases are drawn by using publicly available, off-the-shelf object detectors to provide action signatures for video surveillance. We combine this with our activity models to provide a true *de novo* model of an activity. We provide both quantitative and qualitative results of our zero-shot framework using these “on the fly” models on the challenging DIVA dataset[^2], which contains fine-grained human-object interactions under a real world video surveillance setting. In summary, the main contributions of the paper are: - A zero-shot classification of complex action sequences with dynamic action signatures which establishes a new state-of-the-art on Olympic Sports [@olympic-sports-2010] dataset. We outperform all other methods for the ZSL evaluation regardless of training assumptions (inductive/transductive). - To the best of our knowledge, we are the first to demonstrate zero-shot decoding of complex action sequences. We present our results on a surgical dataset, JIGSAWS [@jigsaws-2014], to jointly segment and classify fine-grained surgical gestures where we establish an impressive baseline. - A demonstration of zero-shot classification of fine-grained human-object interactions that requires no supervised training of attributes by leveraging off-the-shelf object detectors in video surveillance. Related Work ============ Methodology {#sec:methodology} =========== We first establish a basic hierarchy of concepts. At the highest level we have the *activity*—for example, suturing in robotic surgery. Each activity can be decomposed into a sequence of actions $(y_1, \ldots, y_N)$. Possible examples of actions are “Pushing needle through tissue" in a suturing activity or “throwing javelin" in a sporting event. Zero-shot learning approaches further decompose each action $y$ into a set of $K$ elementary attributes (usually taken to be binary-valued) $y = \{a_1, \ldots, a_K\}$. Given a video recording of an activity (represented as a sequence of frames $X = (x_1, \ldots, x_T)$), our goal is to map each frame $x_t$ to its corresponding action $y_t$ by detecting the presence or absence of each attribute $\hat{a}(x_t)$ in the video, then choosing the action whose signature $a(y_t)$ best fits those attributes. In other words, we choose the action with highest score: $$\label{eq:decode_cost} \hat{y}(x) = \argmax_{y} score(a(y), \hat{a}(x))$$ Our methods focus on defining signatures conveniently, and computing the score efficiently. Dynamic Attribute Labeling {#dynamic_attributes} -------------------------- \[sec:das\] Previous work in zero-shot action recognition defines each signature over a set of attributes that are *static*—*i.e.* each attribute is presumed to be constant through the duration of the action. However, in many scenarios the actions of interest are distinguished by their time evolution rather than the presence or absence of static attributes. Take “person entering a vehicle" and “person exiting a vehicle", for example. Both of these actions share the static attribute “vehicle present". However, they are differentiated from each other by what happens to the person over time—in an “entering" action the person disappears into the vehicle, but in an “exiting" action the person appears out of it. In this section we outline a simple and elegant method for implementing *dynamic* attribute signatures, which also generalizes previous work. Our method is flexible enough to accept a high-level ordering of events, but also permits more temporal information to be provided if it is known. For example, it can implement a signature for “person appears" like “person is absent, then person is present", or one additionally specifying that a person should be absent for the first 75% of a segment, and present for the remaining 25%. Finally, several existing zero-shot learning datasets are annotated with static attributes, but do not have temporal information. Our framework allows new dynamic signatures to be defined quickly and easily by specifying the temporal evolution of relevant attributes on a per-activity basis. Activity Signatures {#sec:activity_signatures} ------------------- Because they are well-studied, flexible, and easily-composable, we implement our methods using finite-state logic (specifically using the OpenFST toolkit [@openfst]). For a comprehensive overview of finite-state methods and their use in sequence models, see [@mohri-2009]. We define action signatures as finite-state acceptors that implement time-varying rules. Each signature accepts a sequence of attribute detections. In figure \[fig:signature-sm\] we have illustrated state machines implementing the two rules for “Person appears" from the previous section. At inference time we receive a sequence of attribute detection scores from some black-box system, and need to determine its compatibility with our set of pre-defined attribute signatures. To do so, for each detection sequence, we first instantiate a finite-state transducer that accepts sample indices as input, gives detections as output, and whose weights correspond to attribute detection scores (figure \[fig:detector-sm\] illustrates a minimal example). We then compose that transducer with its corresponding attribute acceptor, giving a machine that measures detection inputs against the attribute rule. Finally, we align the rule to the detection sequence by computing the best path through the state machine, and take the score to be the resulting weight[^3]. Zero-shot reasoning for complex activities ------------------------------------------ As we established at the beginning of this section, complex activities contain sequences of actions. We can extend the zero-shot classification scenario to perform joint classification and segmentation in a zero-shot manner by defining a sequence-level score function over $M$ hypothesized segments: $$\label{eq:segmental-score} score(y, x) = \sum_{i = 1}^{M} \phi(y_i, t_i, d_i, x)$$ In equation \[eq:segmental-score\], $\phi$ implements the segment-level score function of eq. \[eq:decode\_cost\], defined over the $i$-th hypothesized segment with start time $t_i$, duration $d_i$, and label $y_i$. In many cases, these activities have a structure that is known *a priori*, and which can be exploited to disallow impossible action sequences. For example, the JIGSAWS dataset is composed of surgical suturing videos. In these sequences, only certain gesture sequences are realizable. By adding a pairwise label score, we can impart this knowledge to the system: $$score(y, x) = \sum_{i = 1}^{M} \phi(y_i, t_i, d_i, x) + \psi(y_{i}, y_{i-1})$$ In practice, we use $\psi(y_i, y_{i-1})$ to implement first-principles knowledge by giving a score of 0 to a possible transition, and a score of $-\infty$ to an impossible one. We can still use the principle in equation \[eq:decode\_cost\] when predicting action sequences (*i.e.* decoding). However, since $y$ is now a sequence, it is more efficient to compute the argmax using a dynamic program—for example, one of the algorithms presented in [@sarawagi-cohen-2004] or [@Colin_2016]. Experiments =========== We first evaluate our approach on the Olympic Sports [@olympic-sports-2010] dataset in \[sec:olympic\] which demonstrates the added representational power of dynamic action signatures for zero-shot action classification. In Section \[sec:jigsaws\], we then describe our approach for *zero-shot decoding* of complex action sequences using the JIGSAWS [@jigsaws-2014] dataset for segmenting a sequence of surgical gestures in a video. Finally in Section \[sec:diva\], we demonstrate that fine-grained human-object interactions can be recognized in a zero-shot manner that requires no training at all by harnessing off-the-shelf object detectors. Olympic Sports {#sec:olympic} -------------- JIGSAWS {#sec:jigsaws} ------- We use the JIGSAWS dataset [@jigsaws-2014] to evaluate our attribute learning and gesture classification methods described above. This is a publicly available dataset containing 39 instances of eight surgeons performing a benchtop simulation training task of suturing in a robot-assisted minimally invasive surgical setting using the da Vinci Surgical System. The dataset contains endoscopic video of the performance as well as motion data for instruments and manipulators that the surgeons control on the system. In this work, we do not use the instrument motion data. Each performance has per-frame gesture class labels for 10 types of actions that occur during the task. JIGSAWS only provides annotations for gestures (activities), and so we use the method described in Section \[dynamic\_attributes\] to obtain per-frame attribute annotations. ### Classification Although the JIGSAWS dataset is composed of complex surgical activities made up of action *sequences*, we first evaluate the performance of our method on the traditional action-classification setup. In this experiment we use ground-truth action boundaries to segment each sequence into a set of actions. Table \[tab:jigsaws-classification\] show the results of an ablation experiment studying the effectiveness of our dynamic attribute signatures on the JIGSAWS dataset. For the static-signature system, we map all dynamic signatures to their nearest static counterparts. Specifically, we map signatures (2) “at beginning" and (3) “at end" to (0) “never". Notice that accuracy increases by almost 20% when dynamic signatures are used. This is due to an inherent confusion between gestures 4 and 6—the static-signature model overwhelmingly predicts gesture 4 for gesture 6, because these gestures’ signatures differ only in the temporal duration of a single attribute. Allowing dynamic attribute signatures disambiguates between these two gestures, and improves accuracy for gesture 6 from 0% to 83%. ### Joint classification and segmentation We next turn to the task of zero-shot decoding (*i.e.* joint classification and segmentation) of surgical activity. This task can be performed in a naive way by doing zero-shot classification for individual samples or windows of samples, but frequently practitioners are aware of additional structure that restricts which action sequences are realizable. In this experiment, we compare the performance of a grammar derived from first-principles knowledge of surgical suturing tasks with an unstructured baseline. More specifically, our grammar describes an ideal execution of the suturing task (see fig. \[fig:jigsaws-sm\] for an illustration). The practitioner begins by reaching for the needle (G1), then moves to the work area (G5), then executes a suture (G2 - G6). At this point they can either transfer the needle from the left to the right hand (G4) and perform another suture, or drop the suture and end the activity (G11). Note that not every sequence in the JIGSAWS dataset conforms to this model—there is a small number of rare states (G8, G9, and G10) that correspond to errors made during the suturing process. Since this work addresses zero-shot applications, we focus on modelling reliable and structured correct cases instead of the more variable incorrect ones. Table \[tab:jigsaws-decoding\] compares the performance of our zero-shot decoding system with the unstructured baseline. Our structured model improves frame-level accuracy over the unstructured one by about 8%. However, the improvement in edit score is much more substantial. By providing information about the basic structure of the sequence derived from what we know about the underlying process, we obtain close to a 100% relative improvement in edit score. Table \[tab:jigsaws-decoding-comparison\] compares our method with previous, *fully-supervised* ones. The first three rows represent unstructured, neural baselines established in [@Colin_2016]. Interestingly, we obtain better edit distance than all of them and better accuracy than two of the three *without ever training on gesture-labeled data*. Furthermore, our edit distance comes close to that of the segmental spatiotemporal CNN of [@Colin_2016]—a fully-supervised model that also incorporates a grammar. DIVA {#sec:diva} ---- Both experiments on zero-shot classification of human activities and zero-shot segmentation of surgical gestures involve a supervised training step to obtain attribute detectors using instances from seen categories. In this section, we demonstrate that publicly available, off-the-shelf object detectors can be used to compose a system to classify human-object interactions in a truly supervision-less zero-shot manner. We demonstrate how we encode first-principles temporal logic to define activities using state machines combined with off-the-shelf object detectors. The DIVA dataset is an untrimmed activity detection dataset that provides both spatial and temporal localization of predefined set of activities. The videos originate from the VIRAT dataset [@oh2011large] and annotations more suitable for activity detection were collected by the IARPA DIVA program. It is a challenging activity detection benchmark where state-of-the-art end-to-end methods such as [@Rc3d] performs poorly[^4] [@Gleason2018APS]. For example, the dataset contains instances from five disjoint scenes with varying scale, level of occlusion and camera view angles and [@Kim2019SAFERF] has shown that a state of the art end-to-end activity classification model does not generalize well across scenes. The DIVA dataset exemplifies a problem setting where zero-shot methods might be favored over the end-to-end counterparts. ------- ------ ------- ------ ------ ------- ------ Scene LOSO ALL Ours LOSO ALL Ours 0000 9.52 36.8 77.8 23.8 66.7 100 0002 37.8 3.85 52.6 12.5 23.5 100 0400 42.1 35.3 88.9 58.8 37.5 85.7 0401 28.3 33.4 81.1 7.69 31.4 100 0500 33.4 0 100 14.3 16.7 100 Mean 30.2 21.87 74.6 23.4 35.16 95.4 ------- ------ ------- ------ ------ ------- ------ : Classification accuracy on the DIVA dataset comparing our approach to fully *supervised* baselines under the leave-one-scene-out (LOSO) or ALL evaluation settings. []{data-label="tab:diva"} **Zero-shot classification without any supervised training:** Given detectable objects {`Human`, `Vehicle`} from [@He2017MaskR] and temporal attribute patterns defined in Section \[sec:methodology\], we can define a human *Entering* and *Exiting* a vehicle with state machines shown in Figure \[fig:diva\_states\]. We compare our zero-shot system to a state-of-the-art end-to-end supervised system [@Zhou2017TemporalRR] in both settings where the system is trained and tested on all five camera locations (ALL), and when it is trained on 4 and tested on a held-out 5th scene (LOSO). The LOSO evaluation tests performance of a supervised system when there exists a large domain gap between training and testing data points which is a reasonable assumption for many practical applications. We observe that the end-to-end baseline generalizes poorly across scenes but our zero-shot approach performs well. For the ALL setting where training conditions are more favorable to an end-to-end model, our zero-shot approach still achieves higher classification performance which suggests that the presented zero-shot approach may serve as a promising alternative when training and the use of end-to-end models is prohibited by data. It shows the value of explicitly injecting first-principles knowledge especially when adequate data for sufficiently training bottom up data-driven models is difficult. The experiments on DIVA shows that using our approach, a practitioner can quickly define a competitive zero-shot action classification system by describing activities with state machines over dynamic action signatures computed using off-the-shelf object detectors. Discussion and Conclusion ========================= As shown in Sections \[sec:jigsaws\] and \[sec:diva\], zero-shot methods are particularly useful when data conditions are not suitable for data-driven methods. However, there inevitably exists corner cases outside the distribution captured by the zero-shot system. For example, a person exiting a top-less vehicle will always be visible and the state machine would not score such sequence highly. We believe an interesting future work would be to study how to best utilize labeled training data through our framework. Our formulation actually allows the gradients to be propagated all the way back to the attribute detectors so that they can be finetuned under the structure defined by the state machines. In summary, we presented a framework for modeling fine-grained activities as a state machine of dynamic attributes. We show that temporal attributes define a rich semantic label embedding for zero-shot classification of fine-grained actions and establishes a new state-of-the-art results on the Olympic Sports dataset. Our approach is the first to establish a competitive baseline for a novel task of *zero-shot segmentation* of complex surgical gesture sequences. Finally, we show that supervised training can be eliminated entirely by using off-the-shelf object detectors to recognize activities in the surveillance domain. **Acknowledgements.** We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan V GPUs used for this research. This work is supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (DOI/IBC) contract number D17PC00345. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DOI/IBC, or the U.S. Government. [^1]: Authors contributed equally to this work. [^2]: https://actev.nist.gov/ [^3]: Note that one could alternatively use the weight of *all paths* through the machine. Since the best-path and total-path weights can be computed using the same dynamic program by changing the definitions of $(+, \times)$ [@goodman-1999], we focus on the best path without loss of generality. [^4]: RC3D [@Rc3d] was the leading approach and still serves as a strong end-to-end baseline for multiple large-scale activity detection benchmarks such as THUMOS [@THUMOS14]. Please see the current DIVA evaluation leaderboard at: https://actev.nist.gov/sdl

--- abstract: 'The *containment rate* of query $Q1$ in query $Q2$ over database $D$ is the percentage of $Q1$’s result tuples over $D$ that are also in $Q2$’s result over $D$. We directly estimate containment rates between pairs of queries over a specific database. For this, we use a specialized deep learning scheme, CRN, which is tailored to representing pairs of SQL queries. Result-cardinality estimation is a core component of query optimization. We describe a novel approach for estimating queries’ result-cardinalities using estimated containment rates among queries. This containment rate estimation may rely on CRN or embed, unchanged, known *cardinality* estimation methods. Experimentally, our novel approach for estimating cardinalities, using containment rates between queries, on a challenging real-world database, realizes significant improvements to state of the art cardinality estimation methods.' author: - | Rojeh Hayek\ \ \ Oded Shmueli\ \ \ bibliography: - 'main.bib' nocite: '[@*]' title: Improved Cardinality Estimation by Learning Queries Containment Rates --- Introduction ============ Query $Q1$ is contained in (resp. equivalent to), query $Q2$, analytically, if for all the database states $D$, $Q1$’s result over $D$ is contained in (resp., equals) $Q2$’s result over $D$. Query containment is a well-known concept that has applications in query optimization. It has been extensively researched in database theory, and many algorithms were proposed for determining containment under different assumptions [@cnt1; @cnt2; @cnt3; @cnt4]. However, determining query containment analytically is not practically sufficient. Two queries may be analytically unrelated by containment, although, the execution result on a *specific* database of one query may actually be contained in the other. For example, consider the queries:\ Q1: *select \* from movies where title = ’Titanic’*\ Q2: *select \* from movies where release = 1997 and director = ’James Cameron’*\ Both queries execution results are identical since there is only one movie called Titanic that was released in 1997 and directed by James Cameron (he has not directed any other movie in 1997). Yet, using the analytic criterion, the queries are unrelated at all by containment. To our knowledge, while query containment and equivalence have been well researched in past decades, determining the containment rate between two queries on a *specific* database, has not been considered by past research. By definition, the containment rate of query $Q1$ in query $Q2$ on database $D$ is the percentage of rows in $Q1$’s execution result over $D$ that are also in $Q2$’s execution result over $D$. Determining containment rates allows us to solve other problems, such as determining equivalence between two queries, or whether one query is fully contained in another, on the same *specific* database. In addition, containment rates can be used in many practical applications, for instance, query clustering, query recommendation [@SimTuples; @SimStructure], and in cardinality estimation as will be described subsequently. Our approach for estimating containment rates is based on a specialized deep learning model, CRN, which enables us to express query features using sets and vectors. An input query is converted into three sets, $T$, $J$ and $P$ representing the query’s tables, joins and column predicates, respectively. Each element of these sets is represented by a vector. Using these vectors, CRN generates a single vector that represents the whole input query. Finally, to estimate the containment rate of two represented queries, CRN measures the distance between the representative vectors of both queries, using another specialized neural network. Thus, the CRN model relies on the ability of the neural network to learn the vector representation of queries relative to the *specific* database. As a result, we obtain a small and accurate model for estimating containment rates. In addition to the CRN model, we introduce a novel technique for estimating queries’ cardinalities using estimated query containment rates. We show that using the proposed technique we improve current cardinality estimation techniques significantly. This is especially the case when there are multiple joins, where the known cardinality estimation techniques suffer from under-estimated results and errors that grow exponentially as the number of joins increases [@joinsHard]. Our technique estimates the cardinalities more robustly (x150/x175 with 4 joins queries, and x1650/x120 with 5 joins queries, compared with PostgreSQL and MSCN, respectively). We compare our technique with PostgreSQL [@postgreSQL], and the pioneering MSCN model [@LearnedCrd], by examining, on the real-world IMDb database [@HowGoodCar], join crossing correlations queries which are known to present a tough challenge to cardinality estimation methods [@HowGoodCar; @crdHard; @JoinCross]. We show that by employing known existing cardinality estimation methods for containment estimation, we can improve on their cardinality estimates as well, without changing the methods themselves. Thus, our novel approach is highly promising for solving the cardinality estimation problem, the “Achilles heel” of query optimization [@crdHard2], a cause of many performance issues [@HowGoodCar]. The rest of this paper is organized as follows. In Section \[Containment Rate Definition\] we define the containment rate problem and in Sections \[Learned Containment Rates\]-\[Containment Evaluation\] we describe and evaluate the CRN model for solving this problem. In Sections \[Cardinality Estimation Using Containment Rates\]-\[Cardinality Evaluation\] we describe and evaluate our new approach for estimating cardinalities using containment rates. In Section \[Improving Existing Cardinality Estimation Models\] we show how one can adapt the new ideas to improve existing cardinality estimation models. Sections \[Related work\]-\[Conclusion\] present related work, conclusions and future work. Containment Rate Definition {#Containment Rate Definition} =========================== We define the containment rate between two queries $Q1$, and $Q2$ on a *specific* database $D$. *Query $Q1$ is $x\%$-contained in query $Q2$ on database $D$ if precisely $x\%$ of $Q1$’s execution result rows on database $D$ are also in $Q2$’s execution result on database $D$.* The containment rate is formally a function from **QxQxD** to **R**, where **Q** is the set of all queries, **D** of all databases, and **R** the Real numbers. This function can be directly calculated using the cardinality of the results of queries $Q1$ and $Q2$ as follows: $$x\% = \frac{|Q1(D)\ {\mbox{$\cap$}}\ Q2(D)|}{|Q1(D)|} * 100$$ Where, $Q(D)$ denotes $Q$’s execution result on database $D$. (in case $Q1$’s execution result is empty, then $Q1$ is 0%-contained in $Q2$). Note that the containment rate is defined only on pairs of queries whose SELECT and FROM clauses are *identical*. Containment Rate Operator ------------------------- We denote the containment rate *operator* between queries $Q1$ and $Q2$ on database $D$ as: $$Q1 \subset_{\%}^D Q2$$ Operator $\subset_{\%}^D$ returns the containment rate between the given input queries on database $D$. That is, $Q1 \subset_{\%}^D Q2$ returns $x\%$, if $Q1$ is $x\%$-contained in query $Q2$ on database $D$. For simplicity, we do not mention the *specific* database, as it is usually clear from context. Therefore, we write the containment rate operator as $\subset_{\%}$. Learned Containment Rates {#Learned Containment Rates} ========================= From a high-level perspective, applying machine learning to the containment rate estimation problem is straightforward. Following the training of the CRN model with pairs of queries $(Q1,Q2)$ and the actual containment rates $Q1 \subset_{\%} Q2$, the model is used as an estimator for other, unseen pairs of queries. There are, however, several questions whose answers determine whether the machine learning model (CRN) will be successful. (1) Which supervised learning algorithm/model should be used. (2) How to represent queries as input and the containment rates as output to the model (“featurization”). (3) How to obtain the initial training dataset (“cold start problem”). Next, we describe how we address each one of these questions. Cold Start Problem {#Defining the database and the development set} ------------------ ### Defining the Database {#Defining the database} We generated a training-set, and later on evaluated our model on it, using the IMDb database. IMDb contains many correlations and has been shown to be very challenging for cardinality estimators [@HowGoodCar]. This database contains a plethora of information about movies and related facts about actors, directors, and production companies, with more than 2.5M movie titles produced over 130 years (starting from 1880) by 235,000 different companies with over 4M actors. ### Generating the Development Dataset {#queries generator} Our approach for solving the “cold start problem” is to obtain an initial training corpus using a specialized queries generator that randomly generates queries based on the IMDB schema and the actual columns values. Our queries generator generates the dataset in three main steps. In the first step, it repeatedly generates multiple SQL queries as follows. It randomly chooses a set of tables $t$, that can join with each other in the database. Then, it adds the corresponding join edges to the query. For each base table $bt$ in the chosen set of tables $t$, it uniformly draws the number of query predicates $p_{bt}$ (0 $\leq p_{bt} \leq$ number of non-key columns in table $bt$). Subsequently, for each predicate it uniformly draws a non-key column from the relevant table $bt$, a predicate type $(<,\ =,\ or\ >)$, and a value from the corresponding column values range in the database. To avoid a combinatorial explosion, and to simplify the problem that the model needs to learn, we force the queries generator to create queries with up to two joins and let the model generalize to a larger number of joins. Note that all the generated queries include a SELECT \* clause. They are denoted as *initial-queries*. To create pairs of queries that are contained in each other with different containment rates, we generate, in the second step, queries that are “similar” to the *initial-queries*, but still, different from them, as follows. For each query $Q$ in *initial-queries*, the generator repeatedly creates multiple queries by randomly changing query $Q$’s predicates’ types, or the predicates’ values, and by randomly adding additional predicates to the original query $Q$. This way, we create a “hard” dataset, which includes pairs of queries that look “similar”, but having mutual containment rates that vary significantly. Finally, in the third and last step, using the queries obtained from both previous steps, the queries generator generates pairs of queries whose FROM clauses are identical. (Note that our queries generator’s first step is similar to MSCN’s generator [@LearnedCrd], however, in the second step we create more complicated queries). After generating the dataset, we execute the dataset queries on the IMDb database, to obtain their true containment rates. Using this process, we obtain an initial training set for our model, which consists of 100,000 pairs of queries with zero to two joins. We split the training samples into 80% training samples and 20% validation samples. Model {#Model} ----- Featurizing all the queries’ literals and predicates as one “big hot vector”, over all the possible words that may appear in the queries, is impractical. Also, serializing the queries’ SELECT, FROM, and WHERE clauses elements into an ordered sequence of elements, is not practical, since the order in these clauses is arbitrary. Thus, standard deep neural network architectures such as simple multi-layer perceptrons [@RNNMLPCNN], convolutional neural networks [@RNNMLPCNN], or recurrent neural networks [@RNNMLPCNN], are not directly proper to our problem. Our *Containment Rate Network* (CRN) model uses a specialized vector representation for representing the input queries and the output containment rates. As depicted in Figure \[CRN Model Archeticture\], the CRN model runs in three main stages. Consider an input queries pair $(Q1,Q2)$. In the first stage, we convert $Q1$ (resp., $Q2$) into a set of vectors $V1$ (resp., $V2$). Thus $(Q1,Q2)$ is represented by $(V1,V2)$. In the second stage, we convert set $V1$ (resp., $V2$) into a unique single representative vector $Qvec1$ (resp., $Qvec2$), using a specialized neural network, $MLP_i$, for each set separately. In the third stage, we estimate the containment rate $Q1 \subset_{\%} Q2$, using the representative vectors $Qvec1$ and $Qvec2$, and another specialized neural network, $MLP_{out}$. ![CRN Model Archeticture.[]{data-label="CRN Model Archeticture"}](CRN.PNG){width="\linewidth"} {width="\linewidth"} ### First Stage, from $(Q1,Q2)$ to $(V1,V2)$ In the same way as MSCN model [@LearnedCrd], we represent each query $Q$ as a collection of three sets $(T,J,P)$. $T$ is the set of all the tables in $Q$’s FROM clause. $J$ is the set of all the joins (i.e., join clauses) in $Q$’s WHERE clause. $P$ is the set of all the (column) predicates in $Q$’s WHERE clause. Using sets $T$, $J$, and $P$, we obtain a set of vectors $V$ representing the query, as described later. Unlike MSCN, in our model all the vectors of set $V$ have the same dimension and the same segmentation as depicted in Table \[table:Vector Segmentation\], where $\#T$ is the number of all the tables in the database, $\#C$ is the number of all the columns in all the database tables, and $\#O$ is the number of possible predicates operators. In total, the vector dimension is $\#T + 3*\#C + \#O + 1$, denoted as $L$. The queries tables, joins and column predicates (sets $T$, $J$ and $P$) are inseparable, hence, treating each set individually using different neural networks may disorientate the model. Therefore, we choose to featurize these sets using the same vector format in order to ease learning. Element of sets $T$, $J$, and $P$, are represented by vectors as follows (see a simple example in Figure \[example\]). All the vectors have the same dimension $L$. Each table $t \in T$ is represented by a unique one-hot vector (a binary vector of length $\#T$ with a single non-zero entry, uniquely identifying a specific table) placed in the T-seg segment. Each join clause of the form $(col1,=,col2) \in J$ is represented as follows. $col1$ and $col2$ are represented by a unique one-hot vectors placed in J1-seg and J2-seg segments, respectively. Each predicate of the form $(col,op,val) \in P$ is represented as follows. $col$ and $op$ are represented by a unique one-hot vectors placed in the C-seg and V-seg segments, respectively. $val$ is represented as a normalized value $\in [0, 1]$, normalized using the minimum and maximum values of the respective column, placed in the V-seg segment. For each vector, all the other unmentioned segments are zeroed. Given input queries pair, $(Q1,Q2)$, we convert query $Q1$ (resp., $Q2$) into sets $T$, $J$ and $P$, and *each* element of these sets is represented by a vector as described above, together generating set $V1$ (resp., $V2$). ### Second Stage, from $(V1,V2)$ to $(Qvec1,Qvec2)$ Given set of vectors $V_i$, we present each vector of the set into a fully-connected one-layer neural network, denoted as $MLP_i$, converting each vector into a vector of dimension $H$. The final representation $Qvec_i$ for this set is then given by the average over the individual transformed representations of its elements, i.e., $$Qvec_i = \frac{1}{|V_i|} \sum_{v \in V_i} MLP_i (v)$$ $$MLP_i(v) = Relu(vU_i + b_i)$$ Where $U_i \in R^{LxH}$, $b_i \in R^{H}$ are the learned weights and bias, and $v \in R^{L}$ is the input row vector. We choose an average (instead of, e.g., sum) to ease generalization to different numbers of elements in the sets, as otherwise the overall magnitude of $Qvec$ would vary depending on the number of elements in the set $V_i$. ### Third Stage, from $(Qvec1,Qvec2)$ to $Q1 \subset_{\%} Q2$ Given the representative vectors of the input queries,\ $(Qvec1,Qvec2)$, we aim to predict the containment rate $Q1 \subset_{\%} Q2$ as accurately as possible. Since we do not know what a “natural” distance measure is in the representative queries vector space, encoded by the neural networks of the second step, we use a fully-connected two-layer neural network, denoted as $MLP_{out}$, to compute the estimated containment rate of the input queries, leaving it up to this neural network to learn the correct distance function. $MLP_{out}$ takes as input a vector of size $4H$ which is constructed from $Qvec1$ and $Qvec2$. The first layer converts the input vector into a vector of size $2H$. The second layer converts the obtained vector of size $2H$, into a single value representing the containment rate. $$\hat{y} = MLP_{out}(Expand(Qvec1,Qvec2))$$ $$MLP_{out}(v) = Sigmoid(ReLU(vU_{out1} + b_{out1})U_{out2} + b_{out2})$$ $$Expand(v_1,v_2) = [v_1,\ \ v_2,\ \ abs(v_1 - v_2),\ \ v_1 \odot v_2]$$ Here, $\hat{y}$ is the estimated containment rate, $U_{out1} \in R^{4Hx2H}$, $b_{out1} \in R^{2H}$ and $U_{out2} \in R^{2Hx1}$, $b_{out2} \in R^{1}$ are the learned weights and bias, and $\odot$ is the dot-product function. In order to estimate the containment rates more accurately, we use the $Expand$ function which creates a row concatenated vector of size $4H$ using vectors $Qvec1$ and $Qvec2$. We use the $ReLU$[^1] activation function for hidden layers in all the neural networks, as they show strong empirical performance advantages and are fast to evaluate. In the final step, we apply the $Sigmoid$[^2] activation function in the second layer to output a float value in the range \[0,1\], as the containment rate values are within this interval. Therefore, we do not apply any featurization on the containment rates (the output of the model) and the model is trained with the actual containment rate values without any featurization steps. ### Loss Function Since we are interested in minimizing the ratio between the predicted and the actual containment rates, we use the q-error metric in our evaluation. We train our model to minimize the mean q-error [@qerror], which is the ratio between an estimated and the actual contaminate rate (or vice versa). Let $y$ be the true containment rate, and $\hat{y}$ the estimated rate, then the q-error is defined as follows. $$q-error(y,\hat{y})\ =\ \hat{y} > y\ ?\ \frac{\hat{y}}{y}\ :\ \frac{y}{\hat{y}}$$ In addition to optimizing the mean q-error, we also examined the mean squared error (MSE) and the mean absolute error (MAE) as optimization goals. MSE and MAE would optimize the squared/absolute differences between the predicted and the actual containment rates. Optimizing with theses metrics makes the model put less emphasis on heavy outliers (that lead to large errors). Therefore, we decided to optimize our model using the q-error metric which yielded better results. Training and Testing Interface ------------------------------ Building CRN involves two main steps. (1) Generating a random training set using the schema and data information as described in Section \[Defining the database and the development set\]. (2) Repeatedly using this training data, we train the CRN model as described in Section \[Model\] until the mean q-error of the validation test starts to converges to its best absolute value. That is, we use the early stopping technique [@earlyStopping] and stop the training before convergence to avoid over-fitting. Both steps are performed on an immutable snapshot of the database. After the training phase, to predict the containment rate of an input query pair, the queries first need to be transformed into their feature representation, and then they are presented as input to the model, and the model outputs the estimated containment rate (Section \[Model\]). We train and test our model using the Tensor-Flow framework [@tensorFlow], and make use of the efficient Adam optimizer [@adam] for training the model. Hyperparameter Search {#Hyperparameter Search} --------------------- To optimize our model’s performance, we conducted a search over its hyperparameter space. In particular, we focused on tuning the neural networks hidden layer size (H). Note that the same H value is shared in all the neural networks of the CRN model, as described in section \[Model\]. During the tuning of the size hyperparameter of the neural network hidden layer, we found that increasing the size of our hidden layer generally led to an increase in the model accuracy, till it reached the best mean q-error on the validation test. Afterwards, the results began to decline in quality because of over-fitting (See Figure \[The mean q-error on the validation set with different hidden layer sizes\]). Hence, we choose a hidden layer of size 512, as a good balance between accuracy and training time. Overall, we found that our model performs similarly well across a wide range of settings when considering different batch sizes and learning rates. ![The mean q-error on the validation set with different hidden layer sizes.[]{data-label="The mean q-error on the validation set with different hidden layer sizes"}](H.PNG){width="\linewidth"} Model Computational Costs ------------------------- We analyze the training, prediction, and space costs of the CRN model with the default hyperparameters (H=512, batch size=128, learning rate=0.001). ### Training Time Figure \[Convergence of the mean q-error on the validation set\] shows how the mean q-error of the validation set decreases with additional epochs, until convergence to a mean q-error of around 4.5. The CRN model requires almost 120 passes on the training set to converge. On average, measured across six runs, a training run with 120 epochs takes almost 200 minutes. ### Prediction Time The prediction process is dominated by converting the input queries into the corresponding vectors, and then presenting these vectors as input to the CRN model. On average, the prediction time is 0.5ms per single pair of queries, including the overhead introduced by the Tensor-Flow framework. ### Model Size The CRN model includes all the learned parameters mentioned in Section \[Model\] ($U_1$, $U_2$, $U_{out1}$, $U_{out2}$, $b_1$, $b_2$, $b_{out1}$, $b_{out2}$). In total, there are $2*L*H + 8*H^2 + 6*H +1$ learned parameters. In practice, the size of the model, when serialized to disk, is roughly $1.5MB$. ![Convergence of the mean q-error on the validation set.[]{data-label="Convergence of the mean q-error on the validation set"}](validationConverage.PNG){width="\linewidth"} Containment Evaluation {#Containment Evaluation} ====================== In this section we describe how we compared the CRN model to other (baseline) methods. Since to the best of our knowledge, the problem of determining containment rate has not been addressed till now, we used a transformation as described in Section \[From Cardinality to Containment\] below. From Cardinality to Containment {#From Cardinality to Containment} ------------------------------- To our knowledge, this is the first work to address the problem of containment rate estimation. In order to compare our results with different baseline methods, we used existing cardinality estimation methods to predict the containment rates, using the Crd2Cnt transformation, as depicted in the middle part diagram in Figure \[fig:teaser\]. ### The Crd2Cnt Transformation {#Crd2Cnt Transformation} Given a cardinality estimation model[^3] $M$, we can convert it to a containment rate estimation model using the Crd2Cnt transformation which returns a model $M'$ for estimating containment rates. The obtained model $M'$ functions as follows. Given input queries $Q1$ and $Q2$, whose containment rate $Q1 \subset_{\%} Q2$ needs to be estimated: - Calculate the cardinality of query $Q1{\mbox{$\cap$}}Q2$ using $M$. - Calculate the cardinality of query $Q1$ using $M$. - Then, the containment rate estimate is: $$Q1 \subset_{\%} Q2 = \frac{|Q1{\mbox{$\cap$}}Q2|}{|Q1|}$$ Here, $Q1{\mbox{$\cap$}}Q2$ is the intersection query of $Q1$ and $Q2$ whose SELECT and FROM clauses are identical to $Q1$’s (or $Q2$’s) clauses, and whose WHERE clause is $Q1$’s AND $Q2$’s WHERE clauses. Note that, by definition, if $|Q1|=0$ then $Q1 \subset_{\%} Q2=0$. Given model $M$, we denote the obtained model $M'$, via the Crd2Cnt transformation, as Crd2Cnt($M$). We compared the CRN model predictions to those based on the other examined cardinality estimation models, using the Crd2Cnt transformation. We examined the PostgreSQL version 11 cardinality estimation component [@postgreSQL], a simple and commonly used method for cardinality estimation. In addition, we examined the MSCN model [@LearnedCrd]. MSCN was shown to be superior to the best methods for estimating cardinalities such as Random Sampling (RS) [@RS1; @RS2] and the state-of-the-art Index-Based Join Sampling (IBJS) [@IBJS]. ### Comparing with MSCN {#Comparing with MSCN} In order to make a fair comparison between the CRN model and the MSCN model, we train the MSCN model with the *same* data that was used to train the CRN model. The CRN model takes two queries as input, whereas the MSCN model takes one query as input. Therefore, to address this issue, we created the training dataset for the MSCN model as follows. For each pair of queries $(Q1,Q2)$ used in training the CRN model, we added the following two input queries to the MSCN training set: - $Q1 {\mbox{$\cap$}}Q2$, along with its actual cardinality. - $Q1$, along with its actual cardinality. Finally, we ensure that the training set includes only unique queries without repetition. This way, we both models, MSCN and CRN, are trained with the *same* information. ### Comparing with PostgreSQL Comparing with PostgreSQL does not require generating an appropriate training set, since the PostgreSQL cardinality estimation component is based on database profiling techniques and does not require training. Evaluation Workloads -------------------- We evaluate CRN on the IMDb dataset as described in Section \[Defining the database\], using two different query workloads: - cnt\_test1, a synthetic workload generated by the same queries generator as the one used for generating the training data (using a different random seed) with 1200 unique query pairs, with zero to two joins. - cnt\_test2, a synthetic workload generated by the same queries generator as the one used for generating the training data (using a different random seed) with 1200 unique query pairs, with zero to *five* joins. This dataset is used to examine how CRN generalizes to additional joins. The Quality of Estimates {#mylabel} ------------------------ Figure \[fig:cnt\_test1\] depicts the q-error of the CRN model compared to the Crd2Cnt(PostgreSQL) and Crd2Cnt(MSCN) models on the cnt\_test1 workload. While Crd2Cnt(PostgreSQL)’s errors are more skewed towards the positive spectrum,\ Crd2Cnt(MSCN) performs extremely well as does the CRN model. Observe that we make sure to train MSCN in such a way that it will predict containment rates efficiently, while the primary purpose of the MSCN model, as described in [@LearnedCrd] is estimating cardinalities. That is, had we trained the MSCN model for its main purpose, with “independent” queries, we might have ended up with worse results for MSCN. To provide a fuller picture, we also show the percentiles, maximum, and mean q-errors. As depicted in Table \[table:cnt\_test1\], CRN provides the best results in 75% of the tests, whereas MSCN is more robust in the margins, resulting in a better mean. ![Estimation errors on the cnt\_test1 workload. In all the similar plots presented in this paper, the box boundaries are at the 25th/75th percentiles and the horizontal lines mark the 5th/95th percentiles. Hence, 50% of the tests results are located within the box boundaries, and 90% are located between the horizontal lines. The orange horizontal line mark the 50th percentile. []{data-label="fig:cnt_test1"}](cnt1.PNG){width="\linewidth"} Generalizing to Additional Joins {#Generalizing to More Joins} -------------------------------- In this section we examine how the CRN model generalizes to queries with a higher number of joins without having seen such queries during training. To do so, we use the crd\_test2 workload which includes queries with zero to *five* joins. Recall that we trained both the CRN model and the MSCN model only with query pairs that have between zero and two joins. Examining the results, described in Table \[table:cnt\_test2\] and Figure \[fig:cnt\_test2\], the CRN model is noticeably more robust in generalizing to queries with additional joins. The mean q-error of the CRN model is smaller by a factor of almost 8 than the mean q-errors of the other models. ![Estimation errors on the cnt\_test2 workload.[]{data-label="fig:cnt_test2"}](cnt2.PNG){width="\linewidth"} {width="\linewidth"} Cardinality Estimation Using\ Containment Rates {#Cardinality Estimation Using Containment Rates} ============================= In this section we consider one application of the proposed containment rate estimation model: cardinality estimation. We introduce a novel approach for estimating cardinalities using query containment rates, and we show that using the proposed approach, we improve cardinality estimations significantly, especially in the case when there are multiple joins. Since a traditional query optimizer is crucially dependent on cardinality estimation, which enables choosing among different plan alternatives by using the cardinality estimation of intermediate results within query execution plans. Therefore, the query optimizer must use reasonably good estimates. However, estimates produced by all widely-used database cardinality estimation models are routinely significantly wrong (under/over-estimated), resulting in not choosing the best plans, leading to slow executions [@HowGoodCar]. Two principal approaches for estimating cardinalities have emerged. (1) Using database profiling [@postgreSQL]. (2) Using sampling techniques [@RS1; @RS2; @IBJS]. Recently, deep learning neural networks were also used for solving this problem [@LearnedCrd; @LearnedCrd2]. However, all these approaches, with all the many attempts to improve them, have conceptually addressed the problem *directly* in the same way, as a black box, where the input is a query, and the output is its cardinality estimation, as described in the leftmost diagram in Figure \[fig:teaser\]. In our proposed approach, we address the problem differently, and we obtain better estimates as described in Section \[Cardinality Evaluation\]. In today’s databases, the answer to a previous query is rarely useful for speeding up new queries, and the work performed in answering past queries is often ignored afterwards. Using the CRN model for predicting containment rates, we are able to change this by revealing the underlying relations between the new queries and the previous ones. Our new technique for estimating cardinalities mainly relies on two key ideas. The first one is the new framework in which we solve the problem. The second is the use of a *queries pool* that maintains multiple previously executed queries along with their actual cardinalities, as part of the database meta information. The queries pool provides new information that enables our technique to achieve better estimates. Using a containment rate estimation model, we make use of previously executed queries along with their actual cardinalities to estimate the result-cardinality of a new query. This is done with the help of a simple transformation from the problem of containment rate estimation to the problem of cardinality estimation (see Section \[From Containment to Cardinality\]). From Containment to Cardinality {#From Containment to Cardinality} ------------------------------- Using a containment rate estimation models, we can obtain cardinality estimates using the Cnt2Crd transformation, as depicted in the rightmost diagram in Figure \[fig:teaser\]. ### The Cnt2Crd Transformation {#Cnt2Crd Transformation} Given a containment rate estimation model[^4] $M$, we convert it to a cardinality estimation model using the Cnt2Crd transformation which returns a model $M'$ for estimating cardinalities. The obtained model $M'$ functions as follows. Given a “new” query, denoted as $Q_{new}$, as input to cardinality estimation, and assuming that there is an “old” query denoted as $Q_{old}$, whose FROM clause is the same as $Q_{new}$’s FROM clause, that has already been executed over the database, and therefore $|Q_{old}|$ is known, we: - Calculate $x\_rate = Q_{old} \subset_{\%} Q_{new}$ using $M$. - Calculate $y\_rate = Q_{new} \subset_{\%} Q_{old}$ using $M$. - Then, the cardinality estimate equals to: $$|Q_{new}| = \frac{x\_rate}{y\_rate} * |Q_{old}|$$ provided that $y\_rate = Q_{new} \subset_{\%} Q_{old}\ \neq 0$. Given model $M$, we denote the obtained model $M'$, via the Cnt2Crd transformation, as Cnt2Crd($M$). Queries Pool ------------ Our technique for estimating cardinality mainly relies on a queries pool that includes records of multiple queries. The queries pool is envisioned to be an additional component of the DBMS, along with all the other customary components. It includes multiple queries with their actual cardinalities[^5], without the queries execution results. Therefore, holding such a pool in the DBMS as part of its meta information does not require significant storage space or other computing resources. Maintaining a queries pool in the DBMS is thus a reasonable expectation. The DBMS continuously executes queries, and therefore, we can easily configure the DBMS to store these queries along with their actual cardinalities in the queries pool. In addition, we may generate in advance a queries pool using a queries generator that randomly creates multiple queries with many of the possible joins, and with different column predicates. We then execute these queries on the database to obtain and save their actual cardinalities in the queries pool. Notice that, we can combine both approaches (actual computing and a generator) to create the queries pool. The advantage of the first approach is that in a real-world situation, queries that are posed in sequence by the same user, may be similar and therefore we can get more accurate cardinality estimates. The second approach helps in cases where the queries posed by users are diverse (e.g., different FROM clauses). Therefore, in such cases, we need to make sure, in advance, that the queries pool contains sufficiently many queries that cover all the possible cases. Given a query $Q$ whose cardinality is to be estimated , it is possible that we fail to find any appropriate query, in the queries pool, to match with query $Q$. That happens when all the queries in the queries pool have a different FROM clause than that of query $Q$, or that they are not contained at all in query $Q$. In such cases we can always rely on the *known* basic cardinality estimation models. In addition, we can make sure that the queries pool includes queries with the most frequent used FROM clauses, with empty column predicates. That is, queries of the following form.\ *SELECT \* FROM -set of tables- WHERE TRUE*. In this case, for most of the queries posed in the database, there is at least one query that matches in the queries pool with the given query, and hence, we can estimate the cardinality without resorting to the basic cardinality estimation models. A Cardinality Estimation Technique {#A Cardinality Estimation Technique} ---------------------------------- Consider a new query $Q_{new}$, and assume that the DBMS includes a queries pool as previously described. To estimate the cardinality of $Q_{new}$ accurately, we use *multiple* “old” queries instead of *one* query, using the same Cnt2Crd transformation of Section \[Cnt2Crd Transformation\], as described in Figure \[Cardinality Estimation Technique\]. EstimateCardinality(Query $Q_{new}$, Queries Pool $QP$):\ $results$ = empty list\ \ For every pair $(Q_{old},|Q_{old}|)$ in $QP$:\ if $Q_{old}$’s FROM clause $\neq$ $Q_{new}$’s FROM clause:\ continue\ Calculate $ x\_rate = Q_{old} \subset_{\%} Q_{new}$\ Calculate $ y\_rate = Q_{new} \subset_{\%} Q_{old}$\ if $y\_rate <= epsilon$: /\* y equals zero \*/\ continue\ $results$.append($x\_rate/y\_rate * |Q_{old}|$)\ \ return F($results$) Estimating cardinality considers all the *matching* queries whose FROM clauses are identical to $Q_{new}$’s FROM clause. For each matching query, we estimate $Q_{new}$’s cardinality using the Cnt2Crd transformation and save the estimated result in the $results$ list. The final cardinality is obtained by applying the final function, $F$, that converts all the estimated results recorded in the $results$ list, into a single final estimation value. Note that the technique can be easily parallelized since each iteration in the For loop is independent, and thus can be calculated in parallel. ### Comparing Different Final Functions {#Comparing Different Final Functions} We examined various final functions ($F$), including: - Median, returning the median value of the $results$ list. - Mean, returning the mean value of the $results$ list. - Trimmed mean, returning the trimmed mean of the $results$ list without the 25% outliers (trimmed mean removes a designated percentage of the largest and smallest values before calculating the mean). Experimentally, the cardinality estimates using the various functions were very similar in terms of q-error. But the Median function yielded the best estimates (we do not detail these experiments due to limited space). Cardinality Evaluation {#Cardinality Evaluation} ====================== \[Evalutaion Crd\] We evaluate our proposed technique for estimating cardinality, with different test sets, while using the CRN model as defined in Section \[Model\] for estimating containment rates. We compare our cardinality estimates with those of the PostgreSQL version 11 cardinality estimation component [@postgreSQL], and the MSCN model [@LearnedCrd]. We train both the CRN model and the MSCN model with the *same* training set as described in Section \[Comparing with MSCN\]. Also, we create the test workloads using the same queries generator used for creating the training set of the CRN and the MSCN models (described in Section \[queries generator\]), while skipping its last step. That is, we only run the first two steps of the generator. The third step creates query pairs which are irrelevant for the cardinality estimation task. Evaluation Workloads {#Evaluation Workloads} -------------------- We evaluate our approach on the (challenging) IMDb dataset, using three different query workloads: - crd\_test1, a synthetic workload generated by the same queries generator that was used for creating the training data of the CRN model, as described in Section \[Defining the database and the development set\] (using a different random seed) with 450 unique queries, with zero to two joins. - crd\_test2, a synthetic workload generated by the same queries generator as the training data of the CRN model, as described in Section \[Defining the database and the development set\] (using a different random seed) with 450 unique queries, with zero to *five* joins. This dataset is designed to examine how the technique generalizes to additional joins. - scale, another synthetic workload, with 500 unique queries, derived from the MSCN test set as introduced in [@LearnedCrd]. This dataset is designed to examine how the technique generalizes to queries that were *not* created with the same trained queries’ generator. Queries Pool {#Queries Pool} ------------ Our technique relies on a queries pool, we thus created a synthetic queries pool, $QP$, generated by the same queries generator as the training data of the containment rate estimation model, as described in Section \[Defining the database and the development set\] (using a different random seed) with 300 queries, equally distributed among all the possible FROM clauses over the database. In particular, $QP$, covers all the possible FROM clauses that are used in the tests workloads. Note that, there are no shared queries between $QP$ queries and the test workloads queries. Consider a query $Q$ whose cardinality needs to be estimated. On the one hand, the generated $QP$ contains “similar” queries to query $Q$, these can help the machine in predicting the cardinality. On the other hand, it also includes queries that are not similar at all to query $Q$, that may cause erroneous cardinality estimates. Therefore, the generated queries pool $QP$, faithfully represents a real-world situation. Experimental Environment {#Experimenal Environment} ------------------------ In all the following cardinality estimation experiments, for predicting the cardinality of a given query $Q$ in a workload $W$, we use the whole queries pool $QP$ as described in Section \[Queries Pool\] with all its 300 queries. That is, the “old” queries used for predicting cardinalities, are the queries of $QP$. In addition, in all the experiments we use the Median function as the final $F$ function. The Quality of Estimates {#the-quality-of-estimates} ------------------------ Figure \[fig:crd\_test1\] depicts the q-error of the Cnt2Crd(CRN) model as compared to MSCN and PostgreSQL on the crd\_test1 workload. While PostgreSQL’s errors are more skewed towards the positive spectrum, MSCN is competitive with Cnt2Crd(CRN) in all the described values. As can be seen in Table \[table:crd\_test1\], while MSCN provides the best results in the margins, the Cnt2Crd(CRN) model is more accurate in 75% of the test. In addition, we show in the next section (Section \[Generalizing to additional joins-CRD\]) that the Cnt2Crd(CRN) model is more robust when considering queries with more joins than in the training dataset. ![Estimation errors on the crd\_test1 workload.[]{data-label="fig:crd_test1"}](crd1.PNG){width="\linewidth"} Generalizing to Additional Joins {#Generalizing to additional joins-CRD} -------------------------------- We examine how our technique generalizes to queries with additional joins, without having seen such queries during training. To do so, we use the crd\_test2 workload which includes queries with zero to *five* joins. Recall that we trained both the CRN model and the MSCN model only with query pairs that have between zero and two joins. From Tables \[table:crd\_test2\] and \[table:crd\_test2joins3to5\], and Figure \[fig:crd\_test2\], it is clear that Cnt2Crd(CRN) model is significantly more robust in generalizing to queries with additional joins. In terms of mean q-error, the Cnt2Crd(CRN) model reduces the mean by a factor x100 compared with MSCN and by a factor of x1000 compared with PostgreSQL. ![Estimation errors on the crd\_test2 workload.[]{data-label="fig:crd_test2"}](crd2.PNG){width="\linewidth"} As depicted in Figure \[fig:crd\_test2\], the Cnt2Crd(CRN) model generalizes more accurately to additional joins (note that the boxes are still on the same q-error interval). To highlight these improvements, we describe, in Table \[table:Q-error means for each join\] and Figure \[fig:Q-error medians for each join\], the mean and median q-error for each possible number of joins separately (note the logarithmic y-axis scale in Figure \[fig:Q-error medians for each join\]). The known cardinality estimation models suffer from under-estimated results and errors that grow exponentially as the number of joins increases [@joinsHard], as also happens in the cases we examined. The Cnt2Crd(CRN) model was better at handling additional joins (even though CRN was trained only with queries with up to two joins, as was MSCN). ![Q-error medians for each number of joins.[]{data-label="fig:Q-error medians for each join"}](crd2Medians.PNG){width="\linewidth"} Generalizing to Different Kinds of Queries ------------------------------------------ In this experiment, we explore how the Cnt2Crd(CRN) model generalizes to a workload that was not generated by the same queries generator that was used for creating the CRN model training set. To do so, we examine the scale workload that was generated using another queries generator in [@LearnedCrd]. As shown in Table \[table:scale\], clearly Cnt2Crd(CRN) is more robust than MSCN and PostgreSQL in all the described values. Examining Figure \[fig:scaleB\], it is clear that the Cnt2Crd(CRN) model is significantly more robust with queries with 3 and 4 joins. Recall that the $QP$ queries pool in this experiment was not changed, while the scale workload is derived from another queries generator. In summary, this experiment shows that Cnt2Crd(CRN) generalizes to workloads that were created with a different generator than the one used to create the training data. ![Estimation errors on the scale workload.[]{data-label="fig:scaleB"}](scaleB.PNG){width="\linewidth"} To further examine how Cnt2Crd(CRN) generalizes, we conducted the following experiment. We compared the\ Cnt2Crd(CRN) model with an improved version of the MSCN model that combines the deep learning approach and sampling techniques by using samples of 1000 materialized base tables, as described in [@LearnedCrd]. For simplicity we denote this model as MSCN1000. {width="\linewidth"} We make the test easier for MSCN1000 model by training the MSCN1000 model with a training set that was created with the *same* queries generator that was used for generating the scale workload. As depicted in Figure \[fig:scaleB\], while the MSCN1000 model is more robust in queries with zero to two joins, still, the Cnt2Crd(CRN) model was found to be superior on queries with additional joins. Recall that the CRN model training set *was not changed*, while the MSCN1000 model was trained with queries obtained from the *same* queries generator that was used for creating the testing (i.e., scale) workload. In addition, note that MSCN1000 model uses sampling techniques whereas Cnt2Crd(CRN) does not. Thus, this experiment demonstrates the superiority of\ Cnt2Crd(CRN) in generalizing to additional joins. Improving Existing Cardinality Estimation Models {#Improving Existing Cardinality Estimation Models} ================================================ In this section we describe how existing cardinality estimation models can be improved using the idea underlining our proposed technique. The proposed technique for improving existing cardinality estimation models relies on the same technique for predicting cardinalities using a containment rate estimation model, as described in Section \[A Cardinality Estimation Technique\]. In the previous section we used the CRN model in predicting containment rates. CRN can be replaced with *any* other method for predicting containment rates. In particular, it can be replaced with any existing cardinality estimation model after “converting” it to estimating containment rates using the Crd2Cnt transformation, as described in Section \[From Cardinality to Containment\]. At first glance, our proposed technique seems to be more complicated for solving the problem of estimating cardinalities. However, we show that by applying it to known existing models, we improve their estimates, without changing the models themselves. These results indicate that the traditional approach, which directly addressed this problem, straightforwardly, using models to predict cardinalities, can be improved upon. In the remainder of this section, we described the proposed approach, and show how existing cardinality estimation methods are significantly improved upon, by using this technique. Approach Demonstration ---------------------- Given an existing cardinality estimation model $M$, we first convert $M$ to a model $M'$ for estimating containment rates, using the Crd2Cnt transformation, as described in Section \[From Cardinality to Containment\]. Afterwards, given the obtained containment rate estimation model $M'$, we convert it to a model $M''$ for estimating cardinalities, using the Cnt2Crd transformation, as described in Section \[A Cardinality Estimation Technique\], which uses a queries pool. To summarize, our technique converts an existing cardinality estimation model $M$ to an intermediate model $M'$ for estimating containment rates, and then, using $M'$ we create a model $M''$ for estimating cardinalities with the help of the queries pool, as depicted in Figure \[fig:teaser\] from left to right. For simplicity, given cardinality estimation model $M$, we denote the model $M''$ described above, i.e., model\ Cnt2Crd(Crd2Cnt($M$)), as *Improved* $M$ model. Existing Models vs. Improved Models {#Existing Models vs. Improved Models} ----------------------------------- We examine how our proposed technique improves the PostgreSQL and the MSCN models, by using the crd\_test2 workload as defined in Section \[Evaluation Workloads\], as it includes the most number of joins. Table \[table:ImprovedPostgreSQL\] depicts the estimates when using directly the PostgreSQL model, compared with the estimates when adopting our technique with PostgreSQL (i.e., the Improved PostgreSQL model). Similarly, Table \[table:ImprovedMSCN\] depicts the estimates when using directly the MSCN model, compared with the Improved MSCN model. From both tables, it is clear that the proposed technique significantly improves the estimates (by factor x7 for PostgreSQL and x122 for MSCN in terms of mean q-error) without changing the models themselves (embedded within the Improved version). These results highlight the power of our proposed approach that provides an effective and simple technique for improving existing cardinality estimation models. By adopting our approach and crating a queries pool in the database, cardinality estimates can be improved significantly. Improved Models vs. Cnt2Crd(CRN) -------------------------------- Using the crd\_test2 workloade, we examine how our technique improves PostgreSQL and MSCN, compared with\ Cnt2Crd(CRN). Adopting our technique improves the existing models as described in Section \[Existing Models vs. Improved Models\]. Examining Table \[table:Improvedall\], it is clear that in 90% of the tests, the best estimates are those obtained when directly using the CRN model to estimate the containment rates, instead of converting existing cardinality estimation models to obtain containment rates (Improved MSCN and Improved PostgreSQL). Cardinality Prediction Computation Time {#Cardinality Prediction Time} --------------------------------------- Using the proposed idea of using containment rates estimations to predict cardinalities, the cardinality prediction process is dominated by calculating the containment rates of the given input query with the relevant queries in the queries pool, and calculating the final function $F$ on these results to obtain the predicted cardinality, as described in Section \[A Cardinality Estimation Technique\]. Therefore, the larger the queries pool is, the more accurate the predictions are, and the longer the prediction time is. Table \[table:poolSizes\], shows the medians and the means estimation errors on the crd\_test2 workload, along with the average prediction time for a single query, when using the Cnt2Crd(CRN) model for estimating cardinalities, with different sizes of $QP$ (equally distributed over all the possible FROM clauses in the database) while using the same final function $F$ (the Median function)). In table \[table:time\], we compare the average prediction time for estimating the cardinality of a single query using all the examined models (when using the whole $QP$ queries pool of size 300). While the default MSCN model is the fastest model, since it directly estimates the cardinalities without using a queries pool, the Cnt2Crd(CRN) model is the fastest among all the models that use a queries pool. That is, the Cnt2Crd(CRN) model is faster than the Improved MSCN model and the Improved PostgreSQL model. This is the case, since in the Improved MSCN model or the Improved PostgreSQL model, to obtain the containment rates, both models need to estimate cardinalities of two different queries as described in Section \[From Cardinality to Containment\], whereas the CRN model directly obtains a containment rate in one pass. Although the prediction time of the models that use queries pools is higher than the most common cardinality estimation model (PostgreSQL), the prediction time is still in the order of a few milliseconds. In particular, it is similar to the average prediction time of models that use sampling techniques, such as the MSCN version with 1000 base tables samples. Recall that for the results in Table \[table:time\], we used a queries pool ($QP$) of size 300. We could have used a smaller pool, resulting in faster prediction time, and still obtaining better results using the models which employ a queries pool, as depicted in Table \[table:poolSizes\]. Furthermore, all the the models that use queries pools may be easily parallelized as discussed in Section \[A Cardinality Estimation Technique\], and thus, reducing the prediction time (in our tests we ran these models serially). Related Work {#Related work} ============ Over the past five decades, conjunctive queries have been studied in the contexts of database theory and database systems. Conjunctive queries constitute a broad class of frequently used queries. Their expressive power is roughly equivalent to that of the Select-Join-Project queries of relational algebra. Therefore, several problems and algorithms have been researched in depth in this context. Chandra and Merlin [@ChandraMerlin] showed that determining containment of conjunctive queries is an NP-complete problem. Finding the minimal number of conditions that need to be added to a query in order to ensure containment in another query is also an NP-complete problem [@ulmanBook]. This also holds under additional settings involving inclusion and functional dependencies [@ulmanBook; @foundationBook; @dependencies]. Although determining whether query $Q1$ is contained in query $Q2$ (analytically) in the case of conjunctive queries is an intractable problem in its full generality, there are many tractable cases for this problem. For instance, in [@Newcnt2; @Newcnt22] it was shown that query containment in the case of conjunctive queries could be solved in linear time, if every database (edb) predicate occurs at most twice in the body of $Q1$. In [@Newcnt3] it was proved that for every $k \geq 1$, conjunctive query containment could be solved in polynomial time, if $Q2$ has querywidth smaller than $k+1$. In addition to the mentioned cases, there are many other tractable cases [@cnt1; @cnt2; @cnt3; @cnt4]. Such cases are obtained by imposing syntactic or structural restrictions on the input queries $Q1$ and $Q2$. Whereas this problem was well researched in the past, to our knowledge, the problem of determining the containment rate on a *specific* database has not been investigated. In this paper, we address this problem using ML techniques. Lately, we have witnessed extensive adoption of machine learning, and deep neural networks in particular, in many different areas and systems, and in particular in databases. Recent research investigates machine learning for classical database problems such as join ordering [@MLjoinOrder], index structures [@MLindex], query optimization [@MLoptimiztion1; @MLoptimiztion2], concurrency control [@concurrency], and recently in cardinality estimation [@LearnedCrd; @LearnedCrd2]. In this paper, we propose a deep learning-based approach for predicting containment rates on a *specific* database and show how containment rates can be used to predict cardinalities more accurately. There were many attempts to tackle the problem of cardinality estimation; for example, Random Sampling techniques [@RS1; @RS2], Index based Sampling [@IBJS], and recently deep learning [@LearnedCrd; @LearnedCrd2]. However, all these attempts have addressed, conceptually, the problem directly in the same way, as a black box, where the input is a query, and the output is the cardinality estimate. We address this problem differently by using information about queries that have already been executed in the database, together with their actual result cardinalities, and the predicted containment rates between them and the new examined query. Using this new approach, we improve cardinality estimates significantly. Conclusions and Future Work {#Conclusion} =========================== We introduced a new problem, that of estimating containment rates between queries over a *specific* database, and introduced the CRN model, a new deep learning model for solving it. We trained CRN with generated queries, uniformly distributed within a constrained space, and showed that CRN usually obtains the best results in estimating containment rates as compared with other examined models. We introduced a novel approach for cardinality estimation, based on the CRN-based containment rate estimation model, and with the help of a queries pool. We showed the superiority of our new approach in estimating cardinalities more accurately than state-of-the-art approaches. Further, we showed that it addresses the weak spot of existing cardinality estimation models, which is handling multiple joins. In addition, we proposed a technique for improving *any* existing cardinality estimation model without the need to change the model itself, by embedding it within a three step method. Given that the estimates of state-of-the-art models are quite fragile, and that our new approach for estimating cardinalities is simple, has low overhead, and is quite effective, we believe that it is highly promising and practical for solving the cardinality estimation problem. To make our containment based approach suitable for more general queries, the CRN model for estimating containment rates can be extended to support other types of queries, such as the union queries, and queries that include complex predicates. In addition, the CRN model can be configured to support databases that are updated from time to time. Next, we discuss some of these extensions, and sketch possible future research directions. *Strings.* A simple addition to our current implementation may support equality predicates on strings. To do so, we could hash all the possible string literals in the database into the integer domain (similarly to MSCN). This way, an equality predicate on strings can be converted to an equality predicate on integers, which the CRN model can handle. *Complex predicates.* Complex predicates, such as LIKE, are not supported since they are not represented in the CRN model. To support such predicates we need to change the model architecture to handle such predicates. Note that predicates such as BETWEEN and IN, may be converted to ordinary predicates. *SELECT clause.* In this work we addressed only queries with a SELECT \* clause. We can handle queries with SELECT clauses that include specific columns. Given such a query $Q$, $Q$’s cardinality is equivalent to the cardinality of the query with a SELECT \* clause instead, as long as the DISTINCT keyword is not used. *EXCEPT Operator.* Given a query $Q$ of the form\ $Q1$ EXCEPT $Q2$, the CRN model can handle $Q$. In terms of containment rates: $$(Q1\ EXCEPT\ Q2) \subseteq_{\%} Q3 = Q1 \subseteq_{\%} Q3 - (Q1 {\mbox{$\cap$}}Q2) \subseteq_{\%} Q3$$ Using the same idea, we can handle the opposite containment direction case, and the case where there are more than two queries, recursively. The case when considering cardinalities is similar. $$|Q1\ EXCEPT\ Q2| = |Q1| - |Q1 {\mbox{$\cap$}}Q2|$$ *Union Queries.* Given a query $Q$ of the from $Q1$ UNION $Q2$, the CRN model can handle $Q$ as follows: $$(Q1\ UNION\ Q2) \subseteq_{\%} Q3$$ $$= Q1 \subseteq_{\%} Q3 + Q2 \subseteq_{\%} Q3 - (Q1 {\mbox{$\cap$}}Q2) \subseteq{\%} Q3$$ Using the same idea, we can handle the opposite containment direction case, and the case where there are more than two queries, recursively. The case when considering cardinalities is similar. $$|Q1\ UNION\ Q2| = |Q1| + |Q2|$$ *The OR operator.* Given queries that include the operator OR in their WHERE clause, the CRN model does not handle such queries straightforwardly. But, we can handle such queries using a recursive algorithm that converts the queries into multiple conjunctive queries by converting the WHERE clause to DNF, and considering every conjunctive clause as a separate query. $$(Q1\ OR\ Q2) \subseteq_{\%} Q3 = (Q1\ UNION\ Q2) \subseteq_{\%} Q3$$ Using the same idea, we can handle the opposite containment direction case, and the case where there are more than two queries, recursively. The case when considering cardinalities is similar. $$|Q1\ OR\ Q2| = |Q1\ UNION\ Q2| - |Q1 {\mbox{$\cap$}}Q2|$$ *Database updates.* Thus far, we assumed that the database is static (read-only database). However, in many real world databases, updates occur frequently. In addition, the database schema itself may be changed. To handle updates we can use one of the following approaches: \(1) We can always completely re-train the CRN model with a new updated training set. This comes with a considerable compute cost for re-executing queries pairs to obtain up-to-date containment rates and the cost for re-training the model itself. In this approach, we can easily handle changes in the database schema, since we can change the model encodings prior to re-training it. \(2) We can incrementally train the model starting from its current state, by applying new updated training samples, instead of re-training the model from scratch. While this approach is more practical, a key challenge here is to accommodate changes in the database schema. To handle this issue, we could hold, in advance, additional place holders in our model to be used for future added columns or tables. In addition, the values ranges of each column may change when updating the database, and thus, the normalized values may be modified as well. Ways to handle this problem are the subject of current research. [^1]: ReLU(x) = max(0,x); see [@ActivationFunc]. [^2]: Sigmoid(x) = $1/(1+e^{-x})$; see [@ActivationFunc]. [^3]: Here “model” may refer to an ML model or simply to a method. [^4]: Here “model” may refer to an ML model or simply to a method. [^5]: Due to limited space, we do not detail the efficient hash-based data structures used to implement the queries pool.

$$$$ **Heaviside transform of the effective potential** **in the Gross-Neveu model** 1.5cm Hirofumi Yamada *Department of Mathematics, Chiba Institute of Technology* *2-1-1 Shibazono, Narashino-shi, Chiba 275* *Japan* *e-mail:yamadah@cc.it-chiba.ac.jp* **Abstract** Unconventional way of handling the perturbative series is presented with the help of Heaviside transformation with respect to the mass. We apply Heaviside transform to the effective potential in the massive Gross-Neveu model and carry out perturbative approximation of the massless potential by dealing with the resulting Heaviside function. We find that accurate values of the dynamical mass can be obtained from the Heaviside function already at finite orders where just the several of diagrams are incorporated. We prove that our approximants converges to the exact massless potential in the infinite order. Small mass expansion of the effective potential can be also obtained in our approach. [**1 Introduction**]{} Even if the proof of dynamical massless symmetry breaking requires genuine non-perturbative approaches, it does not necessarily mean that the perturbative expansion is totally useless. There is the possibility that non-perturbative quantities in the massless limit may be approximately calculated via perturbative approach. The purpose of this paper is to explore the possibility and show a concrete affirmative result by re-visiting the Gross-Neveu model${^{1}}$. Let us consider the effective potential of the Gross-Neveu model. As is well known, ordinary massless perturbation expansion gives infra-red divergences and to cure the problem one must sum up all the one-loop diagrams. Then the summed result reveals the non-trivial vacuum configuration of $<\bar \psi \psi>$ and the dynamical generation of the mass. The point we like to note is whether such a non-perturbative effect needs, in the approximate evaluation, the infinite sum of perturbative contributions. To resolve the issue, we deal with a truncated series $V_{pert}$, without conventional loop summation, and study the approximate calculation of the effective potential $V$ at $m=0$. A naive way of approximation would go as the following: To get around the infrared singularity we turn to the massive case and probe $V_{pert}(\sigma,m)$ at small $m$. Since the limit, $m\rightarrow 0$, cannot be taken in $V_{pert}(\sigma,m)$, we may choose some non-zero $m$ ($=m^{*}$) and approximate the effective potential $V(\sigma, m=0)$ by $V_{pert}(\sigma, m^{*})$. However, the problem is that $V_{pert}(\sigma, m)$ is not valid for small enough $m$. This is the place where the Heaviside function comes in. Our suggestion to resolve the problem is to contact the Heaviside transformation of $V(\sigma,m)$ with respect to the mass$^{2,3}$. Heaviside transform of the effective potential, $\hat V$, is a function of $\sigma$ and $x$ which is conjugate with $m$. Then, the key relation is that $\lim_{m\rightarrow 0}V(\sigma, m)=\lim_{x\rightarrow \infty}\hat V(\sigma, x)$. Of course this is valid only when the both limits exist and do not apply for $V_{pert}$ and its Heaviside function, $\hat V_{pert}$, because those functions diverge in the limits. However there arises the possibility that $\hat V(\sigma,\infty)$ and hence $V(\sigma,0)$ may be well approximated by putting some finite value of $x$ into $\hat V_{pert}$. This is because $\hat V_{pert}$ has the convergence radius much larger than that of $V_{pert}$. Although $\hat V_{pert}$ shares the similar infra-red problems with $V_{pert}$, we will find that $\hat V_{pert}$ is much more convenient in this kind of massless approximation. Actually we will demonstrate that, at finite perturbative orders where just the several of Feynman diagrams are taken into account, the accurate dynamical mass is obtained via the Heaviside transform approach. Throughout this paper, we use dimensional regularization$^{4}$. We confine ourselves with the leading order of large $N$ expansion and $N$ is omitted for the sake of simplicity. [**2. Heaviside transform with respect to the mass**]{} In this section we summarize basic features of the Heaviside transform and illustrate our strategy by taking a simple example. Let $\Omega(m)$ be a given function of the mass $m$. The Heaviside transform of $\Omega(m)$ is given by the Bromwich integral, $$\hat \Omega(x)=\int^{s+i\infty}_{s-i\infty}{dm \over 2\pi i}{\exp(m x) \over m}\Omega(m),$$ where the vertical straight contour should lie in the right of all the possible poles and the cut of $\Omega(m)/m$ (In (1), the real parameter $s$ specifies the location of the contour). Since $\Omega(m)/m$ is analytic in the domain, $Re(m)>s$, $\hat \Omega(x)$ is zero when $x<0$. It is known that the Laplace transformation (of the second kind) gives the original function as, $$\Omega(m)=m\int^{\infty}_{-\infty}dx\exp(-m x)\hat\Omega(x).$$ Since $\hat \Omega(x)=0$ for $x<0$, the region of the integration effectively reduces to $[0,\infty)$. It is easy to derive the relation, $$\lim_{m\rightarrow +0}\Omega(m)=\lim_{x\rightarrow +\infty} \hat\Omega(x),$$ where the both limits are assumed to exist. As noted before, the point of our scheme consists in utilizing $\hat \Omega$ to approximate the massless value of $\Omega$, $\Omega(0)$, by relying upon (3). To illustrate our strategy based on (3), let us consider a simple example. Given a following truncated series in $1/m$, f\_[L]{}(m)=\^[L]{}\_[n=0]{}[(-1)\^[n]{} m\^[n+1]{}]{}, we try to approximate the value of $f(m)=f_{\infty}(m)=(1+m)^{-1}$ at $m=0$, $f(0)=1$, by using information just contained in the truncated series (4). Since the convergence radius, $\rho$, of $f_{\infty}(m)$ is unity, we cannot have approximation better than $1/2$ from $f_{L}(m)$. However, the state changes if we deal with its Heaviside function. The Heaviside transform of $f_{L}(m)$ is given by f\_[L]{}(x)=\^[L]{}\_[n=1]{}(-1)\^n\^[s+i]{}\_[s-i]{} [dm 2i]{}[(m x) m]{} [1 m\^[n+1]{}]{}=\^[L]{}\_[n=0]{}(-1)\^n [x\^[n+1]{} (n+1)!]{}(x), where $$\theta(x)=\left\{ \begin{array}{@{\,}ll} 1 & \mbox{$(x>0)$}\\ 0 & \mbox{$(x<0)$.} \end{array} \right.$$ From (5) it is easy to find that $\hat f(x)=(1-e^{-x})\theta(x)$ and (3) holds for $f$ and $\hat f$. For our purpose it is crucial that $\rho=\infty$ for $\hat f_{\infty}$ while $\rho=1$ for $f_{\infty}$. The infinite convergence radius ensures us to probe the large $x$ behavior of $\hat f$ by $\hat f_{L}$ to arbitrary precision by increasing perturbative order. Due to the truncation, however, $\hat f_{L}$ diverges as $x\rightarrow \infty$. Then, in approximating $\hat f(\infty)$ and therefore $f(0)$, we stop taking the limit and input some finite value into $x$. The input value of $x$, say $x^{*}$, should be taken as large as possible in the reliable perturbative region in $x$. At this place we understand that the good convergence property of $\hat f_{L}$ is one of the advantage of Heaviside function. Since the upper limit of perturbative region is not a rigorously defined concept, we determine the input value $x^{*}$ in heuristic way. Our suggestion to fix $x^{*}$ is as the following: The series (5) is valid for small $x$ but breaks down at large $x$. The breaking appears as the domination of the highest term in $\hat f_{L}$ which leads to the unlimited growth or decreasing of the function (see Fig.1). Thus $x^{*}$ is located somewhere around the beginning of the dominating behavior. Then, for odd $L$ and large even $L$, we find the plateau region just before the domination and that the region represents the end of the perturbative regime. Thus, we choose the stationary point in the plateau region as representing the typical violation of the perturbation expansion. Hence we fix $x^{*}$ by the stationarity condition, =0. The condition (7) reads as (x\^[\*]{})\^[L]{}\_[n=0]{}[(-x\^[\*]{})\^n n!]{}+(x)\^[L]{}\_[n=0]{}[(-1)\^n (x\^[\*]{})\^[n+1]{} (n+1)!]{}=0, and reduces to \^[L]{}\_[n=0]{}[(-x\^[\*]{})\^n n!]{}=0. The solution exists for odd $L$ and it varies with $L$. We find from (9) that the solution $x^{*}$ tends to $\infty$ as $L\rightarrow \infty$. More precisely the solution scales for large $L$ as x\^[\*]{}\~[1 3]{}L. We have explicitly done the numerical experiment to several higher orders and obtained the result for $L=1,5,9,13,17$, f\_[L]{}(x\^[\*]{})=0.5, 0.850675, 0.953301, 0.985166, 0.995251, at $x^{*}=1, 2.18061, 3.33355, 4.47541, 5.6112$, respectively. Thus, the sequence gives good approximation of the exact value. In this toy model, one finds that $\hat f_{L}(x^{*})$ converges in the $L\rightarrow \infty$ limit by using the scaling relation (10). Up to now we have concentrated on approximating the massless value. We here point out that our scheme is capable of constructing the small $m$ expansion of $\Omega(m)$, that is, the scheme allows the approximation of the function itself when $m$ is small. Consider in general the approximation of the derivatives at $m=0$, \^[(k)]{}(0)=[\^[k]{}m\^[k]{}]{}|\_[m=0]{}, which is needed when one constructs the small $m$ expansion of $\Omega(m)$, (m)=(0)+[\^[(1)]{}(0) 1!]{}m+[\^[(2)]{}(0) 2!]{}m\^[2]{}+. The coefficients, $\Omega^{(k)}(0)\hskip 3pt (k=1,2,3,\cdots)$, can be approximated as follows. The starting formula is that \^[(k)]{}(m)\^[x]{}\_[-]{}dt(-t)\^[k]{}[(t) t]{}\_[k]{}(x), where we used m[m]{} -x[(x) x]{},\^[x]{}\_[-]{}dt(t). Here ${\cal H}$ above the arrow represents the Heaviside transformation. Hence, from the agreement condition (3) we find \_[m0]{}\^[(k)]{}(m)=\_[x]{}\_[k]{}(x)=\^\_[-]{}dt(-t)\^k[(t) t]{}. Now in our perturbative approach we use $\Omega_{L}$ (the truncated series at the order $L$) for real $\Omega$. Then, we show that we can simulate $\alpha_{k}(\infty)$ by $\alpha_{k}(x^{*}_{k})$ where $x^{*}_{k}$ may be fixed following the same reasoning we presented for the case of $f(0)$ approximation. Namely we guess that the break down of the perturbative expansion is represented by the plateau region, if it exists, just before the unlimited growth of the size of the function. Therefore we use stationarity condition which reads, |\_[x=x\^[\*]{}\_[k]{}]{}=(-x\^[\*]{}\_[k]{})\^[k]{}[\_[L]{} x\^[\*]{}\_[k]{}]{}=0, and find that $x^{*}_{k}$ satisfies the same condition as that for $x^{*}$. Thus the solution of (17) is universal for all $k$ and fixes the coefficients of the small $m$ expansion to all orders. This is desirable since the uncertainty connected with the choice of $x^{*}$ and $x^{*}_{k}$ is minimized. We note that for $\alpha_{k}(x^{*})$ $(k=1,2,3,\cdots)$ the integration is necessary and $\theta(x)$ and $\delta$ functions should be kept in the integrand in general. As an example let us calculate the small $m$ expansion of $f(m)$. The coefficient $\alpha_{k}$ is given at order $L$ by, \_[k]{}=(-1)\^k\^[L]{}\_[n=0]{}[(-1)\^n x\^[k+n+1]{} n!(k+n+1)]{}. By substituting $x^{*}$ at $L=17$ into $\alpha_{k}$, we have the following satisfactory approximant of $f(m)=1-m+m^{2}-\cdots$; 0.995251-0.970003 m+0.90271 m\^2-0.78284 m\^3+0.62233 m\^4+. [**3 Application to the effective potential**]{} Having prepared basic analysis, we turn to a model field theory which is of our main interest. Consider the Gross-Neveu model at the leading order of large $N$ expansion$^{1}$. The Lagrangian is given within dimensional regularization$^{4}$ at $D=4-2\epsilon$ by $$\begin{aligned} {\cal L}&=& \bar \psi(i\gamma^{\mu}\partial_{\mu}-m)\psi-{1 \over 2}\sigma^{2}-{g \over \sqrt{N}}\mu^{\epsilon}\sigma \bar\psi\psi +{\cal L}_{ct},\nonumber\\ {\cal L}_{ct}&=&A\sigma-B{1 \over 2}\sigma^{2}, \end{aligned}$$ where $$\psi=(\psi_{1},\cdots,\psi_{N}), \hskip 3mm A=-{\sqrt{N}mg\mu^{-\epsilon} \over 2\pi}{\hat{1 \over \epsilon}}, \hskip 3mm B={g^{2} \over 2\pi}{\hat{1 \over \epsilon}}, \hskip 3mm {\hat{1 \over \epsilon}}={1 \over \epsilon}-\gamma+\log(4\pi).$$ Here ${\overline {MS}}$ scheme$^{5}$ was used for the subtraction. It is well known that the model generates the dynamical fermion mass, $m_{dyn}=\Lambda$, where $\Lambda$ denotes the renormalization group invariant scale in ${\overline {MS}}$ scheme. At the leading order of $1/N$ expansion, the effective potential is given by the sum of diagrams shown in Fig.2. The straightforward calculation gives V(, m)=[m\^[2]{} 4]{}(-1)+[mg2]{}+[g\^[2]{}\^[2]{} 4]{}(+2)-\^\_[n=3]{}[(-g)\^[n]{} n(n-1)(n-2)]{}m\^[-n+2]{}. We note that although the naive power counting with respect to $N$ leads that the contribution with many $\sigma$-legs corresponds to higher order in $1/N$, they must be included since the vacuum value of $\sigma$ is of order $\sqrt{N}$. The series (22) converges only when $|g\sigma/m|<1$ and hence the small $m$ behavior relevant to the dynamical mass generation is not known from (22). However, Heaviside transformation enlarges the convergence radius and enables us to study the large $x$ behavior of the corresponding Heaviside function, $\hat V(\sigma,x)$, as we can see below. To obtain $\hat V(\sigma,x)$ we need to know the transform of $m^{k}(k=0,1,2,\cdots), m\log m, m^{2}\log m$ and $1/m^{k}$. Here the following formula is basic, $$m\Omega(m)\stackrel{{\cal H}}{\rightarrow} {\partial \hat \Omega(1/x) \over \partial x}.$$ For example from (1) we have $$\log(m)\stackrel{{\cal H}}{\rightarrow}(-\gamma-\log(x))\theta(x).$$ The use of (23) on (24) then leads to $$\begin{aligned} m\log m &\stackrel{{\cal H}}{\rightarrow}& -{1 \over x}\theta(x)-(\gamma+\log x )\delta(x),\\ m^{2}\log m &\stackrel{{\cal H}}{\rightarrow}& {1 \over x^{2}}\theta(x)-{2 \over x}\delta(x)-(\gamma+\log x )\delta^{'}(x). \end{aligned}$$ The transformation of $m^{k}$ is easily obtained from $$1 \stackrel{{\cal H}}{\rightarrow} \theta(x),$$ as $$m^{k} \stackrel{{\cal H}}{\rightarrow} \delta^{(k-1)}(x)\quad (k=1,2,3,\cdots).$$ The $\delta$ functions are needed when one carries out Laplace integrals for the Heaviside functions. This is because the $\delta$ function terms cancel out the divergences coming from the first terms of (25) and (26), for example. Since the integration over $x$ is however not necessary as long as the approximation in the massless limit is concerned, we omit, for a while, $\delta$ functions and set $\theta(x)=1$ in the transformed functions. Now, using the results, (24), (25), (26), (27), (28) and ${\cal H}m^{-k}=x^{k}/k!$, we find V(, x)=[1 2x\^2]{}-[gx]{}+[g\^[2]{}\^[2]{} 2]{}(-x-+1)-\^\_[n=3]{}[(-g)\^[n]{}x\^[n-2]{} n!(n-2)]{}. Note that, due to the creation of $1/k!$ in ${\cal H}[1/m^{k}]$, the series converges for any large $x$. Therefore the large $x$ behavior of $\hat V$ can be easily accessed by increasing the order of expansion. This is one of the advantages of $\hat V$ over $V$. We turn to the approximation of the massless effective potential by perturbative series at order $L$, $\hat V_{L}$. At $L$-th order, we have just first $L+1$ terms of (29) and find V\_[L]{}(, x)=[1 2x\^2]{}-[gx]{}+[g\^[2]{}\^[2]{} 2]{}(-x-+1)-\^[L]{}\_[n=3]{}[(-g)\^[n]{}x\^[n-2]{} n!(n-2)]{}. The input $x^{*}$ will be determined as in the previous section. Actually the break down appears as the domination of the last term in $\hat V_{L}$ which shows up as its unlimited behavior for large $x$. This can be seen in Fig. 3. And before the domination the function experiences a stationary behavior for odd $L$ and large even $L$. We find that the plateau region represents the end of the reliable perturbative regime and thus we fix $x^{*}$ by the equation, $${\partial \hat V_{L}(\sigma, x^{*}) \over \partial x^{*}}=0.$$ If there are several solutions we should input the largest one into $x^{*}$ due to the obvious reason. Now, the condition (31) gives $x^{*}$ as $constant/g\sigma$ for odd $L$ . For odd $L$, the substitution of the solution into $\hat V_{L}$ gives the optimized potential $V_{opt}(\sigma)$. For example for $L=3$, we have the solution, $g\sigma x^{*}=1.59607$, and this gives the optimized potential, $$V_{opt}={g^{2}\sigma^{2} \over 2\pi}\Bigl(\log{g\sigma \over \Lambda}-0.373264\Bigl).$$ The dynamical mass is given from $V_{opt}$ in the standard way. Note that, since $x$ must be positive (see section 2), the region of $V_{opt}$ thus approximated is restricted to the positive $\sigma$. In the following, we summarize the result of the approximate calculation of the dynamical mass for $L=3,5,7,9,11$; $$\begin{aligned} L=3, \hskip 3mm m_{dyn}/\Lambda &=& 0.880966\hskip 3mm({\rm at}\hskip 3pt g\sigma x^{*}=1.59607),\nonumber\\ L=5, \hskip 3mm m_{dyn}/\Lambda &=& 0.97760 \hskip 3mm({\rm at}\hskip 3pt g\sigma x^{*}=2.18061),\nonumber\\ L=7, \hskip 3mm m_{dyn}/\Lambda &=& 0.99401\hskip 3mm({\rm at}\hskip 3pt g\sigma x^{*}=2.75900),\nonumber\\ L=9, \hskip 3mm m_{dyn}/\Lambda &=& 0.99809\hskip 3mm({\rm at}\hskip 3pt g\sigma x^{*}=3.33355),\nonumber\\ L=11, \hskip 3mm m_{dyn}/\Lambda &=& 0.999326 \hskip 3mm({\rm at}\hskip 3pt g\sigma x^{*}=3.90545). \end{aligned}$$ The above result is quite good. Thus, via Heaviside transform approach, the dynamical mass is approximated only from perturbative information. The small mass expansion can be also obtained. Our task is just to substitute the solution of (31) into the approximate coefficients, \_[k]{}(x\^[\*]{})=\^[x\^[\*]{}]{}\_[-]{}dx(-x)\^[k]{}[V\_[L]{}(,x) x]{}. To perform integration, we need the full form of $\hat V_{L}$ including the $\theta$ and $\delta$ functions. The full form is given by $$\begin{aligned} \hat V_{L}(x)&=&{1 \over 2\pi}\biggl[{1 \over x^2}\theta(x)-{2 \over x}\delta(x)-(\gamma+\log x\mu+1/2)\delta^{'}(x)\biggl] -{g\sigma \over \pi}\biggl[{1 \over x}\theta(x)+(\gamma+\log x\mu)\delta(x)\biggl]\nonumber\\ &-&{g^{2}\sigma^{2} \over 2\pi}(\gamma+\log x\Lambda-1)\theta(x)-\sum^{L}_{n=3}{(-g\sigma)^n x^{n-2} \over \pi n!(n-2)}\theta(x).\end{aligned}$$ From (35) $\alpha_{k}$ is given at $L$-th order as $$\begin{aligned} \alpha_{1}&=&{g\sigma \over \pi}\biggl[-{1 \over X}+1-\gamma+\log(g\sigma/\mu)-\log X+\sum_{n=2}^{L}{(-1)^nX^{n-1} \over n!(n-1)}\biggl],\nonumber\\ \alpha_{2}&=&{1 \over \pi}\biggl[\log(g\sigma/\mu)-\log X+2+\sum_{n=1}^{L}{(-1)^nX^{n} \over n!n}\biggl],\nonumber\\ \alpha_{k}&=&{(-1)^{k+1} \over \pi (g\sigma)^{k-2}}\sum_{n=0}^{L}{(-1)^nX^{n+k-2} \over n!(n+k-2)}\quad (k>2),\end{aligned}$$ where $X=x^{*}g\sigma$. At $L=11$, for example, we have $$\begin{aligned} V(\sigma,m)&\sim & {g^{2}\sigma^{2} \over 2\pi}\Bigl(\log{g\sigma \over \Lambda}-0.499326\Bigl)+m{g\sigma \over \pi}(\log g\sigma-0.00139409)+ {m^2 \over 2}{1 \over \pi}(\log g\sigma+1.00595)\nonumber\\ &+&{m^3 \over 6g\sigma}0.973714-{m^4 \over 24g^2\sigma^2}0.878954+O(m^5).\end{aligned}$$ This is quite accurate because the exact result reads V(,m)=[g\^[2]{}\^[2]{} 2]{}(-[1 2]{})+ [1 ]{}mg+ [m\^2 2]{}(+1)+[m\^3 6g]{}-[m\^4 24g\^2\^2]{}+O(m\^5). Before closing this section, we prove that our approximants for the massless potential converges to the exact result in the $L\rightarrow \infty$ limit. That is, $\lim_{L\rightarrow \infty}\hat V_{L}(\sigma,x^{*})= V(\sigma,0)$. From (30) we find that $\lim_{L\rightarrow \infty}\partial\hat V_{L}/\partial x$ can be easily summed up and, using $\lim_{L\rightarrow \infty }\hat V_{L}=\hat V$ (given by (29)), =-[1 x\^3]{}. The perturbative truncation of (39) is given by expanding $\exp[-g\sigma x]$ to relevant orders. Since $\rho=\infty$ for the series expansion of (39), the solution of the truncated version of $(39)=0$ approaches to $\infty$ in the $L\rightarrow \infty$ limit. More precisely we find the scaling of the solution for large $L$, gx\^[\*]{} \~[1 3]{}L. Now consider the reminder $\hat R_{L}$, defined by R\_[L]{}=-\^\_[n=L+1]{}[(-g)\^[n]{} n!(n-2)]{}x\^[n-2]{}. Since $\hat V_{L}+\hat R_{L}=\hat V_{\infty}$ and $\rho=\infty$ for $\hat V_{\infty}$, it is sufficient to show that R\_[L]{}(,x\^[\*]{})0(L). Note here that $x^{*}$ depends on $L$ and behaves at large $L$ as (40). Now, using the Stirling’s formula and (40), we have |R\_[L]{}|&lt;[1 x\^[\*]{}]{}[(egx\^\*/L)\^[L+1]{} 1-egx\^\*/L]{}&lt;[9(g)\^2 (3/e-1)]{}L\^[-7/2]{}(e/3)\^L0(L), which proves the convergence. [**4 Discussion**]{} One reason of the success of our approximate calculation is that the transformed function has infinite radius of convergence. The other reason is that, as the order increases, $\hat V_{L}(\sigma,x)$ quickly approaches to the value at $x=\infty$ for fixed $\sigma$. If one uses the closed form, $$\hat V= {g^{2}\sigma^{2} \over 2\pi}\Bigl(\log{g\sigma \over \Lambda} -{1 \over 2}\Bigl)+{g^{2}\sigma^2 \over \pi}\int^{\infty}_{g\sigma x}dx{e^{-x} \over x^3},$$ one finds the reason by expanding (44) for large $g\sigma x$, $$\hat V(\sigma,x)={g^{2}\sigma^{2} \over 2\pi}\Bigl(\log{g\sigma \over \Lambda} -{1 \over 2}\Bigl)+{1 \over \pi x^{2}}\exp\biggl[-g\sigma x\biggl]\Biggl({1 \over g\sigma x}+(-3)({1 \over g\sigma x})^2+ (-3)(-4)({1 \over g\sigma x})^3+O(({1 \over g\sigma x})^4)\Biggl).$$ By contrast, the original function has the power-like expansion as shown in (38). Thus it is obvious that the approximation of the massless potential is more convenient in $\hat V_{L}$ since it approaches to the “massless” value much faster than $V_{L}$. The reason behind why the transformed function behaves so good is not known to us. We have shown that the perturbative series at finite orders produces the approximate massless effective potential and the dynamical mass. The deformation of the effective potential was made by Heaviside transform with respect to the mass and the stationarity prescription to fix the input value $x^{*}$ has found to work good. It was also shown that our scheme is capable of approximating the small $m$ behavior. Thus the Heaviside transform drastically improves the status of the perturbative approximation of physical quantities. We are under the study of full approximation method by which the general case where the explicit mass is not small can be treated. The result of investigation will be reported elsewhere. The author thanks Dr. H. Suzuki for the critics on the primitive version of this work and stimulating discussion. [**References**]{} [1]{} : D.J.Gross and A.Neveu, Phys. Rev. D10 (1974) 3235. [2]{} : S. Moriguchi et al, Suugaku Koushiki II, Iwanami Shoten (in Japanese). [3]{} : H. Yamada, Mod. Phys. Lett. A11 (1996) 1001. [4]{} : G. t’Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189;\ C. G. Bollini and J. J. Giambiagi, Phys. Lett. B40 (1972) 566;\ G. M. Cicuta and E. Montaldi, Nuovo Cimento Lett. 4 (1972) 329. [5]{} : W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev. D18 (1978) 3998. [**Figure Captions**]{} [Figure 1]{} : Model series $\hat f_{L}(x)$ is shown for $L=1,6,11$. [Figure 2]{} : The Feynman diagrams contributing to the effective potential at the leading order of $1/N$ expansion. [Figure 3]{} : The Heaviside functions of the effective potential with fixed $\sigma$ at $L=3,6$ and $9$. For the sake of the simplicity, we have set that $g\sigma=1$ and $\Lambda=1$.

--- abstract: 'We investigate quantum transport through a two-terminal nanoscale device characterized by a model peaked transmission function of the energy carriers. The device is in contact with two reservoirs held at different temperatures and chemical potentials. The above ideal model introduced by Mahan and Sofo for the search of the electronic structure of a thermoelectric material which maximizes the figure of merit, is here addressed in the non linear regime starting from the general expressions of particle-, electric charge-, and heat- currents. We individuate the parameters region where the electron system acts as energy pump (thermal machine) or heat pump (refrigerator machine). We provide contour plots of the power and heat currents involved in the two regions of the parameter space, and evaluate the corresponding thermal efficiency and coefficient of performance. The present transmission model sheds light on the implications of quantum bounds in nanostructures and provides a wealth of precious information on general aspects of transport. Our results can be a guide for the design of realistic thermoelectric devices with sharp density of states near the chemical potentials.' author: - 'G. Bevilacqua$^{1}$, G. Grosso$^{2,3}$, G. Menichetti$^{4,2}$, G. Pastori Parravicini$^{2,5}$' title: Thermoelectric regimes of materials with peaked transmission function --- INTRODUCTION ============ The development of nanotechnology trained new strategies to increase the efficiency of thermoelectric (TE) processes [@WHITNEY18]. The pioneering papers by Hicks and Dresselhaus [@DRESS93a; @DRESS93b; @DRESS07] evidenced the importance of investigation of nanoscale quantum transport for the enhancement of the thermoelectric dimensionless figure of merit $ZT$. In the linear regime $ZT$ is defined as $ZT=\sigma S^2 T/(\kappa_{el}+\kappa_{ph})$, where $\sigma$ is the electronic conductance, $S$ the Seebeck coefficient, $T$ the absolute temperature, and $\kappa_{el}$ ($\kappa_{ph}$) the electronic (phononic) thermal conductance. Several ideas and strategies where reported to maximise the TE figure of merit by suitable choice of device design and appropriate material (see e.g. Refs. ). Most attempts proposed the increase of phonon scattering so to decrease the lattice thermal conductivity, which can be reached by engineering nanostructured devices; other attempts proposed to increase the power factor, $\sigma S^2$, varying the concentration of charge carriers.[@DMITRIEV10] As alternative approach Mahan and Sofo [@SOFO96] addressed the problem in a formal way, looking for the material with suitable shape of the carrier energy levels distribution, i.e. with transport distribution function $\mathcal T(E)$, which guarantees, at given lattice thermal conductivity, the highest figure of merit. The authors demonstrate that for this goal the carriers in the material should possess energy distribution as narrow as possible, i.e. a $\delta$-like shape. Along this line, the impact of energy spectrum width [@LINKE05; @LUO13] and of other shapes in $\mathcal T(E)$, as step-, box-, lorentzian[@BW36] and Fano[@MIRO10; @SSP] like features, have been successively considered [@BEVI16] depending on specific problems or suggested by quantum broadening effects due the contacts. In particular, sharp features in $\mathcal T(E)$ approaching $\delta$-shape have been realised and analysed in terms of single lorentzian peaks of vanishing width $\Gamma$ [@LIU18; @LUO16], in quantum dots weakly interacting with the contacts [@RSAN15; @TALBO17; @MENI18] and in the presence of electron-electron interaction[@KRO18], single molecule junctions [@TORRES15], molecular electronics [@REDDY07; @LAMBERT16; @HANGGI84], resonant tunneling devices [@PATIL17]. The subject of this paper is the analysis of the effects of a peaked transmission function on the thermoelectric transport properties of a nanostructured system, in the absence of lattice contribution to the thermal conductivity, and beyond the linear response regime. Overcome of linear response condition is commonly reached in low-dimensional systems where large values of temperature and electrical potential gradients may easily occur due to their small dimension which can be smaller than the electronic scattering length (see e.g. Refs. ). We consider a thermoelectric system composed of two reservoirs of particles obeying the Fermi-Dirac statistics, connected to the device through left and right perfect leads. $T_L$ is the temperature of the left (hot) reservoir and $T_R$ is the temperature of the right (cold) reservoir, with $\mu_L$ and $\mu_R$ chemical potentials, respectively. For a system characterised by two electron reservoirs connected through perfect leads to a conductor with peaked transmission function at the resonance energy $E_d$, we show, for each difference of temperatures and chemical potentials between the two reservoirs, when the system behaves as good thermal machine, or as good refrigerator, or as useless energy dissipator, according to the position of the resonance energy on the energy axis. We provide contour plots of the power and of heat currents which highlight different thermoelectric behaviors of the system as function of the thermodynamic parameters $T_L , T_R,\mu_L , \mu_R$, and of the transmission filter energy. The above result allows to individuate regions of high performance, when the system works as thermal machine or as refrigerator. The paper is organised as follows: in Section II we provide some definitions and expressions concerning TE transport in the non linear response regime. In Sections III we analyse the cases of transport through a peaked transmission function in the case $\mu_L < \mu_R$ and $\mu_L > \mu_R$, under the condition $T_L > T_R$. Section IV contains contour plots of exchanged power and of heat currents which define the thermoelectric behavior of the considered device, with a discussion of the results. Section V contains conclusive remarks. General expressions of thermoelectric transport equations in the non-linear response regime =========================================================================================== In this section we consider transport through a two-terminal mesoscopic electronic system characterized by the transmission function ${\mathcal T}(E)$. A most general tool to address the transmission function in nanostructures is the non-equilibrium Keldysh Green’s function approach[@KELD64; @AGG06; @WANG08; @RYNDYKD09; @DO18; @MEIR94; @DARE16; @FRED014]; this formalism is exact (i.e. without conceptual approximations: all Feynman diagrams summed out at any order) in the particular case of non-interacting systems. In realistic cases, one needs to go through [*ab initio*]{} evaluation of the transmission function; often one can directly focus on special functional shapes of the transmission ( Lorentzian resonances and antiresonances, Fano profiles) generally encountered in the actual transmission features of thermoelectric materials, due to quantum interference effects.[@GOO09; @DATTA97; @DUBI11] The purpose of this paper is the study of the thermoelectric regimes linked to the presence of a peaked transmission function. This study is of relevance in its own right, and most importantly because it paves the way to the understanding of a variety of peaked transmission functions of wide impact in the nano-material world. Following a well established convention, we assume without loss of generality that the temperature of the left reservoir is hotter than the one of the right reservoir, namely $T_L > T_R$; no a priori assumption is done on the chemical potentials $\mu_L , \mu_R$ of the particle reservoirs. The $left$ or $right$ particle number current $I_N^{(left,right)}$, charge (electric) current $I_e^{(left,right)}$, and heat (thermal) currents $I_Q^{(left,right)}$, are given respectively by the expressions: $$\begin{aligned} I_N &=& I_N^{(left)} = I_N^{(right)} = \frac{1}{h} \int_{-\infty}^{+\infty} dE \, {\mathcal T} (E) \left[ f_{L}(E) - f_{R}(E) \right] % Eq.(1a) \\ [2mm] I_e &=& I_e^{(left)} = I_e^{(right)} = -eI_N % Eq.(1b) \\ [2mm] I_{Q}^{(left)} &=& \frac{1}{h} \int_{-\infty}^{+\infty} dE (E- \mu_{L}) \, {\mathcal T}(E) \left[ f_{L}(E) - f_{R}(E) \right] % Eq.(1c) \\[2mm] I_{Q}^{(right)} &=& \frac{1}{h} \int_{-\infty}^{+\infty} dE (E- \mu_{R}) \, {\mathcal T}(E) \left[ f_{L}(E) - f_{R}(E) \right]\, ; % Eq.(1d) \end{aligned}$$ ($-e$) is the electric charge and $h$ the Planck constant. The output or input power ${\mathcal P}$, due to the transport of spinless electrons across the device in any regime (power generator regime, refrigeration regime, dissipative regime) is given by $$\begin{aligned} {\mathcal P} &=& I_{Q}^{(left)} - I_{Q}^{(right)} = \frac{1}{h} \, (\mu_R - \mu_L) \int dE \, {\mathcal T} (E) \left[ f_{L}(E) - f_{R}(E) \right] . % Eq.(1e) \end{aligned}$$ In steady conditions the number current, $I_N$, and charge currents, $I_e$, in the left and right leads are equal, while, in general, heat currents, $I_{Q}$, have different values in the left and right leads. Equations (1) are general, and apply both in the linear situation (small difference of chemical potentials and temperatures of the two reservoirs), and in the non-linear situation (arbitrary difference of the thermodynamic parameters of the two reservoirs). The standard relation between the applied bias potential and the reservoir chemical potentials is given by $(-e)(V_L-V_R) = (-e) \Delta V = \Delta \mu = \mu_L -\mu_R$. In the present case of two terminal devices we adopt the choice of positive direction for the currents, from the left reservoir to the central device, and then from the central device towards the right reservoir. In the case of three or more reservoirs the assumption of positive directions going from the reservoirs to the central device is preferable. A thermoelectric device can work as heat engine, or as refrigerator, or simply becomes a useless dissipative apparatus, depending on the direction of heat and energy flux. The power production mode (the system behaves as thermal machine) is characterised by the fact that the three quantities, [*left thermal current, right thermal current and power, are all positive*]{}, $$thermal \ machine \quad \Longleftrightarrow \quad I_Q^{(left)} > I_Q^{(right)} > 0 \ .$$ It is apparent that in this mode heat flows from the hot reservoir to the cold one, and part of the thermal energy is converted into power, as schematically shown in Fig.1a. The efficiency of the device in the thermal machine mode is defined as $$\eta^{(tm)} = \frac{ {\mathcal P} } { I_{Q}^{(left)} } = \frac{ I_{Q}^{(left)} - I_{Q}^{(right)} } { I_{Q}^{(left)} } \le \frac{ T_L - T_R } { T_L} \equiv \eta_c^{(tm)} \ ;$$ where $ \eta_c^{(tm)}$ indicates the Carnot thermal efficiency. The thermodynamic bounds of the thermal machine efficiency range from zero (for $T_L \approx T_R$) to unity for ($T_R \ll T_L$).  The refrigeration mode of the system is characterized by the fact that the three quantities, [*left thermal current, right thermal current and absorbed power, are all negative*]{}, thus we can write $$refrigerator \ machine \quad \Longleftrightarrow \quad I_Q^{(left)} < I_Q^{(right)} < 0 \ .$$ It is apparent that heat is extracted from the cold reservoir $(T_R<T_L)$ and pumped into the hot reservoir, with the absorption of external energy as schematically shown in Fig.1b. In the refrigeration mode there is a general consensus to define the efficiency of the refrigerator machine, the so called “[*coefficient of performance*]{}" (COP) as the “formal counterpart" of Eq.(2): $$\eta^{(refr)} = \frac{ I_{Q}^{(right)} }{ \mathcal P } = \frac{ I_{Q}^{(left)} }{ I_{Q}^{(left)} - I_{Q}^{(right)} } - 1 \le \frac{T_L}{T_L-T_R} -1 = \frac{T_R}{T_L-T_R} \equiv \eta_c^{(refr)} \ .$$ With the above definition the upper thermodynamic bound of the refrigeration machine COP is not unity, in general, and can vary from zero to $T_R/(T_L-T_R)$, a quantity that approaches infinity for equal (or nearly equal) reservoirs temperatures. The above efficiency expressions refer exclusively to thermal machines and refrigeration machines. Neither $\eta^{(tm)}$ nor $\eta^{(refr)}$ have a clear physical meaning when the system is working in dissipative modes: the technological interest of thermoelectric devices is either to convert heat into power or to use power for refrigeration. [*Difference of two Fermi functions*]{} From the transport Eqs.(1), it is apparent the basic role played by ${\mathcal T}(E)$ and by the difference of the Fermi functions of the two reservoirs. It is thus important to look closely at the difference $f_{LR}(E)$ of the electronic Fermi distribution functions $f_L(E)$ and $f_R(E)$ kept at temperatures $T_L$ and $T_R$ and chemical potentials $\mu_L$ and $\mu_R$, respectively: $$f_{LR}(E) \equiv f_{L}(E) - f_{R}(E) = \frac{1}{ e^{(E-\mu_L)/ k_BT_L} + 1} - \frac{1}{ e^{(E-\mu_R)/ k_BT_R} + 1} \ .$$ We wish now to determine the energy regions where the [*$f_{LR}(E)$-function is positive or negative*]{}. From Eq.(4) it is easy to verify that $$f_{LR}(E) >0 \qquad {\rm if} \qquad E > \varepsilon_0 \equiv \frac{\mu_R \,T_L - \mu_L \,T_R}{T_L - T_R} \ .$$ The function $f_{LR}(E)$ has a unique zero at the value $E=\varepsilon_0$. For what concerns the position of $\varepsilon_0$ on the energy axis, it is seen by inspection that $\varepsilon_0$ is at the right of both chemical potentials in the case $\mu_L < \mu_R$, while it is at the left of both chemical potentials if $\mu_L > \mu_R$ (having systematically assumed $T_L > T_R)$. In fact it holds $$\frac{\varepsilon_0 - \mu_L}{k_BT_L} = \frac{\varepsilon_0 - \mu_R}{k_BT_R } = \frac{\mu_R - \mu_L}{k_B(T_L -T_R) } \equiv x_0 \ , \hspace{2cm} % Eq.(6)$$ where the dimensionless parameter $x_0$ is positive for $\mu_R>\mu_L$ and negative for $\mu_R<\mu_L$. Good thermal machines or refrigerators have $x_0$-values in the range of unity or so. The behavior of the difference of two Fermi functions with $T_L>T_R$, and $\mu_L < \mu_R$ or $\mu_L > \mu_R$ is reported in Fig.2a and Fig.2b. Throughout this paper, we choose as exemplification $T_L = 600$ K ($k_BT_L \approx 0.05$ eV) and $T_R = 300$ K ($k_BT_L \approx 0.025$ eV), a choice often adopted in the literature.[@DRESS12; @WHITNEY15; @HER13]  Transport through a peaked transmission function ================================================ [**III A. General considerations**]{} We focus now on transport through a device characterized by a narrow peaked transmission function at the resonance energy $E_d$. This situation typically occurs when the leads have strictly a single propagation channel, or in the case of quantum systems operating as energy filters [@HUMP02] as single level quantum dots [@LUO16; @NAP10] or quantum wells [@SOTH13], for which $0< {\mathcal T}(E) \leq 1$. The profile of a single sharp transmission function is generally described by a resonant lorentzian shape with half-maximum width $\Gamma_d$. For convenience we describe such resonance with a narrow rectangular model of the type $${\mathcal T} (E) = \left\{ \begin{array}{rcl} 1 &\quad& \,\, {\rm for } \,\,\, E_d-\dfrac {\Gamma_d}{2} <E < E_d+\dfrac {\Gamma_d}{2} \\[3mm] 0 &\quad& \,\,{\rm otherwise}\,. \end{array} \right.$$ In the case of $N_c$ allowed transmission channels in the narrowest part of the system, the upper bound of the total transmission function is $N_c$. We also assume that in the resonance region $f_{LR}(E)\approx f_{LR}(E_d)$, and that the transmission function ${\mathcal T}(E)$ is rigid with respect to charge injection due to temperature and voltage gradients. In realistic cases, in the presence of electron-electron and electron-phonon interactions ${\mathcal T}(E)$ must be determined as a self-consistent function [@SANCEZ15] of $T_L, T_R$ and $V$. In the forthcoming expressions (eV)$^2/h = 3.874$ nW is assumed as the unit of power and thermal currents, and $\Gamma_d$ (or better $N_c \Gamma_d$) is measured in eV. The general transport equations (1), in the particular case that the transmission is given by Eq.(7), greatly simplify. The particle current of Eq.(1a) becomes $$I_N = \frac{1}{h} \, \Gamma_d \, f_{LR}(E_d) \ .$$ Similarly the microscopic charge current of Eq.(1b) becomes $$I_e = \frac{-e}{h} \, \Gamma_d \, f_{LR}(E_d) \ .$$ The particle current and the associate electric current are proportional to the $f_{LR}$ function at the resonance energy. The left and right heat currents of Eq.(1c) and Eq.(1d) become $$I_{Q}^{(left)} = \frac{1}{h} (E_d - \mu_{L}) \, \Gamma_d \,f_{LR}(E_d)$$ and $$I_{Q}^{(right)} = \frac{1}{h} (E_d - \mu_{R}) \, \Gamma_d \,f_{LR}(E_d) \ .$$ The left (right) heat current is proportional to the $f_{LR}$ function at the resonance energy, as well as to the difference between the resonance energy and the left (right) chemical potential. We notice that [*left heat current and right heat current have different signs if the resonance $E_d$ lies in the interval between the chemical potentials, and the same sign otherwise*]{}. This automatically means that the location of $E_d$ between the two chemical potentials is not useful either for refrigeration or for power production. Equation (1e) for the power takes the expression $${\mathcal P} = \frac{1}{h} \, (\mu_R- \mu_L) \Gamma_d f_{LR}(E_d) = (\mu_R- \mu_L) I_N \ ,$$ i.e. for fixed $\mu_L$ and $ \mu_R$, ${\mathcal P}$ is proportional to the $f_{LR}$ function at the resonance energy. Notice that for $E_d = \varepsilon_0$, the left and right thermal currents, the particle current and the power are all equal to zero. [**III B. Transport properties in the case $\mu_L <\mu_R$**]{} In this subsection we discuss specifically the case $\mu_L <\mu_R$ and $f_{LR}(E)$ given in Fig.2a. In this situation, from Eq.(12) it is seen that the thermoelectric device generates energy (i.e. ${\mathcal P}>0)$ if $E_d>\varepsilon_0.$ On the contrary the thermoelectric device takes in energy (i.e. ${\mathcal P}<0)$ if $E_d<\varepsilon_0.$ The power exchange vanishes exactly at $E_d=\varepsilon_0,$ and is small as the resonance energy becomes much higher than the chemical potentials. The useful regimes of the filter device in the configuration $(T_L>T_R ; \mu_L<\mu_R)$ occur when $E_d>\mu_R$: in this case we find the refrigeration regime for $\mu_R < E_d < \varepsilon_0$, and the thermal machine regime for $E_d>\varepsilon_0$. No useful thermoelectric behavior occurs for $E_d<\mu_R$. Entering in details, we can distinguish the following four regimes. ### Regime I: The power generation mode region $E_d > \varepsilon_0$ {#regime-i-the-power-generation-mode-region-e_d-varepsilon_0 .unnumbered} Consider the situation in which $E_d$ extends at the right of the two chemical potentials $\mu_L, \mu_R$ and also at the right of $\varepsilon_0$, see Fig.3a. In this region, the function $f_{LR}(E_d) >0 $, and from Eqs.(8-12) we have: $ I_Q^{(left)} > 0, I_Q^{(right)} > 0, {\mathcal P} > 0$ and $I_N > 0$, as schematically indicated in Fig.3a. The thermal efficiency parameter for heat-to-power conversion, using Eq.(10) and Eq.(12) becomes $$\eta^{(tm)}(E_d) \equiv \frac{ \mathcal P }{ I_{Q}^{(left)} } = \frac{ \mu_R -\mu_L } { E_d -\mu_L } \qquad \varepsilon_0 < E_d < \infty \ .$$ The maximum value of the efficiency parameter occurs for $E_d \equiv \varepsilon_0$; in fact $$\eta^{(tm)}(E_d =\varepsilon_0) = \frac{ \mu_R -\mu_L } { \varepsilon_{0} -\mu_L } = \ [{\rm using \ Eq.}(5)] \ = \frac{ T_L -T_R } { T_L} \equiv \eta_c^{(tm)} \ .$$ Thus the maximum efficiency occurs at $ E_d=\varepsilon_0$, when the power production ${\mathcal P} (E_d)$ vanishes, and it equals the efficiency of the Carnot cycle. ### Regime II: The refrigeration mode region $\mu_R <E_d<\varepsilon_0$ {#regime-ii-the-refrigeration-mode-region-mu_r-e_dvarepsilon_0 .unnumbered} Consider now the situation with the resonance energy $E_d$ at the right of both chemical potentials, but at the left of $\varepsilon_0$: $$T_L>T_R \quad {\rm with} \quad \ \mu_L < \mu_R <E_d<\varepsilon_0\ \ .$$ In this region $f_{LR}(E_d) <0$, and we have $I_Q^{(left)} <0, I_Q^{(right)} <0, {\mathcal P} <0$ and $I_N <0$, as schematically indicated in Fig.3b. Using Eq.(11) and Eq.(12), the coefficient of performance of the refrigeration regime becomes $$\eta^{(refr)}(E_d) \equiv \frac{ I_{Q}^{(right)}(E_d) }{ \mathcal P \ \ \ } = \frac{E_d- \mu_R} {\mu_R- \mu_L} \qquad {\rm for} \qquad \mu_R \le E_d \le \varepsilon_0 \ .$$ It is apparent that the efficiency of the refrigeration machine is zero for $E_d=\mu_R$, and takes the maximum value at $E_d=\varepsilon_0$. In fact: $$\eta^{(refr)}(E_d= \varepsilon_0) = \frac{\varepsilon_0 - \mu_R} {\mu_R- \mu_L} \ = \ [{\rm using \ Eq.} (5)] \ = \frac { T_R} { T_L -T_R } \equiv \eta_c^{(refr)} \ ,$$ where $ \eta_c^{(refr)} $ is the coefficient of performance of the Carnot cycle for refrigeration. Notice that the efficiency is large where the power absorption ${\mathcal P} (E_d)$ is small. ### Regime III: The dissipative intermediate region $\mu_L<E_d < \mu_R$ {#regime-iii-the-dissipative-intermediate-region-mu_le_d-mu_r .unnumbered} We have seen that the useful thermoelectric behaviors occur when $E_d$ is at the right of both chemical potentials. When $E_d$ is at the left of one or both chemical potentials, energy is absorbed and waisted into heat, and nothing useful is reached. Consider now specifically the region where the resonance energy $E_d$ is intermediate between the two chemical potentials: $$T_L>T_R \quad {\rm with} \quad \mu_L < E_d < \mu_R < \varepsilon_0 \ .$$ In this region $f_{LR}(E_d)<0 $, and we have: $I_Q^{(left)} < 0, I_Q^{(right)}> 0,{\mathcal P}< 0$ and $ I_N < 0$, as schematically indicated in Fig.3c. It is evident that power is absorbed and fully dissipated into heat transferred to both reservoirs. ### Regime IV: The dissipative semi-infinite region $E_d < \mu_L$ {#regime-iv-the-dissipative-semi-infinite-region-e_d-mu_l .unnumbered} When the resonance energy $E_d$ is located in the energy interval \[$-\infty, \mu_L $\] it holds $$T_L>T_R \quad {\rm with} \quad E_d < \mu_L < \mu_R < \varepsilon_0 \ .$$ In this region $f_{LR}(E_d)<0 $ , and we have: $I_Q^{(left)} >0, I_Q^{(right)} >> 0, {\mathcal P} <0$ and $I_N <0$, as schematically indicated in Fig.3d. It is seen that power is absorbed, and waisted into heat transferred to the right reservoir.  [**III C. Transport properties in the case $\mu_L >\mu_R$**]{} In this subsection we study transport in the situation $$T_L>T_R \qquad {\rm and} \qquad \mu_L > \mu_R\ .$$ This case needs only a cursory treatment, since the only formal change concerns the position of $\varepsilon_0$, now at the left of both chemical potentials.  When the resonance energy is smaller than both chemical potentials and also smaller than $\varepsilon_0 $, i.e. $E_d < \varepsilon_0 < \mu_R <\mu_L ,$ we have $f_{LR}(E_d) <0$, as shown in Fig.2b. It follows: $ I_Q^{(left)} > 0, I_Q^{(right)} > 0, {\mathcal P} > 0$ and $I_N <0$. The flow of heat and particles is schematically indicated in Fig.4a. In this region the efficiency parameter for heat-to-power conversion becomes $$\eta^{(tm)} \equiv \frac{ \mathcal P }{ I_{Q}^{(left)} } = \frac{ \mu_R -\mu_L } { E_d -\mu_L } \qquad \qquad - \infty < E_d < \varepsilon_0 \ .$$ The maximum value of the efficiency parameter occurs for $E_d \equiv \varepsilon_0$ where $\eta^{(tm)}(E_d=\varepsilon_0)$ reaches the efficiency of the Carnot cycle. In the case $\varepsilon_0 <E_d < \mu_R < \mu_L,$ we have $f_{LR}(E_d) > 0$, and thus $ I_Q^{(left)} < 0, I_Q^{(right)} < 0, {\mathcal P} < 0$ and $I_N > 0$, as schematically indicated in Fig.4b. The coefficient of performance of the refrigeration mode becomes $$\eta^{(refr)}(E_d) \equiv \frac{ I_{Q}^{(right)} }{ \mathcal P \ \ } = \frac{E_d- \mu_R} {\mu_R - \mu_L} \qquad \varepsilon_0 \le E_d \le \mu_R \ .$$ It is apparent that the coefficient of performance is zero at the boundary $E_d=\mu_R$, and takes the maximum value at $E_d=\varepsilon_0$ where it reaches the value of the Carnot coefficient of performance $ \eta_c^{(refr)}$. When the resonance energy $E_d$ is intermediate between the two chemical potentials: $\varepsilon_0 < \mu_R < E_d < \mu_L $, we have $f_{LR}(E_d)>0$, and thus $I_Q^{(left)} < 0, I_Q^{(right)} > 0, {\mathcal P} < 0$ and $I_N <0$ as schematically indicated in Fig.4c. It is seen that power is absorbed and fully waisted into heat transferred to both reservoirs. Finally, in the case where the resonance energy $E_d$ is larger than the two chemical potentials: $\varepsilon_0 < \mu_R < \mu_L < E_d$, we have $f_{LR}(E_d)>0 $, and thus $I_Q^{(left)} > 0, I_Q^{(right)} >> 0, {\mathcal P} < 0$ and $I_N >0$, as schematically indicated in Fig.4d. It is seen that power is absorbed and waisted into heat transferred to the right reservoir at lower temperature. Thermoelectric regimes of the ideal peaked-filtering device: results and discussion =================================================================================== In Section III we have shown that for $T_L>T_R$ the peaked-filtering device works as thermal generator for $ \mu_L < \mu_R <\varepsilon_0 < E_d $, when $ \mu_L < \mu_R $, and for $ E_d < \varepsilon_0 < \mu_R <\mu_L $ when $ \mu_L > \mu_R $ (region (I)); it works as refrigerator in the parameter region $ \mu_L < \mu_R <E_d<\varepsilon_0$, when $ \mu_L < \mu_R $, and for $\varepsilon_0 < E_d < \mu_R < \mu_L $ when $ \mu_L > \mu_R $ (region (II)). The device operates in dissipative regime in the remaining regions. This is pictorially shown in Fig.5 where the contour plot illustrates the power generated or absorbed in the thermoelectric device as the variables $E_d$ and $\mu_R$ vary in a two dimensional plane. In Fig.5 (and in the following ones) we assume $\mu_L$ as the reference energy and, without loss of generality, we set $\mu_L=0$.  In Fig.5 (and in the following ones) we indicate two particularly important lines in the $(E_d,\mu_R)$ plane. The bisector line $E_d = \mu_R$ signs the border between the refrigeration regime and the dissipative (intermediate) regime. The steeper line $E_d = \mu_R/\eta_c = \varepsilon_0$ signs the border between the refrigeration and the thermal regimes. We can add that the $E_d$-axis signs the natural border between the power production and the semi-infinite dissipative regime, while the $\mu_R$-axis signs the other natural border between the two dissipative regimes. Due to the symmetry aspect of Fig.5, we can restrict our considerations to the first quadrant above the bisector line, on this figure and the following ones, too. For $\mu_R>0$ the power generation region (region (I)) is delimited by the constraints $$\mu_R > 0 \quad , \quad E_d> \varepsilon_0 = \frac{\mu_R}{\eta_c^{(tm)}} \ , %Eq. (20)$$ while the refrigeration region is delimited by the constraints $$\mu_R > 0 \quad , \quad \mu_R < E_d < \varepsilon_0 = \frac{\mu_R}{\eta_c^{(tm)}} \ . %Eq. (20)$$ The expression of Eq.(1e) for the power, in the case of transmission function given in Eq.(7) (and $\mu_L$ set to zero), reads $${\mathcal P} = \frac{\Gamma_d}{h} \, \mu_R \, [f_L(E_d) - f_R(E_d)] \ .$$ It is seen by inspection that [*in the region (I) the produced power ${\mathcal P}$ is bounded*]{}. In fact by virtue of the constraints (20) it holds $${\mathcal P} < \frac{\Gamma_d}{h} \, \mu_R f_L(E_d) < \frac{\Gamma_d}{h} \, \mu_R f_L\!\!\left( \frac{\mu_R}{\eta_c^{(tm)}}\right) .$$ The last quantity in the above equation is evidently bounded as the chemical potential $\mu_R$ is varied in the interval $[0,+\infty]$, and so is the power production, in agreement with the general findings in the literature[@WHITNEY14; @WHITNEY15; @HANGGI76; @LUO18].  Fig.6 reports the contour plot of the power in the positive part of the $\mu_R$ variable and positive $E_d$. We find that in the region (I), the maximum value of the output power occurs for $E_d =0.102 \ {\rm eV \, and} \, \mu_R = 0.030 \ {\rm eV}. $ It should be noticed that the value of ${\mathcal P}$ is zero along the $E_d$-axis, where $\mu_R=0$, and also along the line $E_d=\varepsilon_0 = \mu_R/\eta_c^{(tm)}$, where $f_L - f_R = 0$. Conversely, in the refrigeration region the absorbed power ${\mathcal P}$ is negative and [*is not bounded*]{}. In fact, from Eq.(1e) the absorbed power along the $E_d = \mu_R$ line (setting $\mu_L=0$) reads $${\mathcal P}(E_d=\mu_R) = \frac{\Gamma_d}{h} \, \mu_R \left[ f_L(\mu_R) - f_R(\mu_R)\right] = \frac{\Gamma_d}{h} \, \mu_R \left[ \frac{1}{e^{\mu_R/k_BT_L} + 1} - \frac{1}{2}\right] \ .$$ It is apparent that $${\mathcal P}(E_d=\mu_R) \rightarrow - \frac{1}{2} \frac{\Gamma_d}{h} \, \mu_R \qquad for \qquad \mu_R \rightarrow +\infty \ .$$ The above expression is evidently not bounded for large values of the chemical potential $\mu_R$ (with respect to $\mu_L=0$). In summary: the value of ${\mathcal P}$ is zero along the line $E_d=\varepsilon_0 = \mu_R/\eta_c^{(tm)}$, while it is given approximately by the value $(-1/2) (\Gamma_d/h) \mu_R$ along the line $E_d = \mu_R$, when $\mu_R$ exceeds few $k_B T_L$. The above considerations are qualitatively well represented by the contour plot of Fig.6 in the refrigeration region (II). We report in Fig 7, in the first quadrant of the $(\mu_R,E_d)$-plane, the contour plot of the efficiency (red region) and the performance coefficient (blue region) of the considered ideal filtering nanostructure in the power production and refrigeration regimes. The line $E_d = \mu_R \cdot T_L/(T_L- T_R)$ provides the maximum efficiency both for heat-energy conversion and for refrigeration, where the input or output power is zero. Around this line an optimal trade off for power generation or refrigeration can be established. Notice that the efficiency at the maximum power output in region (I) is $\eta^{(tm)}_{\mathcal P_{Max}}= \mu_R/E_d$=0.296, and that for the chosen temperature ranges $\eta_c^{(tm)}$=0.5 and $\eta_c^{(refr)}=1$.  Further information can be obtained from the study of the heat current flowing from the left reservoir, and reported in Fig.8. The left heat current of Eq.(1c) for a peaked transmission function (and $\mu_L$ set to zero) reads $$I_Q^{(left)} = \frac{\Gamma_d}{h} \, E_d \, [ f_L(E_d) - f_R(E_d) ] \ . \hspace{2cm} % Eq.(24)$$ Within the region (I), $I_Q^{(left)}$ is positive, and becomes zero along the border line $E_d= \varepsilon_0 = \mu_R/\eta_c^{(tm))}.$ Most importantly it can be noticed that [*the left thermal current is bounded in the region (I).*]{} In fact, from Eq.(24), in the region under consideration, we have $$I_Q^{(left)} < \frac{\Gamma_d}{h} \, E_d \, f_L(E_d) = \frac{\Gamma_d}{h} \, E_d \, \frac{1}{e^{E_d/k_BT_L} +1} \ .$$ The last expression of the above inequality is bounded as the value of $E_d$ is varied in the interval $[0,+\infty].$ This entails that also $I_Q^{(left)}$ is bounded in the whole power generation region, in agreement with the general findings in the literature[@WHITNEY14; @WHITNEY15; @HANGGI76; @LUO18]. From Fig.8, it is seen that the maximum of the left thermal current occurs at the point ${\overline E}_d \approx 0.091$ eV, along the border line $\mu_R=0$. This last feature is indeed expected: for $\mu_R=0$ ($\equiv \mu_L$) no chemical potential barrier is of obstacle to the carrier diffusion. For the same token, the contour curves of the thermal current are expected to bend upwards starting from values of $E_d< {\overline E}_d$, and bend downwards starting from values of $E_d> {\overline E}_d$. All the described features are well visualized in Fig.8, whose physical contents can now be better appreciated.  We comment now on the features of $I_Q^{(left)}$ in the refrigeration region, also reported in Fig.8. The refrigeration region (II) is delimited by the border line $E_d = \mu_R/\eta_c^{(tm)}$ and the other border line $E_d=\mu_R$ , bisector of the first quadrant. In the refrigeration region $I_Q^{(left)}<0$, and a most relevant feature to be noticed is that $| I_Q^{(left)}|$ is not bounded in the region (II). Consider in fact Eq.(24) for the left thermal current along the line $E_d=\mu_R$. One obtains $$I_Q^{(left)}(E_d=\mu_R) = \frac{\Gamma_d}{h} \, \mu_R \left[ f_L(\mu_R) - f_R(\mu_R)\right] = \frac{\Gamma_d}{h} \, \mu_R \left[ \frac{1}{e^{\mu_R/k_BT_L} + 1} - \frac{1}{2}\right] \ ,$$ In analogy to the discussion of Eq.(22), the last expression is evidently [*not bounded*]{}, and so is the left heat current flowing from the left reservoir. \[As we shall see, the coincidence of Eq.(22) and Eq.(25) is due to the fact that the right thermal current is rigorously zero on the boundary line $E_d=\mu_R$. We can anticipate that the right thermal current is bounded also in the refrigeration regime, contrary to the left thermal current that is bounded only in the power production regime\]. It should be noticed that the value of $I_Q^{(left)}$ is zero along the line $E_d=\varepsilon_0 = \mu_R/\eta_c$, while it is given approximately by the value $(-1/2) (\Gamma_d/h) \mu_R$ along the line $E_d = \mu_R$. The above considerations are qualitatively well pictured in Fig.8 by the structure of the curves of $I_Q^{(left)}$ in the refrigeration region (II) of the thermoelectric machine.  We consider now the heat current flowing from the right reservoir, and reported in Fig.9. The right thermal current of Eq.(1d) for a peaked transmission function gives $$I_Q^{(right)} = \frac{\Gamma_d}{h} \, (E_d -\mu_R)\, [ f_L(E_d) - f_R(E_d) ] \ . % Eq.(26)$$ In the region (I), $I_Q^{(right)}$ is positive and bounded. In fact in Eq.(26) the factor $(E_d -\mu_R)$ is always positive; it follows $$I_Q^{(right)} < \frac{\Gamma_d}{h} \, (E_d -\mu_R)\, f_L(E_d) < \frac{\Gamma_d}{h} \, E_d \, f_L(E_d) = \frac{\Gamma_d}{h} \, E_d \frac{1} { e^{E_d/k_BT_L} + 1 } \ . \hspace{1cm} % Eq.(27)$$ The last expression is evidently bounded and so is the right thermal current flow, in agreement with the general findings in the literature [@WHITNEY14; @WHITNEY15; @HANGGI76; @LUO18]. In Fig.9 it is seen pictorially that the right heat current from the right reservoir is bounded. Notice that $I_Q^{(right)}$ of Fig.9 and $I_Q^{(left)}$ of Fig.8 are perfectly equal on the $E_d$ axis since the power generated vanishes there. The above considerations emerge with evidence from the structure of the curves of Fig.9 in the power generation region. In the refrigeration region $I_Q^{(right)} <0$ and it is seen by inspection that [*the right heat current is bounded.* ]{} In fact, from Eq.(27) we obtain $$I_Q^{(right)} > - \frac{\Gamma_d}{h} \, (E_d - \mu_R) f_R(E_d) = - \frac{\Gamma_d}{h} \, (E_d - \mu_R) \, \frac{1} {e^{(E_d-\mu_R)/k_BT_R} + 1} \ .$$ The last expression (in absolute value) is evidently bounded, and so is the right thermal current. It can also be noticed that $I_Q^{(right)}$ vanishes along the line $E_d=\varepsilon_0 = \mu_R/\eta_c^{(tm)}$, and also along the line $E_d = \mu_R$. The above considerations are well visualized in Fig.9 by the structure of the curves in the refrigeration region (II) of the thermoelectric machine. In summary the contour plots of $I_Q^{(left)}$ in Fig.8 and $I_Q^{(right)}$ in Fig.9 are rather similar in the power generation region: in particular they are both bounded, and present a single maximum of the same value located on the same position along the $E_d$-axis. Completely different are instead the contour plots of Fig.8 and Fig.9 in the refrigeration region: the basic feature is the absence of bounds in $I_Q^{(left)}$ (similarly to the absence of bounds for the power absorbed), while $I_Q^{(right)}$ is strictly bounded. This restriction is linked to the fact that $I_Q^{(right)}$ represents the thermal current extracted from the cold reservoir in the refrigeration mode: according to the general findings in the literature[@WHITNEY14; @WHITNEY15; @HANGGI76; @LUO18], this useful “cooling power" performance cannot exceed appropriate bounds. Conclusions =========== We have investigated the thermoelectric transport properties of a nanoscale system characterised by a peaked transmission function of the energy carriers around a resonant energy $ E_d$ near the chemical potential. 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--- abstract: 'In hierarchical models, where spheroidal galaxies are primarily produced via a continuous merging of disk galaxies, the number of intrinsically red systems at faint limits will be substantially lower than in “traditional” models where the bulk of star formation was completed at high redshifts. In this paper we analyse the optical–near-infrared colour distribution of a large flux-limited sample of field spheroidal galaxies identified morphologically from archival [*Hubble Space Telescope*]{} data. The $I_{814}-HK''$ colour distribution for a sample jointly limited at $I_{814}<$23 mag and $HK''<$19.5 mag is used to constrain their star formation history. We compare visual and automated methods for selecting spheroidals from our deep HST images and, in both cases, detect a significant deficit of intrinsically red spheroidals relative to the predictions of high-redshift monolithic collapse models. However the overall space density of spheroidals (irrespective of colour) is not substantially different from that seen locally. Spectral synthesis modelling of our results suggests that high redshift spheroidals are dominated by evolved stellar populations polluted by some amount of subsidiary star formation. Despite its effect on the optical-infrared colour, this star formation probably makes only a modest contribution to the overall stellar mass. We briefly discuss the implications of our results in the context of earlier predictions based on models where spheroidals assemble hierarchically.' author: - | F. Menanteau $^1$, R. S. Ellis$^1$, R. G. Abraham$^{1,2}$, A. J. Barger$^{3}$, and L. L. Cowie$^3$\ $^1$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 OHA, England\ $^2$Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 0EZ, England\ $^3$Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\ date: 'Received:   Accepted: ' title: 'The Optical-Infrared Colour Distribution of a Statistically-Complete Sample of Faint Field Spheroidal Galaxies' --- \[firstpage\] INTRODUCTION ============ The age distribution of elliptical galaxies is a controversial issue central to testing hierarchical models of galaxy formation. The traditional viewpoint (Baade 1957, Sandage 1986) interprets the low specific angular momentum and high central densities of elliptical galaxies with their dissipationless formation at high redshift. In support of this viewpoint, observers have cited the small scatter in the colour-magnitude relation for cluster spheroidals at low redshifts (Sandage & Visvanathan 1978, Bower et al 1992) and, more recently, such studies have been extended via HST imaging to high redshift clusters (Ellis et al 1997, Stanford et al 1997). Examples of individual massive galaxies with established stellar populations can be found at quite significant redshifts (Dunlop 1997). In contrast, hierarchical models for the evolution of galaxies (Kauffmann et al 1996, Baugh et al 1996) predict a late redshift of formation for most galactic-size objects because of the need for gas cooling after the slow merger of dark matter halos. These models propose that most spheroidal galaxies are produced by subsequent mergers of these systems, the most massive examples of which accumulate since $z\simeq$1. Although examples of apparently old ellipticals can be found in clusters to quite high redshift, this may not be at variance with expectations for hierarchical cold dark matter (CDM) models since clusters represent regions of high density where evolution might be accelerated (Governato et al 1998). By restricting evolutionary studies to high density regions, a high mean redshift of star formation and homogeneous rest-frame UV colours would result; such characteristics would not be shared by the field population. Constraints on the evolution of field spheroidals derived from optical number counts as a function of morphology (Glazebrook et al 1995, Im et al 1996, Abraham et al 1996a) are fairly weak, because of uncertainties in the local luminosity function. Nonetheless, there is growing evidence of differential evolution when their properties are compared to their clustered counterparts. Using a modest field sample, Schade et al (1998) find a rest-frame scatter of $\delta(U-V)$=0.27 for distant bulge-dominated objects in the HST imaging survey of CFRS/LDSS galaxies, which is significantly larger than the value of $\simeq$0.07-0.10 found in cluster spheroidals at $z\simeq$0.55 by Ellis et al 1997. Likewise, in their study of a small sample of galaxies of known redshift in the [*Hubble Deep Field*]{} (HDF), Abraham et al (1998) found a significant fraction ($\simeq$40%) of distant ellipticals showed a dispersion in their internal colours indicating they had suffered recent star formation possibly arising from dynamical perturbations. Less direct evidence for evolution in the field spheroidal population has been claimed from observations which attempt to isolate early-type systems based on predicted colours, rather than morphology. Kauffmann et al (1995) claimed evidence for a strong drop in the volume density of early-type galaxies via a $V/V_{max}$ analysis of colour-selected galaxies in the [*Canada-France Redshift Survey*]{} (CFRS) sample (Lilly et al 1995). Their claim remains controversial (Totani & Yoshii 1998, Im & Castertano 1998) because of the difficulty of isolating a robust sample of field spheroidals from $V-I$ colour alone (c.f. Schade et al 1998), and the discrepancies noted between their analyses and those conducted by the CFRS team. In addition to small sample sizes, a weakness in most studies of high redshift spheroidals has been the paucity of infrared data. As shown by numerous authors (eg. Charlot & Silk 1994), near-IR observations are crucial for understanding the star formation history of distant galaxies, because at high redshifts optical data can be severely affected by both dust and relatively minor episodes of star-formation. Recognizing these deficiencies, Moustakas et al (1997) and Glazebrook et al (1998) have studied the optica-infrared colours of small samples of of morphologically-selected galaxies. Zepf (1997) and Barger et al (1998) discussed the extent of the red tail in the optical-IR colour distribution of HDF galaxies. Defining this tail ($V_{606}$-$K>$7 and $I_{814}$-$K>$4) in the context of evolutionary tracks defined by Bruzual & Charlot’s (1993) evolutionary models, they found few sources in areas of multicolour space corresponding to high redshift passively-evolving spheroidals. The ultimate verification of a continued production of field ellipticals as required in hierarchical models would be the observation of a decrease with redshift in their comoving space density. Such a test requires a large sample of morphologically-selected ellipticals from which the luminosity function can be constructed as a function of redshift. By probing faint in a few deep fields, Zepf (1997) and Barger et al (1998) were unable to take advantage of the source morphology; constraints derived from these surveys relate to the entire population. Moreover, there is little hope in the immediate term of securing spectroscopic redshifts for such faint samples. The alternative adopted here is to combine shallower near-infrared imaging with more extensive HST archival imaging data, allowing us to isolate a larger sample of [*brighter, morphologically-selected*]{} spheroidals where, ultimately, redshifts and spectroscopic diagnostics will become possible. Our interim objective here is to analyse the optical-infrared colour distribution of faint spheroidals which we will demonstrate already provides valuable constraints on a possible early epoch of star formation. A plan of the paper follows. In $\S$2.1 we discuss the available HST data and review procedures for selecting morphological spheroidals from the images. In $\S$2.2 we discuss the corresponding ground-based infrared imaging programme and the reduction of that data. The merging of these data to form the final catalogue is described in $\S$2.3. In $\S$3 we discuss the optical-infrared colour distribution for our sample in the context of predictions based on simple star formation histories and consider the redshift distribution of our sample for which limited data is available. We also examine constraints based on deeper data available within the Hubble Deep Field. In $\S$4 we summarise our conclusions. CONSTRUCTION OF THE CATALOGUE ============================= THE HST SAMPLE -------------- In searching the HST archive for suitable fields, we adopted a minimum $I$ F814W-band exposure time of 2500 sec and a Galactic latitude of $|b|$=19$^{\circ}$ so that stellar contamination would not be a major concern. These criteria led to 48 fields accessible from the Mauna Kea Observatory comprising a total area of 0.0625 deg$^2$(225 arcmin$^2$). Table 1 lists the fields adopted, including several for which limited redshift data is available e.g. the HDF and its flanking fields (Williams et al 1996), the Groth strip (Groth et al 1994) and the CFRS/LDSS survey fields (Brinchmann et al 1997). F606W imaging is available for 25 of the fields in Table 1. Object selection and photometry for each field was performed using the [SExtractor]{} package (Bertin & Arnouts 1996). Although the detection limit varies from field to field, the -band data is always complete to $\sim 24$ mag and the -band to $\sim 25$ mag[^1]. The morphologies of galaxies in the sample were investigated independently using visual classifications made by one of us (RSE), and automated classifications based on the central concentration ($C$) and asymmetry ($A$) parameters defined in Abraham et al. (1996b). In the case of visual classifications we adoped the MDS scheme defining spheroidal to include E: E/S0: S0 and S0/a (MDS types 0,1,2). As shown below, the visual and automated classifications compare quite favourably, with particularly satisfactory agreement for the regular spheroidal galaxies that are the focus of this paper. The appropriate limiting magnitude of our survey is set by that at which we believe we can robustly isolate spheroidal galaxies from compact HII galaxies, stars and bulge-dominated spirals. The Medium Deep Survey (MDS) analyses adopted a limiting magnitude for morphological classification (using nine classification bins) of =22 mag, although some MDS papers extended this further to =23 mag (see Windhorst et al 1996 for a summary). In Abraham et al (1996b) and Brinchmann et al (1997), HST data similar to that in the present paper was also used to classify galaxies to =22 mag. However, by restricting our analysis in the present paper to spheroidal systems, we are able to extend classifications to slightly deeper limits (=23.0 mag). This is possible because the chief diagnostic for discriminating spheroidals is central concentration, rather than asymmetry which is sensitive to lower surface brightness features. Because the classifications based on $A$ and $C$ are objective, the classification limits for the present dataset have been investigated using simulations, as described below. Figure 1 shows a typical set of morphologically-identified spheroidals at various magnitudes down to =23 mag. Figure 2 compares the $A-C$ and visual morphological distributions at a range of magnitude intervals, down to the limits of our survey. The demarcation between early and late-types on the basis of $A$ and $C$ is made using bright galaxies ($<20$ mag) and shifted slightly as a function of magnitude on the basis of simulations made using the [IRAF]{} package [artdata]{}, which model the apparent change in the central concentration of an $r^{1/4}$ law elliptical galaxy as a function of decreasing signal-to-noise. Random errors on central concentration are also determined on the basis of simulations, and representative error bars are shown in Figure 2. The general agreement between the visual and automated classifications is remarkably good, particularly to $=$22 mag. Between $=$22 and 23 mag the agreement worsens, mostly because of the great increase in the number of visually-classified “compact” systems. We define compacts to be those systems where there is no clear distinction between small early-type galaxies, faint stars and/or HII regions. In order to quantify the concordance between the visual and automated classifications, the $A-C$ distribution was analysed using a statistical bootstrap technique (Efron & Tibshirani 1993). The $A-C$ distribution was resampled 500 times in order to determine the uncertainties in both the number of systems classified as early-type, and the uncertainties on the colour distribution for these systems. These measurements will be discussed further below in §3.3. The somewhat larger number (323 vs 266) of $A/C$-classified early-types relative to the visually classified galaxies is significant at the 3$\sigma$ level. However, if the compact systems are included in the tally of visually classified early-type systems, then the number of visually and A/C classified ellipticals agree closely (to within 1 sigma). It is clear that the distinction between compact galaxies and early-type systems is an important consideration when determining the number counts of early type systems at the faint limits of these data. However, it is perhaps worth noting at this stage that another bootstrap analysis (presented in §3.3) shows that the uncertainty introduced by compact systems into the number counts at $22<I<23$ does [*not*]{} manifest itself as a large uncertainty in the colour histograms of the early-type population. GROUND-BASED INFRARED IMAGING ----------------------------- Although some of the fields in Table 1 have and HST data, such a wavelength baseline is not very useful at thesed depths. As discussed by Moustakas et al (1997) and Zepf (1997), the addition of infrared photometry is especially helpful in distinguishing between passively-evolving systems and those undergoing active star formation, [*regardless of redshift*]{}, primarily because of its reduced sensitivity to K-dimming, small amounts of star formation and dust reddening. Our infrared imaging was mainly conducted using the QUIRC 1024$^2$ infrared imager on the University of Hawaii 2.2-m telescope. The log of observations is summarised in Table 2. In order to improve the observing efficiency in securing deep infrared photometry for a large number of WFPC-2 fields, we used the notched $H+K'$ 1.8$\mu$m filter (which we refer to hereafter as the $HK'$ filter) (Wainscoat & Cowie 1998, Figure 3) which offers a gain in sensitivity of typically a factor of $\simeq$2 over a conventional $K'$ filter. At the f/10 focus of the UH 2.2m, the field of view is $193''\times 193''$ with a scale of $0.1886''\,$pixel$^{-1}$ ensuring that the 3 WFPC2 chips can be comfortably contained within a single exposure. Each $HK'$ exposure was composed of 13 sub-exposures of $\simeq$100 sec duration (depending on the background level) spatially-shifted by increments of 5-20 arcsec in all directions. This dithering pattern was repeated 2-3 times during the exposure. The data was processed using median sky images generated from the disregistered exposures and calibrated using the UKIRT faint standards system. Most of the data was taken under photometric conditions; deep non-photometric data was calibrated via repeated short exposures taken in good conditions. The limiting magnitude of the infrared data varies slightly from field to field and is deepest for the HDF and flanking fields which were taken in a separate campaign (Barger et al 1998). In order to determine the detection limit of our $HK'$ data we performed extensive Monte Carlo simulations. Using the IRAF [artdata]{} package we created simulated data sets, which were subsequently analysed using the same extraction and measurement methods as for the real data. With the task [mkobjects]{} we generated artificial galaxies assuming exponential disk profiles with no internal absorption for spirals and de Vaucouleurs profiles for spheroidals. The profile scales were chosen to be magnitude-dependent converging to the image seeing at faint limits. Figure 4 shows the results of this exercise. The 80% completeness limit for spheroidals is $HK'$=19.5 mag for most of the survey extending to $HK'$=20.5 mag for the HDF and flanking fields. COMPLETENESS OF THE COMBINED OPTICAL-INFRARED CATALOGUE ------------------------------------------------------- The final photometric catalogue of spheroidals was obtained by matching the HST $I_{814}$-band and the ground-based IR [SExtractor]{} catalogues using the adopted magnitude limits of $HK'<$19.5 mag and $<$23.0 mag. In the final matched catalogue, we retained the SExtractor ‘$m_{best}$’ magnitudes but measured $I-HK'$ colours within a fixed 3 arcsec diameter. This aperture size, together with the fairly good seeing of the IR data, ensures that when calculating colours we are looking at the same physical region of the galaxy. Of the 818 sources in the matched catalogue, 266 systems were visually classified as spheroidals (defined to be one of ‘E, E/S0, S0, or S0/a’ in the MDS scheme) and 50 as compact objects. Automated classifications result in 323 sources classed as spheroidals (with no distinction between spheroidals and compacts). Clearly the joint selection by and $HK'$ necessary to exploit HST’s morphological capabilities and establish optical-infrared colours could lead to complications when interpreting $I-HK'$ colour distributions. As a major motivation for this study is to identify as completely as possible the extent of any red tail in the colour distribution, incompleteness caused by the various magnitude limits is an important concern. Figure 5 shows that, within the [*optical, morphologically-selected*]{} sample with $<$23 mag, virtually all of the $HK'<$19.5 mag sample can be matched; only a small fraction (18/818=2.2%) of red $I-HK'>$3.5-5 mag objects are missed. We return to the nature of these sources in $\S$3.3. ANALYSIS ======== Strategy -------- Our analysis is motivated by the two principal differences we might expect between models where ellipticals underwent a strong initial burst of activity with subsequent passive evolution (which we will term the ‘monolithic collapse’ model) and those associated with a hierarchical assembly of ellipticals from the dynamical merger of gas-rich disks (Baugh et al 1996). We recognise at the outset that these models represent extreme alternatives with a continuum of intermediate possibilities (c.f. Peacock et al 1998; Jimenez et al. 1998). Our strategy in this paper, however, will be to discuss our field elliptical data in the context of the simplest models proposed to explain the star formation history of distant [*cluster*]{} ellipticals (Ellis et al 1997, van Dokkum et al 1998). More elaborate analyses are reserved until spectroscopic data is available for a large sample. Firstly, in the monolithic collapse model, the comoving number density of luminous ellipticals should be conserved to the formation redshift (say, $z\simeq$3-5), whereas in hierarchical models we can expect some decline in number density at moderate redshift depending on the cosmological model and other structure formation parameters (Kauffmann et al 1996, Kauffmann & Charlot 1998a). Such a change in the absolute number density would be difficult to convincingly detect without spectroscopic data. The number of faint HST-identified ellipticals has been discussed by Glazebrook et al (1995), Driver et al (1995), Abraham et al (1996b) and Im et al (1996) with fairly inconclusive results because of uncertainties arising from those in the local luminosity function used to make predictions (Marzke et al 1998). Secondly, there will be a redshift-dependent colour shift associated with merger-driven star formation in the hierarchical models whereas, for the monolithic case, the sources will follow the passive evolution prediction. Kauffmann et al (1996) initially claimed that both signatures would combine in the hierarchical picture to produce a factor 3 reduction in the abundance of passively-evolving sources by $z\simeq$1, but a more recent analysis (Kauffmann & Charlot 1998b) shows that the decline is dependent on the input parameters. For example in a model with non-zero cosmological constant (the so-called ’$\Lambda$CDM’), little decline is expected until beyond z$\simeq$1. In contemplating these hypotheses in the context of our HST data, it must be remembered that although HST can be used very effectively to isolate spheroidals morphologically to =23 mag (representing a considerable advance on earlier colour-selected ground-based samples which could be contaminated by dusty later types), in the case of merger models, the predictions will depend critically on the time taken before a system becomes a recognisable spheroidal. However, any hypothesis which postulates a constant comoving number density of well-established spheroidals is well suited for comparison with our data, the outcome being important constraints on the past star formation history and luminosity evolution. Colour Distributions -------------------- Figure 6 shows -$HK'$ colour histograms for both the visual and A/C-selected spheroidals alongside those for the compacts and the remainder. The automated and visual catalogues have nearly identical colour distributions, confirming earlier tests on the reliability of the automated classifier. In fact, the differences between the automated and visually defined histograms are almost completely attributable to the compact systems, which cannot be segregated from other early-types on the basis of central concentration. The colour histogram for compacts spans the range seen for early-type galaxies, with a peak slightly redward of that for visually classified early-types. It is clear from the similarity between the colour histograms for visual and automated classifications that contamination of spheroidals by compacts (expected in the automated catalogue) does not pose a significant uncertainty in determining the colour distribution. The histogram of colours for late-type galaxies peaks at nearly the same colour as that for the early-types, which at first seems somewhat surprising. As we will later see, this is largely a reflection of the very wide redshift range involved. However, the distribution for spirals and later types is skewed toward the blue, although redward of $I-HK'=$2.5 mag the shapes of the distributions are similar (see also §3.5 below). Single Burst Model Predictions ------------------------------ Figure 7 compares the observed colour histograms with a range of model predictions based on the GISSEL96 spectral synthesis code (Bruzual & Charlot 1996) for a range of star-formation histories. Observed and predicted total counts for each of the models are also given in Table 3. At this stage we concentrate on ‘single burst’ or ‘monolithic collapse’ models which conserve the comoving number density at all epochs, and defer discussion of alternative scenarios until §3.5. Our model predictions take into account the joint and $HK'$ selection criteria for our sample, and are based on the present-day optical E/S0 luminosity function (LF) from Pozzetti et al. (1996), ie. a standard Schechter function with $\phi^\star=0.95\times 10^{-3}$ Mpc$^{-3}$, $M^{\star}_{b_{j}}=-20.87$ and a faint-end slope of $\alpha=-0.48$. When making model predictions, this luminosity function is tranformed into one appropriate for the photometric band via a single colour shift, resulting in $M^{\star}_{I_{814}}=-23.12$. For comparison, we also show predictions assuming a suitably transformed luminosity function with a flat faint-end slope ($\alpha = -1$) and $\phi^\star=0.55\times 10^{-3}$Mpc$^{-3}$ as suggested by Marzke et at (1998). Throughout this paper we adopt $H_0=50$ Km s$^{-1}$Mpc$^{-1}$. Given the elementary nature of the comparisons currently possible, and the fact that the expected dispersion in $I-HK'$ from the present-day colour-luminosity relation is minimal, we have avoided the temptation to model a [*distribution*]{} of metallicities within the galaxy population, preferring instead to explore the effects of fixing the metallicity of the entire population to a single value within a large range (40%-250% solar) in the simple predictions discussed below. Other variables in the single burst hypothesis include the redshift of formation, $z_F$ (fixed at $z_f=5$), the burst-duration (represented as a top hat function of width 1.0 Gyr) and the cosmological parameters ($\Omega_M$ and $\Omega_\Lambda$). As shown in Appendix A, the luminosity weighted metallicities of the present sample are not biased strongly by the limiting isophotes of the our observations, and fair comparisons can be made using individual single-metallicity tracks over a broad range of redshifts. Clearly the most important input parameter in the model predictions shown in Figure 7 (summarized in Table 3) is the metallicity. Although our spheroidals are almost exclusively luminous ($>L^{\ast}$) galaxies which, in the context of single-burst models would imply a metallicity of at least solar (cf. Appendix A, and Arimoto et al 1997), here we will explore the possibility that part of the colour distribution could arise from a wider metallicity range than that found locally. ### A Deficit of Red Spheroidals The predicted colour distributions show a characteristic asymmetry. This is caused by the blending of the $I_{814}-HK'$ K-correction and the passive bluing of systems to $z\simeq$1.5. In contrast, the observations reveal a clear excess of blue ($-HK'<$2 mag) spheroidals not predicted by even the lowest metallicity models. More significantly, solar and super-solar metallicity models over-predict the extent of the red tail in the colour distribution. Both discrepancies are consistent with recent star-formation in our sample of faint spheroidals. In order to quantify these discrepancies, we performed a Kolmogorov-Smirnov test (K-S) to check whether the observed spheroidal colours could be drawn from any of the model distributions (allowing a measurement error $\sigma_{I-HK'}=0.25$ mag). In all cases the observed distribution differs from the models at a confidence level higher than 99.99%. Evidently monolithic collapse models with constant co-moving density fail to reproduce the colour distribution of high redshift spheroidals. It will be convenient in the following to quantify the strength of the red tail in the colour distribution by defining a “red fraction excess”, shown in Table 3, constructed as the ratio of the predicted number of early type systems with $I_{814}-HK' > 3.0$ mag to the observed number. The statistical uncertainties on the red fraction excess in this table are based on 500 bootstrap resamplings of the original catalogue, each realization of which was subjected to the same selection criteria applied to the original data. As discussed by many authors (Glazebrook et al 1995, Marzke et al 1998), the absolute numbers depends sensitively on the normalisation and shape of the local LF. Table 3 includes a summary for the range in LF parameters discussed earlier. For a declining faint-end slope ($\alpha=-0.48$) and solar-and-above metallicity, the red fraction excess is more than 5 times that observed. Adopting a metallicity substantially below solar results in closer agreement but assuming such metallicities for the entire population may be unreasonable given local values (see Appendix, and Arimoto et al 1997). Even so, the red fraction excess is only reduced from $\sim 8$ to $\sim 3$ if the adopted metallicity is varied between 250% solar and 40% solar. Models with a flat faint-end slope ($\alpha=-1$) improve the agreement further and in the very lowest metallicity model with $\Omega_M=0$ there is almost no deficit. In Table 3 we have also included models with $\Omega_{\Lambda}>0$ for both slopes of the LF. The effect of $\Omega_{\Lambda}$ is to increase the apparent deficit of red spheroidals (because of the rapid increase in the differential volume element with redshift for $\Omega_{\Lambda}>0$ cosmologies at $z<1$). Although models where the redshift of the initial burst, $z_F$, is reduced to $z=3$ are not shown in the table, these do not result in significant changes in the above discussions. We conclude that we cannot reconcile the number of galaxies in the red end of the observed colour histogram to the corresponding predictions of a constant co-moving density high-redshift monolithic collapse model. Alternative scenarios, which may explain the relatively blue colours of some observed spheroidals, will be considered in §3.5. ### A Declining Number of High Redshift Spheroidals? While monolithic collapse models fail to reproduce the observed colour distributions, Table 3 indicates that the overall number is in reasonable agreement. Specifically, for a low $\Omega_M$ and $\Omega_\Lambda$=0, the Marzke et al. luminosity function and luminosity-weighted metallicities of solar and above, we see no significant evolution in the space density of spheroidals. For the currently popular spatially-flat universe with low $\Omega_M$ and high $\Omega_\Lambda$ (Perlmutter et al 1999), the data imply a deficit of no more than 30%. Stronger evolution ($\sim$ 60% decline) would occur if we adopted the Pozetti et al. luminosity function. We therefore conclude that the colour offset described earlier is more likely the result of star-formation activity in well-formed spheroidals at high redshifts rather than evidence for evolution in their space density. At this point, we return to the nature of those 18 sources identified in the infrared images which are fainter than =23 mag. Although they could formally be included in the colour distributions, they are too faint in the WFPC-2 images for reliable morphological classification. A montage of these sources is shown in Figure 8. At most 3 of the 18 are [*possible*]{} spheroidals. As such, their addition to the colour distribution would have a negligible impact on conclusions drawn from Figure 7. Constraints from Redshift Distributions --------------------------------------- While our principal conclusions are based on the enlarged size of our HST sample and addition of infrared photometry compared to earlier work, it is interesting to consider what can be learnt from the (incomplete) redshift data currently available for our sample. We have collated the published spectroscopic redshift data from the CFRS/LDSS surveys (Brinchmann et al 1998), the MDS survey (Glazebrook et al 1998) and the HDF and its flanking fields (tabulated by Cowie 1997) and matched these with our $HK'$-selected sample. In total, 97 of our galaxies have published redshifts. As the bulk of these surveys were themselves magnitude-limited in $I$, the magnitude and colour distribution of this subset should be representative of that for an appropriate subset of our primary photometric sample. Figure 9 (upper panel) shows the colour-redshift relation for the 97 objects in the subsample with redshift information. Also shown are evolutionary predictions based on spectral synthesis models adopting ranges in metallicity and star-formation history as before. The 47 spheroidals in this set are clearly redder at a given redshift than their spiral and later-type counterparts and span a wide redshift range with median value $\overline{z}\simeq$0.7. However, as Schade et al (1998) discussed in the context of -colours for their smaller sample of HST field galaxies, there is some overlap between the classes at a given redshift. The colour scatter for morphological spheroidals appears somewhat larger than the $\sim 0.2$ mag dispersion expected from slope of the infrared-optical colour-luminosity relations for early-type systems[^2]. In the specific case of the HDF, it is interesting to exploit the increased depth of both the $HK'$ data and the HST optical morphologies (Abraham et al 1996a) as well as to consider the abundant photometric redshift estimates. For this purpose we constructed a 19.5$<HK'<$20.5 mag extension to our HDF sample, with morphological classifications from the deep ($< 25$ mag) morphological catalogue of van den Bergh (1996). For this sample we can take advantage of the apparently rather good precision in photometric redshift estimates for early-type galaxies (Connolly et al 1997, Wang et al 1998)[^3]. By going deeper in the HDF we extend our sample by 20 objects, of which 8 are classed visually as E/S0s (none are compact). Adding this extension to those HDF galaxies already in our catalog, the corresponding numbers with $HK'<20.5$ mag; $I_{814} < 25$ mag become 50 and 26 respectively. The colour-redshift relation for the combined HDF sample is shown in Figure 9 (lower panel). For the same sample, Figure 10 shows the colour-magnitude diagram of the visually classified E/S0s. The inset shows the colour histogram and the arrow indicates the peak in the distribution for the primary sample. As expected, the peak of the HDF colour histogram in Figure 10 lies redward of the colour histogram for our entire sample (by $\sim 0.2$ mag). But the redshift data in Figure 9 makes it clear that this peak is still substantially bluer than expected for the simple monolithic collapse model at solar metallicity. For the 26 HDF spheroidals, spectroscopic redshifts are available for 19, the rest being photometric. Importantly, the spectroscopically-confirmed galaxies include 3 ellipticals beyond z$\simeq$0.9, all of which are substantially bluer than the passive evolution predictions. While based on small numbers of galaxies, the figure lends strong support to the conclusions of the previous subsection, particularly when it is realised there is an in-built bias in favour of photometric redshifts matching the passively-evolving spectral energy distributions. These conclusions based on the HDF are consistent with those of Zepf (1997) and Barger et al (1998) who analysed optical-infrared colours of much fainter sources without taking into account morphological and redshift information. Alternative Star Formation Histories ------------------------------------ The single burst models ruled out by the colours of speroidals in the previous sections are idealised representations of spheroidal history. We now consider alternative histories which could be more consistent with our various datasets. At its most fundamental level, the deficit of red spheroidals at faint limits appears to eliminate models with very short epochs of star formation at high redshifts followed by long quiescent periods. In the context of modelling distant red radio galaxies, Peacock et al (1998) have shows that models with continuous star formation truncated at later times avoid the peak luminosities associated with initial bursts and can produce a significant bluing at redshifts where the red tail would otherwise be seen[^4]. In an important recent paper, Jimenez et al. (1998) argue that monolithic models have been rejected prematurely by some authors: only [*extreme*]{} scenarios with very short duration bursts (eg. $10^7$ Myr) of star-formation followed by absolute quiescence can be ruled out, while bursts with fairly low-levels of extended star-formation activity may still be compatible with the observed data. On the basis of a simple one-dimension chemo-dynamical model for the evolution of spheroids, these authors predict that the integrated star-formation history of early-type galaxies should resemble a quasi-monolithic collapse model. A concrete prediction of this model is that the bulk of the star-formation in high-$z$ spheroids is occuring near their cores, an effect that appears to have been seen (Abraham et al. 1999), and which may be responsible for the blue colours of some ellipticals in our sample. In the light of the above, it is therefore interesting to calculate the amount of recent star formation which must be added to a single burst model in order for the model to match the observed colours. We seek to determine the timescales over which such a system is bluer than the passive evolution model by at least $\delta_{colour} = (I_{814}-HK')_{model} - (I_{814}-HK')_{obs} \simeq 0.3$ mag, i.e. consistent with the typical colour offset from the solar metallicity tracks shown in Figure 9. To model this we added a short burst of star-formation (of duration 0.1 Gyr, forming 15% of the stellar mass) occurring at $z=z_{burst}$ to an underlying solar metallicity monolithic collapse model at a given redshift using the population synthesis models of Bruzual & Charlot (1996). We then computed the redshift range over which the burst at $z_{burst}$ results in $\delta_{color}>0.3$ for a given redshift of observation, $z_o$. The result of this exercise is shown in Figure 11 for several redshifts of observation $z_0$. The range in redshift space where a 15% burst produces a blueing $\delta_{colour} \ge 0.3$ is shown as the shaded area. (For example, a galaxy observed at redshift $z_o=1$ would be at least 0.3 mag bluer than the monolithic collapse models between points A and B in the figure. At $z_o=0.5$ a burst would have to occur at $0.5 <z_{burst}<0.65$, while for $z_o=0.7$ it would have to occur in the range $z_{burst}\sim 0.7-0.9$. Since almost all points in Figure 9 lie blueward of the solar metallicity model tracks, and since galaxies in our sample lying in the redshift range $0<z<1$ have “memory” of bursts over $\delta z \sim 0.1-0.2$, it seems improbable that moderate intensity (ie. 15% mass) single burst events can explain the colour offsets shown in Figure 9. It seems more probable that the duty-cycle of star-formation is more extended, indicative of either a low level of roughly continuous star-formation underlying the old stellar populations, or perhaps of a succession of lower mass bursts. The preceeding analysis indicates the extent to which a modest “polluting” star-forming population superposed on a dominant old stellar population reconciles our observations with traditional monolithic collapse scenarios. In contrast to this, it is interesting to consider what sort of hierarchical formation models may also be consistent with the present data. The rather mild density evolution in our sample is in sharp disagreement with the predicted factor of three decline in the abundance of spheroidals at $z \sim 1$, based on the present generation of “semi-analytical” models in high-density, matter-dominated cosmologies (eg. Kauffmann & Charlot 1998a). However, in a flat $\Lambda$-CDM model (with $\Omega_M=0.3$, $\Omega_\Lambda=0.7$) the decline in the abundance of spheroidals is only 30% at $z \sim 1$ (Kauffmann & Charlot 1998b), and may be consistent with the present observations. In the latter model the oft-quoted factor of three decline in the space density of ellipticals occurs at $z=2$ instead of $z=1$, so future work extending the present sample to higher redshifts may allow us to distinguish between “extended” monolithic collapse scenarios and $\Lambda$-CDM hierarchical models. However, more detailed modelling in this particular case, e.g. of the colours distribution, is beyond the scope of this paper. If star-formation in extended monolithic-collapse scenarios is centrally concentrated (as suggested by the simple one-dimensional models of Jimenez et al. 1998), the two scenarios may also be distinguishable at lower redshifts on the basis of [*resolved*]{} colour distributions (Menanteau et al. 1999, in preparation). CONCLUSIONS =========== We have constructed a new catalogue of $\simeq$300 faint field spheroidal galaxies using HST images for morphology to a limit of =23 mag. Follow-up infrared photometry has enabled us to consider the optical-infrared colour distribution of an infrared magnitude limited sample. We have modelled expected colours using various star formation histories, metallicities and cosmologies. For a limited subset, spectroscopic redshift data is available and within the HDF it is possible to construct a deeper catalogue. Our main results can be summarised as follows: $\bullet$ There is little evidence for strong evolution in the space density of luminous field spheroidal systems out to $z \sim 1$. Within the uncertainties introduced by poorly constrained values for the local spheroidal luminosity function and by $\Omega_M$ and $\Omega_\Lambda$, our data is consistent with no evolution, or with modest evolution (at the level of $\sim 30\%$) in terms of a decline in the space density of spheroidals by $z \sim 1$. $\bullet$ Although we detect little evidence for strong evolution in the space density of luminous spheroidals, we find a marked deficit in the number of red spheroidals compared to predictions where the bulk of star formation was completed prior to $z\simeq$3. $\bullet$ Where redshift data is available it suggests that the duty cycle for star-formation in high-redshift spheroidals is indicative of a low level of extended star-formation underlying a passively evolving population, rather than of relic star-formation following from a massive burst episode. $\bullet$ The apparently mild density evolution and blue colours of high-redshift spheroidals in the present sample are consistent with the predictions of “extended” monolithic collapse scenarios, in which the existing star-formation pollutes the colours of a dominant, underlying old stellar population. The data is [*not*]{} consistent with the predictions of semi-analytical hierarchical models in high-density, matter-dominated cosmologies. However, the observed weak density evolution may be consistent with the predictions of hierarchical $\Lambda$-CDM models (Kauffmann & Charlot 1998b). Spectroscopic redshifts for a complete sub-sample of our catalogue will enable models for the star-formation history of high redshift spheroidals to be rigorously tested, eliminating some of the ambiguities present in the current analysis. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== FM would like to thank PPARC and Fundanción Andes for financial support. RGA acknowledges support from a PPARC Advanced Fellowship. We acknowedge substantial comments from an anonymous referee which transformed an earlier version of this paper. We thank Simon Lilly and Chuck Steidel for valuable input. [1]{} Abraham, R. G., Tanvir, N. R., Santiago, B. X., Ellis, R. 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Zepf, S.E. 1997, Nature, 390, 377 --------------------------------------------------------------------------------------------------------------------------------------------------- Field $I_{814}$ $V_{606}$ $H$+$K'$ FWHM $\alpha$(J2000) $\delta$(J2000) N$_z$ N$_{E/S0}$ --------- ----------- ----------- ---------- --------------------------------------------- ----------------- ----------------- ------- ------------ 0029+13 6300 3300 3900 0.6“ & 00:29:06.2 & 13:08:12.9 & ... & ...\ 01:44:10.6 02:17:51.2 ... ... 0144+2 & 4200 & ... & 2340 & 0.8” 0939+41 4600 5400 3120 0.9“ & 09:39:33.3 & 41:32:47.9 & ... & ...\ 12:10:33.3 39:29:01.6 ... ... 1210+39 & 4899 & 3999 & 3120 & 0.6” 1404+43 8700 5800 3120 0.7“ & 14:04:29.3 & 43:19:15.2 & ... & ...\ 12:37:02.0 62:12:23.4 ... ... East & 5300 & ... & 10985 & 0.8” HDF 123600 109050 10530 0.8“ & 12:36:47.5 & 62:13:04.4 &30 &18\ 12:37:03.7 62:15:13.0 ... ... NEast & 2500 & ... & 10595 & 0.8” NWeast 2500 ... 10985 0.8“ & 12:36:49.4 & 62:15:53.8 & ... & ...\ 12:37:16.2 62:11:42.5 ... ... OutEast & 3000 & ... & 10985 & 0.8” OutWest 2500 ... 10270 0.8“ & 12:36:19.3 & 62:14:25.5 & ... & ...\ 12:36:46.2 62:10:14.6 ... ... SEast & 2500 & ... & 10465 & 0.8” SWest 2500 ... 10395 0.8“ & 12:36:32.0 & 62:10:55.3 & ... & ...\ 12:36:33.6 62:13:44.9 ... ... West & 5300 & ... & 11375 & 0.8” cfrs031 6700 ... 3120 0.8“ & 03:02:57.4 & 00:06:04.9 &8 &4\ 03:02:33.0 00:05:55.2 5 1 cfrs033 & 6700 & ... & 3120 & 0.7” cfrs034 6400 ... 3120 0.8“ & 03:02:49.8 & 00:13:09.2 &5 &1\ 03:02:40.8 00:12:34.3 2 1 cfrs035 & 6400 & ... & 3120 & 1.0” cfrs101 6700 ... 3120 0.8“ & 10:00:23.2 & 25:12:45.1 &7 &2\ 10:00:36.5 25:12:40.1 9 6 cfrs102 & 6700 & ... & 3120 & 0.7” cfrs103 5302 ... 3120 0.6“ & 10:00:46.7 & 25:12:26.3 &12 &7\ 14:15:20.1 52:02:49.9 ... ... u26x1 & 4400 & 2800 & 4420 & 0.8” u26x2 4400 2800 3120 0.7“ & 14:15:13.7 & 52:01:39.6 & ... & ...\ 14:18:03.2 52:32:10.4 ... ... u26x4 & 4400 & ... & 3120 & 0.7” u26x5 4400 ... 3315 0.7“ & 14:17:56.7 & 52:31:00.6 & ... & ...\ 14:17:50.3 52:29:50.8 ... ... u26x6 & 4400 & ... & 3120 & 0.5” u26x7 4400 ... 3120 0.7“ & 14:17:36.9 & 52:27:31.2 & ... & ...\ 14:17:42.7 52:28:31.3 ... ... u2ay0 & 25200 & 24399 & 4160 & 1.4” u2iy 7400 ... 3120 0.6“ & 14:17:43.1 & 52:30:23.3 & ... & ...\ 20:29:39.6 52:39:22.3 ... ... u3d3 & 4200 & 3300 & 3120 & 1.3” ubi1 6300 3300 2600 0.6“ & 01:10:00.7 &-02:27:22.2 &5 &1\ 01:24:41.6 03:51:24.2 ... ... uci1 & 10800 & 4800 & 3900 & 0.4” ueh0 12600 8700 3900 0.5“ & 00:53:24.1 & 12:34:01.9 &4 &2\ 03:55:31.4 09:43:31.9 5 1 uim0 & 11800 & ... & 4090 & 0.6” umd4 9600 2400 3900 0.7“ & 21:51:06.9 & 28:59:57.0 & ... & ...\ 07:50:47.9 14:40:39.1 ... ... uop0 & 4200 & 7200 & 3120 & 0.9” uqc01 7200 7800 3120 1.0“ & 18:07:07.3 & 45:44:33.5 & ... & ...\ 17:12:24.0 33:35:51.2 ... ... usa0 & 6300 & 5400 & 3120 & 0.7” usa1 6300 5400 3900 0.5“ & 17:12:24.7 & 33:36:03.0 & ... & ...\ 08:54:16.9 20:03:35.5 ... ... usp0 & 4200 & 3300 & 3120 & 0.9” ust0 23100 16500 3120 0.9“ & 10:05:45.8 &-07:41:21.0 & ... & ...\ 16:01:13.2 05:36:03.2 ... ... ut21 & 11999 & 3600 & 3900 & 0.8” ux40 7500 3300 3900 0.6“ & 15:19:40.3 & 23:52:09.9 &5 &3\ 15:19:54.6 23:44:57.8 ... ... ux41 & 6000 & 3300 & 3120 & 0.8” uy402 5400 1400 3120 0.8“ & 14:35:16.0 & 24:58:59.9 & ... & ...\ 12:11:12.3 39:27:04.8 ... ... uzk0 & 8700 & 8400 & 3120 & 0.9” uzp0 6300 3300 3120 0.8“ & 11:50:28.9 & 28:48:35.2 & ... & ...\ 12:30:52.7 12:18:52.5 ... ... uzx05 & 4700 & 2600 & 3120 & 0.7” --------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------ Program Date Fields Observed $\langle$Seeing$\rangle$ Telescope ---------------- ----------------- ---------------------------------------- ----------- -- Feb 6-8 1996 HDF  1.0“ & UH-2.2m\ CFHT Apr 5-8 1996 & HDF &  1.0” Apr 17-22 1996 8 HDF-FF  0.8“ & UH-2.2m\ UH-2.2m Aug 20-24 1997 & 21 MDS fields &  0.8” Jan 18-20 1998 23 MDS fields  0.8“ & UH-2.2m\ UH-2.2m Feb 12-13 1998 & 11 MDS fileds &  0.7” ------------------------------------------------------------------------------------------ [@lllll]{} Data Set & $N_{early}$ & $N_{early}^{I_{814}-HK'> 3.0}$ & Total Excess$^{b}$ & Red Fraction Excess$^{c}$\ \ \ Visually Classified & $266 \pm 13$ & $26 \pm 5$\ Visually Classified + Compacts & $316 \pm 14$ & $40 \pm 5$\ A/C classified & $323 \pm 13$ & $40 \pm 6$\ \ \ \  40% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 357.17 & 72.63 & $ 1.34_{ 1.28}^{ 1.41}$ & $ 2.79_{ 2.34}^{ 3.46}$\ 100% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 366.57 & 135.26 & $ 1.38_{ 1.31}^{ 1.45}$ & $ 5.20_{ 4.36}^{ 6.44}$\ 250% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 372.85 & 198.09 & $ 1.40_{ 1.34}^{ 1.47}$ & $ 7.62_{ 6.39}^{ 9.43}$\ \  40% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 433.94 & 90.37 & $ 1.63_{ 1.56}^{ 1.72}$ & $ 3.48_{ 2.92}^{ 4.30}$\ 100% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 449.58 & 168.27 & $ 1.69_{ 1.61}^{ 1.78}$ & $ 6.47_{ 5.43}^{ 8.01}$\ 250% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 458.74 & 247.12 & $ 1.72_{ 1.64}^{ 1.81}$ & $ 9.50_{ 7.97}^{11.77}$\ \ \ \  40% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 240.29 & 28.56 & $ 0.90_{ 0.86}^{ 0.95}$ & $ 1.10_{ 0.92}^{ 1.36}$\ 100% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 259.81 & 65.18 & $ 0.98_{ 0.93}^{ 1.03}$ & $ 2.51_{ 2.10}^{ 3.10}$\ 250% Solar $\Omega_M=0.0$ $\Omega_{\Lambda}=0.0$ & 286.76 & 113.26 & $ 1.08_{ 1.03}^{ 1.13}$ & $ 4.36_{ 3.65}^{ 5.39}$\ \  40% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 284.40 & 35.43 & $ 1.07_{ 1.02}^{ 1.12}$ & $ 1.36_{ 1.14}^{ 1.69}$\ 100% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 311.23 & 80.37 & $ 1.17_{ 1.12}^{ 1.23}$ & $ 3.09_{ 2.59}^{ 3.83}$\ 250% Solar $\Omega_M=0.3$ $\Omega_{\Lambda}=0.7$ & 344.91 & 139.39 & $ 1.30_{ 1.24}^{ 1.36}$ & $ 5.36_{ 4.50}^{ 6.64}$\ \ Calculation of the Luminosity-weighted Mean Metallicity of a Spheroidal Galaxy Interior to an Isophotal Limit ============================================================================================================= Elliptical galaxies are known to contain highly enriched cores with strong metallicity gradients. Because of cosmological dimming, in the present paper we sample ellipticals at a range of rest-frame limiting isophotes, and it is important to have at least a qualitative understanding of the effects of metallicity gradients on the sampled starlight probed by our data. It is shown in Arimoto et al (1997) that the mean luminosity-weighted iron abundance of spheroidal systems with $R^{1/4}$ profiles integrated to infinity is similar to the abundance measured at the effective radius. It is straightforward to generalize the integral in equation (6) of Arimoto et al (1997) to calculate $\langle Z(R)\rangle$, the mean metallicity of an $R^{1/4}$ law spheroid within a given isophotal radius $R$, as a fraction of the metallicity measured within the effective radius, $R_e$. For a circularly symmetric galaxy the resulting expression is analytically tractable. The $R^{1/4}$ law giving surface brightness $I(R)$ as a function of radius $R$ is parameterized as follows: $$I(R) = \exp \left\{ -b \left[ {\left( R\over R_e \right)}^{1/4} -1 \right] \right\}$$ where $R_e$ is the half-light radius, and $b=3.33\ln(10)$. Metallicity gradients can be parameterized by a power-law index $c$ as a function of radius: $$Z(R) = Z(0)\left({R \over R_e}^{-c} \right).$$ Essentially all of the ellipticals studied by Arimoto et al. (1997) have gradients parameterized by $0<c<1.3$, with metallicities of up to several hundred percent solar in their cores decreasing to roughly solar abundance at the effective radius. Integrating equation (6) of Arimoto et al (1997) to a radius $R$ instead of to infinity yields the mean metallicity interior to a radius R, expressed in units of the mean metallicity interior to radius $R_e$: $$\langle Z(R) \rangle = -e^{b (-1 + R^{1/4})} {A / B}\\$$ where $$\begin{aligned} A & = & [-5040 b - 2520 b^2 - 840 b^3 - 210 b^4 - 42 b^5 \\ & & - 7 b^6 - b^7 + 5040 (-1 + E^b)] \cdot \nonumber \\ & & [\Gamma(8 - 4 c) - \Gamma(8 - 4 c, b R^{1/4})] \\ B & = & [-5040 (-1 + e^{b R^{1/4}}) + 5040 b R^{1/4}\\ & & + 2520 b^2 \sqrt{R} + 840 b^3 R^{3/4} + 210 b^4 R + \nonumber \\ & & 42 b^5 R^{5/4} + 7 b^6 R^{3/2} + b^7 R^{7/4}] \cdot \\ & & [\Gamma(8 - 4 c) - \Gamma(8 - 4 c,b)].\end{aligned}$$ (Note that in the preceeding expression, $\Gamma(x)$ is the Euler gamma function, and $\Gamma(x,y)$ is the incomplete gamma function). Mean metallicities as a function of radius are shown in Figure A1 for values of $c=0.4$ and $c=1.3$, spanning the range of metallicity gradients typically seen in luminous ellipticals. As expected, the luminosity-weighted metallicity as a function of isophotal radius rises sharply at $R \ll R_e$, but for our present purposes the important feature to note in Figure 12 is the shallow decline beyond $R > 0.5 R_e$ in the luminosity-weighted mean metallicity, regardless of the strength of the power-law slope $c$ parameterizing the metallicity gradients. This behaviour was also noted by Arimoto et al. 1997 in their simplified calculation. [*The overall metallicity of the galaxy can be well-characterized by the mean metallicity interior to the effective radius (ie. $\sim$ solar in the Arimoto et al. sample) over an extraordinarily large range in isophotal radius.*]{} We can thus make fair comparisons between ellipticals over a broad range of redshifts in spite of cosmological dimming of the rest-frame isophote, provided only the ellipticals are observed with sufficient signal-to-noise that their isophotal radii are at least as large as their effective radii. This is the case for the vast majority of ellipticals in the present sample — only at very high redshifts, where compact ellipticals have limiting isophotes comparable to their effective radii, is the mean luminosity-weighted metallicity expected to deviate from that at the effective radius. \[lastpage\] [^1]: These and subsequent detection limits refer to near-total magnitudes in the Vega system based on profile fitting within the SExtractor package (‘$m_{best}$’). [^2]: Note however that the slope of the infrared-optical colour-magnitude relation for early-type systems is rather uncertain, particularly for $I-K$. On the basis of quite strongly model-dependent conversions based on the cluster $V-K$ data of Bower et al. (1992), Peletier & de Grijs (1998) obtain a slope of $-0.0438 \pm 0.0041$ for the $I-K$ slope of local early-type systems, using the models presented in Vazdekis et al. (1996). [^3]: We note that the good accuracy in photometric redshifts for these galaxies appears to be due to the presence of strong continuum features which are well-explored with the addition of the Kitt Peak JHK photometry to the HDF filter bands. [^4]: On the basis of model predictions used to calculate the density of post-starburst “$H\delta$ strong” systems seen in clusters, Couch & Sharples (1986), Barger et al. (1996) and Abraham et al. (1996c) note that a sharp truncation in the star-formation rate of actively star-forming systems results in a synchronization of optical colours with those expected of early type galaxies after only $\sim 1.5$ Gyr, so it is not too surprising that truncated star formation histories and monolithic collapse models predict similar colours [*provided*]{} the epoch of observation is several Gyr after the period of truncation.

--- abstract: 'We give an algorithm, based on the $\varphi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is to solve an inverse problem for the dynamical systems $\beta x +\alpha\mod 1$. The first part is an exposition of the $\varphi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $\varphi$-expansion, necessary and sufficient conditions, which are different from those in Parry’s paper [@P2].' author: - | B. Faller[^1] and C.-E. Pfister[^2]\ EPF-L, Institut d’analyse et calcul scientifique\ CH-1015 Lausanne, Switzerland date: '16.05.2008\' title: | Computation of Topological Entropy via $\varphi$-expansion,\ an Inverse Problem for the Dynamical Systems $\beta x+\alpha\mod1$ --- Introduction {#section1} ============ In 1957 Rényi published his paper [@R] about representations for real numbers by $f$-expansions, called hereafter $\varphi$-expansions, which had tremendous impact in Dynamical Systems Theory. The ideas of Rényi were further developed by Parry in [@P1] and [@P2]. See also the book of Schweiger [@Sch]. The first part of the paper, section \[section2\], is an exposition of the theory of $\varphi$-expansions in the setting of piecewise monotone dynamical systems. Although many of the results of section \[section2\] are known, for example see [@Bo] chapter 9 for Theorem \[thm2.5\], we state necessary and sufficient conditions for the validity of the $\varphi$-expansion, which are different from those in Parry’s paper [@P2], Theorem \[thm2.1bis\] and Theorem \[thm2.1ter\]. We then use $\varphi$-expansions to study two interesting and related problems in sections \[section3\] and \[section4\]. When one applies the method of section \[section2\] to the dynamical system $\beta x+\alpha\mod1$, one obtains a symbolic shift which is entirely described by two strings $\ud{u}^\ab$ and $\ud{v}^\ab$ of symbols in a finite alphabet $\tA=\{0,\ldots,k-1\}$. The shift space is given by $$\label{1.1} \BSigma(\ud{u}^\ab,\ud{v}^\ab)=\big\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}^\ab\preceq\sigma^n\ud{x}\preceq\ud{v}^\ab\;\,\forall n\geq 0 \big\}\,,$$ where $\preceq$ is the lexicographic order and $\sigma$ the shift map. The particular case $\alpha=0$ has been much studied from many different viewpoints ($\beta$-shifts). For $\alpha\not=0$ the structure of the shift space is richer. A natural problem is to study all shift spaces $\Sigma(\ud{u},\ud{v})$ of the form when we replace $\ud{u}^\ab$ and $\ud{v}^\ab$ by a pair of strings $\ud{u}$ and $\ud{v}$. In section \[section3\] we give an algorithm, Theorem \[thm3.1\], based on the $\varphi$-expansion, which allows to compute the topological entropy of shift spaces $\Sigma(\ud{u},\ud{v})$. One of the essential tool is the follower-set graph associated to the shift space. This graph is presented in details in subsection \[subsectionfollower\]. The algorithm is given in subsection \[subsectionalgo\] and the computations of the topological entropy in subsection \[topological\]. The basic idea of the algorithm is to compute two real numbers $\bar{\alpha}$ and $\bar{\beta}$, given the strings $\ud{u}$ and $\ud{v}$, and to show that the shift space $\BSigma(\ud{u},\ud{v})$ is a modification of the shift space $\Sigma(\ud{u}^\abb,\ud{v}^\abb)$ obtained from the dynamical system $\bar{\beta}x+\bar{\alpha}\mod1$, and that the topological entropies of the two shift spaces are the same. In the last section we consider the following inverse problem for the dynamical systems $\beta x+\alpha \mod1$: given $\ud{u}$ and $\ud{v}$, find $\alpha$ and $\beta$ so that $$\ud{u}=\ud{u}^\ab\quad\text{and}\quad\ud{v}=\ud{v}^\ab\,.$$ The solution of this problem is given in Theorems \[thm4.1\] and \[thm4.2\] for all $\beta>1$. $\varphi$-expansion for piecewise monotone dynamical\ systems {#section2} ===================================================== Piecewise monotone dynamical systems {#subsection2.1} ------------------------------------ Let $X:=[0,1]$ (with the euclidean distance). We consider the case of piecewise monotone dynamical systems of the following type. Let $0=a_0<a_1<\cdots<a_k=1$ and $I_j:=(a_j,a_{j+1})$, $j\in\tA$. We set $\tA:=\{0,\ldots,k-1\}$, $k\geq 2$, and $$S_0:=X\backslash \bigcup_{j\in\tA}I_j\,.$$ For each $j\in\tA$ let $$f_j:I_j\mapsto J_j:=f_j(I_j)\subset [0,1]$$ be a strictly monotone continuous map. When necessary we also denote by $f_j$ the continuous extension of the map on the closure $\overline{I}_j$ of $I_j$. We define a map $T$ on $X\backslash S_0$ by setting $$T(x):=f_j(x)\quad \text{if $x\in I_j$}\,.$$ The map $T$ is left undefined on $S_0$. We also assume that $$\label{2.1} \big(\bigcup_{i\in\tA}J_i\big)\cap I_j=I_j\quad\forall j\,.$$ We introduce sets $X_j$, $S_j$, and $S$ by setting for $j\geq 1$ $$X_0:=[0,1]\,,\quad X_j:=X_{j-1}\backslash S_{j-1}\,,\quad S_j:=\{x\in X_j\colon T(x)\in S_{j-1}\}\,,\quad S:=\bigcup_{j\geq 0}S_j\,.$$ \[lem2.1\] Under the condition , $T^n(X_{n+1})= X_1$ and $T(X\backslash S)=X\backslash S$. $X\backslash S$ is dense in $X$. Condition is equivalent to $T(X_1)\supset X_1$. Since $X_2=X_1\backslash S_1$ and $S_1=\{x\in X_1\colon T(x)\not\in X_1\}$, we have $T(X_2)=X_1$. Suppose that $T^n(X_{n+1})=X_1$; we prove that $T^{n+1}(X_{n+2})=X_{1}$. One has $X_{n+1}=X_{n+2}\cup S_{n+1}$ and $$X_1=T^n(X_{n+1})=T^n(X_{n+2})\cup T^n(S_{n+1})\,.$$ Applying once more $T$, $$X_1\subset T(X_1)=T^{n+1}(X_{n+2})\cup T^{n+1}(S_{n+1})\,.$$ $T^{n+1}$ is defined on $X_{n+1}$ and $S_{n+1}\subset X_{n+1}$. $$T^{n+1}S_{n+1}=\{x\in X_{n+1}\colon T^{n+1}(x)\in S_0\}= \{x\in X_{n+1}\colon T^{n+1}(x)\not\in X_1\}\,.$$ Hence $T^{n+1}(X_{n+2})=X_{1}$. Clearly $T(X\backslash S)\subset X\backslash S$ and $T(S\backslash S_0)\subset S$. Since $X_1$ is the disjoint union of $X\backslash S$ and $S\backslash S_0$, and $TX_1\supset X_1$, we have $T(X\backslash S)=X\backslash S$. The sets $X\backslash S_k$ are open and dense in $X$. By Baire’s Theorem $X\backslash S=\bigcap_{k}(X\backslash S_k)$ is dense. Let $\Z_+:=\{0,1,2,\ldots\}$ and $\tA^{\Z_+}$ be equipped with the product topology. Elements of $\tA^{\Z_+}$ are called [strings]{} and denoted by $\ud{x}=(x_0,x_1,\ldots)$. A finite string $\ud{w}=(w_0,\cdots,w_{n-1})$, $w_j\in\tA$, is a [word]{}; we also use the notation $\ud{w}=w_0\cdots w_{n-1}$. The [length of $\ud{w}$]{} is $|\ud{w}|=n$. A [$n$-word]{} is a word of length $n$. There is a single word of length $0$, the [empty-word]{} $\epsilon$. The set of all words is $\tA^*$. The shift-map $\sigma\colon \tA^{\Z_+}\ra\tA^{\Z_+}$ is defined by $$\sigma(\ud{x}):=(x_1,x_2,\ldots)\,.$$ We define two operations $\tp$ and $\s$ on $\tA^*\backslash\{\epsilon\}$, $$\begin{aligned} \tp{\ud{w}}&:=\begin{cases} w_0\cdots w_{n-2}& \text{if $\ud{w}=w_0\cdots w_{n-1}$ and $n\geq 2$}\label{opp}\\ \epsilon & \text{if $\ud{w}=w_0$} \end{cases}\\ \s{\ud{w}}&:=\begin{cases} w_1\cdots w_{n-1}& \text{if $\ud{w}=w_0\cdots w_{n-1}$ and $n\geq 2$}\label{ops}\\ \epsilon & \text{if $\ud{w}=w_0$.} \end{cases}\end{aligned}$$ On $\tA^{\Z_+}$ we define a total order, denoted by $\prec$. We set $$\delta(j):=\begin{cases} +1 &\text{if $f_j$ is increasing}\\ -1 & \text{if $f_j$ is decreasing,} \end{cases}$$ and for a word $\ud{w}$, $$\delta(\ud{w}):=\begin{cases} \delta(w_0)\cdots \delta(w_{n-1}) &\text{if $\ud{w}=w_0\cdots w_{n-1}$}\\ 1& \text{if $\ud{w}=\epsilon$.} \end{cases}$$ Let $\ud{x}^\prime\not =\ud{x}^{\prime\prime}$ belong to $\tA^{\Z_+}$; define $j$ as the smallest integer with $x^\prime_j\not=x^{\prime\prime}_j$. By definition $$\ud{x}^\prime\prec\ud{x}^{\prime\prime}\iff \begin{cases} x^\prime_j<x^{\prime\prime}_j &\text{if $\delta(x^\prime_0\cdots x^\prime_{j-1})=1$}\\ x^\prime_j>x^{\prime\prime}_j &\text{if $\delta(x^\prime_0\cdots x^\prime_{j-1})=-1$}\,. \end{cases}$$ As usual $\ud{x}^\prime\preceq \ud{x}^{\prime\prime}$ if and only if $\ud{x}^\prime\prec\ud{x}^{\prime\prime}$ or $\ud{x}^\prime=\ud{x}^{\prime\prime}$. When all maps $f_j$ are increasing this order is the lexicographic order. $\varphi$-expansion {#subsection2.2} ------------------- We give an alternative description of a piecewise monotone dynamical system as in Parry’s paper [@P2]. In this description, when all maps $f_j$ are increasing, one could use instead of the intervals $I_j$ the intervals $I^\prime_j:=[a_j,a_{j+1})$, $j\in\tA$. In that case $S_0=\{a_k\}$ and $S_j=\emptyset$ for all $j\geq 1$. This would correspond to the setting of Parry’s paper [@P2]. We define a map $\varphi$ on the disjoint union $${\rm dom}\varphi:=\bigcup_{j=0}^{k-1}j+J_j\subset \R\,,$$ by setting $$\label{2.3} \varphi(x):= f^{-1}_j(t) \quad \text{if $x=j+t$ and $t\in J_j$}\,.$$ The map $\varphi$ is continuous, injective with range $X_1$. On $X_1$ the inverse map is $$\varphi^{-1}(x)=j+Tx\quad\text{if $x\in I_j$}\,.$$ For each $j$, such that $f_j$ is increasing, we define $\overline{\varphi}^j$ on $j+[0,1]$ (using the extension of $f_j$ to $[a_j,a_{j+1}]$) by $$\label{2.4} \overline{\varphi}^j(x):=\begin{cases} a_j & \text{if $x=j+t$ and $t\leq f_j(a_j)$}\\ f^{-1}_j(t) & \text{if $x=j+t$ and $t\in J_j$}\\ a_{j+1} & \text{if $x=j+t$ and $f_j(a_{j+1})\leq t$.} \end{cases}$$ For each $j$, such that $f_j$ is decreasing, we define $\overline{\varphi}^j$ on $j+[0,1]$ by $$\label{2.5} \overline{\varphi}^j(x):=\begin{cases} a_{j+1} & \text{if $x=j+t$ and $t\leq f_j(a_{j+1})$}\\ f^{-1}_j(t) & \text{if $x=j+t$ and $t\in J_j$}\\ a_{j} & \text{if $x=j+t$ and $f_j(a_{j})\leq t$.} \end{cases}$$ It is convenient below to consider the family of maps $\overline{\varphi}^j$ as a single map defined on $[0,k]$, which is denoted by $\overline{\varphi}$. In order to avoid ambiguities at integers, where the map may be multi-valued, we always write a point of $[j,j+1]$ as $x=j+t$, $t\in[0,1]$, so that $$\overline{\varphi}(j+t)\equiv\overline{\varphi}(x):=\overline{\varphi}^j(t)\,.$$ We define the [coding map]{} $\i:X\backslash S\ra \tA^{\Z_+}$ by $$\i(x):=(\i_0(x),\i_1(x),\ldots)\quad\text{with $\i_n(x):=j$ if $T^nx\in I_j$}\,.$$ The [$\varphi$-code]{} of $x\in X\backslash S$ is the string $\i(x)$, and we set $$\Sigma=\{\ud{x}\in\tA^{\Z_+}\colon \text{$\ud{x}=\i(x)$ for some $x\in X\backslash S$}\}\,.$$ For $x\in X\backslash S$ and any $n\geq 0$, $$\label{2.7} \varphi^{-1}(T^nx)=\i_n(x)+T^{n+1}x\quad\text{and}\quad \i(T^nx)=\sigma^n\i(x) \,.$$ Let $z_j\in\tA$, $1\leq j \leq n$, and $t\in [0,1]$; we set $$\overline{\varphi}_1(z_1+t):=\overline{\varphi}(z_1+t)$$ and $$\label{formule} \overline{\varphi}_n(z_1,\ldots,z_n+t):= \overline{\varphi}_{n-1}(z_1,\ldots,z_{n-1}+\overline{\varphi}(z_n+t))\,.$$ For $n\geq 1$ and $m\geq 1$ we have $$\label{formulegenerale} \overline{\varphi}_{n+m}(z_1,\ldots,z_{n+m}+t) = \overline{\varphi}_n(z_1,\ldots,z_{n}+ \overline{\varphi}_m(z_{n+1},\ldots,z_{n+m}+t))\,.$$ The map $t\mapsto \overline{\varphi}_n(x_0,\ldots,x_{n-1}+t)$ is increasing if $\delta(x_0\cdots x_{n-1})=1$ and decreasing if $\delta(x_0\cdots x_{n-1})=-1$. We also write $\overline{\varphi}_n(\ud{x})$ for $\overline{\varphi}_n(x_0,\ldots,x_{n-1})$. \[defn2.2\] The real number $s$ has a [$\varphi$-expansion]{} $\ud{x}\in\tA^{\Z_+}$ if the following limit exists, $$s=\lim_{n\ra\infty}\overline{\varphi}_n(\ud{x})\equiv \overline{\varphi}\big(x_0+\overline{\varphi}(x_1+\ldots)\big)\equiv \overline{\varphi}_\infty(\ud{x})\,.$$ The [$\varphi$-expansion is well-defined]{} if for all $\ud{x}\in\tA^{\Z_+}$, $\lim_{n\ra\infty}\overline{\varphi}_n(\ud{x})= \overline{\varphi}_\infty(\ud{x})$ exists.\ The [$\varphi$-expansion is valid]{} if for all $x\in X\backslash S$ the $\varphi$-code $\i(x)$ of $x$ is a [$\varphi$-expansion]{} of $x$. If the $\varphi$-expansion is valid, then for $x\in X\backslash S$, using , and the continuity of the maps $\overline{\varphi}^j$, $$\begin{aligned} \label{2.8} x&=\lim_{n\ra\infty} \overline{\varphi}_n\big(\i_0(x),\ldots,\i_{n-1}(x)\big)\nonumber\\ &= \lim_{m\ra\infty} \overline{\varphi}_{n}\big(\i_0(x),\ldots,\i_{n-1}(x)+ \overline{\varphi}_{m}\big(\i_n(x),\ldots,\i_{n+m-1}(x)\big)\big) \\ &= \overline{\varphi}_{n}\big(\i_0(x),\ldots,\i_{n-1}(x)+ \overline{\varphi}_\infty(\i(T^nx)\big)\,.\nonumber\end{aligned}$$ The basic and elementary fact of the $\varphi$-expansion is $$\label{2.9} \text{$a,b\in[0,1]$ and $x_0<x_0^\prime$} \implies \overline{\varphi}(x_0+a)\leq \overline{\varphi}(x_0^\prime+b)\,.$$ We begin with two lemmas on the $\varphi$-code (for Lemma \[lem2.4\] see e.g. [@CoE]). \[lem2.4\] The $\varphi$-code $\i$ is $\preceq$-order-preserving on $X\backslash S$: $x\leq y$ implies $\i(x)\preceq\i(y)$. Let $x<y$. Either $\i_0(x)<\i_0(y)$, or $\i_0(x)=\i_0(y)$; in the latter case, the strict monotonicity of $f_{\i_0(x)}$ implies $$\begin{aligned} \varphi^{-1}(x)&=\i_0(x) +T(x)< \varphi^{-1}(y)=\i_0(x) +T(y)\quad\text{if $\delta(\i_0(x))=+1$}\\ \varphi^{-1}(x)&=\i_0(x) +T(x)> \varphi^{-1}(y)=\i_0(x) +T(y)\quad\text{if $\delta(\i_0(x))=-1$.}\end{aligned}$$ Repeating this argument we get $\i(x)\preceq\i(y)$. \[lem2.4bis\] The $\varphi$-code $\i$ is continuous[^3] on $X\backslash S$. Let $x\in X\backslash S$ and $\{x^n\}\subset X\backslash S$, $\lim_n x^n=x$. Let $x\in I_{j_0}$. For $n$ large enough $x^n\in I_{j_0}$ and $\i_0(x^n)=\i_0(x)=j_0$. Let $j_1:=\i_1(x)$; we can choose $n_1$ so large that for $n\geq n_1$ $Tx_{n}\in I_{j_1}$. Hence $\i_0(x^n)=j_0$ and $\i_1(x^n)=j_1$ for all $n\geq n_1$. By induction we can find an increasing sequence $\{n_m\}$ such that $n\geq n_m$ implies $\i_j(x)=\i_j(x^{n})$ for all $j=0,\ldots,m$. The next lemmas give the essential properties of the map $\overline{\varphi}_\infty$. \[lem2.2\] Let $\ud{x}\in\tA^{\Z_+}$. Then there exist $y_\uparrow(\ud{x})$ and $y_\downarrow(\ud{x})$ in $[0,1]$, such that $y_\uparrow(\ud{x})\leq y_\downarrow(\ud{x})$; $y_\uparrow(\ud{x})$ and $y_\downarrow(\ud{x})$ are the only possible cluster points of the sequence $\{\overline{\varphi}_n(\ud{x})\}_n$.\ Let $x\in X\backslash S$ and set $\ud{x}:=\i(x)$. Then $$a_j\leq y_\uparrow(\ud{x})\leq x\leq y_\downarrow(\ud{x})\leq a_{j+1}\quad \text{if $x_0=j$}\,.$$ If the $\varphi$-expansion is valid, then each $y\in X\backslash S$ has a unique $\varphi$-expansion[^4], $$y=\overline{\varphi}_\infty(\ud{x})\in X\backslash S\iff \ud{x}=\i(y)\,.$$ Consider the map $$t\mapsto \overline{\varphi}_n(x_0,\ldots,x_{n-1}+t)\,.$$ Suppose that $\delta(x_0\cdots x_{n-1})=-1$. Then it is decreasing, and for any $m$ $$\begin{aligned} \overline{\varphi}_{n+m}(x_0,\ldots,x_{n+m-1}) &= \overline{\varphi}_n(x_0,\ldots,x_{n-1}+ \overline{\varphi}_m(x_n,\ldots,x_{n+m-1}))\\ &\leq \overline{\varphi}_n(x_0,\ldots,x_{n-1})\,.\end{aligned}$$ In particular the subsequence $\{\overline{\varphi}_n(\ud{x})\}_n$ of all $n$ such that $\delta(x_0\cdots x_{n-1})=-1$ is decreasing with limit[^5] $y_\downarrow(\ud{x})$. When there is no $n$ such that $\delta(x_0\cdots x_{n-1})=-1$, we set $y_\downarrow(\ud{x}):=a_{x_0+1}$. Similarly, the subsequence $\{\overline{\varphi}_n(\ud{x})\}_n$ of all $n$ such that $\delta(x_0\cdots x_{n-1})=1$ is increasing with limit $y_\uparrow(\ud{x})\leq y_\downarrow(\ud{x})$. When there is no $n$ such that $\delta(x_0\cdots x_{n-1})=1$, we set $y_\uparrow(\ud{x}):=a_{x_0}$. Since any $\overline{\varphi}_n(\ud{x})$ appears in one of these sequences, there are at most two cluster points for $\{\overline{\varphi}_n(\ud{x})\}_n$. Let $x\in X\backslash S$; $x=\varphi(\varphi^{-1}(x))$ and by $$\begin{aligned} \label{identity} x&=\varphi(\i_0(x)+Tx)= \varphi(\i_0(x)+\varphi(\varphi^{-1}(Tx)))= \varphi(\i_0(x)+\varphi(\i_1(x)+T^2x))=\cdots \nonumber\\ &=\varphi\big(\i_0(x)+\varphi(\i_1(x)+\ldots +\varphi(\i_{n-1}(x)+T^{n}x))\big)\,.\end{aligned}$$ By monotonicity $$\label{id1} \big(\text{$x\in X\backslash S$ and $\delta(\i_0(x)\cdots\i_{n-1}(x))=-1$}\big)\implies \overline{\varphi}_n(\i_0(x),\cdots,\i_{n-1}(x))\geq x\,,$$ and $$\label{id2} \big(\text{$x\in X\backslash S$ and $\delta(\i_0(x)\cdots\i_{n-1}(x))=1$}\big)\implies \overline{\varphi}_n(\i_0(x),\cdots,\i_{n-1}(x))\leq x\,.$$ The inequalities of Lemma \[lem2.2\] follow from , and $\overline{\varphi}(\i_0(x)+t)\in [a_{x_0}, a_{x_0+1}]$. Suppose that the $\varphi$-expansion is valid and that $\overline{\varphi}_\infty(\ud{x})=y\in X\backslash S$. We prove that $\ud{x}=\i(y)$. By hypothesis $y\in I_{x_0}$; using and the fact that $I_{x_0}$ is open, we can write $$y=\overline{\varphi}\big(x_0+\overline{\varphi}(x_1 +\overline{\varphi}(x_{2}+\ldots))\big) =\varphi\big(x_0+\overline{\varphi}(x_1 +\overline{\varphi}(x_{2}+\ldots))\big)\,.$$ This implies that $$\varphi^{-1}(y)=\i_0(y)+Ty= x_0+ \overline{\varphi}(x_1 +\overline{\varphi}(x_{2}+\ldots))\,.$$ Since $Ty \in X\backslash S$, we can iterate this argument. \[lem2.3\] Let $\ud{x},\ud{x}^\prime\in\tA^{\Z_+}$ and $\ud{x}\preceq\ud{x}^\prime$. Then any cluster point of $\{\overline{\varphi}_n(\ud{x})\}_n$ is smaller then any cluster point of $\{\overline{\varphi}_n(\ud{x}^\prime)\}_n$. In particular, if $\overline{\varphi}_\infty$ is well-defined on $\tA^{\Z_+}$, then $\overline{\varphi}_\infty$ is order-preserving. Let $\ud{x}\prec\ud{x}^\prime$ with $x_k=x^\prime_k$, $k=0,\ldots,m-1$ and $x_m\not=x^\prime_m$. We have $$\overline{\varphi}_{m+n}(\ud{x})= \overline{\varphi}_m(x_0,\ldots,x_{m-1}+ \overline{\varphi}_n(\sigma^m\ud{x}))\,.$$ By , if $\delta(x_0\cdots x_{m-1})=1$, then $x_m<x^\prime_m$ and for any $n\geq 1$, $\ell\geq 1$, $$\overline{\varphi}_n(\sigma^m\ud{x})=\overline{\varphi}_1(x_m+ \overline{\varphi}_{n-1}(\sigma^{m+1}\ud{x}))\leq \overline{\varphi}_\ell(\sigma^m\ud{x}^\prime)= \overline{\varphi}_1(x^\prime_m+ \overline{\varphi}_{\ell-1}(\sigma^{m+1}\ud{x}^\prime))\,;$$ if $\delta(x_0\cdots x_{m-1})=-1$, then $x_m>x^\prime_m$ and $$\overline{\varphi}_n(\sigma^m\ud{x})=\overline{\varphi}_1(x_m+ \overline{\varphi}_{n-1}(\sigma^{m+1}\ud{x}))\geq \overline{\varphi}_\ell(\sigma^m\ud{x}^\prime)= \overline{\varphi}_1(x^\prime_m+ \overline{\varphi}_{\ell-1}(\sigma^{m+1}\ud{x}^\prime))\,.$$ Therefore, in both cases, for any $n\geq 1$, $\ell\geq 1$, $$\overline{\varphi}_{m+n}(\ud{x})\leq \overline{\varphi}_{m+\ell}(\ud{x}^\prime)\,.$$ \[lem2.4ter\] Let $\ud{x}\in\tA^{\Z_+}$ and $x_0=j$.\ 1) Let $\delta(j)=1$ and $y_\uparrow(\ud{x})\in \overline{I}_j$ be a cluster point of $\{\overline{\varphi}_n(\ud{x})\}$. Then $f_j\big(y_\uparrow(\ud{x})\big)\geq y_\uparrow(\sigma\ud{x})$ if $y_\uparrow(\ud{x})=a_j$, $f_j\big(y_\uparrow(\ud{x})\big)\leq y_\uparrow(\sigma\ud{x})$ if $y_\uparrow(\ud{x})=a_{j+1}$ and $f_j\big(y_\uparrow(\ud{x})\big)=y_\uparrow(\sigma\ud{x})$ otherwise. The same conclusions hold when $y_\downarrow(\ud{x})$ is a cluster point of $\{\overline{\varphi}_n(\ud{x})\}$.\ 2) Let $\delta(j)=-1$ and $y_\uparrow(\ud{x})\in \overline{I}_j$ be a cluster point of $\{\overline{\varphi}_n(\ud{x})\}$. Then $f_j\big(y_\uparrow(\ud{x})\big)\leq y_\downarrow(\sigma\ud{x})$ if $y_\uparrow(\ud{x})=a_j$, $f_j\big(y_\uparrow(\ud{x})\big)\geq y_\downarrow(\sigma\ud{x})$ if $y_\uparrow(\ud{x})=a_{j+1}$ and $f_j\big(y_\uparrow(\ud{x})\big)=y_\downarrow(\sigma\ud{x})$ otherwise. The same conclusions hold when $y_\downarrow(\ud{x})$ is a cluster point of $\{\overline{\varphi}_n(\ud{x})\}$. Set $f_j(\overline{I}_j):=[\alpha_j,\beta_j]$. Suppose for example that $\delta(j)=-1$ and that $n_k$ is the subsequence of all $m$ such that $\delta(x_0,\ldots,x_m)=1$. Since $\delta(j)=-1$ the sequence $\{\oph_{n_k-1}(\sigma\ud{x})\}_k$ is decreasing. Hence by continuity $$\label{mon} y_\uparrow(\ud{x})=\lim_k\overline{\varphi}_{n_k}(\ud{x})= \overline{\varphi}(j+\lim_k\overline{\varphi}_{n_k-1}(\sigma\ud{x}))= \overline{\varphi}(j+y_\downarrow(\sigma\ud{x}))\,.$$ If $y_\uparrow(\ud{x})=a_j$, then $f_j(a_j)=\beta_j\leq y_\downarrow(\sigma\ud{x})$; if $y_\uparrow(\ud{x})=a_{j+1}$, then $f_j(a_{j+1})=\alpha_j\geq y_\downarrow(\sigma\ud{x})$; if $a_j<y_\uparrow(\ud{x})<a_{j+1}$, then $$j+f_j\big(y_\uparrow(\ud{x})\big)= \varphi^{-1}\big(\varphi(j+\lim_k\overline{\varphi}_{n_k-1}(\sigma\ud{x}))\big)= j+y_\downarrow(\sigma\ud{x})\,.$$ Similar proofs for the other cases. \[lem2.4quatro\] Let $\ud{x}\in\tA^{\Z_+}$.\ 1) If $\{\overline{\varphi}_n(\ud{x})\}$ has two cluster points, and if $y\in \big(y_\uparrow(\ud{x}), y_\downarrow(\ud{x})\big)$, then $y\in X\backslash S$, $\i(y)=\ud{x}$ and $y$ has no $\varphi$-expansion.\ Let $x\in X\backslash S$ and set $\ud{x}:=\i(x)$.\ 2) If $\lim_n\overline{\varphi}_n(\ud{x})=y_\uparrow(\ud{x})$ and if $y\in (y_\uparrow(\ud{x}), x)$, then $y\in X\backslash S$, $\i(y)=\ud{x}$ and $y$ has no $\varphi$-expansion.\ 3) If $\lim_n\overline{\varphi}_n(\ud{x})=y_\downarrow(\ud{x})$ and if $y\in(x,y_\downarrow(\ud{x}))$, then $y\in X\backslash S$, $\i(y)=\ud{x}$ and $y$ has no $\varphi$-expansion. Suppose that $y_\uparrow(\ud{x})<y< y_\downarrow(\ud{x})$. Then $y\in I_{x_0}$ and $\i_0(y)=x_0$. From Lemma \[lem2.4ter\] $$y_\uparrow(\sigma\ud{x})<Ty<y_\downarrow(\sigma\ud{x})\quad \text{if $\delta(x_0)=1$}\,,$$ and $$y_\downarrow(\sigma\ud{x})>Ty>y_\uparrow(\sigma\ud{x})\quad \text{if $\delta(x_0)=-1$}\,.$$ Iterating this argument we prove that $T^ny\in I_{x_n}$ and $\i_n(y)=x_n$ for all $n\geq 1$. Suppose that $y$ has a $\varphi$-expansion, $y=\overline{\varphi}_\infty(\ud{x}^\prime)$. If $\ud{x}^\p\prec\ud{x}$, then by Lemma \[lem2.3\] $\ophi(\ud{x}^\p)\leq y_\uparrow(\ud{x})$ and if $\ud{x}\prec \ud{x}^\p$, then by Lemma \[lem2.3\] $y_\downarrow(\ud{x})\leq \ophi(\ud{x}^\p)$, which leads to a contradiction. Similar proofs in cases 2 and 3. \[lem2.4quinto\] Let $\ud{x}^\prime\in\tA^{\Z_+}$ and $x\in X\backslash S$. Then $$\text{$y_\downarrow(\ud{x}^\prime)<x$ $\implies$ $\ud{x}^\prime\preceq \i(x)$} \quad\text{and}\quad\text{$x<y_\uparrow(\ud{x}^\prime)$ $\implies$ $\i(x)\preceq \ud{x}^\prime$.}$$ Suppose that $y_\downarrow(\ud{x}^\prime)<x$ and $y_\downarrow(\ud{x}^\prime)$ is a cluster point. Either $x^\prime_0<\i_0(x)$ or $x^\prime_0=\i_0(x)$ and by Lemma \[lem2.4ter\] $$y_\downarrow(\sigma\ud{x}^\prime) <Tx\quad\text{if $\delta(x_0^\prime)=1$,}$$ or $$y_\uparrow(\sigma\ud{x}^\prime) >Tx\quad\text{if $\delta(x_0^\prime)=-1$.}$$ Since $y_\downarrow(\sigma\ud{x}^\prime)$ or $y_\uparrow(\sigma\ud{x}^\prime)$ is a cluster point we can repeat the argument and conclude that $\ud{x}^\prime\preceq\i(x)$. If $y_\downarrow(\ud{x}^\prime)$ is not a cluster point, then we use the cluster point $y_\uparrow(\ud{x}^\prime)<y_\downarrow(\ud{x}^\prime)$ for the argument. \[thm2.1\][[@P2]]{} A $\varphi$-expansion is valid if and only if the $\varphi$-code $\i$ is injective on $X\backslash S$. Suppose that the $\varphi$-expansion is valid. If $x\not=z$, then $$x=\overline{\varphi}\big(\i_0(x)+ \overline{\varphi}(\i_1(x)+\ldots)\big)\not= \overline{\varphi}\big(\i_0(z)+ \overline{\varphi}(\i_1(z)+\ldots)\big)=z\,,$$ and therefore $\i(x)\not=\i(z)$. Conversely, assume that $x\not=z$ implies $\i(x)\not=\i(z)$. Let $x\in X\backslash S$, $\ud{x}=\i(x)$, and suppose for example that $y_\uparrow(\ud{x})< y_\downarrow(\ud{x})$ are two cluster points. Then by Lemma \[lem2.4quatro\] any $y$ such that $y_\uparrow(\ud{x})<y< y_\downarrow(\ud{x})$ is in $X\backslash S$ and $\i(y)=\ud{x}$, contradicting the hypothesis. Therefore $z:=\lim_n\overline{\varphi}_n(\ud{x})$ exists. If $z\not=x$, then we get again a contradiction using Lemma \[lem2.4quatro\]. Theorem \[thm2.1\] states that the validity of the $\varphi$-expansion is equivalent to the injectivity of the map $\i$ defined on $X\backslash S$. One can also state that the validity of the $\varphi$-expansion is equivalent to the surjectivity of the map $\overline{\varphi}_\infty$. \[thm2.1bis\] A $\varphi$-expansion is valid if and only if $\overline{\varphi}_\infty\colon \tA^{\Z_+}\ra [0,1]$ is well-defined on $\tA^{\Z_+}$ and surjective. Suppose that the $\varphi$-expansion is valid. Let $\ud{x}\in\tA^{\Z_+}$ and suppose that $\{\overline{\varphi}_n(\ud{x})\}_n$ has two different accumulation points $y_\uparrow<y_\downarrow$. By Lemma \[lem2.4quatro\] we get a contradiction. Thus $\overline{\varphi}_\infty(\ud{x})$ is well-defined for any $\ud{x}\in\tA^{\Z_+}$. To prove the surjectivity of $\overline{\varphi}_\infty$ it is sufficient to consider $s\in S$. The argument is a variant of the proof of Lemma \[lem2.4quatro\]. Let $\ud{x}^\prime$ be a string such that for any $n\geq 1$ $$f_{x^\prime_{n-1}}\cirk \cdots \cirk f_{x^\prime_0}(s)\in\overline{I}_{x^\prime_n}\,.$$ We use here the extension of $f_j$ to $\overline{I}_j$; we have a choice for $x^\prime_n$ whenever $f_{x^\prime_{n-1}}\cirk \cdots \cirk f_{x^\prime_0}(s)\in S_0$. Suppose that $\overline{\varphi}_\infty(\ud{x}^\prime)<s$ and that $\overline{\varphi}_\infty(\ud{x}^\prime)<z<s$. Since $s, \overline{\varphi}_\infty(\ud{x}^\prime)\in \overline{I}_{x_0^\prime}$, we have $z\in I_{x_0^\prime}$ and therefore $\i(z)=x_0^\prime$. Moreover, $$\overline{\varphi}_\infty(\sigma\ud{x}^\prime)<Tz<f_{x^\prime_0}(s)\quad\text{if $\delta(x_0^\prime)=1$}$$ or $$f_{x^\prime_0}(s)<Tz<\overline{\varphi}_\infty(\sigma\ud{x}^\prime)\quad\text{if $\delta(x_0^\prime)=-1$}\,.$$ Iterating the argument we get $z\in X\backslash S$ and $\i(z)=\ud{x}^\prime$, contradicting the validity of the $\varphi$-expansion. Similarly we exclude the possibility that $\overline{\varphi}_\infty(\ud{x}^\prime)>s$, thus proving the surjectivity of the map $\overline{\varphi}_\infty$. Suppose that $\overline{\varphi}_\infty\colon \tA^{\Z_+}\ra [0,1]$ is well-defined and surjective. Let $x\in X\backslash S$ and $\ud{x}=\i(x)$. Suppose that $x< \overline{\varphi}_\infty(\ud{x})$. By Lemma \[lem2.4quatro\] any $z$, such that $x<z<\overline{\varphi}_\infty(\ud{x})$, does not have a $\varphi$-expansion. This contradicts the hypothesis that $\overline{\varphi}_\infty$ is surjective. Similarly we exclude the possibility that $x> \overline{\varphi}_\infty(\ud{x})$. \[thm2.1ter\] A $\varphi$-expansion is valid if and only if $\overline{\varphi}_\infty: \tA^{\Z_+}\ra [0,1]$ is well-defined, continuous and there exist $\ud{x}^+$ with $\overline{\varphi}_\infty(\ud{x}^+)=1$ and $\ud{x}^-$ with $\overline{\varphi}_\infty (\ud{x}^-)=0$. Suppose that the $\varphi$-expansion is valid. By Theorem \[thm2.1bis\] $\overline{\varphi}_\infty$ is well-defined and surjective so that there exist $\ud{x}^+$ and $\ud{x}^-$ with $\overline{\varphi}_\infty(\ud{x}^+)=1$ and $\overline{\varphi}_\infty(\ud{x}^-)=0$. Suppose that $\ud{x}^n\downarrow\ud{x}$ and set $y:=\overline{\varphi}_\infty(\ud{x})$, $x_n:=\overline{\varphi}_\infty(\ud{x}^n)$. By Lemma \[lem2.3\] the sequence $\{x_n\}$ is monotone decreasing; let $x:=\lim_nx_n$. Suppose that $y<x$ and $y<z<x$. Since $y<z<x_n$ for any $n\geq 1$ and $\lim_n\ud{x}^n=\ud{x}$, we prove, as in the beginning of the proof of Lemma \[lem2.4quatro\], that $z\in X\backslash S$. The validity of the $\varphi$-expansion implies that $z=\overline{\varphi}_\infty(\i(z))$. By Lemma \[lem2.4quinto\] $$\ud{x}\preceq\i(z)\preceq\ud{x}^n\,.$$ Since these inequalities are valid for any $z$, with $y<z<x$, the validity of $\varphi$-expansion implies that we have strict inequalities, $\ud{x}\prec\i(z)\prec \ud{x}^n$. This contradicts the hypothesis that $\lim_{n\ra\infty}\ud{x}^n=\ud{x}$. A similar argument holds in the case $\ud{x}^n\uparrow\ud{x}$. Hence $$\lim_{n\ra\infty}\ud{x}^n=\ud{x}\implies \lim_{n\ra\infty} \overline{\varphi}_\infty(\ud{x}^n)=\overline{\varphi}_\infty(\ud{x})\,.$$ Conversely, suppose that $\overline{\varphi}_\infty \colon \tA^{\Z_+}\ra [0,1]$ is well-defined and continuous. Then, given $\delta>0$ and $\ud{x}\in\tA^{\Z_+}$, $\exists n$ so that $$0\leq\sup\{\overline{\varphi}_\infty(\ud{x}^\prime)\colon x^\prime_j=x_j\;j=0,\ldots,n-1\}- \inf\{\overline{\varphi}_\infty(\ud{x}^\prime)\colon x^\prime_j=x_j\;j=0,\ldots,n-1\}\leq \delta\,.$$ We set $$\ud{x}^{n,-}:=x_0\cdots x_{n-1}\ud{x}^-\quad\text{and}\quad \ud{x}^{n,+}:=x_0\cdots x_{n-1}\ud{x}^+\,.$$ For any $x\in X\backslash S$ we have the identity , $$x=\varphi\big(\i_0(x)+\varphi(\i_1(x)+\ldots +\varphi(\i_{n-1}+T^{n}x))\big)= \overline{\varphi}_{n}(\i_0(x),\ldots,\i_{n-1}(x)+T^{n}x)\,.$$ If $\delta(\i_0(x)\cdots\i_{n-1}(x))=1$, then $$\begin{aligned} \overline{\varphi}_\infty(\ud{x}^{n,-}):&= \overline{\varphi}_n( \i_0(x),\ldots,\i_{n-1}(x)+\overline{\varphi}_\infty(\ud{x}^-))\\ &=\overline{\varphi}_n( \i_0(x),\ldots,\i_{n-1}(x))\\ &\leq \overline{\varphi}_{n}(\i_0(x),\ldots,\i_{n-1}(x)+T^{n}x)\\ &\leq \overline{\varphi}_{n}( \i_0(x),\ldots,\i_{n-1}(x)+1)\\ &=\overline{\varphi}_{n}( \i_0(x),\ldots,\i_{n-1}(x)+\overline{\varphi}_\infty(\ud{x}^+))=: \overline{\varphi}_\infty(\ud{x}^{n,+})\,.\end{aligned}$$ If $\delta(\i_0(x)\cdots\i_{n-1}(x))=-1$, then the inequalities are reversed. Letting $n$ going to infinity, we get $\overline{\varphi}_\infty(\i(x))=x$. \[rem2.4\] When the maps $f_0$ and $f_{k-1}$ are increasing, then we can take $$\ud{x}^+=(k-1,k-1,\ldots)\quad\text{and}\quad \ud{x}^{-}=(0,0,\ldots)\,.$$ \[thm2.2\][[@P2]]{} A necessary and sufficient condition for a $\varphi$-expansion to be valid is that $S$ is dense in $[0,1]$. A sufficient condition is $\sup_t|\varphi^\prime(t)|<1$. For each $j\in\tA$ we define (the limits are taken with $x\in X\backslash S$) $$\label{2.12} \ud{u}^j:=\lim_{x\downarrow a_{j}}\i(x)\quad\text{and}\quad \ud{v}^j :=\lim_{x\uparrow a_{j+1}}\i(x)\,.$$ The strings $\ud{u}^j$ and $\ud{v}^j$ are called [virtual itineraries]{}. Notice that $\underline{v}^j \prec\underline{u}^{j+1}$ since $v^j_0<u_0^{j+1}$. $$\label{2.12b} \sigma^k\ud{u}^j=\sigma^k(\lim_{x\downarrow a_{j}}\i(x))= \lim_{x\downarrow a_{j}}\sigma^k\i(x)=\lim_{x\downarrow a_{j}}\i(T^kx)\qquad(x\in X\backslash S)\,.$$ \[prof2.2\] Suppose that $\ud{x}^\prime\in\tA^{\Z_+}$ verifies $\ud{u}^{x_n^\prime}\prec \sigma^n\ud{x}^\prime\prec \ud{v}^{x_n^\prime}$ for all $n\geq 0$. Then there exists $x\in X\backslash S$ such that $\i(x)=\ud{x}^\prime$. Notice that we do not assume that the $\varphi$-expansion is valid or that the map $\overline{\varphi}_\infty$ is well-defined. For unimodal maps see e.g. Theorem II.3.8 in [@CoE]. Our proof is different. If $y_\uparrow(\ud{x}^\prime)<y_\downarrow(\ud{x}^\prime)$ are two cluster points, then this follows from Lemma \[lem2.4quatro\]. Therefore, assume that $\lim_{n}\overline{\varphi}_n(\ud{x}^\prime)$ exists. Either there exists $m>1$ so that $y_\uparrow(\sigma^m\ud{x}^\prime)<y_\downarrow(\sigma^m\ud{x}^\prime)$ are two cluster points, or $\lim_{n}\overline{\varphi}_n(\sigma^m\ud{x}^\prime)$ exists for all $m\geq 1$. In the first case, there exists $z_m\in X\backslash S$, $$y_\uparrow(\sigma^m\ud{x}^\prime)<z_m<y_\downarrow(\sigma^m\ud{x}^\prime) \quad\text{and}\quad \i(z_m)=\sigma^m\ud{x}^\prime\,.$$ Let $$z_{m-1}:=\overline{\varphi}(x^\prime_{m-1}+z_m)\,.$$ We show that $a_{x^\prime_{m-1}}< z_{m-1}<a_{x^\prime_{m-1}+1}$. This implies that $z_m\in {\rm int(dom} \varphi)$ so that $$\varphi^{-1}(z_{m-1})=x^\prime_{m-1}+Tz_{m-1}=x^\prime_{m-1}+z_m\,.$$ Suppose that $\delta(x^\prime_{m-1})=1$ and $a_{x^\prime_{m-1}}=z_{m-1}$. Then for any $y\in X\backslash S$, $y>a_{x^\prime_{m-1}}$, we have $Ty>z_{m}$. Therefore, by Lemma \[lem2.4\], $\i(Ty)\succeq \i(z_m)=\sigma^m\ud{x}^\prime$; $\i_0(y)=x^\prime_{m-1}$ when $y$ is close to $a_{x^\prime_{m-1}}$, so that $$\lim_{y\downarrow a_{x^\prime_{m-1}}}\i(y)=\ud{u}^{x^\prime_{m-1}}\succeq \sigma^{m-1}\ud{x}^\prime\,,$$ which is a contradiction. Similarly we exclude the cases $\delta(x^\prime_{m-1})=1$ and $a_{x^\prime_{m-1}+1}=z_{m-1}$, $\delta(x^\prime_{m-1})=-1$ and $a_{x^\prime_{m-1}}=z_{m-1}$, $\delta(x^\prime_{m-1})=-1$ and $a_{x^\prime_{m-1}+1}=z_{m-1}$. Iterating this argument we get the existence of $z_0\in X\backslash S$ with $\i(z_0)=\ud{x}^\prime$. In the second case, $\lim_{n}\overline{\varphi}_n(\sigma^m\ud{x}^\prime)$ exists for all $m\geq 1$. Let $x:=\lim_{n}\overline{\varphi}_n(\ud{x}^\prime)$. Suppose that $x^\prime_0=j$, so that $\ud{u}^{j}\prec\ud{x}^\prime\prec \ud{v}^{j}$. By Lemma \[lem2.4\] and definition of $\ud{u}^{j}$ and $\ud{v}^{j}$ there exist $z_1,z_2\in I_j$ such that $$z_1<x<z_2\quad\text{and}\quad\ud{u}^{j}\preceq\i(z_1)\prec \ud{x}^\prime\prec\i(z_2)\preceq \ud{v}^{j}\,.$$ Therefore $a_j< x<a_{j+1}$, $\i_0(x)=x_0^\prime$ and $Tx= \overline{\varphi}_\infty(\sigma\ud{x}^\prime)$ (Lemma \[lem2.4ter\]). Iterating this argument we get $\ud{x}^\prime=\i(x)$. \[thm2.5\] Suppose that the $\varphi$-expansion is valid. Then 1. $\Sigma:=\{\i(x)\in\tA^{\Z_+}\colon x\in X\backslash S\}=\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}^{x_n}\prec \sigma^n\ud{x}\prec \ud{v}^{x_n}\quad\forall\,n\geq 0\}$. 2. The map $\i\colon X\backslash S\ra \Sigma$ is bijective, $\overline{\varphi}_\infty\cirk\i={\rm id}$ and $\i\cirk\overline{\varphi}_\infty={\rm id}$.\ Both maps $\i$ and $\overline{\varphi}_\infty$ are order-preserving. 3. $\sigma(\Sigma)=\Sigma$ and $\overline{\varphi}_\infty(\sigma\ud{x})=T\overline{\varphi}_\infty(\ud{x})$ if $\ud{x}\in\Sigma$. 4. If $\ud{x}\in\tA^{\Z_+}\backslash\Sigma$, then there exist $m\in\Z_+$ and $j\in\tA$ such that $\overline{\varphi}_\infty(\sigma^m\ud{x})=a_j$. 5. $\forall n\geq 0\,,\,\forall j\in\tA\colon \quad\ud{u}^{u^j_n}\preceq \sigma^n\ud{u}^j\prec \ud{v}^{u^j_n}$ if $\delta(\ud{u}^j_0\cdots\ud{u}^j_{n-1})=1$ and $\ud{u}^{u^j_n}\prec \sigma^n\ud{u}^j\preceq \ud{v}^{v^j_n}$ if $\delta(\ud{u}^j_0\cdots\ud{u}^j_{n-1})=-1$. 6. $\forall n\geq 0\,,\,\forall j\in\tA\colon \quad\ud{u}^{u^j_n}\preceq \sigma^n\ud{v}^j\prec \ud{v}^{u^j_n}$ if $\delta(\ud{v}^j_0\cdots\ud{v}^j_{n-1})=-1$ and $\ud{u}^{u^j_n}\prec \sigma^n\ud{v}^j\preceq \ud{v}^{v^j_n}$ if $\delta(\ud{u}^j_0\cdots\ud{u}^j_{n-1})=1$. Let $x\in X\backslash S$. Clearly, by monotonicity, $$\ud{u}^{\i_k(x)}\preceq \sigma^k\i(x)\preceq \ud{v}^{\i_k(x)}\quad \forall\;k\in\Z_+\,.$$ Suppose that there exist $x\in X\backslash S$ and $k$ such that $\sigma^k\i(x)= \ud{v}^{\i_k(x)}$. Since $(\sigma^k\i(x))_0=\i_0(T^kx)$, we can assume, without restricting the generality, that $k=0$ and $\i_0(x)=j$. Therefore $x\in (a_{j}, a_{j+1})$, and for all $y\in X\backslash S$, such that $x\leq y<a_{j+1}$, we have by Lemma \[lem2.4\] that $\i(y)=\i(x)=\ud{v}^{j}$. By Theorem \[thm2.1\] this contradicts the hypothesis that the $\varphi$-expansion is valid. The other case, $\sigma^k\i(x)= \ud{u}^{\i_k(x)}$, is treated similarly. This proves half of the first statement. The second half is a consequence of Proposition \[prof2.2\]. The second statement also follows, as well as the third, since $T(X\backslash S)=X\backslash S$ (we assume that holds). Let $\ud{x}\in\tA^{\Z_+}\backslash\Sigma$ and $m\in\Z_+$ be the smallest integer such that one of the conditions defining $\Sigma$ is not verified. Then either $\sigma^m\ud{x}\preceq\underline{u}^{x_m}$, or $\sigma^m\ud{x}\succeq \underline{v}^{x_m}$. The map $\overline{\varphi}_\infty$ is continuous (Theorem \[thm2.1ter\]). Hence, for any $j\in\tA$, $$\overline{\varphi}_\infty(\ud{u}^j)=a_{j}\quad\text{and}\quad \overline{\varphi}_\infty(\ud{v}^j)=a_{j+1}\,.$$ Let $\sigma^m\ud{x}\preceq\underline{u}^{x_m}$. Since $\ud{v}^{x_m-1}\prec\sigma^m\ud{x}$, $$a_{x_m}=\overline{\varphi}_\infty(\ud{v}^{x_m-1}) \leq\overline{\varphi}_\infty(\sigma^m\ud{x})\leq \overline{\varphi}_\infty(\ud{u}^{x_m})=a_{x_m}\,.$$ The other case is treated in the same way. From definition $\ud{u}^{u^j_n}\preceq \sigma^n\ud{u}^j\preceq \ud{v}^{u^j_n}$. Suppose that $\delta(\ud{u}^j_0\cdots\ud{u}^j_{n-1})=1$ and $\sigma^n\ud{u}^j=\ud{v}^{u^j_n}$. By continuity of the $\varphi$-code there exists $x\in X\backslash S$ such that $x>a_j$ and $\i_k(x)=\ud{u}^j_k$, $k=0,\ldots,n$. Let $a_j<y<x$. Since $\delta(\ud{u}^j_0\cdots\ud{u}^j_{n-1})=1$, $T^ny<T^nx$ and consequently $$\lim_{y\downarrow a_j}\i(T^ny)=\sigma^n\ud{u}^j\preceq\i(T^nx)\preceq \ud{v}^{u^j_n}\,.$$ Hence $\sigma^n\i(x)= \ud{v}^{x_n}$, which is a contradiction. The other cases are treated similarly. Dynamical system $\beta x+\alpha\mod 1$ {#subsection2.3} --------------------------------------- We consider the family of dynamical systems $\beta x+\alpha\mod 1$ with $\beta >1$ and $0\leq \alpha<1$. For given $\alpha$ and $\beta$, the dynamical system is described by $k=\lceil \alpha+\beta\rceil$ intervals $I_j$ and maps $f_j$, $$I_0=\Big(0,\frac{1-\alpha}{\beta}\Big)\,,\,I_j=\Big(\frac{j-\alpha}{\beta}, \frac{j+1-\alpha}{\beta}\Big) \,,\,j=1,\ldots,k-2\,,\,I_{k-1}=\Big(\frac{k-1-\alpha}{\beta},1\Big)$$ and $$f_j(x)=\beta x+\alpha-j\,,\,j=0,\ldots k-1\,.$$ The maps $T_{\alpha,\beta}$, $\varphi^\ab$ and $\overline{\varphi}^\ab$ are defined as in subsection \[subsection2.1\]. $$\overline{\varphi}^\ab(x)=\begin{cases} 0 & \text{if $0\leq x\leq\alpha$}\\ \beta^{-1}(x-\alpha) & \text{if $\alpha\leq x\leq \alpha+\beta$} \\ 1 & \text{if $\alpha+\beta\leq x\leq \lceil \alpha+\beta\rceil$} \end{cases}$$ and $$\label{eqso} S_0=\{a_j\colon j=1,\ldots,k-1\}\cup\{0,1\}\quad\text{with}\quad a_j:=\beta^{-1}(j-\alpha)\,.$$ Since all maps are increasing the total order on $\tA^{\Z_+}$ is the lexicographic order. We have $2k$ virtual orbits, but only two of them are important. Indeed, if we set $$\ud{u}^{\alpha,\beta}:=\ud{u}^0\quad\text{and} \quad\ud{v}^{\alpha,\beta}:=\ud{v}^{k-1}\,,$$ then $$\ud{u}^j=j\ud{u}^{\alpha,\beta}\,,\,j=1,\ldots k-1$$ and $$\ud{v}^j=j\ud{v}^{\alpha,\beta}\,,\,j=0,\ldots,k-2\,.$$ \[pro3.1\] Let $\beta >1$ and $0\leq \alpha<1$. The $\varphi$-expansion for the dynamical system $\beta x+\alpha\mod1$ is valid. $$\Sigma^{\alpha,\beta}:=\big\{\i(x)\in\tA^{\Z_+}\colon x\in X\backslash S\big\}=\big\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}^{\alpha,\beta}\prec\sigma^n\ud{x}\prec\ud{v}^{\alpha,\beta}\quad\forall n\geq 0\big\}\,.$$ Moreover $$\ud{u}^{\alpha,\beta}\preceq\sigma^n\ud{u}^{\alpha,\beta}\prec\ud{v}^{\alpha,\beta} \quad\text{and}\quad \ud{u}^{\alpha,\beta}\prec\sigma^n\ud{v}^{\alpha,\beta}\preceq\ud{v}^{\alpha,\beta} \quad\forall n\geq0\,.$$ The closure of $\Sigma^{\alpha,\beta}$ is the shift space $$\label{3.2} \BSigma(\ud{u}^{\alpha,\beta},\ud{v}^{\alpha,\beta}):=\big\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}^{\alpha,\beta}\preceq\sigma^n\ud{x}\preceq\ud{v}^{\alpha,\beta}\quad\forall n\geq 0\big\}\,.$$ We define the [orbits of $0$, resp. $1$]{} as, (the limits are taken with $x\in X\backslash S$) $$T^k_\ab(0):=\lim_{x\downarrow 0}T_\ab^k(x)\,,\,k\geq 0\quad\text{resp.}\quad T^k_\ab(1):=\lim_{x\uparrow 1}T_\ab^k(x)\,,\,k\geq 0\,.$$ From and the coding of these orbits is $\ud{u}^\ab$, resp. $\ud{v}^\ab$, $$\label{virtual} \sigma^k\ud{u}^\ab=\lim_{x\downarrow 0}\i(T_\ab^k(x))\quad\text{and}\quad \sigma^k\ud{v}^\ab=\lim_{x\uparrow 1}\i(T_\ab^k(x))\,.$$ Notice that $T^k_\ab(0)<1$ and $T^k_\ab(1)>0$ for all $k\geq 0$. The virtual itineraries $\ud{u}\equiv\ud{u}^\ab$ and $\ud{v}\equiv\ud{v}^\ab$ of the dynamical system $\beta x+\alpha \mod 1$ verify the conditions $$\label{condition} \ud{u}\preceq\sigma^n\ud{u}\preceq \ud{v}\quad \forall\,n\geq 0\quad\text{and}\quad \ud{u}\preceq \sigma^n\ud{v}\preceq \ud{v}\quad \forall\,n\geq 0\,.$$ By Theorem \[thm2.1ter\], and Theorem \[thm2.5\] we have $(x\in X\backslash S)$ $$\begin{aligned} \ophi^\ab(\sigma^k\ud{u})&=\lim_{x\downarrow 0}\ophi^\ab(\i(T^k_\ab(x))=\lim_{x\downarrow 0}T^k_\ab(x)\equiv T^k_\ab(0) \label{validity1}\\ \ophi^\ab(\sigma^k\ud{v})&=\lim_{x\uparrow 1}\ophi^\ab(\i(T^k_\ab(x))=\lim_{x\uparrow 1}T^k_\ab(x)\equiv T^k_\ab(1)\,. \label{validity2}\end{aligned}$$ Hence $\ud{u}$ and $\ud{v}$ verify the equations[^6] $$\label{equation} \ophi^\ab(\ud{u})=0\,,\; \ophi^\ab(\sigma\ud{u})=\alpha\quad\text{and}\quad \ophi^\ab(\ud{v})=1\,,\; \ophi^\ab(\sigma\ud{v})=\gamma\,,$$ with $$\label{gamma} \gamma:=\alpha+\beta-k+1\in(0,1]\,.$$ The strings $\ud{u}^\ab$ and $\ud{v}^\ab$ are $\varphi$-expansions of $0$ and $1$. Because of the presence of discontinuities for the transformation $T_\ab$ at $a_1,\ldots a_{k-1}$, there are other strings $\ud{u}$, $\ud{v}$ which verify and , and which are also $\varphi$-expansions of $0$ and $1$. For latter purposes we need to decribe these strings; this is the content of Proposition \[pro3.4\], Proposition \[pro3.4bis\] and Proposition \[pro3.4ter\]. We also take into consideration the borderline cases $\alpha=1$ and $\gamma=0$. When $\alpha=1$ or $\gamma=0$ the dynamical system $T_\ab$ is defined using formula . The orbits of $0$ and $1$ are defined as before. For example, if $\alpha=1$ it is the same dynamical system as $T_{0,\beta}$, but with different symbols for the coding of the orbits. The orbit of $0$ is coded by $\ud{u}^{1,\beta}=(1)^\infty$, that is $\ud{u}^{1,\beta}_j=1$ for all $j\geq 0$. Similarly, if $\gamma=0$ the orbit of $1$ is coded by $\ud{v}^{\alpha,\beta}=(k-2)^\infty$. [*We always assume that $\alpha\in[0,1]$, $\gamma\in[0,1]$ and $\beta\geq1 $*]{}. \[lemelementary\] The equation $$y=\oph^\ab(x_k+t)\,, \quad y\in[0,1]$$ can be solved uniquely if $y\not\in S_0$, and its solution is $x_k=\i_0(y)$ and $t=T_\ab(y)\in(0,1)$.\ If $y<y^\prime$, then the solutions of the equations $$y=\oph^\ab(x_k+t)\quad\text{and}\quad y^\prime=\oph^\ab(x^\prime_k+t^\p)$$ are such that either $x_k=x^\prime_k$ and $T_\ab(y^\prime)-T_\ab(y)=\beta(y^\prime-y)$, or $x_k<x^\prime_k$. The proof is elementary. It suffices to notice that $$y\not\in S_0\implies y= \varphi^\ab(x_k+t)\,.$$ The second statement follows by monotonicity. \[pro3.4\] Let $0\leq\alpha< 1$ and assume that the $\varphi$-expansion is valid. The following assertions are equivalent.\ 1) There is a unique solution ($\ud{u}=\ud{u}^\ab$) of the equations $$\label{eqalpha} \ophi^\ab(\ud{u})=0\quad\text{and}\quad\ophi^\ab(\sigma\ud{u})=\alpha\,.$$ 2) The orbit of $0$ is not periodic or $x=0$ is a fixed point of $T_\ab$.\ 3) $\ud{u}^\ab$ is not periodic or $\ud{u}^\ab=\ud{0}$, where $\ud{0}$ is the string $\ud{x}$ with $x_j=0$ $\forall j\geq 0$. \[pro3.4bis\] Let $0<\gamma\leq 1$ and assume that the $\varphi$-expansion is valid. The following assertions are equivalent.\ 1) There is a unique solution ($\ud{v}=\ud{v}^\ab$) of the equations $$\label{eqbeta} \ophi^\ab(\ud{v})=1\quad\text{and}\quad\ophi^\ab(\sigma\ud{v})=\gamma\,.$$ 2) The orbit of $1$ is not periodic or $x=1$ is a fixed point of $T_\ab$.\ 3) $\ud{v}^\ab$ is not periodic or $\ud{v}^\ab=(k-1)^\infty$. We prove Proposition \[pro3.4\]. Assume 1. The validity of the $\varphi$-expansion implies that $\ud{u}^\ab$ is a solution of . If $\alpha=0$, then $\ud{u}^{0,\beta}=\ud{0}$ is the only solution of since $\ud{x}\not=\ud{0}$ implies $\ophi^{0,\beta}(\ud{x})>0$ and $x=0$ is a fixed point of $T_{0,\beta}$. Let $0<\alpha<1$. Using Lemma \[lemelementary\] we deduce that $u_0=0$ and $$\alpha=T_\ab(0)=\oph^\ab(u_1+\ophi^\ab(\sigma^2\ud{u}))\,.$$ If $\alpha=a_j$, $j=1,\ldots,k-1$ (see ), then has at least two solutions, which are $0j(\sigma^2\ud{u}^\ab)$ with $\ophi^\ab(\sigma^2\ud{u}^\ab)=T^2(0)=0$ (see ), and $0(j-1)\ud{v}^\ab$ with $\ophi^\ab(\ud{v}^\ab)=1$. Therefore, by our hypothesis we have $\alpha\not\in \{a_1,\ldots,a_{k-1}\}$, $u_1=u_1^\ab$ and $\ophi^\ab(\sigma^2\ud{u}^\ab)=T^2(0)\in (0,1)$. Iterating this argument we conclude that $1\implies 2$.\ Assume 2. If $x=0$ is a fixed point, then $\alpha=0$ and $\ud{u}^{0,\beta}=\ud{0}$. If the orbit of $0$ is not periodic, and the validity of the $\varphi$-expansion imply $$\sigma^k\ud{u}^\ab=\lim_{x\downarrow 0}\i(T^k_\ab(x))\succ\lim_{x\downarrow 0}\i(x)=\ud{u}^\ab\,.$$ Assume 3. From and the validity of the $\varphi$-expansion we get $$\ophi^\ab(\sigma^k\ud{u}^\ab)=T^k_\ab(0)> \ophi^\ab(\ud{u}^\ab)=0\,,$$ so that the orbit of $0$ is not periodic. The orbit of $0$ is not periodic if and only if $T^k_{\ab}(0)\not\in\{a_1,\ldots,a_{k-1}\}$ for all $k\geq 1$. Using Lemma \[lemelementary\] we conclude that has a unique solution. Propositions \[pro3.4\] and \[pro3.4bis\] give necessary and sufficient conditions for the existence and uniqueness of the solution of equations . In the following discussion we consider the case when there are several solutions. The main results are summarize in Proposition \[pro3.4ter\]. We assume the validity of the $\varphi$-expansion. Suppose first that the orbit of $1$ is not periodic and that the orbit of $0$ is periodic, with minimal period $p:=\min\{k\colon T^k(0)=0\}>1$. Hence $0<\gamma<1$ and $0<\alpha<1$. Let $\ud{u}$ be a solution of equations and suppose furthermore that $\ud{w}$ is a $\varphi$-expansion of $1$ such that $$\forall n\colon\; \ud{u}\preceq\sigma^n\ud{u}\preceq \ud{w}\quad\text{with}\quad \ophi^\ab(\ud{w})=1\;,\; \ophi^\ab(\sigma\ud{w})\leq\gamma\,.$$ By Lemma \[lemelementary\] we conclude that $$u_j=u_j^\ab\quad\text{and}\quad T_\ab^{j+1}(0)=\ophi^\ab(\sigma^{j+1}\ud{u})\,,\quad j=1,\ldots,p-2\,.$$ Since $T^p(0)=0$, $T^{p-1}(0)\in \{a_1,\ldots,a_{k-1}\}$ and the equation $$T_\ab^{p-1}(0)=\ophi^\ab\big(u_{p-1}+\ophi^\ab(\sigma^{p}\ud{u})\big)$$ has two solutions. Either $u_{p-1}=u_{p-1}^\ab$ and $\ophi^\ab(\sigma^{p}\ud{u})=0$ or $u_{p-1}=u_{p-1}^\ab-1$ and $\ophi^\ab(\sigma^{p}\ud{u})=1$. Let $\ud{a}$ be the prefix of $\ud{u}^\ab$ of length $p$ and $\ud{a}^\prime$ the word of length $p$ obtained by changing the last letter of $\ud{a}$ into[^7] $u_{p-1}^\ab-1$. We have $\ud{a}^\prime<\ud{a}$. If $u_{p-1}=u_{p-1}^\ab$, then we can again determine uniquely the next $p-1$ letters $u_i$. The condition $\ud{u}\leq\sigma^k\ud{u}$ for $k=p$ implies that we have $u_{2p-1}=u_{p-1}^\ab$ so that, by iteration, we get the solution $\ud{u}=\ud{u}^\ab$ for the equations . If $u_{p-1}=u_{p-1}^\ab-1$, then $$1=\ophi^\ab(\sigma^{p}\ud{u})=\ophi^\ab\big(u_p+\ophi^\ab(\sigma^{p+1}\ud{u})\big)\,.$$ When $\ophi^\ab(\sigma^{p}\ud{u})=1$, by our hypothesis on $\ud{u}$ we also have $\ophi^\ab(\sigma^{p+1}\ud{u})=\gamma$. By Proposition \[pro3.4bis\] the equations $$\ophi^\ab(\sigma^{p}\ud{u})=1\quad\text{and}\quad \ophi^\ab(\sigma^{p+1}\ud{u})=\gamma$$ have a unique solution, since we assume that the orbit of $1$ is not periodic. The solution is $\sigma^{p}\ud{u}=\ud{v}^\ab$, so that $\ud{u}=\ud{a}^\prime\ud{v}^\ab\prec\ud{u}^\ab$ is also a solution of . In that case there is no other solution for . The borderline case $\alpha=1$ corresponds to the periodic orbit of the fixed point $0$, $\ud{u}^{1,\beta}=(1)^\infty$. Notice that $\ophi^{1,\beta}(\sigma\ud{u}^{1,\beta})\not=1$. We can also consider $\ophi^{1,\beta}$-expansions of $0$ with $u_0=0$ and $\ophi^{1,\beta}(\sigma\ud{u})=1$. Our hypothesis on $\ud{u}$ imply that $\ophi^{1,\beta}(\sigma^2\ud{u})=\gamma$. Hence, $\ud{u}=0\ud{v}^{1,\beta}=\ud{a}^\prime\ud{v}^\ab\prec\ud{u}^\ab$ is a solution of and a $\ophi^{1,\beta}$-expansion of $0$. We can treat similarly the case when $\ud{u}^\ab$ is not periodic, but $\ud{v}^\ab$ is periodic. When both $\ud{u}^\ab$ and $\ud{v}^\ab$ are periodic we have more solutions, but the discussion is similar. Assume that $\ud{u}^\ab$ has (minimal) period $p>1$ and $\ud{v}^\ab$ has (minimal) period $q>1$. Define $\ud{a}$, $\ud{a}^\prime$ as before, $\ud{b}$ as the prefix of length $q$ of $\ud{v}^\ab$, and $\ud{b}^\prime$ as the word of length $q$ obtained by changing the last letter of $\ud{b}$ into $v_{q-1}^\ab+1$. When $0<\alpha<1$ and $0<\gamma<1$, one shows as above that the elements $\ud{u}\not=\ud{u}^\ab$ and $\ud{v}\not=\ud{v}^\ab$ which are $\oph^\ab$-expansions of $0$ and $1$ are of the form $$\ud{u}=\ud{a}^\prime\ud{b}^{n_1}\ud{b}^\prime\ud{a}^{n_2}\cdots\,,\,n_i\geq 0 \quad\text{and}\quad \ud{v}=\ud{b}^\prime\ud{a}^{m_1}\ud{a}^\prime\ud{b}^{m_2}\cdots\,,\,m_i\geq 0 \,.$$ The integers $n_i$ and $m_i$ must be such that is verified. The largest solution of is $\ud{u}^\ab$ and the smallest one is $\ud{a}^\prime\ud{v}^\ab$. \[pro3.4ter\] Assume that the $\varphi$-expansion is valid.\ 1) Let $\ud{u}$ be a solution of , such that $\ud{u}\preceq\sigma^n\ud{u}$ for all $n\geq 1$, and let $\ud{v}$ be a solution of , such that $\sigma^n\ud{v}\preceq\ud{v}$ for all $n\geq 1$. Then $$\ud{u}\preceq\ud{u}^\ab\quad\text{and}\quad \ud{v}^\ab\preceq\ud{v}\,.$$ 2) Let $\ud{u}$ be a solution of , and let $\ud{u}^\ab=(\ud{a})^\infty$ be periodic with minimal period $p>1$, and suppose that there exists $\ud{w}$ such that $$\forall n\colon\; \ud{u}\preceq\sigma^n\ud{u}\preceq \ud{w}\quad\text{with}\quad \ophi^\ab(\ud{w})=1\;,\; \ophi^\ab(\sigma\ud{w})\leq\gamma\,.$$ Then $$\label{3.3.3} \ud{u}^{\ab}_*\preceq\ud{u}\preceq\ud{u}^\ab\quad\text{where}\quad \ud{u}^{\ab}_*:=\ud{a}^\prime\ud{v}^\ab\;\text{and}\; \ud{a}^\p:=(\tp\ud{a})(a_{p-1}-1)\,.$$ Moreover, $\ud{u}=\ud{u}^\ab$ $\iff$ $\ud{a}$ is a prefix of $\ud{u}$ $\iff$ $\ophi^\ab(\sigma^p\ud{u})<1$.\ 3) Let $\ud{v}$ be a solution of , and let $\ud{v}^\ab=(\ud{b})^\infty$ be periodic with minimal period $q>1$, and suppose that there exists $\ud{w}$ such that $$\forall n\colon\; \ud{w}\preceq\sigma^n\ud{v}\preceq \ud{v}\quad\text{with}\quad \ophi^\ab(\ud{w})=0\;,\; \ophi^\ab(\sigma\ud{w})\geq\alpha\,.$$ Then $$\label{3.3.4} \ud{v}^\ab\preceq\ud{v}\preceq \ud{v}^{\ab}_*\quad\text{where}\quad \ud{v}^{\ab}_*:=\ud{b}^\prime\ud{u}^\ab\;\text{and}\; \ud{b}^\p:=(\tp\ud{b})(b_{q-1}+1)\,.$$ Moreover, $\ud{v}=\ud{v}^\ab$ $\iff$ $\ud{b}$ is a prefix of $\ud{v}$ $\iff$ $\ophi^\ab(\sigma^q\ud{v})>0$. Shift space ${\mathbf\Sigma}(\ud{u},\ud{v})$ {#section3} ============================================ Let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-1$ ($k\geq 2$) and holds. These assumptions are valid for the whole section, except subsection \[subsectionalgo\]. We study the shift-space $$\label{3.2.2} \BSigma(\ud{u},\ud{v}):=\{\ud{x}\in\tA^{\Z_+}\colon \ud{u}\preceq \sigma^n\ud{x}\preceq \ud{v}\quad \forall\,n\geq 0\}\,.$$ It is useful to extend the relation $\prec$ to words or to words and strings. We do it only in the following case. Let $\ud{a}$ and $\ud{b}$ be words (or strings). Then $$\ud{a}\prec\ud{b}\quad\text{iff $\exists$ $\ud{c}\in\tA^*$, $\exists$ $k\geq 0$ such that $\ud{a}=\ud{c}a_k\cdots$, $\ud{b}=\ud{c}b_k\cdots$ and $a_k<b_k$.}$$ If $\ud{a}\prec\ud{b}$ then neither $\ud{a}$ is a prefix of $\ud{b}$, nor $\ud{b}$ is a prefix of $\ud{a}$. In subsection \[subsectionfollower\] we introduce one of the main tool for studying the shift-space $\BSigma(\ud{u},\ud{v})$, the follower-set graph. In subsection \[subsectionalgo\] we give an algorithm which assigns to a pair of strings $(\ud{u},\ud{v})$ a pair of real numbers $(\bar{\alpha},\bar{\beta})\in [0,1]\times[1,\infty)$. Finally in subsection \[topological\] we compute the topological entropy of the shift space $(\ud{u},\ud{v})$. Follower-set graph $\cG(\ud{u},\ud{v})$ {#subsectionfollower} --------------------------------------- We associate to $\BSigma(\ud{u},\ud{v})$ a graph $\cG(\ud{u},\ud{v})$, called the [follower-set graph]{} (see [@LiM]), as well as an equivalent graph $\overline{\cG}(\ud{u},\ud{v})$. The graph $\overline{\cG}(\ud{u},\ud{v})$ has been systematically studied by Hofbauer in his works about piecewise monotone one-dimensional dynamical systems; see [@Ho1], [@Ho2] and [@Ho3] in the context of this paper, as well as [@Ke] and [@BrBr]. Our presentation differs from that of Hofbauer, but several proofs are directly inspired by [@Ho2] and [@Ho3]. We denote by $\cL(\ud{u},\ud{v})$ the [language of]{} $\BSigma(\ud{u},\ud{v})$, that is the set of words, which are factors of $\ud{x}\in \BSigma(\ud{u},\ud{v})$ (including the empty word $\epsilon$). Since $\sigma\BSigma(\ud{u},\ud{v})\subset \BSigma(\ud{u},\ud{v})$, the language is also the set of prefixes of the strings $\ud{x}\in \BSigma(\ud{u},\ud{v})$. To simplify the notations we set in this subsection $\BSigma:=\BSigma(\ud{u},\ud{v})$, $\cL:=\cL(\ud{u},\ud{v})$, $\cG:=\cG(\ud{u},\ud{v})$. Let $\cC_u$ be the set of words $\ud{w}\in\cL$ such that $$\ud{w}=\begin{cases} \text{$\ud{w}^\prime$\colon $\ud{w}^\prime\not=\epsilon$, $\ud{w}^\prime$ is a prefix of $\ud{u}$}\\ \text{$w_0\ud{w}^\prime$\colon $w_0\not=u_0$, $\ud{w}^\prime$ is a prefix of $\ud{u}$, possibly $\epsilon$.} \end{cases}$$ Similarly we introduce $\cC_v$ as the set of words $\ud{w}\in\cL$ such that $$\ud{w}=\begin{cases} \text{$\ud{w}^\prime$\colon $\ud{w}^\prime\not=\epsilon$, $\ud{w}^\prime$ is a prefix of $\ud{v}$}\\ \text{$w_0\ud{w}^\prime$\colon $w_0\not=v_0$, $\ud{w}^\prime$ is a prefix of $\ud{v}$, possibly $\epsilon$.} \end{cases}$$ \[defn3.1\] Let $\ud{w}\in\cL$. The longest suffix of $\ud{w}$, which is a prefix of $\ud{v}$, is denoted by $v(\ud{w})$. The longest suffix of $\ud{w}$, which is a prefix of $\ud{u}$, is denoted by $u(\ud{w})$. The [$u$-parsing of $\ud{w}$]{} is the following decomposition of $\ud{w}$ into $\ud{w}=\ud{a}^1\cdots\ud{a}^k$ with $\ud{a}^j\in\cC_u$. The first word $\ud{a}^1$ is the longest prefix of $\ud{w}$ belonging to $\cC_u$. If $\ud{w}=\ud{a}^1\ud{w}^\prime$ and $\ud{w}^\prime\not=\epsilon$, then the next word $\ud{a}^2$ is the longest prefix of $\ud{w}^\prime$ belonging to $\cC_u$ and so on. The [$v$-parsing of $\ud{w}$]{} is the analogous decomposition of $\ud{w}$ into $\ud{w}=\ud{b}^1\cdots\ud{b}^\ell$ with $\ud{b}^j\in\cC_v$. \[lem3.2.0\] Let $\ud{w}\ud{c}$ and $\ud{c}\ud{w}^\prime$ be prefixes of $\ud{u}$ (respectively of $\ud{v}$). If $\ud{w}\ud{c}\ud{w}^\prime\in\cL$, then $\ud{w}\ud{c}\ud{w}^\prime$ is a prefix of $\ud{u}$ (respectively of $\ud{v}$). Let $\ud{w}\in\cL$. If $\ud{a}^1\cdots\ud{a}^k$ is the $u$-parsing of $\ud{w}$, then only the first word $\ud{a}^1$ can be a prefix of $\ud{u}$, otherwise $u(\ud{a}^j)=\s\ud{a}^j$. Moreover $u(\ud{a}^k)=u(\ud{w})$. Analogous properties hold for the $v$-parsing of $\ud{w}$. Suppose that $\ud{w}\ud{c}$ and $\ud{c}\ud{w}^\prime$ are prefixes of $\ud{u}$. Then $\ud{w}$ is a prefix of $\ud{u}$. Assume that $\ud{w}\ud{c}\ud{w}^\prime\in\cL$ is not a prefix of $\ud{u}$. Then $\ud{u}\prec\ud{w}\ud{c}\ud{w}^\prime$. Since $\ud{w}$ is a prefix of $\ud{u}$, $\sigma^{|\ud{w}|}\ud{u}\prec\ud{c}\ud{w}^\prime$. This contradicts the fact that $\ud{c}\ud{w}^\prime$ is a prefix of $\ud{u}$. By applying this result with $\ud{c}=\epsilon$ we get the result that only the first word in the $u$-parsing of $\ud{w}$ can be a prefix of $\ud{u}$. Suppose that the $u$-parsing of $\ud{w}$ is $\ud{a}^1\cdots\ud{a}^k$. Let $k\geq 2$ and assume that $u(\ud{w})$ is not a suffix of $\ud{a}^k$ (the case $k=1$ is obvious). Since $\ud{a}^k$ is not a prefix of $\ud{u}$, $u(\ud{w})$ has $\ud{a}^k$ as a proper suffix. By the first part of the lemma this contradicts the maximality property of the words in the $u$-parsing. \[lem3.2.1\] Let $\ud{w}\in\cL$. Let $p=|u(\ud{w})|$ and $q=|v(\ud{w})|$. Then $$\big\{\ud{x}\in\BSigma\colon \text{$\ud{w}$ is a prefix of $\ud{x}$}\big\}= \big\{\ud{x}\in \tA^{\Z_+}\colon \ud{x}=\ud{w}\ud{y}\,,\;\ud{y}\in\BSigma\,,\; \sigma^p\ud{u}\preceq\ud{y}\preceq\sigma^q\ud{v}\big\}\,.$$ Moreover, $$\big\{\ud{y}\in\BSigma\colon \ud{w}\ud{y}\in\BSigma\big\}=\big\{\ud{y}\in\BSigma\colon u(\ud{w})\ud{y}\in\BSigma\big\}\quad\text{if $p>q$}$$ $$\;\big\{\ud{y}\in\BSigma\colon \ud{w}\ud{y}\in\BSigma\big\}=\big\{\ud{y}\in\BSigma\colon v(\ud{w})\ud{y}\in\BSigma\big\}\quad\text{if $q>p$.}$$ Suppose that $\ud{x}\in\BSigma$ and $\ud{w}$, $|\ud{w}|=n$, is a prefix of $\ud{x}$. Let $n\geq 1$ (the case $n=0$ is trivial). We can write $\ud{x}=\ud{w}\ud{y}$. Since $\ud{x}\in\BSigma$, $$\ud{u}\preceq \sigma^{\ell+n}\ud{x}\preceq\ud{v}\quad\forall\,\ell\geq 0\,,$$ so that $\ud{y}\in\BSigma$. We have $$\ud{u}\preceq \sigma^{n-p}\ud{x}= u(\ud{w})\ud{y}\,.$$ Since $u(\ud{w})$ is a prefix of $\ud{u}$ of length $p$, we get $\sigma^p\ud{u}\preceq \ud{y}$. Similarly we prove that $\ud{y}\preceq\sigma^q\ud{v}$. Suppose that $\ud{x}=\ud{w}\ud{y}$, $\ud{y}\in\BSigma$ and $\sigma^p\ud{u}\preceq\ud{y}\preceq\sigma^q\ud{v}$. To prove that $\ud{x}\in\BSigma$, it is sufficient to prove that $\ud{u}\preceq\sigma^m\ud{x}\preceq \ud{v}$ for $m=0,\ldots, n-1$. We prove $\ud{u}\preceq\sigma^m\ud{x}$ for $m=0,\ldots, n-1$. The other case is similar. Let $\ud{w}=\ud{a}^1\cdots\ud{a}^\ell$ be the $u$-parsing of $\ud{w}$, $|\ud{w}|=n$ and $p=|u(\ud{w})|$. We have $$\sigma^p\ud{u}\preceq\ud{y}\implies \ud{u}\preceq \sigma^j\ud{u}\preceq \sigma^ju(\ud{w})\ud{y}\quad\forall \,j=0,\ldots,p\,.$$ If $\ud{a}^\ell$ is not a prefix of $\ud{u}$, then $p=n-1$ and we also have $\ud{u}\preceq \ud{a}^k\ud{y}$. If $\ud{a}^\ell$ is a prefix of $\ud{u}$, then $p=n$ (and $\ell=1$). This proves the result for $\ell=1$. Let $\ell\geq 2$. Then $\ud{a}^\ell$ is not a prefix of $\ud{u}$ and $\ud{a}^{\ell-1}\ud{a}^\ell\in\cL$. Suppose that $\ud{a}^{\ell-1}$ is not a prefix of $\ud{u}$. In that case $\ud{u}\preceq\ud{a}^{\ell-1}\ud{a}^\ell\ud{y}$ and we want to prove that $\ud{u}\preceq\sigma^{j}\ud{a}^{\ell-1}\ud{a}^\ell\ud{y}$ for $j=1,\ldots, |\ud{a}^{\ell-1}|$. We know that $\sigma\ud{a}^{\ell-1}$ is a prefix of $\ud{u}$, and by maximality of the words in the $u$-parsing and Lemma \[lem3.2.0\] $\ud{u}\prec\sigma\ud{a}^{\ell-1}\ud{a}^\ell$; hence $\ud{u}\prec\sigma\ud{a}^{\ell-1}\ud{a}^\ell\ud{y}$. Therefore $$\ud{u}\preceq \sigma^j\ud{u}\preceq \sigma^j\ud{a}^{\ell-1}\ud{a}^\ell\ud{y}\quad\forall \,j=0,\ldots,|\ud{a}^{\ell-1}|\,.$$ Similar proof if $\ell=2$ and $\ud{a}^{\ell-1}$ is a prefix of $\ud{u}$. Iterating this argument we prove that $\ud{u}\preceq\sigma^m\ud{x}$ for $m=0,\ldots,n-1$. Suppose that $|u(\ud{w})|>|v(\ud{w})|$ and set $\ud{a}=u(\ud{w})$. We prove that $v(\ud{a})=v(\ud{w})$. By definition $v(\ud{w})$ is the longest suffix of $\ud{w}$ which is a prefix of $\ud{v}$; it is also a suffix of $\ud{a}$, whence it is also the longest suffix of $\ud{a}$ which is a prefix of $\ud{v}$. Therefore, from the first part of the lemma we get $$\big\{\ud{y}\in\BSigma\colon \ud{w}\ud{y}\in\BSigma\big\}=\big\{\ud{y}\in\BSigma\colon u(\ud{w})\ud{y}\in\BSigma\big\}\,.$$ \[defn3.2.1\] Let $\ud{w}\in\cL$. The [follower-set]{}[^8] of $\ud{w}$ is the set $$\cF_{\ud{w}}:=\big\{\ud{y}\in\BSigma\colon \ud{w}\ud{y}\in\BSigma\big\}\,.$$ Lemma \[lem3.2.1\] gives the important results that $\cF_{\ud{w}}=\cF_{u(\ud{w})}$ if $|u(\ud{w})|>|v(\ud{w})|$, and $\cF_{\ud{w}}=\cF_{v(\ud{w})}$ if $|v(\ud{w})|>|u(\ud{w})|$. Moreover, $$\label{3.2.3} \cF_\ud{w}=\big\{\ud{y}\in\BSigma\colon \sigma^p\ud{u}\preceq\ud{y}\preceq\sigma^q\ud{v}\big\}\quad\text{where $p=|u(\ud{w})|$ and $q=|v(\ud{w})|$.}$$ We can define an equivalence relation between words of $\cL$, $$\ud{w}\sim \ud{w}^\prime \iff \cF_{\ud{w}}=\cF_{\ud{w}^\prime}\,.$$ The collection of follower-sets is entirely determined by the strings $\ud{u}$ and $\ud{v}$. Moreover, the strings $\ud{u}$ and $\ud{v}$ are eventually periodic if and only if this collection is finite. Notice that $\BSigma=\cF_\epsilon=\cF_{\ud{w}}$ when $p=q=0$. \[defn3.2.2\] The [follower-set graph]{} $\cG$ is the labeled graph whose set of vertices is the collection of all follower-sets. Let $\cC$ and $\cC^\prime$ be two vertices. There is an edge, labeled by $a\in\tA$, from $\cC$ to $\cC^\prime$ if and only if there exists $\ud{w}\in\cL$ so that $\ud{w}a\in\cL$, $\cC=\cF_{\ud{w}}$ and $\cC^\prime=\cF_{\ud{w}a}$. $\cF_\epsilon$ is called the [root]{} of $\cG$. The following properties of $\cG$ are immediate. From any vertex there is at least one out-going edge and at most $|\tA|$. If $\tA=\{0,1,\ldots,k-1\}$ and $k\geq 3$, then for each $j\in \{1,\ldots,k-2\}$ there is an edge labeled by $j$ from $\cF_\epsilon$ to $\cF_\epsilon$. The out-going edges from $\cF_{\ud{w}}$ are labeled by the first letters of the strings $\ud{y}\in \cF_{\ud{w}}$. The follower-set graph $\cG$ is right-resolving. Given $\ud{w}\in\cL$, there is a unique path labeled by $\ud{w}$ from $\cF_\epsilon$ to $\cF_{\ud{w}}$. \[lem3.2.2\] Let $\ud{a}$ be a $u$-prefix and suppose that $\ud{b}=v(\ud{a})$. Let $p=|\ud{a}|$ and $q=|\ud{b}|$ so that $\cF_\ud{a}=\big\{\ud{y}\in\BSigma\colon \sigma^p\ud{u}\preceq\ud{y}\preceq\sigma^q\ud{v}\big\}$. Then there are more than one out-going edges from $\cF_{\ud{a}}$ if and only if $u_p<v_q$. Assume that $u_p<v_q$. Then there is an edge labeled by $v_q$ from $\cF_{\ud{a}}$ to $\cF_{\ud{b}v_q}$, an edge labeled by $u_p$ from $\cF_{\ud{a}}$ to $\cF_{\ud{a}u_p}$ and $v(\ud{a}u_p\ud{c})=v(\ud{c})$. If there exists $u_p<\ell<v_q$, there is an edge labeled by $\ell$ from $\cF_{\ud{a}}$ to $\cF_\epsilon$. Moreover, there are at least two out-going edges from $\cF_{\ud{b}}$, one labeled by $v_q$ to $\cF_{\ud{b}v_q}$ and one labeled by $\ell^\prime=u_{|u(\ud{b})|+1}<v_q$ to $\cF_{u(\ud{b})\ell^{\prime}}$. Furthermore $u(\ud{b}v_q\ud{c})=u(\ud{c})$. The first part of the lemma is immediate. Suppose that there is only one out-going edge from $\cF_{\ud{b}}$, that is from $\cF_{\ud{b}}$ to $\cF_{\ud{b}v_q}$. This happens if and only if $u(\ud{b})v_q$ is a prefix of $\ud{u}$. By Lemma \[lem3.2.0\] we conclude that $\ud{a}v_q$ is a prefix of $\ud{u}$, which is a contradiction. Therefore $u(\ud{b}v_q)=\epsilon$; hence $u(\ud{b}v_q\ud{c})=u(\ud{c})$. \[lem3.2.3\] Let $\ud{b}$ be a $v$-prefix and suppose that $\ud{a}=u(\ud{b})$. Let $p=|\ud{a}|$ and $q=|\ud{b}|$ so that $\cF_\ud{b}=\big\{\ud{y}\in\BSigma\colon \sigma^p\ud{u}\preceq\ud{y}\preceq\sigma^q\ud{v}\big\}$. Then there are more than one out-going edges from $\cF_{\ud{b}}$ if and only if $u_p<v_q$. Assume that $u_p<v_q$. Then there is an edge labeled by $u_p$ from $\cF_{\ud{b}}$ to $\cF_{\ud{a}u_p}$, an edge labeled by $v_q$ from $\cF_{\ud{b}}$ to $\cF_{\ud{b}v_q}$ and $u(\ud{b}v_q\ud{c})=u(\ud{c})$. If there exists $u_p<\ell<v_q$, there is an edge labeled by $\ell$ from $\cF_{\ud{b}}$ to $\cF_\epsilon$. Moreover, there are at least two out-going edges from $\cF_{\ud{a}}$, one labeled by $u_p$ to $\cF_{\ud{a}u_p}$ and one labeled by $\ell^\prime=v_{|v(\ud{a})|+1}>u_p$ to $\cF_{v(\ud{a})\ell^{\prime}}$. Furthermore $v(\ud{a}u_p\ud{c})=v(\ud{c})$. \[sch\] The picture below illustrates the main properties of the graph $\cG$. The vertices of the graph are labeled by prefixes of $\ud{u}$ and $\ud{v}$. The above line represents a prefix of the string $\ud{u}$ which is written $\ud{w}\,e^{\p\p}$, and the bottom line a prefix of the string $\ud{v}$, which is written $\ud{b}\,e^\p$. Here $u(\ud{b})=\ud{a}$, $v(\ud{w})=\ud{b}$ and we assume that $e^{\p\p}\prec e^\prime$. Therefore, there is an edge labeled by $e^\prime$ from $\cF_{\ud{w}}$ to $\cF_{\ud{b}e^\prime}$ and there is an edge labeled by $e$ from $\cF_{\ud{b}}$ to $\cF_{\ud{a}e}$ with $e\prec e^\p$. Moreover, we also have $e\preceq e^{\p\p}$. Only these two labeled edges are drawn in the picture. (160,35) (15,25)[(10,3)\[b\][$\ud{a}$]{}]{} (40,25)[(40,3)[$\cdots\cdots\cdots$]{}]{} (80,25)[(45,3)[$\ud{b}$]{}]{}(115,25)[(10,3)[$\ud{a}$]{}]{} (15,10)[(45,3)[$\ud{b}$]{}]{}(50,10)[(10,3)\[b\][$\ud{a}$]{}]{} (60,11)[(-2,1)[30]{}]{} (125,26)[(-4,-1)[60]{}]{} (25,25) (60,10) (125,25) (48,18) (88,18) We introduce a variant of the follower-set graph denoted below by $\overline{\cG}(\ud{u},\ud{v})$ or simply by $\overline{\cG}$. We introduce a vertex for each (nontrivial) prefix $\ud{a}$ of $\ud{u}$ and for each (nontrivial) prefix of $\ud{b}$ of $\ud{v}$. We add the vertex $\cF_\epsilon$. Here we do not use the equivalence relation $\sim$. The root is denoted by $[0,0]$; let $\ud{a}$ be a prefix of $\ud{u}$ and let $p=|\ud{a}|$, $q=|v(\ud{a})|$. Then this vertex is denoted by $[p,q]$. Notice that $p>q$. Similarly, let $\ud{b}$ be a prefix of $\ud{v}$ and let $p=|u(\ud{b})|$, $q=|\ud{b}|$. Then the corresponding vertex is denoted by $[p,q]$. Notice that $p<q$. The [upper branch]{} of $\overline{\cG}$ is the set of all vertices $[p,q]$ with $p>q$ and the [lower branch]{} of $\overline{\cG}$ is the set of all vertices $[p,q]$ with $p<q$. There is a single out-going edge from $[p,q]$ if and only if $u_p=v_q$. In that case the edge is labeled by $u_p$ (or $v_q$) and goes from $[p,q]$ to $[p+1,q+1]$. Otherwise there are several out-going edges. If $u_p<v_q$ there is an edge labeled by $u_p$ from $[p,q]$ to $[p+1,0]$, an edge labeled by $v_q$ from $[p,q]$ to $[0,q+1]$, and if $u_p<j<v_q$ then there is an edge labeled by $j$ from $[p,q]$ to $[0,0]$. We define the [level of the vertex]{} $[p,q]$ of $\overline{\cG}$ as $\ell([p,q]):=\max\{p,q\}$. \[defn3.2.3\] Let $\BSigma$ be a shift-space and $\cL$ its language. We denote by $\cL_n$ the set of all words of $\cL$ of length $n$. The [ entropy]{} of $\BSigma$ is $$h(\BSigma):=\lim_{n\ra\infty}\frac{1}{n}\log_2 {\rm card}(\cL_n)\,.$$ The number $h(\BSigma)$ is also equal to the topological entropy of the dynamical system $(\BSigma,\sigma)$ [@LiM]. In our case we can give an equivalent definition using the graph $\cG$ or the graph $\overline{\cG}$. We set $$\ell(n):={\rm card}\{\text{$n$-paths in $\overline{\cG}$ starting at the root $\cF_\epsilon$}\}\,.$$ Since the graph is right-resolving and for any $\ud{w}\in\cL_n$ there is a unique path labeled by $\ud{w}$, starting at the root $[0,0]$, so that $h(\BSigma)=h(\overline{\cG})$ where $$h(\overline{\cG})=\lim_{n\ra\infty}\frac{1}{n}\log_2 \ell(n)\,.$$ Let $K\in\N$ and $\overline{\cG}_K$ be the sub-graph of $\overline{\cG}$ whose set of vertices is the set of all vertices of $\overline{\cG}$ of levels smaller or equal to $K$. The following result is Proposition 9.3.15 in [@BrBr]. \[proBB\] Given $\varepsilon>0$ there exists a $K(\varepsilon)<\infty$ such that for any $K\geq K(\varepsilon)$, $$h(\overline{\cG}_K)\leq h(\overline{\cG})\leq h(\overline{\cG}_K)+\varepsilon\,.$$ \[corBB\] Let $(\ud{u},\ud{v})$ be a pair of strings of $\tA^{\Z_+}$ verifying . Given $\varepsilon>0$ there exists $N(\varepsilon)$ such that if $(\ud{u}^\prime,\ud{v}^\prime)$ is a pair of strings verifying , $\ud{u}$ and $\ud{u}^\prime$ have a common prefix of length larger $N(\varepsilon)$ and $\ud{v}$ and $\ud{v}^\prime$ have a common prefix of length larger than $N(\varepsilon)$, then $$\big|h(\BSigma(\ud{u}^\prime,\ud{v}^\prime))- h(\BSigma(\ud{u},\ud{v}))\big|\leq\varepsilon\,.$$ The algorithm for finding $(\bar{\alpha},\bar{\beta})$ {#subsectionalgo} ------------------------------------------------------ We describe an algorithm, which assigns to a pair of strings $(\ud{u},\ud{v})$, such that $u_0=0$ and $v_0=k-1$, a pair of real numbers $(\bar{\alpha},\bar{\beta})\in [0,1]\times[1,\infty)$. We assume tacitly that for the pair $(\ab)$ one has $\alpha\in[0,1]$, $\beta\leq k$, and that the map $\oph^\ab$ verifies $$0<\oph^\ab(t)<1\quad\forall t\in(1,k-1)\,.$$ In particular $\beta\geq k-2$. When $k=2$ we assume that $\beta\geq 1$. Recall that $$\gamma=\alpha+\beta-k+1\,,$$ and notice that our assumptions imply that $0\leq\gamma\leq 1$. \[defndominate\] The map $\oph^\ab$ [dominates]{} the map $\oph^\abp$ if and only if $\oph^\ab(t)\geq \oph^\abp(t)$ for all $t\in [0,k]$ and there exists $s\in [0,k]$ such that $\oph^\ab(s)>\oph^\abp(s)$. (160,50) (25,15)[(30,30)]{} (55,15)[(30,30)]{} (85,15)[(30,30)]{} (35,15)[(5,2)[75]{}]{} (25,23)[(30,0)]{} (25,35)[(60,0)]{} (20,22) (20,34) (32,10) (108,10) (34,14)[$\bullet$]{} (109,14)[$\bullet$]{} (55,5) \[lem3.3.1\] If $\oph^\ab$ dominates $\oph^\abp$, then, for all $\ud{x}\in\tA^{\Z_+}$, $\ophi^\ab(\ud{x})\geq \ophi^\abp(\ud{x})$. If $$0<\ophi^\ab(\ud{x})<1\quad\text{or}\quad 0<\ophi^\abp(\ud{x})<1\,,$$ then the inequality is strict. If $\oph^\ab$ dominates $\oph^\abp$, then by our implicit assumptions we get by inspection of the graphs that $$\forall t\geq t^\p \colon\;\oph^\ab(t)>\oph^\abp(t^\p)\quad \text{if}\quad t,t^\p\in (\alpha,\alpha^\prime+\beta^\prime)= (\alpha,\alpha+\beta)\cup(\alpha^\prime,\alpha^\prime+\beta^\prime)\,,$$ otherwise $\oph^\ab(t)\geq\oph^\abp(t^\p)$. Therefore, for all $n\geq 1$, $$\oph_n^\ab(\ud{x})\geq\oph_n^\abp(\ud{x})\,.$$ Suppose that $0<\ophi^\ab(\ud{x})<1$. Then $x_0+\ophi^\ab(\sigma\ud{x})\in(\alpha,\alpha+\beta)$ and $$\ophi^\ab(\ud{x})=\oph^\ab(x_0+\ophi^\ab(\sigma\ud{x}))> \oph^\abp(x_0+\ophi^\abp(\sigma\ud{x}))=\ophi^\abp(\ud{x})\,.$$ Similar proof for $0<\ophi^\abp(\ud{x})<1$. \[lem3.3.2\] Let $\alpha=\alpha^\prime\in [0,1]$ and $1\leq \beta<\beta^\prime$. Then, for $\ud{x}\in\tA^{\Z_+}$, $$0\leq \ophi^{\alpha,\beta}(\ud{x})-\ophi^{\alpha,\beta^\prime}(\ud{x})\leq \frac{|\beta-\beta^\prime|}{\beta^\prime-1}\,.$$ Let $\gamma=\gamma^\prime\in [0,1]$, $0\leq\alpha^\prime<\alpha\leq 1$ and $\beta^\prime>1$. Then, for $\ud{x}\in\tA^{\Z_+}$, $$0\leq \ophi^{\abp}(\ud{x})-\ophi^{\alpha,\beta}(\ud{x})\leq \frac{|\alpha-\alpha^\prime|}{\beta^\prime-1}\,.$$ The map $\beta\mapsto \ophi^{\alpha,\beta}(\ud{x})$ is continuous at $\beta=1$. Let $\alpha=\alpha^\prime\in [0,1]$ and $1\leq \beta<\beta^\prime$. For $t,t^\prime\in [0,k]$, $$|\oph^{\alpha,\beta^\prime}(t^\prime)-\oph^\ab(t)|\leq |\oph^{\alpha,\beta^\prime}(t^\prime)-\oph^{\alpha,\beta^\prime}(t)|+ |\oph^{\alpha,\beta^\prime}(t)-\oph^\ab(t)|\leq \frac{|t-t^\prime|}{\beta^\prime}+\frac{|\beta-\beta^\prime|}{\beta^\prime}\,.$$ (The maximum of $|\oph^{\alpha,\beta^\prime}(t)-\oph^\ab(t)|$ is taken at $\alpha+\beta$). By induction $$|\oph_n^{\alpha,\beta^\prime}(x_0,\ldots,x_{n-1})- \oph_n^{\alpha,\beta}(x_0,\ldots,x_{n-1})|\leq |\beta-\beta^\prime|\sum_{j=1}^n(\beta^\prime)^{-j}\,.$$ Since $\beta^\prime>1$ the sum is convergent. This proves the first statement. The second statement is proved similarly using $$|\oph^\abp(t^\prime)-\oph^\ab(t)|\leq |\oph^\abp(t^\prime)-\oph^\abp(t)|+|\oph^\abp(t)-\oph^\ab(t)|\leq \frac{|t-t^\prime|}{\beta^\prime}+\frac{|\alpha-\alpha^\prime|}{\beta^\prime}$$ which is valid for $\gamma=\gamma^\prime\in [0,1]$ and $0\leq\alpha^\prime<\alpha\leq 1$. We prove the last statement. Given $\varepsilon>0$ there exists $n^*$ $$\oph_{n^*}^{\alpha,1}(\ud{x})\geq \ophi^{\alpha,1}(\ud{x})-\varepsilon\,.$$ Since $\beta\mapsto \oph_{n^*}^{\alpha,\beta}(\ud{x})$ is continuous, there exists $\beta^\prime$ so that for $1\leq\beta\leq\beta^\p$, $$\oph_{n}^{\alpha,\beta}(\ud{x})\geq \oph_{n^*}^{\alpha,\beta^\p}(\ud{x})\geq \oph_{n^*}^{\alpha,1}(\ud{x})-\varepsilon\quad\forall n\geq n^*\,.$$ Hence $$\ophi^{\alpha,1}(\ud{x})-2\varepsilon\leq \ophi^\ab(\ud{x})\leq \ophi^{\alpha,1}(\ud{x})\,.$$ \[cor3.2\] Given $\ud{x}$ and $0\leq\alpha^*\leq 1$, let $$g_{\alpha^*}(\gamma):=\ophi^{\alpha^*,\beta(\gamma)}(\ud{x})\quad\text{with}\quad \beta(\gamma):=\gamma-\alpha^*+k-1\,.$$ For $k\geq 3$ the map $g_{\alpha^*}$ is continuous and non-increasing on $[0,1]$. If $0<g_{\alpha^*}(\gamma_0)<1$, then the map is strictly decreasing in a neighborhood of $\gamma_0$. If $k=2$ then the same statements hold on $[\alpha^*,1]$. \[cor3.3\] Given $\ud{x}$ and $0<\gamma^*\leq 1$, let $$h_{\gamma^*}(\alpha):=\ophi^{\alpha,\beta(\alpha)}(\ud{x})\quad\text{with}\quad \beta(\alpha):=\gamma^*-\alpha+k-1\,.$$ For $k\geq 3$ the map $h_{\gamma^*}$ is continuous and non-increasing on $[0,1]$. If $0<h_{\gamma^*}(\alpha_0)<1$, then the map is strictly decreasing in a neighborhood of $\alpha_0$. If $k=2$ then the same statements hold on $[0,\gamma^*)$. \[pro3.3\] Let $k\geq 2$, $\ud{u}, \ud{v}\in\tA^{\Z_+}$ verifying $u_0=0$ and $v_0=k-1$ and $$\sigma\ud{u}\preceq\ud{v}\quad\text{and}\quad \ud{u}\preceq\sigma\ud{v}\,.$$ If $k=2$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Then there exist $\bar{\alpha}\in[0,1]$ and $\bar{\beta}\in[1,\infty)$ so that $\bar{\gamma}\in[0,1]$. If $\bar{\beta}>1$, then $$\ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{u})=\bar{\alpha}\quad\text{and}\quad \ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{v})=\bar{\gamma}\,.$$ We consider separately the cases $\sigma\ud{v}=\ud{0}$ and $\sigma\ud{u}=(k-1)^\infty$ (i.e. $u_j=k-1$ for all $j\geq 1$). If $\sigma\ud{v}=\ud{0}$, then $\ud{u}=\ud{0}$ and $\ud{v}=(k-1)\ud{0}$; we set $\bar{\alpha}:=0$ and $\bar{\beta}:=k-1$ ($\bar{\gamma}=0$). If $\sigma\ud{u}=(k-1)^\infty$, then $\ud{v}=(k-1)^\infty$ and $\ud{u}=0(k-1)^\infty$; we set $\bar{\alpha}:=1$ and $\bar{\beta}:=k$. From now on we assume that $\ud{0}\prec\sigma\ud{v}$ and $\sigma\ud{u}\prec (k-1)^\infty$. Set $\alpha_0:=0$ and $\beta_0:=k$. We consider in details the case $k=2$, so that we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. [Step $1$.]{} Set $\alpha_1:=\alpha_0$ and solve the equation $$\ophi^{\alpha_1,\beta}(\sigma\ud{v})=\beta+\alpha_1-k+1\,.$$ There exists a unique solution, $\beta_1$, such that $k-1<\beta_1\leq k$. Indeed, the map $$G_{\alpha_1}(\gamma):=g_{\alpha_1}(\gamma)-\gamma\quad\text{with}\quad g_{\alpha_1}(\gamma):=\ophi^{\alpha_1,\beta(\gamma)}(\sigma\ud{v}) \;\,\text{and}\;\,\beta(\gamma):=\gamma-\alpha_1+k-1$$ is continuous and strictly decreasing on $[\alpha_1,1]$ (see Corollary \[cor3.2\]). If $\sigma\ud{v}=(k-1)^\infty$, then $G_{\alpha_1}(1)=0$ and we set $\beta_1:=k$ and we have $\gamma_1=1$. If $\sigma\ud{v}\not=(k-1)^\infty$, then there exists a smallest $j\geq 1$ so that $v_j\leq (k-2)$. Therefore $\ophi^{\alpha_1,k}(\sigma^j\ud{v})<1$ and $$\ophi^{\alpha_1,k}(\sigma\ud{v})=\oph^{\alpha_1,k}_{j-1}\big(v_1,\ldots,v_{j-1}+ \ophi^{\alpha_1,k}(\sigma^j\ud{v})\big)<1\,,$$ so that $G_{\alpha_1}(1)<0$. On the other hand, since $\sigma\ud{v}\not=\ud{0}$, $\ophi^{\alpha_1,k-1}(\sigma\ud{v})>0$, so that $G_{\alpha_1}(0)>0$. There exists a unique $\gamma_1\in(0,1)$ with $G_{\alpha_1}(\gamma_1)=0$. Define $\beta_1:=\beta(\gamma_1)=\gamma_1-\alpha_1+k-1$. [Step $2$.]{} Solve in $[0,\gamma_1)$ the equation $$\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})=\alpha\quad\text{with}\quad\beta(\alpha):= \gamma_1-\alpha+k-1=\beta_1+\alpha_1-\alpha\,.$$ If $\sigma\ud{u}=0$, then set $\bar{\alpha}:=0$ and $\bar{\beta}:=\beta_1$. Let $\sigma\ud{u}\not=0$. There exists a smallest $j\geq 1$ such that $u_j\geq 1$. This implies that $\ophi^{\alpha_1,\beta_1}(\sigma^j\ud{u})>0$ and consequently $$\ophi^{\alpha_1,\beta(\alpha_1)}(\sigma\ud{u})= \oph^{\alpha_1,\beta_1}_{j-1}\big(u_1,\ldots,u_{j-1}+ \ophi^{\alpha_1,\beta_1}(\sigma^j\ud{u})\big)>0\,.$$ Since $\sigma\ud{u}\preceq\sigma\ud{v}$, $$0<\ophi^{\alpha_1,\beta_1}(\sigma\ud{u}) \leq\ophi^{\alpha_1,\beta_1}(\sigma\ud{v})=\gamma_1\,.$$ We have $\gamma_1=1$ only in the case $\sigma\ud{v}=(k-1)^\infty$; in that case we also have $\ophi^{\alpha_1,\beta_1}(\sigma\ud{u})<1$. By Corollary \[cor3.3\], for any $\alpha>\alpha_1$ we have $\ophi^{\alpha_1,\beta_1}(\sigma\ud{u})>\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})$. Therefore, the map $$H_{\gamma_1}(\alpha):=h_{\gamma_1}(\alpha)-\alpha\quad\text{with}\quad h_{\gamma_1}(\alpha):=\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})$$ is continuous and strictly decreasing on $[0,\gamma_1)$, $H_{\gamma_1}(\alpha_1)>0$ and $\lim_{\alpha\uparrow\gamma_1}H_{\gamma_1}(\alpha)<0$. There exists a unique $\alpha_2\in (\alpha_1,\gamma_1)$ such that $H_{\gamma_1}(\alpha_2)=0$. Set $\beta_2:=\gamma_1-\alpha_2+k-1=\alpha_1+\beta_1-\alpha_2$ and $\gamma_2:=\alpha_2+\beta_2-k+1=\gamma_1$. Since $\alpha_2\in [0,\gamma_1)$, we have $\beta_2>1$. Hence $$\label{step2} \alpha_1<\alpha_2<\gamma_1\quad\text{and}\quad 1<\beta_2<\beta_1\quad\text{and}\quad \gamma_2=\gamma_1\,.$$ If $\sigma\ud{v}=(k-1)^\infty$, $\gamma_2=1$ and we set $\bar{\alpha}:=\alpha_2$ and $\bar{\beta}:=\beta_2$. [Step $3$.]{} From now on $\sigma\ud{u}\not=\ud{0}$ and $\sigma\ud{v}\not=(k-1)^\infty$. Set $\alpha_3:=\alpha_2$ and solve in $[\alpha_3,1]$ the equation $$\ophi^{\alpha_3,\beta(\gamma)}(\sigma\ud{v})=\gamma\quad\text{with}\quad \beta(\gamma):=\gamma-\alpha_3+k-1\,.$$ By Lemma \[lem3.3.1\] ($k=2$), $$\ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{v})= \ophi^{\alpha_2,1}(\sigma\ud{v})\geq \ophi^{\alpha_2,1}(\sigma\ud{u})>\ophi^{\alpha_2,\beta_2}(\sigma\ud{u})=\alpha_2\,,$$ since $0<\alpha_2<1$. On the other hand by Corollary \[cor3.3\], $$\label{esti3} \ophi^{\alpha_3,\beta(\gamma_1)}(\sigma\ud{v})= \ophi^{\alpha_3,1+\gamma_1-\alpha_3}(\sigma\ud{v})< \ophi^{\alpha_1,1+\gamma_1-\alpha_1}(\sigma\ud{v})=\ophi^{\alpha_1,\beta_1}(\sigma\ud{v}) =\gamma_1\,,$$ since $0<\gamma_1<1$. Therefore, the map $G_{\alpha_3}$ is continuous and strictly decreasing on $[\alpha_3,1]$, $G_{\alpha_3}(\alpha_3)>0$ and $G_{\alpha_3}(\gamma_1)<0$. There exists a unique $\gamma_3\in (\alpha_3,\gamma_1)$ such that $G_{\alpha_3}(\gamma_3)=0$. Set $\beta_3:=\gamma_3-\alpha_3+k-1$, so that $\beta_3<\gamma_1-\alpha_2+k-1=\beta_2$. Hence $$\label{step3} \alpha_3=\alpha_2\quad\text{and}\quad 1<\beta_3<\beta_2\quad\text{and}\quad 0<\gamma_3<\gamma_2<1\,.$$ [Step $4$.]{} Solve in $[0,\gamma_3)$ the equation $$\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})=\alpha\quad\text{with}\quad\beta(\alpha):= \gamma_3-\alpha+k-1=\beta_3+\alpha_3-\alpha\,.$$ By Lemma \[lem3.3.1\] $$\label{esti4} \ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{u})=\ophi^{\alpha_3,\beta_3}(\sigma\ud{u}) >\ophi^{\alpha_3,\beta_2}(\sigma\ud{u})=\ophi^{\alpha_2,\beta_2}(\sigma\ud{u})=\alpha_2\,,$$ since $0<\alpha_2<1$. On the other hand, $$0<\ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{u})= \ophi^{\alpha_3,\beta_3}(\sigma\ud{u})\leq\ophi^{\alpha_3,\beta_3}(\sigma\ud{v}) =\gamma_3<1\,.$$ By Corollary \[cor3.3\] $$\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})<\ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{u}) \quad\forall \alpha\in (\alpha_3,\gamma_3)\,.$$ Therefore, the map $$H_{\gamma_3}(\alpha):=h_{\gamma_3}(\alpha)-\alpha\quad\text{with}\quad h_{\gamma_3}(\alpha):=\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})$$ is continuous and strictly decreasing on $[\alpha_3,\gamma_3)$, $H_{\gamma_3}(\alpha_3)>0$ and $\lim_{\alpha\uparrow\gamma_3}H_{\gamma_3}(\alpha) <0$. There exists a unique $\alpha_4\in (\alpha_3,\gamma_3)$. Set $\beta_4:=\gamma_3-\alpha_4+k-1=\alpha_3+\beta_3-\alpha_4$ and $\gamma_4:=\alpha_4+\beta_4-k+1=\gamma_3$. Hence $$\label{step4} \alpha_3<\alpha_4<\gamma_3\quad\text{and}\quad 1<\beta_4<\beta_3\quad\text{and}\quad \gamma_4=\gamma_3\,.$$ Repeating steps 3 and 4 we get two monotone sequences $\{\alpha_n\}$ and $\{\beta_n\}$. We set $\bar{\alpha}:=\lim_{n\ra\infty}\alpha_n$ and $\bar{\beta}:=\lim_{n\ra\infty}\beta_n$. We consider briefly the changes which occur when $k\geq 3$. Step $1$ remains the same. In step $2$ we solve the equation $H_{\gamma_1}(\alpha)=0$ on $[0,1)$ instead of $[0,\gamma_1)$. The proof that $H_{\gamma_1}(\alpha_1)>0$ remains the same. We prove that $\lim_{\alpha\uparrow 1}H_{\gamma_1}(\alpha)<0$. Corollary \[cor3.3\] implies that $$\gamma_1=\ophi^{\alpha_1,\beta_1}(\sigma\ud{v})= \ophi^{\alpha_1,\beta(\alpha_1)}(\sigma\ud{v})> \ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\quad \forall\alpha>\alpha_1\,.$$ Since $\sigma\ud{u}\preceq \ud{v}$ and $\beta(\alpha_1)=\beta_1$, $$\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})\leq \oph^{\alpha,\beta(\alpha)}\big(v_0+\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\big) \leq \oph^{\alpha_1,\beta(\alpha_1)} \big(v_0+\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\big)<1\,.$$ Instead of we have $$\alpha_1<\alpha_2<1\quad\text{and}\quad 1<\beta_2<\beta_1\quad\text{and}\quad \gamma_2=\gamma_1\,.$$ Estimate is still valid in step 3 with $k\geq 3$. Hence $G_{\alpha_3}(\gamma_1)<0$. We solve the equation $G_{\alpha_3}(\gamma)=0$ on $[0,\gamma_1]$. We have $$\ophi^{\alpha_3,\beta(\gamma_1)}(\sigma\ud{u})= \ophi^{\alpha_2,\beta_2}(\sigma\ud{u})=\alpha_2\,.$$ By Corollary \[cor3.2\] we get $$\ophi^{\alpha_3,\beta(\gamma)}(\sigma\ud{u})> \ophi^{\alpha_2,\beta_2}(\sigma\ud{u})=\alpha_2 \quad \forall\gamma<\gamma_1\,.$$ Since $\ud{u}\preceq\sigma\ud{v}$, $$\ophi^{\alpha_3,\beta(0)}(\sigma\ud{v})\geq \oph^{\alpha_3,\beta(0)}\big(u_0+\ophi^{\alpha_3,\beta(0)}(\sigma\ud{u})\big) \geq \oph^{\alpha_2,\beta(\gamma_1)}\big(u_0+\ophi^{\alpha_3,\beta(0)}(\sigma\ud{u})\big)>0\,.$$ Estimate is still valid in step 4 so that $H_{\gamma_3}(\alpha_3)>0$. Corollary \[cor3.3\] implies that $$\gamma_3=\ophi^{\alpha_3,\beta_3}(\sigma\ud{v})= \ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{v})> \ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\quad \forall\alpha>\alpha_3\,.$$ Therefore $$\ophi^{\alpha_3,\beta(\alpha_3)}(\sigma\ud{u})\leq \oph^{\alpha,\beta(\alpha)}\big(v_0+\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\big) \leq \oph^{\alpha_3,\beta(\alpha_3)} \big(v_0+\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{v})\big)<1\,.$$ Instead of we have $$\alpha_3<\alpha_4<1\quad\text{and}\quad 1<\beta_4<\beta_3\quad\text{and}\quad \gamma_4=\gamma_3\,.$$ Assume that $\bar{\beta}>1$. Then $1<\bar{\beta}\leq\beta_n$ for all $n$. We have $$\ophi^{\alpha_n,\beta_n}(\sigma\ud{v})=\gamma_n\,,\quad\text{$n$ odd}$$ and $$\ophi^{\alpha_n,\beta_n}(\sigma\ud{u})=\alpha_n\,,\quad\text{$n$ even}\,.$$ Let $\bar{\gamma}=\bar{\alpha}+\bar{\beta}-k+1$. For $n$ odd, let $\beta_n^*:=\bar{\gamma}-\alpha_n+k-1$; using Lemma \[lem3.3.2\] we get $$\begin{aligned} |\ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{v})-\bar{\gamma}|&\leq |\ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{v})- \ophi^{\alpha_n,\beta_n^*}(\sigma\ud{v})|+ |\ophi^{\alpha_n,\beta_n^*}(\sigma\ud{v})- \ophi^{\alpha_n,\beta_n}(\sigma\ud{v})|+|\gamma_n-\gamma|\\ &\leq\frac{1}{\bar{\beta}-1}(2|\bar{\alpha}-\alpha_n|+ |\bar{\beta}-\beta_n|)+|\gamma_n-\gamma|\,,\end{aligned}$$ since $\beta_n^*=\bar{\beta}+\bar{\alpha}-\alpha_n$. Letting $n$ going to infinity we get $\ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{v})=\bar{\gamma}$. Similarly we prove $\ophi^{\bar{\alpha},\bar{\beta}}(\sigma\ud{v})=\bar{\alpha}$. \[cor3.4\] Suppose that $(\ud{u},\ud{v})$, respectively $(\ud{u}^\p,\ud{v}^\p)$, verify the hypothesis of Proposition \[pro3.3\] with $k\geq 2$, respectively with $k^\prime\geq 2$. If $k\geq k^\prime$, $\ud{u}\preceq\ud{u}^\p$ and $\ud{v}^\p\preceq\ud{v}$, then $\bar{\beta}^\p\leq\bar{\beta}$ and $\bar{\alpha}^\p\geq \bar{\alpha}$. We consider the case $k=k^\p$, whence $\sigma\ud{v}^\p\preceq\sigma\ud{v}$. From the proof of Proposition \[pro3.3\] we get $\gamma^\p_1\leq\gamma_1$ and $\alpha_1^\p\geq\alpha_1$. Suppose that $\gamma^\p_j\leq\gamma_j$ and $\alpha_j^\p\geq\alpha_j$ for $j=1,\ldots,n$. If $n$ is even, then $\alpha_{n+1}^\p=\alpha_{n}^\p$ and $\alpha_{n+1}=\alpha_{n}$. We prove that $\gamma_{n+1}^\p\leq\gamma_{n+1}$. We have $$\gamma_{n+1}^\p=\ophi^{\alpha_{n+1}^\p,\beta(\gamma_{n+1}^\p)}(\sigma\ud{v}^\p) \leq \ophi^{\alpha_{n+1}^\p,\beta(\gamma_{n+1}^\p)}(\sigma\ud{v})\leq \ophi^{\alpha_{n+1},\beta(\gamma_{n+1}^\p)}(\sigma\ud{v})\implies \gamma_{n+1}\geq\gamma_{n+1}^\p\,.$$ If $n$ is odd, then $\gamma_{n+1}^\p=\gamma_{n}^\p$ and $\gamma_{n+1}=\gamma_{n}$. We prove that $\alpha_{n+1}^\p\geq\alpha_{n+1}$. We have $$\begin{aligned} \alpha_{n+1}&=\ophi^{\alpha_{n+1},\beta(\alpha_{n+1})}(\sigma\ud{u}) \leq \ophi^{\alpha_{n+1},\beta(\alpha_{n+1})}(\sigma\ud{u}^\p)= \ophi^{\alpha_{n+1},\gamma_{n+1}-\alpha_{n+1}+k-1)}(\sigma\ud{u}^\p)\\ & \leq \ophi^{\alpha_{n+1},\gamma_{n+1}^\p-\alpha_{n+1}+k-1)}(\sigma\ud{u}^\p) \implies \alpha_{n+1}^\p\geq\alpha_{n+1}\,.\end{aligned}$$ We state a uniqueness result. The proof uses Theorem \[thm3.1\]. \[prounicity\] Let $k\geq 2$, $\ud{u}, \ud{v}\in\tA^{\Z_+}$, $u_0=0$ and $v_0=k-1$, and assume that holds. Then there is at most one solution $(\ab)\in[0,1]\times[1,\infty)$ for the equations $$\ophi^{\alpha,\beta}(\sigma\ud{u})=\alpha\quad\text{and}\quad \ophi^{\alpha,\beta}(\sigma\ud{v})=\gamma\,.$$ Assume that there are two solutions $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$ with $\beta_1\leq \beta_2$. If $\alpha_2>\alpha_1$, then $$\alpha_2-\alpha_1=\ophi^{\alpha_2,\beta_2}(\sigma\ud{u})- \ophi^{\alpha_1,\beta_1}(\sigma\ud{u})\leq 0\,,$$ which is impossible. Therefore $\alpha_2\leq\alpha_1$. If $\beta_1=\beta_2$, then $$0\geq\alpha_2-\alpha_1=\ophi^{\alpha_2,\beta_2}(\sigma\ud{u})- \ophi^{\alpha_1,\beta_1}(\sigma\ud{u})\geq 0\,,$$ which implies $\alpha_2=\alpha_1$. Therefore we assume that $\alpha_2\leq\alpha_1$ and $\beta_1<\beta_2$. However, Theorem \[thm3.1\] implies that $$\log_2\beta_1=h(\BSigma(\ud{u},\ud{v}))=\log_2\beta_2\,,$$ which is impossible. Computation of the topological entropy of $\BSigma(\ud{u},\ud{v})$ {#topological} ------------------------------------------------------------------ We compute the entropy of the shift space $\BSigma(\ud{u},\ud{v})$ where $\ud{u}$ and $\ud{v}$ is a pair of strings verifying $u_0=0$, $v_0=k-1$ and . The main result is Theorem \[thm3.1\]. The idea for computing the topological entropy is to compute $\bar{\alpha}$ and $\bar{\beta}$ by the algorithm of section \[subsectionalgo\] and to use the fact that $h\big(\BSigma(\ud{u}^\abb,\ud{v}^\abb)\big)=\log_2\bar{\beta}$ (see e.g. [@Ho1]). The most difficult case is when $\ud{u}$ and $\ud{v}$ are both periodic. Assume that the string $\ud{u}:=\ud{a}^\infty$ has minimal period $p$, $|\ud{a}|=p$, and that the string $\ud{v}:=\ud{b}^\infty$ has minimal period $q$, $|\ud{b}|=q$. If $a_0=a_{p-1}=0$, then $\ud{u}=\ud{0}$ and $p=1$. Indeed, if $a_0=a_{p-1}=0$, then $\ud{a}\ud{a}=(\tp\ud{a})00(\s\ud{a})$; the result follows from . Similarly, if $b_0=b_{q-1}=k-1$, then $\ud{v}=(k-1)^\infty$ and $q=1$. These cases are similar to the case when only one of the strings $\ud{u}$ and $\ud{v}$ is periodic and are simpler than the generic case of two periodic strings, which we treat in details. The setting for subsection \[topological\] is the following one. The string $\ud{u}:=\ud{a}^\infty$ has minimal period $p\geq 2$ with $u_0=0$, or $\ud{u}=(1)^\infty$. The string $\ud{v}:=\ud{b}^\infty$ has minimal period $q\geq 2$ with $v_0=k-1$, or $\ud{v}=(k-2)^\infty$. We also consider the strings $\ud{u}^*=\ud{a}^\p\ud{b}^\infty$ and $\ud{v}^*=\ud{b}^\p\ud{a}^\infty$ with $\ud{a}^\p=\tp\ud{a}(u_{p-1}-1)$ and $\ud{b}^\p=\tp\ud{b}(v_{q-1}+1)$. We write $\BSigma\equiv\BSigma(\ud{u},\ud{v})$, $\BSigma^*\equiv \BSigma(\ud{u}^*,\ud{v}^*)$, $\cG\equiv \cG(\ud{u},\ud{v})$ and $\cG^*\equiv \cG(\ud{u}^*,\ud{v}^*)$. The main point is to prove that $h(\cG)=h(\cG^*)$ by comparing the follower-set graphs $\cG$ and $\cG^*$. \[lemgraph\] 1) In the above setting the vertices of the graph $\cG$ are $\cF_\epsilon$, $\cF_{\ud{w}}$ with $\ud{w}$ a prefix of $\tp\ud{a}$ or of $\tp\ud{b}$, $\tp\ud{a}$ and $\tp\ud{b}$ included.\ 2) Let $r:=|v(\tp\ud{a})|$. If $u_{p-1}\not=v_r$, then $\cF_{\ud{a}}=\cF_\epsilon$ and there is an edge labeled by $u_{p-1}$ from $\cF_{\tp\ud{a}}$ to $\cF_\epsilon$. If $u_{p-1}=v_r$, then $\cF_{\ud{a}}=\cF_{v(\tp\ud{a})v_r}$ and there is a single edge, labeled by $u_{p-1}=v_r$, from $\cF_{\tp\ud{a}}$ to $\cF_{v(\tp\ud{a})v_r}$. If $k=2$ the first possibility is excluded.\ 3) Let $s:=|u(\tp\ud{b})|$. If $v_{q-1}\not=u_s$, then $\cF_{\ud{b}}=\cF_\epsilon$ and there is an edge labeled by $v_{q-1}$ from $\cF_{\tp\ud{b}}$ to $\cF_\epsilon$. If $v_{q-1}=u_s$, then $\cF_{\ud{b}}=\cF_{u(\tp\ud{b})u_s}$ and there is a single edge, labeled by $v_{q-1}=u_s$, from $\cF_{\tp\ud{b}}$ to $\cF_{u(\tp\ud{b})u_s}$. If $k=2$ the first possibility is excluded. Suppose that $\ud{w}$ and $\ud{w}\ud{w}^\p$ are two prefixes of $\tp\ud{a}$. We show that $\cF_{\ud{w}}\not=\cF_{\ud{w}\ud{w}^\p}$. Write $\ud{u}=\ud{w}\ud{x}$ and $\ud{u}=\ud{w}\ud{w}^\p\ud{y}$ and suppose that $\cF_{\ud{w}}=\cF_{\ud{w}\ud{w}^\p}$. Then (see ) $\ud{x}=\ud{y}=\sigma^p\ud{u}$, so that $\ud{u}=(\ud{w}^\prime)^\infty$, contradicting the minimality of the period $p$. Consider the vertex $\cF_{\tp\ud{a}}$ of $\cG$. We have $$\cF_{\tp\ud{a}}=\{\ud{x}\in\BSigma\colon \sigma^{p-1}\ud{u}\preceq\ud{x}\preceq \sigma^{r}\ud{v}\}\quad\text{where $r=|v(\tp\ud{a})|$}\,.$$ Let $\ud{d}$ be the prefix of $\ud{v}$ of length $r+1$, so that $\tp\ud{d}=v(\tp\ud{a})$. If $u_{p-1}\not=v_{r}$, then there are an edge labeled by $u_{p-1}$ from $\cF_{\tp\ud{a}}$ to $\cF_{\ud{a}}=\cF_{\epsilon}$ (since $\sigma^p\ud{u}=\ud{u}$) and an edge labeled by $v_{r}$ from $\cF_{\tp\ud{a}}$ to $\cF_{\ud{d}}$. There may be other labeled edges from $\cF_{\tp\ud{a}}$ to $\cF_\epsilon$ (see Lemma \[lem3.2.2\]). If $u_{p-1}=v_{r}$, then there is a single out-going edge labeled by $u_{p-1}$ from $\cF_{\tp\ud{a}}$ to $\cF_{\ud{a}}$ and $v(\ud{a})=\ud{d}$. We prove that $\cF_{\ud{a}}=\cF_{\ud{d}}$. If $u(\ud{d})=\epsilon$, the result is true, since in that case $$\cF_{\ud{d}}=\{\ud{x}\in\BSigma\colon \ud{u}\preceq\ud{x}\preceq \sigma^{r+1}\ud{v}\}=\{\ud{x}\in\BSigma\colon \sigma^p\ud{u}\preceq\ud{x}\preceq \sigma^{r+1}\ud{v}\}=\cF_{\ud{a}}\,.$$ We exclude the possibility $u(\ud{d})\not=\epsilon$. Suppose that $\ud{w}:=u(\ud{d})$ is non-trivial ($|u(\ud{d})|<p$). We can write $\ud{a}=\ud{a}^{\p\p}\ud{w}$ and $\ud{a}=\ud{w}\widehat{\ud{a}}$ since $\ud{w}$ is a prefix of $\ud{u}$, and consequently $\ud{a}\ud{a}=\ud{a}^{\p\p}\ud{w}\ud{w}\widehat{\ud{a}}$. From Lemma \[lem3.2.0\] we conclude that $\ud{w}\ud{w}$ is a prefix of $\ud{u}$, so that $\ud{a}\ud{u}=\ud{a}^{\p\p}\ud{w}\ud{w}\ud{w}\cdots$, proving that $\ud{u}$ has period $|\ud{w}|$, contradicting the hypothesis that $p$ is the minimal period of $\ud{u}$. If $k=2$ the first possibility is excluded because $u_{p-1}\not=0$ and we have $u_{p-1}\preceq v_r$ by $\sigma^{p-r}\ud{u}\preceq\ud{v}$. The discussion concerning the vertex $\cF_{\tp\ud{b}}$ is similar. \[pro3.6\] Consider the above setting. If $h(\BSigma)>0$, then $h(\BSigma)=h(\BSigma^*)$. Consider the vertex $\cF_{\tp\ud{a}}$ of $\cG^*$. In that case $(u_{p-1}-1)\not=v_{r}$ so that we have an additional edge labeled by $u_{p-1}-1$ from $\cF_{\tp\ud{a}}$ to $\cF_{\ud{a}^\p}$ (see proof of Lemma \[lemgraph\]), otherwise all out-going edges from $\cF_{\tp\ud{a}}$, which are present in the graph $\cG$, are also present in $\cG^*$. Let $v^*(\ud{w})$ be the longest suffix of $\ud{w}$, which is a prefix of $\ud{v}^*$. Then $$\cF_{\ud{a}^\p}=\{\ud{x}\in\BSigma^*\colon \sigma^{p}\ud{u}^*\preceq\ud{x}\preceq \ud{v}^*\}= \{\ud{x}\in\BSigma^*\colon \ud{v}\preceq\ud{x}\preceq \ud{v}^*\}\,.$$ Similarly, there is an additional edge labeled by $v_{q-1}+1$ from $\cF_{\tp\ud{b}}$ to $\cF_{\ud{b}^\p}$. Let $u^*(\ud{w})$ be the longest suffix of $\ud{w}$, which is a prefix of $\ud{u}^*$. Then $$\cF_{\ud{b}^\p}=\{\ud{x}\in\BSigma^*\colon \ud{u}^*\preceq\ud{x}\preceq \sigma^q\ud{v}^*\}= \{\ud{x}\in\BSigma^*\colon \ud{u}^*\preceq\ud{x}\preceq \ud{u}\}\,.$$ The structure of the graph $\cG^*$ is very simple from the vertices $\cF_{\ud{a}^\p}$ and $\cF_{\ud{b}^\p}$. There is a single out-going edge from $\cF_{\ud{a}^\p}$ to $\cF_{\ud{a}^{\p}v_0}$, from $\cF_{\ud{a}^{\p}v_0}$ to $\cF_{\ud{a}^{\p}v_0v_1}$ and so on, until we reach the vertex $\cF_{\ud{a}^{\p}\tp\ud{b}}$. From that vertex there are an out-going edge labeled by $v_{q-1}$ to $\cF_{\ud{a}^\p}$ and an out-going edge labeled by $v_{q-1}+1$ to $\cF_{\ud{b}^\p}$. Similarly, there is a single out-going edge from $\cF_{\ud{b}^\p}$ to $\cF_{\ud{b}^{\p}u_0}$, from $\cF_{\ud{b}^{\p}u_0}$ to $\cF_{\ud{b}^{\p}u_0u_1}$ and so on, until we reach the vertex $\cF_{\ud{b}^{\p}\tp\ud{a}}$. From that vertex there are an out-going edge labeled by $u_{p-1}$ to $\cF_{\ud{b}^\p}$ and an out-going edge labeled by $u_{p-1}-1$ to $\cF_{\ud{a}^\p}$. Let us denote that part of $\cG^*$ by $\cG^*\backslash\cG$. This subgraph is strongly connected. The graph $\cG^*$ consists of the union of $\cG$ and $\cG^*\backslash\cG$ with the addition of the two edges from $\cF_{\tp\ud{a}}$ to $\cF_{\ud{a}^\p}$ and $\cF_{\tp\ud{b}}$ to $\cF_{\ud{b}^\p}$. Using Theorem 1.7 of [@BGMY] it easy to compute the entropy of the subgraph $\cG^*\backslash\cG$ (use as rome $\{\cF_{\ud{a}^\p}, \cF_{\ud{b}^\p}\}$). It is the largest root, say $\lambda^*$, of the equation $$\lambda^{-q}+\lambda^{-p}-1=0\,.$$ Hence $\lambda^*$ is equal to the entropy of a graph with two cycles of periods $p$ and $q$, rooted at a common point. To prove Proposition \[pro3.6\] it sufficient to exhibit a subgraph of $\cG$ which has an entropy larger or equal to that of $\cG^*\backslash\cG$. If $k\geq 4$, then there is a subgraph with two cycles of length $1$ rooted at $\cF_\epsilon$. Hence $h(\cG)\geq \log_22>\lambda^*$. If $\cF_{\ud{a}}=\cF_\epsilon$ or $\cF_{\ud{b}}=\cF_\epsilon$, which could happen only for $k\geq 3$ (see Lemma \[lemgraph\]), then there is a subgraph of $\cG$ consisting of two cycles rooted at $\cF_\epsilon$, one of length $p$ or of length $q$ and another one of length $1$. This also implies that $h(\cG)\geq\lambda^*$. Since the minimal periods of $\ud{u}$ and $\ud{v}$ are $p$ and $q$, it is impossible that $\cF_{\ud{w}}=\cF_\epsilon$ for $\ud{w}$ a non trivial prefix of $\tp\ud{a}$ or $\tp\ud{b}$. Therefore we assume that $k\leq 3$, $\cF_{\ud{a}}\not=\cF_\epsilon$ and $\cF_{\ud{b}}\not=\cF_{\epsilon}$. Let $\cH$ be a strongly connected component of $\cG$ which has strictly positive entropy. If $\cF_\epsilon$ is a vertex of $\cH$, which happens only if $k=3$, then we conclude as above that $h(\cG)\geq\lambda^*$. Hence, we assume that $\cF_\epsilon$ is not a vertex of $\cH$. The vertices of $\cH$ are indexed by prefixes of $\tp\ud{a}$ and $\tp\ud{b}$. Let $\cF_{\ud{c}}$ be the vertex of $\cH$ with $\ud{c}$ a prefix of $\ud{u}$ and $|\ud{c}|$ minimal; similarly, let $\cF_{\ud{d}}$ be the vertex of $\cH$ with $\ud{d}$ a prefix of $\ud{v}$ and $|\ud{d}|$ minimal. By our assumptions $r:=|\ud{c}|\geq 1$ and $s:=|\ud{d}|\geq 1$. The following argument is a simplified adaptation of the proof of Lemma 3 in [@Ho3]. The core of the argument is the content of the Scholium \[sch\]. Consider the $v$-parsing of $\ud{a}$ from the prefix $\tp\ud{c}$, and the $u$-parsing of $\ud{b}$ from the prefix $\tp\ud{d}$, $$\ud{a}=(\tp\ud{c})\ud{a}^1\cdots\ud{a}^k\quad\text{and}\quad \ud{b}=(\tp\ud{d})\ud{b}^1\cdots\ud{b}^\ell\,.$$ (From $\tp\ud{c}$ the $v$-parsing of $\ud{a}$ does not depend on $\tp\ud{c}$ since there is an in-going edge at $\cF_{\ud{c}}$.) We claim that there are an edge from $\cF_{(\tp\ud{c})\ud{a}^1}$ to $\cF_{\ud{d}}$ and an edge from $\cF_{(\tp\ud{d})\ud{b}^1}$ to $\cF_{\ud{c}}$. Suppose that this is not the case, for example, there is an edge from $\cF_{(\tp\ud{c})\ud{a}^1\cdots \ud{a}^j}$ to $\cF_{\ud{d}}$, but no edge from $\cF_{(\tp\ud{c})\ud{a}^1\cdots \ud{a}^i}$ to $\cF_{\ud{d}}$, $1\leq i<j$. This implies that $v(\ud{a}^j)=\s\ud{a}^j=\tp(\ud{d})$ and $(\tp\ud{c})\ud{a}^1\cdots \ud{a}^jf^\prime$ is a prefix of $\ud{u}$ with $f^\p\prec f$ and $f$ defined by $\ud{d}=(\tp\ud{d})f$. On the other hand there exists an edge from $\cF_{(\tp\ud{c})\ud{a}^1\cdots \ud{a}^{j-1}}$ to $\cF_{v((\tp\ud{c})\ud{a}^1\cdots \ud{a}^{j-1})*}= \cF_{v(\ud{a}^{j-1})*}$ with $*$ some letter of $\tA$ and $v(\ud{a}^{j-1})*\not=\ud{d}$ by hypothesis. Let $e$ be the first letter of $\ud{a}^j$. Then $*=(e+1)$ since we assume that $\cF_\epsilon$ is not a vertex of $\cH$ and consequently there are only two out-going edges from $\cF_{(\tp\ud{c})\ud{a}^1\cdots \ud{a}^{j-1}}$. There exists an edge from $\cF_{v(\ud{a}^{j-1})}$ to $\cF_{u(v(\ud{a}^{j-1}))*}$, where $*$ is some letter of $\tA$ (see Scholium \[sch\]). Again, since $\cF_\epsilon$ is not a vertex of $\cH$ we must have $*=e$. Either $u(v(\ud{a}^{j-1}))e=\ud{c}$ or $u(v(\ud{a}^{j-1}))e\not=\ud{c}$. In the latter case, by the same reasoning, there exists an edge from $\cF_{u(v(\ud{a}^{j-1}))}$ to $\cF_{v(u(v(\ud{a}^{j-1})))(e+1)}$ and $v(u(v(\ud{a}^{j-1})))(e+1)\not=\ud{d}$ by hypothesis; there exists also an edge from $\cF_{v(u(v(\ud{a}^{j-1})))}$ to $\cF_{u(v(u(v(\ud{a}^{j-1}))))e}$. After a finite number of steps we get $$u(\cdots v(u(v(\ud{a}^{j-1}))))e=\ud{c}\,.$$ This implies that $\tp\ud{c}$ is a suffix of $\ud{a}^{j-1}$, and the last letter of $\ud{c}$ (or the first letter of $\ud{a}^1$) is $e$. Hence $\ud{a}^1=e\ud{d}\cdots$. If we write $\ud{a}^{j-1}=\ud{g}(\tp\ud{c})$ we have $$(\tp\ud{c})\ud{a}^1=\ud{c}\ud{d}\cdots=\ud{c}(\tp\ud{d})f\cdots \quad\text{and}\quad\ud{a}^{j-1}\ud{a}^j f=\ud{g}(\tp\ud{c})e(\tp\ud{d})f^\prime=\ud{g}\ud{c}(\tp\ud{d})f^\prime\,.$$ We get a contradiction with since $\ud{c}(\tp\ud{d})f^\p\prec\ud{c}(\tp\ud{d})f$. Consider the smallest strongly connected subgraph $\cH^\p$ of $\cH$ which contains the vertices $\cF_{\ud{c}}$, $\cF_{(\tp\ud{c})\ud{a}^1}$, $\cF_{\ud{d}}$ and $\cF_{(\tp\ud{d})\ud{b}^1}$. Since $\cH$ has strictly positive entropy, there exists at least one edge from some other vertex $A$ of $\cH$ to $\cF_{\ud{c}}$ or $\cF_{\ud{d}}$, say $\cF_{\ud{c}}$. Define $\cG^\p$ as the smallest strongly connected subgraph of $\cH$, which contains $\cH^\p$ and $A$. This graph has two cycles: one passing through the vertices $\cF_{\ud{c}}$, $\cF_{(\tp\ud{c})\ud{a}^1}$, $\cF_{\ud{d}}$, $\cF_{(\tp\ud{d})\ud{b}^1}$ and $\cF_{\ud{c}}$, the other one passing through the vertices $\cF_{\ud{c}}$, $\cF_{(\tp\ud{c})\ud{a}^1}$, $\cF_{\ud{d}}$, $\cF_{(\tp\ud{d})\ud{b}^1}$, $A$ and $\cF_{\ud{c}}$. The first cycle has length $|\ud{a}^1|+|\ud{b}^1|$, and the second cycle has length $|\ud{a}^1|+|\ud{b}^1|+\cdots+|\ud{b}^j|$ if $A=\cF_{\tp(\ud{d})\ud{b}^1\cdots\ud{b}^j}$. We also have $$|\ud{c}|=|\ud{b}^1|=|\ud{b}^j|\quad\text{and}\quad |\ud{a}^1|=|\ud{d}|\,.$$ Therefore one cycle has period $$|\ud{a}^1|+|\ud{b}^1|\leq |\ud{a}^1|+|\ud{c}|\leq p\,,$$ and the other one has period $$|\ud{d}|+|\ud{b}^1|+\cdots+|\ud{b}^j|\leq q\,.$$ \[thm3.1\] Let $k\geq 2$ and let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-1$ and $$\ud{u}\preceq\sigma^n\ud{u}\preceq \ud{v}\quad \forall\,n\geq 0\quad\text{and}\quad \ud{u}\preceq \sigma^n\ud{v}\preceq \ud{v}\quad \forall\,n\geq 0\,.$$ If $k=2$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Let $\bar{\alpha}$ and $\bar{\beta}$ be the two real numbers defined by the algorithm of Proposition \[pro3.3\]. Then $$h(\BSigma(\ud{u},\ud{v}))=\log_2\bar{\beta}\,.$$ If $k=2$ and $\sigma\ud{v}\prec \sigma\ud{u}$, then $h(\BSigma(\ud{u},\ud{v}))=0$. Let $\bar{\beta}>1$. By Propositions \[pro3.3\] and \[pro3.4ter\] we have $$\BSigma(\ud{u}^\abb,\ud{v}^\abb)\subset \BSigma(\ud{u},\ud{v})\subset\BSigma(\ud{u}^{\abb}_*,\ud{v}^{\abb}_*) \,.$$ From Proposition \[pro3.6\] we get $$h(\BSigma(\ud{u}^\abb,\ud{v}^\abb))= h(\BSigma(\ud{u}^{\abb}_*,\ud{v}^{\abb}_*))=\log_2\bar{\beta}\,.$$ Let $\lim_n\alpha_n=\bar{\alpha}$ and $\lim_n\beta_n=\bar{\beta}=1$. We have $\alpha_n<1$ and $\beta_n>1$ (see proof of Proposition \[pro3.3\]). Let $$\ud{u}^n:=\ud{u}^{\alpha_n,\beta_n}_*\quad\text{and}\quad \ud{v}^n:=\ud{v}^{\alpha_n,\beta_n}_*\,.$$ By Proposition \[pro3.4ter\] point 3, $$\ud{v}^{\alpha_1,\beta_1}\preceq \ud{v}\preceq \ud{v}^1\,.$$ By monotonicity, $$\ophi^{\alpha_2,\beta_2}(\sigma\ud{v}^1)\leq \ophi^{\alpha_1,\beta_1}(\sigma\ud{v}^1)=\gamma_1=\gamma_2 \ophi^{\alpha_2,\beta_2}(\sigma\ud{v}^2)\,.$$ Therefore $\ud{v}^1\preceq\ud{v}^2$ ($v^1_0=v^2_0$) and by Proposition \[pro3.4ter\] point 2, $$\ud{u}^2\preceq \ud{u}\preceq \ud{u}^{\alpha_2,\beta_2}\quad\text{and}\quad \ud{v}\preceq\ud{v}^2\,.$$ By monotonicity, $$\ophi^{\alpha_3,\beta_3}(\sigma\ud{u}^3)=\alpha_3=\alpha_2= \ophi^{\alpha_2,\beta_2}(\sigma\ud{u}^2)\leq \ophi^{\alpha_3,\beta_3}(\sigma\ud{u}^2)\,.$$ Therefore $\ud{u}^3\preceq\ud{u}^2$ and $$\ud{u}^3\preceq \ud{u}\quad\text{and}\quad \ud{v}^{\alpha_3,\beta_3}\preceq\ud{v}\preceq\ud{v}^3\,.$$ Iterating this argument we conclude that $$\ud{u}^n\preceq\ud{u}\quad\text{and}\quad \ud{v}\preceq\ud{v}^n\,.$$ These inequalities imply $$h(\BSigma(\ud{u},\ud{v}))\leq h(\BSigma(\ud{u}^n,\ud{v}^n))=\log_2\beta_n\ra 0\quad\text{for $n\ra\infty$}\,.$$ Finally let $k=2$ and $\sigma\ud{v}\prec\sigma\ud{u}$. If $\sigma\ud{u}=(1)^\infty$, then $\ud{v}_j=0$ for a single value of $j$, so that $h(\BSigma(\ud{u},\ud{v}))=0$. Suppose that $\sigma\ud{u}\not=(1)^\infty$ and fix any $\beta>1$. The function $\alpha\mapsto\ophi^{\alpha,\beta}(\sigma\ud{u})$ is continuous and decreasing since $\oph^{\alpha,\beta}$ dominates $\oph^{\alpha^\p,\beta}$ if $\alpha<\alpha^\p$. There exists $\alpha\in(0,1)$ such that $\ophi^{\alpha,\beta}(\sigma\ud{u})=\alpha$. If $\ud{v}_0< \ud{v}^\ab_0$, then $\ud{v}\prec \ud{v}^\ab$ and $\BSigma(\ud{u},\ud{v})\subset\BSigma(\ud{u},\ud{v}^\ab)$, whence $h(\BSigma(\ud{u},\ud{v}))\leq\log_2\beta$. If $\ud{v}_0= \ud{v}^\ab_0=1$, then $$\ophi^\ab(\sigma\ud{v})\leq \ophi^\ab(\sigma\ud{u})=\alpha<\gamma= \ophi^\ab(\sigma\ud{v}^\ab)\,.$$ The map $\ophi^\ab$ is continuous and non-decreasing on $\tA^{\Z_+}$ so that $\sigma\ud{v}\prec \sigma\ud{v}^\ab$, whence $\ud{v}\prec\ud{v}^\ab$ and $h(\BSigma(\ud{u},\ud{v}))\leq\log_2\beta$. Since $\beta>1$ is arbitrary, $h(\BSigma(\ud{u},\ud{v}))=0$. Inverse problem for $\beta x+\alpha \mod 1$ {#section4} =========================================== In this section we solve the inverse problem for $\beta x+\alpha\mod 1$, namely the question: [*given two strings $\ud{u}$ and $\ud{v}$ verifying $$\label{4.1} \ud{u}\preceq\sigma^n\ud{u}\prec\ud{v} \quad\text{and}\quad \ud{u}\prec\sigma^n\ud{v}\preceq\ud{v} \quad\forall n\geq0\,,$$ can we find $\alpha\in [0,1)$ and $\beta\in(1,\infty)$ so that $\ud{u}=\ud{u}^{\alpha,\beta}$ and $\ud{v}=\ud{v}^{\alpha,\beta}$?* ]{} \[pro3.5\] Let the $\varphi$-expansion be valid. Let $\ud{u}$ be a solution of and $\ud{v}$ a solution of . If holds, then $$\ud{u}^\ab=\ud{u}\iff \forall n\geq 0\colon\;\ophi^\ab(\sigma^n\ud{u})<1 \,\iff\, \forall n\geq 0\colon\;\ophi^\ab(\sigma^n\ud{v})>0\iff \ud{v}^\ab=\ud{v}\,.$$ The $\varphi$-expansion is valid, so that is true, $$\forall n\geq 0\colon\;\ophi^\ab(\sigma^n\ud{u}^\ab)=T^n_\ab(0)<1\,.$$ Proposition \[pro3.4\] and Proposition \[pro3.4ter\] point 2 imply $$\ud{u}=\ud{u}^\ab\,\iff\,\forall n\geq 0\colon\;\ophi^\ab(\sigma^n\ud{u})<1\,.$$ Similarly $$\ud{v}=\ud{v}^\ab\,\iff\,\forall n\geq 0\colon\;\ophi^\ab(\sigma^n\ud{v})>0\,.$$ Let $\ud{x}\prec\ud{x}^\p$, $\ud{x}, \ud{x}^\p\in \BSigma(\ud{u},\ud{v})$. Let $\ell:=\min\{m\geq 0\colon x_m\not=x_m^\p\}$. Then $$\ophi^\ab(\ud{x})=\ophi^\ab(\ud{x}^\p)\implies \ophi^\ab(\sigma^{\ell+1}\ud{x})=1\quad\text{and}\quad \ophi^\ab(\sigma^{\ell+1}\ud{x})=0\,.$$ Indeed, $$\oph_{\ell+1}^\ab\big(x_0,\ldots,x_{\ell-1}, x_\ell+\ophi^\ab(\sigma^{\ell+1}\ud{x})\big)= \oph_{\ell+1}^\ab\big(x_0,\ldots,x_{\ell-1}, x_\ell^\p+\ophi^\ab(\sigma^{\ell+1}\ud{x}^\p)\big)$$ Therefore $x^\p_\ell=x_\ell+1$, $\ophi^\ab(\sigma^{\ell+1}\ud{x})=1$ and $\ophi^\ab(\sigma^{\ell+1}\ud{x})=0$. Suppose that $\ophi^\ab(\sigma^k\ud{u})=1$, and apply the above result to $\sigma^k\ud{u}$ and $\ud{v}$ to get the existence of $m$ with $\ophi^\ab(\sigma^m\ud{v})=0$. Let $\ud{u}\in\tA^{\Z_+}$ with $u_0=0$ and $\ud{u}\preceq\sigma^n\ud{u}$ for all $n\geq 0$. We introduce the quantity $$\widehat{\ud{u}}:=\sup\{\sigma^n\ud{u}\colon n\geq 0\}\,.$$ We have $$\sigma^n\widehat{\ud{u}}\leq\widehat{\ud{u}}\quad\forall n\geq 0\,.$$ Indeed, if $\widehat{u}$ is periodic, then this is immediate. Otherwise there exists $n_j$, with $n_j\uparrow\infty$ as $j\ra\infty$, so that $\widehat{\ud{u}}=\lim_j\sigma^{n_j}\ud{u}$. By continuity $$\sigma^n\widehat{\ud{u}}= \lim_{j\ra\infty}\sigma^{n+n_j}\ud{u}\leq\widehat{\ud{u}}\,.$$ [**Example.**]{} We consider the strings $\ud{u}^\p=(01)^\infty$ and $\ud{v}^\p=(110)^\infty$. One can prove that $\ud{u}^\p=\ud{u}^\ab$ and $\ud{v}^\p=\ud{v}^\ab$ where $\beta$ is the largest solution of $$\beta^6-\beta^5-\beta=\beta(\beta^2-\beta+1)(\beta^3-\beta-1)=0$$ and $\alpha=(1+\beta)^{-1}$. With the notations of Proposition \[pro3.4ter\] we have $$\ud{a}=01\quad\ud{a}^\p=00\quad\ud{b}=110\quad\ud{b}^\p=111\,.$$ Let $$\ud{u}:=(00110111)^\infty=(\ud{a}^\p\ud{b}\ud{b}^\prime)^\infty\,.$$ We have $$\widehat{\ud{u}}=(11100110)^\infty=(\ud{b}^\p\ud{a}^\p\ud{b})^\infty\,.$$ By definition $\ophi^\ab(\sigma\ud{u})=\alpha$. We have $$(\ud{b})^\infty\preceq \widehat{\ud{u}}\preceq \ud{b}^\p(\ud{a})^\infty\,.$$ From Proposition \[pro3.4ter\] point 3 and Proposition \[pro3.6\] we conclude that $\log_2\beta=h(\BSigma(\ud{u},\widehat{\ud{u}}))$. \[thm4.1\] Let $k\geq 2$ and let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-1$ and holds. If $k=2$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Set $\log_2\widehat{\beta}:=h(\BSigma(\ud{u},\widehat{\ud{u}}))$. Let $\bar{\alpha}$ and $\bar{\beta}$ be defined by the algorithm of Proposition \[pro3.3\]. Then\ 1) If $\widehat{\beta}<\bar{\beta}$, then $\ud{u}=\ud{u}^\abb$ and $\ud{v}=\ud{v}^\abb $.\ 2) If $\widehat{\beta}=\bar{\beta}>1$ and $\ud{u}^{\abb}$ and $\ud{v}^{\abb}$ are not both periodic, then $\ud{u}=\ud{u}^{\abb}$ and $\ud{v}=\ud{v}^{\abb}$.\ 3) If $\widehat{\beta}=\bar{\beta}>1$ and $\ud{u}^{\abb}$ and $\ud{v}^{\abb}$ are both periodic, then $\ud{u}\not=\ud{u}^{\abb}$ and $\ud{v}\not=\ud{v}^{\abb}$. Let $\widehat{\beta}<\bar{\beta}$. Suppose that $\ud{u}\not=\ud{u}^\abb$ or $\ud{v}\not=\ud{v}^\abb $. By Proposition \[pro3.5\] $\ud{u}\not=\ud{u}^\abb$ and $\ud{v}\not=\ud{v}^\abb $, and there exists $n$ such that $\ophi^\abb(\sigma^n\ud{u})=1$. Hence $\ophi^\abb(\widehat{\ud{u}})=1$. If $\bar{\gamma}>0$, then $\widehat{\ud{u}}_0=v_0=k-1$ whence $\sigma\widehat{\ud{u}}\preceq\sigma\ud{v}$, so that $\ophi^\abb(\sigma\widehat{\ud{u}})=\bar{\gamma}$. By Propositions \[pro3.4ter\] and \[pro3.6\] we deduce that $$\log_2\widehat{\beta}=h(\BSigma(\ud{u},\widehat{\ud{u}})) =h(\BSigma(\ud{u},\ud{v}))=\log_2\bar{\beta}\,,$$ a contradiction. If $\bar{\gamma}=0$, either $\widehat{\ud{u}}_0=k-1$ and $\ophi^\abb(\sigma\widehat{\ud{u}})=\bar{\gamma}$, and we get a contradiction as above, or $\widehat{\ud{u}}_0=k-2$ and $\ophi^\abb(\sigma\widehat{\ud{u}})=1$. In the latter case, since $\sigma\widehat{\ud{u}}\preceq \widehat{\ud{u}}$, we conclude that $\widehat{\ud{u}}_1=k-2$ and $\ophi^\abb(\sigma^2\widehat{\ud{u}})=1$. Using $\sigma^n\widehat{\ud{u}}\preceq \widehat{\ud{u}}$ we get $\widehat{\ud{u}}=(k-2)^\infty=\ud{v}^{\abb}$, so that $h(\BSigma(\ud{u},\widehat{\ud{u}})) =h(\BSigma(\ud{u},\ud{v}))$, a contradiction.\ We prove 2. Suppose for example that $\ud{u}^\abb$ is not periodic. This implies that $\bar{\alpha}<1$, so that Proposition \[pro3.4\] implies that $\ud{u}=\ud{u}^\abb$. We conclude using Proposition \[pro3.5\]. Similar proof if $\ud{v}^\abb$ is not periodic.\ We prove 3. By Proposition \[pro3.5\], $\ud{u}=\ud{u}^{\abb}$ or $\ud{v}=\ud{v}^{\abb}$ if and only if $\ud{u}=\ud{u}^{\abb}$ and $\ud{v}=\ud{v}^{\abb}$. Suppose $\ud{u}=\ud{u}^{\abb}$, then $\ud{u}$ is periodic so that $\widehat{u}=\sigma^p\ud{u}$ for some $p$. This implies that $$\ophi^\abb(\sigma\widehat{\ud{u}})\leq\ophi^\abb(\widehat{\ud{u}}) =\ophi^\abb(\sigma^p\ud{u})<1\,.$$ by Proposition \[pro3.5\]. Let $\widehat{\ud{u}}_0\equiv \widehat{k}-1$. We can apply the algorithm of Proposition \[pro3.3\] to the pair $(\ud{u},\widehat{\ud{u}})$ and get two real numbers $\widetilde{\alpha}$ and $\widetilde{\beta}$ (if $\widehat{k}=2$, using $\widehat{\beta}>1$ and Theorem \[thm3.1\], we have $\sigma\ud{u}\preceq\sigma\widehat{\ud{u}}$). Theorem \[thm3.1\] implies $\widehat{\beta}=\widetilde{\beta}$, whence $\widetilde{\beta}=\bar{\beta}$. The map $\alpha\mapsto\ophi^{\alpha,\bar{\beta}}(\sigma\ud{u})$ is continuous and decreasing, so that $\alpha\mapsto\ophi^{\alpha,\bar{\beta}}(\sigma\ud{u})-\alpha$ is strictly decreasing, whence there exists a unique solution to the equation $\ophi^{\alpha,\bar{\beta}}(\sigma\ud{u})-\alpha=0$, which is $\bar{\alpha}=\widetilde{\alpha}$. Therefore $\ophi^\abb(\sigma\widehat{\ud{u}})<1$ and we must have $\widehat{k}=k$, whence $$\ophi^\abb(\sigma\widehat{\ud{u}})=\bar{\alpha}+\bar{\beta}-k+1= \ophi^\abb(\sigma\ud{v})\,.$$ But this implies $\ophi^\abb(\widehat{\ud{u}})=1$, a contradiction \[thm4.2\] Let $k\geq 2$ and let $\ud{u}\in\tA^{\Z_+}$ and $\ud{v}\in\tA^{\Z_+}$, such that $u_0=0$, $v_0=k-1$ and holds. If $k=2$ we also assume that $\sigma\ud{u}\preceq\sigma\ud{v}$. Let $\bar{\alpha}$ and $\bar{\beta}$ be defined by the algorithm of Proposition \[pro3.3\]. If $h(\BSigma(\ud{u},\widehat{\ud{u}}))>1$, then there exists $\ud{u}_*\succeq\widehat{\ud{u}}$ such that $$\begin{aligned} \ud{u}_*\prec\ud{v}&\implies \text{$\ud{u}=\ud{u}^\abb$ and $\ud{v}=\ud{v}^\abb$}\\ \ud{u}_*\succ\ud{v}&\implies \text{$\ud{u}\not=\ud{u}^\abb$ and $\ud{v}\not=\ud{v}^\abb$}\end{aligned}$$ As in the proof of Theorem \[thm4.1\] we define $\widetilde{k}$ and, by the algorithm of Proposition \[pro3.3\] applied to the pair $(\ud{u},\widehat{\ud{u}})$, two real numbers $\widetilde{\alpha}$ and $\widetilde{\beta}$. By Theorem \[thm3.1\], $\log_2\widetilde{\beta}=h(\BSigma(\ud{u},\widehat{\ud{u}}))$. We set $$\ud{u}_*:= \begin{cases}\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}_*& \text{if $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}$ is periodic}\\ \ud{v}^{\widetilde{\alpha},\widetilde{\beta}}&\text{if $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}$ is not periodic}\,. \end{cases}$$ It is sufficient to show that $\ud{u}_*\prec\ud{v}$ implies $\bar{\beta}>\widetilde{\beta}$ (see Theorem \[thm4.1\] point 1). Suppose the contrary, $\bar{\beta}=\widetilde{\beta}$. Then $$1=\ophi^{\widetilde{\alpha},\bar{\beta}}(\widehat{\ud{u}})\leq \ophi^{\widetilde{\alpha},\bar{\beta}}(\ud{v})\,.$$ We have $\ophi^{\bar{\alpha},\bar{\beta}}(\ud{v})=1$ and for $\alpha>\bar{\alpha}$, $\ophi^{\alpha,\bar{\beta}}(\ud{v})<1$ (see Lemma \[lem3.3.1\]). Therefore $\widetilde{\alpha}\leq\bar{\alpha}$. On the other hand, applying Corollary \[cor3.4\] we get $\widetilde{\alpha}\geq\bar{\alpha}$ so that $\widetilde{\alpha}=\bar{\alpha}$ and $\widetilde{k}=k$. From Propositions \[pro3.4bis\] or \[pro3.4ter\] we get $\ud{v}\preceq \ud{u}_*$, a contradiction. Suppose that $\ud{u}_*\succ\ud{v}$. We have $\widehat{u}\preceq\ud{v}\prec \ud{u}_*$, whence $h(\BSigma(\ud{u},\widehat{\ud{u}}))=h(\BSigma(\ud{u},\ud{u}_*))$ and therefore $\bar{\beta}=\widetilde{\beta}$. As above we show that $\bar{\alpha}=\widetilde{\alpha}$. Notice that if $\ud{u}^{\widetilde{\alpha},\widetilde{\beta}}$ is not periodic, then by Proposition \[pro3.4\] $\ud{u}^{\widetilde{\alpha},\widetilde{\beta}}=\ud{u}$. If $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}$ is not periodic, then by Proposition \[pro3.4bis\] $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}=\ud{v}$. If $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}$ is periodic, then inequalities imply that we must have $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}_*\prec \ud{v}$. Therefore we may have $\ud{u}_*\succ\ud{v}$ and inequalities only if $\ud{u}^{\widetilde{\alpha},\widetilde{\beta}}$ and $\ud{v}^{\widetilde{\alpha},\widetilde{\beta}}$ are periodic. Suppose that it is the case. If $\ud{u}$ is not periodic, then using Proposition \[pro3.5\] the second statement is true. If $\ud{u}$ is periodic, then $\widehat{\ud{u}}=\sigma^p\ud{u}$ for some $p$, whence $\ophi^{\widetilde{\alpha},\widetilde{\beta}}(\sigma^p\ud{u})=1$; by Proposition \[pro3.5\] $\ud{u}\not=\ud{u}^{\widetilde{\alpha},\widetilde{\beta}}$. [xxxxxxxx]{} L. Block, J. Guckenheimer, M. Misiurewicz, L.S. YoungPeriodic points and topological entropy of one dimensional maps, in Global Theory of Dynamical Systems, Eds. Z. Nitecki, C. Robinson, LNM 819, 18-34 (1980). R. BowenOn Axiom A Diffeomorphisms, [*Regional Conference Series in Mathematics*]{} vol. [**35**]{}, A.M.S. (1977). K.M. Brucks, H. BruinTopics from One-Dimensional Dynamics, London Mathematical Society, Student Texts [**62**]{}, Cambridge University Press (2004). P. Collet, J.-P. EckmannIterated maps on the interval as dynamical systems, Birkhäuser, Basel (1980). F. HofbauerMaximal measures for piecewise monotonically increasing transformations on $[0,1]$, in Ergodic Theory, eds. M. Denker and K. Jacobs, Lecture Notes in Mathematics 729, Springer, Berlin, 66-77 (1979). F. HofbauerOn intrinsic ergodicity of piecewise monotonic transformations with positive entropy, [*Is. J. Math.*]{} [**34**]{}, 213-237 (1979). F. HofbauerMaximal Measures for Simple Piecewise Monotonic Transformations, [*Z. Wahrscheinlichkeitstheorie*]{} [**52**]{}, 289-300 (1980). G. KellerMarkov extensions, zeta functions, and fredholm theory for piecewise invertible dynamical systems, [*Trans. Amer. Math. Soc.*]{} [**314**]{}, 433-497 (1989). D. Lind, B. MarcusSymbolic Dynamics and Coding, Cambridge University Press, Cambridge (1995). W. ParryOn the $\beta$-expansions of real numbers, [*Acta Math. Acad. Sci. Hung.*]{} [**11**]{}, 401-416 (1960). W. ParryRepresentations for real numbers, [*Acta Math. Acad. Sci. Hung.*]{} [**15**]{}, 95-105 (1964). A. RényiRepresentations for real numbers and their ergodic properties, [*Acta Math. Acad. Sci. Hung.*]{} [**8**]{}, 477-493 (1957). F. SchweigerErgodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, Clarendon Press, Oxford (1995). [^1]: e-mail: bastien.faller@epfl.ch [^2]: e-mail: charles.pfister@epfl.ch [^3]: \[f2.3\] If we use the intervals $I^\prime_j=[a_j,a_{j+1})$, then we have only right-continuity [^4]: \[f2.4\] If we use the intervals $I^\prime_j=[a_j,a_{j+1})$, this statement is not correct. [^5]: \[f2.limit\] If the subsequence is finite, then $y_\downarrow(\ud{x})$ is the last point of the subsequence. [^6]: If the $\varphi$-expansion is not valid, which happens when $\beta=1$ and $\alpha\in\Q$, then and are not necessarily true, as simple examples show. Hence $\ud{u}^\ab$ and $\ud{v}^\ab$ do not necessarily verify . [^7]: $u_{p-1}^\ab\geq 1$. $u_{p-1}^\ab=0$ if and only if $p=1$ and $\alpha=0$. [^8]: Usually the follower-set is defined as $\cF_{\ud{w}}=\big\{\ud{y}\in\cL\colon \ud{w}\ud{y}\in\cL\big\}$. Since $\cL$ is a dynamical language, i.e. for each $\ud{w}\in\cL$ there exists a letter $e\in\tA$ such that $\ud{w}e\in\cL$, the two definitions agree.

--- abstract: 'Interfacing a ferromagnet with a polarized ferroelectric gate generates a non-uniform, interfacial spin density coupled to the ferroelectric polarization.' author: - Yaojin Li$^1$ - Min Chen$^1$ - Jamal Berakdar$^2$ - 'Chenglong Jia$^{1,2}$' title: 'Gate-controlled magnon-assisted switching of magnetization in ferroelectric/ferromagnetic junctions' --- Electrical control of magnetism has the potential to boost spintronic devices with a number of novel functionalities [@Eerenstein:2006km; @M.; @Weisheit:2007; @T.; @Maruyama:2009; @D.; @Chiba:2011; @Jia:2015iz]. To mention an example, magnetization switching can be achieved via a spin-polarized electric current due to the spin-transfer torque or the spin-orbital torque in the presence of a spin orbital interaction [@J.; @C.; @Slonczewski:1996; @L.; @Berger:1996; @J.; @A.; @Katine:2000; @M.; @D.; @Stiles:2002; @Y.; @Tserkovnyak:2008; @Brataas:2012fb; @Fan:2014hb; @Brataas:2014dla; @Oh:2016ev; @Fukami:2016kq]. One may also use an electric field to manipulate the magnetization dynamics [@Vaz:2012dp; @T.; @Y.; @Liu:2011; @T.; @Nozaki:2012; @Brovko:2014gsb; @Schueler2017; @Matsukura:2015hya; @Y.; @Shiota:2016] in which case the electric field may lead to modulations in the charge carrier density or may affect the magnetic properties such as the magnetic moment, the exchange interaction and/or the magnetic anisotropy [@Vaz:2012dp; @Brovko:2014gsb; @Schueler2017; @Matsukura:2015hya]. Compared to driving magnetization via a spin-polarized current, an electric field governing the magnetization has a clear advantage as it allows for non-volatile device concepts with significantly reduced energy dissipation. On the other hand, an external electric field applied to an itinerant ferromagnet (FM) is shielded by charge accumulation or depletion caused by spin-dependent screening charge that extends on a length scale of only a few angstroms into the FM [@Zhang:1999cx]. This extreme surface confinement of screening hinders its utilization to steer the magnetic dynamics of bulk or a relatively thick nanometer-sized FM [@Shiota:2012dh; @Wang:2012jf]. Experimentally, ultra-thin metallic FM films were thus necessary to observe an electric field influence on the dynamic of an FM [@Vaz:2012dp; @Brovko:2014gsb; @Nan:2014ck]. In this work we show that while the spin-polarized screening charge is surface confined, in the spin channel a local non-uniform spiral spin density builds up at the interface and goes over into the initial uniform (bulk) magnetization away from the interface. Hence, this interfacial spin spiral acts as a topological defect in the initial uniform magnetization vector field. The range of the spiral defect is set by the spin diffusion length $\lambda_m$ [@J.; @Bass:2007] which is much larger than the charge screening length.This spin-spiral constitutes a magnetoelectric effect that has a substantial influence on the traversal magnetization dynamics of FM layers with thickness over tens of nanometers [@footnot]. The interfacial spiral spin density can be viewed as a magnonic accumulation stabilized by the interfacial, spin-dependent charge rearrangement at the contact region between the FM and the ferroelectrics (having the FE polarization ${\mbox{\boldmath$\mathrm{P}$}}$) and by the uniform (bulk) magnetization of FM far away from the interface [@Jia; @C.; @L:2014]. ${\mbox{\boldmath$\mathrm{P}$}}$ responds to an external electric field and so does the magnetic dynamics. As shown below, this magnonic-assisted magnetoelectric coupling arising when using a dielectric FE gate, allows a (ferro)electric field control of the effective driving field that governs the magnetization switching of a FM layer with a thickness on the range of the spin diffusion length $\lambda_m$, which is clearly of an advantage for designing spin-based, non-volatile nanoelectronic devices.\ In Sec. \[sec1\] we discuss the mathematical details of the spin-spiral magnetoelectric coupling, followed by its implementation into the equations of motion for the magnetization dynamics in Sec. \[sec2\]. In Sec. \[sec3\] results of numerical simulations are presented and discussed showing to which extent the spin-spiral magnetoelectric coupling can allow for the electric field control of the magnetization in FE/FM composites. Ways to enhance the effects are discussed and brief conclusions are made in Sec. \[sec4\]. Theoretically, the above magnon accumulation scenario maybe viewed as follows: $$\mathcal{F}_{sd}=J_{sd}\frac{M}{M_{s}}{\mbox{\boldmath$\mathrm{s}$}}\cdot {{\mbox{\boldmath$\mathrm{m}$}}}, \label{eq:sd}$$ Within the Stoner mean-field theory [@Soulen; @R.J:1998] the spin polarization $\eta$ of the electron density in transition FM metals is usually less than 1, $${\mbox{\boldmath$\mathrm{s}$}}={\mbox{\boldmath$\mathrm{s}$}}_{\parallel}+{\mbox{\boldmath$\mathrm{s}$}}_{\perp}$$ where ${\mbox{\boldmath$\mathrm{s}$}}_{\parallel}$ represents the spin density whose direction follows adiabatically the intrinsic magnetization ${\mbox{\boldmath$\mathrm{M}$}}$ at an instantaneous time t. ${\mbox{\boldmath$\mathrm{s}$}}_{\perp}$ describes the transverse deviation from ${\mbox{\boldmath$\mathrm{M}$}}$. Given that the steady-state charge accumulation entails much higher energy processes than spin excitations, $$\begin{aligned} &&\frac{\partial{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}}{\partial t}{\mbox{\boldmath$\mathrm{m}$}}+{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}\frac{\partial{\mbox{\boldmath$\mathrm{m}$}}}{\partial t}+\frac{\partial{\mbox{\boldmath$\mathrm{s}$}}_{\perp}}{\partial t} -D_{0}\nabla^{2}_{z}{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}-D_{0}\nabla^{2}_{z}{\mbox{\boldmath$\mathrm{s}$}}_{\perp} \nonumber \\ &&= -\frac{{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}}{\tau_{sf}}-\frac{{\mbox{\boldmath$\mathrm{s}$}}_{\perp}}{\tau_{sf}}-\frac{{\mbox{\boldmath$\mathrm{s}$}}_{\perp}\times{\mbox{\boldmath$\mathrm{m}$}}}{\tau_{ex}}\end{aligned}$$ where $D_0$ is the diffusion constant and $\tau_{ex} \approx \hbar/(2J_{sd})$. $\tau_{sf}$ is the spin-flip relaxation time due to scattering with impurities, electrons, and phonons; $\tau_{sf}\sim 10^{-12}-10^{-14}$ s [@L.; @Piraux:1998] and $\tau_{ex}/\tau_{sf}\sim10^{-2}$ in typical FM metals [@J.; @Bass:2007]. The time-derivative terms $\frac{\partial{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}}{\partial t}$, $\frac{\partial{\mbox{\boldmath$\mathrm{m}$}}}{\partial t}$ and $\frac{\partial{\mbox{\boldmath$\mathrm{s}$}}_{\perp}}{\partial t}$ below THz are negligible compared with ${\mbox{\boldmath$\mathrm{s}$}}/\tau_{sf}$ and ${\mbox{\boldmath$\mathrm{s}$}}/\tau_{ex}$. Thus the steady state is set by [@Jia; @C.; @L:2014] $$D_{0}\nabla^{2}_{z}{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}=\frac{{\mbox{\boldmath$\mathrm{s}$}}_{\parallel}}{\tau_{sf}} ~~~\text{and}~~~ D_{0}\nabla^{2}_{z}{\mbox{\boldmath$\mathrm{s}$}}_{\perp}=\frac{{\mbox{\boldmath$\mathrm{s}$}}_{\perp}\times{\mbox{\boldmath$\mathrm{m}$}}}{\tau_{ex}},$$ implying an exponentially decaying spiral spin density, [@Jia; @C.; @L:2014] $$\begin{gathered} \label{eq:14} s_{\parallel}=\eta\frac{\sigma_{FM}}{\lambda_{m}e}e^{-z/\lambda_{m}}, \\ {\mbox{\boldmath$\mathrm{s}$}}_{\perp}=(1-\eta)Q_{m}\frac{\sigma_{FM}}{e}e^{-(1-i){\mbox{\boldmath$\mathrm{Q}$}}_{m}\cdot{\mbox{\boldmath$\mathrm{r}$}}}.\end{gathered}$$ Here $\sigma_{FM} = \sigma_{FE} \approx \epsilon_{FE} E$ is the surface charge density due to the electric neutrality constraint at the interface, $\epsilon_{FE}$ and $E$ are the dielectric permittivity of FE and an applied normal electric field, respectively. $\lambda_{m}=\sqrt{D_{0}\tau_{sf}}$ is the effective spin-diffusion length and the normal spin spiral wave vector ${\mbox{\boldmath$\mathrm{Q}$}}_{m}=\frac{1}{\sqrt{2D_{0}\tau_{ex}}}\hat{{\mbox{\boldmath$\mathrm{e}$}}}_{z}$. Clearly, in the presence of the exchange interaction with long-range FM ordering, the accumulated (magnonic) spin density extends in the FM system over a [nanometer]{} characteristic length ($\sim \lambda_m$ being 38 $\pm$ 12 nm in Co [@J.; @Bass:2007]) which is much larger [than]{} the electrostatic screening length (a few angstroms), albeit both are associated by largely different energy scales. As we [are]{} interested in the effect of the low-energy accumulated magnonic density on the spin dynamic in FM we can safely assume that the spin-dependent charge excitations are frozen (because of the higher energy scale) during the (GHz-THz) spin dynamics in the FM. $${\mbox{\boldmath$\mathrm{H}$}}^{\rm{me}}= - \delta \mathcal{F}_{sd}/\delta{{\mbox{\boldmath$\mathrm{M}$}}} = -\frac{J_{\text{sd}}}{M_{s}}{\mbox{\boldmath$\mathrm{s}$}}.$$ We choose nanometer thick layers Co and BaTiO$_3$ as prototypical FM and FE layers for estimating the characteristics of ${\mbox{\boldmath$\mathrm{H}$}}^{\rm me}$. The density of surface charges [@J.; @Hlinka:2006] reads $\sigma_{\text{FE}} = 0.27$ C/$m^{2}$ and the parameters of Co are [@J.; @M.; @D.; @Coey:2010]: $M_{s}=1.44\times10^{6}$ A/m, $K_{1}=4.1\times 10^{5}$ J/$m^{3}$, $\lambda_{m}= 40$ nm [@J.; @Bass:2007], and $\eta=0.45$ [@Soulen; @R.J:1998]. We find thus $|{\mbox{\boldmath$\mathrm{H}$}}^{\rm{me}}| \approx 0.2$ T with [$J_{sd} \approx 0.1$ eV/atom]{} and the FM thickness $d_{FM} = 40$ nm. Such a strong magnetoelectric field is comparable with the uniaxial anisotropic field $\frac{K_{1}}{M_{s}} \approx 0.3$ T of Co. More importantly, note that the non-adiabatical component ${\mbox{\boldmath$\mathrm{H}$}}^{\rm{me}}_{\perp}$ is always perpendicular to the direction of magnetization ${\mbox{\boldmath$\mathrm{M}$}}$, acting as a field-like torque and a damping-like torque at all time (c.f. Fig.\[axis\]), which would play a key role for electric-field assisted magnetization switching. We start from the Landau-Lifshitz-Baryakhtar equation (LLBar) [@V.G.Baryakhtar:1984; @M.Dvornik:2013; @Weiwei; @Wang:2015] for the magnetization dynamics at the FM interface, $$\label{eq:1} \frac{\partial{\mbox{\boldmath$\mathrm{M}$}}}{\partial t}=-\gamma{\mbox{\boldmath$\mathrm{M}$}}\times{\mbox{\boldmath$\mathrm{H}$}}_{\text{eff}}+\hat{\Lambda}_{r}\cdot{\mbox{\boldmath$\mathrm{H}$}}_{\text{eff}}-\hat{\Lambda}_{e,ij}\frac{\partial^{2}{\mbox{\boldmath$\mathrm{H}$}}_{\text{eff}}}{\partial x_{i}\partial x_{j}}$$ where $\gamma$ is the gyromagnetic ratio. The last two terms describe the local and nonlocal relaxations. $\hat{\Lambda}_{r}$ and $\hat{\Lambda}_{e}$ are generally the relaxation tensors of relativistic and exchange natures, respectively. In contrast to the Landau-Lifshitz-Gilbert equation, the LLBar equation does not conserve the magnitude of the magnetization capturing the magnetic relaxations in metals, especially the case for FM metal interfaces. This is necessary in our case to ensure that the local magnetic order which is in equilibrium with the interface region relaxes to the asymptotic bulk magnetization. By introducing ${\mbox{\boldmath$\mathrm{M}$}} = M {{\mbox{\boldmath$\mathrm{m}$}}}$ into the LLBar equation, we infer the following equation for the direction of magnetization [@Weiwei; @Wang:2015], $$\label{eq:6} \frac{\partial{\mbox{\boldmath$\mathrm{m}$}}}{\partial t}=-\gamma{\mbox{\boldmath$\mathrm{m}$}}\times{\mbox{\boldmath$\mathrm{H}$}}_{eff}+\frac{1}{M_{s}}{\mbox{\boldmath$\mathrm{R}$}}_{\perp}$$ with ${\mbox{\boldmath$\mathrm{R}$}} = \lambda_{r}{\mbox{\boldmath$\mathrm{H}$}}_{eff}-\lambda_{e}\nabla^{2}_z{\mbox{\boldmath$\mathrm{H}$}}_{eff}$ and ${\mbox{\boldmath$\mathrm{R}$}}_{\perp}=-{\mbox{\boldmath$\mathrm{m}$}}\times({\mbox{\boldmath$\mathrm{m}$}}\times{\mbox{\boldmath$\mathrm{R}$}}).$ Here $${\mbox{\boldmath$\mathrm{H}$}}_{eff} = {\mbox{\boldmath$\mathrm{H}$}}_{eff}^0 + {\mbox{\boldmath$\mathrm{H}$}}^{\rm me}$$ is the effective magnetic field, in which ${\mbox{\boldmath$\mathrm{H}$}}_{eff}^0 $ follows from the functional derivative of the free energy density via [@Sukhov:2014] $$\begin{aligned} {\mbox{\boldmath$\mathrm{H}$}}_{eff}^0& =& -\delta \mathcal{F}_{0}/\delta {\mbox{\boldmath$\mathrm{M}$}}, \nonumber \\ \mathcal{F}_{0}&=&-K_{1}(\sin^{2}\theta\cos^{2}\phi\sin^{2}\theta_{u}+\cos^{2}\theta\cos^{2}\theta_{u}) \nonumber \\ & &-\frac{K_{1}}{2}\sin2\theta\sin2\theta_{u}\cos\phi-(K_{s}/{d_{FM}}-\mu_{0}M^{2}_{s}/2)\cos^{2}\theta \nonumber \\ & &-{\mbox{\boldmath$\mathrm{M}$}}\cdot{\mbox{\boldmath$\mathrm{B}$}}. \label{eq:F0}\end{aligned}$$ $K_{1}$ is the uniaxial magnetocrystalline anisotropy energy, $K_{s}$ is the magnetic surface anisotropy contribution which is significant for relatively thin magnetic film and favors magnetization out of the $xy$ plane. $\mu_{0}M^{2}_{s}$ denotes the demagnetizing field contribution, which favors a magnetization in plane. ${\mbox{\boldmath$\mathrm{M}$}}\cdot{\mbox{\boldmath$\mathrm{B}$}}$ is the Zeemann interaction and $\theta_{u}$ is the tilted angle of the easy axis from the $z$ direction. Clearly, the non-uniform effective field $ {\mbox{\boldmath$\mathrm{H}$}}^{\rm me}$ due to the *s-d* interaction with the exponentially decaying spiral spins would give rise to nonlocal damping of the magnetization dynamics. Considering that the contribution of the induced spin density to [the spatial distribution of]{} local ferromagnetic moments is small, we have $$\langle \nabla^{2}_{z}{\mbox{\boldmath$\mathrm{s}$}}_{\perp}\rangle=2Q^{2}_{m}\left( \langle s^{\phi}_{\perp}\rangle \hat{{\mbox{\boldmath$\mathrm{e}$}}}_{\theta}-\langle s^{\theta}_{\perp}\rangle \hat{{\mbox{\boldmath$\mathrm{e}$}}}_{\phi} \right).$$ Without loss of generality one can take $\langle s^{\phi}_{\perp}\rangle =\langle s^{\theta}_{\perp}\rangle =\frac{1}{\sqrt{2}}\langle s_{\perp}\rangle $. It is also convenient to redefine some dimensionless parameters which are $\tilde{d}_{FM}=\frac{d_{FM}}{\lambda_{m}}$, $\tilde{t}$= t$\gamma$T$\approx$ 28t GHz, and $\tilde{J}_{sd}=\frac{J_{sd}}{\rm eV}\frac{\sigma_{FM}}{P_{s}}\frac{\lambda_{m}}{d_{FM}}$ with the FE spontaneous polarization $P_s$. In the following $\tilde{J}_{sd}$ is taken as an adjustable parameter in view of ferroelectric tuning of magnetoelectric field $ {\mbox{\boldmath$\mathrm{H}$}}^{\rm me}$. For the surface anisotropy $K_{s}\approx10^{-3}$ J/$m^{2}$ and $\mu_{0}M^{2}_{s}/2 \approx$ 1.3$\times10^{6}$ J/$m^{3}$ of Co sample [@J.; @M.; @D.; @Coey:2010], the dominant contribution of the anisotropic term $(K_{s}/{d_{FM}}-\mu_{0}M^{2}_{s}/2)$ in Eq.(\[eq:F0\]) has the form either $K_{s}/{d_{FM}}$, or $-\mu_{0}M^{2}_{s}/2$ depending on the thickness $d_{FM}$, i.e., the magnetization will be either normal to the FM interface ($\theta_u =0$) or in the interface plane ($\theta_u = \pi/2$). *Case I*: Normal FM magnetization with $\theta_u =0$: The free energy density is $$\mathcal{F}_{0}=-K_{eff}\cos^{2}\theta-{\mbox{\boldmath$\mathrm{M}$}}\cdot{\mbox{\boldmath$\mathrm{B}$}},\quad K_{eff}= K_1 + \frac{K_s}{d_{FM}} - \frac{\mu_0 M_s^2}{2}$$ which leads to $${\mbox{\boldmath$\mathrm{H}$}}^{0}_{eff}=\frac{2K_{eff}}{M_{s}}\cos\theta \hat{{\mbox{\boldmath$\mathrm{e}$}}}_z$$ without an applied magnetic field ${\mbox{\boldmath$\mathrm{B}$}}$. The LLBar equation reads then, $$\begin{aligned} & \frac{\partial\theta}{\partial \tilde{t}}=\frac{\gamma^{e}_{+}}{\sqrt{2}} {H}^{\rm me}_{\perp}-\alpha \frac{K_{eff}}{M_{s}}\sin2\theta, \\ &\sin\theta\frac{\partial\phi}{\partial \tilde{t}}=-\frac{\gamma^{e}_{-}}{\sqrt{2}} {H}^{\rm me}_{\perp}+\frac{K_{eff}}{M_{s}}\sin2\theta\end{aligned}$$ [with ]{} $\gamma^{e}_{\pm}=1 - 2Q^{2}_{m}\frac{\lambda_{e}}{\gamma M_{s}}\pm \frac{\lambda_{r}}{\gamma M_{s}}= \gamma^{e}\pm \alpha$ . Clearly, under a weak interfacial ME field, the condition $${H}^{\rm me}_{\perp} = \sqrt{2} \frac{\alpha}{\gamma^{e}_{+}} \frac{K_{eff}}{M_{s}}\sin2\theta$$ can be satisfied, the polar angle $\theta$ ends up processionally in the equilibrium state \[c.f. Fig.\[fig2\](a) with ${\partial\theta}/{\partial \tilde{t}}=0$\]. Otherwise, the strong transversal field ${H}^{\rm me}_{\perp}$ results in a magnetization flip over the normal $\hat{{\mbox{\boldmath$\mathrm{e}$}}}_z$-direction \[Fig.\[fig2\](b)\]. Considering that the ME field depends linearly on the applied electric field and the reciprocal of FM thickness, one would expect a transition from the magnetization procession around the z axis (for a small electric field $E$ and/or relatively thick FM layers) to the magnetization [flip over]{} the normal [direction]{} (for a strong electric field and/or ultra-thin FM film) at the critical points, as demonstrated in Figs. \[fig2\](c) and \[fig2\](d). *Case II*: In-plane magnetization with $\theta_u = \pi/2$: Disregarding the surface anisotropy ($K_{s}/d_{FM} \ll \mu_{0}M^{2}_{s}/2$) for a thick FM film, the effective magnetic field reads $${\mbox{\boldmath$\mathrm{H}$}}^{0}_{eff}=2K_{1}/M_{s} \sin\theta\cos\phi \hat{{\mbox{\boldmath$\mathrm{e}$}}}_{x} -\mu_{0}M_{s}\cos\theta\hat{{\mbox{\boldmath$\mathrm{e}$}}}_{z}+{\mbox{\boldmath$\mathrm{B}$}}$$ and the magnetization favors an in-plane $\hat{{\mbox{\boldmath$\mathrm{e}$}}}_x$ axis, which means $\phi(0) =0$ with the external magnetic field ${\mbox{\boldmath$\mathrm{B}$}} =0$. Upon some simplifications the LLBar equation reads $$\begin{aligned} \frac{\partial\theta}{\partial \tilde{t}}&=&\frac{\gamma^{e}_{+}}{\sqrt{2}}{H}^{\rm me}_{\perp}+\alpha \frac{\mu_{0}M_{s}}{2}\sin2\theta + \alpha \frac{K_{1}}{M_{s}}\sin2\theta\cos^{2}\phi \nonumber \\ &-& \frac{K_{1}}{M_{s}}\sin\theta\sin2\phi + \alpha B\cos\theta\cos\phi- B\sin\phi, \\ \sin\theta\frac{\partial\phi}{\partial \tilde{t}}&=&-\frac{\gamma^{e}_{-}}{\sqrt{2}}{H}^{\rm me}_{\perp}-\frac{\mu_{0}M_{s}}{2}\sin2\theta - \alpha \frac{K_{1}}{M_{s}}\sin\theta\sin2\phi \nonumber \\ &-& \frac{K_{1}}{M_{s}}\sin2\theta\cos^{2}\phi -B\cos\theta\cos\phi- \alpha B\sin\phi.\end{aligned}$$ In the absence of external magnetic field ${\mbox{\boldmath$\mathrm{B}$}}$, the magnetization dynamics is determined by three parameters: $\alpha$, $H^{\rm me}_{\perp}$, and $K_1/M_s$. Firstly, let us ignore the damping terms for small Gilbert damping coefficient $\alpha$, the weak ME field $H^{\rm me}_{\perp}$ would satisfy $\frac{\partial\theta}{\partial \tilde{t}}=0$ and $\frac{\partial\phi}{\partial \tilde{t}} =0$, resulting in a relocation of the magnetization with an equilibrium tilted angle [in the vicinity of $x$ axis]{}, as shown in Fig.\[f3\](a). However, when $H^{\rm me}_{\perp}$ is stronger than the anisotropic field $K_1/M_s$ and the demagnetization field $\mu_0 M_s$, no solutions exist for $\partial \theta/\partial \tilde{t} =0$ at all time, the magnetization possesses [a $z$]{}-axial [flip]{} mode in the [whole spin space]{} \[c.f. Fig.\[f3\](b)\] similar to the case of normal FM magnetization. On the other hand, after accounting for terms containing $\alpha$ in the LLBar equations, we would have additional magnetization rotation around the $z$ axis \[Fig.\[f3\](c)\]. Further insight into the detailed characterization of magnetization dynamics is delivered by numerics for a varying strength of the ME field $\tilde{J}_{sd}$ and the uniaxial anisotropy $K_1/M_s$ in Fig.\[fig7A1\] with $\alpha=0.1$. There are two new phases, the [$z$-axial flip]{} mode and the $z$-axial rotational mode, which were unobserved in the FM systems in the absence of interface ME interaction. With decreasing the damping $\alpha$, the area of the $z$-axial rotational mode shrinks vanishing eventually. By applying an external magnetic field ${\mbox{\boldmath$\mathrm{B}$}}$ along the $x$-direction, only slight modifications are found in the phase diagram. However, the initial azimuthal angle $\phi(0)$ deviates from the easy axis with a rotating magnetic field ${\mbox{\boldmath$\mathrm{B}$}}$ in the $xy$ interface plane. Considering the LLBar equations with the initial condition $\theta(t=0) = \pi/2$, we have $$\frac{\partial\theta}{\partial \tilde{t}}|_{\tilde{t}=0} \approx \frac{\gamma^{e}_{+}}{\sqrt{2}}{\mbox{\boldmath$\mathrm{H}$}}^{s-d}_{\perp}-\frac{K_{1}}{M_{s}}\sin2\phi(0)$$ with a small damping $\alpha$. As the dynamic equation is sensitive to the initial azimuthal angle $\phi(0)$, the calculations show that the magnetization dynamics may change between the processional mode around the $x$ axis and the [$z$-axial flip or $z$-axial rotational]{} mode, depending on the initial value of $\phi(0)$. 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--- abstract: 'Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely conjectured that $M_n$ is singular with probability at most $(2+o(1))^{-n}$. On the other hand, the best known upper bound on the singularity probability of $M_n$, due to Vershynin (2011), is $2^{-n^c}$, for some unspecified small constant $c > 0$. This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of $M_n$ is at most $2^{-n^{1/4}\sqrt{\log{n}}/1000}$ for all sufficiently large $n$. The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.' author: - 'Asaf Ferber [^1]' - 'Vishesh Jain[^2]' bibliography: - 'symmetric.bib' title: 'Singularity of random symmetric matrices – a combinatorial approach to improved bounds' --- § ¶ Ł Introduction ============ The invertibility problem for Bernoulli matrices is one of the most outstanding problems in discrete random matrix theory. Letting $A_n$ denote a random $n\times n$ matrix, whose entries are independent and identically distributed (i.i.d.) Bernoulli random variables which take values $\pm 1$ with probability $1/2$ each, this problem asks for the value of $c_n$, which is the probability that $A_n$ is singular. By considering the event that two rows or two columns of $A_n$ are equal (up to a sign), it is clear that $$c_n \geq (1+o(1))n^{2}2^{1-n}.$$ It has been widely conjectured that this bound is, in fact, tight. On the other hand, perhaps surprisingly, it is non-trivial even to show that $c_n$ tends to $0$ as $n$ goes to infinity; this was accomplished in a classical work of Komlós in 1967 [@komlos1967determinant] which showed that $$c_n = O\left(n^{-1/2}\right)$$ using the classical Erdős-Littlewood-Offord anti-concentration inequality. Subsequently, a breakthrough result due to Kahn, Komlós, and Szemerédi in 1995 [@kahn1995probability] showed that $$c_n = O(0.999^{n}).$$ Improving upon an intermediate result by Tao and Vu [@tao2007singularity], the current ‘world record’ is $$c_n \leq (2+o(1))^{-n/2},$$ due to Bourgain, Vu, and Wood [@bourgain2010singularity]. Another widely studied model of random matrices is that of random *symmetric* matrices; apart from being important for applications, it is also very interesting from a technical perspective as it is one of the simplest models with nontrivial correlations between its entries. Formally, let $M_n$ denote a random $n\times n$ symmetric matrix, whose upper-diagonal entries are i.i.d. Bernoulli random variables which take values $\pm 1$ with probability $1/2$ each, and let $q_n$ denote the probability that $M_n$ is singular. Despite its similarity to $c_n$, much less is known about $q_n$. The problem of whether $q_n$ tends to $0$ as $n$ goes to infinity was first posed by Weiss in the early 1990s and only settled in 2005 by Costello, Tao, and Vu [@costello2006random], who showed that $$q_n = O(n^{-1/8 + o(1)}).$$ In order to do this, they introduced and studied a quadratic variant of the Erdős-Littlewood-Offord inequality. Subsequently, Nguyen [@nguyen2012inverse] developed a quadratic variant of *inverse* Littlewood-Offord theory to show that $$q_n = O_{C}(n^{-C})$$ for any $C>0$, where the implicit constant in $O_{C}(\cdot)$ depends only on $C$. This so-called quadratic inverse Littlewood-Offord theorem in [@nguyen2012inverse] builds on previous work of Nguyen and Vu [@nguyen2011optimal], which is itself based on deep Freiman-type theorems in additive combinatorics (see [@tao2008john] and the references therein). The current best known upper bound on $q_n$ is due to Vershynin [@vershynin2014invertibility], who used a sophisticated and technical geometric framework pioneered by Rudelson and Vershynin [@rudelson2008littlewood; @rudelson2010non] to show that $$q_n = O(2^{-n^c})$$ for some unspecified small constant $c > 0$. As far as lower bounds on $q_n$ are concerned, once again, by considering the event that the first and last rows of $M_n$ are equal (up to a sign), we see that $q_n \geq (2+o(1))^{-n}$. It is commonly believed that this lower bound is tight. \[conjecture:prob-singularity\] We have $$q_n = (2+o(1))^{-n}.$$ In this paper, we obtain a much stronger upper bound on $q_n$, thereby making progress towards \[conjecture:prob-singularity\]. \[thm:main-thm\] There exists a natural number $n_0 \in \N$ such that for all $n\geq n_0$, $$q_n \leq 2^{-n^{1/4}\sqrt{\log{n}}/1000}.$$ Apart from providing a stronger conclusion, our proof of the above theorem is considerably shorter than previous works, and introduces and extends several novel combinatorial tools and ideas in discrete random matrix theory (some of which are based on joint work of the authors with Luh and Samotij [@FJLS2018]). We believe that these ideas allow for a unified approach to the singularity problem for many different discrete random matrix models, which have previously been handled in an ad-hoc manner. For completeness and for the convenience of the reader, we have included full proofs of all the simple background lemmas that we use from other papers, making this paper completely self contained. Outline of the proof and comparison with previous work ------------------------------------------------------ In this subsection, we provide a very brief, and rather imprecise, outline of our proof, and compare it to previous works of Nguyen [@nguyen2012inverse] and Vershynin [@vershynin2014invertibility]; for further comparison with the work of Costello, Tao, and Vu, see [@nguyen2012inverse]. Let $x:=(x_1,\ldots,x_n)$ be the first row of $M_n$, let $M^{1}_{n-1}$ denote the bottom-right $(n-1)\times (n-1)$ submatrix of $M_n$, and for $2\leq i,j \leq n$, let $c_{ij}$ denote the cofactor of $M^{1}_{n-1}$ obtained by removing its $(i-1)^{st}$ row and $(j-1)^{st}$ column. Then, Laplace’s formula for the determinant gives $$\det(M_n)=x_1\det(M_{n-1})-\sum_{i,j=2}^n c_{ij}x_ix_j,$$ so that our goal is to bound the probability (over the randomness of $x$ and $c_{ij}$) that this polynomial is zero. By a standard reduction due to [@costello2006random] (see \[lem:rank reduction,lem:second reduction,corollary:remove-first-row\]), we may further assume that $M^{1}_{n-1}$ has rank either $n-2$ or $n-1$. In this outline, we will only discuss the case when $M^{1}_{n-1}$ has rank $n-1$; the other case is easier, and is handled exactly as in [@nguyen2012inverse] (see \[lemma:reduction-to-linear,eqn:conclusion-degenerate-case\]). A decoupling argument due to [@costello2006random] (see \[lemma:decoupling-CTV\]) further reduces the problem (albeit in a manner incurring a loss) to bounding from above the probability that $$\sum_{i\in U_1}\sum_{j \in U_2}c_{ij}(x_i - x_i')(x_j - x_j')=0,$$ where $U_1 \sqcup U_2 $ is an arbitrary non-trivial partition of $[n-1]$, and $x_i', x_j'$ are independent copies of $x_i, x_j$ (see \[corollary:decoupling-conclusion\]). For the remainder of this discussion, the reader should think of $|U_2|$ as ‘small’(more precisely, $|U_2| \sim n^{1/4}\sqrt{\log{n}}$). We remark that a similar decoupling based reduction is used in [@vershynin2014invertibility] as well, whereas [@nguyen2012inverse] also uses a similar decoupling inequality in proving the so-called quadratic inverse Littlewood-Offord theorem. The advantage of decoupling is that for any given realization of the variables $(c_{ij})_{2\leq i,j \leq n}$ and $(x_j - x_j')_{j\in U_2}$, the problem reduces to bounding from above the probability that the *linear sum* $$\sum_{i\in U_1}R_i(x_i - x_i')=0,$$ where $R_i:= \sum_{j \in U_2}c_{ij}(x_j - x_j')$. Problems of this form are precisely the subject of standard (linear) Littlewood-Offord theory. Broadly speaking, Littlewood-Offord theory applied to our problem says that the less ‘additive structure’ the vector $(R_1,\dots,R_{|U_1|})$ possesses, the smaller the probability of the above sum being zero. Quantifying this in the form of ‘Littlewood-Offord type theorems’ has been the subject of considerable research over the years; we refer the reader to [@nguyen2013small; @rudelson2010non] for general surveys on the Littlewood-Offord problem with a view towards random matrix theory. Hence, our goal is to show that with very high probability, the vector $(R_1,\dots,R_{|U_1|})$ is additively ‘very unstructured’. This is the content of our structural theorem (\[thm:structural\]), which is at the heart of our proof. The statement (and usefulness) of our structural theorem is based on the following simple, yet powerful, observations. - The $(n-1)$-dimensional vector $R:= \sum_{j\in U_2}c_{ij}(x_j-x_j')$ is zero if and only if $x_j = x_j'$ for all $j \in |U_2|$, which happens with probability exponentially small in $|U_2|$; the if and only if statement holds since the matrix $(c_{ij})_{2\leq i,j \leq n}$ is proportional to the matrix $(M^{1}_{n-1})^{-1}$, which is assumed to be invertible. - The vector $R$ is orthogonal to at least $n-1-|U_2|$ rows of $M^{1}_{n-1}$ (\[lemma:R-orthogonal\]). This follows since for every $j$, the $n-1$ dimensional vector $(c_{ij})_{2\leq i\leq n}$ is orthogonal to all but the $j^{th}$ row of $M^{1}_{n-1}$, again since the matrix $(c_{i,j})_{2\leq ij \leq n}$ is proportional to the matrix $(M^{1}_{n-1})^{-1}$. - The probability of the linear sum $\sum_{i \in U_1} R_i(x_i - x_i')$ being zero is ‘not much more’ than the probability of the linear sum $\sum_{2\leq i \leq n} R_i (x_i - x_i')$ being zero (\[lemma:restriction-atom-prob\]). Taken together, these observations show that it suffices to prove a structural theorem of the following form: *every* non-zero integer vector which is orthogonal to ‘most’ rows of $M^{1}_{n-1}$ is ‘very unstructured’. In [@nguyen2012inverse], a structural theorem along similar lines is also proven. However, it suffers from two drawbacks. First, the notion of ‘very unstructured’ in the conclusion there is much weaker, leading to the bound $O_C(n^{-C})$ for any constant $C>0$, as opposed to our bound from \[thm:main-thm\]. Second, such a conclusion is not obtained for every non-zero integer vector, but only for those non-zero integer vectors for which ‘most’ coefficients satisfy the additional additive constraint of being contained in a ‘small’ generalized arithmetic progression (GAP) of ‘low complexity’. Consequently, the simple observations mentioned above no longer suffice, and the rest of the proof in [@nguyen2012inverse] is necessarily more complicated. The structural theorem in [@vershynin2014invertibility] is perhaps closer in spirit to ours, although there are many key differences, of which we mention here the most important one. Roughly speaking, both [@vershynin2014invertibility] and the present work prove the respective structural theorems by taking the union bound, over the choice of a non-zero (integer) vector which is not ‘very unstructured’, that the matrix-vector product of $M^{1}_{n-1}$ with this vector is contained in a small prescribed set. *A priori*, this union bound is over an infinite collection of vectors. In order to overcome this obstacle, [@rudelson2008littlewood; @vershynin2014invertibility] adopts a geometric approach of grouping vectors on the unit sphere into a finite number of clusters based on Euclidean distances; using the union bound and a non-trivial estimate of the number of clusters to show that with very high probability, the matrix-vector product of $M^{1}_{n-1}$ with a representative of each cluster is ‘far’ from the small prescribed set; and then, using estimates on the operator norm of $M^{1}_{n-1}$ to deduce a similar result for all other vectors in each cluster. Naturally, this geometric approach is very involved, and leads to additional losses at various steps (which is why [@vershynin2014invertibility] obtains a worse bound on $q_n$ than \[thm:main-thm\]); however, it is worth mentioning that [@vershynin2014invertibility] also provides bounds not just for the probability of singularity of $M_{n}$, but also for the probability that the ‘least singular value’ of $M_{n}$ is ‘very small’. In contrast, we overcome this obstacle with a completely novel and purely combinatorial approach of clustering vectors based on the residues of their coordinates modulo a large prime, and using a combinatorial notion due to Halász [@halasz1977estimates] to quantify the amount of additive structure in a vector (\[prop:structural\]). In particular, with our approach, the analogue of the problem of ‘bounding the covering number of sub-level sets of regularized LCD’ – which constitutes a significant portion of [@vershynin2014invertibility] (see Section 7.1 there), is one of the key contributions of that work, and is also a major contributor to the sub-optimality of the final result – can be solved more efficiently and with a short double-counting argument (see \[thm:counting-lemma\], which is based on joint work of the authors with Luh and Samotij in [@FJLS2018], and \[corollary:counting\]). The rest of this paper is organized as follows. In \[sec:proof-strategy\], we discuss in detail the overall proof strategy leading to the reduction to the structural theorem; in \[sec:structural theorem\], we state and prove our structural theorem; and in \[sec:proof-main-thm\], we put everything together to quickly complete our proof. \[app:counting-lemma\] reproduces the proof of the ‘counting lemma’ from [@FJLS2018], and \[app:halasz\] contains a proof of Halász’s inequality over $\F_p$, which follows the outline of the original proof of Halász [@halasz1977estimates].\ [**Notation:** ]{} Throughout the paper, we will omit floors and ceilings when they make no essential difference. For convenience, we will also say ‘let $p = x$ be a prime’, to mean that $p$ is a prime between $x$ and $2x$; again, this makes no difference to our arguments. As is standard, we will use $[n]$ to denote the discrete interval $\{1,\dots,n\}$. All logarithms are natural unless noted otherwise. Proof strategy: reduction to the structural theorem {#sec:proof-strategy} =================================================== In this section, we discuss the strategy underlying our proof of \[thm:main-thm\]. The key conclusions are \[eqn:split-into-deg-nondeg\] \[eqn:conclusion-degenerate-case\], and \[eqn:conclusion-nondegerate\], which show that it suffices to prove the structural theorem in \[sec:structural theorem\] in order to prove \[thm:main-thm\]. Preliminary reductions ---------------------- For any $n \in \N$ and $k\in [n]$, let $\Ev_{k}(n)$ denote the event that $M_n$ has rank exactly $k$, and let $\Ev_{\leq k}(n)$ denote the event that $M_n$ has rank at most $k$. Thus, our goal is to bound the probability of $\Ev_{\leq n-1}(n)$. The next lemma, which is due to Nguyen [@nguyen2012inverse], shows that it suffices to bound the probability of $\Ev_{n-1}(n)$. \[lem:rank reduction\] For any $\ell \in [n-2]$, $$\Pr\left[\Ev_{\ell}(n)\right] \leq 0.1 \times \Pr\left[\Ev_{2n-\ell-2}(2n-\ell-1)\right].$$ The proof of this lemma uses the following simple observation due to Odlyzko [@Odl]: \[obs:odl\] Let $V$ be any subspace of $\mathbb{R}^n$ of dimension at most $\ell$. Then, $|V\cap \{\pm 1\}^n|\leq 2^\ell$. \[Proof of \[lem:rank reduction\]\] It suffices to show that for any $\ell \leq n-2$, $$\label{eqn:rank-n+1} \Pr\left[\Ev_{\ell+2}(n+1) \mid \Ev_{\ell}(n)\right]\geq 1-2^{-n+\ell}.$$ Indeed, iterating this equation shows that $$\begin{aligned} \Pr[\Ev_{2n-\ell-2}(2n-\ell-1)\mid \Ev_{\ell}(n)] & \geq \prod_{j=1}^{n-\ell -1}\Pr\left[\Ev_{\ell + 2j}(n+j) \mid \Ev_{\ell + 2j-2}(n+j-1)\right] \\ & \geq \prod_{j=1}^{n-\ell-1}(1-2^{-n+\ell+j}) \geq 0.1,\end{aligned}$$ which gives the desired conclusion. In order to prove \[eqn:rank-n+1\], consider the coupling of $M_n$ and $M_{n+1}$ where $M_n$ is the top left $n\times n$ sub-matrix of $M_{n+1}$. Suppose $M_n$ has rank $\ell$, and let $V(M_n)$ be the ($\ell$-dimensional) subspace spanned by its rows. By \[obs:odl\], $|V(M_n)\cap \{\pm 1\}^{n}|\leq 2^\ell$. Therefore, the probability that the vector formed by the first $n$ coordinates of the last row of $M_{n+1}$ lies in $V(M_n)$ is at most $2^{-n+\ell}$. If this vector does not lie in $V(M_n)$, then the symmetry of the matrix also shows that the last column of $M_{n+1}$ does not lie in the span of the first $n$ columns of $M_{n+1}$, so that the rank of $M_{n+1}$ exceeds the rank of $M_{n}$ by $2$. The following lemma, also due to Nguyen, allows us to reduce to the case where the rank of the $(n-1) \times (n-1)$ symmetric matrix obtained by removing the first row and the first column of $M_n$ is at least $n-2$. \[lem:second reduction\] Assume that $M_{n}$ has rank $n-1$. Then, there exists $i\in[n]$ such that the removal of the $i^{th}$ row and the $i^{th}$ column of $M_{n}$ results in a symmetric matrix $M_{n-1}$ of rank at least $n-2$. Without loss of generality, we can assume that the last $n-1$ rows of $M_n$ are independent. Therefore, the matrix $M_{n-1}$, which is obtained by removing the first row and first column of $M_n$ has rank at least $n-2$. As a simple corollary of the above lemma, we obtain the following: \[corollary:remove-first-row\] For $i \in [n]$, let $\Ev^{i}_{n-1}(n)$ denote the event that $M_n$ has rank $n-1$, and the symmetric matrix obtained by removing the $i^{th}$ row and the $i^{th}$ column of $M_n$ has rank at least $n-2$. Then, $$\Pr\left[\Ev_{n-1}(n)\right] \leq n\Pr\left[\Ev^{1}_{n-1}(n)\right].$$ Suppose that $M_n$ has rank $n-1$. By \[lem:second reduction\], there exists an $i\in [n]$ for which the $(n-1)\times (n-1)$ matrix obtained by deleting the $i^{th}$ row and $i^{th}$ column has rank at least $n-2$. Moreover, by symmetry, $$\Pr[\Ev^i_{n-1}(n)]=\Pr[\Ev^1_{n-1}(n)] \text{ for all }i\in [n].$$ Therefore, by the union bound, $$\Pr[\Ev_{n-1}(n)]= \Pr\left[\cup_{i=1}^{n}\Ev^{i}_{n-1}(n)\right] \leq \sum_{i=1}^n\Pr[\Ev^i_{n-1}(n)]=n\Pr[\Ev_{n-1}^1(n)].$$ Let $M^{1}_{n-1}$ denote the $(n-1)\times (n-1)$ symmetric matrix obtained by deleting the first row and first column of $M_n$. Let $\Dg(n-1)$ denote the ‘degenerate’ event that $M^{1}_{n-1}$ has rank $n-2$, and let $\Ndg(n-1)$ denote the ‘non-degenerate’ event that $M^{1}_{n-1}$ has full rank $n-1$. By definition, $$\Ev^{1}_{n-1}(n) = \left(\Ev^{1}_{n-1}(n)\cap\Dg(n-1)\right) \sqcup \left(\Ev^{1}_{n-1}(n)\cap \Ndg(n-1)\right),$$ and hence, $$\label{eqn:split-into-deg-nondeg} \Pr\left[\Ev^{1}_{n-1}(n)\right] = \Pr\left[\Ev^{1}_{n-1}(n)\cap\Dg(n-1)\right]+ \Pr\left[\Ev^{1}_{n-1}(n)\cap\Ndg(n-1)\right].$$ It is thus enough to bound each of the above two summands. Bounding $\Pr\left[\Ev^{1}_{n-1}(n)\cap\Dg(n-1)\right]$ {#sec:degenerate-case} ------------------------------------------------------- Let $x := (x_1,\dots,x_n)$ denote the first row of $M_n$. It follows from Laplace’s formula for the determinant that $$\label{eqn:laplace-expansion} \det(M_n) = x_1\det\left(M^{1}_{n-1}\right) - \sum_{2\leq i,j \leq n}c_{ij}x_i x_j,$$ where $c_{ij}$ denotes the cofactor of $M^{1}_{n-1}$ obtained by removing its $(i-1)^{st}$ row and $(j-1)^{st}$ column. In order to deal with $M_n \in \Ev^{1}_{n-1}(n)\cap\Dg(n-1)$, we use the following observation due to Nguyen (see Section 9 in [@nguyen2012inverse]). \[lemma:reduction-to-linear\] For every $M_n \in \Ev^{1}_{n-1}(n)\cap\Dg(n-1)$, there exists some $\lambda:=\lambda\left(M^{1}_{n-1}\right) \in \Q\setminus\{0\}$ and some $a:= a\left(M^{1}_{n-1}\right)=(a_2,\dots,a_n)\in \Z^{n-1}\setminus\{\operatorname{\textbf{0}}\}$ such that $$\label{eqn:conclusion-1} M^{1}_{n-1} a =0,$$ and $$\label{eqn:conclusion-2} \det(M_n) = \lambda \left(\sum_{2\leq i \leq n} a_ix_i\right)^{2}.$$ Let $\operatorname{adj}\left(M^{1}_{n-1}\right)$ denote the adjugate matrix of $M^{1}_{n-1}$; note that this is an integer-valued symmetric matrix since $M^{1}_{n-1}$ is an integer-valued symmetric matrix. Since $M^{1}_{n-1}$ is of rank $n-2$, its kernel is of rank $1$. Moreover, the equation $$\label{eqn: A adj(A) = det(A)I} M^{1}_{n-1} \operatorname{adj}\left(M^{1}_{n-1}\right) = \det\left(M^{1}_{n-1}\right)I_{n-1}$$ shows that every column of $\operatorname{adj}\left(M^{1}_{n-1}\right)$ is in the kernel of $M^{1}_{n-1}$ as $\det(M^{1}_{n-1}) = 0$ by assumption. It follows that the matrix $\operatorname{adj}\left(M^{1}_{n-1}\right)$ is an integer-valued symmetric matrix of rank $1$, which cannot be zero since $M^{1}_{n-1}$ is of rank $n-2$. Hence, there exists some $\lambda \in \Q\setminus \{0\}$ and a vector $a = (a_2,\dots, a_n)^{T} \in \Z^{n-1}\setminus \{\operatorname{\textbf{0}}\}$ such that $$\label{eqn:rank-1-symmetric-adjoint} \operatorname{adj}\left(M^{1}_{n-1}\right) = \lambda a a^{T}.$$ In particular, every column of $\operatorname{adj}\left(M^{1}_{n-1}\right)$ is equal to a multiple of the vector $a$. By considering any column which is a non-zero multiple of $a$, \[eqn: A adj(A) = det(A)I\] along with $\det\left(M^{1}_{n-1}\right) = 0$ gives \[eqn:conclusion-1\]. Moreover, by writing the entries of the adjugate matrix in terms of the cofactors, we see that \[eqn:rank-1-symmetric-adjoint\] is equivalent to the following: for all $2 \leq i,j \leq n$: $$c_{ij} =\lambda a_i a_j. $$ Substituting this in \[eqn:laplace-expansion\] and using $\det\left(M^{1}_{n-1}\right)=0$ gives \[eqn:conclusion-2\]. Before explaining how to use \[lemma:reduction-to-linear\], we need the following definition. \[defn:atom-prob\] Let $\ring$ be an arbitrary ring (with a unit element). For a vector $a := (a_1,\dots,a_n) \in \ring^{n}$, we define its $\mu$-atom probability by $$\rho^{\ring}_{\mu}(a) := \sup_{c \in \ring}\Pr_{x_1^{\mu},\dots,x_n{\mu}}\left[a_1 x_1^{\mu} + \dots + a_n x_n^{\mu} = c\right],$$ where the $x_i^{\mu}$’s are i.i.d. random variables taking on the value $0$ with probability $\mu$ and the values $\pm 1$, each with probability $(1-\mu)/2$. We will often refer to the $0$-atom probability simply as the atom probability, and denote it by $\rho^{\ring}(a)$ instead of $\rho^{\ring}_{0}(a)$. Similarly, we will denote $x_i^0$ simply as $x_i$. Although we will not need them in this subsection, we will later make use of the following two simple lemmas about the atom probability. The first lemma shows that the $\mu$-atom probability of a vector is bounded above by the $\mu$-atom probability of any of its restrictions. \[lemma:sbp-monotonicity\] Let $a \in \ring^{n}$, and let $a|_{U_1}$ denote the restriction of $a$ to $U_1 \subseteq [n]$. Then, $$\rho^{\ring}_{\mu}\left(a\right) \leq \rho^{\ring}_{\mu}\left(a|_{U_1}\right).$$ Let $c^{*}:=\arg\max_{c \in \ring}\Pr_{x^{\mu}}\left[\sum_{i\in [n]}a_{i}x^{\mu}_{i}=c\right]$. Then, $$\begin{aligned} \rho^{\ring}_{\mu}(a) &= \Pr_{x^{\mu}}\left[\sum_{i\in[n]}a_ix_i^{\mu}=c^{*}\right] = \Pr_{x^{\mu}}\left[\sum_{i\in[U_1]}a_ix_i^{\mu}=c^{*}-\sum_{i\in [\overline{U_1}]}a_ix_i^{\mu}\right]\\ &= \E_{(x^{\mu}_i)_{i\in \overline{U_1}}}\left[\Pr_{(x_i^{\mu})_{i\in [U_1]}}\left[\sum_{i\in[U_1]}a_ix_i^{\mu}=c^{*}-\sum_{i\in [\overline{U_1}]}a_ix_i^{\mu}\right]\right]\\ &\leq \E_{(x^{\mu}_i)_{i\in \overline{U_1}}}\left[\rho^{\ring}_{\mu}(a|_{U_1})\right] = \rho^{\ring}_{\mu}(a|_{U_1}),\end{aligned}$$ where the third equality follows from the law of total probability, and the fourth inequality follows from the definition of $\rho^{\ring}_{\mu}(a|_{U_1})$. The second lemma complements \[lemma:sbp-monotonicity\], and shows that the $\mu$-atom probability cannot increase too much if, instead of the original vector, we work with its restriction to a sufficiently large subset of coordinates. \[lemma:restriction-atom-prob\] Let $a \in \ring^{n}$, and let $a|_{U_1}$ denote the restriction of $a$ to $U_1$. Then, $$\rho^{\ring}_{\mu}\left(a|_{U_1}\right) \leq \max\left\{\mu,\frac{1-\mu}{2}\right\}^{-|U_2|}\rho^{\ring}_{\mu}\left(a\right).$$ Let $c_{0}:=\arg\max_{c \in \ring}\Pr_{x^{\mu}}\left[\sum_{i\in U_{1}}a_{i}x^{\mu}_{i}=c\right]$ where the $x^{\mu}_{i}$’s are as in \[defn:atom-prob\], and let $c_{1}:=c_{0}+\sum_{i\in U_{2}}a_{i}$. Then, $$\begin{aligned} \Pr_{x^{\mu}}\left[\sum_{i\in[n]}a_{i}x^{\mu}_{i}=c_{0}\right] & \geq & \Pr_{(x^{\mu}_{i})_{i\in U_{1}}}\left[\sum_{i\in U_{1}}a_{i}x^{\mu}_{i}=c_{0}\right]\prod_{j\in U_{2}}\Pr_{x^{\mu}_{j}}\left[x^{\mu}_{j}=0\right]\\ & \geq & \rho^{\ring}_{\mu}\left(a|_{U_{1}}\right)\mu^{|U_{2}|},\end{aligned}$$ and $$\begin{aligned} \Pr_{x^{\mu}}\left[\sum_{i\in[n]}a_{i}x^{\mu}_{i}=c_{1}\right] & \geq & \Pr_{(x^{\mu}_{i})_{i\in U_{1}}}\left[\sum_{i\in U_{1}}a_{i}x^{\mu}_{i}=c_{0}\right]\prod_{j\in U_{2}}\Pr_{x^{\mu}_{j}}\left[x^{\mu}_{j}=1\right]\\ & \geq & \rho^{\ring}_{\mu}\left(a|_{U_{1}}\right)\left(\frac{1-\mu}{2}\right)^{|U_{2}|}.\end{aligned}$$ Taking the maximum of the two expressions gives $$\rho^{\ring}_{\mu}(a)\geq\max\left\{ \mu,\frac{1-\mu}{2}\right\} ^{|U_{2}|}\rho^{\ring}_{\mu}\left(a|_{U_{1}}\right),$$ and by rearranging we obtain the desired conclusion. Returning to the goal of this subsection, for $0 < \rho \leq 1$, let $\Gv_{\rho}(n-1)$ denote the event – depending only on $M^{1}_{n-1}$ – that *every* non-zero integer null vector of $M^{1}_{n-1}$ has atom probability (in $\Z$) at most $\rho$. Then, we have $$\begin{aligned} \label{eqn:conclusion-degenerate-case} \Pr_{M_n}\left[\Ev^{1}_{n-1}(n) \cap \Dg(n-1)\right] &\leq \Pr_{M_n}\left[\Ev^{1}_{n-1}(n) \cap \Dg(n-1) \cap \Gv_\rho(n-1) \right] + \Pr_{M^{1}_{n-1}}\left[\overline{\Gv_{\rho}(n-1)}\right] \nonumber \\ &\leq \Pr_{M^{1}_{n-1},x}\left[\left(\sum_{2\leq i \leq n}a_i\left(M^{1}_{n-1}\right)x_i = 0\right) \cap \Gv_{\rho}(n-1) \right] + \Pr_{M^{1}_{n-1}}\left[\overline{\Gv_{\rho}(n-1)}\right] \nonumber \\ \leq \sum_{A_{n-1}\in \Gv_{\rho}(n-1)}&\Pr_{x}\left[\left(\sum_{2\leq i \leq n}a_i\left(A_{n-1}\right)x_i = 0\right)\right]\Pr_{M^{1}_{n-1}}\left[M^{1}_{n-1} = A_{n-1}\right] + \Pr_{M^1_{n-1}}\left[\overline{\Gv_{\rho}(n-1)}\right]\nonumber \\ &\leq \rho + \Pr_{M^{1}_{n-1}}\left[\overline{\Gv_\rho(n- 1)}\right],\end{aligned}$$ where the second line follows from \[eqn:conclusion-2\]; the third line is trivial; and the last line follows from the definition of $\Gv_\rho(n-1)$. \[thm:structural\] shows that ’typically’, every non-zero integer null vector of $M^{1}_{n-1}$ has ‘small’ atom probability, and will be used to bound the right hand side of \[eqn:conclusion-degenerate-case\]. Bounding $\Pr\left[\Ev^{1}_{n-1}(n)\cap\Ndg(n-1)\right]$ -------------------------------------------------------- Once again, we start with \[eqn:laplace-expansion\]. However, for $M_{n-1} \in \Ndg(n-1)$, $\operatorname{adj}\left(M^{1}_{n-1}\right)$ is invertible, and we no longer have the factorization of the determinant in \[lemma:reduction-to-linear\] available to us. In this case, in order to reduce to a problem involving the anti-concentration of a linear form, we will follow an idea by Costello, Tao and Vu [@costello2006random]. The basic tool is the following decoupling inequality from [@costello2006random]. \[lemma:decoupling-CTV\] Let $Y$ and $Z$ be independent random variables, and $E=E(Y,Z)$ be an event depending on $Y$ and $Z$. Then, $$\Pr[E(Y,Z)]^{4} \leq \Pr[E(Y,Z) \cap E(Y',Z) \cap E(Y,Z') \cap E(Y',Z')],$$ where $Y'$ and $Z'$ denote independent copies of $Y$ and $Z$, respectively. For simplicity, and since this is the case of interest to us, we may assume that $Y,Z$ take only finitely many values; for the general case, see [@costello2006random]. Suppose that $Y$ takes the values $y_1,\ldots,y_n$ and $Z$ takes the values $z_1,\ldots,z_m$. Note that one can write $$\Pr[E(Y,Z)]=\sum_{i=1}^n\Pr[E(y_i,Z)]\Pr[Y=y_i],$$ and $$\Pr[E(Y,Z)\cap E(Y,Z')]=\sum_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i],$$ since $Z$ and $Z'$ are i.i.d. Therefore, by Jensen’s inequality, we obtain $$\label{eqn:Jensen-1} \Pr[E(Y,Z)]^2\leq \sum_{i=1}^n\Pr[E(y_i,Z)]^2\Pr[Y=y_i]=\Pr[E(Y,Z)\cap E(Y,Z')].$$ We also have $$\Pr[E(Y,Z)\cap E(Y,Z')]=\sum_{i=1}^m\sum_{j=1}^m\Pr[E(Y,z_i)\cap E(Y,z_j)]\Pr[Z=z_i]\Pr[Z=z_j],$$ and $$\Pr[E(Y,Z)\cap E(Y,Z')\cap E(Y',Z)\cap E(Y',Z')]= \sum_{i=1}^n\sum_{j=1}^n \Pr[E(Y,z_i)\cap E(Y,z_j)]^2\Pr[Z=z_i]\Pr[Z=z_j].$$ Once again, by Jensen’s inequality, we obtain $$\label{eqn:Jensen-2} \Pr[E(Y,Z)\cap E(Y,Z')]^2\leq \Pr[E(Y,Z)\cap E(Y,Z')\cap E(Y',Z)\cap E(Y',Z')].$$ By combining \[eqn:Jensen-1\] and \[eqn:Jensen-2\], we obtain the desired conclusion. Next, we explain how to use the above decoupling lemma for our purpose. For this discussion, recall \[eqn:laplace-expansion\]. Fix a non-trivial partition $[n] = U_1 \sqcup U_2$. Let $Y:= (x_i)_{i \in U_1} $ and $Z:= (x_i)_{i \in U_2}$. Let $E_{\alpha, \textbf{c}} := E_{\alpha, \textbf{c}}(Y,Z)$ denote the event that $$Q_{\alpha, \textbf{c}}(Y,Z):=\alpha - \sum_{2 \leq i,j\leq n}c_{ij}x_ix_j = 0,$$ where $\alpha$ and $\textbf{c}:=(c_{ij})_{2\leq i,j \leq n}$ are fixed. Then, the previous lemma shows that $$\Pr\left[E_{\alpha,\textbf{c}}(Y,Z)\right]^{4} \leq \Pr\left[E_{\alpha,\textbf{c}}(Y,Z) \cap E_{\alpha,\textbf{c}}(Y',Z) \cap E_{\alpha,\textbf{c}}(Y,Z') \cap E_{\alpha,\textbf{c}}(Y',Z')\right].$$ On the other hand, whenever the event on the right holds, we also have $$Q_{\alpha, \textbf{c}}(Y,Z) - Q_{\alpha, \textbf{c}}(Y',Z) - Q_{\alpha,\textbf{c}}(Y,Z') + Q_{\alpha,\textbf{c}}(Y,Z)=0.$$ Direct computation shows that the left hand side equals $$\begin{aligned} R_{\textbf{c}} &:= \sum_{i \in U_1}\sum_{j \in U_2}c_{ij}(x_i - x'_i)(x_j' - x_j)= \sum_{i \in U_1} R_i(x_i - x'_i),\end{aligned}$$ where $x'_i$ denotes an independent copy of $x_i$, and $R_i$ denotes the random sum $\sum_{j \in U_2}c_{ij}(x_j' - x_j)$. To summarize, we have deduced the following. \[corollary:decoupling-conclusion\] Let $U_1 \sqcup U_2$ be an arbitrary non-trivial partition of $[n]$. Let $w = (w_1,\dots,w_{|U_1|})$ be the random vector with coordinates $w_i := x_i - x'_i$. Then, with notation as above, and for any $(n-1) \times (n-1)$ symmetric matrix $A_{n-1}$, we have $$\Pr_{M_n}\left[\Ev^{1}_{n-1}(n) \big\vert M^{1}_{n-1} = A_{n-1}\right] \leq \Pr_{x,x'}\left[\sum_{i \in U_1}R_i w_i =0 \big\vert M^{1}_{n-1} = A_{n-1}\right]^{1/4}.$$ Using this corollary, we thus see that $$\begin{aligned} \label{eqn:decoupling-conclusion} \Pr_{M_{n}}\left[\Ev_{n-1}^{1}(n)\cap\Ndg(n-1)\right]^{4} & = \left(\sum_{A_{n-1}\in\Rv_{n-1}^{1,n-1}}\Pr_{M_{n}}\left[\Ev_{n-1}^{1}(n)|M_{n-1}^{1}=A_{n-1}\right]\Pr\left[M_{n-1}^{1}=A_{n-1}\right]\right)^{4} \nonumber \\ & \leq \sum_{A_{n-1}\in\Ndg(n-1)}\Pr_{M_{n}}\left[\Ev_{n-1}^{1}(n)|M_{n-1}^{1}=A_{n-1}\right]^{4}\Pr\left[M_{n-1}^{1}=A_{n-1}\right] \nonumber \\ & \leq \sum_{A_{n-1}\in\Ndg(n-1)}\Pr_{x,x'}\left[\sum_{i\in U_{1}}R_{i}w_{i}=0|M_{n-1}^{1}=A_{n-1}\right]\Pr\left[M_{n-1}^{1}=A_{n-1}\right] \nonumber \\ & = \Pr_{x,x',M^{1}_{n-1}}\left[\left(\sum_{i\in U_{1}}R_{i}w_{i}=0\right)\cap \Ndg(n-1)\right],\end{aligned}$$ where the second line follows from Jensen’s inequality. Hence, we have reduced the problem of bounding $\Pr\left[\Ev_{n-1}^{1}(n)\cap\Ndg(n-1)\right]$ to a linear anti-concentration problem. In order to use \[eqn:decoupling-conclusion\] profitably, we will rely on the following simple, but crucial, observation about the vector $R:=(R_1,\dots,R_n) \in \Z^{n}$, where $R_i$ is defined as above. \[lemma:R-orthogonal\] R is orthogonal to at least $n - 1 - |U_2|$ rows of $M^{1}_{n-1}$. Observe that $R$ is a linear combination of the columns of $\operatorname{adj}\left(M^{1}_{n-1}\right)$ corresponding to the indices in $U_2$. By \[eqn: A adj(A) = det(A)I\], each of these columns is orthogonal to each of the rows with indices in $[n-1] \cap U_1$; therefore, the same is true for $R$. Since $|[n-1] \cap U_1| \geq n-1 - |U_2|$, we are done. For $0<\delta,\gamma \leq 1$, let $\Hv_{\delta,\gamma n}(n-1)$ denote the event – depending only on $M^{1}_{n-1}$ – that *every* integer non-zero vector which is orthogonal to at least $(1-\gamma)n$ rows of $M^{1}_{n-1}$ has $\mu$-atom probability (in $\Z$) at most $\delta$, uniformly for all $0\leq \mu \leq 1/2$. Let $U_1 \sqcup U_2$ be a partition of $[n]$ where $U_2:= [\gamma n - 1]$. Then, with the vector $R$ defined as above, we have $$\begin{aligned} \label{eqn: breaking-according-to-event-H} \Pr_{x,x', M^{1}_{n-1}}\left[\left(\sum_{i \in U_1}R_i w_i=0\right) \cap \Ndg(n-1)\right] & \leq \Pr_{x,x', M^{1}_{n-1}}\left[\left(\sum_{i \in U_1}R_i w_i=0\right) \cap \Hv_{\delta, \gamma n}(n-1)\cap \Ndg(n-1) \right] \nonumber \\ & \hspace{.5cm} + \Pr_{ M^{1}_{n-1}}\left[\overline{\Hv_{\delta,\gamma n}(n-1)}\right]\nonumber \\ \leq \sum_{A_{n-1} \in \Hv_{\delta, \gamma n}(n-1)\cap \Ndg(n-1)}&\Pr_{w}\left[\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right]\Pr_{ M^{1}_{n-1}}\left[M^{1}_{n-1} = A_{n-1}\right] \nonumber \\ & \hspace{.5cm}+ \Pr_{ M^{1}_{n-1}}\left[\overline{\Hv_{\delta,\gamma n}(n-1)}\right]. \end{aligned}$$ As in \[sec:degenerate-case\], we will provide an upper bound on $\Pr_{w}\left[\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right]$ which is uniform in the choice of $A_{n-1} \in \Hv_{\delta,\gamma n}(n-1)\cap \Ndg(n-1)$. We start by observing that $$\begin{aligned} \label{eqn:linear-independendence-consequence} \Pr_{w}\left[\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right] & \leq \Pr_{w}\left[\left(\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right) \cap \left(R(A_{n-1})\neq 0\right) \right] + \Pr_{w}\left[R(A_{n-1})=0\right] \nonumber \\ &= \Pr_{w}\left[\left(\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right) \cap \left(R(A_{n-1})\neq 0\right) \right] + 2^{-|U_2|} \nonumber \\ &\leq \Pr_{w}\left[\left(\sum_{i\in U_1}R_i(A_{n-1})w_i=0\right) \cap \left(R(A_{n-1})\neq 0\right) \right] + 2^{-\gamma n+1}.\end{aligned}$$ To see why the second equality holds, observe as before that $$R(A_{n-1}):= \sum_{j \in U_2}w_j\text{col}_j\left(\operatorname{adj}\left(M^{1}_{n-1}\right)\right),$$ where $\text{col}_j\left(\operatorname{adj}\left(M^{1}_{n-1}\right)\right)$ denotes the $j^{th}$ column of $\operatorname{adj}\left(M^{1}_{n-1}\right)$. Since $A_{n-1} \in \Ndg(n-1)$, it follows that these columns are linearly independent, and hence $R(A_{n-1}) = 0$ if and only if $w_j = 0$ for all $j \in |U_2|$, which happens precisely with probability $2^{-|U_2|}$. It remains to bound the first summand in \[eqn:linear-independendence-consequence\]. For this, note that since $A_{n-1} \in \Hv_{\delta,\gamma n}(n-1)$ and $|U_2| = \gamma n -1$, \[lemma:R-orthogonal\], together with $R(A_{n-1})\neq 0$, shows that $\rho^{\Z}_{1/2}\left(R(A_{n-1})\right) \leq \delta$. Then, by \[lemma:restriction-atom-prob\], it follows that $\rho^{\Z}_{1/2}\left(R(A_{n-1})|_{U_1}\right) \leq 2^{|U_2|}\delta \leq 2^{\gamma n}\delta$. Finally, combining this with \[eqn:decoupling-conclusion\] and \[eqn: breaking-according-to-event-H\], we have $$\label{eqn:conclusion-nondegerate} \Pr_{M_{n}}\left[\Ev_{n-1}^{1}(n)\cap\Ndg(n-1)\right] \leq \left(2^{\gamma n}\delta + 2^{-\gamma n + 1} + \Pr_{M^{1}_{n-1}}\left[\overline{\Hv_{\delta,\gamma n}(n-1)}\right] \right)^{\frac{1}{4}}.$$ The structural theorem {#sec:structural theorem} ====================== This section is devoted to the proof of our structural theorem, which is motivated by \[eqn:conclusion-degenerate-case,eqn:conclusion-nondegerate\]. Statement and initial reductions -------------------------------- In order to state the structural theorem, we need the following definition. Let $M_{n}$ be an $n\times n$, $\{\pm 1\}$-valued symmetric matrix, chosen uniformly at random from the set of all such matrices. For $0\leq \alpha := \alpha(n), \beta := \beta(n) \leq 1$, let $\Hv_{\alpha, \beta n}(n)$ denote the event that every integer non-zero vector which is orthogonal to at least $(1-\beta)n$ many rows of $M_{n}$ has $\mu$-atom probability (in $\Z$) at most $\alpha$, uniformly for all $0 \leq \mu \leq 1/2$. \[thm:structural\] Let $\alpha(n) = 2^{-n^{1/4}\sqrt{\log{n}}/64}$, $\beta(n) = n^{-3/4}\sqrt{\log{n}}/128$, and $n \in \N$ be sufficiently large. Then, $$\Pr_{M_n}\left[\overline{\Hv_{\alpha,\beta n}(n)}\right] \leq 2^{-n/32}.$$ Roughly, we will prove \[thm:structural\] by taking a union bound, over the choice of the non-zero integer vector with large $\mu$-atom probability, of the probability that this vector is orthogonal to at least $(1-\beta)n$ many rows of $M_n$. However, there is an obstacle since, *a priori*, this union bound is over an infinite collection of vectors. In order to overcome this, we will work instead with the coordinate-wise residues of the vector modulo a suitably chosen prime $p(n)$. In the next proposition, we make use of the event $\Hv^{p}_{\alpha, \beta n}(n)$, which is defined exactly as $\Hv_{\alpha, \beta n}(n)$, except that we work over $\F_p$ instead of the integers. \[prop:structural\] Let $\alpha(n) = 2^{-n^{1/4}\sqrt{\log{n}}/64}$ and $\beta(n) = n^{-3/4}\sqrt{\log{n}}/128$. Let $p(n) = 2^{n^{1/4}\sqrt{\log{n}}/32}$ be a prime, and let $n \in \N$ be sufficiently large. Then, $$\Pr_{M_n}\left[\overline{\Hv^{p}_{\alpha,\beta n}(n)}\right] \leq 2^{-n/32} .$$ Before proving \[prop:structural\], let us quickly show how to deduce \[thm:structural\] from it. It suffices to show that $\overline{\Hv_{\alpha,\beta n}(n)} \subseteq \overline{\Hv^{p}_{\alpha, \beta n}(n)}$ for any prime $p$. To see this, suppose $M_n \in \overline{\Hv_{\alpha,\beta n}(n)}$. So, there exists an integer non-zero vector $a$ which is orthogonal to at least $(1-\beta)n$ many rows of $M_n$ and has $\mu$-atom probability (in $\Z$) greater than $\alpha$, for some $0\leq \mu \leq 1/2$. Furthermore, by rescaling $a$ if necessary, we may assume that $\text{gcd}(a_1,\ldots,a_n)=1$. Therefore, letting $a_{p}$ be the image of $a$ under the natural map from $\Z^{n} \to \F_{p}^{n}$, we see that $a_{p} \in \F_{p}^{n} \setminus \{\operatorname{\textbf{0}}\}$ and is orthogonal (over $\F_p$) to (at least) the same $(1-\beta)n$ rows of $M_n$. Finally, $\rho^{\F_p}_{\mu}(a_p) \geq \rho^{\Z}_{\mu}(a) > \beta$, since for any $c \in \Z$, every solution $x \in \{-1,0,1\}^{n}$ of $a_1x_1 + \dots + a_n x_n = c$ over the integers is also a solution of the same equation in $\F_p$. Thus, the vector $a_p$ witnesses that $M_n \in \overline{\Hv^{p}_{\alpha, \beta n}(n)}$. The next lemma is the first step towards the proof of \[prop:structural\] and motivates the subsequent discussion. In its statement, the support of a vector $a=(a_1,\dots,a_n) \in \F_{p}^{n}$, denoted by $\operatorname{supp}(a)$, refers to the set of indices $i\in [n]$ such that $a_i \neq 0 \mod p$. \[lemma:eliminate-small-support\] Let $M_{n}$ be an $n\times n$, $\{\pm 1\}$-valued symmetric matrix, chosen uniformly at random from among all such matrices. Let $1 \leq d \leq n$ be an integer, and let $p$ be a prime. Let $\Sv^{p}_{\geq d, \beta n}(n)$ denote the event that every vector in $\F_{p}^{n}\setminus\{\operatorname{\textbf{0}}\}$ which is orthogonal (over $\F_{p}$) to at least $(1-\beta)n$ many rows of $M_{n}$ has support of size at least $d$. Suppose further that $\beta \leq 1/2$, $d\leq n/2$, $p^{\beta n} \leq 2^{n/2}$, $p^{d} \leq 2^{n/8}$, $H(\beta) \leq 1/4$, and $H(d/n) \leq 1/16$ (where $H(x):= -x\log_{2}(x) - (1-x)\log_{2}(1-x)$ is the binary entropy function for $x\in [0,1]$) Then, $$\Pr_{M_n}\left[\overline{\Sv^{p}_{\geq d,\beta n}(n)}\right] \leq 2^{-n/16}.$$ The proof of this lemma will use the following simple, yet powerful, observation. \[obs:inject-permutation\] Let $\Sigma$ be an $n\times n$ permutation matrix. Then, for a uniformly random $n\times n$ symmetric $\{\pm 1\}$-matrix $M_n$, the random matrix $\Sigma^{-1}M_n\Sigma$ is also a uniformly distributed $n\times n$ symmetric $\{\pm 1\}$-matrix. It is clear than $\Sigma^{-1}M_n\Sigma$ is an $n\times n$ $\{\pm 1\}$-matrix. That it is symmetric follows from $\Sigma^{-1} = \Sigma^{T}$ and $M_n^{T} = M_n$. Finally, $\Sigma^{-1}M_n\Sigma$ is uniformly distributed since conjugation by $\Sigma$ is manifestly a bijection from the set of $n\times n$ $\{\pm 1\}$ symmetric matrices to itself. Let $d$ be as in the statement of the lemma, and for $1\leq s \leq d$, let $\Supp_{=s}(n)$ denote the set of all vectors in $\F_{p}^{n}$ which have support of size exactly $s$. Observe that $|\Supp_{=s}(n)| \leq \binom{n}{s}p^{s}$. We will now bound the probability that any given $a \in \Supp_{=s}(n)$ is orthogonal to at least $(1-\beta)n$ rows of a uniformly chosen $M_n$. For this, let $\Sigma = \Sigma(a)$ denote a fixed, but otherwise arbitrary, permutation matrix for which $\Sigma \1_{\operatorname{supp}(a)} = \1_{[n-s+1,n]}$. In other words, $\Sigma$ permutes the vector $a$ so that its nonzero entries are placed in the last $s$ coordinates. Since \[obs:inject-permutation\] shows that $\Sigma^{-1} M_n \Sigma $ is a uniformly random $n \times n$ symmetric matrix for a uniformly random $n \times n$ symmetric matrix $M_n$, it follows that $$\begin{aligned} \label{eqn:prob-orthog-many} \Pr_{M_n}[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] &= \Pr_{M_n}\left[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } \Sigma^{-1}M_n\Sigma\right]\nonumber \\ &= \Pr_{M_n}\left[\Sigma^{-1}M_n\Sigma a = v \text{ for some } v\in \bigcup_{t=0}^{\beta n}{\Supp_{=t}(n)}\right] \nonumber \\ &\leq \sum_{t=0}^{\beta n} \Pr_{M_n}\left[\Sigma^{-1}M_n \Sigma a = v \text{ for some } v\in \Supp_{=t}(n)\right] \nonumber \\ &= \sum_{t=0}^{\beta n}\Pr_{M_n}\left[M_n \Sigma a = v \text{ for some } v \in \Supp_{=t}(n)\right] \nonumber \\ & \leq \sum_{t=0}^{\beta n} \sum_{v\in \Supp_{=t}(n)}\Pr_{M_n}\left[M_n\Sigma a = v\right],\end{aligned}$$ where the third line follows by the union bound; the fourth line follows since the size of the support of a vector is invariant under the action of $\Sigma$; and the last line follows again by the union bound. Next, we provide a (crude) upper bound on $\Pr_{M_n}\left[M_n(\Sigma a) = v\right]$ for any fixed $v=(v_1,\dots,v_n) \in \F_{p}^{n}$. For this, we isolate the last column of the matrix $M_n$ by rewriting the system of equations $M_n (\Sigma a) = v$ as $$\label{eqn:prob-fixed-rhs} m_{in} = (\Sigma a)_{n}^{-1}\left(v_{i}-\sum_{j=1}^{n-1}m_{ij}(\Sigma a)_{j}\right) \text{ for all } i\in [n],$$ where $m_{ij}$ denotes the $(i,j)^{th}$ entry of the matrix $M_n$, and the equation makes sense since $(\Sigma a)_n \neq 0$ by our choice of $\Sigma$. Note that the right hand side of the equation is completely determined by the top-left $(n-1)\times (n-1)$ submatrix of $M_n$. Further, the entries $m_{in}, i \in [n]$ are mutually independent even after conditioning on any realisation of the top-left $(n-1)\times (n-1)$ submatrix of $M_n$. Since $m_{in}$ takes on any value with probability at most $1/2$, it follows that conditioned on any realisation of the top-left $(n-1)\times (n-1)$ submatrix of $M_n$, \[eqn:prob-fixed-rhs\] is satisfied with probability at most $(1/2)^{n}$. Hence, by the law of total probability, $\Pr_{M_n}[M_n\Sigma a = v]\leq 2^{-n}$. Substituting this in \[eqn:prob-orthog-many\], we see that $$\begin{aligned} \label{eqn:easy-entropy-bound} \Pr_{M_n}[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] &\leq 2^{-n}\sum_{t=0}^{\beta n} |\Supp_{=t}(n)| \nonumber \\ &\leq 2^{-n}\sum_{t=0}^{\beta n}\binom{n}{t}p^{t} \leq 2^{-n}p^{\beta n}\sum_{t=0}^{\beta n}\binom{n}{t} \nonumber \\ & \nonumber \\ &\leq 2^{-n/2}2^{nH(\beta)} \leq 2^{-n/4},\end{aligned}$$ where the fourth inequality follows by the assumption on $p^{\beta n}$ and the standard inequality $\sum_{t=0}^{\beta n}\binom{n}{t} \leq 2^{n H(\beta)}$ for $\beta \leq 1/2$, and the last inequality follows by the assumption on $nH(\beta)$. Finally, we have $$\begin{aligned} \Pr_{M_n}\left[\overline{\Sv^{p}_{\geq d,\beta n}(n)}\right] &\leq \sum_{s=1}^{d}\sum_{a\in \Supp_{=s}(n)}\Pr_{M_n}[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] \\ &\leq 2^{-n/4}\sum_{s=1}^{d}|\Supp_{=s}(n)| \leq 2^{-n/4}\sum_{s=1}^{d}\binom{n}{s}p^{s}\\ &\leq 2^{-n/4}p^{d}\sum_{s=1}^{d}\binom{n}{s} \leq 2^{-n/8}2^{nH(d/n)} \leq 2^{-n/16},\end{aligned}$$ where the fifth inequality follows by the assumption on $p^{d}$ and $d$, and the last inequality follows by the assumption on $H(d/n)$. Tools and auxiliary results --------------------------- Following \[lemma:eliminate-small-support\], we will bound $\Pr_{M_n}\left[\overline{\Hv^{p}_{\alpha,\beta n}(n)}\cap \Sv^{p}_{\geq d,\beta n}(n)\right]$ for suitably chosen parameters. Our proof of this bound will be based on the following two key ingredients. The first is a classical anti-concentration inequality due to Halász, which bounds the atom probability of a vector in terms of the ‘arithmetic structure’ of its coordinates. In order to state it, we need the following definition. Let $a \in \F_{p}^{n}$ and let $k \in \N$. We define $R_k(a)$ to be the number of solutions to $$\pm a_{i_1} \pm a_{i_2}\pm \dots \pm a_{i_{2k}} = 0 \mod p,$$ where repetitions are allowed in the choice of $i_1,\dots,i_{2k} \in [n]$. \[thm:halasz\] Let $p$ be any prime and let $a:=(a_1,\ldots,a_n) \in \F_p^{n}\setminus \{\operatorname{\textbf{0}}\}$. Then, $$\sup_{0\leq \mu \leq \frac{1}{2}}\max_{q\in \F_p}\Pr\left[\sum_ia_ix_i^{\mu} \equiv_p q \right]\leq \frac{1}{p}+\frac{CR_k(a)}{2^{2k} n^{2k}f(|\operatorname{supp}(a)|)^{1/2}} + e^{-f(|\operatorname{supp}(a)|)/2},$$ where $C$ is an absolute constant (which we may assume is at least $1$), $f(|\operatorname{supp}(a)|) \leq |\operatorname{supp}(a)|/100$ and $k \leq n/f(|\operatorname{supp}(a)|)$. Halász’s inequality is typically stated and proved over the integers, but the version over $\F_p$ stated above easily follows using the same ideas. For the reader’s convenience, we provide a complete proof in \[app:halasz\]. The second ingredient is a ‘counting lemma’ due to the authors together with Luh and Samotij [@FJLS2018], which bounds the number of vectors in $\F_{p}^{n}$ with a slightly different (but practically equivalent) notion of ‘rich additive structure’. Let $a\in \F_{p}^{n}$ and let $k \in \N$. We define $R_k^*(a)$ to be the number of solutions to $$\pm a_{i_1}\pm a_{i_2}\dots \pm a_{i_{2\ell}}=0$$ with *at least one non-repeated index* $i_\ell$. As mentioned above, $R_k(a)$ and $R_k^*(a)$ are practically equivalent. This is made precise by the following lemma. \[lemma:R\_k vs R\_k\^\*\] For any vector $a$ and any $k\leq |a|$, $$R_k(a)\leq R_k^*(a) + (16k)^k\cdot |a|^k.$$ By definition, $R_k(a)$ is equal to $R_k^*(a)$ plus the number of solutions to $\pm a_{i_1}\pm a_{i_2}\pm\dots\pm a_{i_{2k}} = 0$ in which every index is repeated at least once. As an easy upper bound on the number of such solutions, note that for each $1\leq \ell \leq k$, we may choose $\ell$ distinct indices in $\binom{|a|}{\ell}$ ways, and then from these indices, form a sum $\pm a_{i_1}\pm a_{i_2}\pm\dots\pm a_{i_{2\ell}}$ in $\ell^{2\ell}2^{2\ell}$ ways. Thus $$\begin{aligned} R_k(a)&\leq R_k^*(a) + \sum_{\ell=1}^{k}\binom{|a|}{\ell}\ell^{2\ell}2^{2\ell} \leq R_k^*(a) + k^{2k}2^{2k}\sum_{\ell=1}^{k}\binom{|a|}{\ell} \leq R_k^*(a) + (16k)^k\cdot |a|^k. \end{aligned}$$ We can now state the ‘counting lemma’ from [@FJLS2018]. \[thm:counting-lemma\] Let $p$ be a prime and let $k \in \N, s\in [n], t\in [p]$. Let $$\Bad_{k,s,\geq t}(n):= \left\{a \in \F_{p}^{n} \mid \forall b\subset a \text{ s.t. } |b|\geq s \text{ we have } R^*_k(b)\geq t\cdot \frac{2^{2k}\cdot |b|^{2k}}{p}\right\}$$ denote the set of ‘$k,s,\geq t$-bad vectors’. Then, $$|\Bad_{k,s,\geq t}(n)| \leq \left(\frac{s}{n}\right)^{2k-1}p^{n}t^{-n+s}.$$ We include the complete proof from [@FJLS2018] in \[app:counting-lemma\] The above theorem shows that there are very few vectors for which every sufficiently large subset has rich additive structure. However, in order to use the strategy of \[lemma:eliminate-small-support\] effectively, we require that there are very few vectors for which every *moderately-sized* subset has rich additive structure (see \[corollary:prob-orth-fixed-vector\]). This is accomplished by the following corollary. \[corollary:counting\] Let $p$ be a prime and let $k \in \N, s_1,s_2,d\in [n], t\in [p]$ such that $s_1 \leq s_2$. Let $$\Bad^{d}_{k,s_1,s_2,\geq t}(n):= \left\{a \in \F_{p}^{n} \big\vert |\operatorname{supp}(a)|=d \text{ and } \forall b\subset \operatorname{supp}(a) \text{ s.t. } s_2\geq|b|\geq s_1 \text{ we have } R^*_k(b)\geq t\cdot \frac{2^{2k}\cdot |b|^{2k}}{p}\right\}.$$ Then, $$|\Bad^{d}_{k,s_1,s_2,\geq t}(n)| \leq {n \choose d}p^{d+s_{2}}t^{-d+\frac{s_{1}}{s_{2}}d}.$$ At the expense of an overall factor of $\binom{n}{d}$, we may restrict our attention to those vectors in $\Bad^{d}_{k,s_1,s_2,\geq t}(n)$ whose support is $[d]$. In order to count the number of such vectors, we begin by decomposing $[d]$ into the intervals $I_1,\dots,I_{m+1}$, where $m:=\lfloor d/s_2 \rfloor$, $I_j:= \{(j-1)d+1,\dots,jd\}$ for $j\in [m]$, and $I_{m+1}:= \{md+1,\dots,d\}$. For a vector with support $[d]$ to be in $\Bad^{d}_{k,s_1,s_2,\geq t}(n)$, it must necessarily be the case that the restriction of the vector to each of the intervals $I_1,\dots,I_m$ is in $\Bad_{k,s_1,\geq t}(s_2)$. Since there are at most $p^{|I_{m+1}|}\leq p^{s_2}$ many choices for the restriction of the vector to $I_{m+1}$, it follows from \[thm:counting-lemma\] that $$\begin{aligned} |\Bad_{k,s_{1},s_{2},\geq t}^{d}(n)| & \leq & {n \choose d}\left|\Bad_{k,s_{1},\geq t}(s_{2})\right|^{m}p^{s_{2}} \leq {n \choose d}\left\{ \left(\frac{s_{1}}{s_{2}}\right)^{2k-1}p^{s_{2}}t^{-s_{2}+s_{1}}\right\} ^{m}p^{s_{2}}\\ & \leq & {n \choose d}\left(p^{s_{2}}t^{-s_{2}+s_{1}}\right)^{\frac{d}{s_{2}}}p^{s_{2}}= {n \choose d}p^{d+s_{2}}t^{-d+\frac{s_{1}}{s_{2}}d}.\end{aligned}$$ We conclude this subsection with a few corollaries of \[thm:halasz\] and \[corollary:counting\]. Let $a \in \Supp_{=d}(n) \setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)$ for $s_1\leq d\leq n$. Then, by definition, there exists $\Lambda=\Lambda(a) \subseteq \operatorname{supp}(a)$ such that $s_1 \leq |\Lambda |=|\operatorname{supp}(a|_\Lambda)|\leq s_2$ and $R_k^*(a|_\Lambda)< (t+1)\cdot 2^{2k}|\Lambda|^{2k}/p$. From now on, fix such a subset $\Lambda(a)$ for every vector $a$. \[corollary:halasz-usable\] Let $p$ be a prime and let $a \in \Supp_{=d}(n) \setminus \Bad^{d}_{k,s_1,s_2,t}(n)$. Suppose $p \leq \min\left\{e^{-s_1/2k}, \left(4k/s_1\right)^{k}\right\}$, $s_2 \geq s_1\geq 1$, $n\geq d\geq s_1$, and $t \geq s_1 \geq k \geq 100$. Then, $$\sup_{0\leq \mu \leq \frac{1}{2}}\rho_{\mu}^{\F_p}(a|_{\Lambda(a)})\leq \frac{2Ct\sqrt{k}}{p\sqrt{s_1}},$$ where $C \geq 1$ is an absolute constant. For convenience of notation, let $b:= a|_{\Lambda(a)}$. By applying \[thm:halasz\] to the vector $b$ with the function $f(|\operatorname{supp}(b)|) = f(|b|) := |b|/k$ (which is a valid choice for $f$ since $k\geq 100$ by assumption), we get $$\begin{aligned} \sup_{0\leq\mu\leq\frac{1}{2}}\rho_{\mu}^{\F_p}(b) & \leq & \frac{1}{p}+\frac{C\left(R_{k}^{*}(b)+(16k)^{k}|b|^{k}\right)}{2^{2k}|b|^{2k}\sqrt{f(|b|)}}+e^{-f(|b|)/2}\\ & \leq & \frac{1}{p}+\frac{Ct}{p\sqrt{f(|b|)}}+\frac{C(4k)^{k}}{|b|^{k}\sqrt{f(|b|)}}+e^{-f(|b|)/2}\\ & \leq & \frac{1}{p}+\frac{Ct\sqrt{k}}{p\sqrt{|b|}}+\frac{C(4k)^{k}}{|b|^{k}}+e^{-|b|/2k}\\ & \leq & \frac{1}{p}+\frac{Ct\sqrt{k}}{p\sqrt{s_{1}}}+C\left(\frac{4k}{s_{1}}\right)^{k}+e^{-s_{1}/2k}\\ & \leq & \frac{(2+C)}{p}+\frac{Ct\sqrt{k}}{p\sqrt{s_{1}}} \leq \frac{2Ct\sqrt{k}}{p\sqrt{s_{1}}},\end{aligned}$$ where the first line follows from \[thm:halasz\], \[lemma:R\_k vs R\_k\^\*\], and the choice of $\Lambda(a)$, the fifth line follows by the assumption on $p$, and the last line follows since $t \geq s_1 \geq 100$. \[corollary:prob-orth-fixed-vector\] Let $p$ be a prime and let $a \in \Bad^{d}_{k,s_1,s_2,t}(n) \setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)$. Suppose $p \leq \min\left\{e^{-s_1/2k}, \left(4k/s_1\right)^{k}\right\}$, $s_2\geq s_1\geq 1$, $n\geq d\geq s_1$, and $t \geq s_1 \geq k \geq 100$. Then, for $0\leq \beta := \beta(n) \leq 1/2$, $$\Pr_{M_n}[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] \leq 2^{n H(\beta)}p^{\beta n}\left(\frac{2C(t+1)\sqrt{k}}{p\sqrt{s_1}}\right)^{n-s_2},$$ where $C\geq 1$ is an absolute constant. The proof is very similar to the proof of \[lemma:eliminate-small-support\]. For the reader’s convenience, we will spell out the details. Let $\Lambda:=\Lambda(a)$ and $b:= a|_{\Lambda}$. As in the proof of \[lemma:eliminate-small-support\], let $\Sigma$ denote a fixed, but otherwise arbitrary, permutation matrix for which $\Sigma \1_{\Lambda} = \1_{[n-|\Lambda|+1,n]}$. Then, by \[eqn:prob-orthog-many\], $$\begin{aligned} \Pr_{M_n}[a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] = \sum_{t=0}^{\beta n}\sum_{v \in \Supp_{=t}}\Pr_{M_n}\left[M_n \Sigma a = v\right].\end{aligned}$$ Next, we provide an upper bound on $\Pr_{M_n}\left[M_n(\Sigma a)=v\right]$ for any fixed $v=(v_1,\dots,v_n)\in \F_{p}^{n}$. For this, note that the system of equations $M_n(\Sigma a)=v$ implies in particular that $$\label{eqn:orthogonal-many-2} \sum_{j=1}^{|\Lambda|}m_{i,n-|\Lambda|+j}b_j = v_i - \sum_{j=1}^{n-|\Lambda|}m_{i,j}(\Lambda a)_j \text{ for all }i\in [n-|\Lambda|],$$ Note that the right hand side is completely determined by the top-left $(n-|\Lambda|)\times (n-|\Lambda|)$ submatrix of $M_n$, and the entries of $M_n$ appearing on the left are mutually independent even after conditioning on any realisation of the top-left $(n-|\Lambda|)\times (n-|\Lambda|)$ submatrix of $M_n$. In particular, after conditioning on any realisation of the top-left submatrix of this size, each of the $n-|\Lambda|$ equations above is satisfied with probability which is at most $\rho^{\F_p}(b)$, and the satisfaction of different equations is mutually independent. Hence, by the law of total probability, the system \[eqn:orthogonal-many-2\] is satisfied with probability at most $$\left(\rho^{\F_p}(b)\right)^{n-|\Lambda|} \leq \left(\frac{2C(t+1)\sqrt{k}}{p\sqrt{s_1}}\right)^{n-|\Lambda|}\leq \left(\frac{2C(t+1)\sqrt{k}}{p\sqrt{s_1}}\right)^{n-s_2} ,$$ where the middle bound follows from \[corollary:halasz-usable\], and the right-hand bound follows since $|\Lambda|\leq s_2$. Finally, substituting this in \[eqn:prob-orthog-many\] and proceeding as in \[eqn:easy-entropy-bound\] gives the desired conclusion. \[corollary:prob-level-set\] Let $p$ be a prime and $k\in \N, s_1,s_2,d\in [n], t\in [p]$ such that $1\\leq s_1 \leq s_2 \leq n/2$, $s_1 \leq d \leq n$, $p \leq \min\left\{e^{-s_1/2k}, \left(4k/s_1\right)^{k}\right\}$, and $t \geq s_1 \geq k \geq 100$. Then, for $0\leq \beta := \beta(n) \leq 1/2$, $$\Pr_{M_n}[\exists a \in \Bad^{d}_{k,s_1,s_2,t}(n)\setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)|a \text{ is orthogonal to $\geq (1-\beta)n$ rows of } M_n] \leq (12C)^{n}p^{\beta n + 2s_2 + \frac{s_1}{s_2}d}\left(\frac{k}{s_1}\right)^{n/4},$$ where $C\geq 1$ is an absolute constant. Using \[corollary:prob-orth-fixed-vector\] to bound the probability that any given $a \in \Bad^{d}_{k,s_1,s_2,t}(n)\setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)$ is orthogonal to at least $(1-\beta)n$ rows of $M_n$, and taking the union bound over all $|\Bad^{d}_{k,s_1,s_2,t}(n)\setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)|$ such vectors $a$, we see that the desired probability is at most $$\begin{aligned} |\Bad^{d}_{k,s_1,s_2,t}(n)\setminus \Bad^{d}_{k,s_1,s_2,t+1}(n)|2^{n H(\beta)}p^{\beta n}\left(\frac{2C(t+1)\sqrt{k}}{p\sqrt{s_1}}\right)^{n-s_2} &\leq |\Bad^{d}_{k,s_1,s_2,\geq t}(n)|2^{n}p^{\beta n}\left(\frac{2C(t+1)\sqrt{k}}{p\sqrt{s_1}}\right)^{n-s_2}\\ \leq 2^{n}\binom{n}{d}p^{d+s_2}t^{-d+\frac{s_1}{s_2}d}p^{\beta n}\left(\frac{3Ct\sqrt{k}}{p\sqrt{s_1}}\right)^{n-s_2} \leq& (12C)^{n}p^{\beta n+s_2+\frac{s_1}{s_2}d}\left(\frac{t}{p}\right)^{n-d-s_2}\left(\frac{k}{s_1}\right)^{n/4}\\ \leq (12C)^{n}p^{\beta n + 2s_2 + \frac{s_1}{s_2}d}\left(\frac{k}{s_1}\right)^{n/4},\end{aligned}$$ where the second inequality follows from \[corollary:counting\], and the third inequality follows from $s_2 \leq n/2$. Proof of \[prop:structural\] ---------------------------- By combining the results of the previous subsection, we can now prove \[prop:structural\]. Consider the following choice of parameters: $k = n^{1/4}$, $s_1 = n^{1/2}\log{n}$, $s_2 = n^{3/4}\sqrt{\log{n}}$, $\beta n= n^{1/4}\sqrt{\log{n}}/128$, $d = n^{2/3}$, $\alpha = 2^{-n^{1/4}\sqrt{\log{n}}/64}$, and $p = 2^{-n^{1/4}\sqrt{\log{n}}/32}$. Throughout, we will assume that $n$ is sufficiently large for various inequalities to hold, even if we do not explicitly mention this. [**Step 1:** ]{}It is readily seen that the assumptions of \[lemma:eliminate-small-support\] are satisfied, so that $\Pr\left[\overline{\Sv^{p}_{\geq d, \beta n}(n)}\right] \leq 2^{-n/16}$. In other words, except with probability at most $2^{-n/16}$, every vector in $\F_{p}^{n}\setminus\{\operatorname{\textbf{0}}\}$ which is orthogonal to at least $(1-\beta)n$ rows of $M_n$ has support of size at least $d = n^{2/3}$. [**Step 2:** ]{}Let $a \in \Supp_{=s}(n)\setminus \Bad^{s}_{k,s_1,s_2,\sqrt{p}}(n)$ for any $s\geq d$. Since the assumptions of \[corollary:halasz-usable\] are satisfied for our choice of parameters, it follows from \[corollary:halasz-usable\] and \[lemma:sbp-monotonicity\] that for any $0\leq \mu \leq 1/2$, $$\rho_{\mu}^{\F_p}(a) \leq \rho_{\mu}^{\F_p}(a|_{\Lambda(a)})\leq \frac{2C\sqrt{k}}{\sqrt{p{s_1}}} \leq \frac{1}{\sqrt{p}} \leq \alpha ,$$ for all $n$ sufficiently large. [**Step 3:** ]{}Therefore, it suffices to bound the probability that for some $s\geq d$, there exists some vector in $\Bad^{s}_{k,s_1,s_2,\geq \sqrt{p}}(n)$ which is orthogonal to at least $(1-\beta)n$ rows of $M_n$. By writing $$\Bad^{s}_{k,s_1,s_2,\geq \sqrt{p}}(n):= \bigcup_{t=\sqrt{p}}^{p}\Bad^{s}_{k,s_1,s_2,t}(n)\setminus \Bad^{s}_{k,s_1,s_2,t+1}(n),$$ noting that the assumptions of \[corollary:prob-level-set\] are satisfied, and taking the union bound over the choice of $s$ and $t$, it follows that this event has probability at most $$\begin{aligned} np(12C)^{n}p^{\beta n + 2s_2 + \frac{s_1}{s_2}s}\left(\frac{k}{s_1}\right)^{n/4} &\leq np(12C)^{n}p^{4s_2}2^{-(n\log{n})/16}\\ &\leq np(12C)^{n}2^{-(n\log{n})/32} \leq 2^{-(n\log{n})/64},\end{aligned}$$ for all $n$ sufficiently large.\ Combining these steps, it follows that $$\Pr_{M_n}\left[\overline{\Hv^{p}_{\alpha, \beta n}}\right] \leq 2^{-n/16} + 2^{-(n\log{n})/64} \leq 2^{-n/32},$$ as desired. Proof of \[thm:main-thm\] {#sec:proof-main-thm} ========================= Our main result is now immediate. By definition, $\overline{\Gv_{\rho}(n-1)} \subseteq \overline{\Hv_{\rho, \beta n}(n-1)}$ for every $\beta \geq 0$. Therefore, from \[eqn:split-into-deg-nondeg\], \[eqn:conclusion-degenerate-case\], and \[eqn:conclusion-nondegerate\], it follows that $$\begin{aligned} \Pr_{M_n}\left[\Ev^{1}_{n-1}\right] \leq \alpha + \Pr_{M^{1}_{n-1}}\left[\overline{\Hv_{\alpha, \beta n}(n-1)}\right] + \left(2^{\beta n}\alpha + 2^{-\beta n + 1} + \Pr_{M^{1}_{n-1}}\left[\overline{\Hv_{\rho, \beta n}(n-1)}\right] \right)^{1/4},\end{aligned}$$ where $\alpha$ and $\beta$ are as in the statement of \[thm:structural\]. From \[thm:structural\], it follows that the right hand side of the above equation is at most $2^{-n^{1/4}\sqrt{\log{n}}/600}$ for all $n$ sufficiently large. Finally, \[lem:rank reduction\] and \[corollary:remove-first-row\] give the desired conclusion. Proof of the ‘counting lemma’ {#app:counting-lemma} ============================= In this appendix, we reproduce the proof of \[thm:counting-lemma\] from [@FJLS2018]. We begin by slightly recasting the problem. Let $a \in \F_{p}^{n}$ be chosen uniformly at random. We are interested in bounding from above the probability that $a \in \Bad_{k,s,\geq t}(n)$. For this, we consider the auxiliary random variable $Z_s(a)$, which is defined to be the number of triples $$\left(I, \left(i_{s+1},\dots,i_{n}\right), \left(F_{j},{\boldsymbol{\epsilon}}^{j}\right)_{j=s+1}^{n} \right)$$ such that: (i) $I \subseteq [n]$ and $|I|=s$, (ii) $(i_{s+1},\ldots,i_n) \in [n]^{n-s}$ is a permutation of $[n]\setminus I$, (iii) $F_{j}:=\{\ell_1,\dots,\ell_{2k}\}$ is a multisubset of $[n]$ of size $2k$, (iv) $\boldsymbol{\epsilon}^{(j)}\in \{\pm 1\}^{2k}$, and which satisfy the following compatibility conditions: (1) $\ell_{2k}= i_{j}$, (2) $F_{j}\setminus \{\ell_{2k}\}\subseteq I\cup \{i_{s+1},\ldots,i_{j-1}\}$, and (3) $\sum_{i=1}^{2k}\boldsymbol\epsilon^{(j)}_ia_{\ell_i}=0$. We will make use of the following two observations: - The number of triples $\left(I, \left(i_{s+1},\dots,i_{n}\right), \left(F_{j},{\boldsymbol{\epsilon}}^{j}\right)_{j=s+1}^{n} \right)$ satisfying $(i)-(iv)$ and $(1),(2)$ is at most $\binom{n}{s}\cdot \prod_{j=s}^{n-1}\left({j^{2k-1}\cdot 2^{2k}\cdot (n-j)}\right)$. Indeed, we can generate any such triple by first choosing the subset $I$ of $[n]$ in $\binom{n}{s}$ ways, and then in the $j^{th}$ step, for each $s+1 \leq j \leq n$, choosing the following: one of the $(n-j+1)$ remaining coordinates to serve as $i_j$, one of the $2^{2k}$ possible sign patterns to serve as $\boldsymbol{\epsilon}^{j}$, and one of the $(j-1)^{2k-1}$ possible multisubsets of size $2k-1$ using $I \cup {i_{s+1},\dots,i_{j-1}}$ to serve as $F_{j}\setminus \{i_{j}\}$. - Let $\left(I, \left(i_{s+1},\dots,i_{n}\right), \left(F_{j},{\boldsymbol{\epsilon}}^{j}\right)_{j=s+1}^{n} \right)$ be a triple satisfying conditions $(i)-(iv)$ and $(1),(2)$. Then, for a uniformly random vector $a \in \F_{p}^{n}$, the probability that this triple also satisfies compatibility condition $(3)$ is $p^{-n+s}$. Indeed, rewriting $(3)$ (using $(1)$) as $$\boldsymbol{\epsilon}^{(j)}_{2k}a_{i_{j+1}} = -\sum_{i=1}^{2k-1}\boldsymbol\epsilon^{(j)}_ia_{\ell_i},$$ it follows by induction, using $(2)$ and the above equation, that given the triple, the coordinates of $a$ corresponding to the indices in $I$ uniquely determine the coordinates of the vector in $[n]\setminus I$. Since there are $p^{n-s}$ many choices for these remaining coordinates, each of which is equally likely, the claim follows. We now estimate the expectation of $Z_s(a)$ in two ways. First, by combining the above two observations with the linearity of expectation, we see that $$\begin{aligned} \mathbb{E}[Z_s(a)] &\leq \binom{n}{s}\cdot \prod_{j=s}^{n-1}\left({j^{2k-1}\cdot 2^{2k}\cdot (n-j)}\right)\cdot p^{-n+s}\\ &= \frac{2^{2k(n-s)n!}}{s!}\cdot \left(\frac{n!}{s!}\right)^{2k-1}\cdot\left(\frac{s}{n}\right)^{2k-1}\cdot p^{-n+s} \\ &\leq \left(\frac{2^{n-s}n!}{s!}\right)^{2k}\cdot p^{-n+s}\cdot \left(\frac{s}{n}\right)^{2k-1}.\end{aligned}$$ To obtain a lower bound on the expectation of $Z_s(a)$, we begin by providing a uniform lower bound on $Z_s(a)$ for all $a\in \Bad_{k,s,\geq t}(n)$. For this, note that for any such $a$, we can obtain a triple satisfying $(i)-(iv)$ and $(1)-(3)$ by choosing, for each $j=s+2,\dots,n+1$, a solution to $(3)$ with at least one non-repeated index which is fully contained in $[n]\setminus \{i_{n},\dots,i_{j}\}$ (this determines $F_{j-1}$ and $\boldsymbol{\epsilon}^{j}$), and choosing $i_{j-1}$ to be an arbitrary such non-repeated index. By definition of $\Bad_{k,s,\geq t}(n)$, there are at least $t\cdot \frac{2^{2k}\cdot (j-1)^{2k}}{p}$ many choices for such a solution in the $j^{th}$ step. Since different sequences of such solutions manifestly lead to different triples, it follows that for any $a \in \Bad_{k,s,\geq t}(n)$, $$Z_s(a)\geq t^{n-s}\cdot \left(\frac{2^{n-s}n!}{s!}\right)^{2k}\cdot p^{-n+s},$$ so that $$\E\left[Z_s(a)\right] \geq \Pr\left[\Bad_{k,s,\geq t}(n)\right]t^{n-s}\cdot \left(\frac{2^{n-s}n!}{s!}\right)^{2k}\cdot p^{-n+s}.$$ Combining this with the upper bound on the expectation, we get $$\Pr\left[\Bad_{k,s,\geq t}(n)\right]\leq \mathbb{E}[Z_s]\leq \left(\frac{s}{n}\right)^{2k-1}t^{-n+s},$$ as desired. Proof of Halász’s inequality over $\F_{p}$ {#app:halasz} ========================================== In this appendix, we prove \[thm:halasz\]. The proof follows Halász’s original proof in [@halasz1977estimates]. We start with the following discrete Fourier identity in $\F_p$: $$\delta_0(x) = \frac{1}{p} \sum_{k \in \F_p}e_p(kx),$$ where $e_p(x) = \exp(2 \pi i x/p)$. Note that for any $q \in \F_p$, $$\begin{aligned} \Pr_{x^{\mu}}\left[\sum_{i=1}^{n} a_i x_i^{\mu} = q\right] &= \E_{x^{\mu}}\left[\delta_0 \left(\sum_{i=1}^{n} a_i x_i^{\mu} - q\right)\right] \\ &= \E_{x^{\mu}}\left[ \frac{1}{p} \sum_{k\in \F_p} e_p\left(k \left(\sum_{j=1}^{n} a_j x_j^{\mu} - q\right)\right)\right] \\ &= \E_{x^{\mu}}\left[\frac{1}{p} \sum_{k \in \F_p} \prod_{j=1}^{n} e_p\left( k a_j x_j^{\mu} \right) e_p(-k q)\right] \\ &\leq \frac{1}{p} \sum_{k\in \F_p} \prod_{j=1}^{n} \left| \mu + (1-\mu)\cos\left(\frac{2 \pi k a_j}{p}\right) \right| \\ &= \frac{1}{p} \sum_{k \in \F_p} \prod_{j=1}^{n} \left|\mu + (1-\mu) \cos\left(\frac{\pi k a_j}{p}\right) \right|.\end{aligned}$$ At this point, we record the useful inequality $$\mu + (1-\mu)\cos\left(\frac{\pi x}{p}\right) \leq \exp\left(-\frac{1}{2}\left\|\frac{x}{p} \right\|^2\right)$$ uniformly for all $0\leq \mu \leq 1/2$, where $\|x\| := \|x\|_{\R/\Z}$ denotes the distance to the nearest integer. Thus, we arrive at $$\label{eqn:halasz-prelim} \max_{q\in \F_p}\Pr_{x^{\mu}}\left[\sum_{i=1}^{n} a_i x_i^{\mu} = q\right] \leq \frac{1}{p} \sum_{k\in \F_p} \exp\left(-\frac{1}{2} \sum_{j=1}^{n}\|k a_j/p \|^2\right).$$ Now, we define the following level sets $$T_t := \left\{k \in \F_p \mid \sum_{j=1}^{n} \|k a_j/p \|^2 \leq t \right\},$$ and note that $$\label{eqn:halasz-integral} \sum_{k\in \F_p} \exp\left(-\frac{1}{2} \sum_{j=1}^{n} \|k a_j/p \|^2\right) = \frac{1}{2}\int_0^{\infty} e^{-t/2}|T_t| dt.$$ We will now use a critical estimate due to Halász. First, note that for any $m\in \N$, the iterated sumset $mT_t$ is contained in $T_{m^2 t}$. Indeed, for $k_1,\dots, k_m \in T_t$, we have from the triangle inequality and the Cauchy-Schwarz inequality that $$\begin{aligned} \sum_{j=1}^{n} \left\| \sum_{i=1}^{m} k_i a_j/ p \right\|^2 \leq \sum_{j=1}^{n} \left(\sum_{i=1}^{m} \left\|k_i a_j/p \right\|\right)^2 \leq \sum_{j=n}^{m} m \sum_{i=1}^{m} \left \|k_i a_j/p\right \|^2 \leq m^2 t.\end{aligned}$$ Next, observe that $$\begin{aligned} \sum_{k \in \F_p} \sum_{j=1}^{n} \|k a_j/p \|^2 &\geq \sum_{j \in \operatorname{supp}(a)} \sum_{k \in \F_p} \|k a_j/p\|^2 \\ &= \sum_{j \in \operatorname{supp}(a)} (1/p)^2 + (2/p)^2+ \dots (p-1/2 / p)^2 + (p-1/2 / p)^2 + ((p-2)/p)^2 + \dots (1/p)^2 \\ &= \frac{2|\operatorname{supp}(a)|}{p^2} \sum_{i=1}^{(p-1)/2} i^2 \\ &\geq \frac{|\operatorname{supp}(a)|p}{50}.\end{aligned}$$ In particular, we see that $|T_t| < p$ if $t < |\operatorname{supp}(a)|/100$. Therefore, by the well-known Cauchy-Davenport theorem (which says that $|A+B| \geq \min\{p,|A|+|B|-1\}$), it follows that $$|T_t|\leq \frac{|T_{m^2t}|}{m} + 1,$$ provided that $m^{2}t \leq |\operatorname{supp}(a)|/100$. In particular, we see that $$\label{eqn:halasz-cd} |T_t| \leq \frac{\sqrt{t}T_{f(|\operatorname{supp}(a)|}}{\sqrt{f(|\operatorname{supp}(a)|)}} + 1,$$ where $f(|\operatorname{supp}(a)|)$ is as in the statement of the theorem.\ We now bound the size of $T_{f(|\operatorname{supp}(a)|)}$. Using the elementary inequality $1-100 \|z\|^2 \leq \cos(2 \pi z) $, which holds for all $z\in \mathbb{R}$, it follows that $|T_{f(|\operatorname{supp}(a)|)}| \leq |T'|$, where $$T' := \left\{k \in \F_p \mid \sum_{j=1}^n \cos(2 \pi k a_j/p) \geq n - 100 f(|\operatorname{supp}(a)|) \right\}.$$ In turn, we will bound the size of $T'$ by computing the moments of the random variable (over the randomness of $k\in \F_p)$ given by $\sum_{j=1}^n \cos\left(\frac{2 \pi k a_j}{p}\right)$. More precisely, by Markov’s inequality, we have for any $\ell \in \N$ that $$\label{eqn:halasz-moment} |T'|\leq \frac{1}{\left(n-100f(|\operatorname{supp}(a)|)\right)^{2\ell} }\sum_{k\in T'}\left|\sum_{j=1}^{n}\cos(\left(\frac{2\pi ka_j}{p}\right)\right|^{2\ell}.$$ Moreover, we also have $$\begin{aligned} \sum_{k \in T'} \left| \sum_{j=1}^n \cos\left( \frac{2 \pi k a_j}{p}\right)\right|^{2 \ell} &\leq \frac{1}{2^{2 \ell}}\sum_{k\in \F_p} \left| \sum_{j=1}^n (\exp( 2 i \pi k a_j/p) + \exp(- 2 i \pi k a_j/p))\right|^{2\ell} \\ &= \frac{1}{2^{2 \ell}}\sum_{\epsilon_1,\dots, \epsilon_{2\ell}} \sum_{j_1, \dots, j_{2 \ell}} \sum_{k \in F_p} \exp\left( 2 \pi i k \sum_{i=1}^{2 \ell} \epsilon_i a_{j_i}\right) \\ &= \frac{1}{2^{2\ell}}\sum_{\epsilon_1,\dots,\epsilon_{2\ell}} \sum_{j_1,\dots, j_{2\ell}} p \mathbbm{1}_{\sum_{i=1}^{2 \ell} \epsilon_i a_{j_i} = 0} \\ &\leq \frac{p R_\ell(a)}{2^{2 \ell}}.\end{aligned}$$ Finally, combining this with \[eqn:halasz-prelim,eqn:halasz-integral,eqn:halasz-cd,eqn:halasz-moment\], we get for any $0\leq \mu \leq 1/2$, $$\begin{aligned} \max_{q\in\F_{p}}\Pr_{x^{\mu}}\left[\sum_{i=1}^{n}a_{i}x_{i}^{\mu}=q\right] & \leq & \frac{1}{2p}\int_{0}^{f(|\operatorname{supp}(a)|)}e^{-t/2}|T_{t}|dt+\frac{1}{2}e^{-f(|\operatorname{supp}(a)|)/2}\\ & \leq & \frac{1}{2p}\int_{0}^{f(|\operatorname{supp}(a)|)}e^{-t/2}\left(\frac{\sqrt{t}|T'|}{\sqrt{f(|\operatorname{supp}(a)|)}}+1\right)dt+\frac{1}{2}e^{-f(|\operatorname{supp}(a)|)/2}\\ & \leq & \frac{|T'|}{2p\sqrt{f(|\operatorname{supp}(a)|)}}\int_{0}^{f(|\operatorname{supp}(a)|)}e^{-t/2}\sqrt{t}dt+\frac{1}{p}+\frac{1}{2}e^{-f(|\operatorname{supp}(a)|)/2}\\ & \leq & \frac{C_{1}|T'|}{p\sqrt{f(|\operatorname{supp}(a)|)}}+\frac{1}{p}+e^{-f(|\operatorname{supp}(a)|)/2}\\ & \leq & \frac{1}{p}+\frac{C_{1}R_{k}(a)}{2^{2k}\left(n-100f(|\operatorname{supp}(a)|)\right)^{2k}\sqrt{f(|\operatorname{supp}(a)|)}}+e^{-f(|\operatorname{supp}(a)|)/2}\\ & \leq & \frac{1}{p}+\frac{CR_{k}(a)}{2^{2k}n^{2k}\sqrt{f(|\operatorname{supp}(a)|)}}+e^{-f(|\operatorname{supp}(a)|)/2},\end{aligned}$$ as desired. [^1]: Massachusetts Institute of Technology. Department of Mathematics. Email: [ferbera@mit.edu]{}. Research is partially supported by NSF 6935855. [^2]: Massachusetts Institute of Technology. Department of Mathematics. Email: [visheshj@mit.edu]{}. Research is partially supported by NSF CCF 1665252, NSF DMS-1737944 and ONR N00014-17-1-2598.

--- abstract: 'We reply to the recent comments on our published papers, Phys. Rev. Lett. 109 (2012) 152005 and Phys. Lett. B717 (2012) 214. We point out that the criticisms about the transverse polarization parton sum rule we obtained are invalid.' author: - Xiangdong Ji - Xiaonu Xiong - Feng Yuan title: | Reply to arXiv:1211.3957 and arXiv:1211.4731 by Leader [*et al.*]{}\ and arXiv:1212.0761 by Harindranath [*et al.*]{} --- The comments by Leader [*et al.*]{} [@Leader:2012md] and by Harindranath [*et al.*]{} [@Harindranath:2012wn] on our Phys. Rev. Lett. paper [@Ji:2012sj] arise from a mis-understanding of our result. We have in fact published a longer paper [@Ji:2012vj] following the Letter, fully explaining what our partonic transverse spin sum rule mean. We reiterate that our result stands following a careful consideration of all pertinent issues. First of all, we remarked the simple fact that the ordinary transverse angular momentum (AM) does not commute with the longitudinal boost, and thus a frame-independent picture for the transverse spin is not the transverse AM alone, but the well-known Pauli-Lubanski (PL) spin $\hat W_\perp$ [@Ji:2012vj]. The PL spin is diagonalized in the transversely-polarized nucleon state with arbitrary longitudinal momentum. The PL spin is defined as $\hat W^\mu \sim \epsilon^{\mu\alpha\beta\gamma} \hat J_{\alpha\beta} \hat P_\gamma$, and we take the nucleon state with $P^\mu=(P^0,P_\perp=0,P^3)$ and replace $\hat P^\mu$ by its eigenvalue, so that $\hat W^\mu$ linearly depends on the angular momentum operator $\hat J^{ij}$, as well as the boost operator $\hat J^{0i}$. In the Letter paper, we restrict ourselves to the light-cone rest frame with residual momentum $P^3=0$, and thus only $P^+$ and $P^-$ do not vanish. If taking $\mu=1$, and $\alpha=2$, $\beta, \gamma$ to be $+$ and $-$, we have $W^1 \sim -\hat J^{2+}P^- + \hat J^{2-}P^+$. In light-cone quantization, the $\hat J^{2-}$ is a higher-twist contribution depending on products of three or four parton operators; however, its matrix element is related to that of $J^{2+}$ by simple Lorentz symmetry, and hence its contribution is considered known. Thus the leading-twist parton picture arises from $J^{2+}$ which is interaction-independent. One can obtain a simple partonic interpretation for this part related to the tensor $T^{++}$, as explained in the Letter paper, and the result is an integral over the intuitive parton transverse AM density $x(q(x)+E(x))/2$ and consistent with Burkardt wave-packet picture [@burkardt]. Thus, the key aspect of finding a partonic picture for transverse PL spin is to focus on the leading twist part and do away the other parts through Lorentz symmetry, a strategy first pointed out by Burkardt. Note that the spin operators of quarks and gluons do not contribute at the leading twist as they are now higher-twist operators in light-cone quantization. The PL vector was also the starting point of Ref. [@Harindranath:2012wn] and an earlier publication of the same authors, Ref. [@Harindranath:2001rc], in which the equations (2.6) and (2.7) reduce to $W^i$ when the external particle has no transverse momentum, $P^i=0$. One can easily find that they agree with our starting point of the discussion, contrary to the claim in their comment, Ref. [@Harindranath:2012wn]. Moreover, our conclusion does not contradict with that in Ref. [@Harindranath:2001rc]: In our longer version of Ref. [@Ji:2012vj], we find twist-3 and twist-4 parts of $W^i$ are interacting-dependent. Our new result [@Ji:2012sj] beyond Ref. [@Harindranath:2001rc] is that there is a twsit-two contribution of the transverse polarization which can be understood in a simple parton picture, related to the generalized parton distributions (GPD), whereas the interaction-dependent part is related to that of the twist-2 GPD contribution by symmetry. Finally, Leader in a separate note [@Leader:2012ar] criticized our light-front result when generalized to an arbitrary residual momentum frame [@Ji:2012vj]. A careful reading of our paper reveals that we have already commented on the role of higher term $\bar C$ in the paragraph following Eqs. (16) and (23). The frame-independence of our result remains to be true for the leading twist part, which is a consequence of the dependence of the transverse spin $\hat W_\perp$ on the very boost operator that serves to cancel the frame dependence of the transverse AM. [99]{} E. Leader and C. Lorce, arXiv:1211.4731 \[hep-ph\]. A. Harindranath, R. Kundu, A. Mukherjee and R. Ratabole, arXiv:1212.0761 \[hep-ph\]. X. Ji, X. Xiong and F. Yuan, Phys. Rev. Lett.  [**109**]{}, 152005 (2012) \[arXiv:1202.2843 \[hep-ph\]\]. X. Ji, X. Xiong and F. Yuan, Phys. Lett. B [**717**]{}, 214 (2012) \[arXiv:1209.3246 \[hep-ph\]\]. M. Burkardt, Phys.Rev. D72, 094020 (2005) \[hep- ph/0505189\]. A. Harindranath, A. Mukherjee and R. Ratabole, Phys. Rev. D [**63**]{}, 045006 (2001). E. Leader, arXiv:1211.3957 \[hep-ph\].

--- abstract: | We present X-ray/-ray spectra of the binary GX 339–4 observed in the hard state simultaneously by and [*CGRO*]{} OSSE during an outburst in 1991 September. The X-ray spectra are well represented by a power law with a photon spectral index of $\Gamma\simeq 1.75$ and a Compton reflection component with a fluorescent Fe K$\alpha$ line corresponding to a solid angle of an optically-thick, ionized, medium of $\sim 0.4\times 2\pi$. The OSSE data ($\geq 50$ keV) require a sharp high-energy cutoff in the power-law spectrum. The broad-band spectra are very well modelled by repeated Compton scattering in a thermal plasma with an optical depth of $\tau\sim 1$ and $kT\simeq 50$ keV. We also study the distance to the system and find it to be $\ga 3$ kpc, ruling out earlier determinations of $\sim 1$ kpc. Using this limit, the observed reddening and the orbital period, we find the allowed range of the mass of the primary is consistent with it being a black hole. We find the data are incosistent with models of either homogenous or patchy coronae above the surface of an accretion disc. Rather, they are consistent with the presence of a hot inner hot disc with the viscosity parameter of $\alpha\sim 1$ accreting at a rate close to the maximum set by advection. The hot disc is surrounded by a cold outer disc, which gives rise to the reflection component and a soft X-ray excess, also present in the data. The seed photons for Comptonization are unlikely to be due to thermal synchrotron radiation. Rather, they are supplied by the outer cold disc and/or cold clouds within the hot disc.  pair production is negligible if electrons are thermal. The hot disc model, which scaled parameters are independent of the black-hole mass, is supported by the similarity of the spectrum of GX 339–4 to those of other black-hole binaries and Seyfert 1s. On the other hand, their spectra in the soft -ray regime are significantly harder than those of weakly-magnetized neutron stars. Based on this difference, we propose that the presence of broad-band spectra corresponding to thermal Comptonization with $kT\ga 50$ keV represents a black-hole signature. author: - | \ $^1$N. Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland\ $^2$Stockholm Observatory, S-133 36 Saltsjöbaden, Sweden\ $^3$Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Cracow, Poland\ $^4$Laboratory for High Energy Astrophysics, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA\ $^5$E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, DC 20375, USA\ date: 'Accepted 1998 July 28. Received 1998 January 2' title: | Broad-band X-ray/$\bmath{\gamma}$-ray spectra and binary parameters\ of GX 339–4 and their astrophysical implications --- = -1cm == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} @mathgroup@group @mathgroup@normal@group[eur]{}[m]{}[n]{} @mathgroup@bold@group[eur]{}[b]{}[n]{} @mathgroup@group @mathgroup@normal@group[msa]{}[m]{}[n]{} @mathgroup@bold@group[msa]{}[m]{}[n]{} =“019 =”016 =“040 =”336 ="33E == == == == \#1[[ \#1]{}]{} \#1[[ \#1]{}]{} == == == == \[firstpage\] accretion, accretion discs – binaries: general – gamma-rays: observations – gamma-rays: theory – stars: individual (GX 339–4) – X-rays: stars. INTRODUCTION {#s:intro} ============ GX 339–4, a bright and well-studied binary X-ray source, is commonly classified as a black hole candidate based on the similarity of its X-ray spectral states and short-time variability to those of Cyg X-1 (e.g. Tanaka & Lewin 1995). However, determinations of the mass of its compact star, $M_{\rm X}$, have been inconclusive (e.g. Cowley, Crampton & Hutchings 1987, hereafter C87; Callanan et al. 1992, hereafter C92), and thus its nature has been uncertain. Therefore, further studies of the properties of GX 339–4 as well as their comparison to those of objects with more direct evidence for harbouring a black hole is of crucial importance. In this work, we present two, very similar, broad-band X-ray/-ray (hereafter X) spectra of GX 339–4 obtained during a strong outburst of the source in September 1991 (Harmon et al. 1994) simultaneously by (Makino et al.1987) and the Oriented Scintillation Spectroscopy Experiment (OSSE) detector (Johnson et al. 1993) on board the [*Compton Gamma Ray Observatory*]{} ([*CGRO*]{}). The source was in the hard (also called ‘low’) spectral state. The and OSSE observations were reported separately by Ueda, Ebisawa & Done (1994, hereafter U94) and Grabelsky et al. (1995, hereafter G95), respectively. However, the data from the two instruments have not been fitted together, and, e.g. G95 found models with Compton reflection of X photons from an accretion disc unlikely whereas U94 found strong evidence in the data for the presence of this process. Here, we re-analyze the simultaneous and OSSE data based on the present accurate calibration of those instruments. This leads to a reconciliation of the apparent discrepancies between the data sets from the two instruments, and allows us to fit the joint data with physical models. We also study the distance, reddening, Galactic column density and the masses of the binary members. Those results are then used in studying radiative processes, geometry and physical models of the source. Finally, we find the X spectrum of GX 339–4 similar to those of black-hole binaries and Seyfert AGNs, and, in particular, virtually identical to that of NGC 4151, the Seyfert brightest in hard X-rays. This favours physical models with scaled parameters independent of the central mass, such as a hot accretion disc with unsaturated thermal Comptonization (Shapiro, Lightman & Eardley 1976, hereafter S76). On the other hand, the spectrum of GX 339–4 is significantly different from those observed from neutron star binaries, which supports the black-hole nature of the compact object in GX 339–4. THE PARAMETERS OF THE BINARY ============================ In order to analyze the X-ray data meaningfully, we need to estimate basic parameters of the binary system. Of importance here are the Galactic column density, $\nh$, the interstellar reddening, $\ebv$, the distance, $d$ (for which published estimates range from 1.3 to 4 kpc), the masses of the primary and secondary, $M_{\rm X}$ and $M_{\rm c}$, respectively, and the inclination (with respect to the normal to the orbital plane), $i$. Reddening and column density ---------------------------- Grindlay (1979) found strong interstellar Na[i]{} D absorption lines and diffuse interstellar bands at $\lambda \sim 5775$–5795, 6010, 6176, 6284, and 6376 Å, while C87 found a strong interstellar Ca[ii]{} K absorption line and diffuse $\lambda 4430$ Å absorption band. The equivalent widths of these features are consistent with $\ebv \simeq 1$–1.3. From the uncertainties of the published estimates, we derive the weighted mean of $$\ebv=1.2 \pm 0.1\,.$$ The most extended all-sky study of the distribution of neutral H based on high-resolution [*IUE*]{} observations of Ly$\alpha$ absorption towards 554 OB stars shows their $\nh$ well correlated with the column density of dust, measured by $\ebv$, with $\langle \nh/\ebv\rangle = 4.93 \times 10^{21}\, {\rm cm^{-2}\, mag^{-1}}$ (Diplas & Savage 1994). $\ebv = 1.2 \pm 0.1$ derived above thus indicates $$\nh = (6.0 \pm 0.6) \times 10^{21} \rm cm^{-2}\,.$$ This $\nh$ is in excellent agreement with that derived from X-ray data. We obtain $\nh=(6.2\pm 0.7) \times 10^{21}$ cm$^{-2}$ from the depth of the O edge of $\tau_{\rm O}=2.6\pm 0.3$ measured by Vrtilek et al. (1991), and assuming the O abundance of Anders & Ebihara (1982). On the other hand, Vrtilek et al.(1991) and Ilovaisky et al. (1986) have obtained $\nh= (6.6\pm 0.3)\times 10^{21}$ cm$^{- 2}$ and $(5.0\pm 0.7)\times 10^{21}$ cm$^{- 2}$ from continuum fits in the soft and hard state, respectively. Those values are less reliable because those authors assumed the continuum models of optically-thin thermal bremsstrahlung and a single power law for the corresponding two states. The first model cannot hold for a luminous source based on the standard efficiency argument, and the second model disagree with data showing the presence of a strong soft X-ray excess in addition to the harder power law over two orders of magnitude of the flux in the hard state (U94). Therefore, we assume hereafter $\nh=6\times 10^{21}$ cm$^{-2}$, which value agrees with both optical and X-ray data. On the other hand, Ilovaisky et al. (1986) obtained $\ebv=0.7\pm 0.1$ by converting their fitted $\nh$ (which appears underestimated, see above) with an old calibration of $\nh/\ebv$, which underestimates $\ebv(\nh)$ according to the presently most extensive study of Diplas & Savage (1994). This value of $\ebv$ is also in disagreement with the results from interstellar features, and it appears incorrect. Distance -------- Obviously, reddening increases with distance. However, the resulting correlation depends sensitively on direction because the distribution of the interstellar matter is very complex and patchy (e.g. Diplas & Savage 1994; Neckel & Klare 1980), especially towards the Galactic Center, in which direction is the line of sight of GX 339–4. To illustrate this problem, we show here examples of a wide dispersion of $\ebv(d)$ with $l,b$ within $\pm 5\degr$ from the direction to GX 339–4. That field includes many OB stars in stellar systems, for which both $\ebv$ and $d$ have been well studied. First, the distribution of OB stars and dust in a 21-deg$^2$ field centered at $(l,b)= (335\degr,0\degr$) was studied in detail by FitzGerald (1987), who found the dust is distributed in two distinct clouds, one in the local arm at $190 \pm 30$ pc and the other in an interarm cloud at $690 \pm 70$ pc with $\langle\ebv\rangle = 0.21$ and 0.76, respectively. Three OB associations along this line of sight found at $\langle d\rangle= 1.34$, 2.41 and 3.69 kpc have $\ebv=0.82$, 0.86 and 0.92, respectively. On the other hand, the open clusters NGC 6200, NGC 6204 and Hogg 22 at $(l,b)\simeq (338.5,-1.2)$ show $\ebv\leq 0.66$ for $d \leq 2.78$, while NGC 6193 at $(l,b) = (336.7,-1.6)$ and NGC 6250 at (340.8,$- 1.8$) show $\ebv = 0.44$–0.49 at $d = 1.4$ kpc, and $\ebv = 0.38$ at $d = 0.95$ kpc, respectively (Moffat & Vogt 1973, 1975; FitzGerald et al. 1977; Vázquez & Feinstein 1992). Finally, the clusters NGC 6208 at (333.5,$- 5.7$), IC 4651 at (341,$-8$) and NGC 6352 at (342,$-7$) at $d=1$, 1.15 and 5 kpc show $\ebv= 0.18$, 0.12 and 0.4, respectively (Lindoff 1972; Alcaino 1971). These results clearly show that $\ebv(d)$ around GX 339–4 depends sensitively on direction. Therefore, we have determined $\ebv$ and $d$ for $\sim 450$ stars with $V\la 10$ within $\pm 5\degr$ around GX 339–4 using their $UBV$ magnitudes and spectral types from the catalogue of Mermilliod & Mermilliod (1994). Our study confirms the strong dependence of $\ebv(d)$ on direction and allows an accurate determination of $\ebv(d,l,b)$ in that field. We have found that $\ebv$ depends much more sensitively on $b$ than on $l$, and thus we present in Fig. 1 our results only for a smaller field centered at GX 339–4 with ($l,b) = (338.9 \pm 5{\degr}, -4.3 \pm 1{\degr}$). Stars with $l \la 338\degr$ and $\ga 340\degr$ are indicated by open circles and crosses, respectively. Dots represent stars in a central field with a radius of $\sim 1{\degr}$. From Fig. 1, it is clear that the extinction increases up to $\langle\ebv\rangle \approx 0.6$ at $d \approx 2$ kpc. There is a possible further increase beyond $\sim 3$ kpc, but to confirm it we would need a deeper survey than that of Mermilliod & Mermilliod (1994). Two stars nearest to GX 339–4 with known $UBV$ magnitudes and spectral types are the early B-type giants HD153222 and CPD$-48{\degr}11361$ at $(l,b) \approx (338{\degr},-4\degr)$, for which we have found $\ebv = 0.53$, $d \ga 2.2$ kpc, and $\ebv = 0.54$, $d \simeq 2.6$ kpc, respectively. Thus, our strong conclusion is that GX 339–4 with the measured $\ebv = 1.2 \pm 0.1$ must be located at $d> 2.6$ kpc. This result rules out the most recent estimate of $d=1.3$ kpc towards GX 339–4, obtained from a [*ROSAT*]{} measurement of its X-ray halo by Predehl et al. (1991). They suggest that the halo is produced by a dust cloud near the source, and identify that cloud with the second of two dense clouds found at $d \sim 100$–250 pc and $\sim 800$–1400 pc, respectively, by Neckel & Klare (1980). However, we have not been able to reproduce their identification. There are indeed two clouds shown in the direction of $l=339\degr$ in Fig. 9 of Neckel & Klare (1980), but that figure gives the cloud distribution [*projected*]{} onto the Galactic plane. In fact, Table IVa of Neckel & Klare (1980) shows that the field of GX 339–4 (numbered 196 in that paper) does not contribute to extinction in those clouds, as well as Fig. 8c of that paper shows that $A_V$ at 1 kpc in that field is $<1.9$ (whereas $A_V\sim 3.4$–4 corresponds to $\ebv=1.2\pm 0.1$ of GX 339–4). These findings are confirmed by a more extensive study of Galactic extinction by FitzGerald (1987). She shows the same clouds are located in the field of Ara OB 1 association at $(l,b) \simeq (335\degr, 0\degr)$, confirming that they [*do not*]{} concide with the line of sight to GX 339–4. Moreover, the total $\ebv \approx 0.8$ at $\sim 1.34$ kpc (FitzGerald 1987) produced by the two clouds is lower than the $\ebv \simeq 1.2$ measured for GX 339–4 (and adopted by Predehl et al. 1991 in their analysis). Similarly, $d=1.33$ kpc derived by Mauche & Gorenstein (1986) for GX 339–4 from [*Einstein*]{} observation of the X-ray halo corresponds just to the Galaxy’s dust layer, and thus it represents a lower limit to the distance. That limit can be actually higher ($d \ga 2$ kpc) since recent studies (Diplas & Savage 1994) show a value of the scaleheight for the dust layer of 150 pc that is larger than 100 pc adopted by Mauche & Gorenstein. Those two scaleheights predict $\ebv \sim 0.5$ and $\sim 0.8$, respectively, both lower than the $\ebv$ of GX 339–4. On the other hand, the distance to GX 339–4 can be estimated from its systemic velocity of $V_0 = -62 \pm 10$ km s$^{-1}$ (C92). Using the $V_0$-$d$ conversion chart from Burton (1992), the kinematic distance is $d = 4 \pm 1$ kpc. (Note that a possible peculiar velocity of the system as well as a systematic uncertainty of the above $V_0$ makes this estimate less secure than the extinction limit.) We note here that both the studies of distribution of matter in the Galaxy as well as the measurements of radial velocities of H[ii]{} regions in the range $l \approx 300{\degr}$–$340{\degr}$ indicate the presence of a spiral arm at a distance of $\sim 4$ kpc (Scheffler & Elsasser 1987 and references therein). Our derived kinematic distance and the relatively large $\ebv$ of GX 339–4 are consistent with its location in that arm. Finally, we can put an upper limit on $d$ from the Eddington limit, which appears to be satisfied in all known black-hole binaries (Tanaka & Lewin 1995). The most luminous state of GX 339–4 reported as yet appears to be a soft state in 1988 (Miyamoto et al. 1991), for which we estimate the bolometric flux (using the disc blackbody model, see Section 5.2) to be $\sim 4\times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$. This corresponds to $d\simeq 10 (L/L_{\rm E})^{1/2} (M_{\rm X}/ 3 M_\odot)^{1/2}$ kpc, where $L_{\rm E}$ is the Eddington luminosity (see Section 2.3 for discussion of $M_{\rm X}$). On the other hand, typical maximum luminosities of black-hole binaries are at least factor of a few below $L_{\rm E}$ (Tanaka & Lewin 1995), and, e.g. $d\simeq 4.5$ kpc if $L=0.2L_{\rm E}$ and $M_{\rm X}=3M_\odot$. Concluding this section, our strongest limit is $d \ga 3$ kpc based on the extinction study. This limit is consistent with the kinematic distance of $d = 4 \pm 1$ kpc and with the Eddington limit. Thus, we adopt $d=4$ kpc hereafter. The masses and geometry ----------------------- With the above results on the $\ebv$ and $d$, we can constrain the mass of the companion star. In general, we expect $M_{\rm c}\la 1M_\odot$ in low-mass X-ray binaries (hereafter LMXBs, e.g. van Paradijs & McClintock 1995), to which class GX 339–4 most likely belongs. At $d \simeq 4$ kpc and $A_V \sim 3.4$–4, a $1 M_\odot$ main sequence star has an observed $V\simeq 20.5$–21 ($B-V \simeq 1.8$, $V-R_{\rm c} \simeq 1$, $V-I_{\rm c} \simeq 2$), which corresponds to fluxes below the faintest ones observed as yet from GX 339–4, as illustrated in Fig. 2. Thus, the presence of a $1 M_\odot$ secondary star is entirely consistent with the present data. We note that our conclusion differs from that of C92, who claimed $M_{\rm c} \la 0.4M_\odot$. The reason for the discrepancy is that those authors assumed $d=1.3$ kpc (from Predehl et al.1991) and $\ebv=0.7$ (from Ilovaisky et al. 1986), which both values have been found by us to be incorrect, see above. =4.5cm C92 detected a 14.8-h flux modulation in both optically-high and low states. They also found the same periodicity in the radial velocities of He[ii]{} 4686 and H[i]{} Balmer emission lines measured in low resolution spectra from 1986 May 7–9 (C87). We caution here that interpreting this periodicity as due to the orbital period remains relatively uncertain. First, the periodicity is absent in the radial velocity data from 1985 March 14–17 of C87, although it is possibly due to their poorer quality. Second, although C87 claimed that the source was in its ‘normal’ (high optically) state of $\sim 16.5$ mag both in 1985 March and 1986 May, C92 noted a displacement in the mean velocity between those data from $\sim +96$ km s$^{-1}$ to $-62 \pm 10$ km s$^{-1}$, as well as Ilovaisky et al. (1986) reported a relatively faint optical state from late 1985 March ($B \sim 18.8$) through 1985 April 29, which contradict this claim of C87. Third, Corbet et al. (1987) shows strong variability around 1986 May. Namely, their optical photometry from 1986 April and June/July shows variability by several tenths of magnitude during each observing night and an overall brightening by 0.5 mag from April to June. On the other hand, the equivalent widths of both He[ii]{} 4686 and H$\beta$ lines for both data sets are practically the same (C87), and similar to the mean equivalent widths observed in LMXBs (Smale 1996). However, these lines can be formed in either the heated hemisphere of the secondary, the accretion disc, and/or the accretion stream, depending both upon the object and/or its state of activity, which precludes an unambiguous interpretation of the spectroscopic periodicity. Furthermore, we note that C92 do not state whether the 14.8-h periodicity found by them in the (optically) high-state photometry of Corbet et al. (1987) was present in both 1986 April and June/July or in only one of those data sets. C92 also claim that if the maximum in the light curve (presented in their Fig. 6) corresponds to transit of the compact object in front of its companion, the radial velocity changes (shown in their Fig. 7) are consistent with an origin near the compact object. However, this is not the case because at both spectroscopic conjunctions (i.e., the transits of the compact star both in front and behind the companion) the observed radial velocity should be just the systemic velocity. Keeping these doubts in mind let us tentatively assume that the 14.8-h modulation is still due to the orbital motion. The semi-amplitude, $K_{\rm X} = 78 \pm 13$ km s$^{-1}$ (C92; the uncertainty is 1-$\sigma$) implies the secondary mass function of $$f\equiv {M_{\rm c} \sin^3 i\over (1+M_{\rm X}/M_{\rm c})^2}= 0.030^{+0.018}_{-0.012} M_\odot\,,$$ which corresponds to the mass of the primary of $$M_{\rm X}= M_{\rm c}\left[ \left(M_{\rm c} /f\right)^{1/2} \sin^{3/2} i -1\right]\,.$$ From the lack of eclipses, the inclination can be constrained to $i\la 60\degr$ (C87). If we adopt $M_{\rm c}=1M_\odot$ (see above), we obtain $M_{\rm X}\la 5M_\odot$, where the equality corresponds to the minimum $f$ and maximum $i$. On the other hand, the lack of X-ray dips and fits of models with Compton reflection (see Section 4) both favour $<60\degr$, and, e.g., $i= 45\degr$ yields $M_{\rm X}\la 3.4M_\odot$. Current theoretical (see, e.g. Haensel 1995 for a review) and observational (e.g. van Paradijs & McClintock 1995) estimates yield the maximum mass of a neutron star of $\la 2M_\odot$. On the other hand, if we assume that the compact object is a black hole, as strongly suggested by the similarity of its spectral and timing behaviour to Cyg X-1, then the presumed $M_{\rm X}\ga 2M_\odot$ corresponds to $M_{\rm c}\ga 0.5M_\odot$ (at the lower limit of $f$). Thus, the mass function of C92 together with the constraints on $M_{\rm c}$ are fully consistent with the presence of a black hole in GX 339–4 (although the presence of a neutron star is also allowed). Hereafter, we will adopt fiducial values of $M_{\rm X}=3M_\odot$ and $i=45\degr$, which are around the middle of the allowed parameter space. Then, $P_{\rm orb} = 14.8$ h and $M_{\rm X}+ M_{\rm c} \sim 4 M_{\odot}$ imply a $1.3 R_{\odot}$ Roche-lobe (tidal) radius for the secondary. A post-main sequence $1 M_{\odot}$ star with $R \sim 1.3 R_{\odot}$ (and thus filling its Roche lobe) would have a luminosity of $\sim 1 L_{\odot}$ and an effective temperature (unheated) of $\sim 5100$ K (Webbink, Rappaport & Savonije 1983). Such star at $d=4$ kpc and $\ebv= 1.2$ would have $V \sim 21.6$, $B-V \sim 2$, $V-R_{\rm c} \sim 1.1$, and $V-I_{\rm c} \sim 2.3$, which correspond to fluxes below those observed from GX 339–4 (similarly to the case of a $1M_\odot$ main-sequence star, see Fig. 2). The expected Roche-lobe overflow rate, $\dot M$, in such a system is several times $10^{-10} M_{\odot}\, {\rm yr}^{-1}$ (Webbink et al. 1983), which is sufficient to power the average X emission from GX 339–4 (Rubin et al. 1998) of $\sim 2\times 10^{36}$ erg s$^{-1}$ assuming an accretion efficiency of $\eta=0.06$. On the other hand, $\dot M$ can be significantly higher due to X irradiation of the companion star (Podsiadlowski 1991; Tavani & London 1993), which would imply a lower $\eta$. In this geometry, the secondary star subtends a solid angle of $\sim 0.018\times 4\pi$. Thus, the secondary provides a negligible contribution to Compton reflection in GX 339–4 (Section 4). AND OSSE DATA {#s:data} ============== GX 339–4 was observed by on 1991 September 11 and 12 (U94), and by OSSE 1991 September 5–12 (G95). We have extracted the OSSE data for September 11 and 12 for periods approximately coinciding with the observations. The log of the (from U94) and OSSE observations used here is given in Table 1. Both the and OSSE fluxes during September 11–12 were approximately constant (U94; G95). \[t:log\] ---------- --------------- ---------------- ---------------- ---------------- --------------------- ---------------- ---------------- ---------------- --------------------- Data set Date Start End Livetime \[s\] Counts \[s$^{-1}$\] Start End Exposure \[s\] Counts \[s$^{-1}$\] 1 1991 Sept. 11 $03^{h}35^{m}$ $05^{h}26^{m}$ 1952 $1716\pm 4$ $03^{h}25^{m}$ $05^{h}47^{m}$ 4795 $13.14\pm 0.26$ 2 1991 Sept. 12 $00^{h}54^{m}$ $01^{h}11^{m}$ 384 $1695\pm 5$ $01^{h}10^{m}$ $01^{h}59^{m}$ 2488 $12.71\pm 0.35$ ---------- --------------- ---------------- ---------------- ---------------- --------------------- ---------------- ---------------- ---------------- --------------------- We adopt the current best estimate of the effective area of the Large Area Counter (LAC) of (Turner et al. 1989) of 4661 cm$^2$ (D. Smith, private communication). The usable energy range of is 1.2–29 keV for these observations. A 1 per cent systematic error is added in quadrature to the statistical error in each channel (as in U94). The OSSE data are from 50 keV to 1000 keV. They include energy-dependent systematic errors estimated from the uncertainties in the low-energy calibration and response of the detectors using both in-orbit and prelaunch calibration data, and correspond to an uncertainty in the effective area in the OSSE response. They are most important at the lowest energies ($\sim 3$ per cent at 50 keV, decreasing to $\sim 0.3$ per cent at $\ga 150$ keV). ELEMENTARY SPECTRAL MODELS {#s:fits} ========================== In this section, we analyze the and OSSE data using idealized models of blackbody emission, and power-law or Comptonization emission from a hot, isotropic, thermal plasma cloud including their Compton reflection from an underlying slab. Here, we assume the reflected component is not Comptonized by the hot plasma. This allows us to characterize spectral components present in the X spectrum in a phenomenological, but relatively model-independent, way. Similar models are widely used in literature to model spectra of other black-hole sources, e.g. Cyg X-1 (Ebisawa et al. 1996) or Seyfert 1s (Nandra & Pounds 1994), and thus the results of this Section allow a direct comparison of the obtained parameters with those for other objects. On the other hand, we introduce more realistic geometries in context of specific models and treat Comptonization of a fraction of the reflected spectrum as well as energy balance in Sections \[s:geo\]–\[s:soft\] below. For spectral fits, we use [xspec]{} (Arnaud 1996) v10. The confidence ranges of each model parameter are given for a 90 per cent confidence interval, i.e., $\Delta \chi^2=2.7$ (e.g. Press et al. 1992). On the other hand, the plotted vertical error bars are 1-$\sigma$, the upper limits, 2-$\sigma$, and the plotted spectral data are rebinned for clarity of the display. Model spectra are attenuated by $\nh=6\times 10^{21}$ cm$^{-2}$ (Section 2.1). Since the exposure is much longer for the data set 1, we discuss below results obtained with that set. However, we give fit results for both data sets in Table 2. As found by U94, the spectra of GX 339–4 in the hard state consist of four main components: an underlying power law, a soft excess below $\sim 4$ keV, a continuum due to reflection from the surface of an ionized accretion disc (e.g. Lightman & White 1988; George & Fabian 1991), and a fluorescent Fe K$\alpha$ line. For modeling Compton reflection, we use inclination-dependent Green’s functions of Magdziarz & Zdziarski (1995) (instead of the angle-averaged reflection spectrum used in U94), which assumes an isotropic point source (or, equivalently, an optically-thin corona) above a slab. The Green’s functions are convolved with an incident continuum (a power law or a thermal Comptonization spectrum). The reflector inclination is kept at $i=45\degr$ (unless stated otherwise). We treat the solid angle, $\Omega$, subtended by the reflector as a free parameter. Values of $\Omega<2\pi$ may correspond either to a truncation of the reflecting disc, or to substantial Comptonization of the reflected spectrum (or both). The ionization parameter of the reflector, $\xi=L_{\rm ion}/nr^2$, is assumed to be uniform in the reflecting material. Here $L_{\rm ion}$ is defined as the 5 eV–20 keV luminosity in a power law spectrum and $n$ is the density of the reflector located at distance $r$ from the illuminating source (Done et al. 1992). The reflector temperature is kept at $10^6$ K, which is the highest temperature consistent with the model of ionization equilibrium used (Done et al. 1992). This temperature is consistent with our estimate of the origin of reflection from a larger area than that giving rise to the observed soft excess, see Section 5.4. The abundances of both the reflector and the interstellar medium are from Anders & Ebihara (1982) except that the relative Fe abundance in the reflector, $\af$, is a free parameter. The ion edge energies and opacities are from Reilman & Manson (1979, used in U94) except that now the Fe K-edge energies are from Kaastra & Mewe (1993). The continuum reflection is accompanied by an Fe K$\alpha$ line, which we model as a Gaussian centered at an energy, $\efe$, with a width, $\sfe$, and a flux, $\ife$. [lccccccccccccc]{} No. & $A$ & $\Gamma$ &$kT$ &$L$ &$\Omega/2\pi$ &$\xi$ &$\af$ &$\tbb$ &$L_{\rm bb}$ &$\efe$ &$\ife$ &$\wfe$ &$\chi^2$/dof\ \ 1a & 0.75 & $1.75^{+0.02}_{-0.03}$ & – & – & $0.37^{+0.06}_{-0.05}$ & $120^{+160}_{-70}$ & $2.5^{+1.2}_{-0.8}$ & – & – & $6.51_{-0.32}^{+0.31}$ & $1.5_{-0.8}^{+0.7}$ & $49^{+26}_{-26}$ & 11.7/27\ 1b & 0.75 & $1.76^{+0.02}_{-0.01}$ &$56^{+6}_{-6}$ & – &$0.43^{+0.03}_{-0.07}$ &$110^{+120}_{-50}$ &$2.9^{+0.1}_{-0.7}$ & – & – & $6.51^{+0.16}_{-0.35}$ &$1.3^{+0.8}_{-0.7}$ & $45^{+26}_{-25}$ &45.2/79\ 1c & 0.60 & $1.77^{+0.01}_{-0.01}$ &$57^{+7}_{-5}$ &3.12 &$0.44^{+0.06}_{-0.06}$ &$90^{+90}_{-40}$ &$3.0^{+0}_{-0.7}$ &$0.25^{+0.02}_{-0.03}$ &0.29 &$6.54^{+0.33}_{-0.36}$ &$1.3^{+0.7}_{-0.8}$ &$44^{+25}_{-26}$ &47.8/82\ \ 2a & 0.73 & $1.74^{+0.03}_{-0.04}$ & – & – & $0.25^{+0.08}_{-0.08}$ & $230^{+510}_{-180}$ & $1.6^{+2.2}_{-0.8}$ & – & – & $6.56_{-0.37}^{+0.34}$ & $1.4_{-0.9}^{+0.9}$ & $48^{+31}_{-32}$ & 22.4/27\ 2b & 0.75 & $1.76^{+0.02}_{-0.02}$ & $53^{+8}_{-7}$ & – &$0.29^{+0.09}_{-0.08}$ &$170^{+310}_{-130}$ &$2.0^{+1.0}_{-0}$ & – & – & $6.57^{+0.37}_{-0.41}$ &$1.3^{+0.9}_{-0.9}$ & $44^{+31}_{-31}$ &61.5/79\ 2c & 0.55 & $1.76^{+0.02}_{-0.03}$ &$52^{+7}_{-6}$ &2.93 &$0.29^{+0.08}_{-0.08}$ &$170^{+300}_{-120}$ &$2.0^{+1.0}_{-0}$ &$0.27^{+0.03}_{-0.03}$ &0.21 &$6.56^{+0.35}_{-0.38}$ &$1.4^{+0.8}_{-1.0}$ &$46^{+30}_{-31}$ &62.5/82\ We first fit the data set 1 from in the 4.1–29 keV range only, where a contribution from the soft excess was found to be negligible by U94. We first use a model consisting of a power law and a line. This model provides a very poor description of the data, with $\chi^2=209/30$ d.o.f. The pattern of residuals is characteristic of the presence of Compton reflection, as shown in Fig. 2a of U94. Indeed, adding a reflection component improves the fit dramatically, reducing $\chi^2$ to $11.7/27$ d.o.f., see fit 1a in Table 2 (which also give corresponding results for the data set 2) and Fig. 3. Compton reflection is present at very high statistical significance, with a probability of $<10^{-16}$ that adding this component were not required by the data (as obtained using the F-test). The statistical significances of the reflector being ionized, overabudant in Fe, and an Fe K$\alpha$ being present in the spectrum correspond to the probability that the resulting fit improvement were by chance of $<10^{-4}$, $3\times 10^{-7}$ and $3\times 10^{-4}$, respectively. The width of the line is only weakly constrained, $\sfe\la 0.5$ keV at 1$\sigma$, and thus we keep it fixed at 0.1 keV hereafter. We comment here on the low reduced $\chi^2$ obtained for the data set 1. This is due to the 1 per cent systematic error added to the statistical error. This value appears to be an overestimate of the residual inaccuracy of the calibration of the LAC, for which 0.5 per cent appears to be a better estimate. However, we have retained here the statistical error adopted by U94 in order to enable direct comparison with their results. This leads to uncertainties on the parameters of our fits being more conservative than necessary, but it affects neither the best-fit values nor the conlusions regarding the spectral components present in the spectrum. E.g.Compton reflection is still found at a very high significance, see above. The addition of the large systematic error also leads, in some cases, to a relatively low reduced $\chi^2$ for models showing systematic (rather than random) departures from the data, which cases we discuss below individually. On the other hand, the reduced $\chi^2$ is significantly larger for the data set 2, see Table 2. This is due to its much shorter exposure, resulting in typical statistical errors $>1$ per cent, which then reduces the relative importance of the systematic errors. We have then tested whether there is any indication in the data for the presence of kinematic and relativistic smearing of the Fe K$\alpha$ and the reflection continuum. Such smearing would be present if those spectral components were formed close to the central black hole, which effect has been found in observations of the X-ray novae GS 2023+338 and Nova Muscae (Życki, Done & Smith 1997, 1998). We have used a model in which the line and the static reflection continuum were convolved with the disc line profile of Fabian et al. (1989), similar to the model of Życki et al. (1997, 1978). However, we have found no fit improvement ($\Delta\chi^2=-0.6$) allowing for the smearing, although the data also do not allow us to rule it out. The relative normalization of the reflected components, $\Omega/2\pi$, corresponds to $\sim 0.6$ of those obtained by U94. This is explained mostly by their use of the angle-averaged spectrum, which underestimates the actual reflection spectrum for angles $<65\degr$, which effect increases at energies $\ga 15$ keV (Magdziarz & Zdziarski 1995). Accordingly, the fit to the data set 1 with the angle-averaged reflection in U94 shows strong positive residuals above $\sim 20$ keV (Fig. 2b in U94), which systematic residuals disappear completely in the present fit, see Fig. 3. Also, we obtain $\xi$ about 2–3 times less than those of U94. However, the ionization state depends on both $\xi$ and the reflector temperature, and the difference is due to the assumption of U94 that the latter is $10^5$ K (which seems unlikely in inner regions of luminous compact objects in binaries). We then investigate the issue of the Fe abundance. We have fitted the earlier four hard-state observations of GX 339–4 by in 1989–90 (U94), and we found all of them consistent with $2\leq \af\leq 3$, which is also the case for the present data. Therefore, we constrain $\af$ to this range hereafter. In agreement with U94, we find a strong excess below 4 keV, with the fluxes in the 1.2–1.7 keV and 1.7–2.3 keV bands about 40 and 20 per cent, respectively, higher than the extrapolation of the model fitted to the 4.1–29 keV range, as shown in Fig. 3. Although the exact form of the excess depends on the adopted $\nh$, we find it to be significant, with the respective 1.2–1.7 keV and 1.7–2.3 keV relative excess of 25 and 12 per cent even for $\nh=4\times 10^{21}$ cm$^{-2}$, which is much below the optical and X-ray estimates (Section 2.1). Thus, we conclude that the presence of the soft X-ray excess is not an artefact of our choice of $\nh$. At 50 keV, the extrapolated fit predict the flux about 10 per cent higher than that observed by OSSE. Above 50 keV, the OSSE data show a strong cutoff, as illustrated in Fig. 3. We test whether the cutoff can be modelled by thermal Comptonization, similarly to the case of Cyg X-1 (Gierliński et al. 1997, hereafter G97) and Seyfert AGNs (e.g. Zdziarski et al. 1997, hereafter Z97). We fit first the data above 4.1 keV jointly with the OSSE data. The incident continuum is fitted by thermal Comptonization in a spherical cloud with a central source of soft seed photons having a blackbody distribution, see Zdziarski, Johnson & Magdziarz (1996). As discussed in that paper, the solution of the Kompaneets equation with relativistic corrections used there leads to an overestimation of the actual plasma temperature, $T$, when the Thomson depth of the plasma, $\tau$, is $\la 2$. Therefore, we correct here the values of $T$ obtained from the Kompaneets equation by a function (R. Misra, private communication) obtained by comparison with corresponding Monte Carlo results (Zdziarski et al. 1996). The second parameter of the model is the asymptotic power-law index in X-rays, $\Gamma$, which is related to the (geometry-dependent) $\tau$ by $$\tau\simeq \Theta^{-1/2} \left[ \left(\Gamma+{1\over 2}\right)^2 - {9\over 4}\right]^{-1/2},$$ where $\Theta\equiv kT/m_{\rm e} c^2$ and $m_{\rm e}$ is the electron mass. In fits, we initially assume the seed photon temperature of $\tbb =0.1$ keV; as long as $\tbb$ is much less than the minimum energy in the fitted data, its choice does not affect the fit results. This model provides a very good description of the data (see fits 1b and 2b in Table 2). The best fit corresponds to $\tau\simeq 1.8$. As an independent check of our results, we have also used a model of Coppi (1992), which treats thermal Comptonization using the formalism of escape probability. That model yields $kT\simeq 48$ keV, $\tau\simeq 1.93$, $\Omega/2\pi=0.42$ as the best fit ($\chi^2=49/79$ d.o.f.), rather similar to the fit 1b in Table 2. On the other hand, we find that the high-energy cutoff seen it the data is poorly modelled by an e-folded power law, which, apart from its mathematical simplicity, does not correspond to any physical model. We find that model fits the data much worse than thermal Comptonization, with $\Delta \chi^2=+19$ resulting in a systematic pattern of residuals in the OSSE data. This argues for thermal Comptonization being indeed the process giving rise to the intrinsic spectrum. =8.4cm We have considered the effect of possible inaccuracy of the relative normalization of the effective area of and OSSE. We find, however, no fit improvement, $\Delta \chi^2=-0.1$, when the relative normalization of OSSE with respect to is allowed to be free (reaching 0.98 for both data sets). Therefore, we keep it at unity hereafter. We then check the effect of changing the disc inclination. We find that allowing a free $i$ leads to a negligible fit improvement: $\chi^2= 44.3/78$ d.o.f. at $i=23_{-23}^{+38}$ deg. Thus, we keep it at $i=45\degr$ (see Section 2.3) hereafter. At the largest $i$ allowed by the fit, $\Omega/2\pi =0.62$ and $\wfe=35$ eV. We then consider the soft X-ray excess. In the hard state of Cyg X-1, the soft excess contains a blackbody component with $\tbb\approx 0.14$ keV (e.g.Ebisawa et al. 1996). In addition, the data for Cyg X-1 show also a break around 3–4 keV with the power law below the break being softer by $\Delta \Gamma \sim 0.4$ on average (Ebisawa et al. 1996). The physical origin of that spectral break is unclear. We find that the soft X-ray excess in our data is much better fitted by an additional blackbody component than by a broken power law ($\Delta \chi^2=5$). (Our data extend down to 1.2 keV only with a low energy resolution, and thus do not allow us to determine the presence of more than one spectral component of the soft excess.) In the model with a blackbody component, we set its temperature equal to the blackbody temperature of the seed photons in the Comptonization source. (We postpone examining the energy balance of the source to Sections 5.1–5.2 below.) This model gives an excellent description of the /OSSE data in the entire energy range, 1.2–1000 keV, see Fig. 4 and fits 1c and 2c in Table 2. Still, according to the above discussion, the fitted value of $\tbb$ is relatively uncertain; if there were a low-energy break in the power-law spectral component (as in Cyg X-1), $\tbb$ would be lower. PHYSICAL IMPLICATIONS ===================== Geometry and energy balance {#s:geo} --------------------------- We now consider thermal Comptonization, Compton reflection and reprocessing in realistic geometries approximating the structure of an accretion flow. We take into account anisotropy of both seed and Compton-scattered photons, Comptonization of the reflected photons that return to the hot plasma, and energy balance. In general, thermal Comptonization takes place in a hot plasma cloud, and the resulting spectrum is Compton-reflected from a cold medium. A major part of the Comptonized flux incident on the cold medium is bound-free absorbed rather than reflected, which gives rise a flux of reprocessed soft photons with a spectrum close to a blackbody. In turn, a geometry-dependent fraction of the reflected and the blackbody emissions returns to the hot plasma. These blackbody photons then serve as seeds for thermal-Compton upscattering (in addition to any other soft photons present). Those reflected photons that return to the hot plasma are upscattered, which leads to a strong suppression of that component for $\tau\ga 1$. A geometry can be ruled out either if the resulting model spectrum does not provide a good fit to the observed spectrum, or if energy balance between the hot plasma and the source of seed photons cannot be achieved. The condition of energy balance can be expressed as, e.g. the power in soft photons incident on the hot plasma (hereafter seed photons) times the Comptonization amplification factor (a function of $kT$ and $\tau$) being equal the power emitted by the hot plasma, $L_{\rm hot}$. The flux in seed photons arises due to both internal dissipation and reprocessing of the Comptonized radiation. Then, a sufficient condition to reject a model is (i) the Comptonized flux (at fitted $kT$ and $\tau$) larger than that observed, due to a strong flux in seed photons arising from reprocessing. The condition (i) is equivalent to either statement: (ii) at the fitted $\tau$ and the observed hard flux, Compton cooling by the seed photons would result in a $kT$ less than that obtained from fitting, or (iii) at the fitted $kT$ and the observed hard flux, energy balance can be achieved for values of $\tau$ less than that fitted to the data (e.g. Haardt & Maraschi 1993; Stern et al. 1995). Below, we first obtain $kT$ and $\tau$ by spectral fitting, and then a posteriori check the energy balance by finding the equilibrium temperature corresponding to the fitted $\tau$ in given geometry \[condition (ii) above\]. In this section, we neglect spectral constraints from the soft X-ray excess (but return to this issue in Section \[s:soft\] below) because the origin of the observed soft excess is relatively uncertain, see the end of Section 4. Therefore, we consider now only the data above 4 keV, and fix the seed-photon temperature at $\tbb=0.25$ keV (as obtained in Table 2), which is about the highest seed-photon temperature allowed by the data regardless of the actual form of the soft excess. For a lower $\tbb$ at a given seed-photon flux from reprocessing, the predicted Comptonized flux would increase, which would in turn lead to rejection of a larger class of models \[see condition (i) above\]. Thus, setting $\tbb= 0.25$ keV is the most conservative assumption for model rejection. To model Comptonization in anisotropic geometries, we use the iterative scattering method of Poutanen & Svensson (1996). For Compton reflection, we use the same method as in Section 4 (Magdziarz & Zdziarski 1995). Fig. 5 shows the geometries considered in Sections \[s:geo\]–\[s:soft\]. =5.2cm We first consider a homogeneous corona (with the vertical optical depth of $\tau$) covering a cold slab (Haardt & Maraschi 1993), see Fig. 5a. In this model, all Compton-reflected radiation emitted by the cold slab is Comptonized in the hot corona, which leads to a suppression of this component to a level much less than that present in the data. Still, the model can provide a good spectral fit if we allow an additional Compton-reflection component (with $\Omega/2\pi\sim 0.3$) due to an outside cold medium, e.g. an outer cold disc. The fitted plasma parameters are then $kT= 55^{+8}_{-7}$ keV and $\tau= 1.2^{+0.2}_{-0.3}$ ($\chi^2= 45/79$ d.o.f.). However, this model is strongly ruled out by the requirement of energy balance. Namely, the reprocessed flux from the slab is so strong that it would cool down the corona to $\sim 27$ keV. Allowing for internal dissipation in the cold slab (presumably a cold accretion disc) would worsen the discrepancy. We then consider a patchy corona geometry (Galeev, Rosner & Vaiana 1979; Haardt, Maraschi & Ghisellini 1994; Stern et al. 1995). The hot plasma is postulated here to form active regions above a cold accretion disc. First, we assume the active regions to form hemispheres located directly on the surface of the cold disc, see Fig. 5b. We obtain a good spectral fit ($\chi^2=47/79$ d.o.f.) for $kT=64^{+9}_{-8}$ keV and the radial $\tau= 2.1^{+0.2}_{-0.3}$. However, similarly to the case of a homogeneous corona, we find cooling by the seed photons arising from reprocessing in an underlying part of disc is so large that the plasma with the fitted $\tau$ would cool to $kT=45$ keV, i.e., it cannot sustain the temperature implied by the data in this model. The cooling, however, can be reduced if the active regions are at some distance above the disc, as shown in Fig. 5c. We find that at the height of about one radius, there is an energy balance between the disc without internal dissipation and the hot region. The height has to be larger if there is internal dissipation in the disc. However, this model implies much more Compton reflection (from disc regions surrounding the active region) than observed. Even if we assume that the underlying disc is truncated, which would reduce the amount of reflection, the spectral fit of this model is much poorer than other models, $\chi^2=53/79$ d.o.f. (for $kT=62^{+14}_{-4}$ keV, $\tau= 2.2^{+0.2}_{-0.3}$, $\Omega/2\pi= 0.71^{+0.10}_{-0.07}$). The cause of the bad fit is the presence of an anisotropy break in the model spectrum, i.e., a defficiency of low-energy photons emitted upward due to an anisotropic suppression of the first order of Compton scattering (Haardt & Maraschi 1993; Poutanen & Svensson 1996). This effect is not seen in the data, and thus this model yields systematic departures from the data at low energies. On the other hand, a good fit can be obtained if $kT_{\rm bb} \la 0.1$ keV is assumed (which yields $kT=57^{+6}_{-4}$ keV, $\tau= 2.5^{+0.3}_{-0.2}$, $\Omega/2\pi= 0.67^{+0.07}_{-0.09}$, $\chi^2=45/79$ d.o.f.), in which case the effect of the anisotropy break on the model spectrum does not extend above 4 keV. The weak Compton reflection required by the model could then correspond to the cold disc being truncated around the active region, and the origin of the soft X-ray excess has to be due to an effect different than the disc blackbody emission. Thus, in this respect, the model is in principle possible. However, U94 found that $\Omega$ in GX 339–4 correlates positively with $\Gamma$, which behaviour is opposite to that expected in the patchy corona model. We first note that since $\tau\ga 2$ in this model, virtually all Compton-reflected photons passing through an active region are removed from the observed reflected spectrum regardless of the exact value of $\tau$. On the other hand, a decrease of the observed $\Omega$ can occur when the height of the active region decreases, due to fewer reflected photons being able to escape without hitting the hot plasma. This also will soften the spectrum due to the increased cooling by more blackbody photons emitted by the disc intercepted by the hot plasma. The observed opposite correlation provides then strong evidence against the patchy corona model. On the other hand, the data can be well modelled in all respects by a geometry with cold clouds inside a hot slab (S76; Celotti, Fabian & Rees 1992; Kuncic, Celotti & Rees 1997; Collin-Souffrin et al. 1996; Krolik 1998) surrounded by a cold disc, see Fig. 5d. We assume that the clouds are located in the slab midplane and cover a fraction of $f_{\rm c}$ of the midplane. If there is neither dissipation in the cold clouds nor outside seed photons, a solution satisfying both the energy balance and spectral constraints corresponds to $f_{\rm c}\simeq 0.3$, see Fig. 6. The plasma parameters are then $kT=51^{+7}_{-3}$ keV and $\tau =0.95^{+0.13}_{-0.15}$ corresponding to the half-thickness of the slab (at $\chi^2=45/79$ d.o.f.). Compton reflection by the cold clouds is attenuated by the hot plasma and thus we need the presence of additional Compton reflection from an outside matter with $\Omega/2\pi \simeq 0.4$ (see Table 2). This can occur due to reflection of the radiation of the hot flow by an outside cold disc. The covering factor by the cold clouds will be $f_{\rm c}<0.3$ if additional soft photons from the outside cold disc and/or from dissipation in the cold clouds (with the luminosity $L_{\rm bb, intr}$) enter the hot flow, see Fig. 6. As argued by S76, the clouds can be formed by the radiation-pressure induced instability of an optically-thick disc (Lightman & Eardley 1974; Shakura & Sunyaev 976). The clouds are in pressure equilibrium with the hot medium. =10.4cm We have also tested a similar geometry with the hot plasma forming a central hot sphere (rather than a slab) surrounded by a cold disc (e.g. Poutanen, Krolik & Ryde 1997), see Fig. 5e. If the cold disc does not penetrate into the hot cloud, $kT=57_{-3}^{+6}$ keV and the radial $\tau =2.0_{-0.2}^{+0.1}$ ($\chi^2=47/79$ d.o.f.), and internal dissipation in the cold disc is required to provide enough seed soft photons. The model also predicts less Compton reflection than observed, which problem may be solved by flaring of the outside disc. The cold-disc solution of Shakura & Sunyaev (1973) does, in fact, correspond to a flared disc, and the illumination by the central hot source will further increase the amount of flaring. We note that a central hot sphere is a poor approximation to the actual geometry of a hot accretion flow with substantial cooling (see Section \[s:accretion\]), in which the scaleheight is maximized at the outer boundary rather than at the center, which effect will also lead to an increase of the amount of reflected X-rays. We also find that models with the cold disc penetrating into the hot cloud down to $\ga 0.7$ of the sphere radius can also fit the data and satisfy the energy balance, but with less or no dissipation in the cold disc. An overlap between the cold and hot phases increases the amount of Compton reflection in the model and thus reduces the $\Omega/2\pi$ needed from flaring of the outside cold disc. The $\Omega$ vs. $\Gamma$ correlation found by U94 may be explained naturally by geometries shown in Figs. 5d, e. In these geometries, an increase of the area of the outside soft X-ray emitter leads to an increase of the seed-photon flux incident on the hot plasma, which in turn leads to more cooling and softening of the X-ray spectrum, i.e., an increase of $\Gamma$. The same effect leads to more Comton reflection, i.e., an increase of $\Omega$. Summarizing this section, our best models consist of a central hot region surrounded by a cold disc (Figs. 5d, e). These models require the presence of an additional Compton-reflection component, e.g. from an outer cold disc. Among those 2 models, a hot disc with coldclouds insidesurrounded by a cold disc (Fig. 5d) is more likely to be formed by an accretion flow with substantial cooling (see Section \[s:accretion\] below). Both models explain the $\Omega$-$\Gamma$ correlation. On the other hand, a model consisting of active regions at some height above a cold disc (Fig. 5c) is marginally possible, but it requires the temperature of blackbody photons lower than that inferred from the soft X-ray excess and a truncation of the underlying disc, and it predicts an $\Omega$-$\Gamma$ correlation opposite to that observed. Models with a homogeneous corona and with the active regions on the disc surface (Figs. 5a, b) are strongly ruled out by the requirement of energy balance. The origin of the soft X-ray excess {#s:soft} ----------------------------------- In this Section, we fit thedata in the full 1.2–1000 keV range, i.e., including the range $<4$ keV showing the soft X-ray excess. Based on the results of Section \[s:geo\], the most likely geometry of the hot plasma is a hot inner disc mixed with cold clouds. We first check whether the soft excess can be accounted for just by the emission of the cold clouds,which we find energetically viable. However, this model fits the data much worse than our baseline model of Table 2, $\Delta\chi^2=+9$, even for the maximum possible covering factor of the cold clouds (see Fig. 6), $f_{\rm c}=0.3$. On the other hand, the hot disc is surrounded by a cold outer disc beyond a transition radius,$R_{\rm tr}$ (Fig. 5d), which contributes to the soft excess. We have thus added to our spectral model an integral of blackbody spectra over the disc surface at $R\geq R_{\rm tr}$. The local color temperature is in general larger than the effective one, and the ratio of the two for best estimates of the system parameters (Section 2) is $T_{\rm color}/T_{\rm eff} \simeq 1.8$ (Shimura & Takahara 1995). Following Makishima et al. (1986), we neglect here the boundary-condition factor $J(r)\equiv 1-(6/r)^{1/2}$, where $r\equiv Rc^2/G M_{\rm X}$, which is a good approximation for a large enough $r_{\rm tr}$. We find that due to the limited energy range of the data, our fits cannot determine the value of $R_{\rm tr}$. On the other hand, $R_{\rm tr}$ can be obtained from the area covered by the cold clouds emitting blackbody spectrum with the fitted color temperature, $\tbb$. This yields $R_{\rm tr} \simeq 6\times 10^7$ cm at the best fit. The main model parameters are given in Table 3, where $\tin$ is the cold disc temperature at $R_{\rm tr}$, and Compton reflection is from the cold disc. The model spectrum is shown in Fig. 7. The bolometric luminosity is $4\times 10^{37}$ erg s$^{-1}$, and the ratio of the blackbody luminosity of the cold disc to the total one (but without including the outside reflection component) is $\simeq 0.30$. ------------------------ ---------------- ------------------------ ------------------------ ------------------------ -------------- $\tau$ $kT$ $f_{\rm c}$ $\tbb$ $kT_{\rm tr}$ $\chi^2/$dof $0.88^{+0.10}_{-0.10}$ $52^{+1}_{-3}$ $0.26^{+0.04}_{-0.11}$ $0.33^{+0.02}_{-0.04}$ $0.21^{+0.01}_{-0.01}$ 48.6/81 ------------------------ ---------------- ------------------------ ------------------------ ------------------------ -------------- =8.4cm We point out that we have neglected the effect of heating by the central source on the radial temperature dependence of the outside cold disc. This may significantly soften the power-law part of the disc-blackbody spectrum shown in Fig. 7 (e.g. Vrtilek et al. 1990). A model alternative to the hot inner disc that was found by us marginally possible (Section \[s:geo\]) is active regions at some height above the disc (Fig. 5c). In this model, the soft excess can be accounted for by the blackbody emission of the disc region underneath the active region. Due to the intensive heating by the X-rays, the local temperature can be much above the temperature of a disc with internal dissipation only. From the strength of the blackbody component (Table 2), the characteristic size of the heated region of the disc is $\sim 10^7 (T_{\rm color}/T_{\rm eff})^2$ cm. However, when we approximate the emission of the underlying region of the disc as a blackbody, a very poor fit to the data is obtained, $\chi^2=93/82$ d.o.f. This provides one more argument against this model. Summarizing this section, the most likely origin of the soft excess we have found is from an outer cold accretion disc with an additional contribution from cold clouds within the hot disc, see Figs. 5d and 7. Accretion disc solutions {#s:accretion} ------------------------ The best overall model found in Sections \[s:geo\]–\[s:soft\] consists of a hot plasma slab with the half-thickness corresponding to $\tau\la 1$, and $kT\ga 50$ keV. Physically, this geometrical model likely corresponds to a hot accretion disc. We note that indeed the best-fit parameters of the hot slab are close to those predicted by the two-temperature, hot disc model of S76. In that model, the local gravitational energy is converted into the thermal energy of ions (at a temperature $T_{\rm i}\gg T$), which is then transferred to electrons by Coulomb interactions. The electrons, in turn, radiate away their energy by Compton upscattering seed photons irradiating the plasma. As found by e.g. Ichimaru (1977), Abramowicz et al. (1995) and Narayan & Yi (1995), advection of hot ions to the black hole leads to another branch of the hot solution as well as it limits the accretion rate possible in the hot flow. Detailed properties of the solution in the vicinity of the maximum accretion rate are studied by Zdziarski (1998, hereafter Z98). That study follows S76 in characterizing the flow by a value of the Compton parameter, $y\equiv 4\Theta \max(\tau, \tau^2)$ (e.g. Rybicki & Lightman 1979), which approximately determines the X-ray spectral index, $\Gamma$. This is equivalent to assuming that flux in the seed photons is such that it gives rise to an X-ray spectrum with that $\Gamma$. Other parameters of the flow are $\dot M$ and the viscosity parameter, $\alpha$ (Shakura & Sunyaev 1973). \[The rate of advection is characterized by a parameter $\xi_{\rm adv}\simeq 1$, e.g. Chen, Abramowicz & Lasota (1997).\] Z98 has obtained that the optical depth (corresponding to the scaleheight, $H$) of the flow weakly depends on $R$, and that $\tau$ at the local maximum rate, $\mmax$, and the maximum $\tau$ (see Fig. 8) are, $$\label{eq:tau_max} \tau(\mmax)\simeq 1.22 y^{3/5} \alpha^{2/5}, \quad \tau_{\rm max}\simeq 1.54 y^{3/5} \alpha^{2/5},$$ respectively. Those values are reached at $r\simeq 13$, at which the rate of dissipation per logarithmic radius interval is also close to maximum. At $\mmax$, advection carries about half the dissipated power and $H/R \sim 1$. The luminosity of GX 339–4 during the observations reported is about the maximum observed in the hard state (Harmon et al. 1994; Rubin et al. 1998). Thus, we can relate the above $\tau$ with that obtained from fitting the slab model in Section \[s:geo\]. We assume here an intermediate value of the covering factor, $f_{\rm c}=0.1$ (see Fig. 6), for which $kT=52^{+7}_{-5}$ keV, $\tau=0.82^{+0.10}_{-0.12}$ ($\chi^2=45/79$ d.o.f.), implying $y=0.33$. Equating the fitted $\tau$ to $\tau_{\rm max}$, we obtain $\alpha\simeq 1$, which is relatively large \[as it appears to be generally the case in luminous black hole systems (Narayan 1996)\]. Fig. 8 shows the relation between $\tau$ and $\dot m$ ($\equiv \dot M c^2 /L_{\rm E}$) for those $y$ and $\alpha$; we see that $\dot m_{\rm max}\simeq 4$. =6.2cm The maximum possible luminosity from the hot flow corresponds to local accretion rate at the maximum set by advection (Fig. 8), for which radial integration implies, $$\label{L_max} L_{\rm hot}\la 0.15 y^{3/5} \alpha^{7/5} L_{\rm E}$$ (Z98). For the luminosity in Table 2, this relation yields $M_{\rm X}\ga 2.5M_\odot$, which is compatible with our estimates of the black hole mass in Section 2. E.g. $L_{\rm hot}\simeq 0.06 L_{\rm E}$ at $M_{\rm X}=3M_\odot$. Based on our data for GX 339–4 alone, we cannot distinguish which branch of the optically thin solution (Fig. 8) the source follows in the hard state at luminosities lower than the maximum one. In the $\alpha$ model, the advective branch is stable (e.g. Wu 1997) whereas the cooling-dominated branch is unstable (Pringle 1976), but the validity of those predictions has not been tested observationally yet. We point out that if the flux variability in the hard state within a factor of $\sim 50$ (see U94) corresponds to the advective branch of the disc solution, the average accretion rate in the hot flow is within a factor of $\sim 2$ of $\mmax$ (compare the left-hand dashed and solid curves in Fig. 8). The corresponding large $\langle \dot M\rangle$ requires then the presence of a radiation-driven outflow from the companion star for our derived system parameters, see Section 2.3. The formation of the hot disc might be related to the instability of the cold disc in the inner part dominated by radiation pressure (Lightman & Eardley 1974; Shakura & Sunyaev 1976). If this is the case, the transition radius is given by, $$\label{eq:trans} {r_{\rm tr}\over J(r_{\rm tr})^{16/21} } \approx 60 (\alpha M_{\rm X}/M_\odot)^{2/21} \dot m^{16/21}\,,$$ (e.g. Svensson & Zdziarski 1994), where $\alpha$ is the viscosity parameter in the cold disc, not necessarily the same as that in the hot one. For $\dot m\simeq 4$ and $\alpha\simeq 1$, $r_{\rm tr}\sim 200$. Our best-fit model has $L_{\rm bb}/L\simeq 0.3$, see Section \[s:soft\]. If the flow is on average $\sim 50$ per cent advective, $L_{\rm bb}/L \sim 50/r_{\rm tr}$ ($r_{\rm tr}\gg 50$), which then implies $r_{\rm tr}\sim 150$, in good agreement with equation (\[eq:trans\]) above. Then, $R_{\rm tr}=6\times 10^7$ cm obtained in Section \[s:soft\] corresponds to $M_{\rm X}\sim 3$, in good agreement with other estimates of $M_{\rm X}$ here. The physical state of the reflecting medium {#s:refl} ------------------------------------------- The region of formation of the Fe K$\alpha$ line is strongly ionized (Section 4, see also U94). At the best-fit ionization parameters obtained in Table 2, we find that the dominant Fe ions are Fe[xix]{} and Fe[xx]{} for the first and second observation, respectively. Those ions cannot, however, produce the observed line because of the very strong resonant absorption (Matt, Fabian & Ross 1993; Życki & Czerny 1994). The observed line is produced by the nonresonant Fe ions $\leq$ [xvi]{}. We have computed that those ions constitute about 20 per cent of all iron in the reflecting medium, which explains the weakness of the line. The observed line equivalent width of $\sim 40$ eV is then fully consistent with theoretical calculations for $\af\sim 3$, $\Omega/2\pi\simeq 0.4$ and the obtained ionization state of the medium, see Życki & Czerny (1994), George & Fabian (1991) and Matt et al. (1993). That we did not require redshifts of the line center energy nor intrinsic line broadening is consistent with the line origin at large distances from the black hole, $r\gg 1$. We can also obtain an estimate of the transition radius from the ionization state of the reflector. The density, $n$, corresponding to unit Thomson optical depth (where most of Compton reflection takes place) can be calculated from the vertical structure of the standard cold disc (Milson, Chen & Taam 1994). E.g.for $M_{\rm X}=3M_\odot$ and $\dot m=3$, $n=1.8\times 10^{20}$ cm$^{-3}$, $6.0\times 10^{19}$ cm$^{-3}$, $1.8\times 10^{19}$ cm$^{- 3}$, and $1.9\times 10^{17}$ cm$^{-3}$ at $r=20$, 60, 200, and 2000, respectively. If there is a central hot disc irradiating the outer cold disc, we can estimate the distance between the illuminating source and the disc region where most of reflection takes place to roughly equal the transition radius. The illuminating luminosity on each side of the disc is $\sim (1/2) (\Omega/2\pi) L_{\rm hot}$. From that we obtain $\xi\simeq 1200$, 500, 120, 120 erg cm s$^{-1}$ at the 4 values of $r$ above, respectively. Thus, we see that $r_{\rm tr}\ga 10^2$ is compatible with the fitted values of $\xi$ (see Table 2). On the other hand, the illuminating luminosity is just $L_{\rm hot}$ in the model of an active region above the disc (Fig. 5c). From energy balance, we found such an active region is located at the height corresponding to its size (Section \[s:geo\]), and this height has to be $<R$. Then, using the values of density from above, we find $\xi\ga 2000$ erg cm s$^{-1}$ at $r\la 100$. Thus, the disc ionization expected in the active region model is much more than that observed. If there are $k>1$ active regions with a typical luminosity of $L_{\rm hot}/k$, its typical size (and thus, the height) must be at least $<R/k^{1/2}$ to fit on the disc surface, and thus the above estimate of $\xi$ remains unchanged. This estimate provides one more argument against the active region model. We caution, however, that the above results have been obtained using our highly simplified model of photoionization (see Done et al. 1992). Our conclusions here differ from those of U94, who estimated that reflection originates in a region at $r\sim 10$. The difference arises because U94 used the average disc density (which is $\gg$ the photosphere density) as well as underestimated $\dot m$ by using the luminosity in the $\ginga$ energy range only and neglecting advection. The role of magnetic fields {#s:magn} --------------------------- So far, we have neglected in our treatment any magnetic fields, which may be present in GX 339–4 (Fabian et al. 1982; Di Matteo, Celotti & Fabian 1997). If present, thermal synchrotron radiation will provide additional seed photons for thermal Comptonization (Zdziarski 1985, 1986), and this process has been suggested to play a major role in luminous black-hole sources (e.g. Narayan 1996; Di Matteo et al. 1997). For $kT$ and $\tau$ typical of compact objects, the synchrotron radiation is self-absorbed up to an energy, $E_s$, corresponding to a high harmonic. The emitted spectrum is of the Rayleigh-Jeans form below $E_s$, and exponentially cut off above it (Petrosian 1981; Takahara & Tsuruta 1982). Zdziarski (1986) has derived an expression for $E_s$ without taking into account the angular dependence of the synchrotron radiation. Here, we modify his expression to take into account the effect of angular averaging as calculated by Mahadevan, Narayan & Yi (1996) to obtain, $$\label{eq:es} \epsilon_{\rm s}={343\over 36}\Theta^2 \epsilon_{\rm c} \ln^3{C\over \ln{C\over \ln{C\over \dots}}},$$ where $\epsilon_{\rm s}\equiv E_s/m_{\rm e} c^2$, $\epsilon_{\rm c}=B/B_{\rm cr}$, the critical magnetic field strength is $B_{\rm cr}= m_{\rm e}^2 c^3/ e\hbar\simeq 4.4\times 10^{13}$ G, $$\label{eq:C} C={3\over 7\Theta} \left[ \pi \tau A_{\rm M} \exp(1/\Theta) \over 3\alpha_{\rm f} \epsilon_{\rm c} \right]^{2/7},$$ $\alpha_{\rm f}$ is the fine-structure constant and $A_{\rm M}(\Theta, \epsilon_{\rm s}/\epsilon_{\rm c})$ is the low-energy and low-temperature correction defined here as the ratio of equations (33) to (31) in Mahadevan et al. (1996) with coefficients given in their Table 1. The factor $C$ above is given in the limit of $\Theta\ll 1$, appropriate for GX 339–4. In general, any magnetic field in the source will have the strength less than that corresponding to pressure equipartion. In a hot accretion disc, the largest contribution to pressure comes from hot ions (Section \[s:accretion\]), and thus $B^2/24\pi \la n kT_{\rm i}$. The ion temperature in the disc is found to be always sub-virial, $T_{\rm i} \la 2\times 10^{11}$ K (e.g. S76; Chen et al. 1997). The strongest magnetic field is achieved in an inner region, where the plasma density is the largest. For estimates, we take a region with the size of $R\sim 15GM_{\rm X}/c^2$, where dissipation per unit logarithmic radius is maximized. In a disc-like geometry, hydrostatic equilibrium implies $H/R\simeq 1$ around $\dot M_{\rm max}$ (e.g. Z98). For $\tau\sim 1$ fitted in the slab geometry (Sections \[s:geo\]–\[s:soft\]), equipartition corresponds then to $B\simeq 2\times 10^7$ G. Equations (\[eq:es\])-(\[eq:C\]) then yield $E_s\simeq 10$ eV, or $\epsilon_{\rm s} \simeq 2\times 10^{-5}$. ($A_{\rm M}\sim 0.1$ for the parameters used above.) Photons in the resulting self-absorbed spectrum undergo Compton upscattering in the thermal plasma. The spectral index of the Comptonized emission, $\Gamma \simeq 1.75$ (see Table 2) implies certain ratio between the Comptonized luminosity to that in the seed photons (see Zdziarski 1986). Assuming the source area of $2\pi R^2$, the Comptonized-synchrotron luminosity is approximately, $$\label{eq:lcs} L_{\rm CS}\simeq { 10 m_{\rm e} c^3 \over \lambda^3 } {(2\Theta)^{3-\Gamma}\over 2-\Gamma} \epsilon_{\rm s}^{\Gamma+1} R^2 \,,$$ where $\lambda \simeq 2.43\times 10^{-10}$ cm is the electron Compton wavelength. For the source parameters obtained here, $L_{\rm CS} \simeq 3\times 10^{34}$ erg s$^{-1}$. This is 3 orders of magnitude below the observed X luminosity of $\sim 3\times 10^{37}$ erg s$^{-1}$. Thus, our conclusion is that the process of thermal synchrotron emission is negligible for the formation of the X spectrum provided the emitting region is similar in size to the region where the accretion energy is dissipated. Di Matteo et al. (1997) have obtained a larger $L_{\rm CS}$ in GX 339–4 due to their adoption of $kT \simeq 80$ keV, a much stronger $B$-field than that derived above, and neglecting a correction to the rate of the synchrotron emission at low $kT$ \[as given in Table 1 of Mahadevan et al. (1996)\]. Still, $L_{\rm CS}\sim L$ for the observations considered here would require a source size much larger than the region where gravitational energy dissipation is efficient and/or magnetic field much above equipartition (equations \[eq:es\]–\[eq:lcs\]). We also note that equations above imply an approximate scaling of $L_{\rm CS}/L_{\rm E} \propto M_{\rm X}^{(1-\Gamma)/2}$. Thus, Comptonized thermal synchrotron emission from inner hot accretion discs around supermassive black holes in Seyfert AGNs (with typical $\Gamma\sim 1.9$, $\Theta\sim 0.2$, $\tau\sim 1$, e.g., Z97) will be even less important than in black-hole binaries. $\bmath{e^\pm}$ pair production {#s:pair} ------------------------------- If the hot plasma is purely thermal, its parameters of $kT\sim 50$ keV and $\tau\sim 1$ imply that  pair production is due to photons in the extreme Wien tail of the thermal spectrum, and thus it is very inefficient. Indeed, performing standard calculations of the pair production rate using the model spectrum as in Table 2 for the data set 1 and balancing that rate against the pair annihilation rate (e.g. Svensson 1984) leads us to the conclusion that there are almost no  pairs in the plasma. Specifically, the compactness, $\ell\equiv L_{\rm hot}\sigma_{\rm T}/R m_{\rm e} c^3$, required for pair production to be able to produce pairs with the fitted $\tau$ is $\sim 10^4$ for the hot source modelled as a sphere with radius $R$. This corresponds to $R\sim 10^5$ cm, i.e., much less than even the Schwarzschild radius of the black hole. We have also performed pair-balance calculations for a slab, which better approximates the disc geometry. Here, we used the model of a slab with cold clouds in the midplane, see Fig. 5d and Section \[s:geo\]. The relevant compactness is then the local one corresponding to the power dissipated within a cube with size equal to the half-thickness, $H$, of the slab. The results are shown in Fig. 6, where we see that $\ell\sim 10^3$–$3\times 10^3$ (depending on the covering factor of cold clouds, $f_{\rm c}$). Approximating the source as a uniformly radiating disc with the outer radius $R$, we obtain $R^2/H=L_{\rm hot}\sigma_{\rm T}/2\pi\ell m_{\rm e} c^3$, which for $H/R\sim 1$ characteristic for hot discs yields $R\sim 10^5$ cm, which is the same value as obtained above for the spherical geometry, and much less than our estimates of the size of the hot plasma. Thus, there are no  in the hot plasma in GX 339–4 provided the plasma is fully thermal. On the other hand, it is also possible that nonthermal acceleration of selected electrons to relativistic energies operates in the source in addition to heating of the plasma. This hybrid model has been applied to NGC 4151 by Zdziarski et al. (1996) and Johnson et al. (1997) and to Cyg X-1 by Poutanen & Coppi (1998) and Poutanen (1998). We have applied that model to the data set 1. The pair-dominated hybrid model yields the same $\chi^2$ as the thermal model (fit 1c in Table 2) for any $\ell \ga 150$, and the 90 per cent confidence lower limit is $\ell_{\rm min}\simeq 70$ (for spherical geometry). The fraction of the power that is supplied to the accelerated electrons and pairs is $\simeq 0.09$ at $\ell_{\rm min}$, and it decreases with increasing $\ell$. This $\ell_{\rm min}$ implies $R< 10^7$ cm, which corresponds to $r\la 20$ for $M_{\rm X}=3M_\odot$. This is in principle compatible with the expected size of a hot accretion disc ($r\gg 6$). We have also applied the hybrid model to the data obtained from the entire OSSE observation of the source, 1991 September 5–12 (G95), which have much better statistical quality than either of our OSSE data sets. We find those data are fitted somewhat better, $\Delta\chi^2=-4$, by the hybrid model than by the pure thermal one (which gives $kT=62^{+13}_{-9}$ keV at fixed $\Omega/2\pi=0.44$). The residuals in the thermal model show a weak high-energy tail on top of the thermal spectrum above $\sim 300$ keV. This may hint for the presence of nonthermal pairs, accounting for the tail. In the hybrid model, $\ell= 150^{+\infty}_{-30}$, and the nonthermal fraction at the best fit is $\simeq 0.05$. On the other hand, G95 show softening of the spectrum of GX 339–4 with time during that observation. Therefore, the tail could also be an artefact of fitting the spectrum averaged over a range of plasma parameter by a single-component thermal model. COMPARISON WITH BLACK-HOLE AND NEUTRON-STAR SOURCES =================================================== In this section, we compare our X spectrum of GX 339–4 with those of established black-hole and neutron-star sources. Our objective is to determine whether the X spectrum of GX 339–4 is indeed similar to those seen in black-hole sources but not to those of accreting neutron stars. We stress that both black-hole binaries and neutron-star binaries exhibit two main spectral states, X-ray low (hard) and high (soft), and only the corresponding states should be directly compared. X$\bmath{\gamma}$ emission of black-hole sources ------------------------------------------------ ### Black-hole binaries in the hard state Hard-state X-ray spectra of the archetypical black-hole binary Cyg X-1 have been fitted by G97. Their fit to the average data of 1991 June 6 with a power-law and reflection model (as in Section 4) yields $\Gamma= 1.59^{+0.03}_{-0.03}$ and $\Omega/2\pi= 0.34^{+0.05}_{-0.05}$, which is rather similar to our spectra of GX 339–4. X-ray spectra with similar power-law indices and moderately weak Compton-reflection components are seen from other black-hole binaries in the hard (low) state, e.g. Nova Muscae (Ebisawa et al.1994; Życki et al. 1998) and GS 2023+338 (Życki et al. 1997). The broad-band X spectra of Cyg X-1 in G97 are also similar to that of GX 339–4, and well modelled by a primary continuum due to thermal Comptonization in a plasma with similar $\tau$, but higher electron temperature, $kT\sim 100$ keV (and with an additional small spectral component from emission of a plasma with $\tau\gg 1$). Soft -ray spectra similar in shape to that of Cyg X-1 have been observed from other black-hole binaries in the hard state (e.g. Grove et al. 1998; Grebenev, Sunyaev & Pavlinsky 1997). Motivated by the similarity, we test here the hot accretion disc model of Z98 (see Section \[s:accretion\]) against the Cyg X-1 data. We have refitted the spectrum of G97 corresponding to the peak flux using the hot slab model used in Section \[s:geo\] for the primary continuum, and obtained $y\simeq 0.4$ and $\tau\simeq 0.5$ (assuming $f_{\rm c}=0.1$). Those parameters imply $\alpha\simeq 0.4$ using $\tau(\mmax)$ of equation (6). The peak luminosity in the hard state is about $4\times 10^{37}$ erg s$^{-1}$, for which equation (7) implies $M_{\rm X}\ga 10M_\odot$. This agrees well with best estimates of the mass of Cyg X-1 (e.g. van Paradijs & McClintock 1995). Thus, it is possible that a hot accretion disc is present in the hard state of black-hole binaries, and that its structure determines both the observed optical depth of the hot plasma and the maximum luminosity in the hard state. ### Seyfert 1s It is of great interest that the X spectra of black-hole binaries in the hard state are similar to the spectra of Seyfert-1 AGNs, which similarity in the X-ray regime was pointed out by Tanaka (1989). Specifically, both classes of objects show power-law X-ray spectra with $\Gamma \sim 1.7$–2, Compton reflection components with Fe K$\alpha$ lines, and high-energy cutoffs above $\sim 100$ keV (e.g.  Z97). The cutoffs can be modelled in both cases by Comptonization with $\tau\sim 1$ and $kT\sim 100$ keV (Z97). Here, we point out that the intrinsic spectrum above 4 keV of GX 339–4 appears virtually identical to that of NGC 4151, which, in soft -rays, is the brightest and best-studied Seyfert AGN. We note here that apart from strong absorption, the X spectrum of NGC 4151 appears rather typical for Seyfert 1s, and the OSSE spectra of NGC 4151 and of all other Seyfert 1s observed by OSSE are very similar (Zdziarski et al. 1996; Z97). Also, the intrinsic Xspectrum of NGC 4151 is very similar to that of a ‘typical’ Seyfert 1, NGC 5548 (Magdziarz et al. 1998). This makes models specially designed for NGC 4151 (e.g. Poutanen et al. 1996) probably unnecessary. The thermal Comptonization fit (after applying the same correction to solutions of the Kompaneets equation as in Section 4 above) to a joint [*ROSAT*]{}//OSSE spectrum of NGC 4151 (Zdziarski et al. 1996) yields $\Gamma= 1.81^{+0.04}_{-0.04}$, $kT= 63^{+19}_{-11}$ keV , $\Omega/2\pi =0.49^{+0.26}_{-0.26}$, which are consistent within errors with those given in Table 2 for GX 339–4. We find that the form of the high-energy cutoff is almost the same in both sources, as shown in Fig.9, which shows the ratio of the OSSE spectrum of GX 339–4 (from the entire observation of 1991 September 5–12) to the spectrum of NGC 4151 averaged over all OSSE observations (from Johnson et al. 1997). X$\bmath{\gamma}$ emission of weakly-magnetized accreting neutron-stars ----------------------------------------------------------------------- Two types of objects whose X emission (if present) is unambiguously connected with neutron stars are X-ray pulsars and isolated neutron stars. Those X sources possess strong magnetic fields, $B\ga 10^{11}$ G, and their emission can be easily distinguished from that of black-hole systems, see Finger & Prince (1997) and Thompson et al. (1997) for recent reviews. On the other hand, weakly-magnetized accreting neutron stars often show X-ray emission very similar to that of black-hole binaries. Some LMXBs classified as black-hole candidates have been subsequently found to emit type-1 X-ray bursts, leading to their identifications as neutron-star binaries. A recent example is discovery of X-ray bursts from GS 1826–238 (Bazzano et al. 1997), which source appeared as a black-hole candidate in Tanaka (1989). Indeed, X-ray emission of X-ray bursters in their hard (low) state can be very similar to that of GX 339–4. For example, two observations of 4U 1608–522 show a power law component with $\Gamma\simeq 1.9$ accompanied by Compton reflection from a strongly ionized medium with $\Omega/2\pi\sim 0.5$ in the $\sim 2$–60 keV range (Yoshida et al. 1993). Similarly, GS 1826–238 observed by has $\Gamma\sim 1.8$ and Compton reflection (Strickman et al. 1996). Thus, X-ray spectra alone appear not sufficient to distinguish a weakly-magnetized neutron star from a black hole. On the other hand, typical spectra of X-ray bursters observed in the $\sim 30$–200 keV range by SIGMA telescope on board [*GRANAT*]{} and the BATSE detector on board [*CGRO*]{} are rather soft, see reviews by Barret & Vedrenne (1994), van der Klis (1994), Vargas et al. (1997) and Tavani & Barret (1997). In most cases, the spectra can be fitted equally well by a power law with $\langle \Gamma\rangle \sim 3$ or thermal Comptonization with $\langle kT\rangle \sim 15$–20 keV. There is only one reported OSSE detection of an X-ray burster, GS 1826–238, which shows a similar $\Gamma=3.1\pm 0.5$ above 50 keV (Strickman et al. 1996). These spectra are all much softer than the corresponding spectra of black hole binaries in the hard state \[but note their similarity to black-hole binary spectra in the soft state (Grove et al.1998)\]. Unfortunately, there have been almost no observations below 30 keV simultaneous with those at $\ga 30$ keV discussed above. Therefore, there is no information on the X-ray spectral state (high or low) of the bursters during most of those observations. However, X-ray bursters are rather often found in the low state when observed in X-rays alone, and thus we can assume that some of the spectra observed at $\ga 30$ keV correspond to the X-ray low, hard, state. Then, this implies that the X spectra of bursters in the low state have spectral breaks at energies significantly below 100 keV, or have thermal-Comptonization temperatures of $kT\la 30$ keV. This conclusion is indeed confirmed by a $\sim 2$–250 keV observation of 4U 1608–522 in the low state simultaneously by and BATSE, which yields a break at $\sim 60$ keV in the broken power-law model or $kT\sim 25$ keV, $\tau\gg 1$, in the thermal Comptonization model (Zhang et al. 1996). A similar spectral break at $\sim 60$ keV was also seen in a 30–200 keV observation of 4U 1728–34 (Claret et al. 1994). Spectral black-hole signatures ------------------------------ As discussed above, X-ray bursters in the low state have X emission with breaks or cutoffs at significantly [*lower*]{} energies than both that seen in GX 339–4 and those found in black-hole binaries in the hard (low) state in general (Section 6.1.1). When the spectra are fitted by the thermal-Comptonization model (with $\tau\ga 1$), $kT\la 30$ keV and $kT\ga 50$ keV for neutron stars and black holes, respectively. We propose this quantitative criterion to be a black-hole signature. We see that these two ranges of temperature are not far apart, and it is important to test this criterion against available data. Those tests should take into account Compton reflection and relativistic effects in thermal Comptonization, and use broad-band X data. These criteria are not satisfied by thermal-Comptonization fits in Mandrou et al. (1994) and Churazov et al.(1994), who find $kT= 33$ keV and 38 keV, respectively, in GRS 1758–258, a black-hole candidate in the hard state (Sunyaev et al. 1991). Those fits use the nonrelativistic model of Sunyaev & Titarchuk (1980), neglect reflection, and are in a 35–250 keV range only. In fact, very similar assumptions lead to a gross underestimate of the temperature in Cyg X-1 in the hard state, $kT=27$ keV (Sunyaev & Trümper 1979), whereas $kT\sim 100$ keV is obtained from broad-band spectra fitted by a relativistic Comptonization model with reflection (e.g. G97). As a caveat, we mention that a 40–200 keV SIGMA spectrum of a source in Terzan 2 globular cluster appears to have no cutoff up to 200 keV and $\Gamma\simeq 1.7\pm 0.5$ (Barret & Vedrenne 1994; Barret et al. 1991). Such a spectrum may imply a Comptonization temperature $\ga 100$ keV, which is characteristic of black-hole sources but not of neutron-star ones, whereas the same cluster contains an X-ray burster, X1724–308. Finally, we point out that the presence of power-law emission with $\Gamma\sim 2.5$–3 in the $\sim 30$–500 keV range, which is seen from black hole binaries in the soft state (Grove et al. 1998), may not constitute a black-hole signature, in spite of a recent claim (Titarchuk 1997). Very similar soft single power-law spectra without detectable high-energy breaks are seen from many X-ray bursters (e.g. Barret et al. 1992; Goldwurm et al. 1996; Harmon et al. 1996). CONCLUSIONS =========== The results of this work can be divided into two main parts as far as dependence on model assumptions is concerned. The results of Sections 2, 3, 4 and 6 are relatively model-independent. In Section 2, we determine the distance to the object to be $>3$ kpc, with 4 kpc being the most likely value. We show the most recent distance determination of $\sim 1.3$ kpc is in error. We also find $\ebv\simeq 1.2$ and $\nh\simeq 6\times 10^{21}$ cm$^{-2}$ as mutually-consistent most likely values. The mass of the compact object appears relatively low, $M_{\rm X}\la 5 M_\odot$. In Sections 3 and 4, we present our data and show that the spectra can be very well fitted by Comptonization in a thermal plasma with $kT\simeq 50$ keV and $\tau\sim 1$. In addition, we find, at very high significance, the presence of Compton reflection from an ionized medium and a soft X-ray excess. In Section 6, we show that this spectrum is similar to that of other black-hole binaries, as well as to those of Seyfert AGNs. After comparison with spectra of neutron-star sources, we propose that a thermal-Comptonization temperature of $kT\ga 50$ keV represents a black-hole signature. Physical interpretation of our results is given in Section 5. Here, we study constraints following from the X spectra and concentrate on accretion disc and corona models. We do not consider, e.g. models with large scale outflows or jets. Since the X observations were not accompanied by observations in any other wavelength, we do not discuss such data (e.g.optical, Motch et al. 1983) obtained at other epochs. We also do not discuss time variability of the source. With these caveats, our best physical model is that of a hot accretion disc within $\sim 100$ gravitational radii surrounded by a cold outer disc, see Figs. 5d, e. The seed photons for thermal Comptonization in the hot disc are supplied by cold clouds within the hot disc. The emission of the hot disc is Compton-reflected by the outer cold disc. The outer disc also emits most of the observed soft X-ray excess. This model is in agreement with the spectra, energy balance, and ionization balance at the surface of the reflecting outer disc. The observed amount of reflection requires that the outer disc is flared. The hot-disc accretion rate is near the maximum set by advection. Based on the spectral fit of the hot slab model, we find the viscosity parameter of $\alpha\sim 1$ and $M_{\rm X}\ga 3M_\odot$, which mass is in agreement with the dynamical mass determination. The hot disc model, which parameters are independent of $M_{\rm X}$, is also supported by the observed similarity of the spectrum of GX 339–4 to those of Seyfert 1s. We find that  pair production photons in the thermal-Comptonization spectrum is negligible and thus the disc is most likely made of electrons and ions (although more complex models with  pairs are possible). Also, synchrotron emission in the hot disc with equipartition magnetic field is negligible as a source of seed photons for Comptonization. We can rule out models with a cold disc covered by a homogeneous corona or by active regions located on the surface of the disc as violating the energy balance. On the other hand, the energy balance is satisfied if there are active regions at some height above the disc surface. 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--- abstract: | We outline our methods for obtaining high precision mass profiles, combining independent weak-lensing distortion, magnification, and strong-lensing measurements. For massive clusters the strong and weak lensing regimes contribute equal logarithmic coverage of the radial profile. The utility of high-quality data is limited by the cosmic noise from large scale structure along the line of sight. This noise is overcome when stacking clusters, as too are the effects of cluster asphericity and substructure, permitting a stringent test of theoretical models. We derive a mean radial mass profile of four similar mass clusters of high-quality [*Hubble Space Telescope*]{} and Subaru images, in the range $R=40$kpc$\,h^{-1}$ to 2800kpc$h^{-1}$, where the inner radial boundary is sufficiently large to avoid smoothing from miscentering effects. The stacked mass profile is detected at $58\sigma$ significance over the entire radial range, with the contribution from the cosmic noise included. We show that the projected mass profile has a continuously steepening gradient out to beyond the virial radius, in remarkably good agreement with the standard Navarro-Frenk-White form predicted for the family of CDM-dominated halos in gravitational equilibrium. The central slope is constrained to lie in the range, $-d\ln\rho/d\ln{r}=0.89^{+0.27}_{-0.39}$. The mean concentration is $c_{\rm vir}=7.68^{+0.42}_{-0.40}$ (at $M_{\rm vir}=1.54^{+0.11}_{-0.10}\times 10^{15}M_\odot\,h^{-1}$), which is high for relaxed, high-mass clusters, but consistent with $\Lambda$CDM when a sizable projection bias estimated from $N$-body simulations is considered. This possible tension will be more definitively explored with new cluster surveys, such as CLASH, LoCuSS, Subaru HSC, and XXM-XXL, to construct the $c_{\rm vir}$–$M_{\rm vir}$ relation over a wider mass range. author: - 'Keiichi Umetsu, Tom Broadhurst, Adi Zitrin, Elinor Medezinski, Dan Coe, Marc Postman' title: 'A Precise Cluster Mass Profile Averaged from the Highest-Quality Lensing Data' --- Introduction {#sec:intro} ============ Clusters of galaxies represent the largest gravitationally-bound objects in the universe, which contain a wealth of astrophysical and cosmological information, related to the nature of dark matter, primordial density perturbations, and the emergence of structure over cosmic time. Observational constraints on the properties and evolution of clusters provide independent and fundamental tests of any viable cosmology, structure formation scenario, and possible modifications of the laws of gravity, complementing large-scale cosmic microwave background and galaxy clustering measurements [e.g., @Komatsu+2011_WMAP7; @Percival+2010_BAO]. A key ingredient of cluster-based cosmological tests is the mass and internal mass distribution of clusters. In this context, the current cosmological paradigm of structure formation, the standard $\Lambda$ cold (i.e., non relativistic) dark matter ($\Lambda$CDM, hereafter) model, provides observationally testable predictions for CDM-dominated halos over a large dynamical range in density and radius. Unlike galaxies where substantial baryonic cooling is present, massive clusters are not expected to be significantly affected by gas cooling [e.g., @Blumenthal+1986; @Broadhurst+Barkana2008]. This is because the majority of baryons ($\sim 80\%$) in massive clusters comprise a hot, X-ray emitting diffuse intracluster medium (hereafter ICM), in which the high temperature and low density prevent efficient cooling and gas contraction, and hence the gas pressure roughly traces the gravitational potential produced by the dominant dark matter [see @Kawaharada+2010; @Molnar+2010_ApJL]. The ICM represents only a minor fraction of the total mass near the centers of clusters [@2008MNRAS.386.1092L; @2009ApJ...694.1643U]. Consequently, for clusters in a state of quasi equilibrium, the form of their total mass profiles reflects closely the distribution of dark matter [@Mead+2010_AGN]. High-resolution $N$-body simulations of collisionless CDM exhibit an approximately “universal” form for the spherically-averaged density profile of virialized dark matter halos [@1997ApJ...490..493N NFW hereafter], with some intrinsic variance in the mass assembly histories of individual halos [@Jing+Suto2000; @Tasitsiomi+2004; @Navarro+2010]. The predicted logarithmic gradient $\gamma_{\rm 3D}(r)\equiv -d\ln{\rho}/d\ln{r}$ of the NFW form flattens progressively toward the center of mass, with a central cusp slope flatter than a purely isothermal structure ($\gamma_{\rm 3D}=2$) interior to the inner characteristic radius $r_s ({\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}300\,$kpc$h^{-1}$ for cluster-sized halos), providing a distinctive prediction for the empirical form of CDM halos in gravitational equilibrium. A useful index of the degree of concentration is $c_{\rm vir}=r_{\rm vir}/r_s$, which compares the virial radius $r_{\rm vir}$ to the characteristic radius $r_s$ of the Navarro-Frenk-White (NFW, hereafter) profile. This empirical NFW profile is characterized by the total mass within the virial radius, $M_{\rm vir}$, and the halo concentration $c_{\rm vir}$. Theoretical progress has been made in understanding of the form of this profile in terms of the dynamical structure using both numerical and analytical approaches [@Taylor+Navarro2001; @Lapi+Cavaliere2009; @Navarro+2010], though we must currently rely on the quality of $N$-body simulations when making comparisons with CDM-based predictions for cluster mass profiles. In the context of standard hierarchical clustering models, the halo concentration should decline with increasing halo mass as dark matter halos that are more massive collapse later when the mean background density of the universe is correspondingly lower [@2001MNRAS.321..559B; @Zhao+2003; @2007MNRAS.381.1450N]. This prediction for the halo mass-concentration relation and its evolution has been established thoroughly with detailed simulations [e.g., @1997ApJ...490..493N; @2001MNRAS.321..559B; @2007MNRAS.381.1450N; @Duffy+2008; @Klypin+2010], although sizable scatter around the mean relation is present due partly to variations in formation epoch of halos [@2002ApJ...568...52W; @2007MNRAS.381.1450N; @Zhao+2009]. Massive clusters are of particular interest in this context because they are predicted to have a relatively shallow mass profile with a pronounced radial curvature. Gravitational lensing of background galaxies offers a robust way of directly obtaining the mass distribution of galaxy clusters [see @2001PhR...340..291B; @Umetsu2010_Fermi and references therein] without requiring any assumptions on the dynamical and physical state of the system . A detailed examination of this fundamental prediction has been the focus of our preceding work [@BTU+05; @Medezinski+07; @BUM+08; @UB2008; @2009ApJ...694.1643U; @Lemze+2009; @Umetsu+2010_CL0024; @Umetsu+2011]. Systematic cluster lensing surveys are in progress to obtain mass profiles of representative clusters over a wide range of radius by combining high-quality strong and weak lensing data. Deep multicolor images of massive cluster cores from Advanced Camera for Surveys (ACS) observations with the [*Hubble Space Telescope*]{} ([*HST*]{}) allow us to identify many sets of multiple images spanning a wide range of redshifts for detailed strong-lens modeling [e.g., @2005ApJ...621...53B; @Zitrin+2009_CL0024; @Zitrin+2010_A1703; @Zitrin+2011_MACS; @Zitrin+2010_MS1358; @Zitrin+2011_A383]. The wide-field prime-focus cameras of Subaru and CFHT have been routinely producing data of sufficient quality to obtain accurate measurements of the weak-lensing signal, providing model-independent cluster mass profiles out to beyond the virial radius [e.g., @BTU+05; @BUM+08; @2007ApJ...668..643L; @UB2008; @2009ApJ...694.1643U; @Umetsu+2010_CL0024; @Umetsu+2011; @Coe+2010]. Our earlier work has demonstrated that without adequate color information, the weak-lensing signal can be heavily diluted particularly toward the cluster center by the presence of unlensed cluster members, leading to biased cluster mass profile measurements with underestimated concentrations and internal inconsistency, with the weak-lensing based profile underpredicting the observed Einstein radius [@BTU+05; @UB2008; @Medezinski+2010]. Careful lensing work on individual clusters has shown that full mass profiles constructed from combined strong and weak lensing measurements show a continuous steepening radial trend consistent with the predicted form for the family of collisionless CDM halos . Intriguingly these initial results from combined strong and weak lensing measurements reveal a relatively high degree of halo concentration in lensing clusters , lying well above the mass-concentration relation for cluster-sized halos predicted by the $\Lambda$CDM model, despite careful attempts to correct for potential projection and selection biases inherent to lensing [@2007ApJ...654..714H; @Meneghetti+2010a]. This apparent overconcentration of lensing clusters is also indicated by the generally large Einstein radii determined from strong-lensing data [@Broadhurst+Barkana2008; @Meneghetti+2010_MARENOSTRUM; @Zitrin+2011_MACS]. In this paper we explore in greater depth the utility of high-quality lensing data for obtaining highest-precision cluster mass profiles by combining all possible lensing information available in the cluster regime. This extends our recent weak-lensing work by @Umetsu+2011, where a Bayesian method was developed for a direct reconstruction of the projected cluster mass profile from complementary weak-lensing distortion and magnification effects [@UB2008], the combination of which can be used to unambiguously determine the absolute mass normalization. For a massive cluster acting as a super-critical lens, the strong and weak lensing regimes contribute equal logarithmic coverage of the radial profile [@Umetsu+2011], so that here we concentrate on those clusters for which we have high-quality data in both these regimes. The high quality of our data is such that we have now become significantly limited by the cosmic noise from large scale structure behind the cluster center, where magnified sources lie at greater distances. This noise is correlated between radial bins, and so can be overcome by stacking clusters, along independent sight lines. Stacking also helps average over the effects of cluster asphericity and substructure, [@Mandelbaum+2006; @Johnston+2007_SDSS1; @Okabe+2010_WL; @Umetsu+2011], allowing a tighter comparison of the averaged profile with theoretical models. Our aim here is to develop and apply comprehensive methods to a sample of four similarly high-mass lensing clusters (A1689, A1703, A370, and Cl0024+17), for which we have previously identified multiply-lensed images and measured weak magnification and distortion effects from deep [*HST*]{} and Subaru observations [@2005ApJ...621...53B; @UB2008; @Umetsu+2010_CL0024; @Zitrin+2010_A1703; @Medezinski+2010; @Medezinski+2011; @Umetsu+2011]. The paper is organized as follows. In Section \[sec:basis\] we briefly summarize the basis of cluster gravitational lensing. In Section \[sec:method\] we outline our comprehensive methods for obtaining projected cluster mass profiles from weak-lensing distortion, magnification, and strong-lensing measurements. In Section \[sec:app\] we apply our methodology to deep [*HST*]{} and Subaru observations of four massive clusters to derive a mean radial mass profile for the entire clusters, demonstrating how stacking the weak and strong lensing signals improves upon the statistical precision of the mass profile determination; we then examine the radial dependence of the stacked cluster mass profile. Finally, we discuss our results and conclusions in § \[sec:discussion\]. Throughout this paper, we adopt a concordance $\Lambda$CDM cosmology with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $h\equiv H_0/(100\, {\rm km\, s^{-1}\, Mpc^{-1}})=0.7$, unless otherwise noted. Basis of Cluster Lensing {#sec:basis} ======================== The gravitational deflection of light rays by a cluster can be described by the thin lens equation, which relates the angular position of a lensed image $\btheta$ to the angular position of the intrinsic source $\bbeta$ as $$\label{eq:lenseq} \bbeta=\btheta-\bnabla\psi,$$ where $\balpha\equiv \bnabla\psi(\btheta)$ is the deflection field, and $\psi(\btheta)$ is the effective lensing potential, which is defined by the two-dimensional Poisson equation as $\triangle \psi(\btheta)=2\kappa(\btheta)$ with the lensing convergence $\kappa$ given as a source term. This equation can be readily inverted to yield: $\psi(\btheta)=2\int\!d^2\theta'\,\triangle^{-1}(\btheta,\btheta')\kappa(\btheta')=(1/\pi)\int\!d^2\theta'\,\ln|\btheta-\btheta'|\kappa(\btheta')$, so that the deflection field is expressed in terms of $\kappa$ as $$\label{eq:deflection} \balpha(\btheta)=\frac{1}{\pi}\int\!d^2\theta'\,\frac{\btheta-\btheta'}{|\btheta-\btheta'|^2} \kappa(\btheta').$$ For gravitational lensing in the cluster regime [e.g., @Umetsu2010_Fermi], $\kappa$ is expressed as $\kappa(\btheta)=\Sigma_{\rm crit}^{-1}\Sigma(\btheta)$, namely the projected mass density $\Sigma(\btheta)$ in units of the critical surface mass density for gravitational lensing, defined as $$\begin{aligned} \label{eq:sigmacrit} \Sigma_{\rm crit} = \frac{c^2}{4\pi G D_l} \beta^{-1}; \ \ \ \beta(z_s) \equiv {\rm max}\left[ 0,\frac{D_{ls}(z_s)}{D_s(z_s)}\right],\end{aligned}$$ where $D_s$, $D_l$, and $D_{ls}$ are the proper angular diameter distances from the observer to the source, from the observer to the lens, and from the lens to the source, respectively, and $\beta=D_{ls}/D_s$ is the angular-diameter distance ratio associated with the population of background sources. The deformation of the image for a background source can be described by the Jacobian matrix $\cal{A}_{\alpha\beta}\equiv (\partial\bbeta/\partial\btheta)_{\alpha\beta}=\delta_{\alpha\beta}-\psi_{,\alpha\beta}$ ($\alpha,\beta=1,2$) of the lens mapping, where $\delta_{\alpha\beta}$ is Kronecker’s delta.[^1] The real, symmetric Jacobian ${\cal A}_{\alpha\beta}$ can be decomposed as ${\cal A}_{\alpha\beta} = (1-\kappa)\delta_{\alpha\beta} -\Gamma_{\alpha\beta}$, where $\Gamma_{\alpha\beta}(\btheta) \equiv (\partial_\alpha\partial_\beta-\delta_{\alpha\beta}\nabla^2/2)\psi(\btheta)$ is the trace-free, symmetric shear matrix, $$\begin{aligned} \label{eq:jacob} \Gamma_{\alpha\beta}&=& \left( \begin{array}{cc} +{\gamma}_1 & {\gamma}_2 \\ {\gamma}_2 & -{\gamma}_1 \end{array} \right),\end{aligned}$$ with $\gamma_{\alpha}$ being the components of spin-2 complex gravitational shear $\gamma:=\gamma_1+i\gamma_2$. In the strict weak-lensing limit where $\kappa,|\gamma|\ll 1$, $\Gamma_{\alpha\beta}$ induces a quadrupole anisotropy of the background image, which can be observed from ellipticities of background galaxy images. Given an arbitrary circular loop of radius $\vartheta$ on the sky, the average tangential shear $\gamma_+(\vartheta)$ around the loop satisfies the following identity [e.g., @Kaiser1995]: $$\gamma_+ (\vartheta) = \bar{\kappa}(<\vartheta)-\kappa(\vartheta),$$ where $\kappa(\vartheta)$ is the azimuthal average of $\kappa(\btheta)$ around the loop, and $\bar{\kappa}(<\vartheta)$ is the average convergence within the loop. The local area distortion due to gravitational lensing, or magnification, is given by the inverse Jacobian determinant, $$\label{eq:mu} \mu = \frac{1}{|{\rm det}{\cal A}|} = \frac{1}{|(1-\kappa)^2-|\gamma|^2|}.$$ which can influence the observed surface density of background sources, expanding the area of sky, and enhancing the observed flux of background sources [@1995ApJ...438...49B]. The former effect reduces the effective observing area in the source plane, decreasing the number of background sources per solid angle; on the other hand, the latter effect amplifies the flux of background sources, increasing the number of sources above the limiting flux. The net effect is known as magnification bias and depends on the intrinsic slope of the luminosity function of background sources. In general, the observable quantity for quadrupole weak lensing is not the gravitational shear $\gamma$ but the complex [*reduced*]{} shear, $$\label{eq:redshear} g(\btheta)=\frac{\gamma(\btheta)}{1-\kappa(\btheta)}$$ in the subcritical regime where ${\rm det}{\cal A}>0$ (or $1/g^*$ in the negative parity region with ${\rm det}{\cal A}<0$). The spin-2 reduced shear $g$ is invariant under the following global linear transformation: $$\label{eq:invtrans} \kappa(\btheta) \to \lambda \kappa(\btheta) + 1-\lambda, \ \ \ \gamma(\btheta) \to \lambda \gamma(\btheta)$$ with an arbitrary scalar constant $\lambda\ne 0$ . This transformation is equivalent to scaling the Jacobian matrix ${\cal A}(\btheta)$ with $\lambda$, $\cal {A}(\btheta) \to \lambda {\cal A}(\btheta)$, and hence leaves the critical curves ${\rm det}{\cal A}(\btheta)=0$ invariant. Furthermore, the curve $\kappa(\btheta)=1$, on which the gravitational distortions disappear, is left invariant under the transformation (\[eq:invtrans\]). This mass-sheet degeneracy can be unambiguously broken by measuring the magnification effects, because the magnification $\mu$ transforms under the invariance transformation (\[eq:invtrans\]) as $$\mu(\btheta) \to \lambda^2 \mu(\btheta).$$ In practice, the lens magnification $\mu$ can be measured from characteristic variations in the number density of background galaxies due to magnification bias [@1995ApJ...438...49B; @Umetsu+2011] as $$n_\mu(\btheta)=n_0\mu(\btheta)^{2.5s-1},$$ where $n_0=dN_0(<m_{\rm cut})/d\Omega$ is the unlensed number density of background sources for a given magnitude cutoff $m_{\rm cut}$, approximated locally as a power-law cut with slope $s=d\log_{10} N_0(<m)/dm$ ($s>0$). In the strict weak-lensing limit, the magnification bias is $\delta n_\mu/n_0\approx (5s-2)\kappa$. For red background galaxies the intrinsic count slope $s$ at faint magnitudes is relatively flat, $s\sim 0.1$, so that a net count depletion results [@BTU+05; @UB2008; @Umetsu+2010_CL0024; @Umetsu+2011]. On the other hand, the faint blue background population tends to have a steeper intrinsic count slope close to the lensing invariant slope ($s=0.4$). Alternatively, the constant $\lambda$ can be determined such that the mean $\kappa$ averaged over the outermost cluster region vanishes, if a sufficiently wide sky coverage is available.[^2] Cluster Lensing Methodology {#sec:method} =========================== In this section we outline our methods for obtaining cluster mass profiles in a continuous radial coverage from the central region to beyond the virial radius, by combining independent weak-lensing distortion, magnification, and strong-lensing measurements. Cluster Weak Lensing {#subsec:wl} -------------------- The relation between observable distortion ($g$) and underlying convergence ($\kappa$) is nonlocal, and the convergence derived from distortion data alone suffers from a mass-sheet degeneracy (§ \[sec:basis\]). However, by combining the complementary distortion ($g$) and magnification ($\mu$) measurements the convergence can be obtained unambiguously with the correct mass normalization. We construct a discrete convergence profile in the weak-lensing regime from observable lens distortion and magnification profiles, $g_+(\theta)=\gamma_+(\theta)/[1-\kappa(\theta)]$ and $n_\mu (\theta) = n_0\mu(\theta)^{2.5s-1}$ [see Section 3 and Appendix B of @Umetsu+2011 for details of weak-lensing profile measurements], following the Bayesian prescription given by @Umetsu+2011. The Bayesian approach allows for a full parameter-space extraction of model and calibration parameters. A proper Bayesian statistical analysis is of particular importance to explore the entire parameter space and investigate the parameter degeneracies, arising in part from the mass-sheet degeneracy. In the Bayesian framework, we sample from the posterior probability density function (PDF) of the underlying signal $\bs$ given the data $\bd$, $P(\bs|\bd)$. Expectation values of any statistic of the signal $\bs$ shall converge to the expectation values of the a posteriori marginalized PDF, $P(\bs|\bd)$. The variance covariance matrix $C$ of $\bs$ is obtained from the resulting posterior sample. In our problem, the signal $\bs$ is a vector containing the discrete convergence profile, $\kappa_i\equiv \kappa(\theta_i)$ with $i=1,2,..,N^{\rm wl}$ in the weak-lensing regime ($\theta_i>\theta_{\rm Ein}$), and the average convergence within the inner radial boundary $\theta_{\rm min}^{\rm wl}$ of the weak lensing data, $\overline{\kappa}_{\rm min}\equiv \overline{\kappa}(<\theta_{\rm min}^{\rm wl})$, so that $\bs =\{\overline{\kappa}_{\rm min},\kappa_i\}_{i=1}^{N^{\rm wl}}$, being specified by $(N^{\rm wl}+1)$ parameters. The Bayes’ theorem states that $$P(\bs|\bd) \propto P(\bs) P(\bd|\bs),$$ where ${\cal L}(\bs)\equiv P(\bd|\bs)$ is the likelihood of the data given the model ($\bs$), and $P(\bs)$ is the prior probability distribution for the model parameters. The ${\cal L}(\bs)$ function for combined weak lensing observations is given as a product of the two separate likelihoods, ${\cal L}_{\rm wl}={\cal L}_g{\cal L}_\mu$, where ${\cal L}_g$ and ${\cal L}_\mu$ are the likelihood functions for distortion and magnification, respectively, as given in @Umetsu+2011. The log-likelihood for combined weak-lensing distortion and magnification observations, $\{g_{+,i}\}_{i=1}^{N^{\rm wl}}$ and $\{n_{\mu,i}\}_{i=1}^{N^{\rm wl}}$, is given as $$-2\ln{\cal L_{\rm wl}}=\displaystyle\sum_{i=1}^{N^{\rm wl}} \frac{[g_{+,i}-\hat{g}_{+,i}(\bs)]^2}{\sigma_{+,i}^2}+ \displaystyle\sum_{i=1}^{N^{\rm wl}} \frac{[n_{\mu,i}-\hat{n}_{\mu,i}(\bs)]^2}{\sigma_{\mu,i}^2},$$ where $(\hat{g}_{+,i},\hat{n}_{\mu,i})$ are the theoretical predictions for the corresponding observations; the errors $\sigma_{+,i}$ for $g_{+,i}$ $(i=1,2,...,N^{\rm wl})$ due primarily to the variance of the intrinsic source ellipticity distribution can be conservatively estimated from the data using bootstrap techniques; the errors $\sigma_{\mu,i}$ for $n_{\mu,i}$ ($i=1,2,...,N^{\rm wl}$) include both contributions from Poisson errors in the counts and contamination due to intrinsic clustering of red background galaxies [@Umetsu+2011]. For each parameter of the model $\bs$, we consider a simple flat prior with a lower bound of $\bs=0$, that is, $\overline{\kappa}_{\rm min}>0$ and $\kappa_i >0$. Additionally, we account for the calibration uncertainty in the observational parameters, such as the normalization and slope parameters ($n_0,s$) of the background counts and the relative lensing depth due to population-to-population variations between the background samples used for the magnification and distortion measurements [see @Umetsu+2011]. Cluster Strong Lensing {#subsec:sl} ---------------------- We apply our well-tested approach to strong-lens modeling, which has previously uncovered large numbers of multiply-lensed galaxies in ACS images of many clusters, such as A1689 at $z=0.183$ [@2005ApJ...621...53B], Cl0024+17 at $z=0.395$ [@Zitrin+2009_CL0024], 12 high-$z$ MACS clusters [@Zitrin+2011_MACS], MS 1358+62 at $z=0.33$ [@Zitrin+2010_MS1358], and A383 at $z=0.188$ [@Zitrin+2011_A383]. Briefly, the basic assumption adopted is that mass approximately traces light, so that the photometry of the red cluster member galaxies is used as the starting point for our model. Cluster member galaxies are identified as lying close to the cluster sequence by [*HST*]{} multiband photometry. In the strong-lensing regime we approximate the large scale distribution of cluster mass by assigning a power-law mass profile to each cluster galaxy, the sum of which is then smoothed. The degree of smoothing ($S$) and the index of the power-law ($q$) are the most fundamental parameters determining the cluster mass profile dominated by dark matter. A worthwhile improvement in fitting the location of the lensed images is generally found by expanding to first order the gravitational potential of this smooth component, equivalent to a coherent external shear $\Gamma^{\rm ex}_{\alpha\beta}$ ($\alpha,\beta=1,2$) describing the overall matter ellipticity. The direction $\phi_{\rm ex}$ of the spin-2 external shear $\Gamma^{\rm ex}_{\alpha\beta}$ and its amplitude $|\gamma_{\rm ex}|$ are free parameters, allowing for some flexibility in the relation between the distribution of dark matter and the distribution of galaxies, which cannot be expected to trace each other in detail. The total deflection field $\balpha(\btheta)=\sum_j\balpha_j(\btheta)=(\Sigma_{\rm crit}^{-1}/\pi)\int\!d^2\theta'\,(\btheta-\btheta')/|\btheta-\btheta'|^2\sum_j\Sigma_j(\btheta')$ consists of the galaxy component $\balpha_{\rm gal}(\btheta)$, scaled by a factor $K$, the smooth cluster dark-matter component $\balpha_{\rm DM}(\btheta)$, scaled by $(1-K)$, and the external-shear component $\balpha_{\rm ex}(\btheta)$, $$\balpha(\btheta)=K\balpha_{\rm gal}(\btheta)+(1-K)\balpha_{\rm DM}(\btheta) +\balpha_{\rm ex}(\btheta),$$ where $\alpha_{{\rm ex},\alpha}(\btheta)=\left(\Gamma^{\rm ex}\right)_{\alpha\beta}\Delta\theta_\beta$ with $\Delta\btheta$ being the displacement vector of the angular position $\btheta$ with respect to a fiducial reference position. The overall normalization ${\cal N}$ of the model and the relative scaling $K$ of the smooth dark matter component versus the galaxy contribution bring the total number of free parameters in the model to 6. This approach to strong lensing is sufficient to accurately predict the locations and internal structure of multiple images, since in practice the number of multiple images uncovered readily exceeds the number of free parameters, so that the fit is fully constrained. We use this preliminary model to delens the more obvious lensed galaxies back to the source plane by subtracting the derived deflection field. We then relens the source plane in order to predict the detailed appearance and location of additional counter images, which may then be identified in the data by morphology, internal structure and color. The best-fit strong-lensing model is assessed by minimizing the $\chi^2$ value in the image plane: $$\chi^2_{\rm sl}=\displaystyle\sum_i{ \frac{[\btheta_i-\hat{\btheta}_i(q,S,{\cal N},K,\Gamma^{\rm ex})]^2}{\sigma_i^2}},$$ where $i$ runs over all lensed images, $\hat{\btheta}_i(q,S,{\cal N},\Gamma^{\rm ex})$ is the position given by the model, $\btheta_i$ is the observed image position, and $\sigma_i$ is the positional measurement error. For each model parameter, we estimate the $1\sigma$ uncertainty by $\Delta\chi^2\equiv \chi^2-\chi^2_{\rm min}=1$ in the six-parameter space. The uncertainties for the $\Sigma(\btheta)$ field and the $\Sigma(\theta)$ profile are estimated by propagating the errors on the strong-lens model parameters, $(q,S,{\cal N},K,\Gamma^{\rm ex})$. Combining Weak and Strong Lensing {#subsec:wl+sl} --------------------------------- We derive a full-radial mass profile on an individual cluster basis by combining independent weak and strong lensing data, which can be compared for consistency in the region of overlap. In order to obtain meaningful radial profiles, one must carefully define the center of the cluster. It is often assumed that the cluster mass centroid coincides with the position of the brightest cluster galaxy (BCG), whereas the BCGs can be offset from the mass centroids of the corresponding dark matter halos [@Johnston+2007_SDSS1; @Oguri+2010_LoCuSS; @Oguri+Takada2011]. @Umetsu+2011 adopted the location of the BCG as the cluster center in their one-dimensional profile analysis of five massive clusters. A small offset of typically ${\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}20$kpc$h^{-1} \equiv d_{\rm off}$ is found by @Umetsu+2011 between the BCG and the dark matter center of mass recovered from strong-lens modeling (Section \[subsec:sl\]). In the following we will adopt the BCG position as the cluster center, and limit our analysis to radii greater than $R_{\rm min}\equiv 2d_{\rm off}=40\,$kpc$h^{-1}$, beyond which the cluster miscentering effects on the $\Sigma$ profile are negligible [see Section 10 of @Johnston+2007_SDSS1]. Having defined the cluster center, we can construct a joint discrete mass profile $\bSigma =\{\Sigma(R_i)\}_{i=1}^{N}$ as a function of the projected radius $R=D_l\theta$ by combining the weak and strong lensing $\kappa$ profiles: $\Sigma(R_i)= w_i^{-1}\kappa(\theta_i)$ ($i=1,2,...,N$), where $w_i$ is the lensing efficiency function, or the inverse critical surface mass density, $w_i \equiv (\Sigma_{{\rm crit},i})^{-1} = (4\pi G/c^2) D_l \beta_i$, Note, the $i$ dependence arises because strong and weak lensing profiles with different depths are combined together. To simplify the analysis, we exclude the strong-lensing data points in the region of overlap (typically, $\theta_{\rm Ein}{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}\theta{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}2\theta_{\rm Ein}$) as well as the central weak-lensing bin $\overline{\kappa}_{\rm min}$, when defining the joint $\Sigma$ profile. The formulation thus far allows us to derive covariance matrices $C^{\rm stat}_{ij}$ of statistical measurement errors for individual cluster $\bkappa$ profiles. Here we take into account the effect of uncorrelated large scale structure projected along the line of sight on the error covariance matrix $C^{\rm lss}_{ij}$ as $C=C^{\rm stat}+C^{\rm lss}$, where $C^{\rm lss}$ is given as [@1998MNRAS.296..873S; @2003MNRAS.339.1155H; @Dalal+2005; @Hoekstra+2011; @Oguri+Takada2011] $$\label{eq:cl} C^{\rm lss}_{ij} = \int\!\frac{l\,dl}{2\pi}\, C^{\kappa\kappa}(l)\, \hat{J}_0(l\theta_i)\hat{J}_0(l\theta_j). $$ Here $C^{\kappa\kappa}(l)$ is the weak-lensing power spectrum as a function of angular multipole $l$ evaluated for a given source population and a cosmology, and $\hat{J}_0(l\theta_i)$ is the Bessel function of the first kind and order zero averaged over the $i$th annulus between $\theta_{i,1}$ and $\theta_{i,2} (>\theta_{i,1})$, given as $$\hat{J}_0(l\theta_i)=\frac{2}{(l\theta_{i,2})^2-(l\theta_{i,1})^2} \left[ l\theta_{i,2}J_1(l\theta_{i,2})-l\theta_{i,1}J_1(l\theta_{i,1}) \right].$$ We will assume the concordance $\Lambda$CDM cosmological model of @Komatsu+2011_WMAP7 and use the fitting formula of @PD96 to compute the nonlinear mass power spectrum that enters in equation (\[eq:cl\]). Stacked Lensing Analysis {#subsec:stack} ------------------------ The utility of high-quality data is ultimately limited by the cosmic noise from large scale structure along the line of sight, producing covariance between radial bins, particularly behind the cluster center, where magnified sources lie at greater distances. This noise is correlated between radial bins, but can be overcome by stacking an ensemble of clusters along independent lines of sight. Stacking also helps average over the effects of cluster asphericity and substructure inherent in projected lensing measurements. The statistical precision can be greatly improved by stacking together a number of clusters, especially on small angular scales [see @Okabe+2010_WL], allowing a tighter comparison of the averaged profile with theoretical models. With the full mass profiles of individual clusters from combined weak and strong lensing (Section \[subsec:wl+sl\]), we can stack the clusters to produce an averaged radial mass profile. Here we re-evaluate the mass profiles of the individual clusters in $M$ logarithmically-spaced radial bins in the range $R=[R_{\rm min},R_{\rm max}]$ following the prescription given in @Umetsu+2011. Since the noise in different clusters is uncorrelated, the mass profiles of individual clusters can be co-added according to [@Umetsu+2011] $$\label{eq:stack} \langle \bSigma\rangle = \left(\displaystyle\sum_n {\cal W}_n \right)^{-1} \, \left( \displaystyle\sum_n{ {\cal W}_n \bSigma_n} \right),$$ where the index $n$ runs over all clusters, $\bSigma_n$ is a vector containing the discrete surface mass density profile for the $n$th cluster, and ${\cal W}_n$ is the window matrix defined as $$({\cal W}_n)_{ij} \equiv \left(C_n^{-1}\right)_{ij} (w_n)_i (w_n)_j$$ with $(C_n)_{ij}$ and $(w_n)_i$ $(i=1,2,...,M)$ being the full covariance matrix and the lensing efficiency function for the $n$th cluster, respectively. The error covariance matrix for the stacked mass profile $\langle\bSigma\rangle$ is obtained as $$\label{eq:covar_stack} {\cal C} =\left( \displaystyle \sum_n {\cal W}_n\right)^{-1}.$$ Applications: Hubble and Subaru Observations of Four High-Mass Clusters {#sec:app} ======================================================================= Cluster Sample and Lensing Data {#subsec:data} ------------------------------- Following the methodology outlined in Section \[sec:method\], we analyze our consistent weak and strong lensing measurements presented in @Umetsu+2011 to examine the underlying projected mass profile $\Sigma(R)$ of a sample of four well-studied high-mass clusters ($M{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}10^{15}M_\odot$) at intermediate redshifts, A1689 ($z=0.183$), A1703 ($z=0.281$), A370 ($z=0.375$), and Cl0024+17 ($z=0.395$)[^3]. The massive clusters we have analyzed are well-known strong lensing clusters, displaying prominent strong-lensing features and large Einstein radii of $\theta_{\rm Ein}{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}30\arcsec$ [e.g., for a fiducial source redshift $z_s\sim 2$; @Broadhurst+Barkana2008; @Oguri+Blandford2009; @Zitrin+2011_Ein]. Table \[tab:data\] gives a summary of the basic properties of the clusters in our sample. For these clusters, the central mass distributions ($R{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}200$kpc$h^{-1}$) have been recovered in detail by our strong-lensing analysis [@2005ApJ...621...53B; @Zitrin+2009_CL0024; @Zitrin+2010_A1703; @Umetsu+2011] based on many sets of multiply-lensed images identified previously in very deep multicolor imaging with [*HST*]{}/ACS [e.g., @2005ApJ...621...53B; @Limousin+2008_A1703; @Richard+2009_A1703; @Richard+2010_A370; @Zitrin+2009_CL0024; @Zitrin+2010_A1703]. @Umetsu+2011 developed and applied a Bayesian method to derive model-independent projected mass profiles for five high-mass clusters (including RXJ1347-11 in addition to the four clusters) from Subaru weak-lensing distortion and magnification measurements, the combination of which can unambiguously break the mass-sheet degeneracy inherent in any mass inversion method based solely on shape distortion data. It was shown that for the four clusters of the present sample our independent strong and weak lensing mass profiles are in full agreement in the region of overlap ($R\sim 150\,$kpc$h^{-1}$), and together can be well described by, within the noise, a generalized form of the NFW profile for CDM-dominated equilibrium halos. This motivates us to reexamine in detail the form of the radial mass profile for the entire clusters. Results {#subsec:results} ------- [ccccccccc]{} A1689 & $0.183$ & $53\pm 3\arcsec (z_s=3.04)$ & $40,125$ & 12 & $129,2325$ & 11 & 35\ A1703 & $0.281$ & $31\pm 3\arcsec (z_s=2.627)$ & $40,177$ & 14 & $179,2859$ & 10 & 29\ A370 & $0.375$ & $37\pm 3\arcsec (z_s=2)$ & $40,149$ & 15 & $152,3469$ & 14 & 29\ Cl0024+17 & $0.395$ & $30\pm 3\arcsec (z_s=1.675)$ & $40,126$ & 14 & $134,3359$ & 12 & 26\ {width="120mm"} {width="90mm"} {width="120mm"} Our weak and strong lensing data together cover a wide range of radius ranging typically from $R\sim 10\,$kpc$\,h^{-1}$ to $2000$–$3500$kpc$\,h^{-1}$ [@Umetsu+2011], depending on the cluster redshift as limited by the field of view of Subaru/Suprime-Cam ($34\arcmin \times 27\arcmin$). Table \[tab:data\] lists for each cluster the radial ranges $R=[R_{\rm min}^{\rm sl},R_{\rm max}^{\rm sl}]$ and $[R_{\rm min}^{\rm wl},R_{\rm max}^{\rm wl}]$ of strong and weak lensing measurements, respectively, used to define a joint discrete mass profile $\bSigma =\{\Sigma(R_i)\}_{i=1}^{N}$, given in a total of $N$ radial bins spanning from $R_{\rm min}=R_{\rm min}^{\rm sl}$ to $R_{\rm max}=R_{\rm max}^{\rm wl}$. In Table \[tab:data\], we also quote values of the total ${\rm S/N}$ ratio in our joint cluster mass profiles ($\bSigma$) obtained using the full covariance matrix $C$. We find that ignoring the cosmic noise contribution (equation \[\[eq:cl\]\]) will underestimate the errors by $\sim 30\%$–$40\%$. To evaluate $C^{\rm lss}$ for strong-lensing observations, we projected the matter power spectrum out to a fiducial depth of $z_s=2$, which is a typical source redshift of strongly-lensed arcs in clusters at intermediate redshifts. We used the estimated mean source redshifts given in Table 3 of @Umetsu+2011 for weak lensing. We show in the top panel of Figure \[fig:stack\] the resulting averaged radial mass profile $\langle \Sigma(R)\rangle$ in $M=15$ logarithmically-spaced bins with its statistical $1\sigma$ uncertainty (given as the square root of the diagonal part of the full covariance matrix ${\cal C}$), obtained by stacking the four clusters using equations (\[eq:stack\]) and (\[eq:covar\_stack\]). Note, no scaling has been applied to match the mass normalizations between the four clusters, which span a relatively narrow range in mass, $1.3{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}M_{\rm vir}/(10^{15} M_\odot\,h^{-1}){\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}2.3$ [see Table 6 of @Umetsu+2011]. For our sample, we find a sensitivity-weighted average cluster redshift of $\langle z_l\rangle \simeq 0.32$, which is fairly close to the simple average of $\overline{z_l}=0.31$ due to the narrow redshift coverage of our cluster sample. The stacked mass profile exhibits a smooth radial trend with a clear radial curvature over a wide range of radius from $R=40$kpc$h^{-1}$ to $2800$kpc$h^{-1} \approx 1.4 r_{\rm vir}$, and is detected at a high significance level of $58\sigma$, with the contribution from cosmic covariance included. Here the maximum radius for the stacking analysis represents approximately the average maximum radius $\langle R_{\rm max}\rangle$ covered by our data. Also shown in Figure \[fig:stack\] is the cosmic noise contribution, which increases toward the cluster center. A noticeable increase of the stacked cosmic noise is seen at $R{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}150\,$kpc$h^{-1}$, within which the averaged profile is dominated by strong lensing measurements with greater depth. In the bottom panel of Figure \[fig:stack\], we plot the logarithmic density slope $\gamma_{\rm 2D}(R)\equiv -d\ln{\langle\Sigma\rangle}/d\ln{R}$ of the stacked mass profile. The logarithmic gradient of the average profile shows a clear continuous steepening with increasing radius in projection. To quantify and characterize the averaged cluster mass distribution, we compare the $\langle\Sigma\rangle$ profile with the physically and observationally motivated NFW model. Here we consider a generalized parametrization of the NFW model (gNFW, hereafter) of the form [@Zhao1996; @Jing+Suto2000]: $$\label{eq:gnfw} \rho(r)=\frac{\rho_s}{(r/r_s)^\alpha(1+r/r_s)^{3-\alpha}},$$ where $\rho_s$ is the characteristic density, $r_s$ is the characteristic scale radius, and $\alpha$ represents the inner slope of the density profile. This reduces to the NFW model for $\alpha=1$. We introduce the radius $r_{-2}$ at which the logarithmic slope of the density is isothermal, i.e., $\gamma_{\rm 3D}=2$. For the gNFW profile, $r_{-2}=(2-\alpha)r_s$, and thus the corresponding concentration parameter reduces to $c_{-2}\equiv r_{\rm vir}/r_{-2}=c_{\rm vir}/(2-\alpha)$. We specify the gNFW model with the central cusp slope, $\alpha$, the halo virial mass, $M_{\rm vir}$, and the concentration, $c_{-2}=c_{\rm vir}/(2-\alpha)$. We employ the radial dependence of the gNFW lensing profiles given by @Keeton2001_mass. First, when the central cusp slope is fixed to $\alpha=1$ (NFW), the best-fit model for the averaged $\langle\Sigma\rangle$ profile is obtained as $M_{\rm vir}=1.54^{+0.11}_{-0.10}\times 10^{15}M_\odot\,h^{-1}$ and $c_{-2}=c_{\rm vir}=7.68^{+0.42}_{-0.40}$ with the minimized $\chi^2$ value ($\chi^2_{\rm min}$) of 5.8 for 13 degrees of freedom (dof), corresponding to a $Q$-value goodness-of-fit of $Q=0.952$. This model yields an Einstein radius of $\theta_{\rm Ein}=39.9\arcsec^{+4.4}_{-4.1}$ for a fiducial source at $z_s=3$. The resulting best-fit NFW parameters from the stacked analysis are consistent with the respective sample weighted means of the individual NFW model fits obtained by @Umetsu+2011 [Table 6]: $\langle M_{\rm vir}\rangle =1.44\pm 0.11 \times 10^{15}M_\odot\,h^{-1}$ and $\langle c_{\rm vir}\rangle = 7.76\pm 0.79$. Next, when $\alpha$ is allowed to vary, a gNFW fit to $\langle\bSigma\rangle$ gives $M_{\rm vir}=1.50^{+0.14}_{-0.13}\times 10^{15}M_\odot\,h^{-1}$, $c_{-2}=7.91^{+0.72}_{-0.75}$, and $\alpha=0.89^{+0.27}_{-0.39}$ with $\chi^2_{\rm min}/{\rm dof}=5.7/12$ and $Q=0.931$ ($\theta_{\rm Ein}=38.4\arcsec^{+12.2}_{-10.2}$ at $z_s=3$), being consistent with a simple NFW model with $\alpha=1$. Thus the addition of the $\alpha$ parameter does not improve the fit substantially, as shown by the quoted $\chi^2$ and $Q$ values [see also @Zitrin+2011_A383]. The two-dimensional marginalized constraints ($68.3\%, 95.4\%$, and $99.7\%$ confidence levels) on $(M_{\rm vir},\alpha)$ and $(c_{-2},\alpha)$ are shown in Figure \[fig:2dconf\]. Finally, a force fit to the singular isothermal sphere (SIS) model ($\rho\propto r^{-2}$) yields a poor fit with $\chi^2_{\rm min}/{\rm dof}=78.5/14$, so that the SIS model is strongly disfavored at $62\sigma$ significance from a likelihood-ratio test, based on the difference between $\chi^2$ values of the best-fit NFW and SIS models: $\Delta\chi^2\equiv \chi^2_{\rm SIS,min}-\chi^2_{\rm NFW,min}=72.6$ for a 1 degree-of-freedom. Discussion and Conclusions {#sec:discussion} ========================== We have developed a method for improving the statistical precision of cluster mass profiles, combining independent weak-lensing distortion, magnification, and strong-lensing measurements. This extends recent weak-lensing work by @Umetsu+2011 to include the central strong-lensing information in a stacking analysis, for full radial coverage. Our methods take into account the cosmic covariance from uncorrelated large scale structure projected along the line of sight [@2003MNRAS.339.1155H; @Hoekstra+2011], as well as the effect of different cluster redshifts, so that error propagation in terms of lensing efficiency of individual clusters can be properly averaged. We have applied our method to a sample of four similarly high-mass lensing clusters (A1689, A1703, A370, and Cl0024+17), for which we have previously identified multiply-lensed images and measured weak magnification and distortion effects from deep [*HST*]{} and Subaru observations [@2005ApJ...621...53B; @UB2008; @Umetsu+2010_CL0024; @Zitrin+2010_A1703; @Medezinski+2010; @Medezinski+2011; @Umetsu+2011]. For our sample of massive clusters the strong and weak lensing regimes contribute equal logarithmic coverage of the radial profile and can be compared for consistency in the region of overlap. We have formed an averaged radial mass profile $\langle\Sigma(R) \rangle$ from stacking the clusters (Figure \[fig:stack\]), which shows a progressive steepening with increasing radius from $R=40\,$kpc$h^{-1}$ to $2800\,$kpc$h^{-1}$. The inner radial boundary is chosen to be sufficiently large to avoid smoothing from cluster miscentering effects [@Johnston+2007_SDSS1], where the typical offset between the BCG and the dark matter center is estimated as $d{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}20\,$kpc$h^{-1}$ for our sample from our detailed strong-lens modeling (see Section \[subsec:wl+sl\]). The stacked full mass profile is detected at a high significance level of $58\sigma$ over the entire radial range. It is found here that ignoring the cosmic noise contribution will underestimate the errors by $\sim 30\%$–$40\%$. This is due to the correlation of this noise between radial bins and can only be reduced by averaging over independent lines of sight, with uncorrelated line of sight structures, i.e. by averaging over well separated clusters. Our stacked projected mass profile with a continuously steepening radial trend is very accurately described by the NFW form predicted for the family of CDM-dominated halos, whereas it strongly disfavors the SIS model at $62\sigma$ significance. In the context of an assumed gNFW profile, the central cusp slope is constrained as $\alpha=0.89^{+0.27}_{-0.39}$ (at $r{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}0.02 r_{\rm vir}$; see Figures \[fig:stack\] and \[fig:2dconf\]), being consistent with, but slightly shallower than, the simple NFW form with $\alpha=1$. Our results are in agreement with recent high-resolution simulations, which find asymptotic inner slopes somewhat shallower than unity, $\gamma_{\rm 3D}(r\to 0){\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.9$, for galaxy- and cluster-sized $\Lambda$CDM halos [e.g., @Merritt+2006; @Graham+2006; @Navarro+2010]. Note NFW define this profile for halos which they identify as in virial equilibrium, in terms of the simulated CDM particles [see Section 2.2.2 of @1997ApJ...490..493N]. The clusters we have selected for our stacked analysis are, in terms of their lensing properties, very well behaved with at most only $\sim 10\%$ perturbations in mass visible locally in the two-dimensional mass distribution, and otherwise very symmetric over most of the radius [@2005ApJ...621...53B; @BTU+05; @BUM+08; @Umetsu+2010_CL0024]. Detailed hydrodynamical simulations show that equilibrium is relatively rapidly achieved in only a few sound crossing times after a major merger, though some dynamical and gas disruption may continue for over a Gyr. This is not important in terms of the central relaxation time of the dark matter [@2001ApJ...561..621R; @Umetsu+2010_CL0024]. An accurate measurement of the cluster mass profile enables us to constrain dark matter models. Recently @BEC2009 examined in detail an extremely light bosonic dark matter (ELBDM) model ($m\sim 10^{22}$eV) as an alternative to CDM in the context of nonlinear cosmic structure formation. ELBDM with a de-Broglie wavelength of astronomical length scales, if it exists, may well be in a ground-state Bose-Einstein condensate and hence well described by a coherent wave function, which may naturally account for the perceived lack of small galaxies relative to the $\Lambda$CDM model [@Klypin+1999; @Peebles+Nusser2010]. @BEC2009 showed that, irrespective of whether halos form through accretion or merger, ELBDM halos can form steepening density profiles of the form similar to the standard CDM, but with perhaps a steeper central cusp slope of $\gamma_{\rm 3D}\simeq 1.4$ and a shallower outer slope of $\gamma_{\rm 3D}\simeq 2.5$. During a merger between condensates interesting large-scale interference occurs which will differ markedly from standard collisionless CDM, and it will be important to explore this class of dark matter further via more extensive and detailed simulations for testing against accurate lensing profiles of both relaxed and merging clusters. The mean concentration for the four massive lensing clusters considered here is found to be $c_{\rm vir}=7.68^{+0.42}_{-0.40}$ (at a mean virial mass $M_{\rm vir}=1.54^{+0.11}_{-0.10}\times 10^{15}M_\odot\,h^{-1}$), which is apparently higher than the standard $\Lambda$CDM predictions evaluated at the mean redshift $\langle z_l\rangle=0.32$ of our sample: $c_{\rm vir}=4.5^{+1.3}_{-1.0}$ (the errors quoted represent a $1\sigma$ lognormal scatter of $\sigma[\log_{10}{c_{\rm vir}}]=0.11$) for relaxed clusters derived by @Duffy+2008 from $N$-body simulations based on the WMAP 5-year data and $c_{\rm vir}\approx 4.4$ by @Klypin+2010 from the recent [*Bolshoi*]{} $\Lambda$CDM $N$-body simulation. More recent results with greater mass resolution based on four large $N$-body simulations (Bolshoi, MultiDark, Millennium-I and II) exhibit a complex mass and redshift dependence of the median concentration, namely a flattening and upturn of concentration at very high mass and redshift [@Prada+2011]. Accordingly, their concentrations derived for cluster-sized halos (i.e., rare objects corresponding to high-$\sigma$ peaks in the primordial density field) are substantially higher than previous results based on smaller simulations. Interestingly, they find a concentration of $c_{\rm vir}\sim 7$ for their most-massive [*relaxed*]{} halos with $M_{\rm vir}\approx 10^{15}M_\odot \,h^{-1}$ at $z=0$ [Figure 15 of @Prada+2011]. A comparison between our results and the $\Lambda$CDM predictions [@Duffy+2008; @Klypin+2010; @Prada+2011] is given in Figure \[fig:cmplot\]. An accurate characterization of the observed sample is crucial for any cluster-based cosmological tests. In the extreme case, those clusters identified by the presence of a giant arc represent the most lensing-biased population. Calculations of the enhancement of the projected mass and hence boosted Einstein radii (say, $\theta_{\rm Ein}>20\arcsec$) find a statistical bias of $\sim 34\%$ derived from $N$-body simulations of the $\Lambda$CDM model [@2007ApJ...654..714H]. Semi-analytical simulations incorporating idealized triaxial halos yield a $\sim 50\%$ bias correction [@Oguri+Blandford2009]. Applying a conservative $50\%$ bias correction, we find a discrepancy of about $1.8\sigma$ with respect to the $\Lambda$CDM predictions by the @Duffy+2008 model for relaxed clusters (see Figure \[fig:cmplot\]). If this large bias ($\sim 50\%$) is coupled to a sizable intrinsic scatter in concentration, estimated for the full halo population to be $\sigma[\log_{10}c_{\rm vir}]=0.11$–$0.15$, then our measurements can come into line with standard $\Lambda$CDM. The results presented here are very favorable in terms of the standard explanation for dark matter, as collisionless and non-relativistic, interacting only via gravity, with a very precise match between our composite mean mass profile, and that of the general form of the mass profile advocated for massive halos in virial equilibrium. The relatively high concentration we obtain for the averaged profile is consistent with previous lensing work which similarly detected a concentration excess in the lensing based measurements for many individual relaxed strong-lensing clusters . This possibly interesting tension between cluster lensing observations and $\Lambda$CDM models can be more definitively addressed with full-lensing data for new cluster surveys, such as CLASH[^4], LoCuSS, Subaru Hyper Suprime-Cam, and XMM-XXL [@XXL2010], to meaningfully examine the $c_{\rm vir}$–$M_{\rm vir}$ relation over a wider mass and redshift range when applied to sizable samples of relaxed clusters. It is highly desirable to cover the full profile by combining accurate weak and strong lensing measurements, requiring several sets of multiple images over a wide range of source redshift, to obtain a meaningful model-independent inner profile and to add weak lensing with sufficient color information to exclude the otherwise sizable dilution effect on the weak lensing signal from foreground and cluster members. The CLASH survey is in particular designed to generate such useful data free of systematics in both the weak and strong regime, with first results for the substantial smaller mass cluster A383 with $M_{\rm vir}= 5.37^{+0.70}_{-0.63} \times 10^{14}M_\odot\,h^{-1}$ [@Zitrin+2011_A383] showing similar behavior ($c_{\rm vir}=8.77^{+0.44}_{-0.42}$). We thank the anonymous referee for a careful reading of the manuscript and and for providing useful comments. 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--- abstract: 'I demonstrate that an effect similar to the Römer delay, familiar from timing radio pulsars, should be detectable in the first eclipsing double white dwarf (WD) binary, [NLTT 11748]{}. By measuring the difference of the time between the secondary and primary eclipses from one-half period (4.6s), one can determine the physical size of the orbit and hence constrain the masses of the individual WDs. A measurement with uncertainty $<0.1\,$s—possible with modern large telescopes—will determine the individual masses to $\pm0.02M_\odot$ when combined with good-quality ($<1\,{\ensuremath{{\rm km\,s}^{-1}}}$) radial velocity data, although the eccentricity must also be known to high accuracy ($\pm 10^{-3}$). Mass constraints improve as $P^{-1/2}$ (where $P$ is the orbital period), so this works best in wide binaries and should be detectable even for non-degenerate stars, but such constraints require the mass ratio to differ from one and undistorted orbits.' author: - 'David L. Kaplan' title: Mass Constraints from Eclipse Timing in Double White Dwarf Binaries --- Introduction ============ Since the discovery of binary pulsars [@ht75], precision timing (typical uncertainties $<1\,\mu$s) has been used to derive a variety of physical constraints [see the discussion in @lk04]. The arrival-time delay across the orbit (the Römer delay) immediately gives the projected semimajor axis of the pulsar [@bt76]. This, especially when coupled with the relatively narrow mass distribution of neutron stars [@tc99], constrains the mass of the companion. I contrast this with eclipse timing of planetary systems (typically uncertainties are $\gtrsim$seconds). Here, with a mass ratio $\approx 10^{-3}$ the radial velocity curve gives a limit on the mass of the companion planet. With transiting systems, $\sin i\approx 1$ and the mass of the planet is further constrained but not known uniquely [@cblm00], although with knowledge of the stellar parameters one can infer the planetary mass and radius [e.g., @bcg+01]. If individual eclipses can be timed to high precision (and here I mean both primary and secondary eclipses, i.e., transits and occultations), one can learn more about the system (e.g., @winn10). Variations in the eclipse times can unveil the presence of additional bodies in the system [e.g., @assc05; @hm05], precession [e.g., @me02], and kinematics of the system [@rafikov09]. With the recent discovery [@sks+10] of , an eclipsing double white dwarf (WD) binary with a tight enough orbit that the binary will merge within a Hubble time, a whole new series of questions may be asked. The initial constraints are the radial velocity amplitude of the lighter object (owing to the inverted mass–radius relation of WDs, this object is the larger and brighter member of the system) and the widths and depths of both transit and occultation. From this, assuming a cold C/O WD for the heavier object, @sks+10 were able to limit the masses and radii of both objects, but could not determine unique constraints. Measurement of spectral lines from the fainter object would determine both masses uniquely, but this is challenging as the fainter object is only $\approx 3.5$% of the flux of the brighter. A number of other close WD binaries have been discovered in the last 2 years (see Table \[tab:wd\] for those with undetermined inclinations). Most of them, like [NLTT 11748]{}, appear to have a low-mass ($\lesssim 0.2\,M_{\odot}$) He WD in orbit with a more massive (0.5–1.0$M_{\odot}$) C/O WD. Such systems are of interest because of their eventual evolution, with mass transfer brought on by gravitational radiation [@nypzv01] and are presumed to be the progenitors of highly variable objects: R CrB stars, AM CVn binaries, and Type Ia supernovae [@it84; @webbink84]. Many of these binaries are also of immediate interest as verification targets for the *Laser Interferometer Space Antenna* (*LISA*) mission [@nelemans09]. [l c c c c c c c l]{} SDSS J1053+5200 & 1.02 & 265 & 0.26 & $>0.017$ & 0.20 & 0.04& 0.2 & 1,2\ SDSS J1436+5010 & 1.10 & 347 & 0.45 & 0.014 & 0.22 & 0.04& 0.7 & 1,2\ SDSS J0849+0445 & 1.89 & 367 & 0.65 & 0.012 & 0.17 & 0.05& 2.0 & 1\ WD 2331+290 & 4.08 & 156 & 0.39 & 0.015 & 0.32 & 0.016 & 0.5 & 3,4,5\ SDSS J1257+5428 & 4.55 & 323 & 0.92 & 0.009 & 0.15 & 0.04& 4.7 & 6,7,8\ [NLTT 11748]{}& 5.64 & 271 & 0.74 & 0.010 & 0.15 & 0.04& 4.6 & 9,10\ SDSS J0822+2753 & 5.85 & 271 & 0.71 & 0.010 & 0.17 & 0.04& 4.7 & 1\ Of the 11 compact WD binaries known, only [NLTT 11748]{} is known to be eclipsing, but searches for the other sources are not uniformly constraining and additional systems may yet be discovered. The flux ratios vary for the systems, and in some cases it may be easier to directly measure the radial velocity curves for both members of the binary. Without two radial velocity curves, mass constraints are limited. Such constraints are invaluable in understanding the detailed formation histories and expected evolution of these systems as well as in determining the mass–radius relation from eclipse measurements. Moreover, their use as *LISA* verification sources is improved by accurate knowledge of the binary parameters. In this [Letter]{}, I discuss an effect that uses precision timing of the eclipses in such double WD systems to help constrain the individual masses of the WDs. This technique is known in other contexts, being common in radio pulsar systems and planetary systems [@kcn+07; @hdd+10; @ack+10], although in the latter it is largely a nuisance parameter and does not constrain the systems. I discuss its applicability to eclipsing double WD systems, the required observational precision and the resulting accuracy. Light Travel Delay and Mass Constraints ======================================= In a system with a circular orbit, one often speaks of the primary and secondary eclipses as occurring exactly $1/2$ period apart, but this is not the case. If the members of the binary are of unequal mass the finite speed of light will cause an apparent shift in the phase of the secondary eclipse from $P/2$, where $P$ is the period of the binary [@loeb05; @fabrycky10]. This is similar to the shifts in eclipse timing caused by a perturbing third body on a binary system [@sd95; @ddj+98; @ddk+00; @sterken05; @lkk+09; @qdl+09], although here one only requires two bodies and the frequency of the shift is known. In the case of a planet with mass $m\ll M$ orbiting a star with mass $M$, one has a primary eclipse when the planet is in front of the star. The light is blocked at time $t=0$. However, that light was emitted earlier by the star, at time $t_1=0-a/c$, since it traveled a distance $a$ (the semimajor axis). For the secondary eclipse, the light is emitted by the planet at time $t=P/2$ but is blocked $a/c$ later, at $t_2=P/2+a/c$. The difference of these times exceeds $P/2$ by ${\ensuremath{\Delta t_{\rm LT}}}=t_2-t_1-P/2=2a/c$, the sought-after quantity. For two finite masses, I consider two objects orbiting their center of mass with period $P$, masses $M_1$ and $M_2$, and semimajor axis $a$. The total mass of the system is $M=M_1+M_2$, and of course $4\pi^2a^3=P^2G M$; the first object orbits at a radius $a_1=a(M_2/M)$ and the second object orbits at a radius $a_2=a(M_1/M)$. Near primary eclipse, the primary is at $[x,y]=[2\pi a_1 t/P, a_1]$ and the secondary is at $[-2\pi a_2 t/P,-a_2]$ at time $t$, with the observer at $[0,-\infty]$. I project the image of the two objects to the barycenter at $y=0$. This gives $x_{\rm B,1}=2\pi a_1 (t-a_1/c)/P$ and $x_{\rm B,2}=-2\pi a_2(t+a_2/c)/P$. Eclipses occur when these are equal, which has the solution $t_1=(a_1-a_2)/c$. Near secondary eclipse, the primary is at $[-2\pi a_1 (t-P/2)/P, a_1]$ and the secondary is at $[2\pi a_2 (t-P/2)/P,-a_2]$. Following the same argument, eclipses occur when $t_2-P/2=(a_2-a_1)/c$. So the eclipses differ by $t_2-t_1=P/2+2(a_2-a_1)/c$. The light-travel delay is again ${\ensuremath{\Delta t_{\rm LT}}}=t_2-t_1-P/2$, $${\ensuremath{\Delta t_{\rm LT}}}=\left(\frac{2}{c}\right)\left(a_2-a_1\right)=\left(\frac{2 a}{c}\right)\left( \frac{M_1-M_2}{M_1+M_2}\right),$$ reaching $2a/c$ when $M_2\ll M_1$, as expected. From Kepler’s laws the mass function ${K_2^3 P}/2\pi G={M_1^3 \sin^3i}M^{-2}$, where $K_2$ is the radial velocity amplitude of object 2, and since it is a transiting system, $\sin i\approx 1$. Substituting for $a$ and the masses, the time delay in terms of observables and the mass ratio $q$ (where $q=M_2/M_1 \leq1$) is: $${\ensuremath{\Delta t_{\rm LT}}}=\frac{P K_2}{\pi c}(1-q).$$ Magnitude and Detectability --------------------------- The eclipse duration for a circular orbit is roughly $T\approx 2R_2P/(2\pi a)\approx 3\,$minutes [@winn10]; the duration of ingress/egress $\tau$ is decreased by a factor of $R_1/2R_2\approx 8$, $\tau\approx R_1P/(2\pi a)\approx 20\,$s (numerical results are for [NLTT 11748]{}); and ingress/egress are sharpest at inclinations of exactly $90\degr$. The accuracy of the eclipse time determination largely depends on the duration of the ingress/egress and the total number of photons accumulated during ingress/egress, since the bottom of the eclipse is only slightly curved (for the primary eclipse) if not flat (secondary eclipse), and one can derive [e.g., @cye+08] $$\sigma_{t_c}=\frac{\sigma}{\delta}\frac{\tau}{\sqrt{2 N_{\rm obs}}},$$ where the eclipse has fractional depth $\delta$, each observation has fractional uncertainty $\sigma$, and there are $N_{\rm obs}$ observations during $\tau$. This holds in the limit that the noise is uncorrelated [cf. @cw09; @skk10], which should be true at the level discussed here (photometric precision of $\gtrsim$mmag). So, $\sigma_{t_c}$ scales as the duration of ingress/egress divided by the total signal-to-noise ratio accumulated during that portion of the orbit. Since $N_{\rm obs}\propto \tau$ for a constant observing cadence, $\sigma_{t_c}\propto \sqrt{\tau}$. A star with $V=16.5\,$mag like [NLTT 11748]{} gives roughly $0.1\,{\rm photon\,s}^{-1}{\rm cm}^{-2}$ or $2\times 10^{5}$photons detected during a 20 s ingress/egress with a 4 m telescope. For an eclipse depth of 5% this means a precision on individual eclipse times of $<1\,$s. For general binary systems, I can rewrite the expression for [$\Delta t_{\rm LT}$]{} in terms of the primary mass $M_1$, $q$, and $P$ (eliminating $K_2$): $${\ensuremath{\Delta t_{\rm LT}}}= \left(\frac{2 G M_1 P^2}{\pi^2 c^3}\right)^{1/3} \frac{(1-q)}{(1+q)^{2/3}}\label{eqn:dt}$$ For systems with primaries that are typical C/O WDs and with secondaries that are He WDs, with $M_1=0.5-1 M_\odot$ and $q=1/6-1/2$, ${\ensuremath{\Delta t_{\rm LT}}}$ goes from 0.5s to 7s for periods of 0.5–10hr (Table \[tab:wd\]). Eclipse depths are functions of the radii and temperatures of the WDs, as well as the bandpass, and are hard to predict with any generality. The main factor that will change systematically for other double WD systems is $\tau$ itself, which is $\propto R_1 P^{1/3}$. This means the signal-to-noise ratio (i.e., detectability) is ${\ensuremath{\Delta t_{\rm LT}}}/\sigma_{t_c}\propto \sqrt{P/R_1}$, so the effect is easiest to see in long-period binaries. As the mass ratio approaches 1 the magnitude of the delay decreases, limiting its utility, but in such systems it may be easier to search for the second set of spectral lines (depending on the temperatures of the objects). This effect should also be present in partially degenerate (sdB+WD) or non-degenerate binary systems. The precision on the eclipse times goes as $\sqrt{\tau}\propto \sqrt{R_{\rm small}}$. If the binary is wide enough that the larger star(s) are undistorted by tides and hence the orbit remains strictly periodic, the overall detectability ${\ensuremath{\Delta t_{\rm LT}}}/\sigma_{t_c}\propto \sqrt{P/R_{\rm small}}$ can actually increase over the double WD case I have been considering. Requiring $a\propto R_{\rm large}$ to minimize tidal distortions, which scale as $(R_{\rm large}/a)^3$, for a system with a $0.2M_{\odot}$ M star one needs periods of $\gtrsim 1\,$day to have tidal effects that are as small as in the double WD systems. For the ingress/egress duration the radius of the smaller object $R_{\rm small}$ increases from $\sim 0.01R_{\odot}$ to $\sim 0.1R_{\odot}$ (for a $0.1M_\odot$ M star companion, for example), so if the period increases by more than a factor of 10 then the wide system is more easily detectable (the probability of eclipse does decrease as $R/a$, though, and both primary and secondary eclipses must be seen). However, the ephemeris must be known sufficiently well with tight enough limits on (or measurements of) eccentricity so that the light-travel delay is the only deviation from regularity (see below). For [NLTT 11748]{}, I recognize that $K_2$ is the radial velocity that was measured since the heavier object is the fainter one. So, $q\approx 0.15/0.71=0.21$, $K_2=271\,{\ensuremath{{\rm km\,s}^{-1}}}$, and $P=5.64\,$hr, which give ${\ensuremath{\Delta t_{\rm LT}}}=4.6\,$s. @sks+10 measured individual eclipse times to $\sim 10$s, making it hard to detect an effect like this. However, this was using 45 s exposures on a 2 m telescope, while the ingress/egress duration was only $\approx 20\,$s. Increasing to 4m or 8m will improve the S/N of individual exposures by a factor of 4–16, and using a cadence better matched to the orbit will help as well, driving eclipse time uncertainties to $\lesssim 1\,$s (as above). This is sufficient to detect [$\Delta t_{\rm LT}$]{}; below I discuss how well one can measure it and what constraints one can get from it. Comparison With Eccentricity ---------------------------- The above discussion considered circular orbits. For eccentricity $e>0$ the situation changes. I note that the objects in Table \[tab:wd\] have orbits that are consistent with circular orbits, although quantitative limits for $e$ are not always given. This follows from their expected evolutionary histories, where common-envelope evolution [@nvypz00] should have circularized orbits. Nonetheless, in case our understanding of these systems is incorrect or some further evolution (such as interaction with another body) may have caused non-zero eccentricity, I consider the effect of a non-zero eccentricity on our detection of [$\Delta t_{\rm LT}$]{}. First, there are changes to the expression for [$\Delta t_{\rm LT}$]{} [@fabrycky10]: $${\ensuremath{\Delta t_{\rm LT}}}=({\ensuremath{\Delta t_{\rm LT}}})_{e=0}\times\left( \frac{1-e^2}{1-e^2 \sin^2 \omega}\right)\approx ({\ensuremath{\Delta t_{\rm LT}}})_{e=0}\times\left(1-e^2\cos^2\omega+{\cal O}(e^4)\right)$$ where $\omega$ is the argument of pericenter. As $({\ensuremath{\Delta t_{\rm LT}}})_{e=0}$ is small to begin with, this is unlikely to be significant. More important, though, is that an additional term changes the relative timing of the primary and secondary eclipses. Following @winn10: $$\Delta t_{e}\approx \frac{P e}{\pi}\cos\omega$$ and the primary eclipse also changes duration relative to the secondary eclipse by the ratio $1+e\sin\omega$. To compare ${\ensuremath{\Delta t_{\rm LT}}}$ and $\Delta t_{e}$ means effectively comparing $K_2(1-q)/c$ and $e\cos\omega$. For $K_2\sim 300\,{\ensuremath{{\rm km\,s}^{-1}}}$, this means that one is sensitive to $e\sim 10^{-3}$ (although $\omega$ is poorly determined for low $e$). If it can be asserted for some independent reason (i.e., evolutionary assumptions) that $e\ll 10^{-3}$ then one can treat any measured $\Delta t$ as coming from light-travel delay. But if not one must be more careful. Fortunately, eccentricity can be constrained from the radial velocities. Adopting the small-$e$ limit as in @lcw+01, $N_{\rm RV}$ spectra can limit the eccentricity to $\sigma_e\approx 2\sigma_v/K_2\sqrt{N_{\rm RV}}$, where $\sigma_v$ is the precision of the individual velocity measurements (see also @gw07). With $>100$ observations with $<1\,{\ensuremath{{\rm km\,s}^{-1}}}$ precision the eccentricity can be limited (independent of $\omega$) to $\ll 10^{-3}$, and hence can identify whether any measured time delay has a contribution from an eccentric orbit. This requires dedicated radial velocity measurements over one or more full orbits, but is achievable with current instrumentation. At this level one must also account for additional effects such as light-travel delay in the [spectroscopic]{} analysis [@za07]. Tidal distortions can also mimic eccentricity in radial velocity fits [@eaton08], but these can be identified photometrically and are expected to be quite small, $\sim 10^{-4}$, except in the most compact systems. Mass Constraints ---------------- Equation (\[eqn:dt\]) gives an independent constraint on the mass ratio $q$, which helps break the degeneracy in the mass function to measure the masses of the stars individually. For the individual masses $$\begin{aligned} M_1 & = & \frac{K_2}{2\pi G P}\left(2 P K_2-{\ensuremath{\Delta t_{\rm LT}}}\pi c\right)^2 \nonumber \\ M_2 & = & \left(2 P K_2-{\ensuremath{\Delta t_{\rm LT}}}\pi c\right)^2\left(\frac{K_2}{2\pi G P}-\frac{{\ensuremath{\Delta t_{\rm LT}}}c}{2 G P^2}\right).\end{aligned}$$ Assume that I measure $K_2 \pm \sigma_K$ and ${\ensuremath{\Delta t_{\rm LT}}}\pm \sigma_\Delta$ ($P$ is typically known to much higher precision); I also assumed $e=0$. How well can I determine the individual masses? I know $q$ to: $$\sigma_q^2=\frac{\pi^2 c^2}{P^2 K_2^4}\left(K_2^2 \sigma_\Delta^2 + {\ensuremath{\Delta t_{\rm LT}}}^2 \sigma_K^2\right).$$ I now wish to see with what precision I can estimate the masses from the observations. Doing standard error propagation, $$\begin{aligned} \frac{\partial M_1}{\partial K_2} & = & \left(\frac{P M^2}{2\pi G}\right)^{1/3}\frac{5+q}{(1+q)} \nonumber \\ \left|\frac{\partial M_1}{\partial {\ensuremath{\Delta t_{\rm LT}}}}\right| & = & \left(\frac{4 \pi^2 M^2c^3}{P^2G}\right)^{1/3}\frac{1}{(1+q)}.\end{aligned}$$ These are the contributions of the $\sigma_K$ and $\sigma_\Delta$ to the uncertainty on the mass, i.e., $\sigma_M^2=\sigma_K^2\left|\partial M/\partial K\right|^2+\sigma_\Delta^2\left|\partial M/\partial {\ensuremath{\Delta t_{\rm LT}}}\right|^2$. For the two terms to be comparable requires $\sigma_\Delta=((q+5)P/2\pi c)\sigma_K\approx 10P_{\rm hr}\sigma_{K,{\rm kms}}\,{\rm ms}$ (where $P_{\rm hr}$ is the period in hr and $\sigma_{K,\rm kms}$ is the uncertainty on $K_2$ in [${\rm km\,s}^{-1}$]{}). With those, I would have $\sigma_{M_1}\sim 0.01 M_\odot$ for periods $P\gtrsim 1\,$hr and mass ratios $q\sim 0.25$. The constraint on $M_2$ is similar. I illustrate this in Figure \[fig:mass\], where I show mass constraints on [NLTT 11748]{} from hypothetical time-delay measurements. However, it is likely that the uncertainty from the radial velocity amplitude will be considerably less than that from [$\Delta t_{\rm LT}$]{}: individual velocity measurements can easily have uncertainties of a few [${\rm km\,s}^{-1}$]{} with a large telescope, and combining enough of them to give a meaningful constraint on the eccentricity will likewise end up with $\sigma_K=\sigma_v/\sqrt{N_{\rm RV}}< 1\,{\ensuremath{{\rm km\,s}^{-1}}}$. Getting a comparable constraint on the time delay seems implausible: with individual times measured to $\sim 1\,$s, $>10^4$ eclipses are needed to get $\sigma_\Delta<10\,$ms, and only one time delay is measurable per orbit. This means one needs significantly higher signal-to-noise per observation than the several hundred I have been assuming here or that I will be limited by $\sigma_\Delta$. In this case, the precision on $M_1$ improves as $\sigma_{M_1}\sim P^{-2/3}\sigma_\Delta$; including the difficulty in detecting the delay $\sigma_\Delta\sim P^{1/6}$, the determination of $M_1$ improves as $P^{-1/2}$. The only ways that long periods are penalized are in terms of observing strategy, as for long periods the time to get enough eclipses measured will grow long as well, and for the probability of detecting an eclipse in the first place since that decreases as $1/a$. As shown in Figure \[fig:mass\], the joint probability distribution for $(M_1,M_2)$ is strongly correlated between $M_1$ and $M_2$, with a much stronger constraint on $M_2-M_1$ than on $M_2+M_1$. However, this would even be true—although to a lesser degree—if the other radial velocity amplitude $K_1$ were measured, especially if it were at lower precision because it is much fainter. It is straightforward although tedious to compute the linear combinations of the masses that minimize/maximize the variance which would be preferred for fitting. While not perfect, with this constraint one would have a much better picture of the system. Conclusions =========== Motivated by the recent discovery of [NLTT 11748]{}, the first eclipsing double WD binary, I have examined a phenomenon that affects precision eclipse timing of such a system. With knowledge of the individual masses, one can put much stronger constraints on the radii of the two WDs, the evolutionary history of the system, and its expected outcome, not to mention WD atmosphere models and models for the interiors of He WDs [e.g., @pach07; @sba10]. This effect, a delay between perfect phasing of primary and secondary eclipses, can be used to constrain the individual masses of the binary, something difficult to do otherwise. I find that the light-travel delay should be detectable in the case of [NLTT 11748]{}, and possibly in some other similar binary systems should they prove to have both primary and secondary eclipses: long periods are favored both for detecting [$\Delta t_{\rm LT}$]{} and for using it to constrain the masses, although long periods do not favor detecting eclipses to begin with. The constraints on the individual masses can approach $\pm0.01M_{\odot}$ for plausible data-sets, and will likely be limited by the precision of the eclipse timing, suggesting that an intensive timing effort on large telescopes is worthwhile. Detection of a second radial velocity amplitude would over constrain the system, leading to even tighter determinations of the masses. In non-degenerate eclipsing binaries, such as those that *Kepler* may discover, the delay should also be detectable for systems with orbital periods of greater than a few days, although it requires that the mass ratio differs from one and that no other unmodeled orbit variations be present to a high degree of confidence. The orbits of the binary members can also be perturbed by other bodies in the systems, either on shorter (planets or other small bodies in close orbits) or longer timescales (a distant body). In both cases, perturbations in transit timing may be visible (see @assc05 for a detailed discussion). Given the wide variety in possible situations it is out of the scope of this Letter to consider, but any perceived variation in transit timing must be compared against the possible presence of additional bodies. There could also be effects that alter the perceived primary versus secondary eclipse times without altering the orbit, such as hot spots due to irradiation [@kcn+07; @ack+10] or accretion. For the former, I note that the incoming radiation in the double WD systems is typically very small, $\sim 10^{-3}$ of the outgoing radiation. As for accretion, both WDs are well inside their Roche lobes and so none is expected. I find the fortunate coincidence that [NLTT 11748]{}, the one object known to be eclipsing, also has the binary parameters that lead to the highest value of [$\Delta t_{\rm LT}$]{} among similar double WD binaries. Hopefully, with dedicated observing this effect will be detected and can constrain the [NLTT 11748]{} system even more than is possible today. I thank the anonymous referee, as well as L. Bildsten, T. Marsh, J. Winn, E. Agol, D. Fabrycky, S. Gaudi, M. van Adelsberg, and R. Cooper for helpful discussions. DLK was supported by NASA through Hubble Fellowship Grant \#01207.01-A awarded by the STScI which is operated by AURA, Inc., for NASA, under contract NAS 5-26555. 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--- abstract: 'We propose that the dispersion management of coherent atomic matter waves can be exploited to overcome quantum back-action in condensate-based optomechanical sensors. The effective mass of an atomic Bose-Einstein condensate modulated by an optical lattice can become negative, resulting in a negative-frequency optomechanical oscillator, negative environment temperature, and optomechanical properties opposite to those of a positive-mass system. This enables a quantum-mechanics-free subsystem insulated from quantum back-action.' author: - 'Keye Zhang$^1$, Pierre Meystre$^2$, and Weiping Zhang$^{1}$' title: 'Back-action-free quantum optomechanics with negative-mass Bose-Einstein condensates' --- Introduction ============ Atomic Bose-Einstein condensates (BECs) present a number of desirable features for precision measurements as well as for a broad spectrum of tests of fundamental physics. These include, for example, thermal-noise-free sensors for atomic clock and interferometry applications [@Dunningham2005] and high-resolution magnetometers [@Vengalattore2007], tests of the Casimir-Polder force [@Obrecht2007], the development of quantum simulators for studies of quantum phase transitions [@Greiner2002] and artificial gauge fields [@Spielman2009], cavity QED experiments [@Brennecke2007], and studies of decoherence and quantum entanglement in many-body systems [@Esteve2008; @Cramer2013]. These applications benefit significantly from the extremely low temperatures, high-order coherence, and bosonic stimulation properties of BECs. However, the quantum nature of the condensates usually results in quantum back-action that randomly disturbs the quantum state to be detected [@Murch2008; @Treutlein2007], resulting, e.g., in the standard quantum limit (SQL) of displacement measurements  [@Braginsky2]. Recent experiments have also demonstrated that in BECs quantum back-action can be suppressed using spin squeezing or particle entanglement caused by atom-atom interactions [@Esteve2008; @Gross2010]. This approach is inspired by ideas originally developed in the context of gravitational wave detection  [@Braginsky; @Braginsky2], where the injection of squeezed light fields in the empty input port of the gravitational wave interferometer was proposed to beat the SQL. However, strong degrees of squeezing and the entanglement of large numbers of particles remain challenging due to their increasing sensitivity to decoherence. In this paper we show that the dispersion management of the Schr[ö]{}dinger field provides a promising alternative to the elimination of quantum back-action effects in BEC-based measurement schemes. When trapped in a weak optical lattice potential, the condensate can be forced into a regime of anomalous dispersion where it acts as a macroscopic quantum object with negative effective mass [@Eiermann2003]. That negative mass can serve as a back-action canceler to a normal, positive mass partner and isolate quantum-mechanics-free subsystems (QMFSs), as discussed in a recent proposal by Tsang and Caves [@Tsang2012]. A similar noise-canceling effect is also expected to be realized by cavity photons with opposite detunings [@Tsang2010] as well as atomic ensembles with opposite spins [@Wasilewski2010]. Cavity optomechanical systems based on the collective motion of BECs [@Brennecke2008] and non-degenerate ultracold atomic gases [@Murch2008] have proven to be particularly well suited to demonstrate a number of quantum effects, including the observation of the quantum back-action of position measurements [@Murch2008], the asymmetry in the power spectrum of displacement noise due to the noncommuting nature of boson creation and annihilation operators [@Brahms2012], and the optomechanical cooling of a collective motional mode of an atomic ensemble down to the quantum regime [@SSmith2011]. These experiments pave the way to promising ultracold-atoms-based quantum metrology schemes, which we use to illustrate the role of the negative effective mass of the condensate in overcoming the quantum back-action. Back-Action-Free Quantum Optomechanics ====================================== Reference [@Tsang2012] showed that a simple setup to implement a QMFS comprises two harmonic oscillators, $A$ and $B$, of identical frequencies and opposite masses. In the following we assume that they are coupled optomechanically to a common optical field mode $\hat c$ as well as to time-dependent external perturbations $f_A$ and $f_B$ through the interaction Hamiltonian $$V= \hbar [\Delta_c + G(\hat{q}+\hat{q}')]\hat{c}^{\dagger}\hat{c}+f_A \hat{q} +f_B \hat{q}'.\label{H1}$$ Considering then the variables $$\begin{aligned} \hat{Q}&=&\hat{q}+\hat{q}'\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\hat{P}=\frac{1}{2}(\hat{p}+\hat{p}')\nonumber\\ \hat{\Phi}&=&\frac{1}{2}(\hat{q}-\hat{q}')\,\,\,\,\,\,\,\,\,\,\,\hat{\Pi}=\hat{p}-\hat{p}'\end{aligned}$$ It is easily verified that $[\hat{Q}, \hat{\Pi}]=0 $ and $$\dot {\hat{Q}} = \frac{\hat{\Pi}}{m},\,\,\,\dot{ \hat{\Pi} }= -m\omega^2 \hat{Q}+f_B-f_A,\,\,\,\dot{\hat{c}} = i\Delta_{c}\hat{c}-iG\hat{Q}\hat{c}.\label{dQ}$$ so that the dynamical pair of observables formed by the collective position $\hat{Q}$ and relative momentum $\hat{\Pi}$ form a QMFS. Equations (\[dQ\]) describe the motion of a particle driven by the difference in the external perturbations, $f_B-f_A$, resulting in a frequency shift of the cavity field that can be detected by interferometry. However, unlike the general optomechanical case, since the radiation pressure effect is absent in the equation for $\hat{\Pi}$, this measurement does not introduce any back-action and hence is not subject to the SQL. Complementary conclusions hold for the QMFS characterized by the pair operators $\hat{\Phi}$ and $\hat{P}$ for an optomechanical coupling of the form $G(\hat{q}-\hat{q}')$. In that case the frequency shift is proportional to $f_B+f_A$. ![(Color online) Relationship diagram for the back-action evading setup in the “bare” (left) and “composite” (right) representations. The displacement of the composite oscillator $E$ results in a change in the phase of the cavity field $C$ that could be measured by homodyne detection, but the measurement back-action only affects the composite oscillator $D$. []{data-label="loop"}](loop.eps){width="3.5in"} Further insight into the underlying physics of this back-action-free measurement scheme can be gained by considering the quantum state dynamics. We assume that the system is initially uncorrelated, with the cavity field in a coherent state and the positive-mass oscillator $A$ and negative-mass oscillator $B$ both in their ground state, $$\left| {\psi (0)} \right\rangle = {\left| \alpha \right\rangle _C} \otimes {\left| 0 \right\rangle _A} \otimes {\left| 0 \right\rangle _B}.$$ As a result of the optomechanical interaction, (\[H1\]), the oscillators $A$ and $B$ become entangled with the cavity field $C$. The correlation loop of the total scheme is shown in Fig.\[loop\]. However, when expressing the state of the system in terms of the composite oscillators $D$ and $E$, described by the operators $\{\hat{Q}, \hat{P}\} $ and $\{\hat{\Phi}, \hat{\Pi}\} $, respectively, we find that it does not suffer three-body entanglement among the subsystems $C$, $D$, and $E$, but only two-body entanglement. Specifically we find, except for an unimportant constant phase factor, $$\begin{aligned} | \psi (t)\rangle &=& e^{ - |\alpha |^2/2}\sum_n \frac{\alpha^n}{\sqrt {n!}} \exp \left [\frac{-i4nGQ_s}{ \omega }\left (\omega t - \sin \omega t\right )\right ]\nonumber \\ &\times&| n \rangle _C|\phi_n(t) \rangle _D \otimes |\varphi(t)\rangle _E, \label{state} \end{aligned}$$ where $$\begin{aligned} \phi_n(t)&=&\frac{-1}{\omega \sqrt{\hbar m \omega}} (f_A+f_B+2\hbar G n)\left ( 1- e^{-i\omega t} \right ),\nonumber \\ \varphi(t)&=&\sqrt{\frac{m\omega}{\hbar}}Q_s\left ( 1- e^{-i\omega t} \right ),\nonumber\end{aligned}$$ $Q_{s}=(f_B-f_A)/m\omega^2$, and $|n\rangle_C$ are photon Fork states. Equation (\[state\]) shows that in contrast to the composite oscillator $D$, which becomes entangled with the cavity mode $C$, the composite oscillator $E$ remains uncorrelated with the rest of the system. Rather, it evolves into a time-dependent coherent sate $|\varphi(t)\rangle _E$ that is independent of both the state of the optical field and the composite oscillator $D$. The states $|\phi_n(t) \rangle _D$ are $n$-dependent coherent states of the composite oscillator $D$. This is similar to the situation encountered in single-mirror optomechanics [@Knight97], except for the important dependence of the phase factors on the steady-state displacement $Q_s$ of the oscillator $E$. That dependence makes it easy to read out $Q_s$ without measurement back-action. For example, for $\omega t=2m\pi$, $m$ integer, the state of the system reduces to $| \alpha \exp[-8im\pi G Q_s/\hbar\omega]\rangle _C \otimes | 0 \rangle _D \otimes | 0 \rangle _E$. That is, the composite oscillators $D$ and $E$ return to the vacuum state—as do the oscillator $A$ and $B$—while the cavity field becomes a coherent state whose phase could be easily measured by homodyne detection. Negative-Mass Oscillators ========================= We now turn to a discussion of possible realizations of negative-mass optomechanical oscillators. Negative masses are of course absent in the physical world, but the concept of effective masses—which can, in principle, be negative as well as positive—is familiar from solid-state physics, where it has proven useful in describing the motion of electrons in nonideal lattice potentials [@Callaway1974]. Not surprisingly, this idea has recently been expanded to describe aspects of the quantum wave dynamics of ultracold atoms in optical lattices [@Pu2003], including Bloch oscillations [@Dahan1996], the lensing effect in the diffraction of the atomic matter waves [@Eiermann2004], and the formation of gap solitons [@Fallani2003]. We show in the following how to implement negative effective masses in BEC-based quantum optomechanics to realize a QMFS that may prove useful in the detection of feeble forces and fields. Consider for concreteness a scalar atomic BEC confined by both an optical lattice potential $V_{0}\cos^{2}(k_{L}x)$ of periodicity $2\pi/k_L$ and an external trapping potential $U(x)$ that is taken to be slowly varying over the lattice period. Restricting the description to one dimension for simplicity, the condensate is described by the Hamiltonian $$H=\int\hat{\Psi}^{\dagger}\left[-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{0}\cos^{2}(k_L x)+U(x) \right]\hat{\Psi}dx,\label{eq:Horigin}$$ where $\hat{\Psi}(x)$ is the bosonic field operator of the atomic system and we have neglected inter-atomic collisions. Assuming that the lattice potential is sufficiently shallow that we are far from the Mott insulator transition [@Jaksch1998] we proceed by expanding $\hat{\Psi}(x)$ in terms of a complete set of basis Bloch functions as $$\hat{\Psi}(x)=\sum_n\int dq\phi_{n,q}(x)\hat{a}_{n,q}$$ where $n$ and $q$ label the band index and the quasimomentum, respectively, and $\hat{a}_{n,q}$ are the associated boson annihilation operators. In the following we assume that the condensate is properly described by the product of an envelope that varies slowly over the period of the lattice, and is characterized by a central wave vector $q_0$, and Bloch functions that capture the rapid oscillations of the condensate caused by optical lattice. We then have approximately [@Pu2003] $$\phi_{n,q}(x)\approx e^{i(q-q_0)x} \phi_{n, q_0}(x)$$ where the mode functions $\phi_{n,q_{0}}(x)$ capture the density oscillations and $$\hat\Psi(x)=\sqrt{2\pi}\sum_n\phi_{n,q_{0}}(x) \hat{\mathcal A}_{n,q_{0}}(x).$$ Here we have introduced the slowly varying bosonic operators (on the scale of the lattice period $2\pi/k_L$) $$\hat{\mathcal{A}}_{n,q_{0}}(x)=(1/\sqrt{2\pi})\int dqe^{i(q-q_{0})x}\hat{a}_{n,q}$$ which describe the dynamics of the condensate envelope in the trapping potential $U(x)$, with $$[\hat{\mathcal{A}}_{n,q_{0}}(x),\hat{\mathcal{A}}_{n^{\prime},q_{0}}^{\dagger}(x^{\prime})]=\delta_{n,n^{\prime}}\delta(x-x^{\prime}).$$ Applying the effective mass method [@Callaway1974] then yields the effective Hamiltonian describing the condensate in the envelope representation as $$\begin{aligned} H_{A}&=&\sum_n\int\hat{\mathcal{A}}_{n,q_{0}}^{\dagger}(x)\Big [-\frac{\hbar^{2}\nabla^{2}}{2m_{n,q_{0}}^{*}}+U(x)+\mathcal{E}_{n}(q_{0}) \nonumber \\ &+& \mathcal{E}_{n}^{\prime}(q_{0})\left(-i\nabla\right) \Big ]\hat{\mathcal{A}}_{n,q_{0}}(x)dx,\label{eq:Ha}\end{aligned}$$ where the kinetic energy term responsible for the dispersion of the wave packet is modified by the single-particle effective mass $$m_{n,q_{0}}^*=\hbar^{2}/\mathcal{E}_{n}^{\prime\prime}(q_{0}).$$ This corresponds to a parabolic approximation of the energy bands and can be precisely managed by controlling the lattice depth $V_0$, see e.g. Ref. [@Eiermann2003]. Here $\mathcal{E}_{n}^{\prime}(q_{0})$ and $\mathcal{E}_{n}^{\prime\prime}(q_{0})$ are the first and second-order derivatives of the $n$th-band Bloch energy with respect to the quasimomentum, evaluated at $q_{0}$. Figure \[band\] shows that the gradient term $\mathcal{E}_{n}^{\prime}(q_{0})$ vanishes at the center $(q_{0}=0)$ and edges $(q_{0}=\pm k_{L})$ of the first Brillouin zone. Anomalous dispersion, characterized by a negative effective mass is achieved at the zone edges for odd-$n$ bands, and at the zone center for even-$n$ bands. For deep enough lattices it is sufficient to consider the first band only. This is the situation that we consider in the remainder of this paper. ![(Color online) Top: (a) Bloch energy $\mathcal{E}(q_0)$, (b) its derivative, and (c) the effective mass ratio for a lattice depth $V_0=4.5E_r$ where $E_r=\hbar^{2}k_L^2/2m$ is the atomic recoil energy. Solid (red) lines and dashed (blue) lines are for the first- and second-band case, respectively. Bottom: Sketch of the density profile of a negative-effective-mass condensate in a trap potential U(x): the density is modulated by the optical lattice and peaks at the maximum of U(x).[]{data-label="band"}](mass.eps){width="3in"} The validity of the negative effective mass description relies on the existence of a narrow momentum distribution of the condensate relative to the central wave vector $\pm k_L$. This can be achieved by giving an initial velocity to the condensate, or by considering a condensate initially at rest and adiabatically switching on a moving optical lattice realized by two counter-propagating fields of frequency difference $\delta\omega=2\hbar k_L^{2}/m$. This permits us to neglect the excitation to upper bands by Landau-Zener transitions [@Eiermann2003; @Fallani2003], such that Hamiltonian (\[eq:Ha\]) describes a quantum field of negative-effective-mass particles trapped in the potential $U(x)+{\cal E}_1(k_L)$. The acceleration of particles with a negative mass is opposite the direction of the forces to which they are subjected, so that stability occurs at the maximum of the potential $U(x)$. We thus consider the situation where a condensate of negative effective mass $m_{1, k_L}^*$ is trapped in a potential $U(x)$ that we approximate as an inverted harmonic potential of frequency $\Omega=\sqrt{\left|U^{\prime\prime}(x_{1})\mathcal{E}_{1}^{\prime\prime}(k_L)\right|}/\hbar$, with $x_{1}$ the position of its maximum. We expand the envelope field operator $\hat{\mathcal{A}}_{1,k_L}(x)$ on the basis of its eigenfunctions $\xi_\ell (x)$, with eigenenergies $\hbar\omega_\ell=-\hbar\Omega(\ell+\frac{1}{2}) < 0$, as $$\hat{\mathcal{A}}_{1,k_L}(x)=\sum_\ell \xi_{\ell}(x)\hat{b}_\ell$$ where the bosonic operator $\hat{b}_\ell$ annihilates a condensed atom from mode $\ell$. In contrast with the situation for positive masses, the “ground mode” $\xi_{0}(x)$, which has the largest population, now has the highest energy. But since the relative probability of a particle occupying state $\ell$ is proportional to the Boltzmann factor $P_{\ell}\varpropto \exp[-\hbar\omega_{\ell}/k_{B}T]$, we conclude that the effective temperature $T$ of a condensate with negative effective mass is also negative. We note that this situation is closely related to the scheme proposed in Ref. [@Mosk2005] and recently demonstrated by Braun *et al.* [@Braun2013] to achieve negative temperatures. In that case a tight optical lattice was used to establish the density profile of the ultracold atoms initially trapped in a loose harmonic potential $U(x)$. This potential was then reversed from $U(x)$ to $-U(x)$, following which the low-energy states of $U(x)$ corresponded to the high-energy states of $-U(x)$. In our approach, in contrast, $U(x)$ is fixed and we move a shallow optical lattice to change the effective mass of the condensate from positive to negative. Optomechanical Setup ==================== Our realization of a negative mass optomechanical system follows closely the approach pioneered in Ref. [@Brennecke2008], with an optical cavity field of wave vector $k_c\ll k_L$ perturbing the center-of-mass motion of the condensate, but with important differences that we discuss later. For short enough times the depletion of the ground mode $\xi_{0}(x)$ remains small and we can describe it classically, $\hat{b}_{0}\approx\sqrt{N}$, with $N$ the total atom number. The cavity-condensate then coupling takes the form of a multimode cavity optomechanical interaction, $$H=\hbar\omega_{c}\hat{c}^{\dagger}\hat{c}+\sum_\ell \left [G_\ell(\hat{b}_\ell+\hat{b}_\ell^{\dagger})\hat{c}^{\dagger}\hat{c}+\hbar\omega_{M,\ell}\hat{b}_\ell^{\dagger}\hat{b}_\ell \right ] \label{multimode}$$ where the effective optomechanical coupling coefficients are $$G_\ell=\sqrt{N}\mathcal{D}\int\xi_0^{*}(x)\cos^{2}(k_{c}x-\theta)\xi_\ell(x)dx.$$ Here $\mathcal{D}$ is the single-photon potential depth of the cavity field with annihilation operator $\hat{c}$ and frequency $\omega_{c}$ (including the frequency shift due to the condensate mean field), $\omega_{M,\ell}=\omega_{\ell}-\omega_{0} <0$ are the oscillation frequencies of the “effective condensate mirrors” [@Brennecke2008], and $\theta$ is a phase that depends on the position of the maximum of $U(x)$ relative to the optical potential of the cavity field. We found numerically that coupling to the first excited mode $\ell = 1$ can be made dominant by an appropriate choice of that phase, the remaining modes acting as a mechanical reservoir for that mode [@Brahms2012]. (Other situations with another single dominant mode $\ell$, or with two or more modes coupled with comparable strengths, can also be arranged.) For the inverted harmonic trap considered here the bath temperature is negative, as already mentioned. One can thus expect a reversed asymmetric displacement spectrum $S_{x}(-\omega)$ for the negative-frequency oscillator $\hat{b}_{\ell}$ and hence a reversed optical output spectrum $n_{c}(-\omega)$ for the cavity field [@Brahms2012]. This means that the optomechanical properties of a negative-effective-mass oscillator are reversed from the usual case. For example optical cooling is realized by a blue-detuned rather than a red-detuned driving field, and stationary bipartite entanglement is optimized in the red-detuned case [@Genes2008], with the situation possibly even more interesting in the multimode case. When mechanical damping is weaker than the cavity decay, these negative-frequency oscillators can serve as a negative temperature bath for a cavity field of positive frequency $\omega_{c}$, so that it will exhibit gain as in usual laser theory—where a negative temperature is provided by the inversion of the active medium [@QN2004]. ![(Color online) Possible arrangement for back-action-free field and force detection. Two condensates $A$ and $B$ are trapped along the axis of an optical cavity by the potentials $U_A$ and $U_B$, respectively, with a moving optical lattice that drives condensate $B$ and adjusts its effective mass to a negative value.[]{data-label="GW"}](GW.eps){width="3.5in"} Figure \[GW\] shows a possible back-action-free optomechanical setup based on our considerations. Two cigar-shaped condensates are separately trapped along the $x$ direction, with one of them interacting with a moving optical lattice. The cavity resonance is sensitive to the motion of both condensates, which form an effective “two-mirror” cavity optomechanical system. This setup would be suitable to probe a external fields with different strength at positions $A$ and $B$. A back-action-free measurement is realized provided that the trapping potentials and the optical lattice depth are such that the optomechanical parameters of the two condensates are identical except for the sign of their masses. This is the major experimental challenge, although while a small mismatch will result in imperfect back-action cancellation, it can still improve on the original SQL. Other ways to produce a pair of optomechanical oscillators with opposite effective masses might be using a two-component condensate, with only one component sensitive to the optical lattice, or exciting the condensate to the edges of two adjacent Bloch bands. Two-component condensates allow for the sensing and measurement of fields that couple differently to the two components, such as, e.g., magnetic fields and spinor condensates. Conclusion ========== Summarizing, we have shown that the anomalous dispersion of BECs in optical lattices permits us to realize situations where quantum back-action is canceled, with potential applications in the measurement of feeble forces and fields. Our discussion centers on the use of collision-less BECs of a spatial extent large compared to the period of the optical lattice. One might ask if it is possible to achieve a similar effect in ultracold, but noncondensed atoms. In that case every atom is localized within a single lattice well, but the phonon mode associated with the center of mass of the sample can still be delocalized and can enter the quantum regime via cavity optomechanical cooling [@SSmith2011]. However, its thermal damping is of the order of $\gamma\sim\omega_{M}$ and is expected to significantly affect the measurement precision of the scheme. In dense condensates, repulsive atom-atom interactions have nontrivial effects and can result, e.g., in the realization of gap solitons, which have been suggested as an attractive potential source for BEC-based quantum metrology [@Lee2005]. Future work will discuss in detail the effects of collisions in the proposed system and consider its usefulness in specific applications such as magnetometry and weak force detection. *Note added in proof.* Recently, we became aware of a current related work by M. 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--- abstract: 'Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu$ on $[0,1)$, every $f\in L^2(\mu)$ possesses a Fourier series of the form $f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\widehat{f}(n) = \int_{0}^{1} f(x) e^{-2\pi i n x} d \mu(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\mu$-bandlimited.' address: 'Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011' author: - 'John E. Herr and Eric S. Weber' bibliography: - 'fssm.bib' title: Fourier Series for Singular Measures --- Introduction ============ For a Borel probability measure $\mu$, a spectrum is a sequence $\{ \lambda_{n} \}_{n\in I}$ such that the functions $\{ e^{2 \pi i \lambda_{n} x} : n \in I \} \subset L^2(\mu)$ constitute an orthonormal basis. If $\mu$ possesses a spectrum, we say $\mu$ is spectral, and then every $f \in L^2(\mu)$ possesses a (nonharmonic) Fourier series of the form $ f(x) = \sum_{n \in I} \langle f(x), e^{2 \pi i \lambda_{n} x} \rangle e^{2 \pi i \lambda_{n} x}$. In [@JP98], Jorgensen and Pedersen considered the question of whether measures induced by iterated function systems on $\mathbb{R}^d$ are spectral. Remarkably, they demonstrated that the quaternary Cantor measure $\mu_4$ is spectral. Equally remarkably, they also showed that no three exponentials are orthogonal with respect to the ternary Cantor measure $\mu_3$, and hence $\mu_3$ is not spectral. The lack of a spectrum for $\mu_3$ motivated subsequent research to relax the orthogonality condition, instead searching for an exponential frame or Riesz basis, since an exponential frame would provide a Fourier series (see [@DS52]) similar to the spectral case. Though these searches have yielded partial results, it is still an open question whether $L^2(\mu_3)$ possesses an exponential frame. It is known that there exist singular measures without exponential frames. In fact, Lai [@Lai12] showed that self-affine measures induced by iterated function systems with no overlap cannot possess exponential frames if the probability weights are not equal. In this paper, we demonstrate that the Kaczmarz algorithm educes another potentially fruitful substitute for exponential spectra and exponential frames: the “effective” sequences defined by Kwapień and Mycielski [@KwMy01]. We show that the nonnegative integral exponentials in $L^2(\mu)$ for any singular Borel probability measure $\mu$ are such an effective sequence and that this effectivity allows us to define a Fourier series representation of any function in $L^2(\mu)$. This recovers a result of Poltoratskiĭ [@Pol93] concerning the normalized Cauchy transform. A sequence $\{f_n\}_{n=0}^{\infty}$ in a Hilbert space $\mathbb{H}$ is said to be *Bessel* if there exists a constant $B>0$ such that for any $x \in \mathbb{H}$, $$\label{besselcond} \sum_{n=0}^{\infty}\lvert\langle x,f_n\rangle\rvert^2\leq B\lVert x\rVert^2.$$ This is equivalent to the existence of a constant $D>0$ such that $$\left\lVert\sum_{n=0}^{K}c_nf_n\right\rVert\leq D\sqrt{\sum_{n=0}^{K}\lvert c_n\rvert^2}$$ for any finite sequence $\{c_0,c_1,\ldots,c_K\}$ of complex numbers. The sequence is called a *frame* if in addition there exists a constant $A>0$ such that for any $x\in\mathbb{H}$, $$\label{framecond}A\lVert x\rVert^2\leq\sum_{n=0}^{\infty}\lvert\langle x,f_n\rangle\rvert^2\leq B\lVert x\rVert^2.$$ If $A=B$, then the frame is said to be *tight*. If $A=B=1$, then $\{f_n\}_{n=0}^{\infty}$ is a *Parseval frame*. The constant $A$ is called the *lower frame bound* and the constant $B$ is called the *upper frame bound* or *Bessel bound*. The *Fourier-Stieltjes transform* of a finite Borel measure $\mu$ on $[0,1)$, denoted $\widehat{\mu}$, is defined by $$\widehat{\mu}(x):=\int_{0}^{1}e^{-2\pi ixy}\,d\mu(y).$$ Effective Sequences ------------------- Let $\{\varphi_n\}_{n=0}^{\infty}$ be a linearly dense sequence of unit vectors in a Hilbert space $\mathbb{H}$. Given any element $x\in\mathbb{H}$, we may define a sequence $\{x_n\}_{n=0}^{\infty}$ in the following manner: $$\begin{aligned} x_0&=\langle x,\varphi_0\rangle \varphi_0\\ x_n&=x_{n-1}+\langle x-x_{n-1},\varphi_n\rangle \varphi_n.\end{aligned}$$ If $\lim_{n\rightarrow\infty}\lVert x-x_n\rVert=0$ regardless of the choice of $x$, then the sequence $\{\varphi_n\}_{n=0}^{\infty}$ is said to be effective. The above formula is known as the Kaczmarz algorithm. In 1937, Stefan Kaczmarz [@Kacz37] proved the effectivity of linearly dense periodic sequences in the finite-dimensional case. In 2001, these results were extended to infinite-dimensional Banach spaces under certain conditions by Kwapień and Mycielski [@KwMy01]. These two also gave the following formula for the sequence $\{x_n\}_{n=0}^{\infty}$, which we state here for the Hilbert space setting: Define $$\begin{aligned} \begin{split}\label{gs}g_0&=\varphi_0\\ g_n&=\varphi_n-\sum_{i=0}^{n-1}\langle \varphi_n,\varphi_i\rangle g_i.\end{split}\end{aligned}$$ Then $$\label{xnsum}x_n=\sum_{i=0}^{n}\langle x,g_i\rangle \varphi_i.$$ As shown by [@KwMy01], and also more clearly for the Hilbert space setting by [@HalSzw05], we have $$\lVert x\rVert^2-\lim_{n\rightarrow\infty}\lVert x-x_n\rVert^2=\sum_{n=0}^{\infty}\lvert\langle x,g_n\rangle\rvert^2,$$ from which it follows that $\{\varphi_n\}_{n=0}^{\infty}$ is effective if and only if $$\label{gnframe}\sum_{n=0}^{\infty}\lvert\langle x,g_n\rangle\rvert^2=\lVert x\rVert^2.$$ That is to say, $\{\varphi_n\}_{n=0}^{\infty}$ is effective if and only if the associated sequence $\{g_n\}_{n=0}^{\infty}$ is a Parseval frame. If $\{\varphi_n\}_{n=0}^{\infty}$ is effective, then $\eqref{xnsum}$ implies that for any $x\in \mathbb{H}$, $\sum_{i=0}^{\infty}\langle x,g_i\rangle \varphi_i$ converges to $x$ in norm, and as noted $\{g_n\}_{n=0}^{\infty}$ is a Parseval frame. This does not mean that $\{g_n\}_{n=0}^{\infty}$ and $\{\varphi_n\}_{n=0}^{\infty}$ are dual frames, since $\{\varphi_n\}_{n=0}^{\infty}$ need not even be a frame. However, $\{\varphi_n\}_{n=0}^{\infty}$ and $\{g_n\}_{n=0}^{\infty}$ are pseudo-dual in the following sense, first given by Li and Ogawa in [@LiOg01]: \[pseudodef\] Let $\mathcal{H}$ be a separable Hilbert space. Two sequences $\{\varphi_n\}$ and $\{\varphi_n^\star\}$ in $\mathcal{H}$ form a pair of *pseudoframes* for $\mathcal{H}$ if for all $x,y\in\mathcal{H}$, $\displaystyle\langle x,y\rangle=\sum_{n}\langle x,\varphi_n^\star\rangle\langle \varphi_n,y\rangle$. All frames are pseudoframes, but not the converse. Observe that if $x,y\in\mathbb{H}$ and $\{\varphi_n\}_{n=0}^{\infty}$ is effective, then $$\begin{aligned} \langle x,y\rangle&=\left\langle \sum_{m=0}^{\infty}\langle x,g_m\rangle \varphi_m,y\right\rangle\\ &=\sum_{m=0}^{\infty}\langle x,g_m\rangle\left\langle \varphi_m,y\right\rangle,\end{aligned}$$ and so $\{\varphi_n\}_{n=0}^{\infty}$ and $\{g_n\}_{n=0}^{\infty}$ are pseudo-dual. Of course, since $\{g_n\}_{n=0}^{\infty}$ is a Parseval frame, it is a true dual frame for itself. Main Results ============ From this point forward, we shall use the notation $e_{\lambda}(x):=e^{2\pi i\lambda x}$. Our main result is as follows: \[mainthm\] If $\mu$ is a singular Borel probability measure on $[0,1)$, then the sequence $\{e_n\}_{n=0}^{\infty}$ is effective in $L^2(\mu)$. As a consequence, any element $f\in L^2(\mu)$ possesses a Fourier series $$f(x)=\sum_{n=0}^{\infty}c_n e^{2\pi inx},$$ where $$c_n=\int_{0}^{1}f(x)\overline{g_n(x)}\,d\mu(x)$$ and $\{g_n\}_{n=0}^{\infty}$ is the sequence associated to $\{e_n\}_{n=0}^{\infty}$ via Equation $\eqref{gs}$. The sum converges in norm, and Parseval’s identity $\lVert f\rVert^2=\sum_{n=0}^{\infty}{\lvert c_n\rvert}^2$ holds. Our proof proceeds in a series of lemmas. First, in order to show completeness of $\{e_n\}_{n=0}^{\infty}$, we appeal to the well-known theorem of Frigyes and Marcel Riesz [@Riesz16]: Let $\mu$ be a complex Borel measure on $[0,1)$. If $$\int_{0}^{1}e^{2\pi inx}\,d\mu(x)=0$$ for all $n\in\mathbb{N}$, then $\mu$ is absolutely continuous with respect to Lebesgue measure. From this theorem, we prove the desired lemma: \[spanlem\] If $\mu$ is a singular Borel measure on $[0,1)$, then $\{e_n\}_{n=0}^{\infty}$ is linearly dense in $L^2(\mu)$. Assume, for the sake of contradiction, that $\overline{\text{span}}(\{e_n\}_{n=0}^{\infty})\neq L^2(\mu)$. Then there exists some $f\in L^2(\mu)$ such that $f\in\overline{\text{span}}(\{e_n\}_{n=0}^{\infty})^\perp$. Then for any $n\in\mathbb{N}$, we have $$\int_{0}^{1}e^{2\pi inx}\overline{f(x)}\,d\mu(x)=0.$$ By the F. and M. Riesz Theorem, this implies that $\overline{f}d\mu$ is absolutely continuous with respect to Lebesgue measure $d\lambda$. Since $\overline{f}d\mu<<d\lambda$ and $\overline{f}d\mu\perp d\lambda$, it follows by uniqueness in Lebesgue’s Decomposition Theorem that $\overline{f}d\mu\equiv0$. Thus, $f=0$ almost everywhere with respect to $\mu$, which is a contradiction. Therefore, $\overline{\text{span}}(\{e_n\}_{n=0}^{\infty})=L^2(\mu)$. \[stationdef\] A sequence $\{\varphi_k\}_{k=0}^{\infty}$ in a Hilbert space is said to be *stationary* if $\langle \varphi_{k+m},\varphi_{l+m}\rangle=\langle \varphi_k,\varphi_l\rangle$ for any nonnegative integers $k$, $l$, and $m$. As noted in [@KwMy01], given a stationary sequence $\{\varphi_n\}_{n=0}^{\infty}$ and $a_m$ defined by $a_m:=\langle \varphi_k,\varphi_{k+m}\rangle$, where $k$ is any nonnegative integer $k\geq-m$, Bochner’s Theorem implies the existence of a unique positive measure $\sigma$ on $\mathbb{T}$ such that $$a_m=\int_{\mathbb{T}}\overline{z}^m\sigma(dz)=\int_{0}^{1}e^{-2\pi imx}\,d\sigma(x)\hspace{.5cm}\text{for each }m\in\mathbb{Z}.$$ This measure $\sigma$ is called the *spectral measure* of the stationary sequence $\{\varphi_n\}$. We shall make use of the following theorem from [@KwMy01]: \[kwmythm\] A stationary sequence of unit vectors that is linearly dense in a Hilbert space is effective if and only if its spectral measure either coincides with the normalized Lebesgue measure or is singular with respect to Lebesgue measure. We are now ready to prove Theorem \[mainthm\]. By Lemma \[spanlem\], the sequence $\{e_n\}_{n=0}^{\infty}$ is linearly dense in $L^2(\mu)$. It consists of unit vectors, because $\mu$ is a probability measure. We see that for all $k,l,m\in\mathbb{N}_0$, $$\langle e_{k+m},e_{l+m}\rangle = \int_{[0,1)}e^{2\pi i(k-l)x}\,d\mu(x) =\langle e_k,e_l\rangle.$$ Thus, $\{e_n\}_{n=0}^{\infty}$ is stationary in $L^2(\mu)$, and moreover, $\mu$ is its spectral measure. It then follows from the theorem of Kwapień and Mycielski that $\{e_n\}_{n=0}^{\infty}$ is effective in $L^2(\mu)$. Since $\{e_n\}_{n=0}^{\infty}$ is effective, given any $f\in L^2(\mu)$, we have that the Kaczmarz algorithm sequence defined recursively by $$\begin{aligned} f_0&=\langle f,e_0\rangle e_0\\ f_{n}&=f_{n-1}+\langle f-f_{n-1},e_n\rangle e_n\end{aligned}$$ satisfies $$\lim_{n\rightarrow\infty}\lVert f-f_n\rVert=0.$$ We recall that $$f_n=\sum_{i=0}^{n}\langle f,g_i\rangle e_i,$$ where the sequence $\{g_n\}_{n=0}^{\infty}$ is the sequence associated to the sequence $\{e_n\}_{n=0}^{\infty}$ by $\eqref{gs}$. Hence, $$f=\sum_{i=0}^{\infty}\langle f,g_i\rangle e_i.$$ Setting $c_n=\langle f,g_n\rangle=\int_{0}^{1}f(x)\overline{g_n(x)}\,d\mu(x)$ yields $$\label{maineq}f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx},$$ where the convergence is in norm. Furthermore, since $\{e_n\}_{n=0}^{\infty}$ is effective, by $\eqref{gnframe}$ $\{g_n\}_{n=0}^{\infty}$ is a Parseval frame. Thus, $$\sum_{n=0}^{\infty}\lvert c_n\rvert^2=\sum_{n=0}^{\infty}\lvert\langle f,g_n\rangle\rvert^2=\lVert f\rVert^2.$$ This completes the proof. Since the ternary Cantor measure $\mu_3$ is a singular probability measure, Theorem $\ref{mainthm}$ demonstrates that any $f\in L^2(\mu_3)$ possesses a Fourier series of the form prescribed by the theorem. This comes despite the fact that $\mu_3$ does not possess an orthogonal basis of exponentials. It is still unknown whether $L^2(\mu_3)$ even possesses an exponential frame. The sequence $\{e_n\}_{n=0}^{\infty}$ of exponentials is effective in $L^2(\mu)$ for all singular Borel probability measures $\mu$, but it is Bessel in none of them. Indeed, if it were Bessel, $\mu$ would be absolutely continuous rather than singular by Theorem 3.10 of [@DHW14]. Therefore, it is not possible for $\{e_{n}\}_{n=0}^{\infty}$ to be a frame in $L^2(\mu)$. However, by a remark in [@LiOg01], since $\{ e_{n} \}_{n=0}^{\infty}$ is pseudo-dual to the (in this case Parseval) frame $\{g_{n}\}_{n=0}^{\infty}$, the upper frame bound for $\{g_n\}_{n=0}^{\infty}$ implies a lower frame bound for $\{e_n\}_{n=0}^{\infty}$. Moreover, some of the examples in [@Lai12] of measures that do not possess an exponential frame are singular, and hence if we normalize them to be probability measures, Theorem \[mainthm\] applies. We shall give a somewhat more explicit formula for the coefficients $c_n$. We will require a lemma to do this, but first we discuss some notation: Recall that a composition of a positive integer $n$ is an ordered arrangement of positive integers that sum to $n$. Whereas for a partition the order in which the terms appear does not matter, two sequences having the same terms but in a different order constitute different compositions. We will think of compositions of $n$ as tuples of positive integers whose entries sum to $n$. The set of compositions of $n$ will be denoted $P_n$. In other words, $$P_n:=\left\{(p_1,p_2,\ldots,p_k)\mid k\in\mathbb{N},(p_1,p_2,\ldots,p_k)\in\mathbb{N}^k, p_1+p_2+\cdots+p_k=n\right\}.$$ Thus, we have $P_1=\{(1)\}$, $P_2=\{(2),(1,1)\}$, $P_3=\{(3),(1,2),(2,1),(1,1,1)\}$, etc. The length of a tuple $p\in P_n$ will be denoted $l(p)$, i.e. $p=(p_1,p_2,\ldots,p_{l(p)})\in\mathbb{N}^{l(p)}$. \[gnformula\] Let $\mu$ be a Borel probability measure on $[0,1)$ with Fourier-Stieltjes transform $\widehat{\mu}$. Define $\alpha_0=1$, and for $n\geq1$, let $$\alpha_n=\sum_{p\in P_n}{(-1)}^{l(p)}\prod_{j=1}^{l(p)}\widehat{\mu}(p_j).$$ Let $\{g_n\}_{n=0}^{\infty}$ be as defined in $\eqref{gs}$. Then for all $n\in\mathbb{N}_0$, $$g_n=\sum_{j=0}^{n}\overline{\alpha_{n-j}}e_{j}.$$ Clearly, $g_0=e_0$ and $g_1=e_1-\langle e_1,e_0\rangle e_0=e_1-\overline{\widehat{\mu}(1)}e_0$. We have that $P_1=\{(1)\}$, so $$\alpha_1=(-1)^{1}\widehat{\mu}(1)=-\widehat{\mu}(1).$$ So, the conclusion holds for $n=0,1$. Suppose that the conclusion holds up to some $n\in\mathbb{N}$. We have that $$\begin{aligned} g_{n+1}&=e_{n+1}-\sum_{j=0}^{n}\langle e_{n+1},e_j\rangle g_j\\ &=e_{n+1}-\sum_{j=0}^{n}\overline{\widehat{\mu}(n+1-j)}g_j\\ &=e_{n+1}-\sum_{j=0}^{n}\overline{\widehat{\mu}(n+1-j)}\left(\sum_{k=0}^{j}\overline{\alpha_{j-k}}e_{k}\right)\\ &=e_{n+1}-\sum_{j=0}^{n}\sum_{k=0}^{j}\overline{\widehat{\mu}(n+1-j)}\overline{\alpha_{j-k}}e_{k}\\ &=e_{n+1}-\sum_{k=0}^{n}\sum_{j=k}^{n}\overline{\widehat{\mu}(n+1-j)}\overline{\alpha_{j-k}}e_k.\end{aligned}$$ Thus, it remains only to show that $$\alpha_{n+1-k}=-\sum_{j=k}^{n}\widehat{\mu}(n+1-j)\alpha_{j-k}.$$ We have: $$\begin{aligned} -\sum_{j=k}^{n}\widehat{\mu}(n+1-j)\alpha_{j-k}&=-\sum_{j=k}^{n}\widehat{\mu}(n+1-j)\sum_{p\in P_{j-k}}{(-1)}^{l(p)}\prod_{w=1}^{l(p)}\widehat{\mu}(p_w)\\ &=\sum_{j=k}^{n}\sum_{p\in P_{j-k}}{(-1)}^{l(p)+1}\widehat{\mu}(n+1-j)\prod_{w=1}^{l(p)}\widehat{\mu}(p_w)\\ &=\sum_{j=1}^{n+1-k}\sum_{p\in P_{n-k+1-j}}{(-1)}^{l(p)+1}\widehat{\mu}(j)\prod_{w=1}^{l(p)}\widehat{\mu}(p_w)\end{aligned}$$ The last equality is obtained by reindexing the sum $j \mapsto n+1-j$. Now, if $p=(p_1,\ldots,p_{l(p)})\in P_{n}$, then it is obvious that $p_1\in\{1,2,\ldots,n\}$ and $(p_2,p_3,\ldots,p_{l(p)})\in P_{n-p_1}$ (where we define $P_0=\varnothing$). Conversely, if $p_1\in\{1,2,\ldots,n\}$ and $(p_2,p_3,\ldots,p_{l(p)})\in P_{n-p_1}$, then clearly $(p_1,p_2,\ldots,p_{l(p)})\in P_{n}$. Thus, it follows that $$-\sum_{j=k}^{n}\widehat{\mu}(n+1-j)\alpha_{j-k}=\sum_{p\in P_{n+1-k}}{(-1)}^{l(p)}\prod_{w=1}^{l(p)}\widehat{\mu}(p_w)=\alpha_{n+1-k}.$$ This completes the proof. Lemma \[gnformula\] can easily be generalized to any Hilbert space setting in which the $\{g_n\}_{n=0}^{\infty}$ are induced by a stationary sequence $\{\varphi_n\}_{n=0}^{\infty}$ simply by replacing $\widehat{\mu}(m)$ with $a_m$ in all instances, where the $a_m$ are as defined after Definition \[stationdef\]. It should be pointed out that sequence of scalars $\{\alpha_n\}_{n=0}^{\infty}$ depends only on the measure $\mu$. In a general Hilbert space setting where we may not have stationarity, an expansion of the $\{g_n\}$ in terms of the sequence $\{\varphi_n\}$ to which they are associated by $\eqref{gnformula}$ can be described by using inversion of an infinite lower-triangular Gram matrix. For a treatment, see [@HalSzw05]. Define a Fourier transform of $f$ by $$\mathcal{F}f(y)=\widehat{f}(y):=\int_{0}^{1}f(x)e^{-2\pi iyx}\,d\mu(x).$$ Observe that $$\left\lvert\mathcal{F}f(y)\right\rvert=\left\lvert\langle f,e_y\rangle\right\rvert\leq\lVert f\rVert_{L^2(\mu)}\cdot\lVert e_y\rVert_{L^2(\mu)}=\lVert f\rVert_{L^2(\mu)}.$$ Thus $\mathcal{F}$ is a linear operator from $L^2(\mu)$ to $L^\infty(\mathbb{R})$ with operator norm $\lVert\mathcal{F}\rVert=1$. \[maincor\] Assume the conditions and definitions of Theorem \[mainthm\]. Then the coefficients $c_n$ may be expressed $$c_n=\sum_{j=0}^{n}\alpha_{n-j}\widehat{f}(j),$$ and as a result $$f(x)=\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\alpha_{n-j}\widehat{f}(j)\right)e^{2\pi inx},$$ where the convergence is in norm. We compute: $$c_n = \langle f,g_n\rangle = \left\langle f,\sum_{j=0}^{n}\overline{\alpha_{n-j}}e_j\right\rangle = \sum_{j=0}^{n}\alpha_{n-j}\widehat{f}(j).$$ The second formula then follows by substitution into $\eqref{maineq}$. While we have Parseval’s identity $\| f \|^2 = \sum_{n=0}^{\infty} | c_{n} |^2 $ as demonstrated by Theorem \[mainthm\], in general the lack of the Bessel condition means that $\| f \|^2 \lesssim \sum_{n=0}^{\infty} |\widehat{f}(n)|^2$ does not hold. In fact, Proposition 3.10 in [@DHSW11] demonstrates an example of a function where $\sum_{n=0}^{\infty}|\widehat{f}(n)|^2=+\infty$. Non-Uniqueness of Fourier Coefficients -------------------------------------- We begin with an example. In [@JP98], it was shown that the quaternary Cantor measure $\mu_4$ possesses an orthonormal basis of exponentials. This basis is $\{e_\lambda\}_{\lambda\in\Lambda}$, where the spectrum $\Lambda$ is given by $$\Lambda=\left\{\sum_{n=0}^{k}\alpha_n4^n:\alpha_n\in\{0,1\},k\in\mathbb{N}_0\right\}=\{0,1,4,5,16,17,20,21,\ldots\}.$$ As a result, any vector $f\in L^2(\mu_4)$ may be written as $$f=\sum_{\lambda\in\Lambda}\langle f,e_\lambda\rangle e_\lambda,$$ where the convergence is in the $L^2(\mu_4)$ norm. Notice that if we define a sequence of vectors $\{h_n\}_{n=0}^{\infty}$ by $$h_n=\begin{cases}e_n&\text{if }n\in\Lambda\\0&\text{otherwise,}\end{cases}$$ we have that $$\sum_{n=0}^{\infty}\langle f,h_n\rangle e_n=\sum_{\lambda\in\Lambda}\langle f,e_\lambda\rangle e_\lambda=f.$$ On the other hand, since $\mu_4$ is a singular probability measure, by Theorem $\ref{mainthm}$ we also have $$f=\sum_{n=0}^{\infty}c_ne_n=\sum_{n=0}^{\infty}\langle f,g_n\rangle e_n.$$ It can easily be checked that $h_0=g_0=e_0$ and $h_1=g_1=e_1$, but that $g_{2} \neq h_2=0$. Thus, the sequences $\{g_{n}\}$ and $\{h_{n} \}$ yield different expansions for general $f \in L^2(\mu_{4})$. We can again use the Kaczmarz algorithm to generate a large class of sequences $\{h_n\}$ such that $\sum\langle f,h_n\rangle e_n=f$ in the $L^2(\mu)$ norm as follows. We use $\langle \cdot , \cdot \rangle_{\mu}$ to denote the scalar product in $L^2(\mu)$. \[T:reproduce\] Let $\mu$ be a singular Borel probability measure on $[0,1)$. Let $\nu$ be another singular Borel probability measure on $[0,1)$ such that $\nu\perp\mu$. Let $0<\eta\leq1$, and define $\lambda:=\eta\mu+(1-\eta)\nu$. Let $\{h_n\}$ be the sequence associated to $\{e_n\}$ in $L^2(\lambda)$ via the Kaczmarz algorithm in Equation . Then for all $f\in L^2(\mu)$, $$\label{Eq:reproduce} f=\sum_{n=0}^{\infty}{\langle f,\eta h_n\rangle}_{\mu} e_n$$ in the $L^2(\mu)$ norm. Moreover, if $\lambda^\prime=\eta^\prime\mu+(1-\eta^\prime)\nu^\prime$ also satisfies the hypotheses, then $\lambda^\prime\neq\lambda$ implies $\{\eta^\prime h_n^\prime\}\neq\{\eta h_n\}$ in $L^2(\mu)$. Because $\nu\perp\mu$, there exist disjoint Borel sets $A$ and $B$ such that $A\cup B=[0,1)$, $\mu(B)=0$, and $\nu(A)=0$. Since $\lambda$ is a singular Borel probability measure, the exponentials $\{e_n\}_{n=0}^{\infty}$ are effective in $L^2(\lambda)$. Let $\{h_n\}$ denote the sequence associated to $\{e_n\}$ in $L^2(\lambda)$ via Equation $\eqref{gs}$. Let $f\in L^2(\mu)$, and define $\tilde{f}=f\cdot\chi_{A}$. Clearly, $\tilde{f}\in L^2(\lambda)$.\ We have that $$\tilde{f}=\sum_{n=0}^{\infty}{\left\langle \tilde{f},h_n\right\rangle}_{\lambda} e_n$$ in the $L^2(\lambda)$ norm. Now, note that $$\begin{aligned} \langle f,\eta h_n\rangle_{\mu} &=\int_{0}^{1}f(x)\overline{\eta h_n(x)}\,d\mu(x)\\ &=\int_{A}f(x)\overline{h_n(x)}\,d\lambda\\ &=\langle\tilde{f},h_n\rangle_{\lambda}.\end{aligned}$$ Therefore, $$\lim_{N \to \infty} \left\lVert \tilde{f} - \sum_{n=0}^{N} \langle f, \eta h_{n} \rangle_{\mu} e_{n} \right\rVert^{2}_{L^2(\lambda)} = 0.$$ Since $$\left\lVert f-\sum_{n=0}^{N}\langle f,\eta h_n\rangle_{\mu}e_n\right\rVert^2_{L^2(\mu)} \leq \frac{1}{\eta}\left\lVert\tilde{f}-\sum_{n=0}^{N}\langle f,\eta h_n\rangle_{\mu} e_n\right\rVert^2_{L^2(\lambda)},$$ Equation (\[Eq:reproduce\]) follows with convergence in $L^2(\mu)$. It remains only to show that different measures $\lambda$ generate different sequences $\{\eta h_n\}$. Therefore, suppose $\nu^\prime$ is another singular Borel probability measure on $[0,1)$ such that $\nu^\prime\perp\mu$, and let $0<\eta^\prime\leq1$. Set $\lambda^\prime=\eta^\prime\mu+(1-\eta^\prime)\nu^\prime$, and let $\{h_n^\prime\}$ be the sequence associated to $\{e_n\}$ in $L^2(\lambda^\prime)$ via Equation . Suppose that $\lambda\neq\lambda^\prime$. We wish to show that $\{\eta h_n\}\neq\{\eta^\prime h^\prime_n\}$ in $L^2(\mu)$. If $\eta\neq\eta^\prime$, then $\eta h_0=\eta e_0\neq\eta^\prime e_0=\eta^\prime h_0^\prime$ in $L^2(\mu)$. Therefore, assume that $\eta=\eta^\prime$. By virtue of the F. and M. Riesz Theorem, since $\lambda\neq\lambda^\prime$, there must exist an integer $n$ such that $\widehat{\lambda}(n)\neq\widehat{\lambda^\prime}(n)$. Following [@HalSzw05], we define a lower-triangular Gram matrix $G$ of the nonnegative integral exponentials by $$(G)_{ij}=\begin{cases}\langle e_i,e_j\rangle=\widehat{\lambda}(j-i)&\text{if }i\geq j\\0&\text{otherwise}\end{cases},$$ and then the inverse of this matrix determines the sequence $\{h_n\}$ associated to $\{e_n\}$ in $L^2(\lambda)$ via $h_n=\sum_{i=0}^{n}\overline{\alpha_{n-i}}e_i$ where $\alpha_{n-i}=\overline{(G^{-1})_{ni}}$. See [@HalSzw05] for details. ($G$ and $G^{-1}$ are stratified since $\{e_n\}$ is stationary.) Therefore, the sequences of scalars $\{\alpha_n\}_{n=0}^{\infty}$ and $\{\alpha^\prime_n\}_{n=0}^{\infty}$ induced by $\lambda$ and $\lambda^\prime$, respectively, in Lemma \[gnformula\] differ. Let $n$ be the smallest positive integer such that $\alpha_n\neq\alpha_n^\prime$. Then since $\eta=\eta^\prime$, we have $$\eta^\prime h_n^\prime-\eta h_n=\eta\sum_{j=0}^{n}\left(\overline{\alpha^\prime_{n-j}}-\overline{\alpha_{n-j}}\right)e_j=\eta(\overline{\alpha_{n}-\alpha^\prime_n})e_0 \neq 0.$$ Thus, $\{\eta h_n\}$ and $\{\eta^\prime h_n^\prime\}$ are distinct sequences in $L^2(\mu)$. We note that any convex combination of sequences $\{h_n\}$ that satisfy Equation (\[Eq:reproduce\]) will again satisfy that equation. In general, for a fixed $f \in L^{2}(\mu)$ the set of coefficient sequences $\{ d_{n}\}$ that satisfy $f = \sum_{n=0}^{\infty} d_{n} e_{n}$ can be parametrized by sequences $\{\gamma_n\}$ of scalars satisfying $\sum_{n=0}^{\infty}\gamma_n e_n=0$ via $d_{n} = \langle f, g_{n} \rangle_{\mu} + \alpha_{n}$. Clearly, Theorem \[T:reproduce\] is not a complete description of all Fourier series expansions for $f$. Connection to the Normalized Cauchy Transform --------------------------------------------- The series $\sum_{n=0}^{\infty}\langle f,g_n\rangle e_n$ given by Theorem \[mainthm\] is the boundary function of the analytic function $\sum_{n=0}^{\infty}\langle f,g_n\rangle z^n$ on $\mathbb{D}$. This function is in the classical $H^2$ Hardy space since the coefficients are square summable. An intriguing connection between the Kaczmarz algorithm and de Branges-Rovnyak spaces is given by the observations that follow. Given a positive Borel measure $\mu$ on $[0,1)$, define a map $V_\mu$, called the normalized Cauchy transform, from $L^1(\mu)$ to the functions defined on $\mathbb{C}\setminus\mathbb{T}$ by $$V_\mu f(z):=\frac{\int_{0}^{1}\frac{f(e^{2\pi ix})}{1-ze^{-2\pi ix}}\,d\mu(x)}{\int_{0}^{1}\frac{1}{1-ze^{-2\pi ix}}\,d\mu(x)}.$$ Poltoratskiĭ proved in [@Pol93] that $V_{\mu}$ maps $L^2(\mu)$ to the de Branges-Rovnyak space $\mathcal{H}(b)$, where $b(z)$ is the inner function associated to $\mu$ via the Herglotz representation theorem. Poltoratskiĭ also proved that $V_{\mu}$ is the inverse of a unitary operator that is a rank one perturbation of the unilateral shift as given by Clark [@Clark72], and hence $V_{\mu}$ is unitary. \[P:mainprop\] Assume the hypotheses of Theorem \[mainthm\]. Then for $z\in\mathbb{D}$, $$V_\mu f(z) = \sum_{n=0}^{\infty}\langle f,g_n\rangle z^n.$$ Define $$\label{Fdef}F(z):=\int_{0}^{1}\frac{1}{1-ze^{-2\pi ix}}\,d\mu(x).$$ That is, $F(z)$ is the Cauchy integral of $\mu$, which is analytic on $\mathbb{D}$. It is easily seen that $$F(z)=\sum_{n=0}^{\infty}\widehat{\mu}(n)z^n.$$ By , $\text{Re}(F(z))>1/2$ for $z\in\mathbb{D}$, and hence, $1/F(z)$ is also analytic on $\mathbb{D}$. Writing $1/F(z)=\sum_{n=0}^{\infty}c_nz^n$, we have $1=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}c_k\widehat{\mu}(n-k)\right)z^n$, and so $\sum_{k=0}^{n}c_k\widehat{\mu}(n-k)=0$ for all $n\geq1$. Then using , an inductive argument shows that $g_n=\sum_{i=0}^{n}\overline{c_{n-i}}e_i$ for all $n$. The $c_n$ are unique by Gaussian elimination, so in fact $c_n=\alpha_n$ for all $n$, the $\alpha_n$ as in Lemma $\ref{gnformula}$. Hence, $$\frac{1}{F(z)}=\sum_{n=0}^{\infty}\alpha_nz^n.$$ It is also clear that $$\int_{0}^{1}\frac{f(e^{2\pi ix})}{1-ze^{-2\pi ix}}\,d\mu(x)=\sum_{n=0}^{\infty}\langle f,e_n\rangle z^n.$$ Therefore, we have $$\begin{aligned} \frac{\int_{0}^{1}\frac{f(e^{2\pi ix})}{1-ze^{-2\pi ix}}\,d\mu(x)}{\int_{0}^{1}\frac{1}{1-ze^{-2\pi ix}}\,d\mu(x)}&=\left(\sum_{n=0}^{\infty}\langle f,e_n\rangle z^n\right)\left(\sum_{m=0}^{\infty}\alpha_mz^m\right)\\ &=\sum_{n=0}^{\infty}\left(\sum_{i=0}^{n}\langle f,\overline{\alpha_{n-i}}e_i\rangle\right)z^n\\ &=\sum_{n=0}^{\infty}\langle f,g_n\rangle z^n.\end{aligned}$$ Two of the main results in [@Pol93] are Theorems 2.5 and 2.7, which together show that the Fourier series of $V_{\mu}f(z)$ converges to $f$ in the $L^2(\mu)$ norm provided that $\mu$ is singular. Combining this together with Proposition \[P:mainprop\] recovers our Theorem \[mainthm\]. Adding Clark’s result that implies that $V_{\mu}$ is unitary, and we recover the Plancherel identity. Poltoratskiĭ’s results are more general than our Theorem \[mainthm\] in the following way: if $\mu$ has an absolutely continuous component and a singular component, then for any $f \in L^2(\mu)$, the Fourier series of $V_{\mu} f$ converges to $f$ in norm with respect to the singular component. The Fourier series cannot in general converge to $f$ with respect to the absolutely continuous component of $\mu$ since the nonnegative exponentials are incomplete. It is unclear whether for such a $\mu$ every $f$ can be expressed in terms of a bi-infinite Fourier series. For singular $\mu$, our Theorem \[mainthm\] guarantees norm convergence of the Fourier series of $V_{\mu} f$ to $f$ as do Poltoratskiĭ’s results. However, Poltoratskiĭ also comments in [@Pol93] that the Fourier series converges pointwise $\mu$-a.e. to $f$. A Shannon Sampling Formula ========================== In [@Str00], Strichartz introduces a sampling formula for functions that are bandlimited in a generalized sense. He considers functions whose spectra are contained in a certain compact set $K$ that is the support of a spectral measure $\mu$. If $F$ is a strongly $K$-bandlimited function, then he shows that it has an expression $$F(x)=\sum_{\lambda\in\Lambda}F(\lambda)\widehat{\mu}(x-\lambda),$$ where $\Lambda$ is a spectrum for $L^2(\mu)$. We will now prove a similar sampling formula for analogously bandlimited functions. Our formula does not rely on an exponential basis and hence holds even for non-spectral singular measures. (Indeed, it even holds for singular measures devoid of exponential frames.) The price paid for not using an exponential sequence dual to itself is that the samples $F(\lambda)$ are replaced by the less tidy $\sum_{j=0}^{n}\alpha_{n-j} F(j)$. Let $\mu$ be a singular Borel probability measure on $[0,1)$. Let $\{\alpha_i\}_{i=0}^{\infty}$ be the sequence of scalars induced by $\mu$ by Lemma $\ref{gnformula}$. Suppose $F:\mathbb{R}\rightarrow\mathbb{C}$ is of the form $$F(y)=\int_{0}^{1}f(x)e^{-2\pi iyx}\,d\mu(x)$$ for some $f\in L^2(\mu)$. Then $$F(y)=\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\alpha_{n-j}F(j)\right)\widehat{\mu}(y-n),$$ where the series converges uniformly in $y$. By Theorem \[mainthm\], $f$ may be expressed $f=\sum_{n=0}^{\infty}c_ne_n$, the convergence occurring in the $L^2(\mu)$ norm. We compute: $$\begin{aligned} F(y)&=\int_{0}^{1}f(x)e^{-2\pi iyx}\,d\mu(x)\\ &=\langle f,e_y\rangle\\ &=\left\langle\sum_{n=0}^{\infty}c_n e_n,e_y\right\rangle\\ &=\sum_{n=0}^{\infty}c_n\langle e_n,e_y\rangle\\ &=\sum_{n=0}^{\infty}c_n\widehat{\mu}(y-n).\end{aligned}$$ Recall from Corollary \[maincor\] that $$c_n=\sum_{j=0}^{n}\alpha_{n-j}\widehat{f}(j)=\sum_{j=0}^{n}\alpha_{n-j}F(j),$$ where the $\alpha_n$ are defined by Lemma \[gnformula\]. Combining these computations, we obtain that for any $y\in\mathbb{R}$, $$\label{thm2maineq}F(y)=\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\alpha_{n-j}F(j)\right)\widehat{\mu}(y-n).$$ Let $S_k:=\sum_{n=0}^{k}c_n e_n$. Since $S_k\rightarrow f$ in the $L^2(\mu)$ norm and the Fourier transform $\mathcal{F}:L^2(\mu)\rightarrow L^\infty(\mathbb{R})$ is bounded, $\{\mathcal{F}S_k\}\rightarrow\mathcal{F}f$ in $L^\infty(\mathbb{R})$. Then because $\mathcal{F}S_k(y)=\sum_{n=0}^{k}c_n\widehat{\mu}(y-n)$, we have that $\sum_{n=0}^{\infty}c_n\widehat{\mu}(y-n)$ and hence $\eqref{thm2maineq}$ converge uniformly in $y$ to $\mathcal{F}f(y)$. It should be noted that, in contradistinction to the sampling formula of Strichartz, the convergence of the series in Equation (\[thm2maineq\]) does not follow from the Cauchy-Schwarz inequality, because it is possible that $\sum_{n=0}^{\infty} | \widehat{\mu}(y-n) |^2 = + \infty$.

--- abstract: | The scientific output 1994-2014 of the University Centre in Svalbard (UNIS) was bibliometrically analysed. It was found that the majority of the papers have been published as international cooperations and rank above world average. Analysis of the papers’ content reveals that UNIS works and publishes in a wide variety of scientific topics.\ **Keywords**: Svalbard, Bibliometry, Pudovkin-Garfield Percentile Rank Index, Content Analysis. author: - 'Johannes Stegmann[^1]' title: 'Research at UNIS - The University Centre in Svalbard. A bibliometric study' --- Introduction ============ The sensitivity of the Arctic to climate changes and the heavy impact of such transformations on other world regions (Post et al., 2009) as well as its presumed richness in oil, gas and other mineral deposits has moved the Arctic into the focus of intensive scientific, economic, political and public attention (Humrich, 2013).\ The University Centre in Svalbard (UNIS) was established in 1993 as “Arctic extension” of Norway’s universities (UNIS, 2009 a). UNIS is to “represent and secure Norwegian polar interests” (UNIS, 2009 a). UNIS’s mission is also to offer an international research platform for all kinds of basic Arctic research (UNIS, 2009 b).\ It seems to be of interest to analyse UNIS’ scientific acitivities from a bibliometric point of view. This communication tries to answer the following questions: (i) What is produced by UNIS in terms of scientic papers? (ii) To what extent is UNIS’ propagated internationality realised in terms of international coauthorships? (iii) What is the standing of UNIS’ publications in terms of appropriate international standards? (iv) What is the content of UNIS authored papers in terms of subfields and subject topics? Methods ======= Papers published since 1994 by UNIS were retrieved and downloaded from the Web of Science (WoS) on January 19, 2014, using an appropriate address search profile.\ For the analysis of UNIS’ paper output and its distribution to different document types all retrieved records were used. For the analysis of UNIS’ research those papers not being research articles (i.e. not of document type “ARTICLE”) were excluded.\ The citation performance of UNIS’ papers was measured applying the Percentile Rank Index (PRI) developed by Pudovking and Garfield (Pudovkin and Garfield, 2009). I call this version of a percentile rank index PG-PRI but use in this paper “PG-PRI” and “PRI” synonymously because no other PRI methods are involved here.\ Prior to PG-PRI calculation of a paper in question the citation rank of this paper among its “paper peers”, i.e. all papers published in the same source journal in the same year must be determined. Because most papers need some time to gather cites it makes no sense to include too recent papers in a PRI analysis. In this study, only research papers (document type “ARTICLE”) of UNIS published before 2013 (i.e. published in the years 1994-2012) were included (723 papers). For each of these 723 research articles its publication year and publishing journal was determined, and all papers (document type “ARTICLE” only) of the corresponding journal-year pair were retrieved and downloaded. In summary, the papers of 514 journal-year pairs were retrieved and downloaded between the 6^th^ and 10^th^ February 2014. Then, the papers of each journal-year set were ranked top-down according to citations received. In case of ties (several papers having the same citation frequency), each of the tied values was assigned the average of the ranks for the tied set (Pudovkin and Garfield, 2009, Pudovkin et al., 2012). The position of each of the UNIS papers in the corresponding paper set was determined. PG-PRI values were calculated according to the formula $$PRI = \frac{N-R+1}{N}*100$$ where N is the number of papers in the year set of the journal and R is the citation rank of the paper (Pudovkin and Garfield, 2009). R=1 is the top rank (most cited paper) with PRI=100 (Pudovkin and Garfield, 2009).\ For determination of the global (expected) average PRI the Svalbard papers were ordered according to the number of papers published in the corresponding journal-year set. The average PRI was calculated according to the formula $$PRI_{globav} = 50 + \frac{50}{N}.$$ where N is the number of papers published in the journal-year pair at the median position of the ordered set (Pudovkin et al., 2012). In the present study, the median N was found to be 150; therefore, $$PRI_{globav} = 50.33$$\ For cluster analysis of keywords the co-word analysis technique described by Callon et al. (1991) was applied. A detailed description of the algorithm can be found in Stegmann and Grohmann (2003).\ Extraction of record field contents, clustering, data analysis and visualisation were done using homemade programs and scripts for perl (version 5.14.2) and the software package R version 2.14.1 (R Core Team, 2013). All operations were performed on a commercial PC run under Ubuntu version 12.04 LTS. Results and Discussion ====================== Output (papers) --------------- Since 1994 UNIS published 875 papers, more than 85% of them being research papers (Figure 1). In UNIS’ starting years only few papers were published but the annual publication numbers gradually increased up to 94 in 2012. In 2013 only 73 UNIS papers were retrieved but probably not yet all papers with 2013 as publication year had been recorded to the WoS database at the time of retrieval (Januar 2014). The number of UNIS papers retrieved from the WoS database are in good agreement with the corresponding numbers derived from UNIS’ annual reports 2009 to 2012 (UNIS 2009 b, 2010, 2011, 2012). For the subsequent analysis of Svalbard’s research papers of document type “ARTICLE” only were included. These amount to 748 papers for the whole time span (1994 - January 2014). These papers have 2331 distinct authors; the most prolific author is (co)author of 66 papers. The average number of authors per paper is 3.1; predominant is the class with 4 authors per paper. Only 4.1% (31 papers) of the research papers are single-authored (not shown). The paper with the highest number of authors (376) is the yearly published [*State of the Climate*]{} report, a special supplement to the [*Bulletin of the American Meteorological Society*]{} (Blunden and Arndt, 2012).\ 67% of UNIS’s research papers are international papers, jointly authored by at least one author of UNIS (i.e. from Norway) and at least one author from another country. In total, 56 different countries (including Norway) are involved in UNIS’ research papers. Table 1 shows the top 15 cooperating countries. Among them are the other (besides Norway) circumpolar countries: Canada, Denmark (due to its autonomous region Greenland), Iceland, Russia, USA.\ UNIS has published its papers in more than 200 journals; the top ten are displayed in Table 2 (see also Table 3 and next section). Benchmarking (PG-PRI) --------------------- For the analysis of the international standing of UNIS’ research the percentile rank indexing method of Pudovkin and Garfield (2009) was applied (PG-PRI, see Methods). Figure 2 displays the PG-PRI value of each of the 723 research articles of UNIS. Table 4 lists some PRI ranges. The average PRI value of UNIS’ 723 research articles published 1994-2012 is 53.9, well above the expected (global) mean of 50.33 (see Methods). In addition, more than one half (392 = 54.2%) of the UNIS papers have PG-PRI values above the global mean (Figure 2, Table 4). The PG-PRI has the inherent capability for international comparison of an author’s/institute’s papers because it compares the citation performance of the research papers in question with their “direct peers”, i.e. papers of the same type published in the same journals in the same time span ((Pudovkin and Garfield, 2009, Pudovkin et al., 2012). From the data in Figure 2 and Table 4 it is concluded that UNIS’ research perform well above the average of comparable world research. This conclusion is supported by the high impact factor ranks of the top ten journals with UNIS papers within their JCR categories (see Table 3). Content (categories, keywords) ------------------------------ Rough indicators of the scientific (sub)fields to which papers contribute are the WoS categories to which journals are assigned. UNIS contributes to 52 WoS categories. The top 15 categories to which UNIS research papers (i.e. the publishing journals) have been assigned are shown in Table 5. Earth, marine, environmental sciences play a role, but also space and evolutionary sciences are important.\ Deeper insights into UNIS’ research areas may be achieved by an analysis of the keywords assigned to the articles. The keywords were extracted from the record fields DE (author keywords) and ID (keyword plus). 3999 distinct keywords were extracted and - in a first step - assigned to WoS categories of the respective articles. Identical keywords (occuring in both fields, DE and ID) were counted only once.\ Table 6 lists for the 10 top categories (see Table 5) frequent keywords (omitting not so informative descriptors like Svalbard, Spitsbergen, Sea, etc.).\ Another possibility to get an overview of the content of a set of papers is cluster analysis of the relevant keywords. Here, the co-word analysis of Callon et al. (1991) was applied (see Methods). The perl scripts developed by Stegmann and Grohmann (2003) were used to perform a cluster analysis of the keywords assigned to UNIS’ research articles. The keywords are clustered according to their mutual similarity based on their co-occurrence strength, determined by calculation of the [*cosine*]{}. Several thresholds were applied: (i) minimal frequency = 4 (occuring of a keyword in at least 4 records, this threshold reduced the number of keywords to 395); (ii) minimal cosine similarity = 0.2; (iii) mininal (maximal) cluster size = 3 (10), i.e. only clusters containing at least 3 keywords were included. The upper threshold of cluster size was applied in order to avoid clusters with very many keywords which makes the cluster readability difficult. When a cluster had accumulated 10 terms a new cluster was started (see Stegmann and Grohmann, 2003).\ Under these restrictions 38 clusters with 338 keywords were found. For their visualisation the form of a density-centrality diagram (Callon et al., 1991) was chosen. Density means the strength of the links between cluster members, centrality means the strength of links between clusters. To label a cluster (besides its number) the keywords contained in it were ranked according to the product of link strengths (cosine values) and frequency; the keyword with the highest product was selected as cluster name.\ The diagram is displayed in Figure 3. Theoretically, this kind of diagram (also called “strategical diagram”, see Callon et al., 1991) displays in its right half “central” topics. i.e. subjects with strong or many links to other topics (clusters). In the upper half the topics are “dense”, i.e. the keywords constituting a topic are mutually strongly linked but do not have necessarily many or strong links outside the cluster. In the left and lower parts of the diagram subjects developing to more centrality and/or density are displayed.\ Tables 7, 8, 9, 10 lists all clusters with their keywords. Table 7 lists the keywords contained in the clusters positioned in the lower left quadrant of the diagram, Table 8 those of the upper left quadrant, and in Tables 9 and 10 the keywords of the lower and upper right quadrants are tabulated. Inspection of the clusters clearly shows that in most cases co-clustered keywords express associated topics. For a non-expert, however, it is difficult to make substantial statements concerning the cluster contents and the position of the research represented by the cluster keywords within the frame of UNIS’s research. Of course, the help of field experts is needed for a detailed analysis of topics and their keywords. Variation (lowering) of the applied thresholds could result in a more detailed view of topics dealt with by UNIS’ research papers. One should, however, not forget that the keywords in WoS records do not belong to a controlled vocabulary. It might happen that the same issue is described with (slightly) different keywords in different records, thus lowering the link strengths from this issue to other ones. Nevertheless, the data displayed in Figure 3 and Tables 7 to 10 show that UNIS has a diversified spectrum of Arctic research topics. Conclusion ========== The University Centre in Svalbard (UNIS) has been found to be a high-level research centre performing well above world average in wide variety of research subjects. Acknowledgements ================ Part of this paper has been submitted for poster presentation at the 15^th^ COLLNET meeting 2014.\ Helpful PRI-related comments of Alexander Pudovkin are gratefully acknowledged. [99]{} Blunden, J., Arndt, D. S. (Eds.) (2012): State of the Climate in 2011. [*Bulletin of the American Meteorological Society, 93 (7), S1–S264.*]{} Callon, M., Courtial, J. P., Laville, F. (1991): Co-word analysis as a tool for describing the network of interactions between basic and technological research: the case of polymer chemistry. [*Scientometrics,22 (1), 155–205.*]{} Humrich, C. (2009): Fragmented International Governance of Arctic Offshore Oil: Governance Challenges and Institutional Improvement. [*Global Environmental Politics, 13 (3), 79–99.*]{} Post, E., et al. (2009): Ecological Dynamics Across the Arctic Associated with Recent Climate Change. [*Science, 325, 1355–1358.*]{} Pudovkin, A. I., Garfield, E. (2009): Percentile rank and author superiority indexes for evaluating individual journal articles and the author’s overall citation performance. [*Collnet Journal of Scientometrics and Information Management, 3(2), 3–10.*]{} Pudovkin, A., Kretschmer, H., Stegmann, J., Garfield, E., 2012: Research evaluation. Part I: productivity and citedness of a German medical research institution. [*Scientometrics, 93 (1), 3–16.*]{} R Core Team (2013): [*R: A language and environment for statistical computing*]{}. R Foundation for Statistical Computing, Vienna. Stegmann, J., Grohmann, G. (2003): Hypothesis Generation Guided by Co-Word Clustering. [*Scientometrics, 56 (1), 111–135*]{}. UNIS (2009 a): The University Centre in Svalbard - [*Strategic plan 2009–2012.*]{} UNIS (2009 b): The University Centre in Svalbard - [*Annual Report 2009.*]{} UNIS (2010): The University Centre in Svalbard - [*Annual Report 2010.*]{} UNIS (2011): The University Centre in Svalbard - [*Annual Report 2011.*]{} UNIS (2012): The University Centre in Svalbard - [*Annual Report 2012.*]{} ![Publications of UNIS 1994-2014^\*^. ^\*^Total no. of papers: 875. (85.5% articles, 5.9% proceedings papers, 5.6% reviews) []{data-label="fig:fig01"}](figure1.eps) ![Percentile Rank Indexes of UNIS’ research papers 1994-2012^\*^. ^\*^Total no. of research papers: 723. PG-PRI: Pudovkin-Garfield Percentile Rank Index (see Methods). Vertical dashed line: median of papers. Horizontal dashed line: expected global mean PRI (see Methods). []{data-label="fig:fig02"}](figure2.eps) ![UNIS’ research papers 1994-2014: centrality-density diagram of keywords clusters. Cluster numbers are centered at cluster positions. Centrality increases from left to right, Density increases bottom-up. Vertical dashed line: median centrality. Horizontal dashed line: median density. []{data-label="fig:fig03"}](figure3.eps) [lccc]{} Country & No. of papers & % of total (748 papers) &\ UNITED KINGDOM & 197 & 26 &\ USA & 146 & 20 &\ DENMARK & 74 & 10 &\ SWEDEN & 66 & 9 &\ GERMANY & 65 & 9 &\ RUSSIA & 46 & 6 &\ POLAND & 40 & 5 &\ CANADA & 39 & 5 &\ JAPAN & 30 & 4 &\ FRANCE & 29 & 4 &\ FINLAND & 24 & 3 &\ ICELAND & 21 & 3 &\ SWITZERLAND & 19 & 3 &\ NETHERLANDS & 18 & 2 &\ ITALY & 17 & 2 &\ [lccc]{} Journal & No. of papers & % of total (748 papers) &\ J GEOPHYS RES & 54 & 7 &\ POLAR BIOL & 50 & 7 &\ GEOPHYS RES LETT & 41 & 6 &\ COLD REG SCI TECHNOL & 25 & 3 &\ POLAR RES & 24 & 3 &\ J GLACIOL & 21 & 3 &\ ANN GEOPHYS GERMANY & 19 & 3 &\ QUATERNARY SCI REV & 19 & 3 &\ MAR ECOL PROG SER & 13 & 2 &\ BOREAS & 12 & 2 &\ [lcll]{} Journal & IF 2012 & JCR Category & IF rank\ J GEOPHYS RES & 3.17 & GEOSCIENCES, & 23 (172)\ & & MULTIDISCIP. &\ POLAR BIOL & 2.01 & BIODIVERSITY & 14 (40)\ & & CONSERVATION &\ GEOPHYS RES LETT & 3.98 & GEOSCIENCES, & 11 (172)\ & & MULTIDISCIP. &\ COLD REG SCI & & &\ TECHNOL & 1.29 & ENGINEERING, & 32 (122)\ & & CIVIL &\ POLAR RES & 1.62 & OCEANOGRAPHY & 30 (60)\ J GLACIOL & 2.88 & GEOGRAPHY, & 12 (45)\ & & PHYSICAL &\ ANN GEOPHYS & & &\ GERMANY & 1.52 & ASTRONOMY & & 31 (56)\ & & ASTROPHYSICS &\ QUATERNARY SCI REV & 4.08 & GEOGRAPHY, & 3 (45)\ & & PHYSICAL &\ MAR ECOL PROG SER & 2.55 & MARINE & & 16 (100)\ & & FRESHWATER BIOL. &\ BOREAS & 2.46 & GEOGRAPHY, & 18 (45)\ & & PHYSICAL &\ [lcc]{} PRI range & No. of papers & % of total (723 papers)\ PRI = 100 & 5 & 0.7\ PRI $\ge$ 99 & 10 & 1.4\ PRI $\ge$ 90 & 86 & 11.9\ PRI $\ge$ 75 & 202 & 27.9\ PRI $\ge$ 50.33 & 392 & 54.2\ [lccc]{} Category & No. of papers & % of total &\ & & (748 papers) &\ GEOLOGY & 289 & 38 &\ ENVIRONMENTAL & & &\ SCIENCES & ECOLOGY & 212 & 28 &\ PHYSICAL GEOGRAPHY & 103 & 14 &\ OCEANOGRAPHY & 97 & 13 &\ ASTRONOMY & ASTROPHYSICS & 82 & 11 &\ METEOROLOGY & & &\ & ATMOSPHERIC SCIENCES & 75 & 10 &\ MARINE & & &\ & FRESHWATER BIOLOGY & 66 & 9 &\ BIODIVERSY & CONSERVATION & 65 & 9 &\ ENGINEERING & 43 & 6 &\ GEOCHEMISTRY & GEOPHYSICS & 30 & 4 &\ ZOOLOGY & 28 & 4 &\ PLANT SCIENCES & 22 & 3 &\ EVOLUTIONARY BIOLOGY& 21 & 3 &\ LIFE SCIENCES & BIOMEDICINE & & &\ - OTHER TOPICS & 15 & 2 &\ PALEONTOLGY & 13 & 2 &\ [lll]{} WoS Category & Keywords &\ GEOLOGY & &\ & &\ ENVIRONMENTAL & &\ SCIENCES & & &\ ECOLOGY & &\ & &\ & &\ & &\ PHYSICAL & &\ GEOGRAPHY & &\ & &\ & &\ OCEANOGRAPHY & &\ & &\ & &\ ASTRONOMY & & &\ ASTROPHYSICS & &\ & &\ & &\ & &\ METEOROLOGY & & &\ ATMOSPHERIC & &\ SCIENCES & &\ & &\ & &\ & &\ MARINE & & &\ FRESHWATER & &\ BIOLOGY & &\ & &\ & &\ BIODIVERSY & & &\ CONSERVATION & &\ & &\ & &\ ENGINEERING & &\ & &\ & &\ GEOCHEMISTRY & & &\ GEOPHYSICS & &\ & &\ & &\ & &\ [lll]{} no. & keywords &\ 21 & &\ & &\ 26 & &\ & &\ 32 & &\ 22 & &\ & &\ 31 & &\ & &\ & &\ 28 & &\ & &\ 35 &\ & &\ 36 & &\ 37 & &\ 27 & &\ 38 & &\ [lll]{} no. & keywords &\ 1 & &\ & &\ 6 & &\ & &\ & &\ 3 & &\ & &\ 2 & &\ & &\ 19 & &\ & &\ 17 & &\ & &\ 14 & &\ & &\ 8 & &\ & &\ [lll]{} no. & keywords &\ 16 & &\ & &\ 11 & &\ & &\ 30 & &\ & &\ 29 & &\ & &\ 25 & &\ & &\ 23 & &\ & &\ &\ 33 & &\ & &\ 34 & &\ & &\ [lll]{} no. & keywords &\ 5 & &\ & &\ & &\ 7 & &\ &\ & &\ 9 & &\ & &\ &\ & &\ 15 & &\ &\ & &\ & &\ 20 & &\ &\ & &\ 10 & &\ &\ & &\ 4 & &\ & &\ 13 & &\ &\ & &\ 12 & &\ & &\ 24 & &\ & &\ 18 & &\ & &\ [^1]: Member of the Ernst-Reuter-Gesellschaft der Freunde, Förderer und Ehemaligen der Freien Universität Berlin e.V., Berlin, Germany, johannes.stegmann@fu-berlin.de

[ **Subderivative-subdifferential duality formula**]{} -------------------------------------------------------------------- Marc Lassonde Université des Antilles, BP 150, 97159 Pointe à Pitre, France; and LIMOS, Université Blaise Pascal, 63000 Clermont-Ferrand, France E-mail: marc.lassonde@gmail.com -------------------------------------------------------------------- **Abstract.** We provide a formula linking the radial subderivative to other subderivatives and subdifferentials for arbitrary extended real-valued lower semicontinuous functions. **Keywords:** lower semicontinuity, radial subderivative, Dini subderivative, subdifferential. **2010 Mathematics Subject Classification:** 49J52, 49K27, 26D10, 26B25. Introduction {#intro} ============ Tyrrell Rockafellar and Roger Wets [@RW98 p. 298] discussing the duality between subderivatives and subdifferentials write > In the presence of regularity, the subgradients and subderivatives of a function $f$ are completely dual to each other. \[…\] For functions $f$ that aren’t subdifferentially regular, subderivatives and subgradients can have distinct and independent roles, and some of the duality must be relinquished. Jean-Paul Penot [@Pen13 p. 263], in the introduction to the chapter dealing with elementary and viscosity subdifferentials, writes > In the present framework, in contrast to the convex objects, the passages from directional derivatives (and tangent cones) to subdifferentials (and normal cones, respectively) are one-way routes, because the first notions are nonconvex, while a dual object exhibits convexity properties. In the chapter concerning Clarke subdifferentials [@Pen13 p. 357], he notes > In fact, in this theory, a complete primal-dual picture is available: besides a normal cone concept, one has a notion of tangent cone to a set, and besides a subdifferential for a function one has a notion of directional derivative. Moreover, inherent convexity properties ensure a full duality between these notions. \[…\]. These facts represent great theoretical and practical advantages. In this paper, we consider arbitrary extended real-valued lower semicontinuous functions and arbitrary subdifferentials. In spite of the above quotes, we show that there is always a duality formula linking the subderivatives and subdifferentials of such functions. Moreover, we show that at points where the (lower semicontinuous) function satisfies a mild regularity property (called radial accessibility), the upper radial subderivative is always a lower bound for the expressions in the duality formula. This lower bound is an equality in particular for convex functions, but also for various other classes of functions. For such functions, the radial subderivative can therefore be recovered from the subdifferential, and consequently the function itself, up to a constant, can be recovered from the subdifferential. This issue is discussed elsewhere. Subderivatives ============== In the sequel, $X$ is a real Banach space with unit ball $B_X$, $X^*$ is its topological dual, and ${\langle}.,. {\rangle}$ is the duality pairing. For $x, y \in X$, we let $[x,y]:=\{ x+t(y-x) {:}t\in[0,1]\}$; the sets $]x,y[$ and $[x,y[$ are defined accordingly. Set-valued operators $T:X\rightrightarrows X^*$ are identified with their graph $T\subset X\times X^*$. For a subset $A\subset X$, $x\in X$ and ${\lambda}>0$, we let $d_A(x):=\inf_{y\in A} \|x-y\|$ and $B_{\lambda}(A):=\{ y\in X{:}d_A(y)\le {\lambda}\}$. All extended-real-valued functions $f : X\to{{]}{-\infty},+\infty]}$ are assumed to be lower semicontinuous (lsc) and *proper*, which means that the set ${{\rm dom} \kern.15em}f:=\{x\in X{:}f(x)<\infty\}$ is non-empty. For a lsc function $f:X\to{{]}{-\infty},+\infty]}$, a point ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and a direction $u\in X$, we consider the following basic subderivatives (we essentially follow the terminology of Penot’s textbook [@Pen13]): - the (lower right Dini) *radial subderivative*: $$\label{Dinisub} f^r({\bar{x}};u):=\liminf_{t\searrow 0}\,\frac{f({\bar{x}}+tu)-f({\bar{x}})}{t},$$ its upper version: $$\label{Dinisub} f^r_+({\bar{x}};u):=\limsup_{t\searrow 0}\,\frac{f({\bar{x}}+tu)-f({\bar{x}})}{t},$$ and its upper strict version (the *Clarke subderivative*): $$\label{Clarkesub} f^0({\bar{x}};u):= \limsup_{t \searrow 0 \atop{(x,f(x)) \to ({\bar{x}},f({\bar{x}}))}}\frac{f(x+tu) -f(x)}{t};$$ - the (lower right Dini-Hadamard) *directional subderivative*: $$\label{Hsubderiv} f^d({\bar{x}};u):= \liminf_{t \searrow 0 \atop{u' \to u}}\frac{f({\bar{x}}+tu')-f({\bar{x}})}{t},$$ and its upper strict version (the Clarke-Rockafellar subderivative): $$\label{Csubderiv} f^\uparrow({\bar{x}};u):= \sup_{\delta>0} \limsup_{t \searrow 0 \atop{(x,f(x)) \to ({\bar{x}},f({\bar{x}}))}} \inf_{u' \in B_{\delta}(u)}\frac{f(x+tu') -f(x)}{t}.$$ It is immediate from these definitions that the following inequalities hold ($\rightarrow$ means $\le$): $$\begin{aligned} f^r({\bar{x}};u) & \rightarrow f^r_+({\bar{x}};u)\rightarrow f^0({\bar{x}};u)\\ \uparrow \quad & \qquad\qquad\qquad\quad\uparrow\\ f^d({\bar{x}};u) & \qquad\longrightarrow \quad\quad f^\uparrow({\bar{x}};u)\end{aligned}$$ It is well known (and easily seen) that for a function $f$ locally Lipschitz at ${\bar{x}}$, we have $f^r({\bar{x}};u)=f^d({\bar{x}};u)$ and $f^0({\bar{x}};u)=f^\uparrow({\bar{x}};u)$, whereas for a lsc convex $f$, we have $f^d({\bar{x}};u)=f^\uparrow({\bar{x}};u)$. A function $f$ satisfying such an equality is called *regular*. However, in general, $f^d({\bar{x}};u)<f^\uparrow({\bar{x}};u)$, and there are many other types of subderivatives $f'$ which lie between $f^d$ and $f^\uparrow$. PS. The inequality stated in the theorem below is (much) less elementary. It is the analytic form of Treiman’s theorem [@Tre83] on the inclusion of the lower limit of Boulingand contingent cones at neighbouring points of ${\bar{x}}$ into the Clarke tangent cone at ${\bar{x}}$ in the context of a Banach space (in finite dimensional spaces, equality holds between these objects, as was shown earlier by Cornet [@Cor81] and Penot [@Pen81]). A proof of this inequality (or equality in finite dimensional spaces) based on this geometrical approach was given by Ioffe [@Iof84] (see also Rockafellar-Wets [@RW98 Theorem 8.18]). For a proof (in the general context of Banach spaces) using a multidirectional mean value inequality rather than the above geometric approach, see Correa-Gajardo-Thibault [@CGT09]. \[Treiman\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$. Then: [ $$f^\uparrow({\bar{x}};u)\le \sup_{{\varepsilon}>0}\limsup_{x\to{\bar{x}}} \inf_{u'\in B(u,{\varepsilon})}f^d(x;u').$$ ]{} Subdifferentials ================ Given a lsc function $f:X\to{{]}{-\infty},+\infty]}$ and a point ${\bar{x}}\in{{\rm dom} \kern.15em}f$, we consider the following two basic subsets of the dual space $X^*$: - the *Moreau-Rockafellar subdifferential* (the subdifferential of convex analysis): $$\label{convex-sdiff} {\partial}_{MR} f ({\bar{x}}) := \{ x^* \in X^* {:}{\langle}x^*,y-{\bar{x}}{\rangle}+ f({\bar{x}}) \leq f(y),\, \forall y \in X \};$$ - the *Clarke subdifferential*, associated to the Clarke-Rockafellar subderivative: $$\begin{aligned} \label{Csub} \partial_{C} f({\bar{x}}) := \{x^* \in X^* {:}\langle x^*,u\rangle \leq f^\uparrow({\bar{x}};u), \, \forall u \in X\}.\end{aligned}$$ All the classical subdifferentials (proximal, Fréchet, Hadamard, Ioffe, Michel-Penot, …) lie between these two subsets. It is well known that for a lsc convex $f$, ${\partial}_{MR} f ={\partial}_C f$, so all the classical subdifferentials coincide in this case. PS. In the sequel, we call *subdifferential* any operator ${\partial}$ that associates a set-valued mapping $\partial f: X \rightrightarrows X^\ast$ to each function $f$ on $X$ so that $$\label{inclusdansClarke} {\partial}_{MR} f\subset \partial f\subset \partial_{C} f$$ and the following *Separation Principle* is satisfied in $X$: (SP) *For any lsc $f,\varphi$ with $\varphi$ convex Lipschitz near ${\bar{x}}\in{{\rm dom} \kern.15em}f $, if $f+\varphi$ admits a local minimum at ${\bar{x}}$, then $0\in {\widehat{{\partial}}}f({\bar{x}})+ {\partial}\varphi({\bar{x}}),$ where $$\begin{gathered} \label{wclosure} {\widehat{{\partial}}}f({\bar{x}}):= \{\, {\bar{x}}^*\in X^*{:}\mbox{there is a net }((x_\nu,x^*_\nu))_\nu\subset {\partial}f \mbox{ with }\\ (x_\nu,f(x_\nu))\to ({\bar{x}},f({\bar{x}})),\ x^*_\nu{{\stackrel{w^*}{\longrightarrow}}\;}{\bar{x}}^*,\ \limsup_\nu\,{\langle}x^*_\nu,x_\nu-{\bar{x}}{\rangle}\le 0\,\}.\end{gathered}$$* \[SP-rem\] (a) In our paper [@JL13], the set ${\widehat{{\partial}}}f({\bar{x}})$ defined in is called the *weak\*-controlled closure* of the set-valued map ${\partial}f$ at point ${\bar{x}}$. The reason to consider such a closure is that, even for a convex lsc function $f$, the a priori simpler $strong\times weak^*$-closure of the graph of ${\partial}f={\partial}_{MR} f$ is too big for the Separation Principle to be meaningful. The graph of ${\partial}_{MR} f$ is not $strong\times weak^*$-closed in general: see, e.g., [@JL02] for a discussion on what would be sufficient to add to the $strong\times weak^*$ topology on $X\times X^*$ to guarantee the closure of such graphs. More precisely, the graph of the convex subdifferential is $strong\times weak^*$-closed for each lsc convex function if and only if X is finite dimensional (see [@BFG03]). It is worth noting (and easily seen) that, as expected, always ${\partial}_{MR} f={\widehat{{\partial}}}_{MR} f$. PS. (b) If we require the net $((x_\nu,x^*_\nu))_\nu\subset{\partial}f$ in to be actually a sequence $((x_{n},x_{n}^*))_n$, $n\in\N$ (in which case the control assertion $\limsup_n\,{\langle}x^*_n,x_n-{\bar{x}}{\rangle}\le 0$ is automatically satisfied), we obtain the so-called ‘limiting subdifferentials’. A widely used such limiting subdifferential is the weak$^*$ sequential closure of the Fréchet subdifferential, known as the *Mordukhovich subdifferential*. PS. (c) The Separation Principle (SP) is a very simple property expected to be satisfied by a subdifferential ${\partial}$ in a Banach space $X$. This property is actually equivalent to various other properties of the subdifferential ${\partial}$ in the Banach space X: see [@JL13]. We recall that the Clarke subdifferential, the Michel-Penot subdifferential and the Ioffe subdifferential satisfy the Separation Principle in any Banach space. The elementary subdifferentials (proximal, Fréchet, Hadamard, …), as well as their viscosity and limiting versions, satisfy the Separation Principle in appropriate Banach spaces: the Fréchet subdifferential in Asplund spaces, the Hadamard subdifferential in separable spaces, the proximal subdifferential in Hilbert spaces. The Moreau-Rockafellar subdifferential does not satisfy the Separation Principle for the whole class of lsc (non necessarily convex) functions: it is not a subdifferential for this wide class. See, e.g. [@Iof12; @JL13; @Pen13] and the references therein. The following link between the radial subderivative and arbitrary subdifferentials was established in [@JL14 Theorem 2.1] (see also [@JL13 Theorem 3.2]): \[JL\] Let $X$ be a Banach space, $f:X\to{{{]}{-\infty},+\infty]}}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$. Then, there is a sequence $((x_n,x^*_n))\subset{\partial}f$ such that $x_n\to {\bar{x}}$, $f(x_n)\to f({\bar{x}})$, $$f^r({\bar{x}};u)\le \liminf_{n}\,\langle x^*_n,u\rangle \text{ and } \limsup_{n}\,\langle x^*_n,x_n-{\bar{x}}\rangle\le 0.$$ Subderivative-subdifferential duality formula ============================================= A sequence $(x_n)\subset X$ is said to be *directionally convergent to ${\bar{x}}$ from the direction $v\in X$*, written $x_n\to_v {\bar{x}}$, if there are two sequences $t_n\searrow 0$ (that is, $t_n\to 0$ with $t_n>0$) and $v_n\to v$ such that $x_n={\bar{x}}+ t_n v_n$ for all $n$; equivalently: for every ${\varepsilon}>0$ the sequence $(x_n-{\bar{x}})$ eventually lies in the open drop ${}]0,{\varepsilon}B(v,{\varepsilon}){[}:=\{\, tv'{:}0<t<{\varepsilon},\ v'\in B(v,{\varepsilon})\,\}$. Observe that for $v=0$, ${}]0,{\varepsilon}B(v,{\varepsilon}){[}=B(0,{\varepsilon}^2)$ so $x_n\to_v{\bar{x}}$ simply means $x_n\to{\bar{x}}$. We let $D({\bar{x}},v,{\varepsilon}):= {\bar{x}}+{}]0,{\varepsilon}B(v,{\varepsilon}){[}$. We call *subderivative associated to a subdifferential ${\partial}f$* at a point $({\bar{x}},u)\in {{\rm dom} \kern.15em}f\times X$, the *support function* of the set ${\partial}f({\bar{x}})$ in the direction $u$, which we denote by $$f^{\partial}({\bar{x}};u):= \sup \,\{{\langle}{\bar{x}}^*,u {\rangle}{:}{\bar{x}}^*\in{\partial}f({\bar{x}})\}.$$ In the theorem below, given a function $f:X\to{{{]}{-\infty},+\infty]}}$, we denote by $f':{{\rm dom} \kern.15em}f\times X\to {\overline{\R}}$ any function lying between the subderivatives $f^d$ and $f^\uparrow$, that is: $$f^d\le f'\le f^\uparrow.$$ Subderivatives and subdifferentials are linked by the following formula: \[formula\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$. Then, for any direction $v\in X$ and any real number $\alpha\ge 0$, one has \[formula0\] $$\begin{aligned} \limsup_{x\to_v{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x))&= \limsup_{x\to_v{\bar{x}}} f'(x;u+\alpha ({\bar{x}}-x)) \label{formula0a}\\ &=\limsup_{x\to_v{\bar{x}}}f^{\partial}(x;u+\alpha ({\bar{x}}-x)). \label{formula0b}\end{aligned}$$ *First step.* We claim that $$\label{formula00} \limsup_{x\to_v{\bar{x}}} f^\uparrow(x;u+\alpha ({\bar{x}}-x))\le \limsup_{x\to_v{\bar{x}}} f^d(x;u+\alpha ({\bar{x}}-x)).$$ To prove this inequality, we take $\lambda \in\R$ such that $$\label{formula000} \limsup_{x\to_v{\bar{x}}} f^d(x;u+\alpha ({\bar{x}}-x))<\lambda$$ and show that $\lambda$ is greater than or equal to the left-hand side of . From we can find $\delta>0$ such that $$\label{formula01} x\in D({\bar{x}}, v,\delta) \Rightarrow f^d(x;u+\alpha ({\bar{x}}-x))<\lambda.$$ Let $z={\bar{x}}+tv'\in D({\bar{x}}, v,\delta/2)$ and let $\mu< f^\uparrow(z;u+\alpha({\bar{x}}-z))$. By Theorem \[Treiman\], there exist ${\varepsilon}>0$ and $x\in B(z,\rho)$, with $0<\rho\le t\delta/2$ and $\alpha\rho\le{\varepsilon}$, such that $$\label{pas1} \mu< f^d(x;u+\alpha({\bar{x}}-z)+w) \text{ for every } w\in B(0,{\varepsilon}).$$ Since $\alpha\|z-x\|\le \alpha\rho\le{\varepsilon}$, putting $w=\alpha(z-x)$ in , we infer that $$\label{pas2} \mu< f^d(x;u+\alpha({\bar{x}}-x)).$$ Since $\|x-{\bar{x}}-tv'\|=\|x-z\|\le \rho\le t\delta/2$, we have $v^{''}:=(x-{\bar{x}})/t\in B(v',\delta/2)\subset B(v,\delta)$, showing that $x={\bar{x}}+tv^{''} \in D({\bar{x}}, v,\delta)$. Therefore, by , $$\label{pas3} f^d(x;u+\alpha({\bar{x}}-x))<\lambda.$$ Combining and , we derive that $\mu<\lambda$. Since $\mu$ was arbitrarily chosen less than $f^\uparrow(z;u+\alpha({\bar{x}}-z))$, we conclude that $$z\in D({\bar{x}}, v,\delta/2) \Rightarrow f^\uparrow(z;u+\alpha({\bar{x}}-z))<\lambda,$$ hence, $$\limsup_{x\to_v{\bar{x}}} f^\uparrow(x;u+\alpha ({\bar{x}}-x))\le\lambda.$$ This completes the proof of . *Second step.* We claim that $$\label{formula00b} \limsup_{x\to_v{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x))\le \limsup_{x\to_v{\bar{x}}} f^{\partial}(x;u+\alpha ({\bar{x}}-x)).$$ As in the first step, to prove this inequality we take $\lambda \in\R$ such that $$\label{formula000b} \limsup_{x\to_v{\bar{x}}}f^{\partial}(x;u+\alpha ({\bar{x}}-x))<\lambda$$ and show that $\lambda$ is greater than or equal to the left-hand side of . From we can find $\delta>0$ such that $$\label{formula02} x\in D({\bar{x}}, v,\delta) \Rightarrow \sup \,\{ {\langle}x^*, u+ \alpha ({\bar{x}}-x){\rangle}{:}{\bar{x}}^*\in{\partial}f(x)\}<\lambda.$$ Let $z={\bar{x}}+tv'\in D({\bar{x}}, v,\delta/2)$. By Theorem \[JL\], for any $\mu< f^r(z;u+\alpha({\bar{x}}-z))$ and ${\varepsilon}>0$ there exist $x\in B(z,t\delta/2)$ and $x^*\in {\partial}f(x)$ such that $$\mu< {\langle}x^*,u+\alpha({\bar{x}}-z){\rangle}\text{ and } {\langle}x^*,x-z{\rangle}\le {\varepsilon}.$$ As above, we can verify that $x\in D({\bar{x}}, v,\delta)$. Therefore, by , $$\mu< {\langle}x^*,u+\alpha({\bar{x}}-z){\rangle}={\langle}x^*,u+\alpha({\bar{x}}-x){\rangle}+\alpha {\langle}x^*,x-z{\rangle}< \lambda +\alpha{\varepsilon}.$$ Since $\mu$ and ${\varepsilon}$ were arbitrary, we derive that $$z\in D({\bar{x}}, v,\delta/2) \Rightarrow f^r(z;u+\alpha({\bar{x}}-z))< \lambda,$$ showing that holds. *Third step.* Since ${\partial}f\subset {\partial}_C f$, we have $f^{\partial}(z;u') \le f^{{\partial}_C} (z;u') \le f^\uparrow(z;u')$ for every $u'\in X$. Hence, the right-hand side of is less than or equal to the left-hand side of . On the other hand, $f^d\le f^r$. So all the expressions in formulas and are equal. The desired set of equalities – follows because $f^d\le f'\le f^\uparrow$. \(a) In the special case $v=0$ and $\alpha= 0$, the formula was proved by Borwein-Strójwas [@BS89 Theorem 2.1 and Corollary 2.3] \(b) For $f$ locally Lipschitz at ${\bar{x}}$, the formulas do not depend on $\alpha\ge 0$ since $$\label{formula0lip} \limsup_{x\to_v{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x))=\limsup_{x\to_v{\bar{x}}} f^r(x;u).$$ But they may depend on the direction $v\in X$: for $f:x\in \R\mapsto f(x):=-|x|$ and $u\ne 0$, one has $$\limsup_{x\to_u 0} f^r(x;u)=-|u|<\limsup_{x\to 0} f^r(x;u)=|u|.$$ \(c) For arbitrary lsc $f$, the value of the expressions in depends on $\alpha\ge 0$ even for convex $f$. Indeed, as was recalled in Remark \[SP-rem\](a), the graph of the subdifferential $${\partial}_{MR}f({\bar{x}}) = \{x^* \in X^* {:}\langle x^*,u\rangle \leq f^r({\bar{x}};u), \, \forall u \in X\}$$ is generally not $strong\times weak^*$-closed. Therefore, for an arbitrary lsc convex $f$ the function $x\mapsto f^r(x;u)$ is generally not upper semicontinuous, that is $$f^r({\bar{x}};u)<\limsup_{x\to{\bar{x}}} f^r(x;u),$$ while always (see Proposition \[convexcase\] below) $$f^r({\bar{x}};u)=\inf_{\alpha\ge 0}\limsup_{x\to{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x)).$$ \[convexcase\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be convex lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$. Then, $$\label{convexformula0} f^r({\bar{x}};u)=\inf_{\alpha\ge 0}\limsup_{x\to{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x)).$$ Of course, $f^r({\bar{x}};u)$ is always not greater than the expression of the right-hand side of . It is not smaller either since, for every $t>0$, $$\limsup_{x\to{\bar{x}}} f^r(x;tu+{\bar{x}}-x)\le \limsup_{x\to{\bar{x}}}\, (f({\bar{x}}+tu)-f(x))=f({\bar{x}}+tu)-f({\bar{x}}),$$ hence, writing $\alpha=1/t$ for $t>0$, $$\begin{aligned} \inf_{\alpha\ge 0}\limsup_{x\to{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x)) &\le \inf_{t> 0}\limsup_{x\to{\bar{x}}} \frac{1}{t} f^r(x;tu+{\bar{x}}-x)\\ &\le \inf_{t> 0}\frac{f({\bar{x}}+tu)-f({\bar{x}})}{t}=f^r({\bar{x}};u). \tag*{\qedhere}\end{aligned}$$ Radially accessible functions ============================= A lsc function $f:X\to{{]}{-\infty},+\infty]}$ is said to be *radially accessible* at ${\bar{x}}\in{{\rm dom} \kern.15em}f$ from a direction $u\in X$ provided $$f({\bar{x}})=\liminf_{t\searrow 0}f({\bar{x}}+tu),$$ or equivalently, provided there exists a sequence $t_n\searrow 0$ such that $f({\bar{x}}+t_n u)\to f({\bar{x}})$. (The case $u=0$ is a tautology.) *Examples.* 1. Every lsc function $f:X\to{{]}{-\infty},+\infty]}$ which is radially upper semicontinuous at ${\bar{x}}\in{{\rm dom} \kern.15em}f$ from $u$ is evidently radially accessible at ${\bar{x}}$ from $u$. This is the case of convex lsc functions $f$ for any $u\in X$ such that ${\bar{x}}+u\in {{\rm dom} \kern.15em}f$. 2. If $f^r({\bar{x}};u)<\infty$, then $f$ is radially accessible at ${\bar{x}}$ from $u$. Indeed, let $\gamma\in\R$ such that $f^r({\bar{x}};u)<\gamma$. Then, there exists $t_n\searrow 0$ such that $f({\bar{x}}+ t_n u)\le f({\bar{x}})+\gamma t_n$, and consequently $\limsup_{n}f({\bar{x}}+ t_n u)\le f({\bar{x}})$. The condition $f^r({\bar{x}};u)<\infty$ however is not necessary: the continuous function $f:\R\to\R$ given by $f(x):=\sqrt{|x|}$ has $f^r(0;u)=\infty$ for any $u\ne 0$. 3. The function $f:\R\to\R$ given by $$f(x):= \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \mbox{ or } x= 1/n, \mbox{ for }n=1,2,\ldots\\ 1 & \mbox{otherwise}. \end{array} \right.$$ is lsc on $\R$, not upper semicontinuous at $0$ along the ray $\R_+u$ for $u>0$ but radially accessible at $0$ from such $u>0$. 4. The function $f:\R\to\R$ given by $$f(x):= \left\{ \begin{array}{ll} 1 & \mbox{if } x>0\\ 0 & \mbox{if } x\le 0 \end{array} \right.$$ is lsc on $\R$ but not radially accessible at $0$ from $u=1$. We notice that $f^r(0;1)=+\infty$, while $f^r(x;1)=0$ for any $x>0$. Radial accessibility is a mild regularity property. Yet this property leads to a more consistent behaviour of subdifferentials and subderivatives. We give two illustrations. Assume the lsc function $f$ is radially accessible at ${\bar{x}}\in{{\rm dom} \kern.15em}f$ from a direction $u$. Then first, ${{\rm dom} \kern.15em}{\partial}f$ contains a sequence graphically and *directionally* convergent to ${\bar{x}}$ (Theorem \[dense\]), and second, the upper radial subderivative $f^r_+({\bar{x}};u)$ is stable with respect to radially convergent sequences (Proposition \[devdir\]). From the latter statement we derive that the upper radial subderivative is a lower bound for the expressions in with $v=u$ (Theorem \[belowap\]). We recall the statement of Ekeland’s variational principle [@Eke74]: PS. *Variational Principle*. For any lsc function $f$ defined on a closed subset $S$ of a Banach space, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and ${\varepsilon}>0$ such that $ f({\bar{x}})\le \inf f(S) +{\varepsilon}, $ and for any $\lambda>0$, there exists $x_\lambda\in S$ such that $\|x_\lambda-{\bar{x}}\|\le\lambda$, $f(x_\lambda)\le f({\bar{x}})$ and the function $x\mapsto f(x)+({\varepsilon}/\lambda)\|x-x_\lambda\|$ attains its minimum on $S$ at $x_\lambda$. \[dense\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$ such that $f$ is radially accessible at ${\bar{x}}$ from $u$. Then, there exists a sequence $((x_n,x_n^*))_n\subset{\partial}f$ such that $x_n\to_u {\bar{x}}$, $f(x_n)\to f({\bar{x}})$ and $\limsup_n {\langle}x_n^*, x_n-{\bar{x}}{\rangle}\le 0$. Let ${\varepsilon}>0$. Since $f$ is [lsc]{} at ${\bar{x}}$, there exists $\delta\in ]0,{\varepsilon}^2[$ such that $$\label{sci0} f({\bar{x}})\le \inf f(B_\delta({\bar{x}})) + {\varepsilon}^2,$$ and since $f({\bar{x}})=\liminf_{t\searrow 0}f({\bar{x}}+tu)$, there exists $\mu>0$ such that $\mu({\varepsilon}+\|u\|)<\delta$ and $f({\bar{x}}+\mu u)\le f({\bar{x}})+{\varepsilon}^2$. Summarizing, we can find real numbers $\delta$ and $\mu$ satisfying \[sci\] $$\begin{gathered} 0< \mu({\varepsilon}+\|u\|)<\delta<{\varepsilon}^2, \mbox{ and} \label{sci1}\\ f({\bar{x}}+\mu u)\le \inf f(B_\delta({\bar{x}})) + {\varepsilon}^2\label{sci2}.\end{gathered}$$ Now we apply Ekeland’s variational principle to $f$ on the set $B_\delta({\bar{x}})$ at point ${\bar{x}}+\mu u$ with $\lambda=\mu{\varepsilon}$. Observe that the ball $B_\lambda({\bar{x}}+\mu u)$ is contained in the ball $B_\delta({\bar{x}})$ by . We therefore obtain a point ${x_{\varepsilon}}\in X$ such that $$\begin{gathered} \|{x_{\varepsilon}}-({\bar{x}}+\mu u)\|< \mu{\varepsilon}, ~f({x_{\varepsilon}})\le f({\bar{x}}+\mu u), \mbox{ and} \label{BP1}\\ y\mapsto f(y)+({\varepsilon}/\mu)\|y-{x_{\varepsilon}}\| \mbox{ admits a local minimum at } {x_{\varepsilon}}. \label{BP2}\end{gathered}$$ In view of (\[BP2\]), we may apply the Separation Principle at point ${x_{\varepsilon}}$ with the convex Lipschitz function $\varphi:y\mapsto ({\varepsilon}/\mu)\|y-{x_{\varepsilon}}\|$ to obtain a subgradient ${x_{\varepsilon}}^*\in {\widehat{{\partial}}}f({x_{\varepsilon}})$ such that $$\label{BP3} \|{x_{\varepsilon}}^*\|\le {\varepsilon}/\mu.$$ Now, take $({\bar{x}}_{\varepsilon},{\bar{x}}_{\varepsilon}^*)\in {\partial}f$ such that $$\begin{gathered} \|{\bar{x}}_{\varepsilon}-x_{\varepsilon}\|<\mu{\varepsilon},~~|f({\bar{x}}_{\varepsilon})-f(x_{\varepsilon})|<{\varepsilon}^2, \mbox{ and} \label{clos1}\\ {\langle}{\bar{x}}_{\varepsilon}^*-x_{\varepsilon}^*,{\bar{x}}-x_{\varepsilon}{\rangle}>-{\varepsilon},~~{\langle}{\bar{x}}_{\varepsilon}^*,{\bar{x}}_{\varepsilon}-x_{\varepsilon}{\rangle}<{\varepsilon}. \label{clos2} $$ It follows from the first parts of and that $$\label{cond1} \|{\bar{x}}_{\varepsilon}-({\bar{x}}+\mu u)\|< 2\mu{\varepsilon},$$ from the second parts of and combined with and that $$\label{cond2} |f({\bar{x}}_{\varepsilon})-f({\bar{x}})|\le 2{\varepsilon}^2,$$ and from and that $$\begin{aligned} \label{cond3} {\langle}{\bar{x}}_{\varepsilon}^*, {\bar{x}}-{\bar{x}}_{\varepsilon}{\rangle}>-{\varepsilon}({\varepsilon}+\|u\|+2),\end{aligned}$$ since $$\begin{aligned} {\langle}{\bar{x}}_{\varepsilon}^*, {\bar{x}}-{\bar{x}}_{\varepsilon}{\rangle}&= {\langle}x_{\varepsilon}^*, {\bar{x}}-x_{\varepsilon}{\rangle}+ {\langle}{\bar{x}}_{\varepsilon}^*-x_{\varepsilon}^*, {\bar{x}}-x_{\varepsilon}{\rangle}+ {\langle}{\bar{x}}_{\varepsilon}^*, x_{\varepsilon}-{\bar{x}}_{\varepsilon}{\rangle}\\ &> -\|{x_{\varepsilon}}^*\|\|{x_{\varepsilon}}-{\bar{x}}\|-2{\varepsilon}\\ &> -({\varepsilon}/\mu)\mu{\varepsilon}+\mu \|u\|-2{\varepsilon}=-{\varepsilon}({\varepsilon}+\|u\|+2).\end{aligned}$$ Therefore, if for every $n\in\N$, we let ${\varepsilon}=1/n$ and choose $\mu_n=\mu$ satisfying , so that $0<\mu_n<1/n$, we obtain a sequence $((x_n,x_n^*))_n$ in ${\partial}f$ by setting $x_n:={\bar{x}}_{\varepsilon}$ and $x_n^*:={\bar{x}}_{\varepsilon}^*$. It follows from , and that this sequence satisfies the requirements of the theorem. \(a) The case $u=0$ in Theorem \[dense\] is known, see, e.g. [@JL13; @JL14]. PS. (b) The case $u\ne 0$ is new even for convex lsc functions (recall that such functions are radially accessible at any point ${\bar{x}}\in{{\rm dom} \kern.15em}f$ from any $u$ such that ${\bar{x}}+u\in{{\rm dom} \kern.15em}f$). \(c) For $u\ne 0$, the conclusion of Theorem \[dense\] can be false at points where the function is not radially accessible. Let $f:\R\to{{]}{-\infty},+\infty]}$ given by $$f(x):= \left\{ \begin{array}{ll} 0 & \mbox{if } x=0 \mbox{ or } x= 1/n, \mbox{ for }n=1,2,\ldots\\ +\infty & \mbox{otherwise}. \end{array} \right.$$ Then, $f$ is lsc on $\R$ but not radially accessible at any point ${\bar{x}}=1/n$ from $u\ne 0$. We observe that all the points $x\ne{\bar{x}}$ close to ${\bar{x}}$ are not in ${{\rm dom} \kern.15em}f$, hence ${\partial}f(x)=\emptyset$. (d) For $u\ne 0$, we cannot claim in the conclusion of Theorem \[dense\] to find a radially convergent sequence $(x_n)$ instead of a directionally convergent one. Consider the function $f:\R^2\to{{]}{-\infty},+\infty]}$ given, for $x=(\xi_1,\xi_2)$, by $$f(x):= \left\{ \begin{array}{ll} -\sqrt{\xi_1}& \mbox{if } (\xi_1, \xi_2)\in \R_+\times \R \\ +\infty & \mbox{otherwise}. \end{array} \right.$$ Then, $f$ is convex lsc on $\R^2$ and $f(0,t)=0$ for every $t\in\R$, so $f$ is radially continuous at ${\bar{x}}=(0,0)$ in the direction $u=(0,1)$. But, for every $t\in\R$ we have ${\partial}f(0,t)=\emptyset$, so there is no sequence $(x_n)$ radially convergent to ${\bar{x}}$ from the direction $u=(0,1)$ with ${\partial}f(x_n)\not=\emptyset$. We recall the statement of the mean value inequality using the radial subderivative [@JL13; @JL14]: \[mvi\] Let $X$ be a Hausdorff locally convex space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in X$ and $x\in{{\rm dom} \kern.15em}f$. Then, for every real number $\lambda\le f({\bar{x}})-f(x)$, there exist $t_0\in [0,1[$ and $x_0:=x+t_0({\bar{x}}-x)\in [x,{\bar{x}}[$ such that $f(x_0)\le f(x)+t_0\lambda$ and\ $\lambda\le f^r(x_0;{\bar{x}}-x).$ \[devdir\] Let $X$ be a Hausdorff locally convex space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$ such that $f$ is radially accessible at ${\bar{x}}$ from $u$. Then, there is a sequence $\mu_n\searrow 0$ such that $f({\bar{x}}+\mu_n u)\to f({\bar{x}})$ and $$\label{applimvi1} f^r_+({\bar{x}};u)\le \liminf_{n\to +\infty}f^r({\bar{x}}+\mu_n u;u).$$ In particular, $$\label{below00} f^r_+({\bar{x}};u)\le \inf_{\alpha\ge 0} \limsup_{x\to_u{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x)).$$ If $f^r_+({\bar{x}};u)=-\infty$, there is nothing to prove. Otherwise, it suffices to show that for every $\lambda< f^r_+({\bar{x}};u)$, there exists a sequence $\mu_n\searrow 0$ such that $f({\bar{x}}+\mu_n u)\to f({\bar{x}})$ and $$\label{applimvi1b} {\lambda}\le \liminf_{n\to +\infty}f^r({\bar{x}}+\mu_n u;u).$$ So, let ${\lambda}< f^r_+({\bar{x}};u)$. By definition of $f^r_+({\bar{x}};u)$ there is a sequence $\tau_n\searrow 0$ such that $${\lambda}\tau_n < f({\bar{x}}+\tau_nu)-f({\bar{x}})\quad\mbox{for every } n\in \N,$$ and by assumption, there is a sequence $t_n\searrow 0$ such that $f({\bar{x}}+t_n u)\to f({\bar{x}})$. For every $n$, let $k_n\in \N$ such that \[dd1\] $$\begin{gathered} {\lambda}\tau_n < f({\bar{x}}+\tau_nu)-f({\bar{x}}+ t_{k_n}u),\label{dd1a}\\ 0<t_{k_n}<\tau_n^2.\label{dd1b}\end{gathered}$$ Applying Lemma \[mvi\] to we obtain $t_0\in [0,1[$ and $x_0:={\bar{x}}+t_{k_n}u+t_0(\tau_n-t_{k_n})u={\bar{x}}+\mu_n u$ with $\mu_n:=t_{k_n}+t_0(\tau_n-t_{k_n})\in [t_{k_n},\tau_n[$ such that \[dd2\] $$\begin{gathered} f({\bar{x}}+\mu_n u)\le f({\bar{x}}+t_{k_n}u)+t_0{\lambda}\tau_n, \label{dd2a}\\ {\lambda}\tau_n/(\tau_n-t_{k_n})< f^r({\bar{x}}+\mu_n u;u). \label{dd2b}\end{gathered}$$ Observe that in view of one has $\tau_n/(\tau_n-t_{k_n})\to 1$, so, letting $n\to +\infty$ we obtain $$\mu_n\searrow 0,\quad \limsup_{n\to +\infty} f({\bar{x}}+\mu_n u) \le f({\bar{x}}),\quad {\lambda}\le \liminf_{n\to +\infty}f^r({\bar{x}}+\mu_n u;u).$$ This completes the proof of the first statement since we also have $$f({\bar{x}})\le \liminf_{n\to +\infty} f({\bar{x}}+\mu_n u)$$ by the lower semicontinuity of $f$ at ${\bar{x}}$. To show the second statement, let $x_n:={\bar{x}}+\mu_n u$ with $\mu_n\searrow 0$ such that $$\label{z1} f^r_+({\bar{x}};u)\le \liminf_{n\to +\infty}f^r(x_n;u).$$ Since $u+\alpha ({\bar{x}}-x_n)=(1-\alpha\mu_n)u$, it follows that, for any $\alpha\ge 0$, $$\label{z2} \liminf_{n\to +\infty}f^r(x_n;u+\alpha ({\bar{x}}-x_n)) =\liminf_{n\to +\infty}\,(1-\alpha\mu_n)f^r(x_n;u)=\liminf_{n\to +\infty}f^r(x_n;u).$$ Since $x_n\to_u {\bar{x}}$, we derive from and that, for every $\alpha\ge 0$, $$\begin{aligned} f^r_+({\bar{x}};u)\le \liminf_{n\to +\infty}f^r(x_n;u+\alpha ({\bar{x}}-x_n))\le \limsup_{x\to_u{\bar{x}}} f^r(x;u+\alpha ({\bar{x}}-x)). \tag*{\qedhere}\end{aligned}$$ Plugging the formula of Theorem \[formula\] into the inequality of Proposition \[devdir\], we immediately obtain that the upper radial subderivative $f^r_+({\bar{x}};u)$ is a lower bound for the directional limit superior of the support of any subdifferential: \[belowap\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$ such that $f$ is radially accessible at ${\bar{x}}$ from $u$. Then, $$\label{below000} f^r_+({\bar{x}};u)\le\inf_{\alpha\ge 0} \limsup_{x\to_u{\bar{x}}}\,{f^{\partial}}(x;u+\alpha ({\bar{x}}-x)).$$ Appendix: a direct proof of Theorem \[belowap\] =============================================== For the sake of completeness, we provide a direct proof Theorem \[belowap\]. In fact, we shall establish an inequality more accurate than , in the same vein as Theorem \[JL\] and Theorem \[dense\]. \[belowapbis\] Let $X$ be a Banach space, $f:X\to{{]}{-\infty},+\infty]}$ be lsc, ${\bar{x}}\in{{\rm dom} \kern.15em}f$ and $u\in X$ such that $f$ is radially accessible at ${\bar{x}}$ from $u$. Then, there is a sequence $((x_n,x_n^*))\subset{\partial}f$ such that $x_n\to_u {\bar{x}}$, $f(x_n)\to f({\bar{x}})$ and $$\label{below0} f^r_+({\bar{x}};u)\le \liminf_{n}\,\langle x^*_n,u+\alpha({\bar{x}}-x_n)\rangle, \quad \forall \alpha\ge 0.$$ The pattern of the proof is similar to that of [@JL14 Theorem 2.1] but the argument has to be refined in order to obtain a directionally convergent sequence $(x_n)$. PS. *First step.* If $u=0$ or if $f^r_+({\bar{x}};u)=-\infty$, the result follows from Theorem \[dense\]. Otherwise, assume $u\ne 0$, let $\gamma<f^r_+({\bar{x}};u)$ and let ${\varepsilon}>0$. We claim that for each $n\in\N$ sufficiently large, there exists $(x_n,x_n^*)\in {\widehat{{\partial}}}f$ such that \[undeux\] $$\begin{gathered} x_{n}\in D({\bar{x}},u,{\varepsilon}), \quad f(x_{n})< f({\bar{x}})+{\varepsilon}, \label{un}\\ \langle x^*_n,u+\alpha({\bar{x}}-x_n)\rangle>\gamma-(\alpha+1){\varepsilon}, \quad \forall \alpha\ge 0.\label{deux}\end{gathered}$$ Let $z^*\in X^*$ such that ${\langle}z^*,u{\rangle}=-\gamma$, set $g:=f+z^*$ and let $K:= [{\bar{x}}, {\bar{x}}+u]$. Let also $0<\delta<1$ such that $g$ is bounded below on $B_\delta(K)$. By Proposition \[devdir\], there exists a sequence $\mu_n\searrow 0$ such that \[etoile0\] $$\begin{gathered} |f({\bar{x}}+\mu_n u)- f({\bar{x}})|<1/n, \label{etoile0a}\\ \gamma< f^r({\bar{x}}+\mu_n u;u).\label{etoile0b}\end{gathered}$$ We may assume $0<\mu_n< \sqrt{\delta}$. By , there exists $t_n\in {]}0,1-\mu_n]$ such that $$\label{etoile} f({\bar{x}}+\mu_n u)\le f({\bar{x}}+\mu_n u+tu) -\gamma t, \quad \forall t\in [0,t_n].$$ Let $K_n:=[{\bar{x}}+\mu_n u,{\bar{x}}+(\mu_n +t_n)u]\subset K$. Then, (\[etoile\]) can be rewritten as $$\label{etoile2} g({\bar{x}}+\mu_n u) \le g(x), \quad \forall x \in K_n.$$ Take $r>0$ such that $$\label{aa} g({\bar{x}}+\mu_n u)<\inf_{B_{r}(K_n)}g+\mu_n^3t_n,$$ and, observing that both $\inf_{B_{r}(K_n)}g$ and $\inf_{B_{\delta}(K_n)}g$ are finite, choose $\alpha_n>0$ such that $$\label{aaa} \inf_{B_{r}(K_n)}g\le \inf_{B_{\delta}(K_n)}g+\alpha_n r^2.$$ Then $$\label{aaaa} \inf_{B_{r}(K_n)}g \le (g +\alpha_n d^2_{K_n})(x), \quad \forall x\in B_{\delta}(K_n),$$ and therefore, by , $$\label{penal} g({\bar{x}}+\mu_n u) \le (g +\alpha_n d^2_{K_n})(x) + \mu_n^3t_n, \quad \forall x\in B_{\delta}(K_n).$$ Now, apply Ekeland’s variational principle to the function $g +\alpha_n d^2_{K_n}$ on the set $B_\delta(K_n)$ at point ${\bar{x}}+\mu_n u\in K_n$ with ${\varepsilon}=\mu_n^3t_n$ and $\lambda = \mu_n^2t_n$. Observe that the ball $B_\lambda({\bar{x}}+\mu_n u)$ is contained in $B_{\delta}(K_n)$ since for every $x\in B_\lambda({\bar{x}}+\mu_n u)$, we have $d_{K_n}(x)\le \|x-({\bar{x}}+\mu_n u)\|\le \lambda= \mu_n^2t_n<\delta$. We then obtain a point $x_n\in X$ satisfying \[reBP\] $$\begin{gathered} \|x_n-({\bar{x}}+\mu_n u)\|< \mu_n^2t_n,~~ g(x_n)+\alpha_n d^2_{K_n}(x_n)\le g({\bar{x}}+\mu_n u) \label{reBP1}\\ y\mapsto f(y)+{\langle}z^*,y{\rangle}+\alpha_n d^2_{K_n}(y)+\mu_n\|y-x_n\| \mbox{ admits a local minimum at } x_n. \label{reBP2}\end{gathered}$$ It follows from the first half of that $$x_n-{\bar{x}}\in B(\mu_n u,\mu_n^2t_n)=\mu_n B(u,\mu_nt_n),$$ showing that $x_{n}\in D({\bar{x}},u,{\varepsilon})$ for $n$ sufficiently large. On the other hand, the second half of and entail $$f(x_n)\le f({\bar{x}}+\mu_n u)+\|z^*\| \|{\bar{x}}+\mu_n u-x_n\| \le f({\bar{x}})+\|z^*\| \mu_n^2t_n+1/n,$$ showing that $f(x_n)< f({\bar{x}})+{\varepsilon}$ for $n$ sufficiently large. PS. In view of (\[reBP2\]), we may apply the Separation Principle at point $x_n$ with the convex Lipschitz function $\varphi:y\mapsto {\langle}z^*,y{\rangle}+\alpha_n d^2_{K_n}(y)+\mu_n\|y-x_n\|$ to obtain points $x_n^*\in {\widehat{{\partial}}}f(x_n)$, $\zeta_n^*\in{\partial}d^2_{K_n}(x_n)$ and $\beta_n^*\in B^*$ with $$\label{recond} 0=x_n^*+z^*+\alpha_n \zeta_n^*+\mu_n \beta_n^*.$$ We claim that the pair $(x_n,x_n^*)\in {\widehat{{\partial}}}f$ satisfies for large $n\in\N$. Assume it can be shown that for all $\alpha\ge 0$ and for large $n\in\N$, $$\label{ff0} \langle \zeta^*_n,u+\alpha({\bar{x}}-x_n)\rangle\le 0.$$ Then, it follows that for all $\alpha\ge 0$ and for large $n\in\N$, $$\begin{aligned} {\langle}x_n^*,u+\alpha({\bar{x}}-x_n){\rangle}&={\langle}-z^*,u+\alpha({\bar{x}}-x_n){\rangle}-\alpha_n{\langle}\zeta_n^*,u+\alpha({\bar{x}}-x_n){\rangle}-2\mu_n{\langle}\beta_n^*,u+ \alpha({\bar{x}}-x_n){\rangle}\\ &\ge\gamma-\alpha\|z^*\|\|{\bar{x}}-x_n\|-2\mu_n\|u+\alpha({\bar{x}}-x_n)\|,\end{aligned}$$ which implies that $\langle x^*_n,u+\alpha({\bar{x}}-x_n)\rangle>\gamma-(\alpha+1){\varepsilon}$ for $n$ sufficiently large, as claimed. PS. *Second step.* To complete the proof of it remains to prove . We first consider the case $\alpha=0$, that is, we show $$\label{claim2} {\langle}\zeta_n^*, u{\rangle}\le 0, \quad \forall n\in\N.$$ Let $P_{K_n}{x_n}\in K_n$ be any point such that $\|x_n -P_{K_n}{x_n}\|= d_{K_n}(x_n)$. We have $$\|{\bar{x}}+ \mu_nu-P_{K_n}{x_n}\|\le \|{\bar{x}}+ \mu_nu-x_n\|+\|x_n-P_{K_n}{x_n}\|<2\mu_n^2t_n.$$ So, $P_{K_n}{x_n}={\bar{x}}+ \mu_nu+\tau_nu$ with $\tau_n\|u\|<2\mu_n^2t_n$. Hence, $t_n-\tau_n>0$ for large $n$, and $$(t_n-\tau_n)u={\bar{x}}+ (\mu_n+t_n)u-P_{K_n}{x_n}.$$ Notice that $\zeta_n^*=2 d_{K_n}(x_n) \xi_n^*$ where $\xi_n^*\in{\partial}d_{K_n}(x_n)$. Then: $$\begin{aligned} (t_n-\tau_n){\langle}\zeta_n^*,u{\rangle}&={\langle}\zeta_n^*,{\bar{x}}+(\mu_n+t_n) u-P_{K_n}{x_n}{\rangle}\\ &={\langle}\zeta_n^*,{\bar{x}}+(\mu_n+t_n)u-x_n{\rangle}+ {\langle}\zeta_n^*,x_n-P_{K_n}{x_n}{\rangle}\\ &= 2 d_{K_n}(x_n) \left({\langle}\xi_n^*,{\bar{x}}+(\mu_n+t_n)u-x_n{\rangle}+ {\langle}\xi_n^*,x_n-P_{K_n}{x_n}{\rangle}\right)\\ &\le 2 d_{K_n}(x_n) (- d_{K_n}(x_n) + \|x_n -P_{K_n}{x_n}\|)=0.\end{aligned}$$ This proves . Now consider the case $\alpha> 0$. Write $\alpha=1/t$. We must show that for large $n\in\N$, $$\label{claim1} \langle \zeta^*_n,u+\alpha({\bar{x}}-x_n)\rangle=(1/t){\langle}\zeta_n^*, {\bar{x}}+t u-x_n{\rangle}\le 0.$$ But, for $n$ so large that $\mu_n<t$, we have $$\begin{aligned} {\langle}\zeta_n^*, {\bar{x}}+t u-x_n{\rangle}&= {\langle}\zeta_n^*, {\bar{x}}+\mu_n u-x_n{\rangle}+{\langle}\zeta_n^*, (t-\mu_n) u{\rangle}\\ &\le d^2_{K_n}({\bar{x}}+\mu_n u)-d^2_{K_n}(x_n)+(t-\mu_n){\langle}\zeta_n^*, u{\rangle}\\ &\le -d^2_{K_n}(x_n)\quad \mbox{by } \eqref{claim2}\\ &\le 0.\end{aligned}$$ This proves . Hence, holds and so also , as we have observed. PS. *Third step.* Every pair $(x_n,x_n^*)$ in ${\widehat{{\partial}}}f$ is close to a pair $({\bar{x}}_n,{\bar{x}}_n^*)$ in ${\partial}f$ in such a way that the sequence $((x_n,x_n^*))_n$ satisfying – for large $n\in\N$ can actually be assumed to lie in ${\partial}f$ (proceed as in Theorem \[dense\]). *Fourth step.* The theorem is derived from – as follows. Let $(\gamma_k)_k$ be an increasing sequence of real numbers such that $\gamma_k\nearrow f^r({\bar{x}};d)$. We have proved that, for each $k\in\N$, there are a sequence $((x_{n,k},x_{n,k}^*))_n \subset{\partial}f$ and an integer $N_k\in\N$ satisfying for every $n\ge N_k$: $$\begin{gathered} x_{n,k}\in D({\bar{x}},u,1/k), \quad f(x_{n,k})< f({\bar{x}})+ 1/k, \label{unfin}\\ \langle x^*_{n,k},u+\alpha({\bar{x}}-x_{n,k})\rangle >\gamma_k-(\alpha+1)/k, \quad \forall \alpha\ge 0.\label{deuxfin}\end{gathered}$$ Clearly, we may assume $N_{k+1}>N_k$. Then, it is immediate from – that the diagonal sequence defined, for $k\in\N$, by $(x_k,x_k^*) := (x_{N_k,k},x_{N_k,k}^*)$ satisfies the assertions of the theorem. [10]{} J. Borwein, S. Fitzpatrick, and R. Girgensohn. Subdifferentials whose graphs are not norm [$\times$]{} weak[$^*$]{} closed. , 46(4):538–545, 2003. J. M. Borwein and H. M. Str[ó]{}jwas. The hypertangent cone. , 13(2):125–144, 1989. B. Cornet. 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--- abstract: | In this paper, the temporal evolution of 3-dimensional relativistic current sheets in Poynting-dominated plasma is studied for the first time. Over the past few decades, a lot of efforts have been conducted on studying the evolution of current sheets in 2-dimensional space, and concluded that sufficiently long current sheets always evolves into the so-called “plasmoid-chain”, which provides fast reconnection rate independent of its resistivity. However, it is suspected that plasmoid-chain can exist only in the case of 2-dimensional approximation, and would show transition to turbulence in 3-dimensional space. We performed 3-dimensional numerical simulation of relativistic current sheet using resistive relativistic magnetohydrodynamic approximation. The results showed that the 3-dimensional current sheet evolve not into plasmoid-chain but turbulence. The resulting reconnection rate is $0.004$ which is much smaller than that of plasmoid-chain. The energy conversion from magnetic field to kinetic energy of turbulence is just 0.01% which is much smaller than typical non-relativistic cases. Using the energy principle, we also showed that the plasmoid is always unstable for a displacement in opposite direction to its acceleration, probably interchange-type instability, and this always results in seeds of turbulence behind the plasmoids. Finally, the temperature distribution along the sheet is discussed, and it is found that the sheet is less active than plasmoid-chain. Our finding can be applied for many high energy astrophysical phenomena, and can provide a basic model of the general current sheet in Poynting-dominated plasma. author: - | M. Takamoto,$^{1}$[^1]\ $^{1}$Department of Earth and Planetary Science, University of Tokyo, Tokyo 113-0033, Japan\ title: 'Evolution of 3-dimensional Relativistic Current Sheets and Development of Self-Generated Turbulence' --- \[firstpage\] Turbulence — MHD — plasmas — methods:numerical. Introduction {#sec:sec1} ============ Recently, the development of many high energy astronomical observation devices has allowed us to find many flare phenomena from various high energy astrophysical phenomena, such as Crab pulsar wind [@2012ApJ...749...26B; @2013MNRAS.436L..20B; @2014RPPh...77f6901B; @2015PPCF...57a4034P; @2015MNRAS.454.2972T] and blazars [@2012ApJ...754..114H; @2015ApJ...808L..18A]. Relativistic magnetic reconnection is considered to be a good candidates for those phenomena. This is because magnetic reconnection efficiently converts the magnetic field energy into plasma kinetic, thermal, photon, and non-thermal particle energy. In addition, it is known that sufficiently long current sheets in 2-dimensional space always evolves into the so-called “plasmoid-chain” in which the current sheets are filled with many plasmoids generated by the secondary tearing instability . The plasmoids experience many collisions to the neighboring ones, and it is expected that the energy released by such collisions can be responsible for flare phenomena observed in the high energy astrophysical phenomena. The research of magnetic reconnection in relativistic plasma, in particular, in Poynting-dominated plasma has been conducted vigorously for these decades. After several initial analytic work [@1994PhRvL..72..494B; @2003MNRAS.346..540L; @2003ApJ...589..893L; @2005MNRAS.358..113L], numerical simulation became the main method for studying relativistic magnetic reconnection due to its strong non-linear effects. Using relativistic magnetohydrodynamic (RMHD) approximation, @2006ApJ...647L.123W [@2011ApJ...739L..53T] have studied the initial phase of the tearing instability in current sheets with low Lundquist number, $S \equiv 4 \pi L_{\rm sheet} c_{\rm A}/\eta \lesssim 10^3$ where $L_{\rm sheet}$ is the sheet length, $c_{\rm A}$ is the Alfvén velocity, and $\eta$ is the resistivity. They observed strong compression in the downstream region predicted by @1994PhRvL..72..494B [@2003ApJ...589..893L; @2005MNRAS.358..113L], though the observed reconnection rate is very similar to the non-relativistic case predicted by @2003ApJ...589..893L [@2005MNRAS.358..113L]. However, it has been shown that relativistic magnetic reconnection results in faster reconnection rate than the non-relativistic case in Poynting-energy dominated plasma if much larger Lundquist number, $S > 10^4$, is considered and the sheet evolved into “plasmoid-chain” [@2013ApJ...775...50T]. On the other hand, there are also several work taking into account plasma effects, such as considering two-fluid approximation and fully collisionless plasma. [@2009ApJ...696.1385Z; @2009ApJ...705..907Z] have performed numerical simulations of relativistic magnetic reconnection by assuming relativistic two-fluid approximation, and observed enhancement of reconnection rate as electromagnetic energy increases. A similar effect were later observed in resistive RMHD simulation of Petschek reconnection [@2010ApJ...716L.214Z]. Collisionless reconnection work using Particle-in-cell (PIC) have also performed in this decade [@2014ApJ...783L..21S; @2015PhRvL.114i5002L; @2015ApJ...806..167G]. In addition to the increase of reconnection rate as electromagnetic field energy, they found that relativistic magnetic reconnection in Poynting-energy dominated plasma is a very efficient accelerator of particles, and can be a good candidate for flare events of high energy astrophysical phenomena. In spite of the above very active researches, there are still only a few work of magnetic reconnection in 3-dimensional space. It is considered that the current sheet in 3-dimensional scale will not evolve into plasmoid-chain but turbulence because of many instabilities in current sheet which can break the symmetry assumed in 2-dimensional work. In this case, it is known that turbulence enhances the magnetic reconnection rate. One reason of this is that turbulent motion of magnetic field increases the dissipation around reconnection point [@2013PhRvL.110y5001H]; In addition, more importantly, it was shown that turbulent eddy motion drives diffusion of magnetic field line separation, resulting in broader exhaust region and faster reconnection rate [@1999ApJ...517..700L; @2009ApJ...700...63K; @2015ApJ...815...16T] The above work assumed an external driven turbulence, and it depends on each phenomenon if there is sufficiently strong turbulence in those environments. However, current sheets have various kinds of instability, and it is expected that such instabilities will evolve into turbulence. Hence, the recent main research interest is to find detailed mechanisms of the self-generated turbulence in sheet, and the resulting reconnection rate. In non-relativistic case, there are a few work on self-generating turbulence in current sheets [@2015ApJ...806L..12O; @2016ApJ...818...20H; @2017ApJ...838...91K]. They reported that the 3-dimensional self-generate turbulent current sheets show smaller reconnection rate $\sim 0.005 - 0.01$, and 1 to 5 % of the magnetic field energy conversion into kinetic energy of the turbulence. In this paper, we report the first study of the temporal evolution of relativistic 3-dimensional current sheets in Poynting-dominated plasma using a new Godunov type scheme of resistive relativistic magnetohydrodynamic simulation. Since recent studies showed that turbulence in Poynting-energy dominated plasma has very different properties from non-relativistic one [@2015ApJ...815...16T; @2016ApJ...831L..11T; @2017MNRAS.472.4542T], we expect that such effects may modify the behavior of self-generated turbulence in relativistic sheets. In Section 2 we introduce the numerical setup. The numerical result is presented in Section 3, and its theoretical discussion is given in Section 4. Section 5 summarizes our conclusions. Numerical Setup {#sec:sec2} =============== In this paper, an evolution of a very long current sheet is modeled using the relativistic resistive magnetohydrodynamic approximation. We use a newly developed resistive relativistic magnetohydrodynamics (RRMHD) scheme explained in the Appendix, which is an extension of our previous work [@2011ApJ...735..113T], and allows us to obtain the full-Godunov solver of RRMHD for the first time. We calculate the RRMHD equations in a conservative fashion, and the mass density, momentum, and energy are conserved within machine round-off error. We use the constrained transport algorithm [@1988ApJ...332..659E] to preserve the divergence free constraint on the magnetic field. The multi-dimensional extension is achieved using the unsplit method [@2005JCoPh.205..509G; @2008JCoPh.227.4123G]. For the equation of state, a relativistic ideal gas with $h = 1 + (\Gamma / (\Gamma - 1))(p / \rho)$, $\Gamma = 4 / 3$ is assumed where $\rho$ is the rest mass density, and $p$ is the gas pressure in the plasma rest frame. The resistivity $\eta$ is determined from the Lundquist number: $S \equiv 4 \pi L_{\rm sheet} c_{\rm A}/ \eta = 2.912 \times 10^5$. For our numerical calculations, we prepare a numerical box, $[-L_{\rm x}, L_{\rm x}] \times [0, L_{\rm y}] \times [-L_{\rm z}, L_{\rm z}] = [-80 L, 80 L] \times [0, 20L] \times [-20L, 20L]$, where $L$ is the initial current sheet width. We divide it into homogeneous numerical meshes as $N_{\rm x} \times N_{\rm y} \times N_{\rm z} = 4096 \times 1024 \times 2048$. Along the boundaries at $x = \pm L_{\rm x}$ and $z = \pm L_{\rm z}$, the free boundary condition is imposed; Along the boundaries at $y = 0, L_{\rm y}$, the periodic boundary condition is imposed. For the initial condition, the static relativistic Harris current sheet [@1966PhFl....9..277H; @2003ApJ...591..366K] is assumed as: $$B_z(x) = B_0 \tanh (z / L) . $$ The uniform temperature, $k_{\rm B} T = m c^2$, is assumed where $k_{\rm B}$ is the Boltzmann constant, $m$ is the particle rest mass, and $c$ is the light velocity. The gas pressure satisfies the pressure balance condition, and the upstream gas pressure is determined by the magnetization parameter: $\sigma \equiv B_0^2 / 4 \pi \rho h c^2 \gamma^2 = 5$ where $\gamma$ is the Lorentz factor which is unity initially. To save numerical simulation time in 3-dimensional space, we first perform a 2-dimensional simulation. In this simulation, to trigger the initial tearing instability at the origin $(x,z) = (0,0)$, we add the following small perturbation to the magnetic field: $$ \delta A_{\rm y} = - 0.01 B_0 L \exp[-(x^2 + z^2) / 4 L^2] .$$ The 2-dimensional simulation is performed until $t = 60 L/c$ when the initial tearing mode is fully developed but the secondary tearing instability is not developed. We copied the numerical result of this 2-dimensional run into y-direction, and added a small white noise perturbation around $(x, z) = (0, 0)$ in velocity and motional electric field: ${\bf E} = - {\bf v} \times {\bf B}$ whose velocity dispersion is 0.1c. Note that the above time, $t = 60 L/c$, is redefined to the initial time $t = 0$ of the 3-dimensional simulation and analysis in the following. In the following, we use the unit: $c = 1$. Results {#sec:sec3} ======= ![The profiles of rest mass density and $|\gamma v|$ at $t = 120 L/c$. []{data-label="fig:0"}](./Turb_sheet_sgm5.eps){width="8.cm"} Figure \[fig:0\] is side view of density and top view of absolute value of velocity $|\gamma v|$ at $t = 120 L/c$. The top panel shows that there are fragmentation and small scale fluctuations. However, no clear small scale plasmoids are observed in the current sheet. In the bottom panel, it shows that there is no coherent structure in y-direction along the current sheet, indicating the appearance of fully evolved turbulence in the sheet. Interestingly, in the bottom panel we found that a region with relativistic velocity, close to the Alfvén velocity in the upstream region, is appeared locally in the sheet. this is in contrast to the results of 2-dimensional tearing instability in Poynting dominated plasma which resulted in a much slower outflow velocity, typically $\sim 0.3 c$. ![Temporal evolution of reconnection rate in 2 and 3 dimensional cases. []{data-label="fig:1"}](./Rec_Rate.eps){width="8.cm"} Figure \[fig:1\] is the temporal evolution of reconnection rate obtained by our simulation in 2 and 3 dimensional cases. In 2-dimensional case, reconnection rate is measured as: $$\label{eq:3.1} v_{\rm R} / c_{\rm A} \equiv \frac{1}{2 B_0 c_{\rm A} L_{\rm x}} \left | \int^{L_{\rm x}}_{-L_{\rm x}} dx E_{\rm y}(x, z=0) \right| .$$ In 3-dimensional case, reconnection rate is measured by a method proposed by [@2009ApJ...700...63K] which is a natural extension of Equation (\[eq:3.1\]) into 3-dimensional space. Figure \[fig:1\] shows that any clear differences cannot be observed until $t = 80 L/c$, and it shows linear and quasi-linear evolution of tearing instability in both 2 and 3 dimensional cases. At $t = 80 L/c$, reconnection rate of 2-dimensional case shows a rapid growth due to the evolution of plasmoid-chain, and it grows up to $0.03$ as observed in [@2013ApJ...775...50T]. The 3-dimensional case does not show such a rapid growth. Reconnection rate increases but more moderately whose saturation value seems around 0.004. Note that this value is similar to non-relativistic work [@2015ApJ...806L..12O; @2016ApJ...818...20H; @2017ApJ...838...91K]. We will discuss the meaning of this value later. ![Top: Temporal evolution of maximum velocity and velocity dispersion of $v_{\rm y}$. Bottom: Distribution function of $v_{\rm y}$ at $t = 120 L/c$. The orange line is the fitting curve with the form: $\exp[- |v_{\rm y}|/\delta v_{\rm y}]$ where $\delta v_{\rm y}$ is the velocity dispersion of $v_{\rm y}$. []{data-label="fig:2"}](./v_disp_y.eps "fig:"){width="8.cm"} ![Top: Temporal evolution of maximum velocity and velocity dispersion of $v_{\rm y}$. Bottom: Distribution function of $v_{\rm y}$ at $t = 120 L/c$. The orange line is the fitting curve with the form: $\exp[- |v_{\rm y}|/\delta v_{\rm y}]$ where $\delta v_{\rm y}$ is the velocity dispersion of $v_{\rm y}$. []{data-label="fig:2"}](./dist_vy.eps "fig:"){width="8.cm"} Top panel of Figure \[fig:2\] is the evolution of maximum value and dispersion of $v_{\rm y}$. Note that $v_{\rm y}$ represents the evolution of turbulence because it is not observed in 2-dimensional plasmoid-chain. The panel shows that the velocity dispersion saturates around 0.02c, which means that only 0.01% of magnetic field energy transferred into kinetic energy of turbulence [^2] . On the other hand, the maximum velocity reaches around 30 to 40 % of light velocity, indicating local appearance of very strong turbulence. Bottom panel of Figure \[fig:2\] is the distribution of $v_{\rm y}$. We found that the distribution can be described by $f(|v_{\rm y}|) \propto \exp(-|v_{\rm y}|/\delta v_{\rm y})$ where $\delta v_{\rm y}$ is the velocity dispersion discussed above. Note that this is different from the usual turbulence whose 1-point velocity distribution function is usually the Gaussian. Here we return to magnetic reconnection rate. @1999ApJ...517..700L found that if there is a sufficiently strong turbulence, magnetic reconnection rate can be described as: $$\frac{v_{\rm in}}{c_A} \simeq \mathrm{min}\left[ \left( \frac{L}{l} \right)^{1/2}, \left( \frac{l}{L} \right)^{1/2} \right] \left( \frac{v_l}{c_A} \right)^2 \label{eq:3.2} ,$$ where $L$ is the sheet length, $l$ is the injection scale of turbulence, and $v_l$ is the injection velocity. This was proven to be valid in relativistic regime with a slight modification in the trans-Alfvénic regime [@2015ApJ...815...16T]. Unfortunately, direct application of the above equation is difficult in this case, because several parameters, $L, l, v_l$, and the numerical coefficient is difficult to know. Instead, the scaling of $v_{\rm in}$ in terms of $v_l$ would be useful for checking Equation (\[eq:3.2\]). Figure \[fig:2.2\] is a plot of the ratio of reconnection rate $v_{\rm in}$ to the velocity dispersion of $v_{\rm y}$. From $t = 60L/c$, the ratio becomes nearly constant after the turbulence started to evolve in the sheet. This indicates that Equation (\[eq:3.2\]) needs a modification in this case if we reads $\delta v_{\rm y}$ as $v_l$. We consider that this is because the turbulence appeared in our simulation is different from the critical balanced one assumed for obtaining Equation (\[eq:3.2\]). This can be seen in the distribution of $v_{\rm y}$ in Figure \[fig:2\] which does not follow the usual Gaussian distribution, $\exp[-(v_{\rm y}/\delta v_{\rm y})^2]$, but $\exp[- |v_y|/\delta v_{\rm y}]$. [^3] Note that the ratio shows a slight increase in that phase, and we consider that this would reflect an enhancement from local plasmoid-chain in the sheet. ![Temporal evolution of the ratio of reconnection rate $v_{\rm in}$ to the velocity dispersion of $v_{\rm y}$. []{data-label="fig:2.2"}](./RecDv.eps){width="8.cm"} Discussion {#sec:sec5} ========== Onset of 3-dimensional Instability ---------------------------------- ![Density profile in the initial phase of the evolution of turbulence: $t = 35L$. Left: density profile in a plane at $z = 0.352L$. A yellow region is a plasmoid region. Right: 1-dimensional profile of square of Fourier transformed density on a line at $x = 17L$ and $z = 0.352$ which is along a plasmoid. []{data-label="fig:3"}](./Pls_Ptb.eps){width="8.cm"} In this section, the origin of turbulence in the sheet is discussed. The left panel of Figure \[fig:3\] is a density profile in a plane located at $z = 0.352 L$ in the evolving phase of turbulence: $t = 35L/c$. The yellow region is a right-moving plasmoid. Note that a coherent structure in y-direction along the plasmoid can still be observed. However, a small density fluctuation in y-direction exists, indicating the onset of an instability. The right panel of Figure \[fig:3\] is square of Fourier transformed density along y-direction at $x = 17L$ and $z = 0.352L$. It shows a peak at $k_{\rm y} L/2 \pi\sim 2$. This indicates that the density fluctuation does not result directly from the initial velocity perturbation with white noise, but some instabilities are responsible for it. In the following, we discuss the instability using the energy principle [@2005ppfa.book.....K] which allows a simpler analysis than the usual linear analysis [^4]. For simplicity, we assume that a plasmoid is static and at rest [^5] . The energy principle discuss the 2nd-order potential energy change by a displacement $\xi$, which is given as: $$\begin{aligned} \delta W^{(2)} &= \frac{1}{2} \int d x^3 \left[ \frac{Q^2}{4 \pi} + {\bf J} \cdot (\vec{\xi} \times {\bf Q}) + \Gamma p_{\rm g} (\nabla \cdot \vec{\xi})^2 \right. \nonumber \\ &+ \left. (\vec{\xi} \cdot \nabla p_{\rm g}) (\nabla \cdot \vec{\xi}) - \vec{\xi} \cdot \nabla \phi \nabla \cdot (e \vec{\xi}) \right] , \label{eq:4.1}\end{aligned}$$ where ${\bf Q} \equiv \nabla \times (\vec{\xi} \times {\bf B})$, ${\bf J} = \nabla \times {\bf B}/4 \pi$ is the current density, and $e$ is the energy density. $\phi$ is a gravitational potential, which is assumed to be very small enough to allow us to apply non-relativistic treatment. The goal of the analysis is to find a displacement vector $\vec{\xi}$ which makes $\delta W^{(2)}$ negative. In this case, the system is unstable. On the other hand, the system is stable if $\delta W^{(2)}$ is positive for all the $\vec{\xi}$. For simplicity, we consider the following trial displacement vector: $$\begin{aligned} &\nabla \cdot \vec{\xi} = 0 , \\ &\vec{\xi} = (- \xi_{\rm x}(y), 0, 0) . \end{aligned}$$ To model the magnetic field structure around the plasmoid, we consider the following magnetic field: $${\bf B} = (B_x(x,z), 0, B_z(x,z)) .$$ Although we neglected the velocity of plasmoid, we take into account the acceleration of plasmoid as: $- \nabla \phi \rightarrow - \tilde{g} {\bf e}_{\rm x}$ where ${\bf e}_{\rm x}$ is the unit vector in x and $\tilde{g}$ is acceleration of the plasmoid. Then, Equation (\[eq:4.1\]) reduces to $$\begin{aligned} \delta W^{(2)} = \frac{1}{8 \pi} \int d x^3 \xi_{\rm x}^2 \left[ (\partial_{\rm z} B_{\rm z})^2 + (\partial_{\rm x} B_{\rm z}) (\partial_{\rm z} B_{\rm x}) - \tilde{g} (\partial_{\rm x} e) \right] . \label{eq:4.2}\end{aligned}$$ Figure \[fig:5\] are the profile of density, pressure, and the gradients of magnetic field in Equation (\[eq:4.2\]). It shows that the 2nd-order potential energy is negative in the left-hand side of plasmoid, indicating that the plasmoid is unstable for a displacement in negative x-direction. We consider that this is a kind of interchange instability or Rayleigh-Taylor type instability. This is because there is no counter force to prevent corrugation of plasmoid. Note that the plasmoid is surrounded by magnetic field with no guide field. It is also known that such a plasmoid can also be unstable for kink type instability. Hence, we can conclude that the initial plasmoid coherent in y-direction is unstable for several instabilities, and it leaves a density perturbation behind the current sheets, resulting in a seed of turbulent motion of secondary plasmoids. ![ Profiles along 1-dimensional cut along $y = 10L, z = 0.352L$ at $t = 35L/c$, which corresponds to a line passing upstream plasma and a part of plasmoid in the early phase of their development. Top: gas pressure and density. Bottom: spatial gradients of magnetic field. []{data-label="fig:5"}](./P_t35.eps){width="8.cm"} To obtain an expression of the growth rate of this instability, we assume that the plasmoid located at the origin with azimuthal magnetic field: $${\bf B} = B_{\theta}(r) {\bf e}_{\theta}, \label{eq:4.3}$$ where $r$ is the radius of the plasmoid, and $\theta$ is the azimuthal angle. Equation (\[eq:4.2\]) reduces to $$\begin{aligned} \delta W^{(2)} = - \frac{1}{8 \pi} \int d x^3 \xi_{\rm x}^2 \left[ \frac{B_{\theta} \partial_r B_{\theta}}{r} + \tilde{g} (\partial_{\rm x} e) \right] . \label{eq:4.4}\end{aligned}$$ To proceed further, we assume that the plasmoid has a constant current density whose magnetic field can be described as: $$\begin{aligned} B_{\theta}(r) = \begin{cases} B_0 (r/R_0) & (\text{if} \ r < R_0), \\ B_0 & (\text{if} \ r > R_0), \end{cases} \label{eq:4.5}\end{aligned}$$ where $R_0$ is the radius of the plasmoid. For simplicity, we assume the pressure balance, and also assume the internal energy $e$ is proportional to the gas pressure as: $$\label{eq:4.6} e \simeq \tilde{e} \left[ 1 - \frac{B_0^2}{8 \pi P_{\rm tot}} \left( \frac{r}{R_0} \right)^2 \right] \quad ({\rm if} \ r < R_0),$$ where $P_{\rm tot}$ is the total pressure in the upstream region. For the displacement vector $\xi_{\rm x}$, we assume the sinusoidal expression as $\xi_{\rm x} = \xi_0 \cos k y$. To make the analysis easier, we assume the displacement vector work only inside of the plasmoid, that is, $\xi_{\rm x} = \xi_0 \cos k y {\rm Hv}(R_0 - r)$ where ${\rm Hv}(x)$ is the Heaviside step function. Then, Equation (\[eq:4.4\]) becomes $$\begin{aligned} \delta W^{(2)} = - \frac{\xi_0^2 L_{\rm y} B_0^2}{16 \pi} (1 + \tilde{e} R_0 \tilde{g} ) , \label{eq:4.7}\end{aligned}$$ where $L_{\rm y}$ is the size of sheet in y-direction. Note that here we performed the integral of x only in the negative-half plane ($x < 0$) to take into account the non-symmetric form of actual plasmoids which is equivalent to assuming the displacement vector only in the back-part of the plasmoid. Following @2005ppfa.book.....K, the approximate growth rate $\Gamma_{\rm grow}$ of the instability can be obtained as: $$\Gamma_{\rm grow} \sim \sqrt{- \frac{\delta W^{(2)}}{K}} , \label{eq:4.8}$$ where $$K \equiv \frac{1}{2} \int d^3 x \rho h |\xi|^2 . \label{eq:4.9}$$ Here we modified the original non-relativistic expression of $K$ by multiplying the specific enthalpy $h$ because this term comes from the equation of motion. After a similar calculation, we obtain $$\Gamma_{\rm grow} \sim \frac{c_A}{R_0} \sqrt{1 + \tilde{e} R_0 \tilde{g}} . \label{eq:4.10}$$ This means that this instability is driven by the Lorentz force and the effective acceleration, indicating the above discussion that interchange and Rayleigh-Taylor type instabilities are responsible. Note that the ratio of the Lorentz force and effective acceleration is absorbed in $\tilde{e}$. From dimensional analysis, Equation (\[eq:4.10\]) also indicates that the characteristic wavelength in y-direction would be $R_0$, which reproduces the result in the right-panel of Figure \[fig:3\]. This will also explain the small turbulent velocity shown in Figure \[fig:2\], because the source of the instability is the acceleration of plasmoid and the difference of energy in the upstream and inside of plasmoid. Both of them have small energy compared with the inertia of the hot plasma in plasmoid. Note that it is known that plasmoid is unstable for another several instabilities, such as the kink instability [@2000mrp..book.....B]. In our simulations, it is found that the above instability leaves strong perturbations in the current sheet, resulting in the seeds non-coherent secondary tearing instability and plasmoids in y-direction; Such a successive process finally generates the turbulent sheet. [^6] Difference Between 2 and 3 Dimensional Cases -------------------------------------------- ![ Density profile on x-z plane at $t = 120 L/c$ obtained by a 2-dimensional simulation with the same set up as the 3-dimensional one. []{data-label="fig:6"}](./den_2D.eps){width="8.cm"} In 2-dimensional case, it is known that current sheets in high Lundquist number flow evolve into plasmoid-chain as discussed in Section \[sec:sec3\]. Figure \[fig:6\] is the temperature profile of plasmoid chain obtained in our 2-dimensional calculation. As is known, the sheet is filled with a lot of plasmoid, and a self-similar structure seems to be observed, allowing a locally very thin sheet. On the other hand, Figure \[fig:3\] shows that plasmoid-chain does not appear in 3-dimensional case, and almost all the sheet is filled with self-generated turbulence. ![ Distribution function of temperature on a plane ($z = 0$) at a saturation time ($t = 120 L/c$). Both 2 and 3-dimensional results are plotted. []{data-label="fig:7"}](./dist_T.eps){width="8.cm"} Figure \[fig:7\] is the temperature distribution in the plane located at the center of current sheet ($z=0$). It shows that 3-dimensional turbulent sheet is much less active than 2-dimensional plasmoid-chain, and the temperature distribution becomes much narrower than the plasmoid-chain case. This is because plasmoid-chain results in very small plasmoid, and they experience collisions to neighboring plasmoids with approximately Alfvén velocity, resulting in frequent energy conversion from its kinetic velocity into thermal energy. On the other hand, 3-dimensional turbulent sheets prohibits the appearance of such small plasmoids, and the energy concentrated in small plasmoids are distributed into turbulent motions. This indicates that relativistic current sheets are not a good candidate for small flares observed in many high energy astrophysical phenomena, because of no existence of small plasmoids and origin of small scale strong fluctuations. However, in our simulations the initially generated plasmoids can survive for a long time, and evolve into a very large ones, so called “*monster plasmoids*” [@2010PhRvL.105w5002U; @2013ApJ...775...50T; @2014ApJ...783L..21S]. We consider that it is still possible to explain large flare phenomena by the appearance of the monster plasmoids. Conclusion {#sec:sec6} ========== In this paper, we studied an evolution of 3-dimensional relativistic current sheets in Poynting-dominated plasma. In 2-dimensional case, it is known that sufficiently long current sheets always evolve into plasmoid-chain, and their reconnection rate is around 0.01. The plasmoid-chain is found to be one of the final steady states in 2-dimensional cases, and observed in both of magnetohydrodynamic and collisionless simulations from non-relativistic to relativistic plasma. For this reason, the plasmoid-chain is well-studied and even applied for several astrophysical phenomena, such as the solar flares and flares in relativistic jets. However, the plasmoid-chain is a 2-dimensional phenomenon, and there is a doubt on the applicability in 3-dimensional cases. Actually, a few non-relativistic simulations reported that the plasmoid-chain was not observed but instead their sheets evolved into turbulence. In our simulation, we found that relativistic current sheets also develop into turbulence, and no clear evidence of plasmoid-chain was observed. The reconnection rate is around 0.004 which is much smaller than that of plasmoid-chain, $0.03$. Using the energy principle, we also found that the plasmoid is unstable for an displacement of plasma in the opposite directions of plasmoid velocity, seemingly the interchange type instability, resulting in turbulence in the sheet. Finally, we discussed the activity of the sheet using the temperature distribution in it. It is found that the 3-dimensional sheet has the smaller number of high temperature regions than the 2-dimensional sheet. This is because the small scale plasmoids responsible for generating high temperature region do not appear in turbulent sheet, and the turbulence itself also increase the sheet width by its eddy motion. This indicates that the realistic relativistic current sheet in Poynting-dominated plasma is not so active, and cannot be a candidate for mini-flares observed in many high-energy astrophysical phenomena. However, the limitation of our numerical resources and high numerical costs of 3-dimensional simulations prohibits us from the sufficient parameter search, in particular, the dependence on the $\sigma$-parameter, and it still remains unknown if the sheet with higher $\sigma$-parameter upstream plasma would be more active. And this is our future work. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Masahiro Hoshino, Takanobu Amano, Masanori Iwamoto, Shuichiro Inutsuka, and Tsuyoshi Inoue for many fruitful comments and discussions. Numerical computations were carried out on the Cray XC30 at Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan. This work is supported by the Postdoctoral Fellowships by the Japan Society for the Promotion of Science No. 201506571 (M. T.). Resolution Study ================ In the main part of this paper, we performed a numerical simulation of relativistic current sheets. This simulation demands us a very high resolution because of the development of turbulence in current sheet scale. In this Appendix, we discuss the effect of resolution. ![A numerical results of a simulation with the same setup but used half mesh number in all directions. Top: reconnection rate, Bottom: maximum and dispersion of $v_{\rm y}$. []{data-label="fig:A1"}](./Rec_Rate_conv.eps "fig:"){width="8.cm"} ![A numerical results of a simulation with the same setup but used half mesh number in all directions. Top: reconnection rate, Bottom: maximum and dispersion of $v_{\rm y}$. []{data-label="fig:A1"}](./v_disp_y_conv.eps "fig:"){width="8.cm"} Figure \[fig:A1\] is a numerical results of the same setup but used half mesh number in all directions, that is, $N_{\rm x} \times N_{\rm y} \times N_{\rm z} = 2048 \times 512 \times 1024$. The top panel of the figure is the reconnection rate. It shows that the lower resolution run shows much larger reconnection rate than the higher resolution run. The bottom panel of the figure is the maximum and dispersion of $v_{\rm y}$. It indicates that the saturation level of maximum and dispersion of $v_{\rm y}$ is similar, though the onset of the growth is a little faster in the lower resolution run. Note that this is consistent with [@2017ApJ...838...91K]. We consider that the higher reconnection rate in the lower resolution run does not result from physical effects but numerical effects. This is because the turbulence appeared in the simulation is very similar, indicating that reconnection rate is not controlled by the turbulence in the sheet. Note that the resistivity in the lower resolution simulation is only a little less than the numerical resistivity, and this also allowed the strong numerical effects on local magnetic reconnection in a region where the sheet width becomes close to the numerical cell size. Unfortunately, the present numerical resource does not allow us to check a higher resolution study. Numerical Scheme ================ In this section, our new method for RRMHD is explained. For simplicity, only the 1-dimensional case is discussed for the treatment of numerical flux. First, the basic equations are discussed. In the RRMHD case, the Maxwell equations are considered: $$\begin{aligned} \nabla \cdot \mathbf{E} &= q \label{eq:Maxwell_1} , \\ \nabla \cdot \mathbf{B} &= 0 \label{eq:Maxwell_2} , \\ \partial_t \mathbf{E} - \nabla \times \mathbf{B} &= - \mathbf{J} \label{eq:Maxwell_3} , \\ \partial_t \mathbf{B} + \nabla \times \mathbf{E} &= \mathbf{0} \label{eq:Maxwell_4} ,\end{aligned}$$ where ${\bf B} \equiv {\bf B}/\sqrt{4 \pi}$ and ${\bf E} \equiv {\bf E}/\sqrt{4 \pi}$ in the Gauss unit. In addition, the plasma part can be described by: $$\begin{aligned} \partial_t \left( \begin{array}{c} D \\ m^i \\ e \end{array} \right) + \partial_j \left( \begin{array}{c} F_D^j \\ F_m^{ij} \\ F_e^j \end{array} \right) = 0, \label{eq:fluid}\end{aligned}$$ where the conserved variables are: $$\begin{aligned} D &= \gamma \rho \label{eq:D} , \\ \mathbf{m} &= \rho h \gamma^2 {\bf v} + \mathbf{E \times B} \label{eq:m} , \\ e &= \rho h \gamma^2 - p + \frac{1}{2} (E^2 + B^2) \label{eq:e} ,\end{aligned}$$ where ${\bf v}$ is the fluid three-velocity, $\gamma = (1 - v^2)^{-1/2}$ is the Lorentz factor, and numerical fluxes are: $$\begin{aligned} F_D^i &= D v^i , \\ F_m^{ij} &= m^i v^j + p \eta^{ij} - E^i E^j - B^i B^j + \frac{1}{2} (E^2 + B^2) \eta^{ij} , \\ F_e^i &= m^i ,\end{aligned}$$ where $\eta^{\mu \nu}$ is the metric tensor. Although obtaining charge density $q$ and current density ${\bf J}$ is non-trivial, which has already been explained in our previous work [@2011ApJ...735..113T]. In the following, we assume the charge density and current density are determined by some method, and concentrate on obtaining numerical flux. In the above equations, we have to update the following variables: $\{D, m_i, e, {\bf E}, {\bf B} \}$. To obtain the Godunov-type exact solution, we have to solve the jump condition on the above equation. However, it is too complicated to solve, and there is no study succeeding in obtaining exact solutions of RRMHD full Godunov scheme. Here, we propose a better method allowing us to reproduce the exact RMHD numerical flux in ideal regime and take into account resistivity, though it is not the exact numerical flux of RRMHD equation itself. In our new scheme, the numerical flux ${\bf F}$ is calculated using HLLEM Riemann solver [@2016JCoPh.304..275D] as: $$\begin{aligned} \label{eq:B1} {\bf F}_{\rm HLLEM} &\equiv \frac{s_{\rm R} {\bf F}_{\rm L} - s_{\rm L} {\bf F}_{\rm R}}{s_{\rm R} - s_{\rm L}} + \frac{s_{\rm R} s_{\rm L}}{s_{\rm R} - s_{\rm L}} ( {\bf Q}_{\rm R} - {\bf Q}_{\rm L}) \nonumber \\ &- \varphi \frac{s_{\rm R} s_{\rm L}}{s_{\rm R} - s_{\rm L}} {\bf R}_*(\bar{\bf Q}) {\bf \delta}_*(\bar{\bf Q}) {\bf L}_*(\bar{\bf Q})( {\bf Q}_{\rm R} - {\bf Q}_{\rm L}) ,\end{aligned}$$ where ${\bf Q}$ is the conservative variables, and ${\rm F}$ is the flux. The subscript L and R mean the left-hand and right-hand side variables. $s_{\rm L/R}$ is the fastest characteristic speed in left and right directions. The first line is the classical HLL flux; the second line is the “anti-diffusion term” which subtracts numerical dissipation stemming from characteristic mode neglected in HLL flux. The detailed explanation is given in the original paper [@2016JCoPh.304..275D]. In our scheme, we calculate the anti-diffusion term using the RMHD eigen-values which has already been obtained, for example, by @2010ApJS..188....1A. Hence, for the electric field $\{ {\bf E} \}$, the numerical flux is calculated by classical HLL flux, and for the other variables $\{D, m_i, e, {\bf B} \}$, the numerical flux is calculated by the HLLEM flux. This allows us to include the internal structure of RMHD in the intermediate HLL state. Note that the resistivity is naturally taken into account from the non-ideal RMHD terms in the above equations (\[eq:D\])-(\[eq:e\]). ![A numerical results of shock tube problems. Top: Scheme dependence. Bottom: electric conductivity dependence. []{data-label="fig:B1"}](./HLLI_cmp.eps "fig:"){width="8.cm"} ![A numerical results of shock tube problems. Top: Scheme dependence. Bottom: electric conductivity dependence. []{data-label="fig:B1"}](./HLLI_sgm_cmp.eps "fig:"){width="8.cm"} In the following, some numerical results obtained by our new scheme will be shown, and discuss the accuracy and validity of the scheme. $\Gamma = 4/3$ is assumed for all the problems, and grid number $N=100$ is used. Spatial and temporal 2nd-order accuracy scheme is assumed. Figure \[fig:B1\] is the numerical results of shock tube problems. In the left panel, a problem including slow shocks is considered. The initial left and right states are given as: $$\begin{aligned} (\rho^L, (v_z)^L, p^L, (B^y)^L) =& (1, 0.2, 10^{-2}, 0.2) \\ (\rho^R, (v_z)^R, p^R, (B^y)^R) =& (1, -0.2, 5 \times 10^{-2}, -0.1)\end{aligned}$$ All the other fields are set to $0$. The electric conductivity $\sigma \equiv c^2 / \eta$ is set to $10^6$ to reproduce ideal results. The left panel of Figure \[fig:B1\] is the density profile of our numerical results of HLL and HLLEM scheme at $t = 3.2$. It shows that the new HLLEM scheme allows us to capture the contact discontinuity ($x=-0.2$) and slow shock ($x=-0.5$) comparing with HLL scheme. Note that the new scheme also allows us to capture the fast shock ($x=-0.8$) by smaller number of grid points. The right panel of Figure \[fig:B1\] is the simple MHD version of the Brio and Wu test. The initial left and right states are given by $$\begin{aligned} (\rho^L, p^L, (B^y)^L) =& (1.0, 1.0, 0.5) \\ (\rho^R, p^R, (B^y)^R) =& (0.125, 0.1, -0.5)\end{aligned}$$ All the other fields are set to $0$. The panel shows the numerical results of the same problem that changes the conductivity $\sigma = 0, 10, 10^2, 10^3, 10^6$. We also plot the exact ideal RMHD solution by the solid line computed by a publicly available code developed by Giacomazzo and Rezzolla [@2006JFM...562..223G]. This result shows that our numerical solution reproduces the results in [@2011ApJ...735..113T], and the new method can take into account resistive effect properly. 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[^3]: Although the temporal evolution of velocity dispersion in Figure \[fig:2\] indicates that the turbulence reached a steady state, there is a possibility that the turbulence would evolve into critical-balanced turbulence if much longer time has passed. Our present numerical resources prohibits us to perform larger simulation, and it will be our future work to check the above statement. [^4]: In order to apply the energy principle analysis, the 0-th state should be a steady state. In our analysis, the background plasma is not a true steady state but transient state. However, the time scale of the background plasmoid evolution is much longer than that of the instability. Hence, we consider that it can be regarded as a quasi-steady state, and we apply the energy principle analysis on it. [^5]: We consider that this is not so bad approximation because the velocity of the plasmoid is sufficiently small comparing with the Alfvén velocity: $v_{\rm plasmoid} \ll c_{\rm A}$, and also the relativistic effects from the Lorentz factor is negligible. [^6]: In the case of relativistic collisionless pair-plasma, the relativistic drift-kink instability can be a seed of 3-dimensional turbulence when the sheet thickness is close to the Debye length [@2005PhRvL..95i5001Z; @2007ApJ...670..702Z; @2014ApJ...783L..21S; @2015PhRvL.114i5002L; @2015ApJ...806..167G]. Although we consider the magnetohydrodynamic approximation, that is, much denser plasma and larger scale physics in this paper, those plasma scale fluctuations can also be a seed of perturbations in magnetohydrodynamic scale phenomena, which support our statement that current sheets in 3-dimensional space naturally evolve into turbulence. Note that such collisionless effects become important in low density plasma, such as pulsar magnetosphere and wind regions.

--- abstract: | [We show that every $p$-fold strictly-cyclic branched covering of a $b$-bridge link in $\S^3$ admits a $p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$. This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of $p$-symmetric Heegaard splittings as $p$-fold strictly-cyclic branched coverings of links.\ \ [*Mathematics Subject Classification 2000:*]{} Primary 57M12, 57R65; Secondary 20F05, 57M05, 57M25.\ [*Keywords:*]{} 3-manifolds, Heegaard splittings, cyclic branched coverings, links, plats, bridge number, braid number.]{} author: - Michele Mulazzani title: 'An intrinsic characterization of $p$-symmetric Heegaard splittings [^1]' --- Introduction ============ The concept of $p$-symmetric Heegard splittings has been introduced by Birman and Hilden (see [@BH]) in an extrinsic way, depending on a particular embedding of the handlebodies of the splitting in the ambient space $\E^3$. The definition of such particular splittings was motivated by the aim to prove that every closed, orientable 3-manifold of Heegaard genus $g\le 2$ is a 2-fold covering of $\S^3$ branched over a link of bridge number $g+1$ and that, conversely, the 2-fold covering of $\S^3$ branched over a link of bridge number $b\le 3$ is a closed, orientable 3-manifold of Heegaard genus $b-1$ (compare also [@Vi]). A genus $g$ Heegaard splitting $M=Y_g\cup_{\f}Y'_g$ is called [*$p$-symmetric*]{}, with $p>1$, if there exist a disjoint embedding of $Y_g$ and $Y'_g$ into $\E^3$ such that $Y'_g=\t(Y_g)$, for a translation $\t$ of $\E^3$, and an orientation-preserving homeomorphism $\P:\E^3\to\E^3$ of period $p$, such that $\P(Y_g)=Y_g$ and, if $\GG$ denotes the cyclic group of order $p$ generated by $\P$ and $\F:\partial Y_g\to\partial Y_g$ is the orientation-preserving homeomorphism $\F=\t^{-1}_{\vert\partial Y'_g}\f$, the following conditions are fulfilled: - $Y_g/\GG$ is homeomorphic to a 3-ball; - $\mbox{Fix}(\P_{\vert Y_g}^h)=\mbox{Fix}(\P_{\vert Y_g})$, for each $1\le h\le p-1$; - $\mbox{Fix}(\P_{\vert Y_g})/\GG$ is an unknotted set of arcs[^2] in the ball $Y_g/{\cal G}$; - there exists an integer $p_0$ such that $\F\P_{\vert\partial Y_g}\F^{-1}=(\P_{\vert\partial Y_g})^{p_0}$. [**Remark 1**]{} By the positive solution of the Smith Conjecture [@MB] it is easy to see that necessarily $p_0\equiv\pm 1$ mod $p$. The map $\P'=\t\P\t^{-1}$ is obviously an orientation-preserving homeomorphism of period $p$ of $\E^3$ with the same properties as $\P$, with respect to $Y'_g$, and the relation $\f\P_{\vert\partial Y_g}\f^{-1}=(\P'_{\vert\partial Y'_g})^{p_0}$ easily holds. The [*$p$-symmetric Heegaard genus*]{} $g_p(M)$ of a 3-manifold $M$ is the smallest integer $g$ such that $M$ admits a $p$-symmetric Heegaard splitting of genus $g$. The following results have been established in [@BH]: 1. Every closed, orientable 3-manifold of $p$-symmetric Heegaard genus $g$ admits a representation as a $p$-fold cyclic covering of $\S^3$, branched over a link which admits a $b$-bridge presentation, where $g=(b-1)(p-1)$. 2. The $p$-fold cyclic covering of $\S^3$ branched over a knot of braid number $b$ is a closed, orientable 3-manifold $M$ which admits a $p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$. Note that statement 2 is not a complete converse of 1, since it only concerns knots and, moreover, $b$ denotes the braid number, which is greater than or equal to (often greater than) the bridge number. In this paper we fill this gap, giving a complete converse to statement 1. Since the coverings involved in 1 are strictly-cyclic (see next section for details on strictly-cyclic branched coverings of links), our statement will concern this kind of coverings. More precisely, we shall prove in Theorem \[Theorem 3\] that a $p$-fold strictly-cyclic covering of $\S^3$, branched over a link of bridge number $b$, is a closed, orientable 3-manifold $M$ which admits a $p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$, and therefore has $p$-symmetric Heegaard genus $g_p(M)\le (b-1)(p-1)$. This result gives an intrinsic interpretation of $p$-symmetric Heegaard splittings as $p$-fold strictly-cyclic branched coverings of links. Main results ============ Let $\b=\{(p_k(t),t)\,\vert\, 1\le k\le 2n\,,\,t\in[0,1]\}\subset\E^2\times[0,1]$ be a geometric $2n$-string braid of $\E^3$ [@Bi], where $p_1,\ldots,p_{2n}:[0,1]\to\E^2$ are continuous maps such that $p_{k}(t)\neq p_{k'}(t)$, for every $k\neq k'$ and $t\in[0,1]$, and such that $\{p_1(0),\ldots,p_{2n}(0)\}=\{p_1(1),\ldots,p_{2n}(1)\}$. We set $P_k=p_k(0)$, for each $k=1,\ldots,2n$, and $A_i=(P_{2i-1},0),B_i=(P_{2i},0),A'_i=(P_{2i-1},1),B'_i=(P_{2i},1)$, for each $i=1,\ldots,n$ (see Figure 1). Moreover, we set $\FF=\{P_1,\ldots,P_{2n}\}$, $\FF_1=\{P_1,P_3\ldots,P_{2n-1}\}$ and $\FF_2=\{P_2,P_4,\ldots,P_{2n}\}$. The braid $\b$ is realized through an ambient isotopy ${\wh\b}:\E^2\times[0,1]\to\E^2\times[0,1]$, ${\wh\b}(x,t)=(\b_t(x),t)$, where $\b_t$ is an homeomorphism of $\E^2$ such that $\b_0=\mbox{Id}_{\E^2}$ and $\b_t(P_i)=p_i(t)$, for every $t\in[0,1]$. Therefore, the braid $\b$ naturally defines an orientation-preserving homeomorphism ${\wti\b}=\b_1:\E^2\to\E^2$, which fixes the set $\FF$. Note that $\b$ uniquely defines ${\wti\b}$, up to isotopy of $\E^2$ mod $\FF$. Connecting the point $A_i$ with $B_i$ by a circular arc $\a_i$ (called [*top arc*]{}) and the point $A'_i$ with $B'_i$ by a circular arc $\a'_i$ (called [*bottom arc*]{}), as in Figure 1, for each $i=1,\ldots,n$, we obtain a $2n$-plat presentation of a link $L$ in $\E^3$, or equivalently in $\S^3$. As is well known, every link admits plat presentations and, moreover, a $2n$-plat presentation corresponds to an $n$-bridge presentation of the link. So, the bridge number $b(L)$ of a link $L$ is the smallest positive integer $n$ such that $L$ admits a representation by a $2n$-plat. For further details on braid, plat and bridge presentations of links we refer to [@Bi]. ![A $2n$-plat presentation of a link.[]{data-label="Fig. 1"}](Figure1.eps) [**Remark 2**]{} A $2n$-plat presentation of a link $L\subset\E^3\subset\S^3=\E^3\cup\{\infty\}$ furnishes a $(0,n)$-decomposition [@MS] $(\S^3,L)=(D,A_n)\cup_{\f'}(D',A'_n)$ of the link, where $D$ and $D'$ are the 3-balls $D=(\E^2\times]-\infty,0])\cup\{\infty\}$ and $D'=(\E^2\times[1,+\infty[)\cup\{\infty\}$, $A_n=\a_1\cup\cdots\cup\a_n$, $A'_n=\a'_1\cup\cdots\cup\a'_n$ and $\f':\partial D\to\partial D'$ is defined by $\f'(\infty)=\infty$ and $\f'(x,0)=({\wti\b}(x),1)$, for each $x\in\E^2$. If a $2n$-plat presentation of a $\m$-component link $L=\bigcup_{j=1}^{\m}L_j$ is given, each component $L_j$ of $L$ contains $n_j$ top arcs and $n_j$ bottom arcs. Obviously, $\sum_{j=1}^{\m}n_j=n$. A $2n$-plat presentation of a link $L$ will be called [*special* ]{} if: - the top arcs and the bottom arcs belonging to $L_1$ are $\a_1,\ldots,\a_{n_1}$ and $\a'_1,\ldots,\a'_{n_1}$ respectively, the top arcs and the bottom arcs belonging to $L_2$ are $\a_{n_1+1},\ldots,\a_{n_1+n_2}$ and $\a'_{n_1+1},\ldots,\a'_{n_1+n_2}$ respectively, $\ldots$ , the top arcs and the bottom arcs belonging on $L_\m$ are $\a_{n_1+\cdots+n_{\m-1}+1},\ldots,\a_{n_1+\cdots+n_{\m}}=\a_{n}$ and $\a'_{n_1+\cdots+n_{\m-1}+1},\ldots,\a'_{n_1+\cdots+n_{\m}}=\a'_{n}$ respectively; - $p_{2i-1}(1)\in\FF_1$ and $p_{2i}(1)\in\FF_2$, for each $i=1,\ldots,n$. It is clear that, because of (2), the homeomorphism $\wti\b$, associated to a $2n$-string braid $\b$ defining a special plat presentation, keeps fixed both the sets $\FF_1$ and $\FF_2$. Although a special plat presentation of a link is a very particular case, we shall prove that every link admits such kind of presentation. \[Proposition special\] Every link $L$ admits a special $2n$-plat presentation, for each $n\ge b(L)$. Let $L$ be presented by a $2n$-plat. We show that this presentation is equivalent to a special one, by using a finite sequence of moves on the plat presentation which changes neither the link type nor the number of plats. The moves are of the four types $I$, $I'$, $II$ and $II'$ depicted in Figure 2. First of all, it is straightforward that condition (1) can be satisfied by applying a suitable sequence of moves of type $I$ and $I'$. Furthermore, condition (2) is equivalent to the following: $(2')$ there exists an orientation of $L$ such that, for each $i=1,\ldots,n$, the top arc $\a_i$ is oriented from $A_i$ to $B_i$ and the bottom arc $\a'_i$ is oriented from $B'_i$ to $A'_i$. Therefore, choose any orientation on $L$ and apply moves of type $II$ (resp. moves of type $II'$) to the top arcs (resp. bottom arcs) which are oriented from $B_i$ to $A_i$ (resp. from $A'_i$ to $B'_i$). ![Moves on plat presentations.[]{data-label="Fig. 2"}](Figure2.eps) A $p$-fold branched cyclic covering of an oriented $\m$-component link $L=\bigcup_{j=1}^{\m}L_j\subset\S^3$ is completely determined (up to equivalence) by assigning to each component $L_j$ an integer $c_j\in{\bf Z}_p-\{0\}$, such that the set $\{c_1,\ldots,c_{\m}\}$ generates the group ${\bf Z}_p$. The monodromy associated to the covering sends each meridian of $L_j$, coherently oriented with the chosen orientations of $L$ and $\S^3$, to the permutation $(1\,2\,\cdots\,p)^{c_j}\in\Si_p$. Multiplying each $c_j$ by the same invertible element of ${\bf Z}_p$, we obtain an equivalent covering. Following [@MM] we shall call a branched cyclic covering: - [*strictly-cyclic*]{} if $c_{j'}=c_{j''}$, for every $j',j''\in\{1,\ldots,\m\}$, - [*almost-strictly-cyclic*]{} if $c_{j'}=\pm c_{j''}$, for every $j',j''\in\{1,\ldots,\m\}$, - [*meridian-cyclic*]{} if $\gcd(b,c_j)=1$, for every $j\in\{1,\ldots,\m\}$, - [*singly-cyclic*]{} if $\gcd(b,c_j)=1$, for some $j\in\{1,\ldots,\m\}$, - [*monodromy-cyclic*]{} if it is cyclic. The following implications are straightforward: $$\text{ a) }\Rightarrow\text{ b) }\Rightarrow\text{ c) }\Rightarrow\text{ d) } \Rightarrow\text{ e) }.$$ Moreover, the five definitions are equivalent when $L$ is a knot. Similar definitions and properties also hold for a $p$-fold cyclic covering of a 3-ball, branched over a set of properly embedded (oriented) arcs. It is easy to see that, by a suitable reorientation of the link, an almost-strictly-cyclic covering becomes a strictly-cyclic one. As a consequence, it follows from Remark 1 that every branched cyclic covering of a link arising from a $p$-symmetric Heegaard splitting – according to Birman-Hilden construction – is strictly-cyclic. Now we show that, conversely, every $p$-fold branched strictly-cyclic covering of a link admits a $p$-symmetric Heegaard splitting. \[Theorem 3\] A $p$-fold strictly-cyclic covering of $\S^3$ branched over a link $L$ of bridge number $b$ is a closed, orientable 3-manifold $M$ which admits a $p$-symmetric Heegaard splitting of genus $g=(b-1)(p-1)$. So the $p$-symmetric Heegaard genus of $M$ is $$g_p(M)\le(b-1)(p-1).$$ Let $L$ be presented by a special $2b$-plat arising from a braid $\b$, and let $(\S^3,L)=(D,A_b)\cup_{\f'}(D',A'_b)$ be the $(0,b)$-decomposition described in Remark 2. Now, all arguments of the proofs of Theorem 3 of [@BH] entirely apply and the condition of Lemma 4 of [@BH] is satisfied, since the homeomorphism $\wti\b$ associated to $\b$ fixes both the sets $\FF_1$ and $\FF_2$. As a consequence of Theorem \[Theorem 3\] and Birman-Hilden results, there is a natural one-to-one correspondence between $p$-symmetric Heegaard splittings and $p$-fold strictly-cyclic branched coverings of links. [5]{} Birman, J.S.: Braids, links and mapping class groups. Ann. of Math. Studies, vol. 82, Princeton Univ. Press, Princeton, N. J., 1975. Birman, J.S., Hilden, H.M.: Heegaard splittings of branched coverings of $\S^3$. Trans. Am. Math. Soc. [**213**]{} (1975), 315–352. Mayberry, J.; Murasugi, K.: Torsion groups of abelian coverings of links. Trans. Amer. Math. Soc. [**271**]{} (1982), 143–173. Morgan, J.W., Bass, H.: The Smith conjecture. Academic Press, Inc., 1984. Morimoto, K., Sakuma, M.: On unknotting tunnels for knots. Math. Ann. [**289**]{} (1991), 143–167. Viro, O.Ja.: Linkings, two-sheeted branched coverings and braids. Math. USSR, Sbornik [**16**]{} (1972), 223–236. [^1]: [*An intrinsic characterization of $p$-symmetric Heegaard splittings*]{}, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 217–222, Topology Atlas, Toronto, 2002. This contribution is extracted from: M. Mulazzani, [*On $p$-symmetric Heegaard splittings*]{}, J. Knot Theory Ramifications [**9**]{} (2000), no. 8, 1059–1067. Reprinted with permission from World Scientific Publishing Co. [^2]: A set of mutually disjoint arcs $\{t_1,\ldots,t_n\}$ properly embedded in a handlebody $Y$ is [*unknotted*]{} if there is a set of mutually disjoint discs $D=\{D_1,\ldots,D_n\}$ properly embedded in $Y$ such that $t_i\cap D_i=t_i\cap\partial D_i=t_i$, $t_i\cap D_j=\emptyset$ and $\partial D_i-t_i\subset\partial Y$ for $1\le i,j\le n$ and $i\neq j$.

--- abstract: 'Motivated by the observation of several molecule candidates in the heavy quark sector, we discuss the possibility of a state with $J^{PC}=3^{-+}$. In a one-boson-exchange model investigation for the S wave $C=+$ $D^*\bar{D}_2^*$ states, one finds that the strongest attraction is in the case $J=3$ and $I=0$ for both $\pi$ and $\sigma$ exchanges. Numerical analysis indicates that this hadronic bound state may exist. If a state around the $D^*\bar{D}_2^*$ threshold ($\approx$4472 MeV) in the channel $J/\psi\omega$ (P wave) is observed, the heavy quark spin symmetry implies that it is not a $c\bar{c}$ meson and the $J^{PC}$ are very likely to be $3^{-+}$.' author: - 'W. Zhu, T. Yao, Yan-Rui Liu' title: 'Possibility of a $J^{PC}=3^{-+}$ state' --- Introduction {#sec1} ============ Mesons with exotic properties play an important role in understanding the nature of strong interactions. The observation of the so called XYZ states in the heavy quark sector has triggered lots of discussions on their quark structures, decays, and formation mechanisms. It also motivates people to study new states beyond the quark model assignments. The X(3872), first observed in the $J\psi\pi^+\pi^-$ invariant mass distribution by Belle collaboration in 2003 [@X3872-belle], is the strangest heavy quark state. Even now, its angular momentum is not determined. Since its extreme closeness to the $D^0\bar{D}^{*0}$ threshold, lots of discussions about its properties are based on the molecule assumption. However, it is very difficult to identify the X(3872) as a shallow bound state of $D^0\bar{D}^{*0}$ since there are no explicitly exotic molecule properties. A charged charmonium- or bottomonium-like meson labeled as $Z$ is absolutely exotic because its number of quarks and antiquarks must be four or more. Such states include the $Z(4430)$ observed in the $\psi'\pi^\pm$ mass distribution [@Z4430-belle], the $Z_1(4050)$ and $Z_2(4250)$ observed in the $\chi_{c1}\pi^+$ mass distribution [@Z1Z2-belle], and the $Z_b(10610)$ and $Z_b(10650)$ in the mass spectra of the $\Upsilon(nS)\pi^\pm$ ($n$=1,2,3) and $\pi^\pm h_b(mP)$ ($m$=1,2) [@Zb-belle]. They are all observed by Belle Collaboration. Though BABAR has not confirmed them [@Z4430-babar; @Z1Z2-babar], the existence signal of multiquark states is still exciting. Since $Z(4430)$ is around the $D^*D_1$ threshold, $Z_b(10610)$ is around the $BB^*$ threshold, and $Z_b(10650)$ is around the $B^*B^*$ threshold, molecular models seem to be applicable to their structure investigations [@LLDZ08-4430mole; @DingHLY09; @SunHLLZ; @ZhangZH11; @OhkodaYYSH; @YangPDZ12; @LiWDZ13]. To identify a state as a molecule is an important issue in hadron studies. One should consider not only bound state problem of two hadrons, but also how to observe a molecular state in possible production processes. In Refs. [@Nstar-dyn; @Nstar-chiqm; @Nstar-obe; @Nstar-cc], bound states of $\Sigma_c\bar{D}$ and $\Sigma_c\bar{D}^*$ were studied. Since the quantum numbers are the same as the nucleon but the masses are much higher, identifying them as multiquark baryons is rather apparent. To obtain a deeper understanding of the strong interaction, it is necessary to explore possible molecules with explicitly exotic quantum numbers. Quark model gives us a constraint on the quantum numbers of a meson, namely, a meson with $J^{PC}=0^{--}$, $0^{+-}$, $1^{-+}$, $2^{+-}$, $3^{-+}$, $\cdots$ could not be a $q\bar{q}$ state, but it may be a multiquark state. So the study on such states may deepen our understanding of nature. If two $q\bar{q}$ mesons can form a molecule with such quantum numbers, one gets the simplest configuration. Next simpler configuration is the baryon-antibaryon case. A possible place to search for them is around hadron-hadron thresholds. There are some discussions on low spin heavy quark exotic states in Refs. [@ShenCLHZYL10; @HuCLHZYL11]. Here we would like to discuss the possibility of a higher spin state, $J^{PC}=3^{-+}$. One will see that identification of it from strong decay is possible. First, we check meson-antimeson systems that can form $3^{-+}$ states, where meson (antimeson) means that its quark structure is $c\bar{q}$ ($\bar{c}q$). The established mesons may be found in the Particle Data Book [@PDG]. One checks various combinations and finds that the lowest S-wave system is $D^*\bar{D}_2^*$. The next S-wave one is $D_s^*D_{s2}^*$. Between these two thresholds, one needs $D$ or $G$ wave to combine other meson-antimeson pairs (see Fig. \[th3-\]). Below the threshold of $D^*\bar{D}_2^*$, the orbital angular momentum is $D$, $F$, or $G$-wave. Above the $D_s^*D_{s2}^*$ threshold, a partial wave of $P$, $F$, or $H$ is needed. Since the difference between these two thresholds is more than 200 MeV, one may neglect the channel coupling and choose the $D^*D_2^*$ system to study. ![Thresholds of $J^{PC}=3^{-+}$ meson-antimeson systems between that of $D^*D_2^*$ and that of $D_s^*D_{s2}^*$. $S$, $D$, $G$, and $I$ are orbital angular momenta.[]{data-label="th3-"}](th3-) Secondly, we check baryon-antibaryon systems. If one combines the established $cqq$ baryons and their antibaryons, one finds that the lowest S-wave threshold is for $\Lambda_c(2880)$ and $\bar{\Lambda}_c$ ($\approx5168$ MeV). Even for the lowest threshold of $\Lambda_c(2595)$ and $\bar{\Lambda}_c$ in F-wave, the value ($\approx4879$ MeV) is still higher than that of $D_s^*D_{s2}^*$. Thus, we may safely ignore the possible baryon-antibaryon contributions in this study. In a $3^{-+}$ $D^*\bar{D}_2^*$ state, partial waves of $S$, $D$, $G$, and $I$ may all contribute. As a first step exploration, we consider only the dominant S-wave interactions. Possible coupled channel effects will be deferred to future works. The present study is organized as follows. After the introduction in Sec. \[sec1\], we present the main ingredients for our study in Sec. \[sec2\]. Then we give the numerical results in Sec. \[sec3\]. The final part is for discussions and conclusions. Wavefunctions, amplitudes, and Lagrangian {#sec2} ========================================= We study the meson-antimeson bound state problem in a meson exchange model. The potential is derived from the scattering amplitudes [@LLDZ08-4430wave] and the flavor wave functions of the system are necessary. Since the states we are discussing have a definite C-parity while the combination of $c\bar{q}$ and $\bar{c}q$ mesons does not, a relative sign problem arises between the two parts of a flavor wave function. One has to find the relation between the flavor wave function and the potential with definite C-parity. There are some discussions about this problem in the literatures [@LLDZ08-4430wave; @LLDZ08-3872wave; @ThomasC08; @Stancu08-wave; @LiuZ09]. Here we revisit it by using the G-parity transformation rule which relates the amplitudes between $NN$ and $N\bar{N}$ [@KlemptBMR]. Since $D$ mesons do not have defined C-parity, we may assume arbitrary complex phases $\alpha$ and $\beta$ under the C-parity transformations $$\begin{aligned} &\bar{D}^{*0}\leftrightarrow \alpha_2 D^{*0},\qquad D^{*-}\leftrightarrow \alpha_1D^{*+},&\nonumber\\ &\bar{D}^{*0}_2\leftrightarrow\beta_2 D^{*0}_2,\qquad D^{*-}_2\leftrightarrow \beta_1D^{*+}_2.&\end{aligned}$$ According to the SU(2) transformation, one finds the following isospin doublets $$\begin{aligned} \left(\begin{array}{c}\bar{D}^{*0}\\D^{*-}\end{array}\right),\quad \left(\begin{array}{c}\alpha_1D^{*+}\\-\alpha_2 D^{*0}\end{array}\right),\quad \left(\begin{array}{c}\bar{D}^{*0}_2\\D^{*-}_2\end{array}\right),\quad \left(\begin{array}{c}\beta_1D^{*+}_2\\-\beta_2 D^{*0}_2\end{array}\right),\end{aligned}$$ from which the G-parity transformations read $$\begin{aligned} \label{G-rule} &\left(\begin{array}{c}\bar{D}^{*0}\\D^{*-}\end{array}\right)\rightarrow \left(\begin{array}{c}\alpha_1D^{*+}\\-\alpha_2 D^{*0}\end{array}\right)\rightarrow\left(\begin{array}{c}-\bar{D}^{*0}\\-D^{*-}\end{array}\right),&\nonumber\\ &\left(\begin{array}{c}\bar{D}^{*0}_2\\D^{*-}_2\end{array}\right)\rightarrow \left(\begin{array}{c}\beta_1D^{*+}_2\\-\beta _2 D^{*0}_2\end{array}\right)\rightarrow\left(\begin{array}{c}-\bar{D}^{*0}_2\\-D^{*-}_2\end{array}\right).&\end{aligned}$$ Similar to the study of the $D^*\bar{D}_1$ bound state problem [@LLDZ08-4430wave], one may construct several states from $D^*$ and $\bar{D}_2^*$. Here we concentrate only on the $C=+$ case. If the system is an isovector (isoscalar), we label it $Z_J$ ($X_J$) where $J$ is the angular momentum. Explicitly, one has the G-parity eigenstates $$\begin{aligned} Z_J^0&=&\frac{1}{2\sqrt2}\Big[(D^{*-}D_2^{*+}+D_2^{*+}D^{*-})-\beta_1^\dag\beta_2(\bar{D}^{*0}D_2^{*0}+D_2^{*0}\bar{D}^{*0})\nonumber\\ &&+c\alpha_1\beta_1^\dag (D_2^{*-}D^{*+}+D^{*+}D_2^{*-})-c\alpha_2\beta_1^\dag(\bar{D}_2^{*0}D^{*0}+D^{*0}\bar{D}_2^{*0})\Big],\nonumber\\ X_J^0&=&\frac{1}{2\sqrt2}\Big[(D^{*-}D_2^{*+}+D_2^{*+}D^{*-})+\beta_1^\dag\beta_2(\bar{D}^{*0}D_2^{*0}+D_2^{*0}\bar{D}^{*0})\nonumber\\ &&+c\alpha_1\beta_1^\dag (D_2^{*-}D^{*+}+D^{*+}D_2^{*-})+c\alpha_2\beta_1^\dag(\bar{D}_2^{*0}D^{*0}+D^{*0}\bar{D}_2^{*0})\Big],\end{aligned}$$ where $c=1$ is the C-parity and the superscript indicates the electric charge. One may check $$\begin{aligned} \hat{G}Z_J^0=-cZ_J^0,\quad \hat{C}X_J^0=cX_J^0.\end{aligned}$$ The procedure to derive the potential is similar to that in Ref. [@LLDZ08-4430wave]. Now we calculate the amplitude $T(Z_J^0)=\langle Z_J^0|\hat{T}|Z_J^0\rangle$ with the G-parity transformation rule (\[G-rule\]). We just present several terms to illustrate the derivation. Together with the above $Z_J^0$ wave function, one has $$\begin{aligned} T(Z_J^0)&=&\frac14\left\{ T_{[D_2^{*+}\to D_2^{*+}, D^{*-}\to D^{*-}]}-\beta_1\beta_2^\dag T_{[D_2^{*+}\to D_2^{*0}, D^{*-}\to \bar{D}^{*0} ]}+c\alpha_1^\dag \beta_1T_{[D_2^{*+}\to D^{*+}, D^{*-}\to D_2^{*-}]}+\cdots\right\}\nonumber\\ &=&\frac{G^\pi}{4}\left\{ T_{[D_2^{*+}\to D_2^{*+}, D^{*0}\to D^{*0}]}+\alpha_1^\dag\alpha_2\beta_1\beta_2^\dag T_{[D_2^{*+}\to D_2^{*0}, D^{*0}\to {D}^{*+}]}+c\alpha_1^\dag\alpha_2\beta_1\beta_2^\dag T_{[D_2^{*+}\to D^{*+}, D^{*0}\to D_2^{*0}]}+\cdots\right\}.\end{aligned}$$ In fact, the convention $\alpha_1\alpha_2^\dag=\beta_1\beta_2^\dag$ is implied in the Lagrangian in Eq. (\[Lag-pi\]). So $\alpha_1\alpha_2^\dag\beta_1^\dag\beta_2=\alpha_1^\dag\alpha_2\beta_1\beta_2^\dag=1$ and one finally gets $$\begin{aligned} \label{T-matrices} T(Z_J^0)&=&\frac12G^\pi\left\{ T_{[D_2^{*+}\to D_2^{*+}, D^{*0}\to D^{*0}]}+ T_{[D_2^{*+}\to D_2^{*0}, D^{*0}\to {D}^{*+}]}+T_{[D_2^{*0}\to D_2^{*+},{D}^{*+}\to D^{*0}]}+T_{[D_2^{*0}\to D_2^{*0},{D}^{*+}\to {D}^{*+}]}\right.\nonumber\\ &&\left.+c T_{[D_2^{*+}\to D^{*+}, D^{*0}\to D_2^{*0}]}+c T_{[D_2^{*+}\to D^{*0}, D^{*0}\to{D}_2^{*+} ]}+c T_{[D_2^{*0}\to D^{*+},{D}^{*+}\to D_2^{*0}]}+c T_{[D_2^{*0}\to D^{*0},{D}^{*+}\to{D}_2^{*+}]}\right\}, \nonumber\\ T(X_J^0)&=&\frac12G^\pi\left\{ T_{[D_2^{*+}\to D_2^{*+}, D^{*0}\to D^{*0}]}- T_{[D_2^{*+}\to D_2^{*0}, D^{*0}\to {D}^{*+} ]}-T_{[D_2^{*0}\to D_2^{*+},{D}^{*+}\to D^{*0}]}+T_{[D_2^{*0}\to D_2^{*0},{D}^{*+}\to {D}^{*+}]}\right.\nonumber\\ &&\left.+c T_{[D_2^{*+}\to D^{*+}, D^{*0}\to D_2^{*0}]}-c T_{[D_2^{*+}\to D^{*0}, D^{*0}\to{D}_2^{*+} ]} -cT_{[D_2^{*0}\to D^{*+},{D}^{*+}\to D_2^{*0}]}+cT_{[D_2^{*0}\to D^{*0},{D}^{*+}\to{D}_2^{*+}]}\right\}.\end{aligned}$$ It is obvious that we may calculate the potential of meson-antimeson interaction from that of meson-meson together with a given Lagrangian for ($c\bar{q}$) meson fields. The arbitrary relative phase in the flavor wave function of a meson-antimeson system is canceled in this procedure. To derive the explicit expression of the potential, one needs interaction Lagrangian. The Lagrangian for pion interactions in the heavy quark limit and chiral limit reads [@hchi-coupling; @FalkL92] $$\begin{aligned} \label{Lag-pi} \mathcal{L}_\pi&=&g {\rm Tr}[H {A}\!\!\!\slash\gamma_5\bar{H}] +g''{\rm Tr}[T_{\mu}A\!\!\!\slash\gamma_5\bar{T}^{\mu}]\nonumber\\ &&+\{\frac{h_1}{\Lambda_{\chi}}{\rm Tr}[T^{\mu}(D_{\mu}{A}\!\!\!\slash)\gamma_5\bar{H}]+h.c.\} +\{\frac{h_2}{\Lambda_{\chi}}{\rm Tr}[T^{\mu}(D\!\!\!\!/A_{\mu})\gamma_5\bar{H}]+h.c.\},\end{aligned}$$ where $$\begin{aligned} H&=&\frac{1+v\!\!\!/}{2 }[P^{*\mu}\gamma_\mu+P \gamma_5],\quad \bar{H}=\gamma^0H^\dag\gamma^0\nonumber\\ T^{\mu}&=&\frac{1+v\!\!\!/}{2}\Big\{P^{*\mu\nu}_{2} \gamma_{\nu}+\sqrt{\frac{3}{2}}P_{1}^{\nu}\gamma_5 [g_{\nu}^{\mu}-\frac{1}{3}\gamma_{\nu}(\gamma^{\mu}-v^{\mu})]\Big\},\quad \bar{T}^{\mu}=\gamma^0T^\dag\gamma^0.\end{aligned}$$ The fields $P^*=(D^{*0},D^{*+})$, and $P_2^*=(D_2^{*0},D_2^{*+})$ annihilate the $c\bar{q}$ mesons. $P$ and $P_1$ have similar form but we do not involve them in the following calculation. The axial vector field $A^{\mu}$ is defined as $A^{\mu}=\frac{i}{2}(\xi^{\dag}\partial^{\mu}\xi-\xi\partial^{\mu}\xi^{\dag})$ with $\xi=\exp(i\mathcal{M}/f)$, $f=132$ MeV and $$\begin{aligned} \mathcal{M}&=& \left(\begin{array}{cc} \frac{\pi^{0}}{\sqrt{2}}&\pi^{+}\\ \pi^{-}&-\frac{\pi^{0}}{\sqrt{2}} \end{array}\right).\end{aligned}$$ If we further consider $\sigma$ exchange, one needs additional interaction terms $$\begin{aligned} \label{Lag-sig} {\cal L}_\sigma&=&g_\sigma{\rm Tr}[H\sigma\bar{H}]+g_\sigma''{\rm Tr}[T^\mu\sigma\bar{T}_\mu]+\frac{h_\sigma'}{f_\pi}{\rm Tr}[T^\mu(\partial_\mu\sigma)\bar{H}+H(\partial_\mu\sigma)\bar{T}^\mu].\end{aligned}$$ The coupling constants must be determined in order for numerical analysis. One extracts the pion coupling constant $g$ from the decay $D^*\to D\pi$: $g=0.59\pm0.07\pm0.01$ [@g-coupling]. For $h_\chi=\frac{h_1+h_2}{\Lambda_\chi}$, we use the value $0.55\text{ GeV}^{-1}$ estimated in Ref. [@hchi-coupling]. To determine the coupling constant $g''$, we turn to the chiral quark model [@ZhangYSDFS97] with which one may get the relation $g''=-g$. For the sigma coupling constants, we can just get estimates from the chiral quark model or the chiral multiplet assumption [@BardeenEH03]. These approaches have been used in the baryon case [@LiuO12] for the purpose of cross checking, where we get consistent results. Now the former method may give the relation $g_\sigma''=-g_\sigma$ and the value $g_\sigma=g_{ch}=2.621$ if one adopts the Lagrangian [@ZhangYSDFS97] $$\begin{aligned} L_I&=&-g_{ch}\bar{\psi}(\sigma+i\gamma_5\pi_a\tau_a)\psi,\end{aligned}$$ where $\psi=(u,d)^T$ is the quark field and $\tau_a$ the Pauli matrix. One should note the normalization problem in this approach [@YanCCLLY92; @FalkL92]. However, if one estimates $g_\sigma$ from the chiral multiple assumption, a value less than 1 is obtained. For the remaining $h_\sigma'$, no available approach may be used. Since the large uncertainties of the coupling constants, we will select several values to see the $\sigma$-exchange effects on the conclusions. In deriving the above relations for the coupling constants, we have used the polarization vectors $\varepsilon_{\pm1}^\mu=\frac{1}{\sqrt2}(0,\pm1,i,0)$ and $\varepsilon_{0}^\mu=(0,0,0,-1)$ for the vector meson $D^*$ and $$\begin{aligned} \varepsilon_{\pm2}^{\mu\nu}&=&\varepsilon_{\pm1}^\mu\varepsilon_{\pm1}^\nu,\nonumber\\ \varepsilon_{\pm1}^{\mu\nu}&=&\sqrt{\frac12}[\varepsilon_{\pm1}^\mu\varepsilon_0^\nu+\varepsilon_0^\mu\varepsilon_{\pm1}^\nu],\nonumber\\ \varepsilon_0^{\mu\nu}&=&\sqrt{\frac16}[\varepsilon_{+1}^\mu\varepsilon_{-1}^\nu+\varepsilon_{-1}^\mu\varepsilon_{+1}^\nu+2\varepsilon_0^\mu\varepsilon_0^\nu],\end{aligned}$$ for the tensor meson $D_2^*$ [@ChengY11]. Potentials and numerical analysis {#sec3} ================================= Now one may derive the potentials through the amplitudes in (\[T-matrices\]). Using the same procedure as Ref. [@LLDZ08-4430wave], one gets the one-pion-exchange potential (OPEP) for S-wave interaction in the case $I=1$ $$\begin{aligned} V^\pi(Z_J)&=&-\frac{gg''}{6f^2}G^\pi C_d\Big[\delta(\vec{r})-\frac{m_\pi^2e^{-m_\pi r}}{4\pi r}\Big] +\frac{|h_\chi|^2}{15f^2}cG^\pi C_e\Big[\nabla^2\delta(\vec{r})-\mu^2\delta(\vec{r})-\frac{\mu^4}{4\pi r}\cos(\mu r)\Big],\end{aligned}$$ where $\mu=\sqrt{(m_{D_2}-m_{D^*})^2-m_\pi^2}$, and $$\begin{aligned} C_d&=&\left\{\begin{array}{rl} -1,&(J=3)\\ \frac{1}{2},&(J=2)\\ \frac{3}{2},&(J=1) \end{array}\right., \quad C_e=\left\{\begin{array}{rl} \frac{1}{2},&(J=3)\\ -\frac{5}{4},&(J=2)\\ -\frac{3}{4},&(J=1) \end{array}\right..\end{aligned}$$ There are two parts in the potential: direct part and spin-exchange part. The later corresponds to the terms containing $c$ in Eq. (\[T-matrices\]). For the case of $I=0$, $V^\pi(X_J)=-3V^\pi(Z_J)$. The singular behavior at small distances needs to be regularized [@Tornqvist94]. If a form factor $FF=\left(\frac{\Lambda^2-m^2}{\Lambda^2-q^2}\right)^2$ is added to each vertex, one finally has $$\begin{aligned} V^\pi(Z_J)&=&-\frac{gg''}{6f^2}G^\pi C_d \Big[-\frac{m_\pi ^2}{4\pi r}(e^{-m_\pi r}-e^{-\Lambda r})+\frac{m_\pi ^2\eta^2}{8\pi\Lambda}e^{-\Lambda r}\nonumber\\ &&+\frac{m_\pi^2\eta^4}{32\pi\Lambda^3}(1+\Lambda r)e^{-\Lambda r} +\frac{\eta^6}{192\pi\Lambda^3}(3+3\Lambda r+\Lambda^2r^2)e^{-\Lambda r}\Big]\nonumber\\ &&+\frac{|h_\chi|^2}{15 f^2}cG^\pi C_e\Big\{-\frac{\mu^4}{4\pi r}(\cos(\mu r)-e^{-\alpha r}) +\frac{\mu^4\eta^2}{8\pi\alpha}e^{-\alpha r}\nonumber\\ &&-\frac{\mu^2\eta^4}{32\pi\alpha}(1+\alpha r) e^{-\alpha r}-\frac{\eta^6}{192\pi\alpha}(3+3\alpha r-\alpha^2r^2)e^{-\alpha r}\Big\},\end{aligned}$$ where $\eta=\sqrt{\Lambda^2-m_\pi^2}$, and $\alpha=\sqrt{\Lambda^2-(m_{D_2}-m_{D^*})^2}$. Similarly, the one-$\sigma$-exchange potential (OsEP) is $$\begin{aligned} V^\sigma(Z_J)&=&g_\sigma g_\sigma''\Big[\frac{1}{4\pi r}(e^{-m_\sigma r}-e^{-\Lambda r})-\frac{\eta_\sigma^2}{8\pi \Lambda}e^{-\Lambda r} -\frac{\eta_\sigma^4}{32\pi\Lambda^3}(1+\Lambda r)e^{-\Lambda r}\nonumber\\ &&-\frac{\eta_\sigma^6}{192\pi\Lambda^5}(3+3\Lambda r+\Lambda^2r^2)e^{-\Lambda r}\Big]\nonumber\\ &&+\frac{|h_\sigma'|^2}{3f_\pi^2}C_\sigma\Big[\frac{\mu_\sigma^2}{4\pi r}(e^{-\mu_\sigma r}-e^{-\alpha r})-\frac{\mu_\sigma^2\eta_\sigma^2}{8\pi\alpha}e^{-\alpha r} -\frac{\mu_\sigma^2\eta_\sigma^4}{32\pi\alpha^3}(1+\alpha r)e^{-\alpha r}\nonumber\\ &&-\frac{\eta_\sigma^6}{192\pi\alpha^3}(3+3\alpha r+\alpha^2r^2)e^{-\alpha r}\Big],\nonumber\\ V^\sigma(X_J)&=&V^\sigma(Z_J),\end{aligned}$$ where $\mu_\sigma=\sqrt{m_\sigma^2-(m_{D_2}-m_{D*})^2}$, $\eta_\sigma=\sqrt{\Lambda^2-m_\sigma^2}$. The coefficient $C_\sigma=1$ for $J=3$, $\frac12$ for $J=2$, and $\frac16$ for $J=1$. The spin-dependent nature of OsEP comes from the third coupling term in the Lagrangian (\[Lag-sig\]). Before the numerical calculation, we take a look at the relative strengthes of potentials. For the meson masses, we use $m_\pi=137.27$ MeV, $m_{D*}=2008.63$ MeV, and $m_{D_2}=2463.5$ MeV [@PDG]. We plot OPEPs with $\Lambda=1$ GeV in Fig. \[potpi\]. It is obvious that $X_3$ is the most attractive case. ----- ----- (a) (b) ----- ----- For the $\sigma$ meson exchange contribution, we use $m_\sigma=600$ MeV for the illustration. In Fig. \[potsig\], we show OsEPs with $g_\sigma''=-g_\sigma=-1.0$, $h_\sigma'=1.0$, and $\Lambda=1$ GeV. It is interesting that the potential for $X_3$ is also the most attractive one. Thus the long-range and medium-range meson-exchanges are both helpful for the formation of a $I^G(J^{PC})=0^-(3^{-+})$ state. [c]{} Now we turn to the numerical results for the OPEP case by solving the Schrödinger equation. In the potential, there is an unknown phenomenological cutoff parameter $\Lambda$. It incorporates the size information of the interacting mesons. If $\Lambda$ goes to infinity, the potential describes the interactions of structureless mesons. A small cutoff is relevant to the real case. In principle, an appropriate value should be around 1 GeV which is realized from the nuclear potential models. Since we do not want to give accurate prediction on the binding energy, this parameter is not fixed. We just tune its value in a range and check whether a bound state exists or not. The results of binding energy (B.E.) and root-mean-square radius ($r_{rms}$) for $X_3$ with various $\Lambda$ are presented in Tab. \[BEpi\], where the cases for $r_{rms}<$0.8 fm or $\Lambda>$4 GeV are neglected. Similarly, one may get numerical results for other possibilities, which are also given in that table. The resultant cutoff much larger than 1 GeV indicates that the attraction is not strong enough for the formation of a hadronic bound state. Comparing the three cases in the table, of course $X_3$ is more likely to be existent. ------- ----------------- ------------ ---------------- State $\Lambda$ (GeV) B.E. (MeV) $r_{rms}$ (fm) $X_3$ 2.3 0.6 3.8 2.4 3.7 1.6 2.5 9.9 1.0 2.6 19.8 0.8 $Z_2$ 3.8 2.5 1.9 $Z_1$ 3.6 1.9 2.2 3.7 8.4 1.1 ------- ----------------- ------------ ---------------- : Cutoff ($\Lambda$), binding energy (B.E.) and root-mean-square radius ($r_{rms}$) for $X$ and $Z$ states with OPEP. We do not show results for $\Lambda>4$ GeV or $r_{rms}<0.8$ fm.[]{data-label="BEpi"} The minimal cutoff for a binding solution is a little larger than 2 GeV if we consider only $\pi$-exchange. This means that additional attraction may lower the value to a more appropriate number. We would like to check how much attraction the sigma meson contributes. Because of the large uncertainty of the coupling constants, we take three sets of them: (1) $g_\sigma$=2.621, $g_\sigma''=-g_\sigma$, $h_\sigma'=0$, which corresponds to neglecting spin-exchange potential; (2) $g_\sigma=1.0$, $g_\sigma''=-g_\sigma$, $h_\sigma'=1$; and (3) $g_\sigma=2.621$, $g_\sigma''=-g_\sigma$, $h_\sigma'=2.621$. The last set is the most attractive case. After the solution of the Schrödinger equation, the cutoff parameters satisfying the condition $r_{rms}>$0.8 fm and $\Lambda<$4 GeV are summarized in Tab. \[Cuts-ps\]. From the resultant cutoff parameters, we see that the existence of $X_3$ is probable. Its mass should be around the $D^*D_2^*$ threshold ($\approx4472$ MeV). -------- -------------- -------------- -------------- States Set 1 Set 2 Set 3 $X_3$ 1.7$\sim$2.2 1.5$\sim$1.7 1.0$\sim$1.1 $X_2$ 1.4$\sim$1.5 $X_1$ $Z_3$ 2.7$\sim$3.0 1.1$\sim$1.2 $Z_2$ 2.6$\sim$3.2 2.5$\sim$2.7 1.2$\sim$1.3 $Z_1$ 2.2$\sim$2.9 2.8$\sim$3.1 1.5$\sim$1.7 -------- -------------- -------------- -------------- : Cutoff values (GeV) for $X$ and $Z$ states with OPEP+OsEP when binding solutions exist. We do not show cutoffs if $\Lambda>4$ GeV or $r_{rms}<0.8$ fm.[]{data-label="Cuts-ps"} Discussions and conclusions {#sec4} =========================== From the meson exchange potentials and the numerical analysis, one has found that the most probable molecule in the $C=+$ $D^*\bar{D}_2^*$ system is $X_3$. If the state really exists, it may decay through its components, i.e. $D^*\to D\pi$, $D_2^*\to D\pi$, or $D_2^*\to D^*\pi$. The $X_3$ may also decay through the quark rearrangement, i.e. the final states are a $c\bar{c}$ meson and a $q\bar{q}$ ($q=u,d$) meson. The later type decay may be used to identify the exotic quantum numbers. Here we focus only on this case. For convenience of discussion, we assume that $L$ is the relative orbital momentum between the $c\bar{c}$ and the $q\bar{q}$ mesons and relax the isospin requirement temporarily. Since the spins of the charm quark and the light quark in both $D^*$ and $D_2^*$ are parallel, the spin of $c\bar{c}$ in $X_3$ must be 1. According to the heavy quark spin symmetry, the spin of the final charmonium after rearrangement should also be $S=1$. Thus the final $c\bar{c}$ state can only be $\psi$ or $\chi_{cJ}$. The decay channels are obtained as follows: \(1) If the final $c\bar{c}$ is $J/\psi$, the $J^{PC}$ of the produced $q\bar{q}$ meson may be $(1\sim5)^{--}$ for $L=1$, $(1,3,5)^{+-}$ for $L=2$, $(1\sim7)^{--}$ for $L=3$, and so on. After some inspections on the meson masses, one finds that kinematically allowed decays for the $X_3$ molecule are just $J/\psi\rho$ and $J/\psi\omega$ with $L=1,3,5$, and $J/\psi h_1(1170)$ and $J/\psi b_1(1235)$ with $L=2,4$. If it is $\psi(2S)$, the kinematically allowed decays are $\psi(2S)\rho$ and $\psi(2S)\omega$ with $L=1,3,5$. \(2) If the $c\bar{c}$ is $\chi_{c0}$, the $J^{PC}$ of the $q\bar{q}$ meson may be $(2\sim4)^{++}$ for $L=1$, $(2,4)^{-+}$ for $L=2$, $(0\sim6)^{++}$ for $L=3$, and so on. The kinematically allowed decays are $\chi_{c0}f_0(500)$, $\chi_{c0}f_0(980)$, and $\chi_{c0}a_0(980)$ with $L=3$. \(3) If the $c\bar{c}$ is $\chi_{c1}$, the $J^{PC}$ of the $q\bar{q}$ meson may be $(2,4)^{-+}$ for $L=0$, $(1\sim5)^{++}$ for $L=1$, $(0,2,4,6)^{-+}$ for $L=2$, $(0\sim7)^{++}$ for $L=3$, and so on. The kinematically allowed decays are $\chi_{c1}\pi$, $\chi_{c1}\eta$, $\chi_{c1}\eta'$ with $L=2,4$, and $\chi_{c1}f_0(500)$ with $L=3$. \(4) If the $c\bar{c}$ is $\chi_{c2}$, the $J^{PC}$ of the $q\bar{q}$ meson may be $(2,4)^{-+}$ for $L=0$, $(0\sim6)^{++}$ for $L=1$, $(0,2,4,6)^{-+}$ for $L=2$, $(0\sim8)^{++}$ for $L=3$, and so on. The kinematically allowed decays are $\chi_{c2}f_0(500)$ with $L=1,3,5$, and $\chi_{c2}\pi$ and $\chi_{c2}\eta$ with $L=2,4$. Therefore, the allowed two-body strong decays for $X_3$ are $J/\psi\omega$ (PFH), $\psi(2S)\omega$ (PFH), $J/\psi h_1(1170)$ (DG), $\chi_{c0}f_0(500)$ (F), $\chi_{c0}f_0(980)$ (F), $\chi_{c1}\eta$ (DG), $\chi_{c1}\eta'$ (DG), $\chi_{c1}f_0(500)$ (F), $\chi_{c2}f_0(500)$ (PFH), and $\chi_{c2}\eta$ (DG). There is no S-wave decay. Because high $L$ processes are suppressed and $\psi(2S)$ and $\chi_{c2}$ are excited states, the simplest way to identify $X_3$ may be through the $J/\psi\omega$ channel. Let us analyze the $J^{PC}$ of an assumed state $X(4472)$ observed in the $J/\psi\omega$ mass distribution. Since $J/\psi$ and $\omega$ are both $J^{PC}=1^{--}$ mesons, the quantum numbers of $J/\psi\omega$ are $(0,1,2)^{++}$ for S-wave combination, $(0\sim3)^{-+}$ for P-wave combination, $(0\sim4)^{++}$ for D-wave combination, and so on. What we are interested in is the case that the partial wave is determined to be $P$. If $X$ were a conventional $c\bar{c}$ meson, the state is $\eta_c(4472)$ and the spin of $c\bar{c}$ must be 0. Because of the heavy quark spin symmetry, the decay $\eta_c(4472)\to J/\psi\omega$ is suppressed. Then $X(4472)$ could be a hadronic state. Although other meson-antimeson pairs may also form molecules with $J^{PC}=(0\sim2)^{-+}$, the masses are smaller. Therefore, based on our numerical analysis, this $X$ around the $D^*\bar{D}_2^*$ threshold is very likely to be a state with the exotic $J^{PC}=3^{-+}$. If one wants to look for $Z_3$, one can use those kinematically allowed decay channels, $J/\psi\rho$ (PFH), $\psi(2S)\rho$ (PFH), $J/\psi b_1(1235)$ (DG), $\chi_{c0}a_0(980)$ (F), $\chi_{c1}\pi$ (DG), and $\chi_{c2}\pi$ (DG). The practical way to identify the $J^{PC}$ is to analyze the partial wave of $J/\psi\rho$. The search is also helpful to test the meson exchange models. We consider only S-wave interactions of $D^*$ and $\bar{D}_2^*$ in this paper. Higher partial waves also have contributions to the $3^{-+}$ molecular bound state. Additional attraction from the channel coupling may reinforce the present preliminary result. Future studies on such effects and production cross section may be helpful for the search on exotic states. Replacing a $c$ quark with a $b$ quark, one may study the bottom case. Because the production of a hidden bottom molecule $B^*\bar{B}_2^*$ needs much higher energy and the production cross section is smaller, it is difficult for experimentalists to explore this case in near future. However, with the replacement $c\to s$, one may study whether there is a bound state or resonance with $J^{PC}=3^{-+}$ near the $K^*K_2(\approx 2322 \text{ MeV})$ threshold. If such a state exists, it may decay into $\omega\phi$ and can be detected. In short summary, we have investigated whether hadronic bound states exist in the $D^*\bar{D}_2^*$ system in a one-boson-exchange model. The $C=+$ case is discussed in this paper. We find that the $I^G(J^{PC})=0^+(3^{-+})$ $X_3$ state is the most probable one. A feasible place to identify it may be in the invariant mass distribution of $J/\psi\omega$ around 4472 MeV. A similar study for a state around 2322 MeV in the $\omega\phi$ mass distribution is also called for. 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--- abstract: 'In textual information extraction and other sequence labeling tasks it is now common to use recurrent neural networks (such as LSTM) to form rich embedded representations of long-term input co-occurrence patterns. Representation of output co-occurrence patterns is typically limited to a hand-designed graphical model, such as a linear-chain CRF representing short-term Markov dependencies among successive labels. This paper presents a method that learns embedded representations of latent output structure in sequence data. Our model takes the form of a finite-state machine with a large number of latent states per label (a latent variable CRF), where the state-transition matrix is factorized—effectively forming an embedded representation of state-transitions capable of enforcing long-term label dependencies, while supporting exact Viterbi inference over output labels. We demonstrate accuracy improvements and interpretable latent structure in a synthetic but complex task based on CoNLL named entity recognition.' bibliography: - 'example\_paper.bib' --- Introduction {#sec:intro} ============ Neural networks have long been used for prediction tasks involving complex structured outputs [@lecun2006tutorial; @collobert11; @DBLP:journals/corr/LampleBSKD16]. In structured prediction, output variables obey local and global constraints that are difficult to satisify using purely local feedforward prediction from an input representation. For example, in sequence tagging tasks such as named entity recognition, the outputs must obey several hard constraints e.g., I-PER cannot follow B-ORG. The results of [@collobert11] show a significant improvement when such structural output constraints are enforced by incorporating a linear-chain graphical model that captures the interactions between adjacent output variables. The addition of a graphical model to enforce output consistency is now common practice in deep structured prediction models for tasks such as sequence tagging [@DBLP:journals/corr/LampleBSKD16] and image segmentation [@chen2014semantic]. From a probabilistic perspective, the potentials of a probabilistic graphical model over the output variables $y$ are often parameterized using a deep neural network that learns global features of the input $x$ [@lecun2006tutorial; @collobert11; @DBLP:journals/corr/LampleBSKD16]. This approach takes advantage of deep architectures to learn robust feature representations for $x$, but is limited to relatively simple pre-existing graphical model structures to model the interactions among $y$. This paper presents work in which feature learning is used not only to learn rich representations of inputs, but also to learn latent output structure. We present a model for sequence tagging that takes the form of a latent-variable conditional random field [@quattoni07; @sutton07; @morency07], where interactions in the latent state space are parametrized by low-rank embeddings. This low-rank structure allows us to use a larger number of latent states learning rich and interpretable substructures in the output space without overfitting. Additionally, unlike LSTMs, the model permits exact MAP and marginal inference via the Viterbi and forward-backward algorithms. Because the model learns large numbers of latent hidden states, interactions among $y$ are not limited to simple Markov dependencies among labels as in most deep learning approaches to sequence tagging. Previous work on representation learning for structured outputs has taken several forms. Output-embedding models such as [@srikumar2014learning] have focused on learning low-rank similarity among label vectors $y$, with no additional latent structure. The input-output HMM [@bengio95] incorporates learned latent variables, parameterized by a neural network, but the lack of low-rank structure limits the size of the latent space. Structured prediction energy networks [@belanger2016structured] use deep neural networks to learn global output representations, but do not allow for exact inference and are difficult to apply in cases when the number of outputs varies independently of the number of inputs, such as entity extraction systems. In this preliminary work, we demonstrate the utility of learning a large embedded latent output space on a synthetic task based on CoNLL named entity recognition (NER). We consider the task synthetic because we employ input features involving only single tokens, which allows us to better examine the effects of both learned latent output variables and low-rank embedding structure. (The use of NER data is preferable, however, to completely synthetically generated data because its real-world text naturally contains easily interpretable complex latent structure.) We demonstrate significant accuracy gains from low-rank embeddings of large numbers of latent variables in output space, and explore the interpretable latent structure learned by the model. These results show promise for future application of low-rank latent embeddings to sequence modeling tasks involving more complex long-term memory, such as citation extraction, resum[' e]{}s, and semantic role labeling. Related Work {#sec:related} ============ The ability of neural networks to efficiently represent local context features sometimes allows them to make surprisingly good independent decisions for each structured output variable [@collobert11]. However, these independent classifiers are often insufficient for structured prediction tasks where there are strong dependencies between the output labels [@collobert11; @DBLP:journals/corr/LampleBSKD16]. A natural solution is to use these neural feature representations to parameterize the factors of a conditional random field [@lafferty01] for joint inference over output variables [@collobert11; @jaderberg14; @DBLP:journals/corr/LampleBSKD16]. However, most previous work restricts the linear-chain CRF states to be the labels themselves—learning no additional output structure. The latent dynamic conditional random field (LDCRF) learns additional output structure beyond the labels by employing hidden states (latent variables) with Markov dependencies, each associated with a label; it has been applied to human gesture recognition [@morency07]. The dynamic conditional random field (DCRF) learns a factorized representation of each state [@sutton07]. The hidden-state conditional random field (HCRF) also employs a Markov sequence of latent variables, but the latent variables are used to predict a single label rather than a sequence of labels; it has been applied to phoneme recognition [@gunawardana05] and gesture recognition [@quattoni07]. All these models learn output representations while preserving the ability to perform exact joint inference by belief propagation. While the above use a log-linear parameterization of the potentials over latent variables, the input-output HMM [@bengio95] uses a separate neural network for each source state to produce transition probabilities to its destination states. Experiments in all of the above parameterizations use only a small hidden state space due to the large numbers of parameters required. In this paper we enable a large number of states by using a low-rank factorization of the transition potentials between latent states, effectively learning distributed embeddings for the states. This is superficially similar to the label embedding model of [@srikumar2014learning], but that work learns embeddings only to model similarity between observable output labels, and does not learn a latent output state structure. Embedded Latent CRF Model {#sec:model} ========================= We consider the task of sequence labeling: given an input sequence $\textbf{x} = \{x_1, x_2, \ldots, x_T\}$, find the corresponding output labels $\textbf{y} = \{y_1, y_2, \ldots, y_T\}$ where each output $y_i$ is one of $N$ possible output labels. Each input $x_i$ is associated with a feature vector $f_i \in \mathbb{R}^n$, such as that produced by a feed-forward or recurrent neural network. The models we consider will associate each input sequence with a sequence of hidden states $\{z_1, z_2, \ldots, z_T\}$. These discrete hidden states capture rich transition dynamics of the output labels. We consider the case where the number of hidden states $M$ is much larger than the number of output labels, $M >> N$. Given the above notation, the energy for a particular configuration is: $$\begin{aligned} \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) = \sum_{t=1}^T ( & \psi_{zf}(f_t, z_t) + \psi_{zy}(z_t, y_t) \nonumber \\ &+ \psi_{zz}(z_t, z_{t+1})) \label{eq:energy}\end{aligned}$$ where $\psi$ are scalar scoring functions of their arguments. $\psi_{zf}(f_t, z_t)$ and $\psi_{zy}(z_t, y_t)$ are the local scores for the interaction between the input features and the hidden states, and the hidden state and the output state, respectively. $\psi_{zz}(z_t, z_{t+1})$ are the scores for transitioning from a hidden state $z_t$ to hidden state $z_{t+1}$. The distribution over output labels is given by: $$\begin{aligned} {\mathbb{P}}({\mathbf{y}}|{\mathbf{x}}) = \frac{1}{Z} \sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right); \label{eq:py}\end{aligned}$$ $Z = \sum_{{\mathbf{y}}} \sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right)$ is the partition function. In the case of our Embedded Latent CRF model, as in the LDCRF model, latent states are deterministically partitioned to correspond to output values. That is, the number of latent states is a multiple of the number of output values, and $\psi_{zy}=0$ for pairs in the partitioning and $-\infty$ otherwise. The other scoring functions are learned as global bilinear parameter matrices. In order to manage large numbers of latent states without overfitting, the Embedded Latent CRF enforces an additional restriction that the scoring function $\psi_{zz}$ possess a low-rank structure: that is, $\psi_{zz}(z_i,z_j)=z_i^\top U V^\top z_j$, where $U$ and $V$ are skinny rectangular matrices and $z_i,z_j$ are represented by one-hot vectors. While inference in this model is tractable using tree belief propagation even when learning $\psi_{zy}$, the deterministic factors make it especially simple to implement. Computing the quantities involved in can be carried out efficiently with dynamic programming using the forward algorithm, as in HMMs. To see this, note that to compute the numerator $\sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right)$, given an output label ${\mathbf{y}}$, we can fold the local scores $\psi_{zf}$ and $\psi_{zy}$ into one score $\psi_{zfy}(f_t, z_t; y_t)$, and summing the resulting energy corresponds exactly to the forward algorithm in a CRF with $M$ states. The partition function can also be computed by dynamic programming: $$\begin{aligned} Z &= \sum_{{\mathbf{y}}} \sum_{{\mathbf{z}}} \exp \left( \mathcal{E}({\mathbf{y}}, {\mathbf{z}}| {\mathbf{x}}) \right) \\ &= \sum_{{\mathbf{z}}} \sum_{{\mathbf{y}}} \prod_t \exp \left( \psi_{zf}(z_t, f_t) + \psi_{zz}(z_t, z_{t+1}) \right) \\ & \exp(\psi_{zy}(z_t, y_t))\end{aligned}$$ At test time we perform MAP inference using the exact Viterbi algorithm, which can be done as in the above dynamic program, replacing sums with maxes. Experiments {#sec:experiments} =========== We demonstrate the benefits of latent and embedded large-cardinality state spaces on a synthetic task based on CoNLL-2003 named entity recognition [@DBLP:conf/conll/SangM03], which provides easily interpretable inputs and outputs to explore. With IOB encoding the dataset has eight output labels representing non-entities, as well as inside and beginning of person, location, organization and miscelaneous enties (B-PER is not needed). We consider the task synthetic because we use relatively impoverished input features to explore the capacity of the output representation. We use the BiLSTM+CRF featurization from [@DBLP:journals/corr/LampleBSKD16], but instead of using BiLSTM, we produce local potentials from a feedforward network conditioned on word- and character-level features from the current time-step only. We demonstrate that performance of this model is improved by increasing the size of the latent state space (perhaps unsurprisingly), and that significant further improvement can be obtained from learning low-rank embeddings of the latent states. Qualitatively, we also show the latent states learn interpretable structure. #### Training Details: All models are implemented using TensorFlow. We use the hyperparameter settings from the LSTM+CRF model of [@DBLP:journals/corr/LampleBSKD16], with the exception that we use minibatches of size 20 and the Adam optimizer [@DBLP:journals/corr/KingmaB14] with a learning rate of 1e-3 and $\epsilon$ of 1e-6. We initialise our word level embeddings using pretrained 100 dimensional skip-n-gram embeddings [@DBLP:conf/emnlp/LingTAFDBTL15] where available, and use Glorot initialisation [@DBLP:journals/jmlr/GlorotB10] otherwise. For the Embedded Latent CRF, we learn 16 dimensional embeddings (the matrices $U$ and $V$ have rank 16) for all but the 16 hidden state ELCRF, for which we learn 8 dimensional embeddings. We train all models for 200 epochs with early-stopping on the validation set. We use the train-dev-test split of the CoNLL-2003 dataset. Quantitative Results -------------------- **Table \[tab:crf-res\]** reports the performance of all the models. We see that the performance on this structured prediction task is improved both by increasing the size of the latent state space, and by embedding the states into a lower dimensional space (i.e. a low-rank factorization of the log-space transition potential). The benefits of a complex output space are especially important when using a restricted set of input features, as noted in [@liang2008structure]. [**LDCRF F1**]{} [**ELCRF F1**]{} ----- ------------------ ------------------ 8 81.88 - 16 83.69 84.03 32 84.31 84.58 64 84.52 85.02 128 84.36 86.29 256 **84.82** 86.83 512 84.78 **86.91** : NER field F1 on the CoNLL’03 test set with synthetically simple input representation. Note that learning an embedded state representation (ELCRF) improves accurcy over not learning an embedded representation (LDCRF).[]{data-label="tab:crf-res"} Qualitative Insight ------------------- We now turn to a qualitative analysis of the ELCRF model. Table \[tab:ner\] gives examples of some hidden states and tokens for which they were activated. First, we observe that the model has discovered separate latent states for surnames (228) and surname nobiliary particles (192). The latter almost always transitions to a surname state, and capturing this special transition signature (as distinct from [Per]{} label generically, as in a traditional CRF without latent states) improves accuracy when the surname is poorly associated with the [Per]{} label. We also observe the model’s ability to detect phrase boundaries. This is true not only for [Per]{} phrases where the model identifies boundaries by identifying first names and last names but also for [Misc]{} phrases and [Org]{} phrases. We observe that state 257 fires everytime an [I-Misc]{} token is followed by an [I-Misc]{} token—signalling the start of an [I-Misc]{} phrase, and state 272 fires at the end of that phrase. Similarly, state 392 signals the start of an [Org]{} phrase and 445 signals the end. [c|S]{} & [**tokens**]{}\ & [Van, Dal, De, Manne, Jan, Den, Della, Der]{}\ [204]{} & [Hendrix, Lien, Werner, Peter, Sylvie, Jack]{}\ [228]{} & [Miller, Cesar, Jensen, Dickson, Abbott]{}\ [283]{} & [British, German, Polish, Australian]{}\ [297]{} & [911, 310, 150, 11]{}\ [269]{} & [Korean-related, Beijing-funded, Richmond-based]{}\ The latent state space also learns block structure in the state transition matrix which gives the joint prediction a long-term bidirectional “memory,” encouraging unusual and beneficial interpretation of other parts of the sequence. For example, in the sentence “Boston ’s Mo Vaughn went 3-for-3 with a walk , stole home ...,” our model with factorized latent states was correctly able to label “Boston” as [I-Org]{} (the team) due to the longer range context of a baseball game, whereas the model without latent states incorrectly labels it [I-Loc]{}. In the phrase “Association for Relations Across the Taiwan Straits” our model correctly labels the entire sequence as an [Org]{} using a special state for “the,” while the traditional model loses context at “the” and labels the last two words as a [Loc]{}. Conclusion and Future Work ========================== We present a method for learning output representations in finite-state sequence modeling. Our Embedded Latent CRF learns state embeddings by representing the transition matrix in a large latent state space with low-rank factorization. Unlike most recent work that learns input representations, but settles with simple Markov dependencies among the given output labels, our approach learns output representations of a large number of memory-providing, expressive and interpretable states, avoids overfitting due to factorization, and maintains tractable exact inference plus maximum likelihood learning using dynamic programming. In future work we will apply this model to non-synthetic sequence labeling tasks involving complex joint predictions and long term memory, such as citation extraction, r[' e]{}sum[' e]{} field extraction, and semantic role labeling.

--- abstract: 'The purpose of this note is two-fold. Firstly, we prove that the variety $\mathbf{RDMSH_1}$ of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity and present (equational) axiomatizations for several subvarieties of $\mathbf{RDMSH_1}$. Secondly, we give a concrete description of the lattice of subvarieties of the variety $\mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras that contains $\mathbf{RDMSH_1}$. Furthermore, we prove that every subvariety of $\mathbf{RDQDStSH_1}$, and hence of $\mathbf{RDMSH_1}$, has Amalgamation Property. The note concludes with some open problems for further investigation.' author: - 'Hanamantagouda P. Sankappanavar' title: 'A Note on Regular De Morgan Semi-Heyting Algebras' --- \[section\] \[Lemma\][**THEOREM**]{} \[Lemma\][**CLAIM**]{} \[Lemma\][**COROLLARY**]{} \[Lemma\][**PROPOSITION**]{} \[Lemma\][**EXAMPLE**]{} \[Lemma\][**FACT**]{} \[Lemma\][**DEFINITION**]{} \[Lemma\][**NOTATION**]{} \[Lemma\][**REMARK**]{} [**Introduction**]{} {#SA} ==================== Semi-Heyting algebras were introduced by us in [@Sa07] as an abstraction of Heyting algebras. They share several important properties with Heyting algebras, such as distributivity, pseudocomplementedness, and so on. On the other hand, interestingly, there are also semi-Heyting algebras, which, in some sense, are “quite opposite” to Heyting algebras. For example, the identity $0 \to 1 \approx 0$, as well as the commutative law $x \to y \approx y \to x$, hold in some semi-Heyting algebras. The subvariety of commutative semi-Heyting algebras was defined in [@Sa07] and is further investigated in [@Sa10]. Quasi-De Morgan algebras were defined in [@Sa87a] as a common abstraction of De Morgan algebras and distributive $p$-algebras. In [@Sa12], expanding semi-Heyting algebras by adding a dual quasi-De Morgan operation, we introduced the variety $\mathbf{DQDSH}$ of dually quasi-De Morgan semi-Heyting algebras as a common generalization of De Morgan Heyting algebras (see [@Sa87] and [@Mo80]) and dually pseudocomplemented Heyting algebras (see [@Sa85]) so that we could settle an old conjecture of ours. The concept of regularity has played an important role in the theory of pseudocomplemented De Morgan algebras (see [@Sa86]). Recently, in [@Sa14] and [@Sa14a], we inroduced and examined the concept of regularity in the context of $\mathbf{DQDSH}$ and gave an explicit description of (twenty five) simple algebras in the (sub)variety $\mathbf{DQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1. The work in [@Sa14] and [@Sa14a] led us to conjecture that the variety $\mathbf{RDMSH_1}$ of regular De Morgan algebras satisfies Stone identity. The purpose of this note is two-fold. Firstly, we prove that the variety $\mathbf{RDMSH_1}$ of regular De Morgan semi-Heyting algebras of level 1 satisfies Stone identity, thus settlieng the above mentioned conjecture affirmatively. As applications of this result and the main theorem of [@Sa14], we present (equational) axiomatizations for several subvarieties of $\mathbf{RDMSH_1}$. Secondly, we give a concrete description of the lattice of subvarieties of the variety $\mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras, of which $\mathbf{RDMSH_1}$ is a subvariety. Furthermore, we prove that every subvariety of $\mathbf{RDQDStSH_1}$, and hence of $\mathbf{RDMSH_1}$, has Amalgamation Property. The note concludes with some open problems for further investigation. **[Dually Quasi-De Morgan Semi-Heyting Algebras]{}** {#SB} ==================================================== The following definition is taken from [@Sa07]. An algebra ${\mathbf L}= \langle L, \vee ,\wedge ,\to,0,1 \rangle$ is a [*semi-Heyting algebra*]{} if\ $\langle L,\vee ,\wedge ,0,1 \rangle$ is a bounded lattice and ${\mathbf L}$ satisfies: 1. $x \wedge (x \to y) \approx x \wedge y$ 2. $x \wedge(y \to z) \approx x \wedge ((x \wedge y) \to (x \wedge z))$ 3. $x \to x \approx 1$. Let ${\mathbf L}$ be a semi-Heyting algebra and, for $x \in {\mathbf L}$, let $x^*:=x \to 0$. ${\mathbf L}$ is a [*Heyting algebra*]{} if ${\mathbf L}$ satisfies: 1. $(x \wedge y) \to y \approx 1$. ${\mathbf L}$ is a [*commutative semi-Heyting algebra*]{} if ${\mathbf L}$ satisfies: 1. $x \to y \approx y \to x$. ${\mathbf L}$ is a [*Boolean semi-Heyting algebra*]{} if ${\mathbf L}$ satisfies: 1. $x \lor x^{*} \approx 1$. ${\mathbf L}$ is a [*Stone semi-Heyting algebra*]{} if ${\mathbf L}$ satisfies: 1. $x^* \lor x^{**} \approx 1$. Semi-Heyting algebras are distributive and pseudocomplemented, with $a^*$ as the pseudocomplement of an element $a$. We will use these and other properties (see [@Sa07]) of semi-Heyting algebras, frequently without explicit mention, throughout this paper. The following definition is taken from [@Sa12]. An algebra ${\mathbf L}= \langle L, \vee ,\wedge ,\to, ', 0,1 \rangle $ is a [*semi-Heyting algebra with a dual quasi-De Morgan operation*]{} or [*dually quasi-De Morgan semi-Heyting algebra*]{} [(]{}$\mathbf {DQDSH}$-algebra, for short[)]{} if\ $\langle L, \vee ,\wedge ,\to, 0,1 \rangle $ is a semi-Heyting algebra, and ${\mathbf L}$ satisfies: - $0' \approx 1$ and $1' \approx 0$ - $(x \land y)' \approx x' \lor y'$ - $(x \lor y)'' \approx x'' \lor y''$ - $x'' \leq x$. Let $\mathbf{L} \in \mathbf {DQDSH}$. Then ${\bf L}$ is a [*dually Quasi-De Morgan Stone semi-Heyting algebra*]{} [(]{}$\mathbf{DQDStSH}$-algebra[)]{} if ${\bf L}$ satisfies (St). $\mathbf {L}$ is a [*De Morgan semi-Heyting algebra*]{} or [*symmetric semi-Heyting algebra*]{} [(]{}$\mathbf{DMSH}$-algebra[)]{} if ${\bf L}$ satisfies: - $x'' \approx x$. $\mathbf{L}$ is a [*dually pseudocomplemented semi-Heyting algebra*]{} [(]{}$\mathbf {DPCSH}$-algebra if $\mathbf{L}$ satisfies: - $x \lor x' \approx 1$. The varieties of $\mathbf {DQDSH}$-algebras, $\mathbf{DQDStSH}$-algebras, $\mathbf{DMSH}$-algebras and $\mathbf {DPCSH}$-algebras are denoted, respectively, by $\mathbf {DQDSH}$, $\mathbf{DQDStSH}$, $\mathbf{DMSH}$ and $\mathbf {DPCSH}$. Furthermore, $\mathbf {DMcmSH}$ denotes the subvariety of $\mathbf {DMSH}$ defined by the commutative identity (Co), and $\mathbf {DQDBSH}$ denotes the one defined by (Bo). If the underlying semi-Heyting algebra of a $\mathbf{DQDSH}$-algebra is a Heyting algebra we denote the algebra by $\mathbf{DQDH}$-algebra, and the corresponding variety is denoted by $\mathbf{DQDH}$. In the sequel, $a'{^*}'$ will be denoted by $a^+$, for $a \in \mathbf{L} \in \mathbf{DQDSH}$. The following lemma will often be used without explicit reference to it. Most of the items in this lemma were proved in [@Sa12], and the others are left to the reader. \[2.2\] Let ${\mathbf L} \in \mathbf{DQDSH}$ and let $x,y, z \in L$. Then 1. $1'^{*}=1$ 2. $x \leq y$ implies $x' \geq y'$ 3. $(x \land y)'^{*}=x'^{*} \land y'^{*}$ 4. $ x''' = x'$ 5. $(x \lor y)' = (x'' \lor y'')'$ \[C.934\] 6. $(x \lor y)' = (x'' \lor y)'$ \[12662\] 7. $x \leq (x \lor y) \to x$ \[B.63\] 8. $x \land [(x \lor y) \to z] = x \land z$. Next, we describe some examples of $\mathbf {DQDSH}$-algebras by expanding the semi-Heyting algebras given in Figure 1. These will play a crucial role in the rest of the note. (5,2) (1.7,1) (1.7,2.3) (1.3,.9)[$0$]{} (1.3,2.1)[$1$]{} (.2,1.5)[$\bf 2$  :]{} (1.7,1)[(0,1)[1.2]{}]{} (2.6,1.5) $\to$    0    1 --------- ------ ---- 0 1 1 1 0 1 (8.7,1) (8.7,2.3) (8.3,.9)[$0$]{} (8.3,2.1)[$1$]{} (7.0,1.5)[${\bf {\bar{2}}}$  :]{} (8.7,1)[(0,1)[1.2]{}]{} (9.5,1.5) $\to$    0    1 --------- ------ ---- 0 1 0 1 0 1 \ (5, 1) (1.5, .2) (1.5,1.0) (1.5,1.8) (1.1, .1)[$0$]{} (1.1,.9)[$a$]{} (1.1,1.7)[$1$]{} (.1,.7)[${\bf L}_1$  :]{} (1.5, .2)[(0,1)[1.5]{}]{} (2.2,1) $\to$    0    $a$    1 --------- ------ -------- ---- 0 1 1 1 $a$ 0 1 1 1 0 $a$ 1 (8.2, .3) (8.2,1.1) (8.2, 1.8) (7.8, .2)[$0$]{} (7.8,.9)[$a$]{} (7.8,2.1)[$1$]{} (6.7,.7)[${\bf L}_2$  :]{} (8.2, .3)[(0,1)[1.5]{}]{} (8.9,1) $\to$    0    $a$    1 --------- ------ -------- ---- 0 1 $a$ 1 $a$ 0 1 1 1 0 $a$ 1 \ (7,.3) (1.5, .2) (1.5,1.0) (1.5,1.8) (1.1, .1)[$0$]{} (1.1,.9)[$a$]{} (1.1,1.7)[$1$]{} (.1,.7)[${\bf L}_3$  :]{} (1.5, .2)[(0,1)[1.5]{}]{} (2.2,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 1 1 $a$ 0 1 $a$ 1 0 $a$ 1 (8.2, .3) (8.2,1.1) (8.2, 1.8) (7.8, .2)[$0$]{} (7.8,.9)[$a$]{} (7.8,2.1)[$1$]{} (6.7,.7)[${\bf L}_4$  :]{} (8.2, .3)[(0,1)[1.5]{}]{} (8.9,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 $a$ 1 $a$ 0 1 $a$ 1 0 $a$ 1 \ (7, 1.7) (1.5, .2) (1.5,1.0) (1.5,1.8) (1.1, .1)[$0$]{} (1.1,.9)[$a$]{} (1.1,1.7)[$1$]{} (.1,.7)[${\bf L}_5$  :]{} (1.5, .2)[(0,1)[1.5]{}]{} (2.2,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 $a$ $a$ $a$ 0 1 1 1 0 $a$ 1 (8.2, .3) (8.2,1.1) (8.2, 1.8) (7.8, .2)[$0$]{} (7.8,.9)[$a$]{} (7.8,2.1)[$1$]{} (6.7,.7)[${\bf L}_6$  :]{} (8.2, .3)[(0,1)[1.5]{}]{} (8.9,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 1 $a$ $a$ 0 1 1 1 0 $a$ 1 \  \  \  \ (7,0.3) (1.5, .2) (1.5,1.0) (1.5,1.8) (1.1, .1)[$0$]{} (1.1,.9)[$a$]{} (1.1,1.7)[$1$]{} (.1,.7)[${\bf L}_7$  :]{} (1.5, .2)[(0,1)[1.5]{}]{} (2.2,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 $a$ $a$ $a$ 0 1 $a$ 1 0 $a$ 1 (8.2, .3) (8.2,1.1) (8.2, 1.8) (7.8, .2)[$0$]{} (7.8,.9)[$a$]{} (7.8,2.1)[$1$]{} (6.7,.7)[${\bf L}_8$  :]{} (8.2, .3)[(0,1)[1.5]{}]{} (8.9,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 1 $a$ $a$ 0 1 $a$ 1 0 $a$ 1 \  \  \ (7,2.0) (1.5, .2) (1.5,1.0) (1.5,1.8) (1.1, .1)[$0$]{} (1.1,.9)[$a$]{} (1.1,1.7)[$1$]{} (.1,.7)[${\bf L}_9$  :]{} (1.5, .2)[(0,1)[1.5]{}]{} (2.2,1) $\to$    0    $a$    1 --------- ------ -------- ---- 0 1 0 0 $a$ 0 1 1 1 0 $a$ 1 (8.2, .3) (8.2,1.1) (8.2, 1.8) (7.8, .2)[$0$]{} (7.8,.9)[$a$]{} (7.8,2.1)[$1$]{} (6.7,.7)[${\bf L}_{10}$  :]{} (8.2, .3)[(0,1)[1.5]{}]{} (8.9,1) $\to$    0    $a$    1 --------- ------ -------- ----- 0 1 0 0 $a$ 0 1 $a$ 1 0 $a$ 1 \ (7,4) (.4,2.5)[$\bf D_1$  :]{} (7.0,2.5)[$\bf D_2$  :]{} (1.5,2.4) $\to$    0    $1$    $a$   $b$ --------- ------ -------- ------- ------ 0 1 0 $b$ $a$ $1$ 0 1 $a$ $b$ $a$ $b$ $a$ 1 0 $b$ $a$ $b$ 0 1 (8.0,2.4) $\to$    0    $1$    $a$   $b$ --------- ------ -------- ------- ------ 0 1 1 1 1 $1$ 0 1 $a$ $b$ a $b$ $1$ 1 $b$ $b$ $a$ 1 $a$ 1 (7, 3)\ (.4,2.0)[$\bf D_3$  :]{} (2.2,2.4) $\to$    0    $1$    $a$   $b$ --------- ------ -------- ------- ------ 0 1 $a$ 1 $a$ $1$ 0 1 $a$ $b$ $a$ $b$ $a$ 1 0 $b$ $a$ 1 $a$ 1 (6,0.2)[Figure 1]{} Let $\mathbf {2^e}$ and $\mathbf{\bar{2}^e}$ be the expansions of the semi-Heyting algebras $\mathbf {2}$ and $\mathbf{\bar{2}}$ (shown in Figure 1) by adding the unary operation $'$ such that $0'=1$, $1'=0$. Let $\mathbf{L}^{dp}_i$, $i=1, \ldots,10$, denote the expansion of the semi-Heyting algebra $\mathbf{L}_i$ (shown in Figure 1) by adding the unary operation $'$ such that $0'=1$, $1'=0$, and $a'=1$. Let $\mathbf{L}^{dm}_i$, $i=1, \ldots, 10$, denote the expansion of ${\bf L}_i$ (in Figure 1) by adding the unary operation $'$ such that $0'=1$, $1'=0$, and $a'=a$. We Let $\mathbf{C^{dp}_{10}} := \{\mathbf{L}^{dp}_i: i =1, \ldots,10 \}$ and $\mathbf{C^{dm}_{10}} := \{\mathbf{L}^{dm}_i: i =1, \ldots,10 \}$. We also let $ \mathbf{C_{20}} := \mathbf{C^{dm}_{10}} \cup \mathbf{C^{dp}_{10}}$. Each of the three $4$-element algebras $\mathbf{D_1}$, $\mathbf{D_2}$ and $\mathbf{D_3}$ has its lattice reduct as the Boolean lattice with the universe $\{0,a,b,1\}$, $b$ being the complement of $a$, has the operation $\to$ as defined in Figure 1, and has the unary operation $'$ defined as follows: $a' = a$, $b' = b$, $0'=1$, $1'=0$. For the variety $\mathbf{V(D_1, D_2, D_3)}$ generated by $\mathbf{\{D_1, D_2, D_3 \}}$, it was shown in [@Sa12] that $\mathbf{V(D_1, D_2, D_3)} = \mathbf{DQDBSH}$. The following is a special case of Definition 5.5 in [@Sa12]. Let $x'{^*}'{^*} := x{^{2(}}{'^{*)}}$. Note that $x{^{2(}}{'^{*)}} \leq x$ in a $\mathbf{DMSH}$-algebra. \[5.5\] The subvariety $\mathbf {DMSH_1}$ of level 1 of $\mathbf {DMSH}$ is defined by the identity: $x \land x'^* \land x^{2{\rm(}}{'^{*{\rm{)}}}} \approx x \land x'^*,$ or equivalently, by the identity: - $(x \land x'^*)'^* \approx x \land x'^*$. It follows from [@Sa12] that the variety $\mathbf {DMSH_1}$, is a discriminator variety. We note here that the algebras described above in Figure 1 are actually in $\mathbf {DMSH_1}$. Regular De Morgan Semi-Heyting algebras of level 1 ================================================== Recall that $a^+:= a'{^*}'$ in $\mathbf{L} \in \mathbf{DMSH_1}$. Let $\mathbf{L} \in \mathbf{DMSH_1}$. Then $\mathbf{L}$ [*is regular*]{} if $\mathbf{L}$ satisfies the following identity: - $ x \land x^+ \leq y \lor y^*$. The variety of regular $\mathbf{DMSH_1}$-algebras will be denoted by $\mathbf{RDMSH_1}$. In the rest of this section, $\mathbf{L}$ denotes an $\mathbf{RDMSH_1}$-algebra and $x,y \in L$. The following lemmas lead us to prove that $\mathbf{RDMSH_1}$ satisfies (St). \[regB\] $(x \lor x{^*}')^* = x' \land x^*$. $$\begin{aligned} x' \land x^* &=& x' \land x''^*\\ &=& (x' \land x'{'^*})'^* \quad \text{ by (L1)}\\ &= & (x'' \lor x''{^*}')^* \\ &=& (x \lor x{^*}')^*, \quad \text{ since $x''=x$}. \end{aligned}$$ \[regC\] $x \lor x^* \lor x{^*}' = 1$. $$\begin{aligned} x \lor x^* \lor x{^*}' &=& (x{^*}' \land x' \land x^*)' \quad \text{ by (DM)}\\ &=& [x{^*}' \land (x \lor x{^*}')^*]' \quad \text{ by Lemma \ref{regB}}\\ &=& (x{^*}' \land 0)' \quad \text{ by Lemma \ref{2.2}(viii)}\\ &=& 0'\\ &=& 1.\end{aligned}$$ \[regE\] We have - $x \land (x^+ \lor y \lor y^*) = x \land (y \lor y^*)$. $$\begin{aligned} x \land (y \lor y^*) &=& x \land [(x \land x^+) \lor (y \lor y^*)] \quad \text{ by (R)}\\ &=& (x \land x^+) \lor [x \land (y \lor y^*)] \\ & =& x \land [x^+ \lor y \lor y^*].\end{aligned}$$ \[regF\] Let $x \neq 1$. Then $x \leq x'$. Since $x \neq 1$, we have $x \land x'^*=0$ by (L1). So, $$\begin{aligned} x \land x' &=&(x \land x')\lor (x\land x'^*) \\ &=& x \land (x' \lor x'^*) \\ &= & x \land (x^+ \lor x' \lor x'^*) \quad \text{by Lemma \ref{regE}}\\ & = & x \land 1 \quad \text{ by Lemma \ref{regC}} $$ So, $x \leq x'$. \[regG\] Let $x^* \neq 0$. Then $x \lor x{^*} =1$. Since $x^* \neq 0$, we have $x{^*}' \neq1$, so $x{^*}' \leq x{^*} $ by Lemma \[regF\] and (DM), implying $x \lor x^* =1$ by Lemma \[regC\]. \[regH\] Let $\mathbf{L} \in \mathbf{RDMSH_1}$. Then $\mathbf{L} \models x^* \lor x^{**} \approx 1$. Let $a \in L$. If $a^* = 0$, Then the theorem is trivially true. So, we can assume that $a^* \neq 0$. Then $a \lor a^* = 1$, in view of the preceding lemma. The conclusion is now immediate. Recall from [@Sa12] that the subvariety $\mathbf {DMSH_2}$ of level 2 of $\mathbf {DMSH}$ is defined by the identity: $x \land x'^* \land x^{2{\rm(}}{'^{*{\rm{)}}}} \approx x \land x'^*\land x^{2{\rm(}}{'^{*{\rm{)}}}} \land x^{3{\rm(}}{'^{*{\rm{)}}}},$ or equivalently, by the identity:\ [(L2)]{} $(x \land x'^*)^{2{\rm(}}{'^{*{\rm{)}}}} \approx (x \land x'^*)^{{\rm(}}{'^{*{\rm{)}}}}$. The above theorem fails in $\mathbf{RDMSH_2}$, as the following example shows: (8,1) (6,0) (6.2,0)[$e$]{} (4,0) (3.5,0)[$d$]{} (5,1) (5.2,-1.1)[$c$]{} (4,-2) (3.5, -2)[$a$]{} (6,-2) (6.2, -2.1)[$b$]{} (5,-3) (5.2,-3.3)[$0$]{} (5,1) (5.2, 1.1)[$1$]{} (4,0)[(1,1)[1]{}]{} (5,-1)[(1,1)[1]{}]{} (4,-2)[(1,1)[1]{}]{} (5,-3)[(1,1)[1]{}]{} (4,0)[(1,-1)[1]{}]{} (5,-1)[(1,-1)[1]{}]{} (4,-2)[(1,-1)[1]{}]{} (5,1)[(1,-1)[1]{}]{} (5,-4.3)[Figure 2]{} Applications ============ Let $\mathbf{V(K)}$ denote the variety generated by the class $\mathbf{K}$ of algebras. The following corollary is immediate from Theorem \[regH\] and Corollary 3.4(a) of [@Sa14a], and hence is an improvement on Corollary 3.4(a) of [@Sa14a]. We have\ - $\mathbf{RDMSH_1} = \mathbf{RDMStSH_1} = \mathbf{V(C_{10}^{dm})} \lor \mathbf{V(D_1, D_2, D_3)}$\ - $\mathbf{RDMH_1} = \mathbf{RDMStH_1} = \mathbf{V(L_1^{dm})} \lor \mathbf{V(D_2)}$\ - $\mathbf{RDMcmSH1} = \mathbf{V(L_{10}^{dm}, D_1)} = \mathbf{V(L_{10}^{dm})} \lor \mathbf{V(D_1)}$. Let $L \in \mathbf{DMSH_1}$. We say $\mathbf{L}$ is pseudocommutative if\ $\mathbf{L} \models (x \to y)^* = (y \to x)^*$.\ Let $\mathbf{V}$ be a subvariety of $ \mathbf{RDMSH_1}$. Then $\mathbf{V}$ is pseudocommutative iff $\mathbf{V}= \mathbf{V(L_9^{dm}, L_{10}^{dm}, D_1)}$. It suffices, in view of (a) of the preceding corollary, to verify that $\mathbf{L_9^{dm}, L_{10}^{dm}}$, and $\mathbf{D_1}$ satisfy the pseudocommutative law, while the rest of the simples in $\mathbf{RDMSH_1}$ do not. The proofs of the following corollaries are similar. The variety $\mathbf{V(L_9^{dm}, L_{10}^{dm}, D_1})$ is also defined, modulo $\mathbf{RDMSH1}$, by - $x^* \to y^* \approx y^* \to x^*$. The variety $\mathbf{V(L_1^{dm}, L_2^{dm}, L_3^{dm}, L_4^{dm}, D_2, D_3})$ is defined, modulo $\mathbf{RDMSH1}$, by - $(0 \to 1)^+ \to (0 \to 1){^*}'^* \approx 0 \to 1$. It was proved in [@Sa12] that $\mathbf{V(D_1, D_2, D_3}) = \mathbf{DQDBSH}$. Here are some more bases for $\mathbf{V(D_1, D_2, D_3})$. Each of the following identities is a base for the variety $\mathbf{V(D_1, D_2, D_3})$ modulo $\mathbf{RDMSH1}$: - $x \to y \approx y^* \to x^*$ [(Law of contraposition)]{} - $x \lor (y \to z) \approx (x \lor y) \to (x \lor z)$ - $ [\{x \lor (x \to y^*)\} \to (x \to y^*)] \lor (x \lor y^*)=1$.\ The variety\ $\mathbf{V(L_1^{dm}, L_2^{dm}, L_5^{dm}, L_6^{dm}, L_9^{dm}, D_1, D_2, D_3})$ is defined, modulo\ $\mathbf{RDMSH1}$, by - $x \to y^* \approx y \to x^*$. The variety $\mathbf{V(L_7^{dm}, L_8^{dm}, L_9^{dm}, L_{10}^{dm}, D_1, D_2, D_3})$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (x \to y) \approx x \lor [(x \to y) \to 1]$. The variety $\mathbf{V(L_7^{dm}, L_8^{dm}, D_2})$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (x \to y) \approx x \lor [(x \to y) \to 1]$ - $ (0 \to 1)^{**} \approx 1$. The variety $\mathbf{V(2^e, L_7^{dm}, L_8^{dm}, L_9^{dm}, L_{10}^{dm}})$ is\ defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (x \to y) \approx x \lor [(x \to y) \to 1]$ - $x{^*}' \approx x^{**}$. The variety\ $\mathbf{V(2^e, L_9^{dm}, L_{10}^{dm}})$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (x \to y) \approx x \lor [(x \to y) \to 1]$ - $x{^*}' \approx x^{**}$ - $ (0 \to 1) \lor (0 \to 1)^* \approx 1$. The variety\ $\mathbf{V(L_9^{dm}, L_{10}^{dm}})$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (x \to y) \approx x \lor [(x \to y) \to 1]$ - $x{^*}' \approx x^{**}$ - $ (0 \to 1)^{*} \approx 1$. The variety\ $\mathbf{V(L_1^{dm}, L_2^{dm}, L_3^{dm}, L_4^{dm}, L_5^{dm}, L_6^{dm}, L_7^{dm}, L_8^{dm}})$ is defined, modulo\ $\mathbf{RDMSH1}$, by - $x{^*}' \approx x^{**}$ - $ (0 \to 1)^{**} \approx 1$. The variety $\mathbf{V(L_1^{dm}, L_2^{dm}, L_3^{dm}, L_4^{dm}, D_2})$ is\ defined, modulo $\mathbf{RDMSH1}$, by - $(0 \to 1) \lor (0 \to 1)^* \approx 1$ - $ (0 \to 1)^{**} \approx 1$. The variety $\mathbf{V(L_1^{dm}, L_3^{dm}, D_1, D_2, D_3})$ is\ defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (y \to x) \approx (x \lor y) \to x$ - $(0 \to 1) \lor (0 \to 1)^* \approx 1 $. The variety $\mathbf{V(L_1^{dm}, L_3^{dm}, D_2})$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor (y \to x) \approx (x \lor y) \to x$ - $(0 \to 1)^{**}=1$ - $(0 \to 1) \lor (0 \to 1)^* \approx 1 $. The variety $\mathbf{V(L_1^{dm}, L_2^{dm}, L_8^{dm}, D_1, D_2, D_3})$ is defined, modulo $\mathbf{RDMSH1}$, by - $y \lor (y \to (x \lor y)) \approx (0 \to x) \lor (x \to y)$. The variety $\mathbf{V(L_8^{dm}, D_1, D_2, D_3)}$ is defined, modulo $\mathbf{RDMSH1}$, by - $x \lor [y \to (0 \to (y \to x))] \approx x \lor y \lor (y \to x)$. $\mathbf{V(D_2)}$ was axiomatized in [@Sa12]. Here are some more bases for it. Each of the following identities is an equational base for $\mathbf{V(D_2)}$, mod $\mathbf{RDMH_1}$:\ - $ y \to [0 \to (y \to x)] \approx y \lor (y \to x)$\ - $x \lor (y \to z) \approx (x \lor y) \to (x \lor z)$\ - $x \lor [y \to (y \to x)^*] \approx x \lor y \lor (y \to x) $\ - $[\{x \lor (x \to y^*)\} \to (x \to y^*)] \lor x \lor y^* \approx 1.$ $\mathbf{V(D_1)}$ was axiomatized in [@Sa12]. Here are more bases for it. Each of the following identities is an equational base for $\mathbf{V(D_1)}$, mod $\mathbf{RDMcmSH_1}$:\ - $y \lor (y \to (x \lor y)) \approx (0 \to x) \lor (x \to y)$\ - $x \lor [y \to (y \to x)^*] \approx x \lor y \lor (y \to x)$\ - $[\{x \lor (x \to y^*)\} \to (x \to y^*)] \lor x \lor y^* \approx 1$\ - $x \lor (y \to z) \approx (x \lor y) \to (x \lor z)$.\ The variety $\mathbf{V(L_1^{dm}, L_2^{dm}, L_3^{dm}, D_1, D_2, D_3))}$ is defined, mod $\mathbf{RDMSH_1}$, by - $x \to (y \to z)=y \to (x \to z).$ The variety $\mathbf{V(C_{10}^{dm})}$ is defined, mod\ $\mathbf{RDMSH_1}$, by - $x \land x' \leq y \lor y'$ [(Kleene identity)]{}. The variety $\mathbf{V(L_{10}^{dm})}$ is defined, mod $\mathbf{RDMSH_1}$, by - $x \land x' \leq y \lor y'$ [(Kleene identity)]{} - $x \to y \approx y \to x$. Lattice of subvarieties of $\mathbf{RDQDStSH_1}$ ================================================= We now turn to describe the lattice of subvarieties of $\mathbf{RDQDStSH_1}$ which contains $\mathbf{RDMSH_1}$ in view of Theorem \[regH\]. For this purpose we need the following theorem which is proved in [@Sa14]. \[Th\] Let $\mathbf{L} \in \mathbf{RDQDStSH1}$. Then TFAE: - $L$ is simple\ - $L$ is subdirectly irreducible\ - $\mathbf{L} \in \mathbf{\{2^e, \bar{2}^e\}} \cup \mathbf{C_{20}} \cup \mathbf{\{D_1, D_2, D_3 \}}$. Let $\mathcal{L}$ denote the lattice of subvarieties of $\mathbf{RDQDStSH_1}$. $\mathbf{T}$ denotes the trivial variety, and, for $n$ a positive integer, $\mathbf{B_n}$ denotes the $n$-atom Boolean lattice. We also let $\mathbf{1+B}$ denote the lattice obtained by adding a new least element $0$ to the Boolean lattice $\mathbf{B}$. $\mathcal{L} \cong \mathbf{(1+B_9)} \times \mathbf{(1+B_5)} \times \mathbf{B_9}$. Let $S_1$ := $\mathbf{\{L_i^{dm}: i =1,2,3,4\}} \cup \mathbf{\{L_i^{dp}: i =1,2,3,4\}} \cup \mathbf{\{D_2 \}}$,\ $S_2$ := $\mathbf{\{L_i^{dm}: i =9, 10\}} \cup \mathbf{\{L_i^{dp}: i =9, 10 \}} \cup \mathbf{\{D_1 \}}$, and\ $S_3$:= $\mathbf{\{L_i^{dm}: i =5,6,7,8 \}} \cup \mathbf{\{L_i^{dp}: i = 5,6,7,8 \}} \cup \mathbf{\{D_3 \}}$. Observe that each of the simples in $S_1$ contains $\mathbf{2^e}$. Let us first look at the interval $\mathbf{[V(2^e), V(S_1)]}$. Since each algebra in $S_1$ is an atom in this interval, we can conclude that the interval is a 9-atom Boolean lattice; thus the interval $\mathbf{[T, V(S_1)]}$ is isomorhic to $\mathbf{1+B_9}$. Similarly, since each of the simples in $S_2$ contains $\mathbf{\bar{2}^e}$, it is clear that the interval $\mathbf{[T, V(S_2)]}$ is isomorphic to $\mathbf{1+B_5}$. Likewise, since each of the simples in $S_3$ has only one subalgebra, namely the trivial algebra, the interval $\mathbf{[T, S_3]}$ is isomprphic to $\mathbf{B_9}$. Observe that the the intersection of the subvarieties $\mathbf{V(S_1)}$, $\mathbf{V(S_2)}$ and $\mathbf{V(S_3)}$ is $\mathbf{T}$ and their join is $\mathbf{RDQDSH_1}$ in $\mathcal{L}$. It, therefore, follows that $\mathcal{L}$ is isomorphic to $\mathbf{(1+B_9)} \times \mathbf{(1+B_5)} \times \mathbf{B_9}$. The lattice of subvarieties of $\mathbf{RDMSH_1}$ is isomorphic to $\mathbf{(1+B_5)} \times \mathbf{(1+B_3)} \times \mathbf{B_5}$. The lattice of subvarieties of $\mathbf{RDPCSH_1}$ is isomorphic to $\mathbf{(1+B_4)} \times \mathbf{(1+B_2)} \times \mathbf{B_4}$. Similar formulas can be obtained for other subvarieties of $\mathbf{RDQDSH_1}$. Amalgamation ============ We now examine the Amalgamation Property for subvarities of the variety $\mathbf{RDQDStSH_1}$. For this purpose we need the following theorem from [@GrLa71]. Let K be an equational class of algebras satisfying the Congruence Extension Property, and let every subalgebra of each subdirectly irreducible algebra in K be subdirectly irreducible. Then K satisfies the Amalgamation Property if and only if whenever A, B, C are subdirectly irreducible algebras in K with A a common subalgebra of B and C, the amalgam (A; B, C) can be amalgamated in K. Every subvariety of $\mathbf{RDQDStSH_1}$ has the Amalgamation Property. It follows from [@Sa12] that $\mathbf{RDQDStSH_1}$ has CEP. Also, it follows from Theorem \[Th\] that every subalgebra of each subdirectly irreducible (= simple) algebra in $\mathbf{RDQDStSH_1}$ is subdirectly irreducible. Therefore, in each subvariety $\mathbf{V}$ of $\mathbf{RDQDStSH_1}$, we need only consider an amalgam $(A: B, C)$, where $A, B, C$ are simple in $\mathbf{RDQDStSH_1}$ and $A$ a subalgebra of $B$ and $C$. Then it is not hard to see, in view of the description of simples in $\mathbf{RDQDStSH_1}$ given in Theorem \[Th\], that $(A: B, C)$ can be amalgamated in $\mathbf{V}$. Concluding Remarks and Open Problems ==================================== We know from [@Sa12] that every simple algebra in $\mathbf{RDQDH_1}$ is quasiprimal. Of all the 25 simple algebras in $\mathbf{RDQDStSH_1}$, $\mathbf{2^e}$, $\mathbf{\bar{2}^e}$, and $\mathbf{L_i}, i = 5,6,7,8$, and $\mathbf{D_3}$ are primal algebras and the rest are semiprimal algebras. We now mention some open problems for further research.\ Problem 1: For each variety $\mathbf{V(L)}$, where $\mathbf{L}$ is a simple algebra in $\mathbf{RDMSH_1}$ (except $\mathbf{V(2^e)}$), find a Propositional Calculus $\mathbf{P(V)}$ such that the equivalent algebraic semantics for $\mathbf{P(V)}$ is $\mathbf{V(L)}$ (with $1$ as the designated truth value, using $\to$ and $'$ as implication and negation respectively). (For the variety $\mathbf{V(2^e)}$, the answer is, of course, well known: Classical Propositional Calculus.) We think such (many-valued) logics will be of interest in computer science and in switcing circuit theory.\ Problem 2: Describe simples in the variety of pseudocommutative $\mathbf{RDQDStSH_1}$-algebras.\ Problem 3: Find equational bases for the remaining subvarieties of $\mathbf{RDMSH_1}$.\ Problem 4: Let $\mathbf{RDmsStSH_1}$ denote the subvariety of $\mathbf{DQDStSH_1}$ defined by: $(x \lor y)' \approx x' \land y'$. Describe simples in $\mathbf{RDmsStSH_1}$.\ [99]{} R. Balbes and PH. Dwinger, [**Distributive lattices**]{}, Univ. of Missouri Press, Columbia, 1974. S. Burris and H.P. Sankappanavar, [**A course in universal algebra**]{}, Springer–Verlag, New York, 1981. The free, corrected version (2012) is available online as a PDF file at [math.uwaterloo.ca/$\sim$snburris]{}. G. Grätzer and H. Lakser, [*The structure of pseudocomplemented distributive lattices II: Congruence extension and amalgamation*]{}, Trans. Amer. Math. Soc., 156 (1971), 343-358. B. Jónsson, [*Algebras whose congruence lattices are distributive*]{}, Math. Scand. 21 (1967), 110–121. A. Monteiro, [*Sur les algebres de Heyting symetriques*]{}, Portugaliae Mathemaica 39 (1980), 1–237. W. McCune, [*Prover9 and Mace 4, http://www.cs.unm.edu/mccune/prover9/*]{} H. Rasiowa, [**An algebraic approach to non-classical logics**]{}, North–Holland Publ.Comp., Amsterdam, (1974). H.P. Sankappanavar, [*Heyting algebras with dual pseudocomplementation*]{}, Pacific J. Math. 117 (1985), 405–415. H.P. Sankappanavar, [*Pseudocomplemented Okham and De Morgan algebras*]{}, Zeitschr. f. math. Logik und Grundlagen d. Math. 32 (1986), 385–394. H.P. Sankappanavar, [*Heyting algebras with a dual lattice endomorphism*]{}, Zeitschr. f. math. Logik und Grundlagen d. Math. 33 (1987), 565–573. H.P. Sankappanavar, [*Semi-De Morgan algebras*]{}, J. Symbolic. Logic 52 (1987), 712–724. H.P. Sankappanavar, [*Semi–Heyting algebras: An abstraction from Heyting algebras*]{}, Actas del IX Congreso Dr. A. Monteiro (2007), 33-66. H.P. Sankappanavar, [*Semi–Heyting algebras II*]{}. In Preparation. H.P. Sankappanavar, [*Expansions of Semi-Heyting algebras. I: Discriminator varieties*]{}, Studia Logica 98 (1-2) (2011), 27-81. H.P. Sankappanavar, [*Dually quasi-De Morgan Stone semi-Heyting algebras I. Regularity*]{}, Categories and General Algebraic Structures with Applications, 2 (2014), 47-64. H.P. Sankappanavar, [*Dually quasi-De Morgan Stone semi-Heyting algebras II. Regularity*]{}, Categories and General Algebraic Structures with Applications, 2 (2014), 65-82. H.P. Sankappanavar, [*Expansions of Semi-Heyting algebras. II*]{}. In Preparation. J, Varlet, [*A regular variety of type $\langle 2,2,1,1,0,0 \rangle$*]{}, Algebra Univ. 2 (1972), 218-223. H. Werner, [**Discriminator algebras**]{}, Studien zur Algebra und ihre Anwendungen, Band 6, Academie–Verlag, Berlin, 1978.  \         \ Department of Mathematics\ State University of New York\ New Paltz, NY 12561\ \ sankapph@newpaltz.edu\

--- abstract: | For any $m \geq 1$, let $H_m$ denote the quantity $\liminf_{n \to \infty} (p_{n+m}-p_n)$, where $p_n$ is the $n^{\operatorname{th}}$ prime. A celebrated recent result of Zhang showed the finiteness of $H_1$, with the explicit bound $H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $H_1 \leq 4680$, and then by Maynard to $H_1 \leq 600$, who also established for the first time a finiteness result for $H_m$ for $m \geq 2$, and specifically that $H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $H_1 \leq 12$, improving upon the previous bound $H_1 \leq 16$ of Goldston, Pintz, and Y[i]{}ld[i]{}r[i]{}m, as well as the bound $H_m \ll m^3 e^{2m}$. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound $H_1 \leq 246$ unconditionally, and $H_1 \leq 6$ under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple $(h_1,h_2,h_3)$, there are infinitely many $n$ for which at least two of $n+h_1,n+h_2,n+h_3$ are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds, or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most $2$, or both. We also modify the “parity problem” argument of Selberg to show that the $H_1 \leq 6$ bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger $m$, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound $H_m \ll m e^{(4-\frac{28}{157})m}$, or $H_m \ll m e^{2m}$ under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for $H_m$ when $m=2,3,4,5$. address: ' , ' author: - title: 'Variants of the Selberg sieve, and bounded intervals containing many primes' --- Introduction ============ For any natural number $m$, let $H_m$ denote the quantity $$H_m \coloneqq \liminf_{n \to \infty} (p_{n+m} - p_n),$$ where $p_n$ denotes the $n^{\operatorname{th}}$ prime. The twin prime conjecture asserts that $H_1=2$; more generally, the Hardy-Littlewood prime tuples conjecture [@hardy] implies that $H_m = H(m+1)$ for all $m \geq 1$, where $H(k)$ is the diameter of the narrowest admissible $k$-tuple (see Section \[subclaim-sec\] for a definition of this term). Asymptotically, one has the bounds $$(\frac{1}{2}+o(1)) k \log k \leq H(k) \leq (1+o(1)) k \log k$$ as $k \to \infty$ (see Theorem \[hk-bound\] below); thus the prime tuples conjecture implies that $H_m$ is comparable to $m \log m$ as $m \to \infty$. Until very recently, it was not known if any of the $H_m$ were finite, even in the easiest case $m=1$. In the breakthrough work of Goldston, Pintz, and Y[i]{}ld[i]{}r[i]{}m [@gpy], several results in this direction were established, including the following conditional result assuming the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ (see Claim \[eh-def\] below) concerning the distribution of the prime numbers in arithmetic progressions: \[gpy-thm\] Assume the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ for all $0 < \vartheta < 1$. Then $H_1 \leq 16$. Furthermore, it was shown in [@gpy] that any result of the form $\operatorname*{EH}[\frac{1}{2} + 2\varpi]$ for some fixed $0 < \varpi < 1/4$ would imply an explicit finite upper bound on $H_1$ (with this bound equal to $16$ for $\varpi > 0.229855$). Unfortunately, the only results of the type $\operatorname*{EH}[\vartheta]$ that are known come from the Bombieri-Vinogradov theorem (Theorem \[bv-thm\]), which only establishes $\operatorname*{EH}[\vartheta]$ for $0 < \vartheta < 1/2$. The first unconditional bound on $H_1$ was established in a breakthrough work of Zhang [@zhang]: \[zhang-thm\] $H_1 \leq \num{70000000}$. Zhang’s argument followed the general strategy from [@gpy] on finding small gaps between primes, with the major new ingredient being a proof of a weaker version of $\operatorname*{EH}[\frac{1}{2}+2\varpi]$, which we call $\operatorname*{MPZ}[\varpi,\delta]$; see Claim \[mpz-claim\] below. It was quickly realized that Zhang’s numerical bound on $H_1$ could be improved. By optimizing many of the components in Zhang’s argument, we were able [@polymath8a; @polymath8a-unabridged] to improve Zhang’s bound to $$H_1 \leq \num{4680}.$$ Very shortly afterwards, a further breakthrough was obtained by Maynard [@maynard-new] (with related work obtained independently in unpublished work of Tao), who developed a more flexible “multidimensional” version of the Selberg sieve to obtain stronger bounds on $H_m$. This argument worked without using any equidistribution results on primes beyond the Bombieri-Vinogradov theorem, and amongst other things was able to establish finiteness of $H_m$ for all $m$, not just for $m=1$. More precisely, Maynard established the following results. Unconditionally, we have the following bounds: - $H_1 \leq 600$. - $H_m \leq C m^3 e^{4m}$ for all $m \geq 1$ and an absolute (and effective) constant $C$. Assuming the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ for all $0 < \vartheta < 1$, we have the following improvements: - $H_1 \leq 12$. - $H_2 \leq 600$. - $H_m \leq C m^3 e^{2m}$ for all $m \geq 1$ and an absolute (and effective) constant $C$. For a survey of these recent developments, see [@granville]. In this paper, we refine Maynard’s methods to obtain the following further improvements. \[main\] Unconditionally, we have the following bounds: - $H_1 \leq 246$. - $H_2 \leq \num{398130}$. - $H_3 \leq \num{24797814}$. - $H_4 \leq \num{1431556072}$. - $H_5 \leq \num{80550202480}$. - $H_m \leq C m \exp( (4 - \frac{28}{157}) m )$ for all $m \geq 1$ and an absolute (and effective) constant $C$. Assume the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ for all $0 < \vartheta < 1$. Then we have the following improvements: - $H_2 \leq 270$. - $H_3 \leq \num{52116}$. - $H_4 \leq \num{474266}$. - $H_5 \leq \num{4137854}$. - $H_m \leq Cme^{2m}$ for all $m \geq 1$ and an absolute (and effective) constant $C$. Finally, assume the generalized Elliott-Halberstam conjecture $\operatorname*{GEH}[\vartheta]$ (see Claim \[geh-def\] below) for all $0 < \vartheta < 1$. Then - $H_1 \leq 6$. - $H_2 \leq 252$. In Section \[subclaim-sec\] we will describe the key propositions that will be combined together to prove the various components of Theorem \[main\]. As with Theorem \[gpy-thm\], the results in (vii)-(xiii) do not require $\operatorname*{EH}[\vartheta]$ or $\operatorname*{GEH}[\vartheta]$ for all $0 < \vartheta < 1$, but only for a single explicitly computable $\vartheta$ that is sufficiently close to $1$. Of these results, the bound in (xii) is perhaps the most interesting, as the parity problem [@selberg] prohibits one from achieving any better bound on $H_1$ than $6$ from purely sieve-theoretic methods; we review this obstruction in Section \[parity-sec\]. If one only assumes the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ instead of its generalization $\operatorname*{GEH}[\vartheta]$, we were unable to improve upon Maynard’s bound $H_1 \leq 12$; however the parity obstruction does not exclude the possibility that one could achieve (xii) just assuming $\operatorname*{EH}[\vartheta]$ rather than $\operatorname*{GEH}[\vartheta]$, by some further refinement of the sieve-theoretic arguments (e.g. by finding a way to establish Theorem \[nonprime-asym\](ii) below using only $\operatorname*{EH}[\vartheta]$ instead of $\operatorname*{GEH}[\vartheta]$). The bounds (ii)-(vi) rely on the equidistribution results on primes established in our previous paper [@polymath8a]. However, the bound (i) uses only the Bombieri-Vinogradov theorem, and the remaining bounds (vii)-(xiii) of course use either the Elliott-Halberstam conjecture or a generalization thereof. A variant of the proof of Theorem \[main\](xii), which we give in Section \[remarks-sec\], also gives the following conditional “near miss” to (a disjunction of) the twin prime conjecture and the even Goldbach conjecture: \[disj\] Assume the generalized Elliott-Halberstam conjecture $\operatorname*{GEH}[\vartheta]$ for all $0 < \vartheta < 1$. Then at least one of the following statements is true: - (Twin prime conjecture) $H_1=2$. - (near-miss to even Goldbach conjecture) If $n$ is a sufficiently large multiple of six, then at least one of $n$ and $n-2$ is expressible as the sum of two primes. Similarly with $n-2$ replaced by $n+2$. (In particular, every sufficiently large even number lies within $2$ of the sum of two primes.) We remark that a disjunction in a similar spirit was obtained in [@bgc], which established (prior to the appearance of Theorem \[zhang-thm\]) that either $H_1$ was finite, or that every interval $[x,x+x^{\varepsilon}]$ contained the sum of two primes if $x$ was sufficiently large depending on ${\varepsilon}>0$. There are two main technical innovations in this paper. The first is a further generalization of the multidimensional Selberg sieve introduced by Maynard and Tao, in which the support of a certain cutoff function $F$ is permitted to extend into a larger domain than was previously permitted (particularly under the assumption of the generalized Elliott-Halberstam conjecture). As in [@maynard-new], this largely reduces the task of bounding $H_m$ to that of efficiently solving a certain multidimensional variational problem involving the cutoff function $F$. Our second main technical innovation is to obtain efficient numerical methods for solving this variational problem for small values of the dimension $k$, as well as sharpened asymptotics in the case of large values of $k$. The methods of Maynard and Tao have been used in a number of subsequent applications [@freiberg], [@consecutive], [@thorner], [@benatar], [@li-pan], [@castillo], [@pollack], [@banks], [@clusters], [@lola], [@pintz-ratio], [@chua], [@pintz-new]. The techniques in this paper should be able to be used to obtain slight numerical improvements to such results, although we did not pursue these matters here. Organization of the paper ------------------------- The paper is organized as follows. After some notational preliminaries, we recall in Section \[dist-sec\] the known (or conjectured) distributional estimates on primes in arithmetic progressions that we will need to prove Theorem \[main\]. Then, in Section \[subclaim-sec\], we give the key propositions that will be combined together to establish this theorem. One of these propositions, Lemma \[crit\], is an easy application of the pigeonhole principle. Two further propositions, Theorem \[prime-asym\] and Theorem \[nonprime-asym\], use the prime distribution results from Section \[dist-sec\] to give asymptotics for certain sums involving sieve weights and the von Mangoldt function; they are established in Section \[sieving-sec\]. Theorems \[maynard-thm\], \[maynard-trunc\], \[epsilon-trick\], \[epsilon-beyond\] use the asymptotics established in Theorems \[prime-asym\], \[nonprime-asym\], in combination with Lemma \[crit\], to give various criteria for bounding $H_m$, which all involve finding sufficiently strong candidates for a variety of multidimensional variational problems; these theorems are proven in Section \[variational-sec\]. These variational problems are analysed in the asymptotic regime of large $k$ in Section \[asymptotics-sec\], and for small and medium $k$ in Section \[h1-sec\], with the results collected in Theorems \[mlower\], \[mlower-var\], \[mke-lower\], \[piece\]. Combining these results with the previous propositions gives Theorem \[main-dhl\], which, when combined with the bounds on narrow admissible tuples in Theorem \[hk-bound\] that are established in Section \[tuples-sec\], will give Theorem \[main\]. (See also Table \[ingredients\] for some more details of the logical dependencies between the key propositions.) Finally, in Section \[parity-sec\] we modify an argument of Selberg to show that the bound $H_1 \leq 6$ may not be improved using purely sieve-theoretic methods, and in Section \[remarks-sec\] we establish Theorem \[disj\] and make some miscellaneous remarks. Notation -------- The notation used here closely follows the notation in our previous paper [@polymath8a]. We use $|E|$ to denote the cardinality of a finite set $E$, and ${\mathbf{1}}_E$ to denote the indicator function of a set $E$, thus ${\mathbf{1}}_E(n)=1$ when $n \in E$ and ${\mathbf{1}}_E(n)=0$ otherwise. In a similar spirit, if $E$ is a statement, we write ${\mathbf{1}}_E=1$ when $E$ is true and ${\mathbf{1}}_E=0$ otherwise. All sums and products will be over the natural numbers ${\mathbb{N}}\coloneqq \{1,2,3,\ldots\}$ unless otherwise specified, with the exceptions of sums and products over the variable $p$, which will be understood to be over primes. The following important asymptotic notation will be in use throughout the paper: \[asym\] We use $x$ to denote a large real parameter, which one should think of as going off to infinity; in particular, we will implicitly assume that it is larger than any specified fixed constant. Some mathematical objects will be independent of $x$ and referred to as *fixed*; but unless otherwise specified we allow all mathematical objects under consideration to depend on $x$ (or to vary within a range that depends on $x$, e.g. the summation parameter $n$ in the sum $\sum_{x \leq n \leq 2x} f(n)$). If $X$ and $Y$ are two quantities depending on $x$, we say that $X = O(Y)$ or $X \ll Y$ if one has $|X| \leq CY$ for some fixed $C$ (which we refer to as the *implied constant*), and $X = o(Y)$ if one has $|X| \leq c(x) Y$ for some function $c(x)$ of $x$ (and of any fixed parameters present) that goes to zero as $x \to \infty$ (for each choice of fixed parameters). We use $X {\llcurly}Y$ to denote the estimate $|X| \leq x^{o(1)} Y$, $X {\asymp}Y$ to denote the estimate $Y \ll X \ll Y$, and $X \approx Y$ to denote the estimate $Y {\llcurly}X {\llcurly}Y$. Finally, we say that a quantity $n$ is of *polynomial size* if one has $n = O(x^{O(1)})$. If asymptotic notation such as $O()$ or ${\llcurly}$ appears on the left-hand side of a statement, this means that the assertion holds true for any specific interpretation of that notation. For instance, the assertion $\sum_{n=O(N)} |\alpha(n)| {\llcurly}N$ means that for each fixed constant $C>0$, one has $\sum_{|n| \leq CN} |\alpha(n)|{\llcurly}N$. If $q$ and $a$ are integers, we write $a|q$ if $a$ divides $q$. If $q$ is a natural number and $a \in {\mathbb{Z}}$, we use $a\ (q)$ to denote the residue class $$a\ (q) \coloneqq \{ a+nq: n \in {\mathbb{Z}}\}$$ and let ${\mathbb{Z}}/q{\mathbb{Z}}$ denote the ring of all such residue classes $a\ (q)$. The notation $b=a\ (q)$ is synonymous to $b \in \, a \ (q)$. We use $(a,q)$ to denote the greatest common divisor of $a$ and $q$, and $[a,q]$ to denote the least common multiple.[^1] We also let $$({\mathbb{Z}}/q{\mathbb{Z}})^\times \coloneqq \{ a\ (q): (a,q)=1 \}$$ denote the primitive residue classes of ${\mathbb{Z}}/q{\mathbb{Z}}$. We use the following standard arithmetic functions: - ${\varphi}(q) \coloneqq |({\mathbb{Z}}/q{\mathbb{Z}})^\times|$ denotes the Euler totient function of $q$. - $\tau(q) \coloneqq \sum_{d|q} 1$ denotes the divisor function of $q$. - $\Lambda(q)$ denotes the von Mangoldt function of $q$, thus $\Lambda(q)=\log p$ if $q$ is a power of a prime $p$, and $\Lambda(q)=0$ otherwise. - $\theta(q)$ is defined to equal $\log q$ when $q$ is a prime, and $\theta(q)=0$ otherwise. - $\mu(q)$ denotes the Möbius function of $q$, thus $\mu(q) = (-1)^k$ if $q$ is the product of $k$ distinct primes for some $k \geq 0$, and $\mu(q)=0$ otherwise. - $\Omega(q)$ denotes the number of prime factors of $q$ (counting multiplicity). We recall the elementary *divisor bound* $$\label{divisor-bound} \tau(n) {\llcurly}1$$ whenever $n \ll x^{O(1)}$, as well as the related estimate $$\label{divisor-2} \sum_{n \ll x} \frac{\tau(n)^C}{n} \ll \log^{O(1)} x$$ for any fixed $C>0$; this follows for instance from [@polymath8a Lemma 1.3]. The *Dirichlet convolution* $\alpha \star \beta \colon {\mathbb{N}}\to {\mathbb{C}}$ of two arithmetic functions $\alpha,\beta \colon {\mathbb{N}}\to {\mathbb{C}}$ is defined in the usual fashion as $$\alpha\star\beta(n) \coloneqq \sum_{d|n} \alpha(d) \beta\left(\frac{n}{d}\right) =\sum_{ab=n}{\alpha(a)\beta(b)}.$$ Distribution estimates on arithmetic functions {#dist-sec} ============================================== As mentioned in the introduction, a key ingredient in the Goldston-Pintz-Y[i]{}ld[i]{}r[i]{}m approach to small gaps between primes comes from distributional estimates on the primes, or more precisely on the von Mangoldt function $\Lambda$, which serves as a proxy for the primes. In this work, we will also need to consider distributional estimates on more general arithmetic functions, although we will not prove any new such estimates in this paper, relying instead on estimates that are already in the literature. More precisely, we will need averaged information on the following quantity: For any function $\alpha \colon {\mathbb{N}}\to {\mathbb{C}}$ with finite support (that is, $\alpha$ is non-zero only on a finite set) and any primitive residue class $a\ (q)$, we define the (signed) *discrepancy* $\Delta(\alpha; a\ (q))$ to be the quantity $$\label{disc-def} \Delta(\alpha; a\ (q)) \coloneqq \sum_{n = a\ (q)} \alpha(n) - \frac{1}{{\varphi}(q)} \sum_{(n,q)=1} \alpha(n).$$ For any fixed $0 < \vartheta < 1$, let $\operatorname*{EH}[\vartheta]$ denote the following claim: \[eh-def\] If $Q {\llcurly}x^\vartheta$ and $A \geq 1$ is fixed, then $$\label{qq} \sum_{q \leq Q} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta(\Lambda {\mathbf{1}}_{[x,2x]}; a\ (q))| \ll x \log^{-A} x.$$ In [@elliott] it was conjectured that $\operatorname*{EH}[\vartheta]$ held for all $0 < \vartheta < 1$. (The conjecture fails at the endpoint case $\vartheta=1$; see [@fg], [@fghm] for a more precise statement.) The following classical result of Bombieri [@bombieri] and Vinogradov [@vinogradov] remains the best partial result of the form $\operatorname*{EH}[\vartheta]$: \[bv-thm\][@bombieri; @vinogradov] $\operatorname*{EH}[\vartheta]$ holds for every fixed $0 < \vartheta < 1/2$. In [@gpy] it was shown that any estimate of the form $\operatorname*{EH}[\vartheta]$ with some fixed $\vartheta > 1/2$ would imply the finiteness of $H_1$. While such an estimate remains unproven, it was observed by Motohashi-Pintz [@mp] and by Zhang [@zhang] that a certain weakened version of $\operatorname*{EH}[\vartheta]$ would still suffice for this purpose. More precisely (and following the notation of our previous paper [@polymath8a]), let $\varpi,\delta > 0$ be fixed, and let $\operatorname*{MPZ}[\varpi,\delta]$ be the following claim: \[mpz-claim\] Let $I \subset [1,x^\delta]$ and $Q {\llcurly}x^{1/2+2\varpi}$. Let $P_I$ denote the product of all the primes in $I$, and let ${\mathcal{S}}_I$ denote the square-free natural numbers whose prime factors lie in $I$. If the residue class $a\ (P_I)$ is primitive (and is allowed to depend on $x$), and $A \geq 1$ is fixed, then $$\label{qq-mpz} \sum_{\substack{q \leq Q\\q \in {\mathcal{S}}_I}} |\Delta(\Lambda {\mathbf{1}}_{[x,2x]}; a\ (q))| \ll x \log^{-A} x,$$ where the implied constant depends only on the fixed quantities $(A,\varpi,\delta)$, but not on $a$. It is clear that $\operatorname*{EH}[\frac{1}{2}+2\varpi]$ implies $\operatorname*{MPZ}[\varpi,\delta]$ whenever $\varpi,\delta \geq 0$. The first non-trivial estimate of the form $\operatorname*{MPZ}[\varpi,\delta]$ was established by Zhang [@zhang], who (essentially) obtained $\operatorname*{MPZ}[\varpi,\delta]$ whenever $0 \leq \varpi, \delta < \frac{1}{1168}$. In [@polymath8a Theorem 2.17], we improved this result to the following. \[mpz-poly\] $\operatorname*{MPZ}[\varpi,\delta]$ holds for every fixed $\varpi,\delta \geq 0$ with $600 \varpi + 180 \delta < 7$. In fact, a stronger result was established in [@polymath8a], in which the moduli $q$ were assumed to be *densely divisible* rather than smooth, but we will not exploit such improvements here. For our application, the most important thing is to get $\varpi$ as large as possible; in particular, Theorem \[mpz-poly\] allows one to get $\varpi$ arbitrarily close to $\frac{7}{600} \approx 0.01167$. In this paper, we will also study the following generalization of the Elliott-Halberstam conjecture for a fixed choice of $0 < \vartheta < 1$: \[geh-def\] Let ${\varepsilon}> 0$ and $A \geq 1$ be fixed. Let $N,M$ be quantities such that $x^{\varepsilon}{\llcurly}N {\llcurly}x^{1-{\varepsilon}}$ and $x^{\varepsilon}{\llcurly}M {\llcurly}x^{1-{\varepsilon}}$ with $NM {\asymp}x$, and let $\alpha, \beta: {\mathbb{N}}\to {\mathbb{R}}$ be sequences supported on $[N,2N]$ and $[M,2M]$ respectively, such that one has the pointwise bounds $$\label{ab-div} |\alpha(n)| \ll \tau(n)^{O(1)} \log^{O(1)} x; \quad |\beta(m)| \ll \tau(m)^{O(1)} \log^{O(1)} x$$ for all natural numbers $n,m$. Suppose also that $\beta$ obeys the Siegel-Walfisz type bound $$\label{sig} | \Delta(\beta {\mathbf{1}}_{(\cdot,r)=1}; a\ (q)) | \ll \tau(qr)^{O(1)} M \log^{-A} x$$ for any $q,r \geq 1$, any fixed $A$, and any primitive residue class $a\ (q)$. Then for any $Q {\llcurly}x^\vartheta$, we have $$\label{qq-gen} \sum_{q \leq Q} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta(\alpha \star \beta; a\ (q))| \ll x \log^{-A} x.$$ In [@bfi Conjecture 1] it was essentially conjectured[^2] that $\operatorname*{GEH}[\vartheta]$ was true for all $0 < \vartheta < 1$. This is stronger than the Elliott-Halberstam conjecture: \[geh-eh\] For any fixed $0 < \vartheta < 1$, $\operatorname*{GEH}[\vartheta]$ implies $\operatorname*{EH}[\vartheta]$. (Sketch) As this argument is standard, we give only a brief sketch. Let $A > 0$ be fixed. For $n \in [x,2x]$, we have Vaughan’s identity[^3] [@vaughan] $$\Lambda(n) = \mu_< \star L(n) - \mu_< \star \Lambda_< \star 1(n) + \mu_\geq \star \Lambda_\geq \star 1(n),$$ where $L(n) \coloneqq \log(n)$, $1(n) \coloneqq 1$, and $$\begin{gathered} \Lambda_\geq(n) \coloneqq \Lambda(n) {\mathbf{1}}_{n \geq x^{1/3}},\quad\quad \Lambda_<(n) \coloneqq \Lambda(n) {\mathbf{1}}_{n < x^{1/3}}\\ \mu_\geq(n) \coloneqq \mu(n) {\mathbf{1}}_{n \geq x^{1/3}},\quad\quad \mu_<(n) \coloneqq \mu(n) {\mathbf{1}}_{n < x^{1/3}}.\end{gathered}$$ By decomposing each of the functions $\mu_<$, $\mu_\geq$, $1$, $\Lambda_<$, $\Lambda_{\geq}$ into $O(\log^{A+1} x)$ functions supported on intervals of the form $[N, (1+\log^{-A} x) N]$, and discarding those contributions which meet the boundary of $[x,2x]$ (cf. [@fouvry], [@fi-2], [@bfi], [@zhang]), and using $\operatorname*{GEH}[\vartheta]$ (with $A$ replaced by a much larger fixed constant $A'$) to control all remaining contributions, we obtain the claim (using the Siegel-Walfisz theorem, see e.g. [@siebert Satz 4] or [@ik Th. 5.29]). By modifying the proof of the Bombieri-Vinogradov theorem Motohashi [@motohashi] established the following generalization of that theorem (see also [@gallagher] for some related ideas): \[gbv-thm\][@motohashi] $\operatorname*{GEH}[\vartheta]$ holds for every fixed $0 < \vartheta < 1/2$. One could similarly describe a generalization of the Motohashi-Pintz-Zhang estimate $\operatorname*{MPZ}[\varpi,\delta]$, but unfortunately the arguments in [@zhang] or Theorem \[mpz-poly\] do not extend to this setting unless one is in the “Type I/Type II” case in which $N,M$ are constrained to be somewhat close to $x^{1/2}$, or if one has “Type III” structure to the convolution $\alpha \star \beta$, in the sense that it can refactored as a convolution involving several “smooth” sequences. In any event, our analysis would not be able to make much use of such incremental improvements to $\operatorname*{GEH}[\vartheta]$, as we only use this hypothesis effectively in the case when $\vartheta$ is very close to $1$. In particular, we will not directly use Theorem \[gbv-thm\] in this paper. Outline of the key ingredients {#subclaim-sec} ============================== In this section we describe the key subtheorems used in the proof of Theorem \[main\], with the proofs of these subtheorems mostly being deferred to later sections. We begin with a weak version of the Dickson-Hardy-Littlewood prime tuples conjecture [@hardy], which (following Pintz [@pintz-polignac]) we refer to as $\operatorname*{DHL}[k,j]$. Recall that for any $k \in \mathbb{N}$, an *admissible $k$-tuple* is a tuple ${\mathcal H} = (h_1,\ldots,h_{k})$ of $k$ increasing integers $h_1 < \ldots < h_{k}$ which avoids at least one residue class $a_p\ (p) := \{ a_p + np: n \in {\mathbb{Z}}\}$ for every $p$. For instance, $(0,2,6)$ is an admissible $3$-tuple, but $(0,2,4)$ is not. For any $k \geq j \geq 2$, we let $\operatorname*{DHL}[k,j]$ denote the following claim: For any admissible $k$-tuple ${\mathcal H}=(h_1,\ldots,h_{k})$ there exist infinitely many translates $n + {\mathcal H} = (n+h_1,\ldots,n+h_{k})$ of ${\mathcal H}$ which contain at least $j$ primes. The full Dickson-Hardy-Littlewood conjecture is then the assertion that $\operatorname*{DHL}[k,k]$ holds for all $k \ge 2$. In our analysis we will focus on the case when $j$ is much smaller than $k$; in fact $j$ will be of the order of $\log k$. For any $k$, let $H(k)$ denote the minimal diameter $h_k-h_1$ of an admissible $k$-tuple; thus for instance $H(3)=6$. It is clear that for any natural numbers $m \geq 1$ and $k \geq m+1$, the claim $\operatorname*{DHL}[k,m+1]$ implies that $H_m \leq H(k)$ (and the claim $\operatorname*{DHL}[k,k]$ would imply that $H_{k-1} = H(k)$). We will therefore deduce Theorem \[main\] from a number of claims of the form $\operatorname*{DHL}[k,j]$. More precisely, we have: \[main-dhl\] Unconditionally, we have the following claims: - $\operatorname*{DHL}[50,2]$. - $\operatorname*{DHL}[\num{35410},3]$. - $\operatorname*{DHL}[\num{1649821},4]$. - $\operatorname*{DHL}[\num{75845707},5]$. - $\operatorname*{DHL}[\num{3473955908},6]$. - $\operatorname*{DHL}[k,m+1]$ whenever $m \geq 1$ and $k \geq C\exp( (4 - \frac{28}{157}) m )$ for some sufficiently large absolute (and effective) constant $C$. Assume the Elliott-Halberstam conjecture $\operatorname*{EH}[\vartheta]$ for all $0 < \vartheta < 1$. Then we have the following improvements: - $\operatorname*{DHL}[54,3]$. - $\operatorname*{DHL}[\num{5511},4]$. - $\operatorname*{DHL}[\num{41588},5]$. - $\operatorname*{DHL}[\num{309661},6]$. - $\operatorname*{DHL}[k,m+1]$ whenever $m \geq 1$ and $k \geq C\exp( 2 m )$ for some sufficiently large absolute (and effective) constant $C$. Assume the generalized Elliott-Halberstam conjecture $\operatorname*{GEH}[\vartheta]$ for all $0 < \vartheta < 1$. Then - $\operatorname*{DHL}[3,2]$. - $\operatorname*{DHL}[51,3]$. Theorem \[main\] then follows from Theorem \[main-dhl\] and the following bounds on $H(k)$ (ordered by increasing value of $k$): \[hk-bound\] - $H(3)=6$. - $H(50) = 246$. - $H(51) = 252$. - $H(54) = 270$. - $H(\num{5511}) \leq \num{52116}$. - $H(\num{35410}) \leq \num{398130}$. - $H(\num{41588}) \leq \num{474266}$. - $H(\num{309661}) \leq \num{4137854}$. - $H(\num{1649821}) \leq \num{24797814}$. - $H(\num{75845707}) \leq \num{1431556072}$. - $H(\num{3473955908}) \leq \num{80550202480}$. - In the asymptotic limit $k \to \infty$, one has $H(k) \leq k \log k + k \log\log k - k + o(k)$, with the bounds on the decay rate $o(k)$ being effective. We prove Theorem \[hk-bound\] in Section \[tuples-sec\]. In the opposite direction, an application of the Brun-Titchmarsh theorem gives $H(k) \geq (\frac{1}{2} + o(1)) k \log k$ as $k \to \infty$; see [@polymath8a-unabridged §3.9] for this bound, as well as with some slight refinements. Theorem \[main-dhl\] Results used ---------------------- ---------------------------------------------------------- (i) Theorems \[bv-thm\], \[epsilon-trick\], \[mke-lower\] (ii)-(vi) Theorems \[mpz-poly\], \[maynard-trunc\], \[mlower-var\] (vii)-(xi) Theorems \[maynard-thm\], \[mlower\] (xii) Theorems \[epsilon-beyond\], \[piece\] (xiii) Theorems \[epsilon-trick\], \[mke-lower\] : Results used to prove various components of Theorem \[main-dhl\]. Note that Theorems \[maynard-thm\], \[maynard-trunc\], \[epsilon-trick\], \[epsilon-beyond\] are in turn proven using Theorems \[prime-asym\], \[nonprime-asym\], and Lemma \[crit\].[]{data-label="ingredients"} The proof of Theorem \[main-dhl\] follows the Goldston-Pintz-Y[i]{}ld[i]{}r[i]{}m strategy that was also used in all previous progress on this problem (e.g. [@gpy], [@mp], [@zhang], [@polymath8a], [@maynard-new]), namely that of constructing a sieve function adapted to an admissible $k$-tuple with good properties. More precisely, we set $$w := \log \log \log x$$ and $$W := \prod_{p \leq w} p,$$ and observe the crude bound $$\label{W-bound} W \ll \log \log^{O(1)} x.$$ We have the following simple “pigeonhole principle” criterion for $\operatorname*{DHL}[k,m+1]$ (cf. [@polymath8a Lemma 4.1], though the normalization here is slightly different): \[crit\] Let $k \geq 2$ and $m \geq 1$ be fixed integers, and define the normalization constant $$\label{bnorm} B := \frac{{\varphi}(W)}{W} \log x.$$ Suppose that for each fixed admissible $k$-tuple $(h_1,\dots,h_k)$ and each residue class $b\ (W)$ such that $b+h_i$ is coprime to $W$ for all $i=1,\dots,k$, one can find a non-negative weight function $\nu \colon {\mathbb{N}}\to {\mathbb{R}}^+$ and fixed quantities $\alpha > 0$ and $\beta_1,\dots,\beta_k \geq 0$, such that one has the asymptotic upper bound $$\label{s1} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \leq (\alpha+o(1)) B^{-k} \frac{x}{W},$$ the asymptotic lower bound $$\label{s2} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_i) \geq (\beta_i-o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ for all $i=1,\dots,k$, and the key inequality $$\label{key} \frac{\beta_1 + \dots + \beta_k}{\alpha} > m.$$ Then $\operatorname*{DHL}[k,m+1]$ holds. Let $(h_1,\ldots,h_{k})$ be a fixed admissible $k$-tuple. Since it is admissible, there is at least one residue class $b\ (W)$ such that $(b+h_i,W)=1$ for all $h_i \in {\mathcal H}$. For an arithmetic function $\nu$ as in the lemma, we consider the quantity $$N:=\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \left(\sum_{i=1}^{k} \theta(n+h_i) - m\log 3x\right).$$ Combining and , we obtain the lower bound $$N\geq (\beta_1+\dots+\beta_k-o(1)) B^{1-k} \frac{x}{{\varphi}(W)} - (m\alpha+o(1)) B^{-k} \frac{x}{W} \log 3x.$$ From and the crucial condition , it follows that $N>0$ if $x$ is sufficiently large. On the other hand, the sum $$\sum_{i=1}^{k} \theta(n+h_i) - m\log 3x$$ can be positive only if $n+h_i$ is prime for *at least* $m+1$ indices $i=1, \ldots, k$. We conclude that, for all sufficiently large $x$, there exists some integer $n \in [x,2x]$ such that $n+h_i$ is prime for at least $m+1$ values of $i=1,\ldots,k$. Since $(h_1,\dots,h_k)$ is an arbitrary admissible $k$-tuple, $\operatorname*{DHL}[k,m+1]$ follows. The objective is then to construct non-negative weights $\nu$ whose associated ratio $\frac{\beta_1+\dots+\beta_k}{\alpha}$ has provable lower bounds that are as large as possible. Our sieve majorants will be a variant of the multidimensional Selberg sieves used in [@maynard-new]. As with all Selberg sieves, the $\nu$ are constructed as the square of certain (signed) divisor sums. The divisor sums we will use will be finite linear combinations of products of “one-dimensional” divisor sums. More precisely, for any fixed smooth compactly supported function $F: [0,+\infty) \to {\mathbb{R}}$, define the divisor sum $\lambda_F: {\mathbb{Z}}\to {\mathbb{R}}$ by the formula $$\label{lambdaf-def} \lambda_F(n) := \sum_{d|n} \mu(d) F( \log_x d )$$ where $\log_x$ denotes the base $x$ logarithm $$\label{logx-def} \log_x n:= \frac{\log n}{\log x}.$$ One should think of $\lambda_F$ as a smoothed out version of the indicator function to numbers $n$ which are “almost prime” in the sense that they have no prime factors less than $x^{\varepsilon}$ for some small fixed ${\varepsilon}>0$; see Proposition \[almostprime\] for a more rigorous version of this heuristic. The functions $\nu$ we will use will take the form $$\label{nuform} \nu(n) = \left( \sum_{j=1}^J c_j \lambda_{F_{j,1}}(n+h_1) \dots \lambda_{F_{j,k}}(n+h_k) \right)^2$$ for some fixed natural number $J$, fixed coefficients $c_1,\dots,c_J \in {\mathbb{R}}$ and fixed smooth compactly supported functions $F_{j,i}: [0,+\infty) \to {\mathbb{R}}$ with $j=1,\dots,J$ and $i=1,\dots,k$. (One can of course absorb the constant $c_j$ into one of the $F_{j,i}$ if one wishes.) Informally, $\nu$ is a smooth restriction to those $n$ for which $n+h_1,\dots,n+h_k$ are all almost prime. Clearly, $\nu$ is a (positive-definite) fixed linear combination of functions of the form $$n \mapsto \prod_{i=1}^k \lambda_{F_i}(n+h_i) \lambda_{G_i}(n+h_i)$$ for various fixed smooth functions $F_1,\dots,F_k,G_1,\dots,G_k: [0,+\infty) \to {\mathbb{R}}$. The sum appearing in can thus be decomposed into fixed linear combinations of sums of the form $$\label{sfg-1} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \prod_{i=1}^k \lambda_{F_i}(n+h_i) \lambda_{G_i}(n+h_i).$$ Also, if $F$ is supported on $[0,1]$, then from we clearly have $$\label{lambdan-prime} \lambda_F(n) = F(0)$$ when $n \geq x$ is prime, and so the sum appearing in can be similarly decomposed in this case into fixed linear combinations of sums of the form $$\label{sfg-2} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \theta(n+h_i) \prod_{1 \leq i' \leq k; i' \neq i} \lambda_{F_{i'}}(n+h_{i'}) \lambda_{G_{i'}}(n+h_{i'}).$$ To estimate the sums , we use the following asymptotic, proven in Section \[sieving-sec\]. For each compactly supported $F: [0,+\infty) \to {\mathbb{R}}$, let $$\label{S-def} S(F) \coloneqq \sup \{ x \geq 0: F(x) \neq 0\}$$ denote the upper range of the support of $F$ (with the convention that $S(0)=0$). \[prime-asym\] Let $k \geq 2$ be fixed, let $(h_1,\dots,h_k)$ be a fixed admissible $k$-tuple, and let $b\ (W)$ be such that $b+h_i$ is coprime to $W$ for each $i=1,\dots,k$. Let $1 \leq i_0 \leq k$ be fixed, and for each $1 \leq i \leq k$ distinct from $i_0$, let $F_{i}, G_{i}: [0,+\infty) \to {\mathbb{R}}$ be fixed smooth compactly supported functions. Assume one of the following hypotheses: - (Elliott-Halberstam) There exists a fixed $0 < \vartheta < 1$ such that $\operatorname*{EH}[\vartheta]$ holds, and such that $$\label{fg-upper} \sum_{1 \leq i \leq k; i \neq i_0} ( S(F_{i}) + S(G_{i}) ) < \vartheta.$$ - (Motohashi-Pintz-Zhang) There exists fixed $0 \leq \varpi < 1/4$ and $\delta > 0$ such that $\operatorname*{MPZ}[\varpi,\delta]$ holds, and such that $$\label{fg-upper-alt} \sum_{1 \leq i \leq k; i \neq i_0} ( S(F_{i}) + S(G_{i}) ) < \frac{1}{2} + 2 \varpi$$ and $$\max_{1 \leq i \leq k; i \neq i_0} \Bigl\{S(F_{i}), S(G_{i}) \Bigr\}< \delta.$$ Then we have $$\label{theta-oo} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \theta(n+h_{i_0}) \prod_{1 \leq i \leq k; i \neq i_0} \lambda_{F_{i}}(n+h_{i}) \lambda_{G_{i}}(n+h_{i}) = (c+o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ where $B$ is given by and $$c := \prod_{1 \leq i \leq k; i \neq i_0} \left(\int_0^1 F'_i(t_{i}) G'_{i}(t_{i})\ dt_{i}\right).$$ Here of course $F'$ denotes the derivative of $F$. To estimate the sums , we use the following asymptotic, also proven in Section \[sieving-sec\]. \[nonprime-asym\] Let $k \geq 1$ be fixed, let $(h_1,\dots,h_k)$ be a fixed admissible $k$-tuple, and let $b\ (W)$ be such that $b+h_i$ is coprime to $W$ for each $i=1,\dots,k$. For each fixed $1 \leq i \leq k$, let $F_{i}, G_{i}: [0,+\infty) \to {\mathbb{R}}$ be fixed smooth compactly supported functions. Assume one of the following hypotheses: - (Trivial case) One has $$\label{easy-upper} \sum_{i=1}^k ( S(F_i) + S(G_i) ) < 1.$$ - (Generalized Elliott-Halberstam) There exists a fixed $0 < \vartheta < 1$ and $i_0 \in \{1,\dots,k\}$ such that $\operatorname*{GEH}[\vartheta]$ holds, and $$\label{eh-upper} \sum_{1 \leq i \leq k; i \neq i_0} ( S(F_{i}) + S(G_{i}) ) < \vartheta.$$ Then we have $$\label{lflg} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \prod_{i=1}^k \lambda_{F_{i}}(n+h_{i}) \lambda_{G_{i}}(n+h_{i}) = (c+o(1)) B^{-k} \frac{x}{W},$$ where $B$ is given by and $$\label{c-def} c := \prod_{i=1}^k \left(\int_0^1 F'_i(t_i) G'_i(t_i)\ dt_i\right).$$ A key point in (ii) is that no upper bound on $S(F_{i_0})$ or $S(G_{i_0})$ is required (although, as we will see in Section \[geh-case\], the result is a little easier to prove when one has $S(F_{i_0})+S(G_{i_0}) < 1$). This flexibility in the $F_{i_0}, G_{i_0}$ functions will be particularly crucial to obtain part (xii) of Theorem \[main-dhl\] and Theorem \[main\]. Theorems \[prime-asym\], \[nonprime-asym\] can be viewed as probabilistic assertions of the following form: if $n$ is chosen uniformly at random from the set $\{ x \leq n \leq 2x: n = b\ (W)\}$, then the random variables $\theta(n+h_i)$ and $\lambda_{F_j}(n+h_j) \lambda_{G_j}(n+h_j)$ for $i,j=1,\dots,k$ have mean $(1+o(1)) \frac{W}{{\varphi}(W)}$ and $(\int_0^1 F'_j(t) G'_j(t)\ dt + o(1)) B^{-1}$ respectively, and furthermore these random variables enjoy a limited amount of independence, except for the fact (as can be seen from ) that $\theta(n+h_i)$ and $\lambda_{F_i}(n+h_i) \lambda_{G_i}(n+h_i)$ are highly correlated. Note though that we do not have asymptotics for any sum which involves two or more factors of $\theta$, as such estimates are of a difficulty at least as great as that of the twin prime conjecture (which is equivalent to the divergence of the sum $\sum_n \theta(n) \theta(n+2)$). Theorems \[prime-asym\], \[nonprime-asym\] may be combined with Lemma \[crit\] to reduce the task of establishing estimates of the form $\operatorname*{DHL}[k,m+1]$ to that of obtaining sufficiently good solutions to certain variational problems. For instance, in Section \[may-sec\] we reprove the following result of Maynard [@maynard-new Proposition 4.2]: \[maynard-thm\] Let $k \geq 2$ and $m \geq 1$ be fixed integers. For any fixed compactly supported square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$, define the functionals $$\label{i-def} I(F) := \int_{[0,+\infty)^k} F(t_1,\dots,t_k)^2\ dt_1 \dots dt_k$$ and $$\label{ji-def} J_i(F) := \int_{[0,+\infty)^{k-1}} \left(\int_0^\infty F(t_1,\dots,t_k)\ dt_i\right)^2 dt_1 \dots dt_{i-1} dt_{i+1} \dots dt_k$$ for $i=1,\dots,k$, and let $M_k$ be the supremum $$\label{mk4} M_k := \sup \frac{\sum_{i=1}^k J_i(F)}{I(F)}$$ over all square-integrable functions $F$ that are supported on the simplex $${\mathcal R}_k := \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq 1 \}$$ and are not identically zero (up to almost everywhere equivalence, of course). Suppose that there is a fixed $0 < \vartheta < 1$ such that $\operatorname*{EH}[\vartheta]$ holds, and such that $$M_k > \frac{2m}{\vartheta}.$$ Then $\operatorname*{DHL}[k,m+1]$ holds. Parts (vii)-(xi) of Theorem \[main-dhl\] (and hence Theorem \[main\]) are then immediate from the following results, proven in Sections \[asymptotics-sec\], \[h1-sec\], and ordered by increasing value of $k$: \[mlower\] - $M_{54} > 4.00238$. - $M_{5511} > 6$. - $M_{41588} > 8$. - $M_{309661} > 10$. - One has $M_k \geq \log k - C$ for all $k \geq C$, where $C$ is an absolute (and effective) constant. For sake of comparison, in [@maynard-new Proposition 4.3] it was shown that $M_5 > 2$, $M_{105} > 4$, and $M_k \geq \log k - 2\log\log k - 2$ for all sufficiently large $k$. As remarked in that paper, the sieves used on the bounded gap problem prior to the work in [@maynard-new] would essentially correspond, in this notation, to the choice of functions $F$ of the special form $F(t_1,\dots,t_k) := f(t_1+\dots+t_k)$, which severely limits the size of the ratio in (in particular, the analogue of $M_k$ in this special case cannot exceed $4$, as shown in [@sound]). In the converse direction, in Corollary \[mk-upper\] we will also show the upper bound $M_k \leq \frac{k}{k-1} \log k$ for all $k \geq 2$, which shows in particular that the bounds in (vii) and (xi) of the above theorem cannot be significantly improved. We remark that Theorem \[mlower\](vii) and the Bombieri-Vinogradov theorem also gives a weaker version $\operatorname*{DHL}[54,2]$ of Theorem \[main-dhl\](i). We also have a variant of Theorem \[maynard-thm\] which can accept inputs of the form $\operatorname*{MPZ}[\varpi,\delta]$: \[maynard-trunc\] Let $k \geq 2$ and $m \geq 1$ be fixed integers. Let $0<\varpi<1/4$ and $0 <\delta < 1/2$ be such that $\operatorname*{MPZ}[\varpi,\delta]$ holds. For any $\alpha>0$, let $M_k^{[\alpha]}$ be defined as in , but where the supremum now ranges over all square-integrable functions $F$ supported in the *truncated* simplex $$\label{ttk} \{ (t_1,\dots,t_k) \in [0,\alpha]^k: t_1+\dots+t_k \leq 1 \}$$ and are not identically zero. If $$M_k^{[\frac{\delta}{1/4+\varpi}]} > \frac{m}{1/4+\varpi},$$ then $\operatorname*{DHL}[k,m+1]$ holds. In Section \[asymptotics-sec\] we will establish the following variant of Theorem \[mlower\], which when combined with Theorem \[mpz-poly\], allows one to use Theorem \[maynard-trunc\] to establish parts (ii)-(vi) of Theorem \[main-dhl\] (and hence Theorem \[main\]): \[mlower-var\] - There exist $\delta,\varpi>0$ with $600 \varpi + 180 \delta < 7$ and $M_{\num{35410}}^{[\frac{\delta}{1/4+\varpi}]} > \frac{2}{1/4+\varpi}$. - There exist $\delta,\varpi>0$ with $600 \varpi + 180 \delta < 7$ and $M_{\num{1649821}}^{[\frac{\delta}{1/4+\varpi}]} > \frac{3}{1/4+\varpi}$. - There exist $\delta,\varpi>0$ with $600 \varpi + 180 \delta < 7$ and $M_{\num{75845707}}^{[\frac{\delta}{1/4+\varpi}]} > \frac{4}{1/4+\varpi}$. - There exist $\delta,\varpi>0$ with $600 \varpi + 180 \delta < 7$ and $M_{\num{3473955908}}^{[\frac{\delta}{1/4+\varpi}]} > \frac{5}{1/4+\varpi}$. - For all $k \geq C$, there exist $\delta,\varpi>0$ with $600 \varpi + 180 \delta < 7$, $\varpi \geq \frac{7}{600} - \frac{C}{\log k}$, and $M_k^{[\frac{\delta}{1/4+\varpi}]} \geq \log k - C$ for some absolute (and effective) constant $C$. The implication is clear for (ii)-(v). For (vi), observe that from Theorem \[mlower-var\](vi), Theorem \[mpz-poly\], and Theorem \[maynard-trunc\], we see that $\operatorname*{DHL}[k,m+1]$ holds whenever $k$ is sufficiently large and $$m \leq (\log k - C) \left(\frac{1}{4} + \frac{7}{600} - \frac{C}{\log k}\right)$$ which is in particular implied by $$m \leq \frac{\log k}{4 - \frac{28}{157}} - C'$$ for some absolute constant $C'$, giving Theorem \[main-dhl\](vi). Now we give a more flexible variant of Theorem \[maynard-thm\], in which the support of $F$ is enlarged, at the cost of reducing the range of integration of the $J_i$. \[epsilon-trick\] Let $k \geq 2$ and $m \geq 1$ be fixed integers, and let $0 < {\varepsilon}< 1$ be fixed also. For any fixed compactly supported square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$, define the functionals $$J_{i,1-{\varepsilon}}(F) := \int_{(1-{\varepsilon}) \cdot {\mathcal R}_{k-1}} \left(\int_0^\infty F(t_1,\dots,t_k)\ dt_i\right)^2 dt_1 \dots dt_{i-1} dt_{i+1} \dots dt_k$$ for $i=1,\dots,k$, and let $M_{k,{\varepsilon}}$ be the supremum $$M_{k,{\varepsilon}} := \sup \frac{\sum_{i=1}^k J_{i,1-{\varepsilon}}(F)}{I(F)}$$ over all square-integrable functions $F$ that are supported on the simplex $$(1+{\varepsilon}) \cdot {\mathcal R}_k = \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq 1+{\varepsilon}\}$$ and are not identically zero. Suppose that there is a fixed $0 < \vartheta < 1$, such that one of the following two hypotheses holds: - $\operatorname*{EH}[\vartheta]$ holds, and $1+{\varepsilon}< \frac{1}{\vartheta}$. - $\operatorname*{GEH}[\vartheta]$ holds, and ${\varepsilon}< \frac{1}{k-1}$. If $$M_{k,{\varepsilon}} > \frac{2m}{\vartheta}$$ then $\operatorname*{DHL}[k,m+1]$ holds. We prove this theorem in Section \[trick-sec\]. We remark that due to the continuity of $M_{k,{\varepsilon}}$ in ${\varepsilon}$, the strict inequalities in (i), (ii) of this theorem may be replaced by non-strict inequalities. Parts (i), (xiii) of Theorem \[main-dhl\], and a weaker version $\operatorname*{DHL}[4,2]$ of part (xii), then follow from Theorem \[bv-thm\] and the following computations, proven in Sections \[mkeps-sec\], \[4d\]: \[mke-lower\] - $M_{50,1/25} > 4.0043$. - $M_{4,0.168} > 2.00558$. - $M_{51,1/50} > 4.00156$. We remark that computations in the proof of Theorem \[mke-lower\](xii$'$) are simple enough that the bound may be checked by hand, without use of a computer. The computations used to establish the full strength of Theorem \[main-dhl\](xii) are however significantly more complicated. In fact, we may enlarge the support of $F$ further. We give a version corresponding to part (ii) of Theorem \[epsilon-trick\]; there is also a version corresponding to part (i), but we will not give it here as we will not have any use for it. \[epsilon-beyond\] Let $k \geq 2$ and $m \geq 1$ be fixed integers, let $0 <\vartheta < 1$ be a fixed quantity such that $\operatorname*{GEH}[\vartheta]$ holds, and let $0 < {\varepsilon}< \frac{1}{k-1}$ be fixed also. Suppose that there is a fixed non-zero square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$ supported in $\frac{k}{k-1} \cdot {\mathcal R}_k$, such that for $i=1,\dots,k$ one has the vanishing marginal condition $$\label{vanishing-marginal} \int_0^\infty F(t_1,\dots,t_k)\ dt_i = 0$$ whenever $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$ are such that $$t_1+\dots+t_{i-1}+t_{i+1}+\dots+t_k > 1+{\varepsilon}.$$ Suppose that we also have the inequality $$\frac{\sum_{i=1}^k J_{i,1-{\varepsilon}}(F)}{I(F)} > \frac{2m}{\vartheta}.$$ Then $\operatorname*{DHL}[k,m+1]$ holds. This theorem is proven in Section \[beyond-sec\]. Theorem \[main-dhl\](xii) is then an immediate consequence of Theorem \[epsilon-beyond\] and the following numerical fact, established in Section \[3d\]. \[piece\] Set ${\varepsilon}:= \frac{1}{4}$. Then there exists a piecewise polynomial function $F: [0,+\infty)^3 \to {\mathbb{R}}$ supported on the simplex $$\frac{3}{2} \cdot {\mathcal R}_3 = \left\{ (t_1,t_2,t_3) \in [0,+\infty)^3: t_1+t_2+t_3 \leq \frac{3}{2}\right\}$$ and symmetric in the $t_1,t_2,t_3$ variables, such that $F$ is not identically zero and obeys the vanishing marginal condition $$\int_0^\infty F(t_1,t_2,t_3)\ dt_3 = 0$$ whenever $t_1,t_2 \geq 0$ with $t_1+t_2 > 1+{\varepsilon}$, and such that $$\frac{3 \int_{t_1+t_2 \leq 1-{\varepsilon}} (\int_0^\infty F(t_1,t_2,t_3)\ dt_3)^2\ dt_1 dt_2}{\int_{[0,\infty)^3} F(t_1,t_2,t_3)^2\ dt_1 dt_2 dt_3} > 2.$$ There are several other ways to combine Theorems \[prime-asym\], \[nonprime-asym\] with equidistribution theorems on the primes to obtain results of the form $\operatorname*{DHL}[k,m+1]$, but all of our attempts to do so either did not improve the numerology, or else were numerically infeasible to implement. Multidimensional Selberg sieves {#sieving-sec} =============================== In this section we prove Theorems \[prime-asym\] and \[nonprime-asym\]. A key asymptotic used in both theorems is the following: \[mul-asym\] Let $k \geq 1$ be a fixed integer, and let $N$ be a natural number coprime to $W$ with $\log{N}=O(\log^{O(1)}{x})$. Let $F_1,\dots,F_k,G_1,\dots,G_k: [0,+\infty)\to {\mathbb{R}}$ be fixed smooth compactly supported functions. Then $$\label{multisum} \sum_{\substack{d_1,\dots,d_k,d'_1,\dots,d'_k \\ [d_1,d'_1],\dots,[d_k,d'_k], W, N \text{ coprime}}} \prod_{j=1}^k \frac{\mu(d_j) \mu(d'_j) F_j( \log_x d_j ) G_j( \log_x d'_j)}{[d_j,d'_j]} = (c+o(1)) B^{-k}\frac{N^k}{{\varphi}(N)^k}$$ where $B$ was defined in , and $$c := \prod_{j=1}^k \int_0^\infty F'_j(t_j) G'_j(t_j)\ dt_j.$$ The same claim holds if the denominators $[d_j,d'_j]$ are replaced by ${\varphi}([d_j,d'_j])$. Such asymptotics are standard in the literature; see e.g. [@gy] for some similar computations. In older literature, it is common to establish these asymptotics via contour integration (e.g. via Perron’s formula), but we will use the Fourier-analytic approach here. Of course, both approaches ultimately use the same input, namely the simple pole of the Riemann zeta function at $s=1$. We begin with the first claim. For $j=1,\dots,k$, the functions $t \mapsto e^t F_j(t)$, $t \mapsto e^t G_j(t)$ may be extended to smooth compactly supported functions on all of ${\mathbb{R}}$, and so we have Fourier expansions $$\label{etf} e^t F_j(t) = \int_{\mathbb{R}}e^{-it\xi} f_j(\xi)\ d\xi$$ and $$e^t G_j(t) = \int_{\mathbb{R}}e^{-it\xi} g_j(\xi)\ d\xi$$ for some fixed functions $f_j, g_j: {\mathbb{R}}\to {\mathbb{C}}$ that are smooth and rapidly decreasing in the sense that $f_j(\xi), g_j(\xi) = O( (1+|\xi|)^{-A} )$ for any fixed $A>0$ and all $\xi \in {\mathbb{R}}$ (here the implied constant is independent of $\xi$ and depends only on $A$). We may thus write $$F_j( \log_x d_j ) = \int_{\mathbb{R}}\frac{f_j(\xi_j)}{d_j^{\frac{1+i\xi_j}{\log x}}}\ d\xi_j$$ and $$G_j( \log_x d'_j ) = \int_{\mathbb{R}}\frac{g_j(\xi'_j)}{(d'_j)^{\frac{1+i\xi'_j}{\log x}}}\ d\xi'_j$$ for all $d_j,d'_j \geq 1$. We note that $$\begin{aligned} \sum_{d_j,d_j'}\frac{|\mu(d_j)\mu(d_j')|}{[d_j,d_j']d_j^{1/\log{x}}(d_j')^{1/\log{x}}} &= \prod_{p}\Bigl(1+\frac{2}{p^{1+1/\log{x}}}+\frac{1}{p^{1+2/\log{x}}}\Bigr) \\ &\leq \zeta\left(1+\frac{1}{\log x}\right)^3 \\ &\ll \log^3 x.\end{aligned}$$ Therefore, if we substitute the Fourier expansions into the left-hand side of , the resulting expression is absolutely convergent. Thus we can apply Fubini’s theorem, and the left-hand side of can thus be rewritten as $$\label{fg-int} \int_{\mathbb{R}}\dots \int_{\mathbb{R}}K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k)\ \prod_{j=1}^k f_j(\xi_j) g_j(\xi'_j) d\xi_j d\xi'_j,$$ where $$K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) := \sum_{\substack{d_1, \dots, d_k, d'_1, \dots, d'_k \\ [d_1,d'_1], \dots, [d_k,d'_k], W, N \text{ coprime}}} \prod_{j=1}^k \frac{\mu(d_j) \mu(d'_j)}{[d_j,d'_j] d_j^{\frac{1+i\xi_j}{\log x}} (d'_j)^{\frac{1+i\xi'_j}{\log x}}}.$$ This latter expression factorizes as an Euler product $$K = \prod_{p\nmid WN} K_p,$$ where the local factors $K_p$ are given by $$\label{kp} K_p(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) := 1 + \frac{1}{p} \sum_{\substack{d_1,\dots,d_k,d'_1,\dots,d'_k \\ [d_1, \dots, d_k, d'_1, \dots, d'_k]=p \\ [d_1,d'_1],\dots,[d_k,d'_k] \text{ coprime}}} \prod_{j=1}^k \frac{\mu(d_j) \mu(d'_j)}{d_j^{\frac{1+i\xi_j}{\log x}} (d'_j)^{\frac{1+i\xi'_j}{\log x}}}.$$ We can estimate each Euler factor as $$\label{kp-est} K_p(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) = \Bigl(1+O(\frac{1}{p^2})\Bigr) \prod_{j=1}^k \frac{\left(1 - p^{-1-\frac{1+i\xi_j}{\log x}}\right)\left(1 - p^{-1-\frac{1+i\xi'_j}{\log x}}\right)}{1 - p^{-1-\frac{2+i\xi_j+i\xi'_j}{\log x}}}.$$ Since $$\prod_{p: p>w} \Bigl(1 + O(\frac{1}{p^2})\Bigr) = 1 + o(1),$$ we have $$K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) = (1+o(1)) \prod_{j=1}^k \frac{ \zeta_{W N}( 1 + \frac{2+i\xi_j+i\xi'_j}{\log x}) }{ \zeta_{W N}(1 + \frac{1+i\xi_j}{\log x}) \zeta_{W N}(1 + \frac{1+i\xi'_j}{\log x}) }$$ where the modified zeta function $\zeta_{WN}$ is defined by the formula $$\zeta_{W N}(s) := \prod_{p\nmid W N} \left(1-\frac{1}{p^s}\right)^{-1}$$ for ${\operatorname{Re}}(s) > 1$. For ${\operatorname{Re}}(s) \geq 1 + \frac{1}{\log x}$ we have the crude bounds $$\begin{aligned} |\zeta_{W N}(s)|, |\zeta_{W N}(s)|^{-1} &\leq \zeta( 1 + \frac{1}{\log x}) \\ &\ll \log x\end{aligned}$$ where the first inequality comes from comparing the factors in the Euler product. Thus $$K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) = O( \log^{3k} x ).$$ Combining this with the rapid decrease of $f_j, g_j$, we see that the contribution to outside of the cube $\{\max(|\xi_1|,\dots,|\xi_k|,|\xi'_1|,\dots,|\xi'_k|) \leq \sqrt{\log x}\}$ (say) is negligible. Thus it will suffice to show that $$\int_{-\sqrt{\log x}}^{\sqrt{\log x}} \dots \int_{-\sqrt{\log x}}^{\sqrt{\log x}} K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k)\ \prod_{j=1}^k f_j(\xi_j) g_j(\xi'_j) d\xi_j d\xi'_j = (c+o(1)) B^{-k}\frac{N^k}{{\varphi}(N)^k}.$$ When $|\xi_j| \leq \sqrt{\log x}$, we see from the simple pole of the Riemann zeta function $\zeta(s) = \prod_p (1-\frac{1}{p^s})^{-1}$ at $s=1$ that $$\zeta\left(1 + \frac{1+i\xi_j}{\log x}\right) = (1+o(1)) \frac{\log x}{1+i\xi_j}.$$ For $-\sqrt{\log{x}}\le \xi_j\le \sqrt{\log{x}}$, we see that $$1-\frac{1}{p^{1+\frac{1+i\xi_j}{\log{x}}}}=1-\frac{1}{p}+O\Bigl(\frac{\log{p}}{p\sqrt{\log{x}}}\Bigr).$$ Since $\log(WN)\ll \log^{O(1)}{x}$, this gives $$\begin{aligned} \prod_{p|WN} \Bigl(1 - \frac{1}{p^{1+\frac{1+i\xi_j}{\log x}}}\Bigr) &= \frac{{\varphi}(WN)}{WN}\exp\Bigl(O\Bigl(\sum_{p|WN}\frac{\log{p}}{p\sqrt{\log{x}}}\Bigr)\Bigr) = (1+o(1))\frac{{\varphi}(WN)}{WN},\end{aligned}$$ since the sum is maximized when $WN$ is composed only of primes $p\ll \log^{O(1)}{x}$. Thus $$\zeta_{WN}\Bigl(1 + \frac{1+i\xi_j}{\log x}\Bigr) = \frac{(1+o(1)) B {\varphi}(N)}{(1 + i\xi_j)N}.$$ Similarly with $1+i\xi_j$ replaced by $1+i\xi'_j$ or $2+i\xi_j+i\xi'_j$. We conclude that $$\label{kt} K(\xi_1,\dots,\xi_k,\xi'_1,\dots,\xi'_k) = (1+o(1)) B^{-k}\frac{N^k}{{\varphi}(N)^k} \prod_{j=1}^k \frac{(1+i\xi_j) (1+i\xi'_j)}{2+i\xi_j+i\xi'_j}.$$ Therefore it will suffice to show that $$\int_{\mathbb{R}}\dots \int_{\mathbb{R}}\prod_{j=1}^k \frac{(1+i\xi_j) (1+i\xi'_j)}{2+i\xi_j+i\xi'_j} f_j(\xi_j) g_j(\xi'_j) d\xi_j d\xi'_j = c,$$ since the errors caused by the $1+o(1)$ multiplicative factor in or the truncation $|\xi_j|, |\xi'_j| \leq \sqrt{\log x}$ can be seen to be negligible using the rapid decay of $f_j,g_j$. By Fubini’s theorem, it suffices to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{(1+i\xi) (1+i\xi')}{2+i\xi+i\xi'} f_j(\xi) g_j(\xi')\ d\xi d\xi' = \int_0^{+\infty} F_j'(t) G_j'(t)\ dt$$ for each $j=1,\dots,k$. But from dividing by $e^t$ and differentiating under the integral sign, we have $$F'_j(t) = - \int_{\mathbb{R}}(1+i\xi) e^{-t(1+i\xi)} f_j(\xi)\ d\xi,$$ and the claim then follows from Fubini’s theorem. Finally, suppose that we replace the denominators $[d_j,d'_j]$ with ${\varphi}([d_j,d'_j])$. An inspection of the above argument shows that the only change that occurs is that the $\frac{1}{p}$ term in is replaced by $\frac{1}{p-1}$; but this modification may be absorbed into the $1+O(\frac{1}{p^2})$ factor in , and the rest of the argument continues as before. The trivial case {#triv-sec} ---------------- We can now prove the easiest case of the two theorems, namely case (i) of Theorem \[nonprime-asym\]; a closely related estimate also appears in [@maynard-new Lemma 6.2]. We may assume that $x$ is sufficiently large depending on all fixed quantities. By , the left-hand side of may be expanded as $$\label{lflg-expand} \sum_{d_1,\dots,d_k,d'_1,\dots,d'_k} \left(\prod_{i=1}^k \mu(d_i) \mu(d'_i) F_i(\log_x d_i) G_i(\log_x d'_i)\right) S(d_1,\dots,d_k,d'_1,\dots,d'_k)$$ where $$S(d_1,\dots,d_k,d'_1,\dots,d'_k) := \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W) \\ n + h_i = 0\ ([d_i,d'_i])\ \forall i\\}} 1.$$ By hypothesis, $b+h_i$ is coprime to $W$ for all $i=1,\dots,k$, and $|h_i-h_j| < w$ for all distinct $i,j$. Thus, $S(d_1,\dots,d_k,d'_1,\dots,d'_k)$ vanishes unless the $[d_i,d'_i]$ are coprime to each other and to $W$. In this case, $S(d_1,\dots,d_k,d'_1,\dots,d'_k)$ is summing the constant function $1$ over an arithmetic progression in $[x,2x]$ of spacing $W [d_1,d'_1] \dots [d_k,d'_k]$, and so $$S(d_1,\dots,d_k,d'_1,\dots,d'_k) = \frac{x}{W [d_1,d'_1] \dots [d_k,d'_k]} + O(1).$$ By Lemma \[mul-asym\], the contribution of the main term $\frac{x}{W [d_1,d'_1] \dots [d_k,d'_k]}$ to is $(c+o(1)) B^{-k} \frac{x}{W}$; note that the restriction of the integrals in to $[0,1]$ instead of $[0,+\infty)$ is harmless since $S(F_i), S(G_i) < 1$ for all $i$. Meanwhile, the contribution of the $O(1)$ error is then bounded by $$O\Bigl( \sum_{d_1,\dots,d_k,d'_1,\dots,d'_k} (\prod_{i=1}^k |F_i(\log_x d_i)| |G_i(\log_x d'_i)|)\Bigr).$$ By the hypothesis in Theorem \[nonprime-asym\](i), we see that for $d_1,\dots,d_k,d'_1,\dots,d'_k$ contributing a non-zero term here, one has $$[d_1,d'_1] \dots [d_k,d'_k] {\llcurly}x^{1-{\varepsilon}}$$ for some fixed ${\varepsilon}>0$. From the divisor bound we see that each choice of $[d_1,d'_1] \dots [d_k,d'_k]$ arises from ${\llcurly}1$ choices of $d_1,\dots,d_k,d'_1,\dots,d'_k$. We conclude that the net contribution of the $O(1)$ error to is ${\llcurly}x^{1-{\varepsilon}}$, and the claim follows. The Elliott-Halberstam case {#eh-case} --------------------------- Now we show case (i) of Theorem \[prime-asym\]. For sake of notation we take $i_0=k$, as the other cases are similar. We use to rewrite the left-hand side of as $$\label{theta-oo2} \sum_{d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}} \Bigl(\prod_{i=1}^{k-1} \mu(d_i) \mu(d'_i) F_i(\log_x d_i) G_i(\log_x d'_i)\Bigr) \tilde S(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1})$$ where $$\tilde S(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) := \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W) \\ n + h_i = 0\ ([d_i,d'_i])\ \forall i=1,\dots,k-1}} \theta(n+h_k).$$ As in the previous case, $\tilde S(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1})$ vanishes unless the $[d_i,d'_i]$ are coprime to each other and to $W$, and so the summand in vanishes unless the modulus $q_{W,d_1,\dots,d'_{k-1}}$ defined by $$\label{q-def} q_{W,d_1,\dots,d'_{k-1}} := W [d_1,d'_1] \dots [d_{k-1},d'_{k-1}]$$ is squarefree. In that case, we may use the Chinese remainder theorem to concatenate the congruence conditions on $n$ into a single primitive congruence condition $$n+h_k = a_{W,d_1,\dots,d'_{k-1}} \ (q_{W,d_1,\dots,d'_{k-1}})$$ for some $a_{W,d_1,\dots,d'_{k-1}}$ depending on $W, d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}$, and conclude using that $$\label{ts} \begin{split} \tilde S(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) &= \frac{1}{{\varphi}(q_{W,d_1,\dots,d'_{k-1}})} \sum_{x+h_k \leq n \leq 2x+h_k} \theta(n)\\ &\quad + \Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a_{W,d_1,\dots,d'_{k-1}}\ (q_{W,d_1,\dots,d'_{k-1}})). \end{split}$$ From the prime number theorem we have $$\sum_{x+h_k \leq n \leq 2x+h_k} \theta(n) = (1+o(1)) x$$ and this expression is clearly independent of $d_1,\dots,d'_{k-1}$. Thus by Lemma \[mul-asym\], the contribution of the main term in to is $(c+o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$. By and , it thus suffices to show that for any fixed $A$ we have $$\label{sosmall} \sum_{d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}} \Bigl(\prod_{i=1}^{k-1} |F_i(\log_x d_i)| |G_i(\log_x d'_i)|\Bigr) |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))| \ll x \log^{-A} x,$$ where $a=a_{W,d_1,\dots,d'_{k-1}}$ and $q=q_{W,d_1,\dots,d'_{k-1}}$. For future reference we note that we may restrict the summation here to those $d_1,\dots,d'_{k-1}$ for which $q_{W,d_1,\dots,d'_{k-1}}$ is square-free. From the hypotheses of Theorem \[prime-asym\](i), we have $$q_{W,d_1,\dots,d'_{k-1}} {\llcurly}x^\vartheta$$ whenever the summand in is non-zero, and each choice $q$ of $q_{W,d_1,\dots,d'_{k-1}}$ is associated to $O( \tau(q)^{O(1)} )$ choices of $d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}$. Thus this contribution is $$\ll \sum_{q {\llcurly}x^\vartheta} \tau(q)^{O(1)} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q) )|.$$ Using the crude bound $$|\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q) )| \ll \frac{x}{q} \log^{O(1)} x$$ and , we have $$\sum_{q {\llcurly}x^\vartheta} \tau(q)^C \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q) )| \ll x \log^{O(1)} x$$ for any fixed $C>0$. By the Cauchy-Schwarz inequality it suffices to show that $$\sum_{q {\llcurly}x^\vartheta} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q) )| \ll x \log^{-A} x$$ for any fixed $A>0$. However, since $\theta$ only differs from $\Lambda$ on powers $p^j$ of primes with $j>1$, it is not difficult to show that $$|\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q) ) - \Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \Lambda; a\ (q) )| {\llcurly}\sqrt{\frac{x}{q}},$$ so the net error in replacing $\theta$ here by $\Lambda$ is ${\llcurly}x^{1 - (1-\vartheta)/2}$, which is certainly acceptable. The claim now follows from the hypothesis $\operatorname*{EH}[\vartheta]$, thanks to Claim \[eh-def\]. The Motohashi-Pintz-Zhang case ------------------------------ Now we show case (ii) of Theorem \[prime-asym\]. We repeat the arguments from Section \[eh-case\], with the only difference being in the derivation of . As observed previously, we may restrict $q_{W,d_1,\dots,d'_{k-1}}$ to be squarefree. From the hypotheses in Theorem \[prime-asym\](ii), we also see that $$q_{W,d_1,\dots,d'_{k-1}} {\llcurly}x^{1/2+2\varpi}$$ and that all the prime factors of $q_{W,d_1,\dots,d'_{k-1}}$ are at most $x^\delta$. Thus, if we set $I := [1,x^\delta]$, we see (using the notation from Claim \[mpz-claim\]) that $q_{W,d_1,\dots,d'_{k-1}}$ lies in ${\mathcal{S}}_I$, and is thus a factor of $P_I$. If we then let ${\mathcal A} \subset {\mathbb{Z}}/P_I{\mathbb{Z}}$ denote all the primitive residue classes $a\ (P_I)$ with the property that $a = b\ (W)$, and such that for each prime $w < p \leq x^\delta$, one has $a + h_i = 0\ (p)$ for some $i=1,\dots,k$, then we see that $a_{W,d_1,\dots,d'_{k-1}}$ lies in the projection of ${\mathcal A}$ to ${\mathbb{Z}}/q_{W,d_1,\dots,d'_{k-1}}{\mathbb{Z}}$. Each $q \in {\mathcal{S}}_I$ is equal to $q_{W,d_1,\dots,d'_{k-1}}$ for $O(\tau(q)^{O(1)})$ choices of $d_1,\dots,d'_{k-1}$. Thus the left-hand side of is $$\ll \sum_{q \in {\mathcal{S}}_I: q {\llcurly}x^{1/2+2\varpi}} \tau(q)^{O(1)} \sup_{a \in {\mathcal A}} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))|.$$ Note from the Chinese remainder theorem that for any given $q$, if one lets $a$ range uniformly in ${\mathcal A}$, then $a\ (q)$ is uniformly distributed among $O( \tau(q)^{O(1)})$ different moduli. Thus we have $$\sup_{a \in {\mathcal A}} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))| \ll \frac{\tau(q)^{O(1)}}{|{\mathcal A}|} \sum_{a \in {\mathcal A}} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))|,$$ and so it suffices to show that $$\sum_{q \in {\mathcal{S}}_I: q {\llcurly}x^{1/2+2\varpi}} \frac{\tau(q)^{O(1)}}{|{\mathcal A}|} \sum_{a \in {\mathcal A}} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))|\ll x \log^{-A} x$$ for any fixed $A>0$. We see it suffices to show that $$\sum_{q \in {\mathcal{S}}_I: q {\llcurly}x^{1/2+2\varpi}} \tau(q)^{O(1)} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \theta; a\ (q))|\ll x \log^{-A} x$$ for any given $a \in {\mathcal A}$. But this follows from the hypothesis $MPZ[\varpi,\delta]$ by repeating the arguments of Section \[eh-case\]. Crude estimates on divisor sums ------------------------------- To proceed further, we will need some additional information on the divisor sums $\lambda_F$ (defined in ), namely that these sums are concentrated on “almost primes”; results of this type have also appeared in [@pintz-szemeredi]. \[almostprime\] Let $k \geq 1$ be fixed, let $(h_1,\dots,h_k)$ be a fixed admissible $k$-tuple, and let $b\ (W)$ be such that $b+h_i$ is coprime to $W$ for each $i=1,\dots,k$. Let $F_1,\dots,F_k: [0,+\infty) \to {\mathbb{R}}$ be fixed smooth compactly supported functions, and let $m_1,\dots,m_k \geq 0$ and $a_1,\dots,a_k \geq 1$ be fixed natural numbers. Then $$\label{lambdatau} \sum_{x \leq n \leq 2x: n = b\ (W)} \prod_{j=1}^k \Bigl(|\lambda_{F_j}(n+h_j)|^{a_j} \tau(n+h_j)^{m_j}\Bigr) \ll B^{-k} \frac{x}{W}.$$ Furthermore, if $1 \leq j_0 \leq k$ is fixed and $p_0$ is a prime with $p_0 \leq x^{\frac{1}{10k}}$, then we have the variant $$\label{lambdatau-fix} \sum_{x \leq n \leq 2x: n = b\ (W)} \prod_{j=1}^k \Bigl(|\lambda_{F_j}(n+h_j)|^{a_j} \tau(n+h_j)^{m_j}\Bigr) {\mathbf{1}}_{p_0|n+h_{j_0}} \ll \frac{\log_x p_0}{p_0} B^{-k} \frac{x}{W}.$$ As a consequence, we have $$\label{lambdatau-fix2} \sum_{x \leq n \leq 2x: n = b\ (W)} \prod_{j=1}^k \Bigl(|\lambda_{F_j}(n+h_j)|^{a_j} \tau(n+h_j)^{m_j}\Bigr) {\mathbf{1}}_{p(n+h_{j_0}) \leq x^{\varepsilon}} \ll {\varepsilon}B^{-k} \frac{x}{W},$$ for any ${\varepsilon}> 0$, where $p(n)$ denotes the least prime factor of $n$. The exponent $\frac{1}{10k}$ can certainly be improved here, but for our purposes any fixed positive exponent depending only on $k$ will suffice. The strategy is to estimate the alternating divisor sums $\lambda_{F_j}(n+h_j)$ by non-negative expressions involving prime factors of $n+h_j$, which can then be bounded combinatorially using standard tools. We first prove . As in the proof of Proposition \[mul-asym\], we can use Fourier expansion to write $$F_j( \log_x d ) = \int_{\mathbb{R}}\frac{f_j(\xi)}{d^{\frac{1+i\xi}{\log x}}}\ d\xi$$ for some rapidly decreasing $f_j: {\mathbb{R}}\to {\mathbb{C}}$ and all natural numbers $d$. Thus $$\lambda_{F_j}(n) = \int_{\mathbb{R}}\Bigl(\sum_{d|n} \frac{\mu(d)}{d^{\frac{1+i\xi}{\log x}}}\Bigr) f_j(\xi)\ d\xi,$$ which factorizes using Euler products as $$\lambda_{F_j}(n) = \int_{\mathbb{R}}\prod_{p|n} \Bigl(1 - \frac{1}{p^{\frac{1+i\xi}{\log x}}}\Bigr) f_j(\xi)\ d\xi.$$ The function $s \mapsto p^{\frac{-s}{\log x}}$ has a magnitude of $O(1)$ and a derivative of $O( \log_x p )$ when ${\operatorname{Re}}(s) > 1$, and thus $$1 - \frac{1}{p^{\frac{1+i\xi}{\log x}}} = O\Bigl( \min( (1+|\xi|) \log_x p, 1) \Bigr).$$ From the rapid decrease of $f_j$ and the triangle inequality, we conclude that $$|\lambda_{F_j}(n)| \ll \int_{\mathbb{R}}\Bigl(\prod_{p|n} O\Bigl( \min( (1+|\xi|) \log_x p, 1)\Bigr)\Bigr) \frac{d\xi}{(1+|\xi|)^A}$$ for any fixed $A > 0$. Thus, noting that $\prod_{p|n}O(1)\ll \tau(n)^{O(1)}$, we have $$|\lambda_{F_j}(n)|^{a_j} \ll \tau(n)^{O(1)}\int_{\mathbb{R}}\dots \int_{\mathbb{R}}\Bigl(\prod_{p|n} \prod_{l=1}^{a_j} \min( (1+|\xi_l|) \log_x p, 1)\Bigr) \frac{d\xi_1 \dots d\xi_{a_j}}{(1+|\xi_1|)^A \dots (1+|\xi_{a_j}|)^A}$$ for any fixed $a_j,A$. However, we have $$\prod_{i=1}^{a_j}\min( (1+|\xi_i|) \log_x p, 1)) \leq \min( (1+|\xi_1|+\dots+|\xi_{a_j}|) \log_x p, 1 ),$$ and so $$|\lambda_{F_j}(n)|^{a_j} \ll \tau(n)^{O(1)}\int_{\mathbb{R}}\dots \int_{\mathbb{R}}\frac{(\prod_{p|n} \min( (1+|\xi_1|+\dots+|\xi_{a_j}|) \log_x p, 1)) d\xi_1 \dots d \xi_{a_j}}{(1+|\xi_1|+\dots+|\xi_{a_j}|)^A}.$$ Making the change of variables $\sigma := 1+|\xi_1|+\dots+|\xi_{a_j}|$, we obtain $$|\lambda_{F_j}(n)|^{a_j} \ll \tau(n)^{O(1)} \int_1^\infty \Bigl(\prod_{p|n} \min(\sigma \log_x p, 1)\Bigr) \frac{d\sigma}{\sigma^A}$$ for any fixed $A>0$. In view of this bound and the Fubini-Tonelli theorem, it suffices to show that $$\sum_{x \leq n \leq 2x: n = b\ (W)} \prod_{j=1}^k \Bigl(\tau(n+h_j)^{O(1)} \prod_{p|n+h_j} \min(\sigma_j \log_x p,1)\Bigr) \ll B^{-k} \frac{x}{W} (\sigma_1+\dots+\sigma_k)^{O(1)}$$ for all $\sigma_1,\dots,\sigma_k \geq 1$. By setting $\sigma := \sigma_1+\dots+\sigma_k$, it suffices to show that $$\label{xnx} \sum_{x \leq n \leq 2x: n = b\ (W)} \prod_{j=1}^k \Bigl(\tau(n+h_j)^{O(1)} \prod_{p|n+h_j} \min(\sigma \log_x p,1)\Bigr) \ll B^{-k} \frac{x}{W} \sigma^{O(1)}$$ for any $\sigma \geq 1$. To proceed further, we factorize $n+h_j$ as a product $$n+h_j = p_1 \dots p_r$$ of primes $p_1 \leq \dots \leq p_r$ in increasing order, and then write $$n+h_j = d_j m_j$$ where $d_j := p_1 \dots p_{i_j}$ and $i_j$ is the largest index for which $p_1 \dots p_{i_j} < x^{\frac{1}{10k}}$, and $m_j := p_{i_j+1} \dots p_r$. By construction, we see that $0 \leq i_j < r$, $d_j \leq x^{\frac{1}{10k}}$. Also, we have $$p_{i_j+1} \geq (p_1 \dots p_{i_j+1})^{\frac{1}{i_j+1}} \geq x^{\frac{1}{10k(i_j+1)}}.$$ Since $n \leq 2x$, this implies that $$r = O( i_j + 1 )$$ and so $$\tau(n+h_j) \leq 2^{O(1+\Omega(d_j))},$$ where we recall that $\Omega(d_j) = i_j$ denotes the number of prime factors of $d_j$, counting multiplicity. We also see that $$p(m_j) \geq x^{\frac{1}{10k(1+\Omega(d_j))}}\geq x^{\frac{1}{10k(1+\Omega(d_1\dots d_k))}}=:R,$$ where $p(n)$ denotes the least prime factor of $n$. Finally, we have that $$\prod_{p|n+h_j} \min(\sigma \log_x p,1) \leq \prod_{p|d_j} \min( \sigma \log_x p, 1 ),$$ and we see the $d_1,\dots,d_k,W$ are coprime. We may thus estimate the left-hand side of by $$\ll \sum_* \Bigl(\prod_{j=1}^k 2^{O(1+\Omega(d_j))} \prod_{p|d_j} \min( \sigma \log_x p, 1 )\Bigr) \sum_{**} 1$$ where the outer sum $\sum_*$ is over $d_1,\dots,d_k \leq x^{\frac{1}{10k}}$ with $d_1,\dots,d_k,W$ coprime, and the inner sum $\sum_{**}$ is over $x \leq n \leq 2x$ with $n = b\ (W)$ and $n + h_j = 0\ (d_j)$ for each $j$, with $p( \frac{n+h_j}{d_j}) \geq R$ for each $j$. We bound the inner sum $\sum_{**} 1$ using a Selberg sieve upper bound. Let $G$ be a smooth function supported on $[0,1]$ with $G(0)=1$, and let $d=d_1\dots d_k$. We see that $$\sum_{**} 1\le \sum_{\substack{x\le n\le 2x\\ n+h_i = 0\ (d_i) \\ n\equiv b\ (W)}}\prod_{i=1}^k\Bigl(\sum_{\substack{e|n+h_i\\ (e,dW)=1}}\mu(e)G(\log_R{e})\Bigr)^2,$$ since the product is $G(0)^{2k}=1$ if $p( \frac{n+h_j}{d_j}) \geq R$, and non-negative otherwise. The right hand side may be expanded as $$\sum_{\substack{e_1,\dots,e_k,e_1',\dots,e_k'\\ (e_ie_i',dW)=1\forall i}}\Bigl(\prod_{i=1}^k\mu(e_i)\mu(e_i')G(\log_R{e_i})G(\log_R{e_i'})\Bigr)\sum_{\substack{x\le n\le 2x \\ n+h_i = 0\ (d_i[e_i,e_i']) \\ n= b\ (W)}}1.$$ As in Section \[triv-sec\], the inner sum vanishes unless the $e_ie_i'$ are coprime to each other and $dW$, in which case it is $$\frac{x}{dW[e_1,e_1']\dots [e_k,e_k']}+O(1).$$ The $O(1)$ term contributes ${\llcurly}R^k{\llcurly}x^{1/10}$, which is negligible. By Lemma \[mul-asym\], if $\Omega(d)\ll \log^{1/2}{x}$ then the main term contributes $$\ll \Bigl(\frac{d}{{\varphi}(d)}\Bigr)^k\frac{x}{d W}(\log{R})^{-k}\ll 2^{\Omega(d)}B^{-k}\frac{x}{d W}.$$ We see that this final bound applies trivially if $\Omega(d)\gg \log^{1/2}{x}$. The bound thus reduces to $$\label{xnx-2} \sum_* \Bigl(\prod_{j=1}^k \frac{2^{O(1+\Omega(d_j))}}{d_j} \prod_{p|d_j} \min( \sigma \log_x p, 1 )\Bigr) \ll \sigma^{O(1)}.$$ Ignoring the coprimality conditions on the $d_j$ for an upper bound, we see this is bounded by $$\prod_{w<p\le x^{\frac{1}{10k}}}\Bigl(1+\frac{O(\min(\sigma\log_x(p),1))}{p}\sum_{j\ge 0}\frac{O(1)^j}{p^j}\Bigr)^k \ll \exp\Bigl(O\Bigl(\sum_{p\le x}\frac{(\min(\sigma\log_x(p),1))}{p}\Bigr)\Bigr).$$ But from Mertens’ theorem we have $$\sum_{p \leq x} \frac{\min(\sigma \log_x p, 1)}{p} = O\Bigl( \log \frac{1}{\sigma} \Bigr),$$ and the claim follows. The proof of is a minor modification of the argument above used to prove . Namely, the variable $d_{j_0}$ is now replaced by $[d_0,p_0]<x^{1/5k}$, which upon factoring out $p_0$ has the effect of multiplying the upper bound for by $O( \frac{\sigma \log_x p_0}{p_0})$ (at the negligible cost of deleting the prime $p_0$ from the sum $\sum_{p \leq x}$), giving the claim; we omit the details. Finally, follows immediately from when ${\varepsilon}> \frac{1}{10k}$, and from and Mertens’ theorem when ${\varepsilon}\leq \frac{1}{10k}$. As in [@pintz-szemeredi], one can use Proposition \[almostprime\], together with the observation that the quantity $\lambda_F(n)$ is bounded whenever $n = O(x)$ and $p(n) \geq x^{\varepsilon}$, to conclude that whenever the hypotheses of Lemma \[crit\] are obeyed for some $\nu$ of the form , then there exists a fixed ${\varepsilon}>0$ such that for all sufficiently large $x$, there are $\gg \frac{x}{\log^k x}$ elements $n$ of $[x,2x]$ such that $n+h_1,\dots,n+h_k$ have no prime factor less than $x^{\varepsilon}$, and that at least $m$ of the $n+h_1,\dots,n+h_k$ are prime. The generalized Elliott-Halberstam case {#geh-case} --------------------------------------- Now we show case (ii) of Theorem \[nonprime-asym\]. For sake of notation we shall take $i_0=k$, as the other cases are similar; thus we have $$\label{ik1} \sum_{i=1}^{k-1} (S(F_{i}) + S(G_{i})) < \vartheta.$$ The basic idea is to view the sum as a variant of , with the role of the function $\theta$ now being played by the product divisor sum $\lambda_{F_k} \lambda_{G_k}$, and to repeat the arguments in Section \[eh-case\]. To do this we rely on Proposition \[almostprime\] to restrict $n+h_i$ to the almost primes. We turn to the details. Let ${\varepsilon}> 0$ be an arbitrary fixed quantity. From and Cauchy-Schwarz one has $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \Bigl(\prod_{i=1}^k \lambda_{F_{i}}(n+h_{i}) \lambda_{G_{i}}(n+h_{i})\Bigr) {\mathbf{1}}_{p(n+h_k) \leq x^{\varepsilon}} = O\left( {\varepsilon}B^{-k} \frac{x}{W} \right)$$ with the implied constant uniform in ${\varepsilon}$, so by the triangle inequality and a limiting argument as ${\varepsilon}\to 0$ it suffices to show that $$\label{lltrunc} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \Bigl( \prod_{i=1}^k \lambda_{F_{i}}(n+h_{i}) \lambda_{G_{i}}(n+h_{i}) \Bigr){\mathbf{1}}_{p(n+h_k) > x^{\varepsilon}} = (c_{\varepsilon}+ o(1)) B^{-k} \frac{x}{W}$$ where $c_{\varepsilon}$ is a quantity depending on ${\varepsilon}$ but not on $x$, such that $$\lim_{{\varepsilon}\to 0} c_{\varepsilon}= \prod_{i=1}^k \int_0^1 F'_i(t) G'_i(t)\ dt.$$ We use to expand out $\lambda_{F_{i}}, \lambda_{G_{i}}$ for $i=1,\dots,k-1$, but *not* for $i=k$, so that the left-hand side of becomes $$\label{ddd} \sum_{d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}} \Bigl(\prod_{i=1}^k \mu(d_{i}) \mu(d'_{i}) F_{i}(\log_x d_{i}) G_{i}(\log_x d'_{i}) \Bigr) S'(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1})$$ where $$S'(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) := \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W) \\ n + h_{i} = 0\ ([d_{i},d'_{i}])\ \forall i=1,\dots,k-1}} \hspace{0pt minus 1fil} \lambda_{F_k}(n+h_k) \lambda_{G_k}(n+h_k) {\mathbf{1}}_{p(n+h_k) > x^{\varepsilon}}.$$ As before, the summand in vanishes unless the modulus[^4] $q_{W,d_1,\dots,d'_{k-1}}$ defined in is squarefree, in which case we have the analogue $$\begin{aligned} \label{tsp} S'(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) &= \frac{1}{{\varphi}(q)} \sum_{\substack{x+h_k \leq n \leq 2x+h_k\\ (n,q)=1}} \lambda_{F_k}(n) \lambda_{G_k}(n) {\mathbf{1}}_{p(n)>x^{\varepsilon}}\nonumber\\ & + \Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}}; a\ (q))\end{aligned}$$ of . Here we have put $q=q_{W,d_1,\dots,d'_{k-1}}$ and $a=a_{W,d_1,\dots,d'_{k-1}}$ for convenience. We thus split $$S' = S'_1 - S'_2 + S'_3,$$ where, $$\begin{aligned} S'_1(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) &= \frac{1}{{\varphi}(q)} \sum_{x+h_k \leq n \leq 2x+h_k} \lambda_{F_k}(n) \lambda_{G_k}(n) {\mathbf{1}}_{p(n)>x^{\varepsilon}} \label{sp1-def},\\ S'_2(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) &= \frac{1}{{\varphi}(q)} \sum_{x+h_k \leq n \leq 2x+h_k; (n,q) > 1} \lambda_{F_k}(n) \lambda_{G_k}(n) {\mathbf{1}}_{p(n)>x^{\varepsilon}} \label{sp2-def},\\ S'_3(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}) &= \Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}}; a\ (q)), \label{sp3-def}\end{aligned}$$ when $q=q_{W,d_1,\dots,d'_{k-1}}$ is squarefree, with $S'_1=S'_2=S'_3=0$ otherwise. For $j\in\{1,2,3\}$, let $$\Sigma_j= \sum_{d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}}\Bigl(\prod_{i=1}^k \mu(d_{i}) \mu(d'_{i}) F_{i}(\log_x d_{i}) G_{i}(\log_x d'_{i})\Bigr) S'_j(d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}). \label{eq:SigmaDef}$$ To show , it thus suffices to show the main term estimate $$\label{ll-main} \Sigma_1= (c_{\varepsilon}+ o(1)) B^{-k} \frac{x}{W},$$ the first error term estimate $$\label{ll-error1} \Sigma_2 {\llcurly}x^{1-{\varepsilon}},$$ and the second error term estimate $$\label{ll-error2} \Sigma_3 \ll x \log^{-A} x$$ for any fixed $A>0$. We begin with . Observe that if $p(n) > x^{\varepsilon}$, then the only way that $(n,q_{W,d_1,\dots,d'_{k-1}})$ can exceed $1$ is if there is a prime $x^{\varepsilon}< p \ll x$ which divides both $n$ and one of $d_1,\dots,d'_{k-1}$; in particular, this case can only occur when $k > 1$. For sake of notation we will just consider the contribution when there is a prime that divides $n$ and $d_1$, as the other $2k-3$ cases are similar. By , this contribution to $\Sigma_2$ can then be crudely bounded (using ) by $$\begin{aligned} \Sigma_2&{\llcurly}\sum_{x^{\varepsilon}< p \ll x} \sum_{d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1} \leq x; p|d_1} \frac{1}{[d_1,d'_1] \dots [d_{k-1},d'_{k-1}]} \sum_{n \ll x: p|n} 1 \\ &{\llcurly}\sum_{x^{\varepsilon}< p \ll x} \frac{x}{p} \Bigl(\sum_{e_1 \leq x^2; p|e_1} \frac{\tau(e_1)}{e_1}\Bigr) \prod_{i=2}^{k-1} \Bigl(\sum_{e_i \leq x^2} \frac{\tau(e_i)}{e_i}\Bigr) \\ &{\llcurly}\sum_{x^{\varepsilon}< p \ll x} \frac{x}{p^2} \\ &{\llcurly}x^{1-{\varepsilon}}\end{aligned}$$ as required, where we have made the change of variables $e_i := [d_i,d'_i]$, using the divisor bound to control the multiplicity. Now we show . From the hypothesis we have $q_{W,d_1,\dots,d'_{k-1}} {\llcurly}x^\vartheta$ whenever the summand in is non-zero. From the divisor bound, for each $q {\llcurly}x^\vartheta$ there are $O( \tau(q)^{O(1)})$ choices of $d_1,\dots,d'_{k-1}$ with $q_{W,d_1,\dots,d'_{k-1}} = q$. We see the product in is $O(1)$. Thus by , we may bound $\Sigma_3$ by $$\Sigma_3 \ll \sum_{q {\llcurly}x^\vartheta} \tau(q)^{O(1)} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}}; a\ (q))|.$$ From we easily obtain the bound $$\Sigma_3\ll \sum_{q {\llcurly}x^\vartheta} \tau(q)^{O(1)} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}}; a\ (q))| \ll x \log^{O(1)} x,$$ so by Cauchy-Schwarz it suffices to show that $$\label{local} \sum_{q {\llcurly}x^\vartheta} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{[x+h_k,2x+h_k]} \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}}; a\ (q))| \ll x \log^{-A} x$$ for any fixed $A>0$. If we had the additional hypothesis $S(F_k) + S(G_k) < 1$, then this would follow easily from the hypothesis $\operatorname*{GEH}[\vartheta]$ thanks to Claim \[geh-def\], since one can write $\lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{p(\cdot) > x^{\varepsilon}} = \alpha \star \beta$ with $$\alpha(n) := {\mathbf{1}}_{p(n) > x^{\varepsilon}} \sum_{d,d': [d,d'] = n} \mu(d) F_k(\log_x d) \mu(d') G_k(\log_x d')$$ and $$\beta(n) := {\mathbf{1}}_{p(n) > x^{\varepsilon}}.$$ But even in the absence of the hypothesis $S(F_k) + S(G_k) < 1$, we can still invoke $\operatorname*{GEH}[\vartheta]$ after appealing to the fundamental theorem of arithmetic. Indeed, if $n \in [x+h_k,2x+h_k]$ with $p(\cdot) > {\varepsilon}$, then we have $$n = p_1 \dots p_r$$ for some primes $x^{\varepsilon}< p_1 \leq \dots \leq p_r \leq 2x+h_k$, which forces $r \leq \frac{1}{{\varepsilon}}+1$. If we then partition $[x^{\varepsilon},2x+h_k]$ by $O( \log^{A+1} x )$ intervals $I_1,\dots,I_m$, with each $I_j$ contained in an interval of the form $[N, (1+\log^{-A} x) N]$, then we have $p_i \in I_{j_i}$ for some $1 \leq j_1 \leq \dots \leq j_r \leq m$, with the product interval $I_{j_1} \cdot \dots \cdot I_{j_r}$ intersecting $[x+h_k, 2x+h_k]$. For fixed $r$, there are $O( \log^{Ar+r} x)$ such tuples $(j_1,\dots,j_r)$, and a simple application of the prime number theorem with classical error term (and crude estimates on the discrepancy $\Delta$) shows that each tuple contributes $O( x \log^{-Ar+O(1)} x)$ to (here, and for the rest of this section, implied constants will be independent of $A$ unless stated otherwise). In particular, the $O(\log^{A(r-1)}{x})$ tuples $(j_1,\dots,j_r)$ with one repeated $j_i$, or for which the interval $I_{j_1} \cdot \dots \cdot I_{j_r}$ meets the boundary of $[x+h_k, 2x+h_k]$, contribute a total of $O(\log^{-A+O(1)}{x})$. This is an acceptable error to , and so these tuples may be removed. Thus it suffices to show that $$\sum_{q {\llcurly}x^\vartheta} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( \lambda_{F_k} \lambda_{G_k} {\mathbf{1}}_{A_{j_1,\dots,j_r}}; a\ (q))| \ll x \log^{-A(r+1)+O(1)} x$$ for any $1 \leq r \leq \frac{1}{{\varepsilon}}+1$ and $1 \leq j_1 < \dots < j_r \leq m$ with $I_{j_1} \cdot \dots \cdot I_{j_r}$ contained in $[x+h_k,x+2h_k]$, where $A_{j_1,\dots,j_r}$ is the set of all products $p_1 \dots p_r$ with $p_i \in I_{j_i}$ for $i=1,\dots,r$, and where we allow implied constants in the $\ll$ notation to depend on ${\varepsilon}$. But for $n$ in $A_{j_1,\dots,j_r}$, the $2^r$ factors of $n$ are just the products of subsets of $\{p_1,\dots,p_r\}$, and from the smoothness of $F_k,G_k$ we see that $\lambda_{F_k}(n)$ is equal to some bounded constant (depending on $j_1,\dots,j_r$, but independent of $p_1,\dots,p_r$), plus an error of $O(\log^{-A} x)$. As before, the contribution of this error is $O( \log^{-A(r+1)+O(1)} x)$, so it suffices to show that $$\sum_{q {\llcurly}x^\vartheta} \sup_{a \in ({\mathbb{Z}}/q{\mathbb{Z}})^\times} |\Delta( {\mathbf{1}}_{A_{j_1,\dots,j_r}}; a\ (q))| \ll x \log^{-A(r+1)+O(1)} x.$$ But one can write ${\mathbf{1}}_{A_{j_1,\dots,j_r}}$ as a convolution ${\mathbf{1}}_{A_{j_1}} \star \dots \star {\mathbf{1}}_{A_{j_r}}$, where $A_{j_i}$ denotes the primes in $I_{j_i}$; assigning $A_{j_r}$ (for instance) to be $\beta$ and the remaining portion of the convolution to be $\alpha$, the claim now follows from the hypothesis $\operatorname*{GEH}[\vartheta]$, thanks to the Siegel-Walfisz theorem (see e.g. [@siebert Satz 4] or [@ik Th. 5.29]). Finally, we show . By Lemma \[mul-asym\] we have $$\sum_{\substack{ d_1,\dots,d_{k-1},d'_1,\dots,d'_{k-1}\\ d_1d'_1,\dots,d_{k-1}d'_{k-1}, W \text{ coprime}}} \hspace{0pt minus 1fil}\frac{\prod_{i=1}^{k-1} \mu(d_{i}) \mu(d'_{i}) F_{i}(\log_x d_{i}) G_{i}(\log_x d'_{i}) }{{\varphi}(q_{W,d_1,\dots,d'_{k-1}})} = \frac{1}{{\varphi}(W)} (c'+o(1)) B^{-k+1},$$ where $$c' := \prod_{i=1}^{k-1} \int_0^1 F'_i(t) G'_i(t)\ dt$$ (note that $F_i, G_i$ are supported on $[0,1]$ by hypothesis), so by it suffices to show that $$\label{cpeps} \sum_{x+h_k \leq n \leq 2x+h_k} \lambda_{F_k}(n) \lambda_{G_k}(n) {\mathbf{1}}_{p(n)>x^{\varepsilon}} = (c''_{\varepsilon}+ o(1)) \frac{x}{\log x},$$ where $c''_{{\varepsilon}}$ is a quantity depending on ${\varepsilon}$ but not on $x$ such that $$\lim_{{\varepsilon}\to 0} c''_{{\varepsilon}} = \int_0^1 F'_k(t) G'_k(t)\ dt.$$ In the case $S(F_k)+S(G_k) < 1$, this would follow easily from (the $k=1$ case of) Theorem \[nonprime-asym\](i) and Proposition \[almostprime\]. In the general case, we may appeal once more to the fundamental theorem of arithmetic. As before, we may factor $n = p_1 \dots p_r$ for some $x^{\varepsilon}\leq p_1 \leq \dots \leq p_r \leq 2x+h_k$ and $r \leq \frac{1}{{\varepsilon}}+1$. The contribution of those $n$ with a repeated prime factor $p_i = p_{i+1}$ can easily be shown to be ${\llcurly}x^{1-{\varepsilon}}$ in the same manner we dealt with $\Sigma_2$, so we may restrict attention to the square-free $n$, for which the $p_i$ are strictly increasing. In that case, one can write $$\lambda_{F_k}(n) = (-1)^r \partial_{(\log_x p_1)} \dots \partial_{(\log_x p_r)} F_k(0)$$ and $$\lambda_{G_k}(n) = (-1)^r \partial_{(\log_x p_1)} \dots \partial_{(\log_x p_r)} G_k(0)$$ where $\partial_{(h)} F(x) := F(x+h)-F(x)$. On the other hand, a standard application of Mertens’ theorem and the prime number theorem (and an induction on $r$) shows that for any fixed $r \geq 1$ and any fixed continuous function $f: {\mathbb{R}}^r \to {\mathbb{R}}$, we have $$\sum_{x^{\varepsilon}\leq p_1 < \dots < p_r: x+h_k \leq p_1 \dots p_r \leq 2x+h_k} f(\log_x p_1,\dots, \log_x p_r) = (c_f + o(1)) \frac{x}{\log x}$$ where $c_f$ is the quantity $$c_f := \int_{{\varepsilon}\leq t_1 < \dots < t_r: t_1 + \dots + t_r = 1} f( t_1,\dots,t_r)\ \frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r}$$ where we lift Lebesgue measure $dt_1 \dots dt_{r-1}$ up to the hyperplane $t_1+\dots+t_r=1$, thus $$\int_{t_1+\dots+t_r=1} F(t_1,\dots,t_r)\ dt_1 \dots dt_{r-1} := \int_{{\mathbb{R}}^{r-1}} F(t_1,\dots,t_{r-1},1-t_1-\dots-t_{r-1}) dt_1 \dots dt_{r-1}.$$ Putting all this together, we see that we obtain an asymptotic with $$c''_{\varepsilon}:= \sum_{1 \leq r \leq \frac{1}{{\varepsilon}}+1} \int_{{\varepsilon}\leq t_1 < \dots < t_r: t_1 + \dots + t_r = 1} \partial_{(t_1)} \dots \partial_{(t_r)} F_k(0) \partial_{(t_1)} \dots \partial_{(t_r)} G_k(0)\ \frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r}.$$ Comparing with the first part of Proposition \[almostprime\] we see that $c''_{\varepsilon}= O(1)$ uniformly in ${\varepsilon}$; subtracting two instances of and comparing with the last part of Proposition \[almostprime\] we see that $|c''_{{\varepsilon}_1} - c''_{{\varepsilon}_2}| \ll {\varepsilon}_1 + {\varepsilon}_2$ for any ${\varepsilon}_1,{\varepsilon}_2 > 0$. We conclude that $c''_{\varepsilon}$ converges to a limit as ${\varepsilon}\rightarrow 0$ for any $F,G$. This implies the absolute convergence $$\label{absconv} \sum_{r>0} \int_{0 < t_1 < \dots < t_r: t_1 + \dots + t_r = 1} |\partial_{(t_1)} \dots \partial_{(t_r)} F_k(0)| |\partial_{(t_1)} \dots \partial_{(t_r)} G_k(0)|\ \frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r} < \infty;$$ indeed, by the Cauchy-Schwarz inequality it suffices to establish this for $F=G$, at which point we may remove the absolute value signs and use the boundedness of $c''_{\varepsilon}$. By the dominated convergence theorem, it therefore suffices to establish the identity $$\label{condconv} \sum_{r>0} \int_{0 < t_1 < \dots < t_r: t_1 + \dots + t_r = 1} \partial_{(t_1)} \dots \partial_{(t_r)} F_k(0) \partial_{(t_1)} \dots \partial_{(t_r)} G_k(0)\ \frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r} = \int_0^1 F'_k(t) G'_k(t)\ dt.$$ It will suffice to show the identity $$\label{depol} \sum_{r>0} \int_{0 < t_1 < \dots < t_r: t_1 + \dots + t_r = 1} |\partial_{(t_1)} \dots \partial_{(t_r)} F(0)|^2\ \frac{dt_1 \dots dt_{r-1}}{t_1 \dots t_r} = \int_0^1 |F'(t)|^2\ dt$$ for any smooth $F: [0,+\infty) \to {\mathbb{R}}$, since follows by replacing $F$ with $F_k+G_k$ and $F_k-G_k$ and then subtracting. At this point we use the following identity: For any positive reals $t_1,\dots,t_r$ with $r \geq 1$, we have $$\label{star} \frac{1}{t_1 \dots t_r} = \sum_{\sigma \in S_r} \frac{1}{\prod_{i=1}^r (\sum_{j=i}^r t_{\sigma(j)})}.$$ Thus, for instance, when $r=2$ we have $$\frac{1}{t_1 t_2} = \frac{1}{(t_1+t_2) t_1} + \frac{1}{(t_1+t_2)t_2}.$$ If the right-hand side of is denoted $f_r( t_1,\dots,t_r )$, then one easily verifies the identity $$f_r(t_1,\dots,t_r) = \frac{1}{t_1+\dots+t_r} \sum_{i=1}^r f_{r-1}(t_1,\dots,t_{i-1},t_{i+1},\dots,t_r)$$ for any $r > 1$; but the left-hand side of also obeys this identity, and the claim then follows from induction. From this lemma and symmetrisation, we may rewrite the left-hand side of as $$\sum_{r>0} \int_{\substack{t_1,\dots,t_r \geq 0\\ t_1+\dots+t_r =1}} |\partial_{(t_1)} \dots \partial_{(t_r)} F(0)|^2\ \frac{dt_1 \dots dt_{r-1}}{\prod_{i=1}^r(\sum_{j=i}^rt_i)}.$$ Let $$I_a(F) := \int_0^a F'(t)^2\ dt,$$ and $$J_a(F) := (\partial_{(a)} F(0))^2.$$ One can then rewrite as the identity $$\label{depol-2} I_1(F) = \sum_{r=1}^\infty K_{1,r}(F),$$ where $$K_{a,r}(F) := \int_{\substack{t_1,\dots,t_r \geq 0\\ t_1+\dots+t_r =a}} J_{t_r}( \partial_{(t_1)} \dots \partial_{(t_{r-1})} F) \frac{dt_1 \dots dt_{r-1}}{a(a-t_1) \dots (a-t_1-\dots-t_{r-1})}.$$ To prove this, we first observe the identity $$I_a(F) = \frac{1}{a} J_a(F) + \int_{0 \leq t \leq a} I_{a-t}( \partial_{(t)} F ) \frac{dt}{a}$$ for any $a>0$; indeed, we have $$\begin{aligned} \int_{0 \leq t \leq a} I_{a-t}( \partial_{(t)} F ) \frac{dt}{a} &= \int_{0 \leq t \leq a; 0 \leq u \leq a-t} |F'(t+u) - F'(t)|^2\ \frac{du dt}{a} \\ &= \int_{0 \leq t \leq s \leq a} |F'(s) - F'(t)|^2\ \frac{ds dt}{a} \\ &= \frac{1}{2} \int_0^a \int_0^a |F'(s) - F'(t)|^2\ \frac{ds dt}{a} \\ &= \int_0^a |F'(s)|^2\ ds - \frac{1}{a} \left(\int_0^a F'(s)\ ds\right) \left(\int_0^a F'(t)\ dt\right) \\ &= I_a(F) - \frac{1}{a} J_a(F),\end{aligned}$$ and the claim follows. Iterating this identity $k$ times, we see that $$\label{iak} I_a(F) = \sum_{r=1}^k K_{a,r}(F) + L_{a,k}(F)$$ for any $k \geq 1$, where $$L_{a,k}(F) := \int_{\substack{t_1,\dots,t_k \geq 0\\ t_1+\dots+t_k \leq a}} I_{1-t_1-\dots-t_k}( \partial_{(t_1)} \dots \partial_{(t_k)} F) \frac{dt_1 \dots dt_k}{a(a-t_1) \dots (a-t_1-\dots-t_{k-1})}.$$ In particular, dropping the $L_{a,k}(F)$ term and sending $k \to \infty$ yields the lower bound $$\label{krsum} \sum_{r=1}^\infty K_{a,r}(F) \leq I_a(F).$$ On the other hand, we can expand $L_{a,k}(F)$ as $$\int_{\substack{t_1,\dots,t_k,t \geq 0\\t_1+\dots+t_k+t \leq a}} |\partial_{(t_1)} \dots \partial_{(t_k)} F'(t)|^2 \frac{dt_1 \dots dt_k dt}{a(a-t_1) \dots (a-t_1-\dots-t_{k-1})}.$$ Writing $s := t_1 + \dots + t_k$, we obtain the upper bound $$L_{a,k}(F) \leq \int_{s,t \geq 0: s+t \leq a} K_{s,k}( F'_t )\ dt,$$ where $F_t(x) := F(x+t)$. Summing this and using and the monotone convergence theorem, we conclude that $$\sum_{k=1}^\infty L_{a,k}(F) \leq \int_{s,t \geq 0: s+t \leq a} I_{s}( F_t )\ dt < \infty,$$ and in particular $L_{a,k}(F) \to 0$ as $k \to \infty$. Sending $k \to \infty$ in , we obtain as desired. Reduction to a variational problem {#variational-sec} ================================== Now that we have proven Theorems \[prime-asym\] and \[nonprime-asym\], we can establish Theorems \[maynard-thm\], \[maynard-trunc\], \[epsilon-trick\], \[epsilon-beyond\]. The main technical difficulty is to take the multidimensional measurable functions $F$ appearing in these functions and approximate them by tensor products of smooth functions, for which Theorems \[prime-asym\] and \[nonprime-asym\] may be applied. Proof of Theorem \[maynard-thm\] {#may-sec} -------------------------------- We now prove Theorem \[maynard-thm\]. Let $k, m, \vartheta$ obey the hypotheses of that theorem, thus we may find a fixed square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$ supported on the simplex $${\mathcal R}_k := \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq 1 \}$$ and not identically zero and with $$\label{jbig} \frac{\sum_{i=1}^k J_i(F)}{I(F)} > \frac{2m}{\vartheta}.$$ We now perform a number of technical steps to further improve the structure of $F$. Our arguments here will be somewhat convoluted, and are not the most efficient way to prove Theorem \[maynard-thm\] (which in any event was already established in [@maynard-new]), but they will motivate the similar arguments given below to prove the more difficult results in Theorems \[maynard-trunc\], \[epsilon-trick\], \[epsilon-beyond\]. In particular, we will use regularisation techniques which are compatible with the vanishing marginal condition that is a key hypothesis in Theorem \[epsilon-beyond\]. We first need to rescale and retreat a little bit from the slanted boundary of the simplex ${\mathcal R}_k$. Let $\delta_1 > 0$ be a sufficiently small fixed quantity, and write $F_1: [0,+\infty)^k \to {\mathbb{R}}$ to be the rescaled function $$F_1(t_1,\dots,t_k) := F( \frac{t_1}{\vartheta/2-\delta_1}, \dots, \frac{t_k}{\vartheta/2-\delta_1} ).$$ Thus $F_1$ is a fixed square-integrable measurable function supported on the rescaled simplex $$(\vartheta/2-\delta_1) \cdot {\mathcal R}_k = \{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq \vartheta/2-\delta_1 \}.$$ From , we see that if $\delta_1$ is small enough, then $F_1$ is not identically zero and $$\label{jbig-2} \frac{\sum_{i=1}^k J_i(F_1)}{I(F_1)} > m.$$ Let $\delta_1$ and $F_1$ be as above. Next, let $\delta_2 > 0$ be a sufficiently small fixed quantity (smaller than $\delta_1$), and write $F_2: [0,+\infty)^k \to {\mathbb{R}}$ to be the shifted function, defined by setting $$F_2(t_1,\dots,t_k) := F_1( t_1-\delta_2, \dots, t_k-\delta_2 )$$ when $t_1,\dots,t_k \geq \delta_2$, and $F_2(t_1,\dots,t_k)=0$ otherwise. As $F_1$ was square-integrable, compactly supported, and not identically zero, and because spatial translation is continuous in the strong operator topology on $L^2$, it is easy to see that we will have $F_2$ not identically zero and that $$\label{jbig-3} \frac{\sum_{i=1}^k J_i(F_2)}{I(F_2)} > m$$ for $\delta_2$ small enough (after restricting $F_2$ back to $[0,+\infty)^k$, of course). For $\delta_2$ small enough, this function will be supported on the region $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: t_1 \dots + t_k \leq \vartheta/2-\delta_2; t_1,\dots,t_k \geq \delta_2 \},$$ thus the support of $F_2$ stays away from all the boundary faces of ${\mathcal R}_k$. By convolving $F_2$ with a smooth approximation to the identity that is supported sufficiently close to the origin, one may then find a *smooth* function $F_3: [0,+\infty)^k \to {\mathbb{R}}$, supported on $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: t_1 \dots + t_k \leq \vartheta/2-\delta_2/2; t_1,\dots,t_k \geq \delta_2/2 \},$$ which is not identically zero, and such that $$\label{jbig-4} \frac{\sum_{i=1}^k J_i(F_3)}{I(F_3)} > m.$$ We extend $F_3$ by zero to all of ${\mathbb{R}}^k$, and then define the function $f_3: {\mathbb{R}}^k \to {\mathbb{R}}$ by $$f_3(t_1,\dots,t_k) := \int_{s_1 \geq t_1, \dots, s_k \geq t_k} F_3(s_1,\dots,s_k)\ ds_1 \dots ds_k,$$ thus $f_3$ is smooth, not identically zero and supported on the region $$\label{ftj-set} \{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/2) \leq \vartheta/2 - \delta_2/2 \}.$$ From the fundamental theorem of calculus we have $$\label{ftj} F_3(t_1,\dots,t_k) := (-1)^k \frac{\partial^k}{\partial t_1 \dots \partial t_k} f_3(t_1,\dots,t_k),$$ and so $I(F_3) = \tilde I(f_3)$ and $J_i(F_3) = \tilde J_i(f_3)$ for $i=1,\dots,k$, where $$\label{tidef} \tilde I(f_3) := \int_{[0,+\infty)^k} \left|\frac{\partial^k}{\partial t_1 \dots \partial t_k} f_3(t_1,\dots,t_k)\right|^2\ dt_1 \dots dt_k$$ and $$\label{tjdef} \tilde J_i(f_3) := \int_{[0,+\infty)^{k-1}} \left|\frac{\partial^{k-1}}{\partial t_1 \dots \partial t_{i-1} \partial t_{i+1} \dots \partial t_k} f_3(t_1,\dots,t_{i-1}, 0, t_{i+1}, \dots, t_k)\right|^2\ dt_1 \dots dt_{i-1} dt_{i+1} \dots dt_k.$$ In particular, $$\label{jbig-5} \frac{\sum_{i=1}^k \tilde J_i(f_3)}{\tilde I(f_3)} > m.$$ Now we approximate $f_3$ by linear combinations of tensor products. By the Stone-Weierstrass theorem, we may express $f_3$ (on $[0,+\infty)^k$) as the uniform limit of functions of the form $$\label{cff} (t_1,\dots,t_k) \mapsto \sum_{j=1}^J c_j f_{1,j}(t_1) \dots f_{k,j}(t_k)$$ where $c_1,\dots,c_J$ are real scalars, and $f_{i,j}: {\mathbb{R}}\to {\mathbb{R}}$ are smooth compactly supported functions. Since $f_3$ is supported in , we can ensure that all the components $f_{1,j}(t_1) \dots f_{k,j}(t_k)$ are supported in the slightly larger region $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/4) \leq \vartheta/2 - \delta_2/4 \}.$$ Observe that if one convolves a function of the form with a smooth approximation to the identity which is of tensor product form $(t_1,\dots,t_k) \mapsto \varphi_1(t_1) \dots \varphi_1(t_k)$, one obtains another function of this form. Such a convolution converts a uniformly convergent sequence of functions to a *uniformly smoothly* convergent sequence of functions (that is to say, all derivatives of the functions converge uniformly). From this, we conclude that $f_3$ can be expressed (on $[0,+\infty)^k$) as the *smooth* limit of functions of the form , with each component $f_{1,j}(t_1) \dots f_{k,j}(t_k)$ supported in the region $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/8) \leq \vartheta/2 - \delta_2/8 \}.$$ Thus, we may find such a linear combination $$\label{f4} f_4(t_1,\dots,t_k) = \sum_{j=1}^J c_j f_{1,j}(t_1) \dots f_{k,j}(t_k)$$ with $J$, $c_j$, $f_{i,j}$ fixed and $f_4$ not identically zero, with $$\label{jbig-6} \frac{\sum_{i=1}^k \tilde J_i(f_4)}{\tilde I(f_4)} > m.$$ Furthermore, by construction we have $$\label{sfg} S(f_{1,j}) + \dots + S(f_{k,j}) < \frac{\vartheta}{2} \leq \frac{1}{2}$$ for all $j=1,\dots,J$, where $S()$ was defined in . Now we construct the sieve weight $\nu: {\mathbb{N}}\to {\mathbb{R}}$ by the formula $$\label{nu-def} \nu(n) := \left( \sum_{j=1}^J c_j \lambda_{f_{1,j}}(n+h_1) \dots \lambda_{f_{k,j}}(n+h_k) \right)^2,$$ where the divisor sums $\lambda_f$ were defined in . Clearly $\nu$ is non-negative. Expanding out the square and using Theorem \[nonprime-asym\](i) and , we see that $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) = (\alpha + o(1)) B^{-k} \frac{x}{\log x}$$ where $$\alpha := \sum_{j=1}^J \sum_{j'=1}^J c_j c_{j'} \prod_{i=1}^k \int_0^\infty f'_{i,j}(t_i) f'_{i,j'}(t_i)\ dt_i$$ which factorizes using , as $$\begin{aligned} \alpha &= \int_{[0,+\infty)^k} \left|\frac{\partial^{k-1}}{\partial t_1 \dots \partial t_k} f_4(t_1,\dots,t_k)\right|^2\ dt_1 \dots dt_k \\ &= \tilde I(f_4).\end{aligned}$$ Now consider the sum $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_k).$$ By , one has $$\lambda_{f_{k,j}}(n+h_k) = f_{k,j}(0)$$ whenever $n$ gives a non-zero contribution to the above sum. Expanding out the square in again and using Theorem \[prime-asym\](i) and (and the hypothesis $\operatorname*{EH}[\vartheta]$), we thus see that $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_k) = (\beta_k + o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ where $$\beta_k := \sum_{j=1}^J \sum_{j'=1}^J c_j c_{j'} f_{i,j}(0) f_{i,j'}(0) \prod_{i=1}^{k-1} \int_0^\infty f'_{i,j}(t_i) f'_{i,j'}(t_i)\ dt_i$$ which factorizes using , as $$\begin{aligned} \beta_k &= \int_{[0,+\infty)^k} \left|\frac{\partial^k}{\partial t_1 \dots \partial t_{k-1}} f_4(t_1,\dots,t_{k-1},0)\right|^2\ dt_1 \dots dt_{k-1} \\ &= \tilde J_k(f_4).\end{aligned}$$ More generally, we see that $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_i) = (\beta_i + o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ for $i=1,\dots,k$, with $\beta_i := \tilde J_i(f_4)$. Applying Lemma \[crit\] and , we obtain $\operatorname*{DHL}[k,m+1]$ as required. Proof of Theorem \[maynard-trunc\] {#trunc-sec} ---------------------------------- Now we prove Theorem \[maynard-trunc\], which uses a very similar argument to that of the previous section. Let $k, m, \varpi, \delta, F$ be as in Theorem \[maynard-trunc\]. By performing the same rescaling as in the previous section (but with $1/2 + 2\varpi$ playing the role of $\vartheta$), we see that we can find a fixed square-integrable measurable function $F_1$ supported on the rescaled truncated simplex $$\{ (t_1,\dots,t_k) \in [0,+\infty)^k: t_1+\dots+t_k \leq \frac{1}{4} + \varpi - \delta_1; t_1,\dots,t_k < \delta - \delta_1 \}$$ for some sufficiently small fixed $\delta_1>0$, such that holds. By repeating the arguments of the previous section we may eventually arrive at a smooth function $f_4: {\mathbb{R}}^k \to {\mathbb{R}}$ of the form , which is not identically zero and obeys , and such that each component $f_{1,j}(t_1) \dots f_{k,j}(t_k)$ is supported in the region $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/8) \leq \frac{1}{4}+\varpi - \delta_2/8; t_1,\dots,t_k < \delta - \delta_2/8 \}$$ for some sufficiently small $\delta_2>0$. In particular, one has $$S(f_{1,j}) + \dots + S(f_{k,j}) < \frac{1}{4}+\varpi \leq \frac{1}{2}$$ and $$S(f_{1,j}),\dots,S(f_{k,j}) < \delta$$ for all $j=1,\dots,J$. If we then define $\nu$ by as before, and repeat all of the above arguments (but use Theorem \[prime-asym\](ii) and $\operatorname*{MPZ}[\varpi,\delta]$ in place of Theorem \[prime-asym\](i) and $\operatorname*{EH}[\vartheta]$), we obtain the claim; we leave the details to the interested reader. Proof of Theorem \[epsilon-trick\] {#trick-sec} ---------------------------------- Now we prove Theorem \[epsilon-trick\]. Let $k, m, {\varepsilon}, \vartheta$ be as in that theorem. Then one may find a square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$ supported on $(1+{\varepsilon}) \cdot {\mathcal R}_k$ which is not identically zero, and with $$\frac{\sum_{i=1}^k J_{i,1-{\varepsilon}}(F)}{I(F)} > \frac{2m}{\vartheta}.$$ By truncating and rescaling as in Section \[may-sec\], we may find a fixed bounded measurable function $F_1: [0,+\infty)^k \to {\mathbb{R}}$ on the simplex $(1+{\varepsilon}) (\frac{\vartheta}{2}-\delta_1) \cdot {\mathcal R}_k$ such that $$\frac{\sum_{i=1}^k J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(F_1)}{I(F_1)} > m.$$ By repeating the arguments in Section \[may-sec\], we may eventually arrive at a smooth function $f_4: {\mathbb{R}}^k \to {\mathbb{R}}$ of the form , which is not identically zero and obeys $$\label{f4-ratio} \frac{\sum_{i=1}^k \tilde J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(f_4)}{\tilde I(f_4)} > m$$ with $$\begin{aligned} \tilde J_{i, (1-{\varepsilon}) \frac{\vartheta}{2}}(f_4) &:= \int_{(1-{\varepsilon}) \frac{\vartheta}{2} \cdot {\mathcal R}_{k-1}} \left|\frac{\partial^{k-1}}{\partial t_1 \dots \partial t_{i-1} \partial t_{i+1} \dots \partial t_k} f_4(t_1,\dots,t_{i-1}, 0, t_{i+1}, \dots, t_k)\right|^2\\ &\quad \ dt_1 \dots dt_{i-1} dt_{i+1} \dots dt_k,\end{aligned}$$ and such that each component $f_{1,j}(t_1) \dots f_{k,j}(t_k)$ is supported in the region $$\left\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/8) \leq (1+{\varepsilon}) \frac{\vartheta}{2} - \frac{\delta_2}{8}\right\}$$ for some sufficiently small $\delta_2>0$. In particular, we have $$\label{sff} S(f_{1,j}) + \dots + S(f_{k,j}) \leq (1+{\varepsilon}) \frac{\vartheta}{2} - \frac{\delta_2}{8}$$ for all $1 \leq j \leq J$. Let $\delta_3 > 0$ be a sufficiently small fixed quantity (smaller than $\delta_1$ or $\delta_2$). By a smooth partitioning, we may assume that all of the $f_{i,j}$ are supported in intervals of length at most $\delta_3$, while keeping the sum $$\label{abs-sum} \sum_{j=1}^J |c_j| |f_{1,j}(t_1)| \dots |f_{k,j}(t_k)|$$ bounded uniformly in $t_1,\dots,t_k$ and in $\delta_3$. Now let $\nu$ be as in , and consider the expression $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n).$$ This expression expands as a linear combination of the expressions $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \prod_{i=1}^k \lambda_{f_{i,j}}(n+h_i) \lambda_{f_{i,j'}}(n+h_i)$$ for various $1 \leq j,j' \leq J$. We claim that this sum is equal to $$\left(\prod_{i=1}^k \int_0^1 f'_{i,j}(t_i) f'_{i,j'}(t_i)\ dt_i + o(1)\right) B^{-k} \frac{x}{W}.$$ To see this, we divide into two cases. First suppose that hypothesis (i) from Theorem \[epsilon-trick\] holds. Then from we have $$\sum_{i=1}^k (S(f_{i,j}) + S(f_{i,j'})) < (1+{\varepsilon}) \vartheta < 1$$ and the claim follows from Theorem \[nonprime-asym\](i). Now suppose instead that hypothesis (ii) from Theorem \[epsilon-trick\] holds, then from one has $$\sum_{i=1}^k (S(f_{i,j}) + S(f_{i,j'})) < (1+{\varepsilon}) \vartheta < \frac{k}{k-1} \vartheta,$$ and so from the pigeonhole principle we have $$\sum_{1 \leq i \leq k: i \neq i_0} (S(f_{i,j}) + S(f_{i,j'})) < \vartheta$$ for some $1 \leq i_0 \leq k$. The claim now follows from Theorem \[nonprime-asym\](ii). Putting this together as in Section \[may-sec\], we conclude that $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) = (\alpha + o(1)) B^{-k} \frac{x}{W}$$ where $$\alpha := \tilde I(f_4).$$ Now we consider the sum $$\label{theta-sum} \sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_k).$$ From Proposition \[geh-eh\] we see that we have $\operatorname*{EH}[\vartheta]$ as a consequence of the hypotheses of Theorem \[epsilon-trick\]. However, this combined with Theorem \[prime-asym\] is not strong enough to obtain an asymptotic for the sum , as there is an epsilon loss in . But observe that Lemma \[crit\] only requires a *lower* bound on the sum , rather than an asymptotic. To obtain this lower bound, we partition $\{1,\dots,J\}$ into ${\mathcal J}_1 \cup {\mathcal J}_2$, where ${\mathcal J}_1$ consists of those indices $j \in \{1,\dots,J\}$ with $$\label{sff-minus} S(f_{1,j}) + \dots + S(f_{k-1,j}) < (1-{\varepsilon}) \frac{\vartheta}{2}$$ and ${\mathcal J}_2$ is the complement. From the elementary inequality $$(x_1 + x_2)^2 = x_1^2 + 2x_1 x_2 + x_2^2 \geq (x_1+2x_2) x_1$$ we obtain the pointwise lower bound $$\nu(n) \geq \left( (\sum_{j \in {\mathcal J}_1} + 2 \sum_{j \in {\mathcal J}_2}) c_j \lambda_{f_{1,j}}(n+h_1) \dots \lambda_{f_{k,j}}(n+h_k) \right) \left( \sum_{j' \in {\mathcal J}_1} c_{j'} \lambda_{f_{1,j'}}(n+h_1) \dots \lambda_{f_{k,j'}}(n+h_k) \right).$$ The point of performing this lower bound is that if $j \in {\mathcal J}_1 \cup {\mathcal J}_2$ and $j' \in {\mathcal J}_1$, then from , one has $$\sum_{i=1}^{k-1} (S(f_{i,j}) + S(f_{i,j'})) < \vartheta$$ which makes Theorem \[prime-asym\](i) available for use. Indeed, for any $j \in \{1,\dots,J\}$ and $i=1,\dots,k$, we have from that $$S(f_{i,j}) \leq (1+{\varepsilon}) \frac{\vartheta}{2} < \vartheta < 1$$ and so by we have $$\label{nutheta} \begin{split} \nu(n) \theta(n+h_k) &\geq \left( (\sum_{j \in {\mathcal J}_1} + 2 \sum_{j \in {\mathcal J}_2}) c_j \lambda_{f_{1,j}}(n+h_1) \dots \lambda_{f_{k-1,j}}(n+h_{k-1}) f_{k,j}(0) \right)\\ &\quad \times \left( \sum_{j' \in {\mathcal J}_1} c_{j'} \lambda_{f_{1,j'}}(n+h_1) \dots \lambda_{f_{k-1,j'}}(n+h_{k-1}) f_{k,j'}(0) \right) \theta(n+h_k) \end{split}$$ for $x \leq n \leq 2x$. If we then apply Theorem \[prime-asym\](i) and the hypothesis $\operatorname*{EH}[\vartheta]$, we obtain the lower bound $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_k) \geq (\beta_k - o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ with $$\beta_k := (\sum_{j \in {\mathcal J}_1} + 2 \sum_{j \in {\mathcal J}_2}) \sum_{j' \in {\mathcal J}_1} c_j c_{j'} f_{k,j}(0) f_{k,j'}(0) \prod_{i=1}^{k-1} \int_0^\infty f'_{i,j}(t_i) f'_{i,j'}(t_i)\ dt_i$$ which we can rearrange as $$\begin{aligned} \beta_k &= \int_{[0,+\infty)^{k-1}} \left(\frac{\partial^{k-1}}{\partial t_1 \dots \partial t_{k-1}} f_{4,1}(t_1,\dots,t_{k-1},0) + 2\frac{\partial^{k-1}}{\partial t_1 \dots \partial t_{k-1}} f_{4,2}(t_1,\dots,t_{k-1},0)\right)\\ &\quad\quad \frac{\partial^{k-1}}{\partial t_1 \dots \partial t_{k-1}} f_{4,1}(t_1,\dots,t_{k-1},0)\ dt_1 \dots dt_{k-1} \end{aligned}$$ where $$f_{4,l}(t_1,\dots,t_k) :=\sum_{j \in {\mathcal J}_l} c_j f_{1,j}(t_1) \dots f_{k,j}(t_k)$$ for $l=1,2$. Note that $f_{4,1}, f_{4,2}$ are both bounded pointwise by , and their supports only overlap on a set of measure $O( \delta_3 )$. We conclude that $$\beta_k = \tilde J_k( f_{4,1} ) + O(\delta_3)$$ with the implied constant independent of $\delta_3$, and thus $$\beta_k = \tilde J_{k, (1-{\varepsilon}) \frac{\vartheta}{2}}( f_4 ) + O(\delta_3).$$ A similar argument gives $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_i) \geq (\beta_i - o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ for $i=1,\dots,k$ with $$\beta_i = \tilde J_{i, (1-{\varepsilon}) \frac{\vartheta}{2}}( f_4 ) + O(\delta_3).$$ If we choose $\delta_3$ small enough, then the claim $\operatorname*{DHL}[k,m+1]$ now follows from Lemma \[crit\] and . Proof of Theorem \[epsilon-beyond\] {#beyond-sec} ----------------------------------- Finally, we prove Theorem \[epsilon-beyond\]. Let $k, m, {\varepsilon}, F$ be as in that theorem. By rescaling as in previous sections, we may find a square-integrable function $F_1: [0,+\infty)^k \to {\mathbb{R}}$ supported on $(\frac{k}{k-1} \frac{\vartheta}{2}-\delta_1) \cdot {\mathcal R}_k$ for some sufficiently small fixed $\delta_1 > 0$, which is not identically zero, which obeys the bound $$\frac{\sum_{i=1}^k J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(F_1)}{I(F_1)} > m$$ and also obeys the vanishing marginal condition whenever $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$ are such that $$t_1+\dots+t_{i-1}+t_{i+1}+\dots+t_k > (1+{\varepsilon}) \frac{\vartheta}{2} - \delta_1.$$ As before, we pass from $F_1$ to $F_2$ by a spatial translation, and from $F_2$ to $F_3$ by a regularisation; crucially, we note that both of these operations interact well with the vanishing marginal condition , with the end product being that we obtain a smooth function $F_3: [0,+\infty)^k \to {\mathbb{R}}$, supported on the region $$\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: t_1 \dots + t_k \leq \frac{k}{k-1} \frac{\vartheta}{2}-\frac{\delta_2}{2}; t_1,\dots,t_k \geq \frac{\delta_2}{2} \}$$ for some sufficiently small $\delta_2>0$, which is not identically zero, obeying the bound $$\frac{\sum_{i=1}^k J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(F_3)}{I(F_3)} > m$$ and also obeying the vanishing marginal condition whenever $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$ are such that $$t_1+\dots+t_{i-1}+t_{i+1}+\dots+t_k > (1+{\varepsilon}) \frac{\vartheta}{2} - \frac{\delta_2}{2}.$$ As before, we now define the function $f_3: {\mathbb{R}}^k \to {\mathbb{R}}$ by $$f_3(t_1,\dots,t_k) := \int_{s_1 \geq t_1, \dots, s_k \geq t_k} F_3(s_1,\dots,s_k)\ ds_1 \dots ds_k,$$ thus $f_3$ is smooth, not identically zero and supported on the region $$\left\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/2) \leq \frac{k}{k-1} \frac{\vartheta}{2} - \frac{\delta_2}{2} \right\}.$$ Furthermore, from the vanishing marginal condition we see that we also have $$f_3(t_1,\dots,t_k) =0$$ whenever we have some $1 \leq i \leq k$ for which $t_i \leq \delta_2/2$ and $$t_1 + \dots + t_{i-1} + t_{i+1} + \dots + t_k \geq (1+{\varepsilon}) \frac{\vartheta}{2} - \frac{\delta_2}{2}.$$ From the fundamental theorem of calculus as before, we have $$\frac{\sum_{i=1}^k \tilde J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(f_3)}{\tilde I(f_3)} > m.$$ Using the Stone-Weierstrass theorem as before, we can then find a function $f_4$ of the form $$\label{cff-again} (t_1,\dots,t_k) \mapsto \sum_{j=1}^J c_j f_{1,j}(t_1) \dots f_{k,j}(t_k)$$ where $c_1,\dots,c_J$ are real scalars, and $f_{i,j}: {\mathbb{R}}\to {\mathbb{R}}$ are smooth functions supported on intervals of length at most $\delta_3>0$ for some sufficiently small $\delta_3>0$, with each component $f_{1,j}(t_1) \dots f_{k,j}(t_k)$ supported in the region $$\left\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: \sum_{i=1}^k \max(t_i, \delta_2/8) \leq \frac{k}{k-1} \frac{\vartheta}{2} - \delta_2/8 \right\}$$ and avoiding the regions $$\left\{ (t_1,\dots,t_k) \in {\mathbb{R}}^k: t_i \leq \delta_2/8; \quad t_1 + \dots + t_{i-1} + t_{i+1} + \dots + t_k \geq (1+{\varepsilon}) \frac{\vartheta}{2} - \delta_2/8 \right\}$$ for each $i=1,\dots,k$, and such that $$\frac{\sum_{i=1}^k \tilde J_{i,(1-{\varepsilon}) \frac{\vartheta}{2}}(f_4)}{\tilde I(f_4)} > m.$$ In particular, for any $j=1,\dots,J$ we have $$\label{sfk-sum} S(f_{1,j}) + \dots + S(f_{k,j}) < \frac{k}{k-1} \frac{\vartheta}{2} < \frac{1}{2} \frac{k}{k-1} \leq 1$$ and for any $i=1,\dots,k$ with $f_{k,i}$ not vanishing at zero, we have $$\label{sfk-sum-2} S(f_{1,j}) + \dots + S(f_{k,i-1}) + S(f_{k,i+1}) + \dots + S(f_{k,j}) < (1+{\varepsilon}) \frac{\vartheta}{2}.$$ Let $\nu$ be defined by . From , the hypothesis $\operatorname*{GEH}[\vartheta]$, and the argument from the previous section used to prove Theorem \[epsilon-trick\](ii), we have $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) = (\alpha + o(1)) B^{-k} \frac{x}{W}$$ where $$\alpha := \tilde I(f_4).$$ Similarly, from (and the upper bound $S(f_{i,j}) < 1$ from ), the hypothesis $\operatorname*{EH}[\vartheta]$ (which is available by Proposition \[geh-eh\]), and the argument from the previous section we have $$\sum_{\substack{x \leq n \leq 2x\\ n = b\ (W)}} \nu(n) \theta(n+h_i) \geq (\beta_i - o(1)) B^{1-k} \frac{x}{{\varphi}(W)}$$ for $i=1,\dots,k$ with $$\beta_i = \tilde J_{i, (1-{\varepsilon}) \frac{\vartheta}{2}}( f_4 ) + O(\delta_3).$$ Setting $\delta_3$ small enough, the claim $\operatorname*{DHL}[k,m+1]$ now follows from Lemma \[crit\]. Asymptotic analysis {#asymptotics-sec} =================== We now establish upper and lower bounds on the quantity $M_k$ defined in , as well as for the related quantities appearing in Theorem \[maynard-trunc\]. To obtain an upper bound on $M_k$, we use the following consequence of the Cauchy-Schwarz inequality. \[cs\] Let $k \geq 2$, and suppose that there exist positive measurable functions $G_i: {\mathcal R}_k \to (0,+\infty)$ for $i=1,\dots,k$ such that $$\label{gi} \int_0^\infty G_i(t_1,\dots,t_k)\ dt_i \leq 1$$ for all $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$, where we extend $G_i$ by zero to all of $[0,+\infty)^k$. Then we have $$\label{mk-bound} M_k \leq \operatorname{ess} \sup_{(t_1,\dots,t_k) \in {\mathcal R}_k} \sum_{i=1}^k \frac{1}{G_i(t_1,\dots,t_k)}.$$ Here $\operatorname{ess} \sup$ refers to essential supremum (thus, we may ignore a subset of ${\mathcal R}_k$ of measure zero in the supremum). Let $F: [0,+\infty)^k \to {\mathbb{R}}$ be a square-integrable function supported on ${\mathcal R}_k$. From the Cauchy-Schwarz inequality and , we have $$\left(\int_0^\infty F(t_1,\dots,t_k)\ dt_i\right)^2 \leq \int_0^\infty \frac{F(t_1,\dots,t_k)^2}{G_i(t_1,\dots,t_k)} \ dt_i$$ for any $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$, with $F^2/G$ extended by zero outside of ${\mathcal R}_k$. Inserting this into and integrating, we conclude that $$J_i(F) \leq \int_{{\mathcal R}_k} \frac{F(t_1,\dots,t_k)^2 }{G_i(t_1,\dots,t_k)}\ dt_1\dots dt_k.$$ Summing in $i$ and using , , we obtain the claim. As a corollary, we can compute $M_k$ exactly if we can locate a positive eigenfunction: \[ef\] Let $k \geq 2$, and suppose that there exists a positive function $F: {\mathcal R}_k \to (0,+\infty)$ obeying the eigenfunction equation $$\label{lf} \lambda F(t_1,\dots,t_k) = \sum_{i=1}^k \int_0^\infty F(t_1,\dots,t_{i-1},t'_i, t_{i+1},\dots,t_k)\ dt'_i$$ for some $\lambda > 0$ and all $(t_1,\dots,t_k) \in {\mathcal R}_k$, where we extend $F$ by zero to all of $[0,+\infty)^k$. Then $\lambda = M_k$. On the one hand, if we integrate against $F$ and use , we see that $$\lambda I(F) = \sum_{i=1}^k J_i(F)$$ and thus by we see that $M_k \geq \lambda$. On the other hand, if we apply Lemma \[cs\] with $$G_i(t_1,\dots,t_k) \coloneqq \frac{F(t_1,\dots,t_k)}{\int_0^\infty F(t_1,\dots,t_{i-1},t'_i, t_{i+1},\dots,t_k)\ dt'_i}$$ we see that $M_k \leq \lambda$, and the claim follows. This allows for an exact calculation of $M_2$: \[m2-comp\] We have $$M_2 = \frac{1}{1 - W(1/e)} = 1.38593\dots$$ where the Lambert $W$-function $W(x)$ is defined for positive $x$ as the unique positive solution to $x = W(x) e^{W(x)}$. If we set $\lambda \coloneqq \frac{1}{1-W(1/e)} = 1.38593\dots$, then a brief calculation shows that $$\label{ll1} 2\lambda - 1 = \lambda \log \lambda - \lambda \log(\lambda-1).$$ Now if we define the function $f: [0,1] \to [0,+\infty)$ by the formula $$f(x) \coloneqq \frac{1}{\lambda-1+x} + \frac{1}{2\lambda-1} \log \frac{\lambda-x}{\lambda-1+x}$$ then a further brief calculation shows that $$\int_0^{1-x} f(y)\ dy = \frac{\lambda-1+x}{2\lambda-1} \log \frac{\lambda-x}{\lambda-1+x} + \frac{\lambda \log \lambda - \lambda \log(\lambda-1)}{2\lambda-1}$$ for any $0 \leq x \leq 1$, and hence by that $$\int_0^{1-x} f(y)\ dy = (\lambda-1+x) f(x).$$ If we then define the function $F: {\mathcal R}_2 \to (0,+\infty)$ by $F(x,y) \coloneqq f(x) + f(y)$, we conclude that $$\int_0^{1-x} F(x',y)\ dx' + \int_0^{1-y} F(x,y')\ dy' = \lambda F(x,y)$$ for all $(x,y) \in {\mathcal R}_2$, and the claim now follows from Corollary \[ef\]. We conjecture that a positive eigenfunction for $M_k$ exists for all $k \geq 2$, not just for $k=2$; however, we were unable to produce any such eigenfunctions for $k>2$. Nevertheless, Lemma \[cs\] still gives us a general upper bound: \[mk-upper\] We have $M_k \leq \frac{k}{k-1} \log k$ for any $k \geq 2$. Thus for instance one has $M_2 \leq 2 \log 2 = 1.38629\dots$, which compares well with Corollary \[m2-comp\]. On the other hand, Corollary \[mk-upper\] also gives $$M_4 \leq \frac{4}{3} \log 4 = 1.8454\dots,$$ so that one cannot hope to establish $\operatorname*{DHL}[4,2]$ (or $\operatorname*{DHL}[3,2]$) solely through Theorem \[maynard-thm\] even when assuming GEH, and must rely instead on more sophisticated criteria for $\operatorname*{DHL}[k,m]$ such as Theorem \[epsilon-trick\] or Theorem \[epsilon-beyond\]. If we set $G_i: {\mathcal R}_k \to (0,+\infty)$ for $i=1,\dots,k$ to be the functions $$G_i(t_1,\dots,t_k) \coloneqq \frac{k-1}{\log k} \frac{1}{1-t_1-\dots-t_k + kt_i}$$ then direct calculation shows that $$\int_0^\infty G_i(t_1,\dots,t_k)\ dt_i \leq 1$$ for all $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k \geq 0$, where we extend $G_i$ by zero to all of $[0,+\infty)^k$. On the other hand, we have $$\sum_{i=1}^k \frac{1}{G_i(t_1,\dots,t_k)} = \frac{k}{k-1} \log k$$ for all $(t_1,\dots,t_k) \in {\mathcal R}_k$. The claim now follows from Lemma \[cs\]. The upper bound arguments for $M_k$ can be extended to other quantities such as $M_{k,{\varepsilon}}$, although the bounds do not appear to be as sharp in that case. For instance, we have the following variant of Lemma \[mk-upper\], which shows that the improvement in constants when moving from $M_k$ to $M_{k,{\varepsilon}}$ is asymptotically modest: \[mkeps\] For any $k \geq 2$ and $0 \leq {\varepsilon}< 1$ we have $$M_{k,{\varepsilon}} \leq \frac{k}{k-1} \log(2k-1).$$ Let $F: [0,+\infty)^k \to {\mathbb{R}}$ be a square-integrable function supported on $(1+{\varepsilon}) \cdot {\mathcal R}_k$. If $i=1,\dots,k$ and $(t_1,\dots,t_{i-1},t_{i+1},\dots,t_k) \in (1-{\varepsilon}) \cdot {\mathcal R}_k$, then if we write $s := 1-t_1-\dots-t_{i-1}-t_{i+1}-\dots-t_k$, we have $s \geq {\varepsilon}$ and hence $$\begin{aligned} \int_0^{1-t_1-\dots-t_{i-1}-t_{i+1}-\dots-t_k+{\varepsilon}} \frac{1}{1-t_1-\dots-t_k + kt_i}\ dt_i &= \int_0^{s+{\varepsilon}} \frac{1}{s+(k-1)t_i}\ dt_i \\ &= \frac{1}{k-1} \log \frac{ks + (k-1){\varepsilon}}{s} \\ &\leq \frac{1}{k-1} \log(2k-1).\end{aligned}$$ By Cauchy-Schwarz, we conclude that $$\left(\int_0^\infty F(t_1,\dots,t_k)\ dt_i\right)^2 \leq \frac{1}{k-1} \log(2k-1) \int_0^\infty (1-t_1-\dots-t_k + k t_i) F(t_1,\dots,t_k)^2\ dt_i.$$ Integrating in $t_1,\dots,t_{i-1},t_{i+1},\dots,t_k$ and summing in $i$, we obtain the claim. The same argument, using the weight $1 + a(-t_1-\dots-t_k + kt_i)$, gives the more general inequality $$M_{k,{\varepsilon}} \leq \frac{k}{a(k-1)} \log\left(k + \frac{(a(1+{\varepsilon})-1)(k-1)}{1-a(1-{\varepsilon})} \right)$$ whenever $\frac{1}{1+{\varepsilon}} < a < \frac{1}{1-{\varepsilon}}$; the case $a=1$ is Proposition \[mkeps\], and the limiting case $a=\frac{1}{1+{\varepsilon}}$ recovers Lemma \[mk-upper\] when one sends ${\varepsilon}$ to zero. One can also adapt the computations in Corollary \[m2-comp\] to obtain exact expressions for $M_{2,{\varepsilon}}$, although the calculations are rather lengthy and will only be summarized here. For fixed $0 < {\varepsilon}< 1$, the eigenfunctions $F$ one seeks should take the form $$F(x,y) \coloneqq f(x) + f(y)$$ for $x,y \geq 0$ and $x+y \leq 1+{\varepsilon}$, where $$f(x) := {\mathbf{1}}_{x \leq 1-{\varepsilon}} \int_0^{1+{\varepsilon}-x} F(x,t)\ dt.$$ In the regime $0 < {\varepsilon}< 1/3$, one can calculate that $f$ will (up to scalar multiples) take the form $$\begin{aligned} f(x) &= {\mathbf{1}}_{x \leq 2{\varepsilon}} \frac{C_1}{\lambda-1-{\varepsilon}+x} \\ &\quad + {\mathbf{1}}_{2{\varepsilon}\leq x \leq 1-{\varepsilon}} \left(\frac{\log(\lambda-x)-\log(\lambda-1-{\varepsilon}+x)}{2\lambda-1-{\varepsilon}} + \frac{1}{\lambda-1-{\varepsilon}+x} \right)\end{aligned}$$ where $$C_1 := \frac{\log(\lambda-2{\varepsilon}) - \log(\lambda-1+{\varepsilon})}{1 - \log(\lambda-1+{\varepsilon}) + \log(\lambda-1-{\varepsilon})}$$ and $\lambda$ is the largest root of the equation $$\begin{aligned} 1 &= C_1 ( \log(\lambda-1+{\varepsilon}) - \log(\lambda-1-{\varepsilon})) - \log(\lambda-1+{\varepsilon}) \\ &\quad + \frac{ (\lambda-1+{\varepsilon}) \log(\lambda-1+{\varepsilon}) - (\lambda-2{\varepsilon}) \log(\lambda-2{\varepsilon}) }{2\lambda-1-{\varepsilon}}.\end{aligned}$$ In the regime $1/3 \leq {\varepsilon}< 1$, the situation is significantly simpler, and one has the exact expressions $$f(x) = \frac{{\mathbf{1}}_{x \leq 1-{\varepsilon}}}{\lambda - 1 - {\varepsilon}+ x}$$ and $$\lambda = \frac{e(1+{\varepsilon})-2{\varepsilon}}{e-1}.$$ In both cases, a variant of Corollary \[ef\] can be used to show that $M_{2,{\varepsilon}}$ will be equal to $\lambda$; thus for instance $$M_{2,{\varepsilon}} = \frac{e(1+{\varepsilon})-2{\varepsilon}}{e-1}$$ for $1/3 \leq {\varepsilon}< 1$. In particular, $M_{2,{\varepsilon}}$ increases to $2$ in the limit ${\varepsilon}\to 1$; the lower bound $\liminf_{{\varepsilon}\to 1} M_{2,{\varepsilon}} \geq 2$ can also be established by testing with the function $F(x,y) := {\mathbf{1}}_{x \leq \delta, y \leq 1+{\varepsilon}-\delta} + {\mathbf{1}}_{y \leq \delta, x \leq 1+{\varepsilon}-\delta}$ for some sufficiently small $\delta>0$. Now we turn to lower bounds on $M_k$, which are of more relevance for the purpose of establishing results such as Theorem \[mlower\]. If one restricts attention to those functions $F: {\mathcal R}_k \to {\mathbb{R}}$ of the special form $F(t_1,\dots,t_k) = f(t_1+\dots+t_k)$ for some function $f: [0,1] \to {\mathbb{R}}$ then the resulting variational problem has been optimized in previous works [@revesz], [@polymath8a-unabridged] (and originally in unpublished work of Conrey), giving rise to the lower bound $$M_k \geq \frac{4k(k-1)}{j_{k-2}^2}$$ where $j_{k-2}$ is the first positive zero of the Bessel function $J_{k-2}$. This lower bound is reasonably strong for small $k$; for instance, when $k=2$ it shows that $$M_2 \geq 1.383\dots$$ which compares well with Corollary \[m2-comp\], and also shows that $M_6 > 2$, recovering the result of Goldston, Pintz, and Y[i]{}ld[i]{}r[i]{}m that $\operatorname*{DHL}[6,2]$ (and hence $H_1 \leq 16$) was true on the Elliott-Halberstam conjecture. However, one can show that $\frac{4k(k-1)}{j_{k-2}^2} < 4$ for all $k$ (see [@sound]), so this lower bound cannot be used to force $M_k$ to be larger than $4$. In [@maynard-new] the lower bound $$\label{klog} M_k \geq \log k - 2\log\log k - 2$$ was established for all sufficiently large $k$. In fact, the arguments in [@maynard-new] can be used to show this bound for all $k \geq 200$ (for $k<200$, the right-hand side of is either negative or undefined). Indeed, if we use the bound [@maynard-new (7.19)] with $A$ chosen so that $A^2 e^A = k$, then $3 < A < \log k$ when $k \geq 200$, hence $e^A = k/A^2 > k / \log^2 k$ and so $A \geq \log k - 2 \log\log k$. By using the bounds $\frac{A}{e^A-1} < \frac{1}{6}$ (since $A >3$) and $e^A/k = 1/A^2 < 1/9$, we see that the right-hand side of [@maynard-new (8.17)] exceeds $A - \frac{1}{(1-1/6 - 1/9)^2} \geq A-2$, which gives . We will remove the $\log\log k$ term in via the following explicit estimate. \[explicit\] Let $k \geq 2$, and let $c,T,\tau > 0$ be parameters. Define the function $g: [0,T] \to {\mathbb{R}}$ by $$\label{g-def} g(t) \coloneqq \frac{1}{c + (k-1) t}$$ and the quantities $$\begin{aligned} m_2 &\coloneqq \int_0^T g(t)^2\ dt \label{m2-def}\\ \mu &\coloneqq \frac{1}{m_2} \int_0^T t g(t)^2\ dt \label{mu-def}\\ \sigma^2 &\coloneqq \frac{1}{m_2} \int_0^T t^2 g(t)^2\ dt - \mu^2.\label{sigma-def}\end{aligned}$$ Assume the inequalities $$\begin{aligned} k\mu &\leq 1-\tau \label{tau-bound}\\ k\mu &< 1-T \label{T-bound}\\ k\sigma^2 &< (1+\tau-k\mu)^2. \label{ksb}\end{aligned}$$ Then one has $$\label{bigbound} \frac{k}{k-1} \log k - M_k^{[T]} \leq \frac{k}{k-1} \frac{Z + Z_3 + WX + VU}{(1+\tau/2) (1 - \frac{k\sigma^2}{(1+\tau-k\mu)^2})}$$ where $Z, Z_3, W, X, V, U$ are the explicitly computable quantities $$\begin{aligned} Z &\coloneqq \frac{1}{\tau} \int_1^{1+\tau}\left( r\left(\log\frac{r-k\mu}{T} + \frac{k\sigma^2}{4(r-k\mu)^2 \log \frac{r-k\mu}{T}} \right) + \frac{r^2}{4kT}\right)\ dr\label{z-def}\\ Z_3 &\coloneqq \frac{1}{m_2} \int_0^T kt \log(1+\frac{t}{T}) g(t)^2\ dt \label{z3-def}\\ W &\coloneqq \frac{1}{m_2} \int_0^T \log(1+\frac{\tau}{kt}) g(t)^2\ dt\label{W-def}\\ X &\coloneqq \frac{\log k}{\tau} c^2 \label{X-def}\\ V &\coloneqq \frac{c}{m_2} \int_0^T \frac{1}{2c + (k-1)t} g(t)^2\ dt \label{V-def} \\ U &\coloneqq \frac{\log k}{c} \int_0^1 \left((1 + u\tau- (k-1)\mu- c)^2 + (k-1) \sigma^2\right)\ du.\label{U-def}\end{aligned}$$ Of course, since $M_k^{[T]} \leq M_k$, the bound also holds with $M_k^{[T]}$ replaced by $M_k$. From we have $$\sum_{i=1}^k J_i(F) \leq M_k^{[T]} I(F)$$ whenever $F: [0,+\infty)^k \to {\mathbb{R}}$ is square-integrable and supported on $[0,T]^k \cap {\mathcal R}_k$. By rescaling, we conclude that $$\sum_{i=1}^k J_i(F) \leq r M_k^{[T]} I(F)$$ whenever $r>0$ and $F: [0,+\infty)^k \to {\mathbb{R}}$ is square-integrable and supported on $[0,rT]^k \cap r \cdot {\mathcal R}_k$. We apply this inequality with the function $$F(t_1,\dots,t_k) \coloneqq {\mathbf{1}}_{t_1+\dots+t_k \leq r} g(t_1) \dots g(t_k)$$ where $r>1$ is a parameter which we will eventually average over, and $g$ is extended by zero to $[0,+\infty)$. We thus have $$I(F) = m_2^k \int_0^\infty \dots \int_0^\infty {\mathbf{1}}_{t_1+\dots+t_k \leq r} \prod_{i=1}^k \frac{g(t_i)^2\ dt_i}{m_2}.$$ We can interpret this probabilistically as $$I(F) = m_2^k {\mathbb{P}}( X_1 + \dots + X_k \leq r )$$ where $X_1,\dots,X_k$ are independent random variables taking values in $[0,T]$ with probability distribution $\frac{1}{m_2} g(t)^2\ dt$. In a similar fashion, we have $$J_k(F) = m_2^{k-1} \int_0^\infty \dots \int_0^\infty\left (\int_{[0,r-t_1-\dots-t_{k-1}]} g(t)\ dt\right)^2 \prod_{i=1}^{k-1} \frac{g(t_i)^2\ dt_i}{m_2},$$ where we adopt the convention that $\int_{[a,b]}$ vanishes when $b<a$. In probabilistic language, we thus have $$J_k(F) = m_2^{k-1} {\mathbb{E}}\left(\int_{[0,r-X_1-\dots-X_{k-1}]} g(t)\ dt\right)^2$$ where we adopt the convention that the expectation operator ${\mathbb{E}}$ applies to the entire expression to the right of that operator unless explicitly restricted by parentheses. Also by symmetry we see that $J_i(F)=J_k(F)$ for all $i=1,\dots,k$. Putting all this together, we conclude that $${\mathbb{E}}\left(\int_0^{r-X_1-\dots-X_{k-1}} g(t)\ dt\right)^2 \leq \frac{m_2 M_k^{[T]} r}{k} {\mathbb{P}}( X_1 + \dots + X_k \geq r )$$ for all $r>1$. Writing $S_i \coloneqq X_1 + \dots + X_i$, we abbreviate this as $$\label{rg1} {\mathbb{E}}\left(\int_{[0,r-S_{k-1}]} g(t)\ dt\right)^2 \leq \frac{m_2 M_k^{[T]} r}{k} {\mathbb{P}}( S_k \geq r ).$$ Now we run a variant of the Cauchy-Schwarz argument used to prove Corollary \[mk-upper\]. If, for fixed $r>0$, we introduce the random function $h: (0,+\infty) \to {\mathbb{R}}$ by the formula $$\label{hdef} h(t) \coloneqq \frac{1}{r-S_{k-1}+(k-1)t} {\mathbf{1}}_{S_{k-1} < r}$$ and observe that whenever $S_{k-1} < r$, we have $$\label{h-int} \int_{[0,r-S_{k-1}]} h(t)\ dt = \frac{\log k}{k-1}$$ and thus by the Legendre identity we have $$\left(\int_{[0,r-S_{k-1}]} g(t)\ dt\right)^2 = \frac{\log k}{k-1} \int_{[0,r-S_{k-1}]} \frac{g(t)^2}{h(t)}\ dt - \frac{1}{2} \int_{[0,r-S_{k-1}]} \int_{[0,r-S_{k-1}]} \frac{(g(s)h(t)-g(t)h(s))^2}{h(s)h(t)}\ ds dt$$ for $S_{k-1} < r$; but the claim also holds when $r \leq S_{k-1}$ since all integrals vanish in that case. On the other hand, we have $$\begin{aligned} {\mathbb{E}}\int_{[0,r-S_{k-1}]} \frac{g(t)^2}{h(t)}\ dt &= m_2 {\mathbb{E}}(r - S_{k-1} + (k-1) X_k) {\mathbf{1}}_{X_k \leq r - S_{k-1} }\\ &= m_2 {\mathbb{E}}(r - S_k + k X_k) {\mathbf{1}}_{S_k \leq r} \\ &= m_2 {\mathbb{E}}r {\mathbf{1}}_{S_k \leq r} \\ &= m_2 r {\mathbb{P}}( S_k \leq r)\end{aligned}$$ where we have used symmetry to get the third equality. We conclude that $${\mathbb{E}}(\int_{[0,r-S_{k-1}]} g(t)\ dt)^2 = \frac{\log k}{k-1} m_2 r {\mathbb{P}}(S_k \leq r) - \frac{1}{2} {\mathbb{E}}\int_{[0,r-S_{k-1}]} \int_{[0,r-S_{k-1}]} \frac{(g(s)h(t)-g(t)h(s))^2}{h(s)h(t)}\ ds dt.$$ Combining this with , we conclude that $$\Delta r {\mathbb{P}}( S_k \leq r) \leq \frac{k}{2m_2} {\mathbb{E}}\int_{[0,r-S_{k-1}]} \int_{[0,r-S_{k-1}]} \frac{(g(s)h(t)-g(t)h(s))^2}{h(s)h(t)}\ ds dt$$ where $$\Delta \coloneqq \frac{k}{k-1} \log k - M_k^{[T]}.$$ Splitting into regions where $s,t$ are less than $T$ or greater than $T$, and noting that $g(s)$ vanishes for $s > T$, we conclude that $$\Delta r {\mathbb{P}}( S_k \leq r) \leq Y_1(r) + Y_2(r)$$ where $$Y_1(r) \coloneqq \frac{k}{m_2} {\mathbb{E}}\int_{[0,T]} \int_{[T,r-S_{k-1}]} \frac{g(t)^2}{h(t)} h(s)\ ds dt$$ and $$Y_2(r) \coloneqq \frac{k}{2m_2} {\mathbb{E}}\int_{[0,\min(T,r-S_{k-1})]} \int_{[0,\min(T,r-S_{k-1})]} \frac{(g(s)h(t)-g(t)h(s))^2}{h(s)h(t)}\ ds dt.$$ We average this from $r=1$ to $r=1+\tau$, to conclude that $$\Delta (\frac{1}{\tau} \int_1^{1+\tau} r {\mathbb{P}}( S_k \leq r)\ dr) \leq \frac{1}{\tau} \int_1^{1+\tau} Y_1(r)\ dr + \frac{1}{\tau} \int_1^{1+\tau} Y_2(r)\ dr.$$ Thus to prove , it suffices (by ) to establish the bounds $$\label{denom-bound} \frac{1}{\tau} \int_1^{1+\tau} r {\mathbb{P}}( S_k \leq r)\ dr \geq (1+\tau/2) \left(1 - \frac{k\sigma^2}{(1+\tau-k\mu)^2}\right),$$ $$\label{y1-bound} \frac{k}{k-1} Y_1(r) \leq Z + Z_3$$ for all $1 < r \leq 1+\tau$, and $$\label{y2-bound} \frac{1}{\tau} \int_1^{1+\tau} Y_2(r)\ dr \leq \frac{k}{k-1}( WX+VU ).$$ We begin with . Since $$\frac{1}{\tau} \int_1^{1+\tau} r\ dr = 1+\frac{\tau}{2}$$ it suffices to show that $$\frac{1}{\tau} \int_1^{1+\tau} r {\mathbb{P}}( S_k > r) \leq (1+\frac{\tau}{2}) \frac{k\sigma^2}{(1+\tau-k\mu)^2}.$$ But, from , , we see that each $X_i$ has mean $\mu$ and variance $\sigma^2$, so $S_k$ has mean $k\mu$ and variance $k\sigma^2$. It thus suffices to show the pointwise bound $$\frac{1}{\tau} \int_1^{1+\tau} r 1_{x>r} \leq (1+\frac{\tau}{2}) \frac{(x-k\mu)^2}{(1+\tau-k\mu)^2}$$ for any $x$. It suffices to verify this in the range $1 \leq x \leq 1+\tau$. But in this range, the left-hand side is convex, equals $0$ at $1$ and $1+\tau/2$ at $1+\tau$, while the right-hand side is convex, and equals $1+\tau/2$ at $1+\tau$ with slope at least $(1+\tau/2)/\tau$ there thanks to . The claim follows. Now we show . The quantity $Y_1(r)$ is vanishing unless $r-S_{k-1} \geq T$. Using the crude bound $h(s) \leq \frac{1}{(k-1)s}$ from , we see that $$\int_{[T,r-S_{k-1}]} h(s)\ ds \leq \frac{1}{k-1} \log_+ \frac{r-S_{k-1}}{T}$$ where $\log_+(x) \coloneqq \max(\log x, 0)$. We conclude that $$Y_1(r) \leq \frac{k}{k-1} \frac{1}{m_2} {\mathbb{E}}\int_{[0,T]} \frac{g(t)^2}{h(t)}\ dt \log_+ \frac{r-S_{k-1}}{T}.$$ We can rewrite this as $$Y_1(r) \leq \frac{k}{k-1} {\mathbb{E}}\frac{{\mathbf{1}}_{S_k \leq r}}{h(X_k)} \log_+ \frac{r-S_{k-1}}{T}.$$ By , we have $$\frac{{\mathbf{1}}_{S_k \leq r}}{h(X_k)} = (r-S_k+kX_k) {\mathbf{1}}_{S_k \leq r}.$$ Also, from the elementary bound $\log_+(x+y) \leq \log_+ x + \log(1+y)$ for any $x,y \geq 0$, we see that $$\log_+ \frac{r-S_{k-1}}{T} \leq \log_+ \frac{r-S_{k}}{T} + \log\left(1+\frac{X_k}{T}\right).$$ We conclude that $$\begin{aligned} Y_1(r) &\leq \frac{k}{k-1} {\mathbb{E}}(r-S_k+kX_k) \left( \log_+ \frac{r-S_{k}}{T} + \log\left(1+\frac{X_k}{T}\right) \right) {\mathbf{1}}_{S_k \leq r}\\ &\leq \frac{k}{k-1} \left( {\mathbb{E}}(r-S_k+kX_k) \log_+ \frac{r-S_{k}}{T} + \max(r-S_k,0) \frac{X_k}{T} + k X_k \log\left(1+\frac{X_k}{T}\right) \right)\end{aligned}$$ using the elementary bound $\log(1+y) \leq y$. Symmetrizing in the $X_1,\dots,X_k$, we conclude that $$\label{y1r} Y_1(r) \leq \frac{k}{k-1} (Z_1(r) + Z_2(r) + Z_3)$$ where $$\begin{aligned} Z_1(r) &\coloneqq {\mathbb{E}}r \log_+ \frac{r-S_k}{T} \\ Z_2(r) &\coloneqq {\mathbb{E}}(r-S_k) {\mathbf{1}}_{S_k \leq r} \frac{S_k}{kT} \end{aligned}$$ and $Z_3$ was defined in . For the minor error term $Z_2$, we use the crude bound $(r-S_k) {\mathbf{1}}_{S_k \leq r} S_k \leq \frac{r^2}{4}$, so $$\label{z2r} Z_2(r) \leq \frac{r^2}{4kT}.$$ For $Z_1$, we upper bound $\log_+ x$ by a quadratic expression in $x$. More precisely, we observe the inequality $$\log_+ x \leq \frac{(x-2a\log a-a)^2}{4a^2 \log a}$$ for any $a > 1$ and $x \in {\mathbb{R}}$, since the left-hand side is concave in $x$ for $x \geq 1$, while the right-hand side is convex in $x$, non-negative, and tangent to the left-hand side at $x=a$. We conclude that $$\log_+ \frac{r-S_k}{T} \leq \frac{(r-S_k-2aT\log a-aT)^2}{4a^2 T^2 \log a}.$$ On the other hand, from , , we see that each $X_i$ has mean $\mu$ and variance $\sigma^2$, so $S_k$ has mean $k\mu$ and variance $k\sigma^2$. We conclude that $$Z_1(r) \leq r \frac{ (r-k\mu-2aT\log a-aT)^2 + k \sigma^2}{4a^2 T^2 \log a}$$ for any $a > 1$. From and the assumption $r > 1$, we may choose $a \coloneqq \frac{r-k\mu}{T}$ here, leading to the simplified formula $$\label{z1r} Z_1(r) \leq r \left( \log \frac{r-k\mu}{T} + \frac{k\sigma^2}{4(r-k\mu)^2 \log \frac{r-k\mu}{T}}\right).$$ From , , , we conclude . Finally, we prove . Here, we finally use the specific form of the function $g$. Indeed, from , we observe the identity $$g(t) - h(t) = (r - S_{k-1} - c) g(t) h(t)$$ for $t \in [0, \min(r-S_{k-1},T)]$. Thus $$\begin{aligned} Y_2(r) &= \frac{k}{2 m_2} {\mathbb{E}}\int_{[0,\min(r-S_{k-1},T)]} \int_{[0,\min(r-S_{k-1},T)]} \frac{((g-h)(s) h(t)-(g-h)(t) h(s))^2}{h(s) h(t)}\ ds dt \\ &= \frac{k}{2 m_2} {\mathbb{E}}(r - S_{k-1} - c)^2 \int_{[0,\min(r-S_{k-1},T)]} \int_{[0,\min(r-S_{k-1},T)]} (g(s)-g(t))^2 h(s) h(t)\ ds dt.\end{aligned}$$ Using the crude bound $(g(s)-g(t))^2 \leq g(s)^2+g(t)^2$ and using symmetry, we conclude $$Y_2(r) \leq \frac{k}{m_2} {\mathbb{E}}(r - S_{k-1} - c)^2 \int_{[0,\min(r-S_{k-1},T)]} \int_{[0,\min(r-S_{k-1},T)]} g(s)^2 h(s) h(t)\ ds dt.$$ From , we conclude that $$Y_2(r) \leq \frac{k}{k-1} Z_4(r)$$ where $$Z_4(r) \coloneqq \frac{\log k}{m_2} {\mathbb{E}}\left( (r - S_{k-1} - c)^2 \int_{[0,\min(r-S_{k-1},T)]} \frac{g(s)^2}{r-S_{k-1}+(k-1)s}\ ds \right) .$$ To prove , it thus suffices (after making the change of variables $r = 1 + u \tau$) to show that $$\label{z4-bound} \int_0^1 Z_4(1+u\tau)\ du \leq WX+VU.$$ We will exploit the averaging in $u$ to deal with the singular nature of the factor $\frac{1}{r-S_{k-1}+(k-1)s}$. By Fubini’s theorem, the left-hand side of may be written as $$\frac{\log k}{m_2} {\mathbb{E}}\int_0^1 Q(u)\ du$$ where $Q(u)$ is the random variable $$Q(u) \coloneqq (1+u\tau - S_{k-1} - c)^2 \int_{[0,\min(1+u\tau-S_{k-1},T)]} \frac{g(s)^2}{1+u\tau-S_{k-1}+(k-1)s}\ ds.$$ Note that $Q(u)$ vanishes unless $1+u\tau-S_{k-1} > 0$. Consider first the contribution of those $Q(u)$ for which $$0 < 1+u\tau-S_{k-1} \leq 2c.$$ In this regime we may bound $$(1+u\tau-S_{k-1}-c)^2 \leq c^2,$$ so this contribution to may be bounded by $$\frac{\log k}{m_2} c^2 {\mathbb{E}}\int_{[0,T]} g(s)^2 \left(\int_0^1 \frac{{\mathbf{1}}_{1+u\tau-S_{k-1} \geq s}}{1+u\tau-S_{k-1}+(k-1)s}\ du\right)\ ds.$$ Observe on making the change of variables $v \coloneqq 1 + u\tau - S_{k-1} + (k-1)s$ that $$\begin{aligned} \int_0^1 \frac{{\mathbf{1}}_{1+u\tau-S_{k-1} \geq s}}{1+u\tau-S_{k-1}+(k-1)s}\ du &= \frac{1}{\tau} \int_{[\max(ks, 1-S_{k-1}+(k-1)s), 1-S_{k-1}+\tau+(k-1)s]} \frac{dv}{v} \\ &\leq \frac{1}{\tau} \log \frac{ks+\tau}{ks}\end{aligned}$$ and so this contribution to is bounded by $WX$, where $W,X$ are defined in , . Now we consider the contribution to when[^5] $$1+u\tau-S_{k-1} > 2c.$$ In this regime we bound $$\frac{1}{1+u\tau-S_{k-1}+(k-1)s} \leq \frac{1}{2c+(k-1)t},$$ and so this portion of $\int_0^1 Z_4[1+u\tau]\ du$ may be bounded by $$\int_0^1 \frac{\log k}{c} {\mathbb{E}}(1 + u\tau - S_{k-1} - c)^2 V\ du = VU$$ where $V, U$ are defined in , . The proof of the theorem is now complete. We can now perform an asymptotic analysis in the limit $k \to \infty$ to establish Theorem \[mlower\](xi) and Theorem \[mlower-var\](vi). For $k$ sufficiently large, we select the parameters $$\begin{aligned} c &\coloneqq \frac{1}{\log k} + \frac{\alpha}{\log^2 k} \\ T &\coloneqq \frac{\beta}{\log k} \\ \tau &\coloneqq \frac{\gamma}{\log k}\end{aligned}$$ for some real parameters $\alpha \in {\mathbb{R}}$ and $\beta,\gamma > 0$ independent of $k$ to be optimized in later. From , we have $$\begin{aligned} m_2 &= \frac{1}{k-1} \left( \frac{1}{c} - \frac{1}{c+(k-1)T} \right) \\ &= \frac{\log k}{k} \left(1 - \frac{\alpha}{\log k} + o( \frac{1}{\log k} ) \right)\end{aligned}$$ where we use $o(f(k))$ to denote a function $g(k)$ of $k$ with $g(k)/f(k) \to 0$ as $k \to \infty$. On the other hand, we have from , that $$\begin{aligned} m_2 (c + (k-1) \mu) &= \int_0^T (c+(k-1)t) g(t)^2\ dt \\ &= \frac{1}{k-1} \log \frac{c+(k-1)T}{c} \\ &= \frac{\log k}{k} \left(1 + \frac{\log \beta}{\log k} + o( \frac{1}{\log k} ) \right)\end{aligned}$$ and thus $$\begin{aligned} k\mu &= \frac{k}{k-1} \left(1 + \frac{\log \beta + \alpha}{\log k} + o(\frac{1}{\log k} ) \right) - \frac{kc}{k-1} \\ &= 1 + \frac{\log \beta + \alpha}{\log k} + o\left( \frac{1}{\log k} \right) - \left( \frac{1}{\log k} + o\left( \frac{1}{\log k} \right) \right)\\ &= 1 + \frac{\log \beta + \alpha - 1}{\log k} + o\left( \frac{1}{\log k} \right).\end{aligned}$$ Similarly, from , , we have $$\begin{aligned} m_2 (c^2 + 2c(k-1)\mu + (k-1)^2 (\mu^2+\sigma^2)) &= \int_0^T (c+(k-1)t)^2 g(t)^2\ dt \\ &= T\end{aligned}$$ and thus $$\begin{aligned} k \sigma^2 &= \frac{k}{(k-1)^2} \left(\frac{T}{m_2} - c^2 - 2c(k-1)\mu\right) - k \mu^2 \\ &= \frac{\beta}{\log^2 k} + o( \frac{1}{\log^2 k} ).\end{aligned}$$ We conclude that the hypotheses , , will be obeyed for sufficiently large $k$ if we have $$\begin{aligned} \log \beta + \alpha + \gamma &< 1 \\ \log \beta + \alpha + \beta &< 1 \\ \beta &< (1 + \gamma - \alpha - \log \beta)^2.\end{aligned}$$ These conditions can be simultaneously obeyed, for instance by setting $\beta=\gamma=1$ and $\alpha = -1$. Now we crudely estimate the quantities $Z,Z_3,W,X,V,U$ in -. For $1 \leq r \leq 1+\tau$, we have $r-k\mu {\asymp}1/\log k$, and so $$\frac{r-k\mu}{T} {\asymp}1; \quad \frac{k \sigma^2}{(r-k\mu)^2} {\asymp}1; \quad \frac{r^2}{4kT} = o(1)$$ and so by $Z = O(1)$. Using the crude bound $\log(1+\frac{t}{T}) = O(1)$ for $0 \leq t \leq T$, we see from , that $Z_3 = O( k \mu ) = O(1)$. It is clear that $X = O(1)$, and using the crude bound $\frac{1}{2c+(k-1)t} \leq \frac{1}{c}$ we see from , that $V = O(1)$. For $0 \leq u \leq 1$ we have $1 + u\tau - (k-1)\mu - c = O(1/\log k)$, so from we have $U=O(1)$. Finally, from and the change of variables $t = \frac{s}{k \log k}$ we have $$\begin{aligned} W &= \frac{\log k}{k m_2} \int_0^{kT\log k} \log\left(1 + \frac{\gamma}{s}\right) \frac{ds}{(1 + \frac{\alpha}{\log k} + \frac{k-1}{k} s)^2} \\ &= O\left( \int_0^\infty \log\left(1+\frac{\gamma}{s}\right) \frac{ds}{(1+o(1))(1+s)^2} \right) \\ &= O(1).\end{aligned}$$ Finally we have $$1 - \frac{k\sigma^2}{(1+\tau-k\mu)^2} {\asymp}1.$$ Putting all this together, we see from that $$M_k \geq M_k^{[T]} \geq \frac{k}{k-1} \log k - O(1)$$ giving Theorem \[mlower\](xi). Furthermore, if we set $$\varpi \coloneqq \frac{7}{600} - \frac{C}{\log k}$$ and $$\delta \coloneqq \left(\frac{1}{4} + \frac{7}{600}\right) \frac{\beta}{\log k}$$ then we will have $600 \varpi + 180 \delta < 7$ for $C$ large enough, and Theorem \[mlower-var\](vi) also follows (as one can verify from inspection that all implied constants here are effective). Finally, Theorem \[mlower\](viii), (ix), (x) follow by setting $$\begin{aligned} c &:= \frac{\theta}{\log k} \\ T &:= \frac{\beta}{\log k} \\ \tau &= 1 - k\mu\end{aligned}$$ with $\theta,\beta$ given by Table \[narrow-table\], with then giving the bound $M_k^{[T]} > M$ with $M$ as given by the table, after verifying of course that the conditions , , are obeyed. Similarly, Theorem \[mlower-var\] (ii), (iii), (iv), (v) follows with $\theta,\beta$ given by the same table, with $\varpi$ chosen so that $$M = \frac{m}{\frac{1}{4}+\varpi}$$ with $m=2,3,4,5$ for (ii), (iii), (iv), (v) respectively, and $\delta$ chosen by the formula $$\delta := T (\frac{1}{4} + \varpi).$$ $k$ $\theta$ $\beta$ M ---------------------- ----------- ----------- ------------- 5511 0.965 0.973 6.000048609 35410 0.99479 0.85213 7.829849259 \[0.5ex\] 41588 0.97878 0.94319 8.000001401 309661 0.98627 0.92091 10.00000032 \[0.5ex\] 1649821 1.00422 0.80148 11.65752556 75845707 1.00712 0.77003 15.48125090 \[0.5ex\] 3473955908 1.0079318 0.7490925 19.30374872 : Parameter choices for Theorems \[mlower\], \[mlower-var\].[]{data-label="narrow-table"} The case of small and medium dimension {#h1-sec} ====================================== In this section we establish lower bounds for $M_k$ (and related quantities, such as $M_{k,{\varepsilon}}$) both for small values of $k$ (in particular, $k=3$ and $k=4$) and medium values of $k$ (in particular, $k=50$ and $k=54$). Specifically, we will establish Theorem \[mlower\](vii), Theorem \[mke-lower\], and Theorem \[piece\]. Bounding Mk for medium k ------------------------ We begin with the problem of lower bounding $M_k$. We first formalize an observation[^6] of Maynard [@maynard-new] that one may restrict without loss of generality to symmetric functions: For any $k \geq 2$, one has $$M_k \coloneqq \sup \frac{k J_1(F)}{I(F)}$$ where $F$ ranges over *symmetric* square-integrable functions on ${\mathcal R}_k$ that are not identically zero. Firstly, observe that if one replaces a square-integrable function $F: [0,+\infty)^k \to {\mathbb{R}}$ with its absolute value $|F|$, then $I(|F|) = I(F)$ and $J_i(|F|) \geq J_i(F)$. Thus one may restrict the supremum in to non-negative functions without loss of generality. We may thus find a sequence $F_n$ of square-integrable non-negative functions on ${\mathcal R}_k$, normalized so that $I(F_n)=1$, and such that $\sum_{i=1}^k J_i(F_n) \to M_k$ as $n \to \infty$. Now let $$\overline{F_n}(t_1,\dots,t_k) \coloneqq \frac{1}{k!} \sum_{\sigma \in S_k} F_n( t_{\sigma(1)},\dots,t_{\sigma(k)} )$$ be the symmetrization of $F_n$. Since the $F_n$ are non-negative with $I(F_n)=1$, we see that $$I( \overline{F_n} ) \geq I( \frac{1}{k!} F_n ) = \frac{1}{(k!)^2}$$ and so $I(\overline{F_n})$ is bounded away from zero. Also, from , we know that the quadratic form $$Q(F) \coloneqq M_k I(F) - \sum_{i=1}^k J_i(F)$$ is positive semi-definite and is also invariant with respect to symmetries, and so from the triangle inequality for inner product spaces we conclude that $$Q( \overline{F_n} ) \leq Q( F_n ).$$ By construction, $Q(F_n)$ goes to zero as $n \to \infty$, and thus $Q(\overline{F_n})$ also goes to zero. We conclude that $$\frac{k J_1(\overline{F_n})}{I(\overline{F_n})} = \frac{\sum_{i=1}^k J_i(\overline{F_n})}{I(\overline{F_n})} \to M_k$$ as $n \to \infty$, and so $$M_k \geq \sup \frac{k J_1(F)}{I(F)}.$$ The reverse inequality is immediate from , and the claim follows. To establish a lower bound of the form $M_k > C$ for some $C > 0$, one thus seeks to locate a symmetric function $F: [0,+\infty)^k \to {\mathbb{R}}$ supported on ${\mathcal R}_k$ such that $$\label{kjaf} k J_1(F) > C I(F).$$ To do this numerically, we follow [@maynard-new] (see also [@gpy] for some related ideas) and can restrict attention to functions $F$ that are linear combinations $$F = \sum_{i=1}^n a_i b_i$$ of some explicit finite set of symmetric square-integrable functions $b_1,\dots,b_n: [0,+\infty)^k \to {\mathbb{R}}$ supported on ${\mathcal R}_k$, and some real scalars $a_1,\dots,a_n$ that we may optimize in. The condition then may be rewritten as $$\label{ama} \mathbf{a}^T \mathbf{M}_2 \mathbf{a} - C \mathbf{a}^T \mathbf{M}_1 \mathbf{a} > 0$$ where $\mathbf{a}$ is the vector $$\mathbf{a} \coloneqq \begin{pmatrix} a_1 \\ \vdots \\ a_n \end{pmatrix}$$ and $\mathbf{M}_1,\mathbf{M}_2$ are the real symmetric and positive semi-definite $n \times n$ matrices $$\begin{aligned} \label{Eq:M1} \mathbf{M}_1 &= \left( \int_{{\mathbb{R}}^k} b_i(t_1,\dots,t_k) b_j(t_1,\dots,t_k)\ dt_1 \dots dt_k \right)_{1 \leq i,j \leq n} \\\label{Eq:M2} \mathbf{M}_2 &= \left( k \int_{{\mathbb{R}}^{k+1}} b_i(t_1,\dots,t_k) b_j(t_1,\dots,t_{k-1},t'_k)\ dt_1 \dots dt_k dt'_k\right)_{1 \leq i,j \leq n}.\end{aligned}$$ If the $b_1,\dots,b_n$ are linearly independent in $L^2({\mathcal R}_k)$, then $\mathbf{M}_1$ is strictly positive definite, and (as observed in [@maynard-new Lemma 8.3]), one can find $\mathbf{a}$ obeying if and only if the largest eigenvalue of $\mathbf{M}_2 \mathbf{M}_1^{-1}$ exceeds $C$. This is a criterion that can be numerically verified for medium-sized values of $n$, if the $b_1,\dots,b_n$ are chosen so that the matrix coefficients of $\mathbf{M}_1,\mathbf{M}_2$ are explicitly computable. In order to facilitate computations, it is natural to work with bases $b_1,\dots,b_n$ of symmetric polynomials. We have the following basic integration identity: \[bfi\] For any non-negative $a,a_1,\dots,a_k$, we have $$\int_{{\mathcal R}_k} (1-t_1-\dots-t_k)^a t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_k = \frac{\Gamma(a+1) \Gamma(a_1+1) \dots \Gamma(a_k+1)}{\Gamma(a_1+\dots+a_k+k+a+1)}$$ where $\Gamma(s) := \int_0^\infty t^{s-1} e^{-t}\ dt$ is the Gamma function. In particular, if $a_1,\dots,a_k$ are natural numbers, then $$\int_{{\mathcal R}_k} (1-t_1-\dots-t_k)^a t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_k = \frac{a! a_1! \dots a_k!}{(a_1+\dots+a_k+k+a)!}.$$ Since $$\int_{{\mathcal R}_k} (1-t_1-\dots-t_k)^a t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_k = a \int_{{\mathcal R}_{k+1}} t_1^{a_1} \dots t_k^{a_k} t_{k+1}^{a-1}\ dt_1 \dots dt_{k+1}$$ we see that to establish the lemma it suffices to do so in the case $a=0$. If we write $$X := \int_{t_1+\dots+t_k=1} t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_{k-1}$$ then by homogeneity we have $$r^{a_1+\dots+a_k+k-1} X = \int_{t_1+\dots+t_k=r} t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_{k-1}$$ for any $r > 0$, and hence on integrating $r$ from $0$ to $1$ we conclude that $$\frac{X}{a_1+\dots+a_k+k} = \int_{{\mathcal R}_k} t_1^{a_1} \dots t_k^{a_k}\ dt_1 \dots dt_k.$$ On the other hand, if we multiply by $e^{-r}$ and integrate $r$ from $0$ to $\infty$, we obtain instead $$\int_0^\infty r^{a_1+\dots+a_k+k-1} X e^{-r}\ dr = \int_{[0,+\infty)^k} t_1^{a_1} \dots t_k^{a_k} e^{-t_1-\dots-t_k}\ dt_1 \dots dt_k.$$ Using the definition of the Gamma function, this becomes $$\Gamma(a_1+\dots+a_k+k) X = \Gamma(a_1+1) \dots \Gamma(a_k+1)$$ and the claim follows. Define a *signature* to be a non-increasing sequence $\alpha = (\alpha_1,\alpha_2,\dots,\alpha_k)$ of natural numbers; for brevity we omit zeroes, thus for instance if $k=6$, then $(2,2,1,1,0,0)$ will be abbreviated as $(2,2,1,1)$. The number of non-zero elements of $\alpha$ will be called the *length* of the signature $\alpha$, and as usual the *degree* of $\alpha$ will be $\alpha_1+\dots+\alpha_k$. For each signature $\alpha$, we then define the symmetric polynomials $P_\alpha = P^{(k)}_\alpha$ by the formula $$P_\alpha(t_1,\dots,t_k) = \sum_{a: s(a)=\alpha} t_1^{a_1} \dots t_k^{a_k}$$ where the summation is over all tuples $a = (a_1,\dots,a_k)$ whose non-increasing rearrangement $s(a)$ is equal to $\alpha$. Thus for instance $$\begin{aligned} P_{(1)}(t_1,\dots,t_k) &= t_1 + \dots + t_k \\ P_{(2)}(t_1,\dots,t_k) &= t_1^2 + \dots + t_k^2 \\ P_{(1,1)}(t_1,\dots,t_k) &= \sum_{1 \leq i < j \leq k} t_i t_j \\ P_{(2,1)}(t_1,\dots,t_k) &= \sum_{1 \leq i < j \leq k} t_i^2 t_j + t_i t_j^2\end{aligned}$$ and so forth. Clearly, the $P_\alpha$ form a linear basis for the symmetric polynomials of $t_1,\dots,t_k$. Observe that if $\alpha = (\alpha',1)$ is a signature containing $1$, then one can express $P_\alpha$ as $P_{(1)} P_{\alpha'}$ minus a linear combination of polynomials $P_\beta$ with the length of $\beta$ less than that of $\alpha$. This implies that the functions $P_{(1)}^a P_\alpha$, with $a \geq 0$ and $\alpha$ avoiding $1$, are also a basis for the symmetric polynomials. Equivalently, the functions $(1-P_{(1)})^a P_\alpha$ with $a \geq 0$ and $\alpha$ avoiding $1$ form a basis. After extensive experimentation, we have discovered that a good basis $b_1,\dots,b_n$ to use for the above problem comes by setting the $b_i$ to be all the symmetric polynomials of the form $(1-P_{(1)})^a P_\alpha$, where $a \geq 0$ and $\alpha$ consists entirely of even numbers, whose total degree $a + \alpha_1+\dots +\alpha_k$ is less than or equal to some chosen threshold $d$. For such functions, the coefficients of $\mathbf{M}_1,\mathbf{M}_2$ can be computed exactly using Lemma \[bfi\]. More explicitly, first we quickly compute a look-up table for the structure constants $c_{\alpha,\beta,\gamma}\in {\mathbb{Z}}$ derived from simple products of the form $$P_{\alpha}P_{\beta}=\sum_{\gamma}c_{\alpha,\beta,\gamma}P_{\gamma}$$ where $\deg(\alpha)+\deg(\beta)\leq d$. Using this look-up table we rewrite the integrands of the entries of the matrices in (\[Eq:M1\]) and (\[Eq:M2\]) as integer linear combinations of nearly “pure” monomials of the form $(1-P_{(1)})^a t_1^{a_1}\dots t_k^{a_k}$. We then calculate the entries of $\mathbf{M}_1$ and $\mathbf{M}_2$, as exact rational numbers, using Lemma \[bfi\]. We next run a generalized eigenvector routine on (real approximations to) $\mathbf{M}_1$ and $\mathbf{M}_2$ to find a vector $\mathbf{a}'$ which nearly maximizes the quantity $C$ in (\[ama\]). Taking a rational approximation $\mathbf{a}$ to $\mathbf{a}'$, we then do the quick (and exact) arithmetic to verify that (\[ama\]) holds for some constant $C>4$. This generalized eigenvector routine is time-intensive when the sizes of $\mathbf{M}_1$ and $\mathbf{M}_2$ are large (say, bigger than $1500\times 1500$), and in practice is the most computationally intensive step of our calculation. When one does not care about an exact arithmetic proof that $C>4$, instead one can run a test for positive-definiteness for the matrix $C\mathbf{M}_1-\mathbf{M}_2$, which is usually much faster and less RAM intensive. Using this method, we were able to demonstrate $M_{54} > 4.00238$, thus establishing Theorem \[mlower\](vii). We took $d=23$ and imposed the restriction on signatures $\alpha$ that they be composed only of even numbers. It is likely that $d=22$ would suffice in the absence of this restriction on signatures, but we found that the gain in $M_{54}$ from lifting this restriction is typically only in the region of $0.005$, whereas the execution time is increased by a large factor. We do not have a good understanding of why this particular restriction on signatures is so inexpensive in terms of the trade-off between the accuracy of $M$-values and computational complexity. The total run-time for this computation was under one hour. We now describe a second choice for the basis elements $b_1,\dots,b_n$, which uses the Krylov subspace method; it gives faster and more efficient numerical results than the previous basis, but does not seem to extend as well to more complicated variational problems such as $M_{k,{\varepsilon}}$. We introduce the linear operator ${\mathcal L}: L^2({\mathcal R}_k) \to L^2({\mathcal R}_k)$ defined by $${\mathcal L} f( t_1,\dots,t_k) \coloneqq \sum_{i=1}^k \int_0^{1-t_1-\dots-t_{i-1}-t_{i+1}-\dots-t_k} f(t_1,\dots,t_{i-1},t'_i,t_{i+1},\dots,t_k)\ dt'_i.$$ This is a self-adjoint and positive semi-definite operator on $L^2({\mathcal R}_k)$. For symmetric $b_1,\dots,b_n \in L^2({\mathcal R}_k)$, one can then write $$\begin{aligned} \mathbf{M}_1 &= \left( \langle b_i, b_j \rangle \right)_{1 \leq i,j \leq n} \\ \mathbf{M}_2 &= \left( \langle {\mathcal L} b_i, b_j \rangle \right)_{1 \leq i,j \leq n}.\end{aligned}$$ If we then choose $$b_i \coloneqq {\mathcal L}^{i-1} 1$$ where $1$ is the unit constant function on ${\mathcal R}_k$, then the matrices $\mathbf{M}_1,\mathbf{M}_2$ take the Hankel form $$\begin{aligned} \mathbf{M}_1 &= \left( \langle {\mathcal L}^{i+j-2} 1, 1 \rangle \right)_{1 \leq i,j \leq n} \\ \mathbf{M}_2 &= \left( \langle {\mathcal L}^{i+j-1} 1, 1 \rangle \right)_{1 \leq i,j \leq n},\end{aligned}$$ and so can be computed entirely in terms of the $2n$ numbers $\langle {\mathcal L}^i 1, 1 \rangle$ for $i=0,\dots,2n-1$. The operator ${\mathcal L}$ maps symmetric polynomials to symmetric polynomials; for instance, one has $$\begin{aligned} {\mathcal L} 1 &= k - (k-1) P_{(1)} \\ {\mathcal L} P_{(1)} &= \frac{k}{2} - \frac{k-1}{2} P_{(2)} - (k-2) P_{(1,1)}\end{aligned}$$ and so forth. From this and Lemma \[bfi\], the quantities $\langle {\mathcal L}^i 1, 1 \rangle$ are explicitly computable rational numbers; for instance, one can calculate $$\begin{aligned} \langle 1, 1 \rangle &= \frac{1}{k!} \\ \langle {\mathcal L} 1, 1\rangle &= \frac{2k}{(k+1)!} \\ \langle {\mathcal L}^2 1, 1\rangle &= \frac{k (5k+1)}{(k+2)!} \\ \langle {\mathcal L}^3 1, 1\rangle &= \frac{2k^2 (7k+5)}{(k+3)!}\end{aligned}$$ and so forth. With Maple, we were able to compute $\langle {\mathcal L}^i 1, 1 \rangle$ for $i \leq 50$ and $k \leq 100$, leading to lower bounds on $M_k$ for these values of $k$, a selection of which are given in Table \[tab\]. $k$ Lower bound on $M_k$ $\frac{k}{k-1} \log k$ --------------- ---------------------- ------------------------ 2 1.38593 1.38630 3 1.64644 1.64792 \[0.5ex\] 4 1.84540 1.84840 5 2.00714 2.01180 \[0.5ex\] 10 2.54547 2.55843 20 3.12756 3.15341 \[0.5ex\] 30 3.48313 3.51849 40 3.73919 3.78347 \[0.5ex\] 50 3.93586 3.99187 53 3.98621 4.04665 \[0.5ex\] 54 4.00223 4.06425 60 4.09101 4.16375 \[0.5ex\] 100 4.46424 4.65169 : Selected lower bounds on $M_k$ obtained from the Krylov subspace method, with the $\frac{k}{k-1} \log k$ upper bound displayed for comparison.[]{data-label="tab"} Bounding Mk,epsilon for medium k {#mkeps-sec} -------------------------------- When bounding $M_{k,{\varepsilon}}$, we have not been able to implement the Krylov method, because the analogue of ${\mathcal L}^i 1$ in this context is piecewise polynomial instead of polynomial, and we were only able to compute it explicitly for very small values of $i$, such as $i=1,2,3$, which are insufficient for good numerics. Thus, we rely on the previously discussed approach, in which symmetric polynomials are used for the basis functions. Instead of computing integrals over the region ${\mathcal R}_k$ we pass to the regions $(1\pm \varepsilon)\mathcal{R}_k$. In order to apply Lemma \[bfi\] over these regions, this necessitates working with a slightly different basis of polynomials. We chose to work with those polynomials of the form $(1+\varepsilon-P_{(1)})^a P_{\alpha}$, where $\alpha$ is a signature with no 1’s. Over the region $(1+\varepsilon)\mathcal{R}_k$, a single change of variables converts the needed integrals into those of the form in Lemma \[bfi\], and we can then compute the entries of $\mathbf{M}_1$. On the other hand, over the region $(1-\varepsilon)\mathcal{R}_k$ we instead want to work with polynomials of the form $(1-\varepsilon-P_{(1)})^aP_{\alpha}$. Since $(1+\varepsilon-P_{(1)})^a=(2\varepsilon +(1-\varepsilon-P_{(1)}))^a$, an expansion using the binomial theorem allows us to convert from our given basis to polynomials of the needed form. With these modifications, and calculating as in the previous section, we find that $M_{50,1/25}>4.00124$ if $d=25$ and $M_{50,1/25}>4.0043$ if $d=27$, thus establishing Theorem \[mke-lower\](i). As before, we found it optimal to restrict signatures to contain only even entries, which greatly reduced execution time while only reducing $M$ by a few thousandths. One surprising additional computational difficulty introduced by allowing $\varepsilon>0$ is that the “complexity” of $\varepsilon$ as a rational number affects the run-time of the calculations. We found that choosing $\varepsilon=1/m$ (where $m\in {\mathbb{Z}}$ has only small prime factors) reduces this effect. A similar argument gives $M_{51,1/50} > 4.00156$, thus establishing Theorem \[mke-lower\](xiii). In this case our polynomials were of maximum degree $d= 22$. Code and data for these calculations may be found at [www.dropbox.com/sh/0xb4xrsx4qmua7u/WOhuo2Gx7f/Polymath8b](www.dropbox.com/sh/0xb4xrsx4qmua7u/WOhuo2Gx7f/Polymath8b). Bounding M4,0.18 {#4d} ---------------- We now prove Theorem \[mke-lower\](xii$'$), which can be established by a direct numerical calculation. We introduce the explicit function $F: [0,+\infty)^4 \to {\mathbb{R}}$ defined by $$F(t_1,t_2,t_3,t_4) \coloneqq (1 - \alpha (t_1+t_2+t_3+t_4)) {\mathbf{1}}_{t_1+t_2+t_3+t_4 \leq 1+{\varepsilon}}$$ with ${\varepsilon}\coloneqq 0.168$ and $\alpha \coloneqq 0.784$. As $F$ is symmetric in $t_1,t_2,t_3,t_4$, we have $J_{i,1-{\varepsilon}}(F) = J_{1,1-{\varepsilon}}(F)$, so to show Theorem \[mke-lower\](xii$'$) it will suffice to show that $$\label{jf} \frac{4 J_{1,1-{\varepsilon}}(F)}{I(F)} > 2.00558.$$ By making the change of variables $s=t_1+t_2+t_3+t_4$ we see that $$\begin{aligned} I(F) &= \int_{t_1+t_2+t_3+t_4 \leq 1+{\varepsilon}} (1-\alpha(t_1+t_2+t_3+t_4))^2\ dt_1 dt_2 dt_3 dt_4 \\ &= \int_0^{1+{\varepsilon}} (1-\alpha s)^2 \frac{s^3}{3!}\ ds\\ &= \alpha^2 \frac{(1+{\varepsilon})^6}{36} - \alpha \frac{(1+{\varepsilon})^5}{15} + \frac{(1+{\varepsilon})^4}{24} \\ &= 0.00728001347\dots\end{aligned}$$ and similarly by making the change of variables $u = t_1+t_2+t_3$ $$\begin{aligned} J_{1,1-{\varepsilon}}(F) &= \int_{t_1+t_2+t_3 \leq 1-{\varepsilon}} (\int_0^{1+{\varepsilon}-t_1-t_2-t_3} (1-\alpha(t_1+t_2+t_3+t_4))\ dt_4)^2 dt_1 dt_2 dt_3 \\ &= \int_0^{1-{\varepsilon}} (\int_0^{1+{\varepsilon}-u} (1-\alpha(u+t_4))\ dt_4)^2 \frac{u^2}{2!} du \\ &= \int_0^{1-{\varepsilon}} (1+{\varepsilon}-u)^2 (1-\alpha \frac{1+{\varepsilon}+u}{2})^2 \frac{u^2}{2} du \\ &= 0.003650160667\dots\end{aligned}$$ and so follows. If one uses the truncated function $$\tilde F(t_1,t_2,t_3,t_4) \coloneqq F(t_1,t_2,t_3,t_4) {\mathbf{1}}_{t_1,t_2,t_3,t_4 \leq 1}$$ in place of $F$, and sets ${\varepsilon}$ to $0.18$ instead of $0.168$, one can compute that $$\frac{4 J_{1,1-{\varepsilon}}(\tilde F)}{I(\tilde F)} > 2.00235.$$ Thus it is possible to establish Theorem \[mke-lower\](xii$'$) using a cutoff function $F'$ that is also supported in the unit cube $[0,1]^4$. This allows for a slight simplification to the proof of $\operatorname*{DHL}[4,2]$ assuming GEH, as one can add the additional hypothesis $S(F_{i_0})+S(G_{i_0}) < 1$ to Theorem \[nonprime-asym\](ii) in that case. By optimising in ${\varepsilon}$ and taking $F$ to be a symmetric polynomial of degree higher than $1$, one can get slightly better lower bounds for $M_{4,{\varepsilon}}$; for instance setting ${\varepsilon}= 5/21$ and choosing $F$ to be a cubic polynomial, we were able to obtain the bound $M_{4,{\varepsilon}} \geq 2.05411$. On the other hand, the best lower bound for $M_{3,{\varepsilon}}$ that we were able to obtain was $1.91726$ (taking ${\varepsilon}= 56/113$ and optimizing over cubic polynomials). Again, see [www.dropbox.com/sh/0xb4xrsx4qmua7u/WOhuo2Gx7f/Polymath8b](www.dropbox.com/sh/0xb4xrsx4qmua7u/WOhuo2Gx7f/Polymath8b) for the relevant code and data. Three-dimensional cutoffs {#3d} ------------------------- In this section we establish Theorem \[piece\]. We relabel the variables $(t_1,t_2,t_3)$ as $(x,y,z)$, thus our task is to locate a piecewise polynomial function $F: [0,+\infty)^3 \to {\mathbb{R}}$ supported on the simplex $$R := \left\{ (x,y,z) \in [0,+\infty)^3: x+y+z \leq \frac{3}{2}\right\}$$ and symmetric in the $x,y,z$ variables, obeying the vanishing marginal condition $$\label{fmor} \int_0^\infty F(x,y,z)\ dz = 0$$ whenever $x,y \geq 0$ with $x+y > 1+{\varepsilon}$, and such that $$\label{tt1} J(F) > 2 I(F)$$ where $$\label{jf-def} J(F) := 3\int_{x+y \leq 1-{\varepsilon}} \left(\int_0^\infty F(x,y,z)\ dz\right)^2\ dx dy$$ and $$\label{if-def} I(F) := \int_R F(x,y,z)^2\ dx dy dz$$ and $${\varepsilon}:= 1/4.$$ Our strategy will be as follows. We will decompose the simplex $R$ (up to null sets) into a carefully selected set of disjoint open polyhedra $P_1,\dots,P_m$ (in fact $m$ will be $60$), and on each $P_i$ we will take $F(x,y,z)$ to be a low degree polynomial $F_i(x,y,z)$ (indeed, the degree will never exceed $3$). The left and right-hand sides of become quadratic functions in the coefficients of the $F_i$. Meanwhile, the requirement of symmetry, as well as the marginal requirement , imposes some linear constraints on these coefficients. In principle, this creates a finite-dimensional quadratic program, which one can try to solve numerically. However, to make this strategy practical, one needs to keep the number of linear constraints imposed on the coefficients to be fairly small, as compared with the total number of coefficients. To achieve this, the following properties on the polyhedra $P_i$ are desirable: - (Symmetry) If $P_i$ is a polytope in the partition, then every reflection of $P_i$ formed by permuting the $x,y,z$ coordinates should also lie in the partition. - (Graph structure) Each polytope $P_i$ should be of the form $$\label{qi-form} \{ (x,y,z): (x,y) \in Q_i; a_i(x,y) < z < b_i(x,y)\},$$ where $a_i(x,y), b_i(x,y)$ are linear forms and $Q_i$ is a polygon. - (Epsilon splitting) Each $Q_i$ is contained in one of the regions $\{ (x,y): x+y < 1-{\varepsilon}\}$, $\{ (x,y): 1-{\varepsilon}< x+y < 1+{\varepsilon}\}$, or $\{ (x,y): 1+{\varepsilon}< x+y < 3/2 \}$. Observe that the vanishing marginal condition now takes the form $$\label{vanish} \sum_{i: (x,y) \in Q_i} \int_{a_i(x,y)}^{b_i(x,y)} F_i(x,y,z)\ dz = 0$$ for every $x,y > 0$ with $x+y > 1+{\varepsilon}$. If the set $\{i: (x,y) \in Q_i\}$ is fixed, then the left-hand side of is a polynomial in $x,y$ whose coefficients depend linearly on the coefficients of the $F_i$, and thus imposes a set of linear conditions on these coefficients for each possible set $\{ i: (x,y) \in Q_i\}$ with $x+y > 1+{\varepsilon}$. Now we describe the partition we will use. This partition can in fact be used for all ${\varepsilon}$ in the interval $[1/4,1/3]$, but the endpoint ${\varepsilon}= 1/4$ has some simplifications which allowed for reasonably good numerical results. To obtain the symmetry property, it is natural to split $R$ (modulo null sets) into six polyhedra $R_{xyz}, R_{xzy}, R_{yxz}, R_{yzx}, R_{zxy}, R_{zyx}$, where $$\begin{aligned} R_{xyz} &:= \{(x,y,z)\in R\ :\ x+y < y+z < z+x\} \\ &= \{(x,y,z): 0 < y < x < z; x+y+z \leq 3/2 \}\end{aligned}$$ and the other polyhedra are obtained by permuting the indices $x,y,z$, thus for instance $$\begin{aligned} R_{yxz} &:= \{(x,y,z)\in R\ :\ y+x < x+z < z+y\} \\ &= \{(x,y,z): 0 < x < y < z; y+x+z \leq 3/2 \}.\end{aligned}$$ To obtain the epsilon splitting property, we decompose $R_{xyz}$ (modulo null sets) into eight sub-polytopes $$\begin{aligned} A_{xyz} & =\{(x,y,z)\in R\ :\ x+y< y+z< z+x< 1-{\varepsilon}\},\\ B_{xyz} & =\{(x,y,z)\in R\ :\ x+y< y+z< 1-{\varepsilon}< z+x< 1+{\varepsilon}\},\\ C_{xyz} & =\{(x,y,z)\in R\ :\ x+y< 1-{\varepsilon}< y+z < z+x< 1+{\varepsilon}\},\\ D_{xyz} & =\{(x,y,z)\in R\ :\ 1-{\varepsilon}< x+y< y+z < z+x< 1+{\varepsilon}\},\\ E_{xyz} & =\{(x,y,z)\in R\ :\ x+y< y+z< 1-{\varepsilon}< 1+{\varepsilon}< z+x\},\\ F_{xyz} & =\{(x,y,z)\in R\ :\ x+y< 1-{\varepsilon}< y+z < 1+{\varepsilon}< z+x\},\\ G_{xyz} & =\{(x,y,z)\in R\ :\ x+y< 1-{\varepsilon}< 1+{\varepsilon}< y+z < z+x\},\\ H_{xyz} & =\{(x,y,z)\in R\ :\ 1-{\varepsilon}< x+y< y+z < 1+{\varepsilon}< z+x\};\end{aligned}$$ the other five polytopes $R_{xzy}, R_{yxz}, R_{yzx}, R_{zxy}, R_{zyx}$ are decomposed similarly, leading to a partition of $R$ into $6 \times 8 = 48$ polytopes. This is almost the partition we will use; however there is a technical difficulty arising from the fact that some of the permutations of $F_{xyz}$ do not obey the graph structure property. So we will split $F_{xyz}$ further, into the three pieces $$\begin{aligned} S_{xyz} & = \{(x,y,z)\in F_{xyz}\ :\ z< 1/2+{\varepsilon}\},\\ T_{xyz} & = \{(x,y,z)\in F_{xyz}\ :\ z> 1/2+{\varepsilon};x> 1/2-{\varepsilon}\}, \\ U_{xyz} & = \{(x,y,z)\in F_{xyz}\ :\ x< 1/2-{\varepsilon}\}.\end{aligned}$$ Thus $R_{xyz}$ is now partitioned into ten polytopes $A_{xyz},$ $B_{xyz},$ $C_{xyz},$ $D_{xyz}$, $E_{xyz}$, $S_{xyz}$, $T_{xyz}$, $U_{xyz}$, $G_{xyz}$, $H_{xyz}$, and similarly for permutations of $R_{xyz}$, leading to a decomposition of $R$ into $6 \times 10 = 60$ polytopes. A symmetric piecewise polynomial function $F$ supported on $R$ can now be described (almost everywhere) by specifying a polynomial function $F\downharpoonright_{P}: P \to {\mathbb{R}}$ for the ten polytopes $P = A_{xyz}, B_{xyz}, C_{xyz}, D_{xyz}, E_{xyz}, S_{xyz}, T_{xyz}, U_{xyz}, G_{xyz}, H_{xyz}$, and then extending by symmetry, thus for instance $$F\downharpoonright_{A_{yzx}}(x,y,z) = F\downharpoonright_{A_{xyz}}(z,x,y).$$ As discussed earlier, the expressions $I(F), J(F)$ can now be written as quadratic forms in the coefficients of the $F\downharpoonright_P$, and the vanishing marginal condition imposes some linear constraints on these coefficients. Observe that the polytope $D_{xyz}$ and all of its permutations make no contribution to either the functional $J(F)$ or to the marginal condition , and give a non-negative contribution to $I(F)$. Thus without loss of generality we may assume that $$F\downharpoonright_{D_{xyz}} = 0.$$ However, the other nine polytopes $A_{xyz}, B_{xyz}, C_{xyz}, E_{xyz}, S_{xyz}, T_{xyz}, U_{xyz}, G_{xyz}, H_{xyz}$ have at least one permutation which gives a non-trivial contribution to either $J(F)$ or to , and cannot be easily eliminated. Now we compute $I(F)$. By symmetry we have $$I(F) = 3! I(F \downharpoonright_{R_{xyz}} ) = 6 \sum_P I( F \downharpoonright_P )$$ where $P$ ranges over the nine polytopes $A_{xyz}, B_{xyz}, C_{xyz}, E_{xyz}, S_{xyz}, T_{xyz}, U_{xyz}, G_{xyz}, H_{xyz}$. A tedious but straightforward computation shows that $$\begin{aligned} I(F\downharpoonright_{A_{xyz}}) & =\int_{x=0}^{1/2-{\varepsilon}/2}\int_{y=0}^{x}\int_{z=x}^{1-{\varepsilon}-x}F\downharpoonright_{A_{xyz}}^2\ dz\ dy\ dx \\ I(F\downharpoonright_{B_{xyz}}) &= \left(\int_{z=1/2-{\varepsilon}/2}^{1/2+{\varepsilon}/2}\int_{x=1-{\varepsilon}-z}^{z} +\int_{z=1/2+{\varepsilon}/2}^{1-{\varepsilon}}\int_{x=1-{\varepsilon}-z}^{1+{\varepsilon}-z}\right) \int_{y=0}^{1-{\varepsilon}-z}F\downharpoonright_{B_{xyz}}^2\ dy\ dx\ dz \\ I(F\downharpoonright_{C_{xyz}}) & = \left(\int_{y=0}^{1/2-3{\varepsilon}/2} \int_{x=y}^{y+2{\varepsilon}} + \int_{y=1/2-3{\varepsilon}/2}^{1/2-{\varepsilon}}\int_{x=y}^{1-{\varepsilon}-y} \right) \int_{z=1-{\varepsilon}-y}^{1+{\varepsilon}-x} \\ &\quad + \int_{y=1/2-{\varepsilon}}^{1/2-{\varepsilon}/2}\int_{x=y}^{1-{\varepsilon}-y}\int_{z=1-{\varepsilon}-y}^{3/2-x-y} F\downharpoonright_{C_{xyz}}^2\ dz\ dx\ dy \\ I(F\downharpoonright_{E_{xyz}}) & = \int_{z=1/2+{\varepsilon}/2}^{1-{\varepsilon}}\int_{x=1+{\varepsilon}-z}^{z}\int_{y=0}^{1-{\varepsilon}-z} F\downharpoonright_{E_{xyz}}^2\ dy\ dx\ dz \\ I(F\downharpoonright_{S_{xyz}}) & = \left(\int_{y=0}^{1/2-3{\varepsilon}/2}\int_{z=1-{\varepsilon}-y}^{1/2+{\varepsilon}} +\int_{y=1/2-3{\varepsilon}/2}^{1/2-{\varepsilon}}\int_{z=y+2{\varepsilon}}^{1/2+{\varepsilon}} \right)\int_{x=1+{\varepsilon}-z}^{1-{\varepsilon}-y} F\downharpoonright_{S_{xyz}}^2\ dx\ dz\ dy \\ I(F\downharpoonright_{T_{xyz}}) & = \left(\int_{z=1/2+{\varepsilon}}^{1/2+2{\varepsilon}}\int_{x=1+{\varepsilon}-z}^{3/2-z} + \int_{z=1/2+2{\varepsilon}}^{1+{\varepsilon}}\int_{x=1/2-{\varepsilon}}^{3/2-z}\right)\int_{y=0}^{3/2-x-z} F\downharpoonright_{T_{xyz}}^2\ dy\ dz\ dx \\ I(F\downharpoonright_{U_{xyz}}) & = \int_{x=0}^{1/2-{\varepsilon}}\int_{y=0}^{x}\int_{z=1+{\varepsilon}-x}^{1+{\varepsilon}-y} F\downharpoonright_{U_{xyz}}^2\ dz\ dy\ dx\\ I(F\downharpoonright_{G_{xyz}}) & = \int_{x=0}^{1/2-{\varepsilon}}\int_{y=0}^{x}\int_{z=1+{\varepsilon}-y}^{3/2-x-y} F\downharpoonright_{G_{xyz}}^2\ dx\ dz\ dy\end{aligned}$$ and $$\begin{aligned} I(F\downharpoonright_{H_{xyz}}) & =\left(\int_{x=1/2+{\varepsilon}/2}^{1-{\varepsilon}}\int_{y=1-{\varepsilon}-x}^{3/2-2x} +\int_{x=1-{\varepsilon}}^{3/4}\int_{y=0}^{3/2-2x} \right) \int_{z=x}^{3/2-x-y}\\ &\quad + \int_{x=1/2}^{1/2+{\varepsilon}/2}\int_{y=1-{\varepsilon}-x}^{1/2-{\varepsilon}}\int_{z=1+{\varepsilon}-x}^{3/2-x-y} F\downharpoonright_{H_{xyz}}^2\ dz\ dy\ dx.\end{aligned}$$ Now we consider the quantity $J(F)$. Here we only have the symmetry of swapping $x$ and $y$, so that $$J(F) = 6 \int_{0 < y < x; x+y < 1-{\varepsilon}} \left(\int_0^{3/2-x-y} F(x,y,z)\ dz\right)^2 dx dy.$$ The region of integration meets the polytopes $A_{xyz}$, $A_{yzx}$, $A_{zyx}$, $B_{xyz}$, $B_{zyx}$, $C_{xyz}$, $E_{xyz}$, $E_{zyx}$, $S_{xyz}$, $T_{xyz}$, $U_{xyz}$, and $G_{xyz}$. Projecting these polytopes to the $(x,y)$-plane, we have the diagram:  This diagram is drawn to scale in the case when $\varepsilon=1/4$, otherwise there is a separation between the $J_5$ and $J_7$ regions. For these eight regions there are eight corresponding integrals $J_1,J_2,\ldots, J_8$, thus $$J(F) = 6( J_1 + \dots + J_8).$$ We have $$\begin{aligned} J_1 & =\int_{x=0}^{1/2-{\varepsilon}}\int_{y=0}^{x} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{x}F\downharpoonright_{A_{zyx}} +\int_{z=x}^{1-{\varepsilon}-x}F\downharpoonright_{A_{xyz}} +\int_{z=1-{\varepsilon}-x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}} \right. \\ &\quad \left. +\int_{z=1-{\varepsilon}-y}^{1+{\varepsilon}-x}F\downharpoonright_{C_{xyz}} + \int_{z=1+{\varepsilon}-x}^{1+{\varepsilon}-y}F\downharpoonright_{U_{xyz}} + \int_{z=1+{\varepsilon}-y}^{3/2-x-y}F\downharpoonright_{G_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ Next comes $$\begin{aligned} J_2 & =\int_{x=1/2-{\varepsilon}}^{1/2-{\varepsilon}/2}\int_{y=1/2-{\varepsilon}}^{x} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{x}F\downharpoonright_{A_{zyx}}+\int_{z=x}^{1-{\varepsilon}-x}F\downharpoonright_{A_{xyz}} +\int_{z=1-{\varepsilon}-x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}}\right. \\ &\quad \left. +\int_{z=1-{\varepsilon}-y}^{3/2-x-y}F\downharpoonright_{C_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ Third is the piece $$\begin{aligned} J_3 & =\int_{x=1/2-{\varepsilon}}^{1/2-{\varepsilon}/2}\int_{y=0}^{1/2-{\varepsilon}} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{x}F\downharpoonright_{A_{zyx}} +\int_{z=x}^{1-{\varepsilon}-x}F\downharpoonright_{A_{xyz}} +\int_{z=1-{\varepsilon}-x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}} \right. \\ &\quad \left. + \int_{z=1-{\varepsilon}-y}^{1+{\varepsilon}-x}F\downharpoonright_{C_{xyz}} + \int_{z=1+{\varepsilon}-x}^{3/2-x-y}F\downharpoonright_{T_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ We now have dealt with all integrals involving $A_{xyz}$, and all remaining integrals pass through $B_{zyx}$. Continuing, we have $$\begin{aligned} J_4 & =\int_{x=1/2-{\varepsilon}/2}^{1/2}\int_{y=1/2-{\varepsilon}}^{1-{\varepsilon}-x} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{1-{\varepsilon}-x}F\downharpoonright_{A_{zyx}} +\int_{z=1-{\varepsilon}-x}^{x}F\downharpoonright_{B_{zyx}} +\int_{z=x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}}\right. \\ &\quad \left. +\int_{z=1-{\varepsilon}-y}^{3/2-x-y}F\downharpoonright_{C_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ Another component is $$\begin{aligned} J_5 & = \int_{x=1/2-{\varepsilon}/2}^{1/2}\int_{y=0}^{1/2-{\varepsilon}} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{1-{\varepsilon}-x}F\downharpoonright_{A_{zyx}} \right. \\ &\quad \left. +\int_{z=1-{\varepsilon}-x}^{x}F\downharpoonright_{B_{zyx}} +\int_{z=x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}} +\int_{z=1-{\varepsilon}-y}^{1+{\varepsilon}-x}F\downharpoonright_{C_{xyz}} + \int_{z=1+{\varepsilon}-x}^{3/2-x-y}F\downharpoonright_{T_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ The most complicated piece is $$\begin{aligned} J_6 & =\left(\int_{x=1/2}^{2{\varepsilon}} \int_{y=0}^{1-{\varepsilon}-x} + \int_{x=2{\varepsilon}}^{1/2+{\varepsilon}/2}\int_{y=x-2{\varepsilon}}^{1-{\varepsilon}-x} \right) \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{1-{\varepsilon}-x}F\downharpoonright_{A_{zyx}} +\int_{z=1-{\varepsilon}-x}^{x}F\downharpoonright_{B_{zyx}} \right. \\ &\quad \left. +\int_{z=x}^{1-{\varepsilon}-y}F\downharpoonright_{B_{xyz}} +\int_{z=1-{\varepsilon}-y}^{1+{\varepsilon}-x}F\downharpoonright_{C_{xyz}} + \int_{z=1+{\varepsilon}-x}^{1/2+{\varepsilon}}F\downharpoonright_{S_{xyz}}+ \int_{z=1/2+{\varepsilon}}^{3/2-x-y}F\downharpoonright_{T_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ Here we use $\left(\int_{x=1/2}^{2{\varepsilon}} \int_{y=0}^{1-{\varepsilon}-x} + \int_{x=2{\varepsilon}}^{1/2+{\varepsilon}/2}\int_{y=x-2{\varepsilon}}^{1-{\varepsilon}-x} \right) f(x,y)\ dy dx$ as an abbreviation for $$\int_{x=1/2}^{2{\varepsilon}} \int_{y=0}^{1-{\varepsilon}-x} f(x,y)\ dy dx + \int_{x=2{\varepsilon}}^{1/2+{\varepsilon}/2}\int_{y=x-2{\varepsilon}}^{1-{\varepsilon}-x} f(x,y)\ dy dx.$$ We have now exhausted $C_{xyz}$. The seventh piece is $$\begin{aligned} J_7 & = \int_{x=2{\varepsilon}}^{1/2+{\varepsilon}/2}\int_{y=0}^{x-2{\varepsilon}} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{1-{\varepsilon}-x}F\downharpoonright_{A_{zyx}} +\int_{z=1-{\varepsilon}-x}^{x}F\downharpoonright_{B_{zyx}} \right. \\ &\quad \left. +\int_{z=x}^{1+{\varepsilon}-x}F\downharpoonright_{B_{xyz}} + \int_{z=1+{\varepsilon}-x}^{1-{\varepsilon}-y}F\downharpoonright_{E_{xyz}} + \int_{1-{\varepsilon}-y}^{1/2+{\varepsilon}}F\downharpoonright_{S_{xyz}}+ \int_{1/2+{\varepsilon}}^{3/2-x-y}F\downharpoonright_{T_{xyz}} \ dz \right)^{2}\ dy\ dx.\end{aligned}$$ Finally, we have $$\begin{aligned} J_8 & = \int_{x=1/2+{\varepsilon}/2}^{1-{\varepsilon}}\int_{y=0}^{1-{\varepsilon}-x} \left(\int_{z=0}^{y}F\downharpoonright_{A_{yzx}}+\int_{z=y}^{1-{\varepsilon}-x}F\downharpoonright_{A_{zyx}} +\int_{z=1-{\varepsilon}-x}^{1+{\varepsilon}-x}F\downharpoonright_{B_{zyx}} \right. \\ &\quad \left. + \int_{z=1+{\varepsilon}-x}^{x}F\downharpoonright_{E_{zyx}} + \int_{z=x}^{1-{\varepsilon}-y}F\downharpoonright_{E_{xyz}} + \int_{1-{\varepsilon}-y}^{1/2+{\varepsilon}}F\downharpoonright_{S_{xyz}}+ \int_{1/2+{\varepsilon}}^{3/2-x-y}F\downharpoonright_{T_{xyz}}\ dz \right)^{2}\ dy\ dx.\end{aligned}$$ In the case ${\varepsilon}=1/4$, the marginal conditions reduce to requiring $$\begin{aligned} \int_{z=0}^{3/2-x-y}F\downharpoonright_{G_{yzx}}\ dz &= 0\label{m1}\\ \int_{z=0}^{y}F\downharpoonright_{G_{yzx}} +\int_{z=y}^{3/2-x-y}F\downharpoonright_{G_{zyx}}\ dz &= 0\label{m2}\\ \int_{z=0}^{1+\varepsilon-x}F\downharpoonright_{U_{yzx}} + \int_{z=1+\varepsilon-x}^{y}F\downharpoonright_{G_{yzx}} +\int_{z=y}^{3/2-x-y}F\downharpoonright_{G_{zyx}}\ dz &= 0\label{m3}\\ \int_{z=0}^{1+\varepsilon-x}F\downharpoonright_{U_{yzx}} + \int_{z=1+\varepsilon-x}^{3/2-x-y}F\downharpoonright_{G_{yzx}}\ dz &= 0\label{m4}\\ \int_{z=0}^{3/2-x-y}F\downharpoonright_{T_{yzx}}\ dz &= 0\label{m5}\\ \int_{z=0}^{1-\varepsilon-x}F\downharpoonright_{E_{yzx}} + \int_{z=1-\varepsilon-x}^{1-\varepsilon-y}F\downharpoonright_{S_{yzx}} + \int_{z=1-\varepsilon-y}^{3/2-x-y}F\downharpoonright_{H_{yzx}}\ dz &= 0.\label{m7}\end{aligned}$$ Each of these constraints is only required to hold for some portion of the parameter space $\{ (x,y): 1+{\varepsilon}\leq x+y \leq 3/2 \}$, but as the left-hand sides are all polynomial functions in $x,y$ (using the signed definite integral $\int_b^a = -\int_a^b$), it is equivalent to require that all coefficients of these polynomial functions vanish. Now we specify $F$. After some numerical experimentation, we have found the simplest choice of $F$ that still achieves the desired goal comes by taking $F(x,y,z)$ to be a polynomial of degree $1$ on each of $E_{xyz}$, $S_{xyz}$, $H_{xyz}$, degree $2$ on $T_{xyz}$, vanishing on $D_{xyz}$, and degree $3$ on the remaining five relevant components of $R_{xyz}$. After solving the quadratic program, rounding, and clearing denominators, we arrive at the choice $$\begin{aligned} F\downharpoonright_{A_{xyz}} &:= -66+96 x-147 x^2+125 x^3+128 y-122 x y+104 x^2 y-275 y^2+394 y^3+99 z\\ &\quad -58 x z+63 x^2 z-98 y z+51 x y z+41 y^2 z-112 z^2+24 x z^2+72 y z^2+50 z^3 \\ F\downharpoonright_{B_{xyz}} &:= -41+52 x-73 x^2+25 x^3+108 y-66 x y+71 x^2 y-294 y^2+56 x y^2+363 y^3\\ &\quad +33 z+15 x z+22 x^2 z-40 y z-42 x y z+75 y^2 z-36 z^2-24 x z^2+26 y z^2+20 z^3 \\ F\downharpoonright_{C_{xyz}} &:= -22+45 x-35 x^2+63 y-99 x y+82 x^2 y-140 y^2+54 x y^2+179 y^3 \\ F\downharpoonright_{E_{xyz}} &:= -12+8 x+32 y \\ F\downharpoonright_{S_{xyz}} &:= -6+8 x+16 y \\ F\downharpoonright_{T_{xyz}} &:= 18-30 x+12 x^2+42 y-20 x y-66 y^2-45 z+34 x z+22 z^2 \\ F\downharpoonright_{U_{xyz}} &:= 94-1823 x+5760 x^2-5128 x^3+54 y-168 x^2 y+105 y^2+1422 x z-2340 x^2 z\\ &\quad -192 y^2 z-128 z^2-268 x z^2+64 z^3 \\ F\downharpoonright_{G_{xyz}} &:= 5274-19833 x+18570 x^2-5128 x^3-18024 y+44696 x y-20664 x^2 y+16158 y^2\\ &\quad -19056 x y^2-4592 y^3-10704 z+26860 x z-12588 x^2 z+24448 y z-30352 x y z\\ &\quad -10980 y^2 z+7240 z^2-9092 x z^2-8288 y z^2-1632 z^3 \\ F\downharpoonright_{H_{xyz}} &:= 8 z.\end{aligned}$$ One may compute that $$I(F) = \frac{62082439864241}{507343011840}$$ and $$J(F) = \frac{9933190664926733}{40587440947200}$$ with all the marginal conditions - obeyed, thus $$\frac{J(F)}{I(F)} = 2 + \frac{286648173}{4966595189139280}$$ and follows. The parity problem {#parity-sec} ================== In this section we argue why the “parity barrier” of Selberg [@selberg] prohibits sieve-theoretic methods, such as the ones in this paper, from obtaining any bound on $H_1$ that is stronger than $H_1 \leq 6$, even on the assumption of strong distributional conjectures such as the generalized Elliott-Halberstam conjecture $\operatorname*{GEH}[\vartheta]$, and even if one uses sieves other than the Selberg sieve. Our discussion will be somewhat informal and heuristic in nature. We begin by briefly recalling how the bound $H_1 \leq 6$ on GEH (i.e., Theorem \[main\](xii)) was proven. This was deduced from the claim $\operatorname*{DHL}[3,2]$, or more specifically from the claim that the set $$\label{A-def} A := \{ n \in {\mathbb{N}}: \hbox{ at least two of } n, n+2, n+6 \hbox{ are prime} \}$$ was infinite. To do this, we (implicitly) established a lower bound $$\sum_n \nu(n) {\mathbf{1}}_A(n) > 0$$ for some non-negative weight $\nu: {\mathbb{N}}\to {\mathbb{R}}^+$ supported on $[x,2x]$ for a sufficiently large $x$. This bound was in turn established (after a lengthy sieve-theoretic analysis, and with a carefully chosen weight $\nu$) from upper bounds on various discrepancies. More precisely, one required good upper bounds (on average) for the expressions $$\label{flip} \left|\sum_{x \leq n \leq 2x: n = a\ (q)} f(n+h) - \frac{1}{{\varphi}(q)} \sum_{x \leq n \leq 2x: (n+h,q)=1} f(n+h)\right|$$ for all $h \in \{0,2,6\}$ and various residue classes $a\ (q)$ with $q \leq x^{1-{\varepsilon}}$ and arithmetic functions $f$, such as the constant function $f=1$, the von Mangoldt function $f = \Lambda$, or Dirichlet convolutions $f = \alpha \star \beta$ of the type considered in Claim \[geh-def\]. (In the presentation of this argument in previous sections, the shift by $h$ was eliminated using the change of variables $n' = n+h$, but for the current discussion it is important that we do not use this shift.) One also required good asymptotic control on the main terms $$\label{flop} \sum_{x \leq n \leq 2x: (n+h,q)=1} f(n+h).$$ Once one eliminates the shift by $h$, an inspection of these arguments reveals that they would be equally valid if one inserted a further non-negative weight $\omega: {\mathbb{N}}\to {\mathbb{R}}^+$ in the summation over $n$. More precisely, the above sieve-theoretic argument would also deduce the lower bound $$\sum_n \nu(n) {\mathbf{1}}_A(n) \omega(n) > 0$$ if one had control on the weighted discrepancies $$\label{flip-weight} \left|\sum_{x \leq n \leq 2x: n= a\ (q)} f(n+h) \omega(n) - \frac{1}{{\varphi}(q)} \sum_{x \leq n \leq 2x: (n+h,q)=1} f(n+h) \omega(n)\right|$$ and on the weighted main terms $$\label{flop-weight} \sum_{x \leq n \leq 2x: (n+h,q)=1} f(n+h) \omega(n)$$ that were of the same form as in the unweighted case $\omega=1$. Now suppose for instance that one was trying to prove the bound $H_1 \leq 4$. A natural way to proceed here would be to replace the set $A$ in with the smaller set $$\label{app} A' := \{ n \in {\mathbb{N}}: n, n+2 \hbox{ are both prime} \} \cup \{ n \in {\mathbb{N}}: n+2, n+6 \hbox{ are both prime} \}$$ and hope to establish a bound of the form $$\sum_n \nu(n) {\mathbf{1}}_{A'}(n) > 0$$ for a well-chosen function $\nu: {\mathbb{N}}\to {\mathbb{R}}^+$ supported on $[x,2x]$, by deriving this bound from suitable (averaged) upper bounds on the discrepancies and control on the main terms . If the arguments were sieve-theoretic in nature, then (as in the $H_1 \leq 6$ case), one could then also deduce the lower bound $$\label{ap-lower} \sum_n \nu(n) {\mathbf{1}}_{A'}(n) \omega(n) > 0$$ for any non-negative weight $\omega: {\mathbb{N}}\to {\mathbb{R}}^+$, provided that one had the same control on the weighted discrepancies and weighted main terms that one did on , . We apply this observation to the weight $$\begin{aligned} \omega(n) &:= (1 - \lambda(n) \lambda(n+2)) (1 - \lambda(n+2) \lambda(n+6)) \\ &= 1 - \lambda(n)\lambda(n+2) - \lambda(n+2)\lambda(n+6) + \lambda(n) \lambda(n+6)\end{aligned}$$ where $\lambda(n) := (-1)^{\Omega(n)}$ is the Liouville function. Observe that $\omega$ vanishes for any $n\in A'$, and hence $$\label{ap-none} \sum_n \nu(n) {\mathbf{1}}_{A'}(n) \omega(n) = 0$$ for any $\nu$. On the other hand, the “Möbius randomness law” (see e.g. [@ik]) predicts a significant amount of cancellation for any non-trivial sum involving the Möbius function $\mu$, or the closely related Liouville function $\lambda$. For instance, the expression $$\sum_{x \leq n \leq 2x: n = a\ (q)} \lambda(n+h)$$ is expected to be very small (of size[^7] $O( \frac{x}{q} \log^{-A} x)$ for any fixed $A$) for any residue class $a\ (q)$ with $q \leq x^{1-{\varepsilon}}$, and any $h \in \{0,2,6\}$; similarly for more complicated expressions such as $$\sum_{x \leq n \leq 2x: n = a\ (q)} \lambda(n+2) \lambda(n+6)$$ or $$\sum_{x \leq n \leq 2x: n = a\ (q)} \Lambda(n) \lambda(n+2) \lambda(n+6)$$ or more generally $$\sum_{x \leq n \leq 2x: n = a\ (q)} f(n) \lambda(n+2) \lambda(n+6)$$ where $f$ is a Dirichlet convolution $\alpha \star \beta$ of the form considered in Claim \[geh-def\]. Similarly for expressions such as $$\sum_{x \leq n \leq 2x: n = a\ (q)} f(n) \lambda(n) \lambda(n+2);$$ note from the complete multiplicativity of $\lambda$ that $(\alpha \star \beta) \lambda = (\alpha \lambda) \star (\beta \lambda)$, so if $f$ is of the form in Claim \[geh-def\], then $f\lambda$ is also. In view of these observations (and similar observations arising from permutations of $\{0,2,6\}$), we conclude (heuristically, at least) that all the bounds that are believed to hold for , should also hold (up to minor changes in the implied constants) for , . Thus, if the bound $H_1 \leq 4$ could be proven in a sieve-theoretic fashion, one should be able to conclude the bound , which is in direct contradiction to . Similar arguments work for any set of the form $$A_H := \{ n \in {\mathbb{N}}: \exists n \leq p_1 < p_2 \leq n+H; p_1,p_2 \hbox{ both prime}, p_2 - p_1 \leq 4 \}$$ and any fixed $H > 0$, to prohibit any non-trivial lower bound on $\sum_n \nu(n) {\mathbf{1}}_{A_H}(n)$ from sieve-theoretic methods. Indeed, one uses the weight $$\omega(n) := \prod_{0 \leq i \leq i' \leq H; (n+i,3) = (n+i',3) = 1; i'-i \leq 4} (1 - \lambda(n+i) \lambda(n+i'));$$ we leave the details to the interested reader. This seems to block any attempt to use any argument based only on the distribution of the prime numbers and related expressions in arithmetic progressions to prove $H_1 \leq 4$. The same arguments of course also prohibit a sieve-theoretic proof of the twin prime conjecture $H_1 = 2$. In this case one can use the simpler weight $\omega(n) = 1 - \lambda(n) \lambda(n+2)$ to rule out such a proof, and the argument is essentially due to Selberg [@selberg]. Of course, the parity barrier could be circumvented if one were able to introduce stronger sieve-theoretic axioms than the “linear” axioms currently available (which only control sums of the form or ). For instance, if one were able to obtain non-trivial bounds for “bilinear” expressions such as $$\sum_{x \leq n \leq 2x} f(n) \Lambda(n+2) = \sum_d \sum_m \alpha(d) \beta(m) {\mathbf{1}}_{[x,2x]}(dm) \Lambda(dm+2)$$ for functions $f = \alpha \star \beta$ of the form in Claim \[geh-def\], then (by a modification of the proof of Proposition \[geh-eh\]) one would very likely obtain non-trivial bounds on $$\sum_{x \leq n \leq 2x} \Lambda(n) \Lambda(n+2)$$ which would soon lead to a proof of the twin prime conjecture. Unfortunately, we do not know of any plausible way to control such bilinear expressions. (Note however that there are some other situations in which bilinear sieve axioms may be established, for instance in the argument of Friedlander and Iwaniec [@fi] establishing an infinitude of primes of the form $a^2+b^4$.) Additional remarks {#remarks-sec} ================== The proof of Theorem \[main-dhl\](xii) may be modified to establish the following variant: Assume the generalized Elliott-Halberstam conjecture $\operatorname*{GEH}[\vartheta]$ for all $0 < \vartheta < 1$. Let $0 < {\varepsilon}< 1/2$ be fixed. Then if $x$ is a sufficiently large multiple of $6$, there exists a natural number $n$ with ${\varepsilon}x \leq n \leq (1-{\varepsilon}) x$ such that at least two of $n, n-2, x-n$ are prime. Similarly if $n-2$ is replaced by $n+2$. Note that if at least two of $n,n-2,x-n$ are prime, then either $n,n+2$ are twin primes, or else at least one of $x,x-2$ is expressible as the sum of two primes, and Theorem \[disj\] easily follows. (Sketch) We just discuss the case of $n-2$, as the $n+2$ case is similar. Observe from the Chinese remainder theorem (and the hypothesis that $x$ is divisible by $6$) that one can find a residue class $b\ (W)$ such that $b, b-2, x-b$ are all coprime to $W$ (in particular, one has $b=1\ (6)$). By a routine modification of the proof of Lemma \[crit\], it suffices to find a non-negative weight function $\nu \colon {\mathbb{N}}\to {\mathbb{R}}^+$ and fixed quantities $\alpha > 0$ and $\beta_1,\beta_2,\beta_3 \geq 0$, such that one has the asymptotic upper bound $$\sum_{\substack{{\varepsilon}x \leq n \leq (1-{\varepsilon}) x\\ n = b\ (W)}} \nu(n) \leq {\mathfrak S} (\alpha+o(1)) B^{-k} \frac{(1-2{\varepsilon}) x}{W},$$ the asymptotic lower bounds $$\begin{aligned} \sum_{\substack{{\varepsilon}x \leq n \leq (1-{\varepsilon}) x\\ n = b\ (W)}} \nu(n) \theta(n) &\geq {\mathfrak S} (\beta_1-o(1)) B^{1-k} \frac{(1-2{\varepsilon}) x}{{\varphi}(W)} \\ \sum_{\substack{{\varepsilon}x \leq n \leq (1-{\varepsilon}) x\\ n = b\ (W)}} \nu(n) \theta(n+2) &\geq {\mathfrak S} (\beta_2-o(1)) B^{1-k} \frac{(1-2{\varepsilon}) x}{{\varphi}(W)} \\ \sum_{\substack{{\varepsilon}x \leq n \leq (1-{\varepsilon}) x\\ n = b\ (W)}} \nu(n) \theta(x-n) &\geq {\mathfrak S} (\beta_3-o(1)) B^{1-k} \frac{(1-2{\varepsilon}) x}{{\varphi}(W)} \end{aligned}$$ and the inequality $$\beta_1+\beta_2+\beta_3 > 2 \alpha,$$ where ${\mathfrak S}$ is the singular series $${\mathfrak S} := \prod_{p|x(x-2); p > w} \frac{p}{p-1}.$$ We select $\nu$ to be of the form $$\nu(n) = \left( \sum_{j=1}^J c_j \lambda_{F_{j,1}}(n) \lambda_{F_{j,2}}(n+2) \lambda_{F_{j,3}}(x-n) \right)^2$$ for various fixed coefficients $c_1,\dots,c_J \in {\mathbb{R}}$ and fixed smooth compactly supported functions $F_{j,i}: [0,+\infty) \to {\mathbb{R}}$ with $j=1,\dots,J$ and $i=1,\dots,3$. It is then routine[^8] to verify that analogues of Theorem \[prime-asym\] and Theorem \[nonprime-asym\] hold for the various components of $\nu$, with the role of $x$ in the right-hand side replaced by $(1-2{\varepsilon}) x$, and the claim then follows by a suitable modification of Theorem \[epsilon-beyond\], taking advantage of the function $F$ constructed in Theorem \[piece\]. It is likely that the bounds in Theorem \[main\] can be improved further by refining the sieve-theoretic methods employed in this paper, with the exception of part (xii) for which the parity problem prevents further improvement, as discussed in Section \[parity-sec\]. We list some possible avenues to such improvements as follows: 1. In Theorem \[mke-lower\], the bound $M_{k,{\varepsilon}} > 4$ was obtained for some ${\varepsilon}>0$ and $k=50$. It is possible that $k$ could be lowered slightly, for instance to $k=49$, by further numerical computations, but we were only barely able to establish the $k=50$ bound after two weeks of computation. However, there may be a more efficient way to solve the required variational problem (e.g. by selecting a more efficient basis than the symmetric monomial basis) that would allow one to advance in this direction; this would improve the bound $H_1 \leq 246$ slightly. Extrapolation of existing numerics also raises the possibility that $M_{53}$ exceeds $4$, in which case the bound of $270$ in Theorem \[main\](vii) could be lowered to $264$. 2. To reduce $k$ (and thus $H_1$) further, one could try to solve another variational problem, such as the one arising in Theorem \[maynard-trunc\] or in Theorem \[epsilon-beyond\], rather than trying to lower bound $M_k$ or $M_{k,{\varepsilon}}$. It is also possible to use the more complicated versions of $\operatorname*{MPZ}[\varpi,\delta]$ established in [@polymath8a] (in which the modulus $q$ is assumed to be densely divisible rather than smooth) to replace the truncated simplex appearing in Theorem \[maynard-trunc\] with a more complicated region (such regions also appear implicitly in [@polymath8a §4.5]). However, in the medium-dimensional setting $k \approx 50$, we were not able to accurately and rapidly evaluate the various integrals associated to these variational problems when applied to a suitable basis of functions. One key difficulty here is that whereas polynomials appear to be an adequate choice of basis for the $M_k$, an analysis of the Euler-Lagrange equation reveals that one should use piecewise polynomial basis functions instead for more complicated variational problems such as the $M_{k,{\varepsilon}}$ problem (as was done in the three-dimensional case in Section \[3d\]), and these are difficult to work with in medium dimensions. From our experience with the low $k$ problems, it looks like one should allow these piecewise polynomials to have relatively high degree on some polytopes, low degree on other polytopes, and vanish completely on yet further polytopes[^9], but we do not have a systematic understanding of what the optimal placement of degrees should be. 3. In Theorem \[epsilon-beyond\], the function $F$ was required to be supported in the simplex $\frac{k}{k-1} \cdot {\mathcal R}_k$. However, one can consider functions $F$ supported in other regions $R$, subject to the constraint that all elements of the sumset $R+R$ lie in a region treatable by one of the cases of Theorem \[nonprime-asym\]. This could potentially lead to other optimization problems that lead to superior numerology, although again it appears difficult to perform efficient numerics for such problems in the medium $k$ regime $k \approx 50$. One possibility would be to adopt a “free boundary” perspective, in which the support of $F$ is not fixed in advance, but is allowed to evolve by some iterative numerical scheme. 4. To improve the bounds on $H_m$ for $m=2,3,4,5$, one could seek a better lower bound on $M_k$ than the one provided by Theorem \[explicit\]; one could also try to lower bound more complicated quantities such as $M_{k,{\varepsilon}}$. 5. One could attempt to improve the range of $\varpi,\delta$ for which estimates of the form $\operatorname*{MPZ}[\varpi,\delta]$ are known to hold, which would improve the results of Theorem \[main\](ii)-(vi). For instance, we believe that the condition $600 \varpi + 180\delta < 7$ in Theorem \[mpz-poly\] could be improved slightly to $1080 \varpi + 330 \delta < 13$ by refining the arguments in [@polymath8a], but this requires a hypothesis of square root cancellation in a certain four-dimensional exponential sum over finite fields, which we have thus far been unable to establish rigorously. Another direction to pursue would be to improve the $\delta$ parameter, or to otherwise relax the requirement of smoothness in the moduli, in order to reduce the need to pass to a truncation of the simplex ${\mathcal R}_k$, which is the primary reason why the $m=1$ results are currently unable to use the existing estimates of the form $\operatorname*{MPZ}[\varpi,\delta]$. Another speculative possibility is to seek $\operatorname*{MPZ}[\varpi,\delta]$ type estimates which only control distribution for a positive proportion of smooth moduli, rather than for all moduli, and then to design a sieve $\nu$ adapted to just that proportion of moduli (cf. [@fouvry-invent]). Finally, there may be a way to combine the arguments currently used to prove $\operatorname*{MPZ}[\varpi,\delta]$ with the automorphic forms (or “Kloostermania”) methods used to prove nontrivial equidistribution results with respect to a fixed modulus, although we do not have any ideas on how to actually achieve such a combination. 6. It is also possible that one could tighten the argument in Lemma \[crit\], for instance by establishing a non-trivial lower bound on the portion of the sum $\sum_n \nu(n)$ when $n+h_1,\dots,n+h_k$ are all composite, or a sufficiently strong upper bound on the pair correlations $\sum_n \theta(n+h_i) \theta(n+h_j)$ (see [@banks §6] for a recent implementation of this latter idea). However, our preliminary attempts to exploit these adjustments suggested that the gain from the former idea would be exponentially small in $k$, whereas the gain from the latter would also be very slight (perhaps reducing $k$ by $O(1)$ in large $k$ regimes, e.g. $k \geq 5000$). 7. All of our sieves used are essentially of Selberg type, being the square of a divisor sum. We have experimented with a number of non-Selberg type sieves (for instance trying to exploit the obvious positivity of $1 - \sum_{p \leq x: p|n} \frac{\log p}{\log x}$ when $n \leq x$), however none of these variants offered a numerical improvement over the Selberg sieve. Indeed it appears that after optimizing the cutoff function $F$, the Selberg sieve is in some sense a “local maximum” in the space of non-negative sieve functions, and one would need a radically different sieve to obtain numerically superior results. 8. Our numerical bounds for the diameter $H(k)$ of the narrowest admissible $k$-tuple are known to be exact for $k \leq 342$, but there is scope for some slight improvement for larger values of $k$, which would lead to some improvements in the bounds on $H_m$ for $m=2,3,4,5$. However, we believe that our bounds on $H_m$ are already fairly close (e.g. within $10\%$) of optimal, so there is only a limited amount of gain to be obtained solely from this component of the argument. Narrow admissible tuples {#tuples-sec} ======================== In this section we outline the methods used to obtain the numerical bounds on $H(k)$ given by Theorem \[hk-bound\], which are reproduced below: 1. $H(3) = 6$, 2. $H(50) = 246$, 3. $H(51) = 252$, 4. $H(54) = 270$, 5. $H(\num{5511}) \leq \num{52116}$, 6. $H(\num{35410}) \leq \num{398130}$, 7. $H(\num{41588}) \leq \num{474266}$, 8. $H(\num{309661}) \leq \num{4137854}$, 9. $H(\num{1649821}) \leq \num{24797814}$, 10. $H(\num{75845707}) \leq \num{1431556072}$, 11. $H(\num{3473955908}) \leq \num{80550202480}$. H(k) values for small k ----------------------- The equalities in the first four bounds (1)-(4) were previously known. The case $H(3)=6$ is obvious: the admissible 3-tuples $(0,2,6)$ and $(0,4,6)$ have diameter $6$ and no $3$-tuple of smaller diameter is admissible. The cases $H(50)=246$, $H(51)=252$, and $H(54)=270$ follow from results of Clark and Jarvis [@clark]. They define $\varrho^*(x)$ to be the largest integer $k$ for which there exists an admissible $k$-tuple that lies in a half-open interval $(y,y+x]$ of length $x$. For each integer $k>1$, the largest $x$ for which $\varrho^*(x)=k$ is precisely $H(k+1)$. Table 1 of [@clark] lists these largest $x$ values for $2\le k\le 170$, and we find that $H(50)=246$, $H(51)=252$, and $H(54)=270$. Admissible tuples that realize these bounds are shown in Figures \[k50tup\],  \[k51tup\] and \[k54tup\]. $$\begin{aligned} &0,4,6,16,30,34,36,46,48,58,60,64,70,78,84,88,90,94,100,106,\\ &108,114,118,126,130,136,144,148,150,156,160,168,174,178,184,\\ &190,196,198,204,210,214,216,220,226,228,234,238,240,244,246.\end{aligned}$$ $$\begin{aligned} &0,6,10,12,22,36,40,42,52,54,64,66,70,76,84,90,94,96,100,106,\\ &112,114,120,124,132,136,142,150,154,156,162,166,174,180,184,\\ &190,196,202,204,210,216,220,222,226,232,234,240,244,246,250,252.\end{aligned}$$ $$\begin{aligned} &0,4,10,18,24,28,30,40,54,58,60,70,72,82,84,88,94,102,108,112,114,\\ &118,124,130,132,138,142,150,154,160,168,172,174,180,184,192,198,202,\\ &208,214,220,222,228,234,238,240,244,250,252,258,262,264,268,270.\end{aligned}$$ H(k) bounds for mid-range k {#secmidk} --------------------------- As previously noted, exact values for $H(k)$ are known only for $k\le 342$. The upper bounds on $H(k)$ for the five cases (5)-(9) were obtained by constructing admissible $k$-tuples using techniques developed during the first part of the Polymath8 project. These are described in detail in Section 3 of [@polymath8a-unabridged], but for the sake of completeness we summarize the most relevant methods here. ### Fast admissibility testing A key component of all our constructions is the ability to efficiently determine whether a given $k$-tuple $\mathcal{H}=(h_1,\ldots, h_k)$ is admissible. We say that $\mathcal{H}$ is *admissible modulo* $p$ if its elements do not form a complete set of residues modulo $p$. Any $k$-tuple $\mathcal{H}$ is automatically admissible modulo all primes $p > k$, since a $k$-tuple cannot occupy more than $k$ residue classes; thus we only need to test admissibility modulo primes $p < k$. A simple way to test admissibility modulo $p$ is to enumerate the elements of $\mathcal{H}$ modulo $p$ and keep track of which residue classes have been encountered in a table with $p$ boolean-valued entries. Assuming the elements of $\mathcal{H}$ have absolute value bounded by $O(k\log k)$ (true of all the tuples we consider), this approach yields a total bit-complexity of $O(k^2/\log k\ \textsf{M}(\log k))$, where $\textsf{M}(n)$ denotes the complexity of multiplying two $n$-bit integers, which, up to a constant factor, also bounds the complexity of division with remainder. Applying the Schönhage-Strassen bound $\textsf{M}(n)=O(n\log n\log\log n)$ from [@schonhage], this is $O(k^2\log\log k\log\log\log k)$, essentially quadratic in $k$. This approach can be improved by observing that for most of the primes $p < k$ there are likely to be many unoccupied residue classes modulo $p$. In order to verify admissibility at $p$ it is enough to find one of them, and we typically do not need to check them all in order to do so. Using a heuristic model that assumes the elements of $\mathcal{H}$ are approximately equidistributed modulo $p$, one can determine a bound $m < p$ such that $k$ random elements of ${\mathbb{Z}}/p{\mathbb{Z}}$ are unlikely to occupy all of the residue classes in $[0,m]$. By representing the $k$-tuple $\mathcal{H}$ as a boolean vector $\mathcal{B}=(b_0,\ldots,b_{h_k-h_1})$ in which $b_i=1$ if and only if $i=h_j-h_1$ for some $h_j\in \mathcal{H}$, we can efficiently test whether $\mathcal{H}$ occupies every residue class in $[0,m]$ by examining the entries $$b_0,\ldots,b_m,b_p,\ldots,b_{p+m},b_{2p},\ldots,b_{2p+m},\ldots$$ of $\mathcal{B}$. The key point is that when $p<k$ is large, say $p > (1+\epsilon)k/\log k$, we can choose $m$ so that we only need to examine a small subset of the entries in $\mathcal{B}$. Indeed, for primes $p > k/c$ (for any constant $c$), we can take $m=O(1)$ and only need to examine $O(\log k)$ elements of $\mathcal{B}$ (assuming its total size is $O(k\log k)$, which applies to all the tuples we consider here). Of course it may happen that $\mathcal{H}$ occupies every residue class in $[0,m]$ modulo $p$. In this case we revert to our original approach of enumerating the elements of $\mathcal{H}$ modulo $p$, but we expect this to happen for only a small proportion of the primes $p < k$. Heuristically, this reduces the complexity of admissibility testing by a factor of $O(\log k)$, making it sub-quadratic. In practice we find this approach to be much more efficient than the straight-forward method when $k$ is large. See [@polymath8a §3.1] for further details. ### Sieving methods Our techniques for constructing admissible $k$-tuples all involve sieving an integer interval $[s,t]$ of residue classes modulo primes $p < k$ and then selecting an admissible $k$-tuple from the survivors. There are various approaches one can take, depending on the choice of interval and the residue classes to sieve. We list four of these below, starting with the classical sieve of Eratosthenes and proceeding to more modern variations. - **Sieve of Eratosthenes**. We sieve an interval $[2,x]$ to obtain admissible $k$-tuples $$p_{m+1},\ldots,p_{m+k}.$$ with $m$ as small as possible. If we sieve the residue class $0(p)$ for all primes $p\le k$ we have $m=\pi(k)$ and $p_{m+1}>k$. In this case no admissibility testing is required, since the residue class $0(p)$ is unoccupied for all $p \le k$. Applying the Prime Number Theorem in the forms $$\begin{aligned} p_k &=k \log k+ k \log \log k - k +O\Bigl(k \frac{ \log \log k}{\log k} \Bigr),\\ \pi(x)&=\frac{x}{\log x}+O\Bigl(\frac{x}{\log^2 x}\Bigr),\end{aligned}$$ this construction yields the upper bound $$\label{hk-eratosthenes} H(k)\le k\log k + k\log\log k - k + o(k).$$ As an optimization, rather than sieving modulo every prime $p \le k$ we instead sieve modulo increasing primes $p$ and stop as soon as the first $k$ survivors form an admissible tuple. This will typically happen for some $p_m < k$. - **Hensley-Richards sieve**. The bound in was improved by Hensley and Richards [@hensley; @hensley-2; @richards], who observed that rather than sieving $[2,x]$ it is better to sieve the interval $[-x/2,x/2]$ to obtain admissible $k$-tuples of the form $$-p_{m+\lfloor k/2\rfloor-1},\ldots,p_{m+1},\ldots,-1,1,\ldots,p_{m+1},\ldots,p_{m+\lfloor(k+1)/2\rfloor-1},$$ where we again wish to make $m$ as small as possible. It follows from Lemma 5 of [@hensley-2] that one can take $m=o(k/\log k)$, leading to the improved upper bound $$\label{hk-hensely-richards} H(k)\le k\log k + k\log\log k -(1+\log 2)k + o(k).$$ - **Shifted Schinzel sieve**. As noted by Schinzel in [@schinzel], in the Hensley-Richards sieve it is slightly better to sieve $1(2)$ rather than $0(2)$; this leaves unsieved powers of $2$ near the center of the interval $[-x/2,x/2]$ that would otherwise be removed (more generally, one can sieve $1(p)$ for many small primes $p$, but we did not). Additionally, we find that shifting the interval $[-x/2,x/2]$ can yield significant improvements (one can also view this as changing the choices of residue classes). This leads to the following approach: we sieve an interval $[s,s+x]$ of odd integers and multiples of odd primes $p\le p_m$, where $x$ is large enough to ensure at least $k$ survivors, and $m$ is large enough to ensure that the survivors form an admissible tuple, with $x$ and $m$ minimal subject to these constraints. A tuple of exactly $k$ survivors is then chosen to minimize the diameter. By varying $s$ and comparing the results, we can choose a starting point $s\in [-x/2,x/2]$ that yields the smallest final diameter. For large $k$ we typically find $s\approx k$ is optimal, as opposed to $s\approx -(k/2)\log k$ in the Hensley-Richards sieve. - **Shifted greedy sieve**. As a further optimization, we can allow greater freedom in the choice of residue class to sieve. We begin as in the shifted Schinzel sieve, but for primes $p \le p_m$ that exceed $2\sqrt{k\log k}$, rather than sieving $0(p)$ we choose a minimally occupied residue class $a(p)$. As above we sieve the interval $[s,s+x]$ for varying values of $s\in [-x/2,x/2]$ and select the best result, but unlike the shifted Schinzel sieve, for large $k$ we typically choose $s\approx -(k/\log k - k)/2$. We remark that while one might suppose that it would be better to choose a minimally occupied residue class at all primes, not just the larger ones, we find that this is generally not the case. Fixing a structured choice of residue classes for the small primes avoids the erratic behavior that can result from making greedy choices to soon (see [@gordon Fig. 1] for an illustration of this). Table \[kmidtable\] lists the bounds obtained by applying each of these techniques (in the online version of this paper, each table entry includes a link to the constructed tuple). To the admissible tuples obtained using the shifted greedy sieve we additionally applied various local optimizations that are detailed in [@polymath8a §3.6]. As can be seen in the table, the additional improvement due to these local optimizations is quite small compared to that gained by using better sieving algorithms, especially when $k$ is large. Table \[kmidtable\] also lists the value $\lfloor k\log k+k\rfloor$ that we conjecture as an upper bound on $H(k)$ for all sufficiently large $k$. $k$ ------------------------------- ------------------------------------------------------------------------- --------------------------------------------------------------------------- --------------------------------------------------------------------------- ----------------------------------------------------------------------------- ------------------------------------------------------------------------------- $k$ primes past $k$ [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_56538.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_433992.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_516586.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4505700.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_26916060.txt)]{} Eratosthenes [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_55160.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_424636.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_505734.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4430212.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_26540720.txt)]{} Hensley-Richards [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_54480.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_415642.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_494866.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4312612.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_25841884.txt)]{} Shifted Schinzel [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_53774.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_411060.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_489056.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4261858.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_25541910.txt)]{} Shifted greedy [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_52296.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_399936.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_476028.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4142780.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_24798306.txt)]{} Best known [[](http://math.mit.edu/~primegaps/tuples/admissible_5511_52116.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_35410_398130.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_41588_474266.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_309661_4137854.txt)]{} [[](http://math.mit.edu/~primegaps/tuples/admissible_1649821_24797814.txt)]{} $\lfloor k\log k + k \rfloor$ : Upper bounds on $H(k)$ for selected values of $k$.[]{data-label="kmidtable"} H(k) bounds for large k ----------------------- . The upper bounds on $H(k)$ for the last two cases (10) and (11) were obtained using modified versions of the techniques described above that are better suited to handling very large values of $k$. These entail three types of optimizations that are summarized in the subsections below. ### Improved time complexity As noted above, the complexity of admissibility testing is quasi-quadratic in $k$. Each of the techniques listed in §\[secmidk\] involves optimizing over a parameter space whose size is at least quasi-linear in $k$, leading to an overall quasi-cubic time complexity for constructing a narrow admissible $k$-tuple; this makes it impractical to handle $k > 10^9$. We can reduce this complexity in a number of ways. First, we can combine parameter optimization and admissibility testing. In both the sieve of Eratosthenes and Hensley-Richards sieves, taking $m=k$ guarantees an admissible $k$-tuple. For $m<k$, if the corresponding $k$-tuple is inadmissible, it is typically because it is inadmissible modulo the smallest prime $p_{m+1}$ that appears in the tuple. This suggests a heuristic approach in which we start with $m=k$, and then iteratively reduce $m$, testing the admissibility of each $k$-tuple modulo $p_{m+1}$ as we go, until we can proceed no further. We then verify that the last $k$-tuple that was admissible modulo $p_{m+1}$ is also admissible modulo all primes $p > p_{m+1}$ (we know it is admissible at all primes $p \le p_m$ because we have sieved a residue class for each of these primes). We expect this to be the case, but if not we can increase $m$ as required. Heuristically this yields a quasi-quadratic running time, and in practice it takes less time to find the minimal $m$ than it does to verify the admissibility of the resulting $k$-tuple. Second, we can avoid a complete search of the parameter space. In the case of the shifted Schinzel sieve, for example, we find empirically that taking $s=k$ typically yields an admissible $k$-tuple whose diameter is not much larger than that achieved by an optimal choice of $s$; we can then simply focus on optimizing $m$ using the strategy described above. Similar comments apply to the shifted greedy sieve. ### Improved space complexity We expect a narrow admissible $k$-tuple to have diameter $d=(1+o(1))k\log k$. Whether we encode this tuple as a sequence of $k$ integers, or as a bitmap of $d+1$ bits, as in the fast admissibility testing algorithm, we will need approximately $k\log k$ bits. For $k > 10^9$ this may be too large to conveniently fit in memory. We can reduce the space to $O(k\log\log k)$ bits by encoding the $k$-tuple as a sequence of $k-1$ gaps; the average gap between consecutive entries has size $\log k$ and can be encoded in $O(\log\log k)$ bits. In practical terms, for the sequences we constructed almost all gaps can be encoded using a single 8-bit byte for each gap. One can further reduce space by partitioning the sieving interval into windows. For the construction of our largest tuples, we used windows of size $O(\sqrt{d})$ and converted to a gap-sequence representation only after sieving at all primes up to an $O(\sqrt{d})$ bound. ### Parallelization With the exception of the greedy sieve, all the techniques described above are easily parallelized. The greedy sieve is more difficult to parallelize because the choice of a minimally occupied residue class modulo $p$ depends on the set of survivors obtained after sieving modulo primes less than $p$. To address this issue we modified the greedy approach to work with batches of consecutive primes of size $n$, where $n$ is a multiple of the number of parallel threads of execution. After sieving fixed residue classes modulo all small primes $p < 2\sqrt{k\log k}$, we determine minimally occupied residue classes for the next $n$ primes in parallel, sieve these residue classes, and then proceed to the next batch of $n$ primes. In addition to the techniques described above, we also considered a modified Schinzel sieve in which we check admissibility modulo each successive prime $p$ before sieving multiples of $p$, in order to verify that sieving modulo $p$ is actually necessary. For values of $p$ close to but slightly less than $p_m$ it will often be the case that the set of survivors is already admissibile modulo $p$, even though it does contain multiples of $p$ (because some other residue class is unoccupied). As with the greedy sieve, when using this approach we sieve residue classes in batches of size $n$ to facilitate parallelization. ### Results for large k Table \[klargetable\] lists the bounds obtained for the two largest values of $k$. For $k=\num{75845707}$ the best results were obtained with a shifted greedy sieve that was modified for parallel execution as described above, using the fixed shift parameter $s=-(k\log k-k)/2$. A list of the sieved residue classes is available at [math.mit.edu/\~drew/greedy\_75845707\_1431556072.txt](math.mit.edu/~drew/greedy_75845707_1431556072.txt). This file contains values of $k$, $s$, $d$, and $m$, along with a list of prime indices $n_i> m$ and residue classes $r_i$ such that sieving the interval $[s,s+d]$ of odd integers, multiples of $p_n$ for $1<n \le m$, and at $r_i$ modulo $p_{n_i}$ yields an admissible $k$-tuple. For $k=\num{3473955908}$ we did not attempt any form of greedy sieving due to practical limits on the time and computational resources available. The best results were obtained using a modified Schinzel sieve that avoids unnecessary sieving, as described above, using the fixed shift parameter $s=k0$. A list of the sieved residue classes is available at [math.mit.edu/\~drew/schinzel\_3473955908\_80550202480.txt](math.mit.edu/~drew/schinzel_3473955908_80550202480.txt). This file contains values of $k$, $s$, $d$, and $m$, along with a list of prime indices $n_i> m$ such that sieving the interval $[s,s+d]$ of odd integers, multiples of $p_n$ for $1<n \le m$, and multiples of $p_{n_i}$ yields an admissible $k$-tuple. Source code for our implementation is available at [math.mit.edu/\~drew/ompadm\_v0.5.tar](math.mit.edu/~drew/ompadm_v0.5.tar); this code can be used to verify the admissibility of both the tuples listed above. $k$ ------------------------------- -- --------------- $k$ primes past $k$ Eratosthenes Hensley-Richards Shifted Schinzel Shifted Greedy not available Best known $\lfloor k\log k + k \rfloor$ : Upper bounds on $H(k)$ for selected values of $k$.[]{data-label="klargetable"} Acknowledgements {#acknowledgements .unnumbered} ================ This paper is part of the *Polymath project*, which was launched by Timothy Gowers in February 2009 as an experiment to see if research mathematics could be conducted by a massive online collaboration. The current project (which was administered by Terence Tao) is the eighth project in this series, and this is the second paper arising from that project, after [@polymath8a]. 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[^1]: When $a,b$ are real numbers, we will also need to use $(a,b)$ and $[a,b]$ to denote the open and closed intervals respectively with endpoints $a,b$. Unfortunately, this notation conflicts with the notation given above, but it should be clear from the context which notation is in use. [^2]: Actually, there are some differences between [@bfi Conjecture 1] and the claim here. Firstly, we need an estimate that is uniform for all $a$, whereas in [@bfi] only the case of a fixed modulus $a$ was asserted. On the other hand, $\alpha,\beta$ were assumed to be controlled in $\ell^2$ instead of via the pointwise bounds , and $Q$ was allowed to be as large as $x \log^{-C} x$ for some fixed $C$ (although, in view of the negative results in [@fg], [@fghm], this latter strengthening may be too ambitious). [^3]: One could also use the Heath-Brown identity [@hb-ident] here if desired. [^4]: In the $k=1$ case, we of course just have $q_{W,d_1,\dots,d'_{k-1}} = W$. [^5]: One could obtain a small improvement to the bounds here by replacing the threshold $2c$ with a parameter to be optimized over. [^6]: The arguments in [@maynard-new] are rigorous under the assumption of a positive eigenfunction as in Corollary \[ef\], but the existence of such an eigenfunction remains open for $k \geq 3$. [^7]: Indeed, one might be even more ambitious and conjecture a square-root cancellation ${\llcurly}\sqrt{x/q}$ for such sums (see [@mont] for some similar conjectures), although such stronger cancellations generally do not play an essential role in sieve-theoretic computations. [^8]: One new technical difficulty here is that some of the various moduli $[d_j,d'_j]$ arising in these arguments are not required to be coprime at primes $p > w$ dividing $x$ or $x-2$; this requires some modification to Lemma \[mul-asym\] that ultimately leads to the appearance of the singular series ${\mathfrak S}$. However, these modifications are quite standard, and we do not give the details here. [^9]: In particular, the optimal choice $F$ for $M_{k,{\varepsilon}}$ should vanish on the polytope $\{ (t_1,\dots,t_k) \in (1+{\varepsilon}) \cdot {\mathcal R}_k: \sum_{i \neq i_0} t_i \ge 1-{\varepsilon}\hbox{ for all } i_0=1,\dots,k\}$.

--- abstract: 'We use first-principles density functional theory total energy and linear response phonon calculations to compute the Helmholtz and Gibbs free energy as a function of temperature, pressure, and cell volume in the flexible metal-organic framework material MIL-53(Cr) within the quasiharmonic approximation. GGA and metaGGA calculations were performed, each including empirical van der Waals (vdW) forces under the D2, D3, or D3(BJ) parameterizations. At all temperatures up to 500 K and pressures from -30 MPa to 30 MPa, two minima in the free energy versus volume are found, corresponding to the narrow pore ($np$) and large pore ($lp$) structures. Critical positive and negative pressures are identified, beyond which there is only one free energy minimum. While all results overestimated the stability of the $np$ phase relative to the $lp$ phase, the best overall agreement with experiment is found for the metaGGA PBEsol+RTPSS+U+J approach with D3 or D3(BJ) vdW forces. For these parameterizations, the calculated free energy barrier for the $np$-$lp$ transition is only 3 to 6 kJ per mole of Cr$_4$(OH)$_4$(C$_8$H$_4$O$_4$)$_4$.' author: - Eric Cockayne title: 'Thermodynamics of the Flexible Metal-Organic Framework Material MIL-53(Cr) From First Principles' --- \#1[[*\#1*]{}]{} \#1[[Eq. (\[eq:\#1\])]{}]{} \#1[[Fig. \[fig:\#1\]]{}]{} \#1[[Sec. \[sec:\#1\]]{}]{} \#1[[Ref. ]{}]{} \#1[[Table \[tab:\#1\]]{}]{} Microporous flexible metal-organic framework materials are fascinating both from a fundamental point of view and for their numerous potential applications such as gas storage, gas separation, sensors, drug delivery, etc.[@Ferey09; @Alhamami14; @Schneemann14; @Coudert15; @Ferey16] A well-studied example is the MIL-53 family,[@Serre02] with formula M(OH)(C$_8$H$_4$O$_4)$, where is M is a trivalent species such as Cr, Sc, Al, Ga or Fe. These structures consist of zigzag M-OH-M-OH$\dots$ chains, crosslinked by 1,4-benzodicarboxylate O$_2$C-C$_6$H$_4$-CO$_2$ (bdc) units (). Each M is coordinated by two oxygens of OH units and four carboxylate oxygens yielding octahedral oxygen coordination. ![Structure of MIL-53(Cr). Cr atoms green, O red, C gray, and H yellow. (a) bdc linkers joining zigzag Cr-OH-Cr-$\dots$ chains. (b) Each zigzag chain is coordinated with four neighboring chains; each Cr is octahedrally coordinated with six O. (c) Narrow pore ($np$) phase showing bdc rotations. (d) Large pore ($lp$) phase. In (c) and (d), the H are not shown.[]{data-label="fig:mil53x"}](mil53x.pdf){width="85mm"} These MIL-53 compounds exhibit a variety of topologically equivalent structures with different volumes, but generally include a narrow pore ($np$) structure and a large pore ($lp$) structure, both with formula M$_4$(OH)$_4$(bdc)$_4$ per conventional unit cell, but with significantly different volumes. In MIL-53(Al), the phase transition between $np$ and $lp$ forms can be reversibly achieved by cycling the temperature;[@Liu08] the cell parameter corresponding to the direction of the short axis of the lozenge pores was found to increase by 87 % in the $np$-$lp$ transformation. By way of comparison, the strain variations achieved or predicted in functional “hard" materials such as (PbMg$_{1/3}$Nb$_{2/3}$O$_3$)$_{(1-x)}$-(PbTiO$_3$)$_{x}$[@Park97] or BiFeO$_3$[@Dieguez11] are much smaller. The large hysteresis[@Liu08] in the $np$-$lp$ phase transition of MIL-53(Al) indicates that the transition is first-order. Taking the transition temperature as the midrange of the hysteresis loop, the transition temperature $T_c$ is approximately 260 K; an estimate based on experimental sorption measurements places the transition at a somewhat lower temperature of 203 K.[@Boutin10] For empty MIL-53(Cr), the $lp$ structure is thermodynamically preferred at all temperatures. In this system, a phase transition to a $np$ structure has instead been observed in the case of (1) sorption of a variety of sorbates; (2) pressure. The hysteresis of the process in each case[@Serre07] indicates again that there is a transition barrier. By fitting sorption isotherms, it was determined that the free energy difference between the $lp$ and $np$ forms of MIL-53(Cr) was only about 12 kJ mol$^{-1}$ of Cr$_4$(OH)$_4$(bdc)$_4$.[@Coudert08; @DevatourVinot09; @Coombes09] An experiment that put the system under hydrostatic pressure[@Beurroies10] came up with a similar free energy difference. The phase transition of MIL-53(Al) was explained by Walker et al.[@Walker10] in 2010. Van der Waals interactions stabilize the $np$ structure at low temperature, and vibrational entropy drives the structural transition to the $lp$ phase above $T_c$. Density functional theory (DFT) phonon calculations were used to quantify the vibrational entropy. In that work, however, the DFT energy and vibrational entropy were determined for only the $np$ and $lp$ structures. However, to build an accurate picture of the $np$-$lp$ phase transition, including the hysteresis and possible coexistence of $np$ and $lp$ phases,[@Triguero12] it is necessary to know the quantitative free energy landscape over the [*full*]{} volume range spanning the $np$ and $lp$ structures. This free-energy landscape of MIL-53 systems has previously been modeled in an [*ad hoc*]{} manner.[@Triguero11; @Ghysels13] This paper uses density functional total energy and phonon linear response calculations to compute the Helmholtz and Gibbs free energy in MIL-53(Cr) as a function of temperature, pressure, and cell volume, under the quasiharmonic approximation. MIL-53(Cr) was chosen because of its relatively simple phase transformation behavior and because it is well-characterized experimentally. The thermodynamic calculations are performed within the quasiharmonic approximation. In the quasiharmonic approximation, the anharmonic lattice dynamics that leads to thermal expansion, etc., is approximated by harmonic lattice dynamics where the phonon frequencies are volume-dependent. Suppose that one has a crystal where the rank-ordered frequencies ${\nu_{\mu} (V)}$ can be determined for an arbitrarily large supercell (equivalently at arbitrary points in the Brillouin zone of the primitive cell). The contribution of phonons to the thermodynamics is then given well-known expressions.[@Maradudin71; @vandeWalle02; @Fultz10; @Huang16] Defining a dimensionless parameter $x_{\mu}(V,T) = \frac{h \nu_{\mu}(V)}{k_B T}$, the molar internal energy as a function of volume and temperature is given by $$\begin{aligned} \frac{U}{N}(V,T) = {\rm Lim}_{|a_{\rm min}|\rightarrow \infty} \frac{1}{N} \bigl(U_0(V) + \nonumber \\ k_B T \sum_{\mu = 4}^{3 N_A} [\frac{x_{\mu}(V,T)}{2} {\rm coth}(\frac {x_{\mu}(V,T)}{2})]\bigr), \label{eq:inten}\end{aligned}$$ the Helmholtz free energy by $$\begin{aligned} \frac{F}{N}(V,T) = {\rm Lim}_{|a_{\rm min}|\rightarrow \infty} \frac{1}{N} \bigl(U_0(V) + \nonumber \\ k_B T \sum_{\mu = 4}^{3 N_A} [\frac{x_{\mu}(V,T)}{2} + {\rm ln} (1 - e^{-x_{\mu}(V,T)})]\bigr), \label{eq:helm}\end{aligned}$$ and the Gibbs free energy is given by $\frac{G}{N}(V,T) = \frac{F}{N}(V,T) + P V$. $U_0(V)$ is the ground state energy neglecting zero-point vibrations, $N$ the number of moles and $N_A$ the number of atoms in the supercell, and the summation begins at $\mu = 4$ to avoid the weak singularity due to the zero-frequency translational modes. First principles density functional theory calculations, as encoded in the [VASP]{} software (), were used to compute $U_0(V)$ and ${\nu_{\mu} (V)}$ for a 152-atom supercell of MIL-53(Cr), doubled along $c$ so as to make $a$, $b$, and $c$ similar in magnitude for the $lp$ phase. Two different sets of calculations were performed: GGA calculations using the PBEsol functional[@Perdew08] and meta-GGA calculations using the PBEsol+RTPSS[@Sun11] functionals. These functionals were chosen because we have had success with them in past studies of microporous materials.[@Cockayne12; @Cockayne15] For each level of DFT, the nonlocal van der Waals interactions were treated using three different approximations of Grimme et al.: DFT-D2,[@Grimme06] DFT-D3,[@Grimme10] and DFT-D3(BJ).[@Grimme11] Anisotropic Hubbard parameters[@Liech95] were used for Cr and O atoms (GGA: U(Cr) = 4.0 eV, J(Cr) = 0.5 eV; metaGGA: U(Cr) = 2.8 eV, J(Cr) = 0.5 eV; U(O) = 7.05 eV). Spin polarized calculations were performed using the most-stable antiferromagnetic arrangement of charges on the Cr$^{3+}$ ions. Further details of the DFT calculations are given in the Supplementary Information (SI). Determination of $U_0(V)$ for each functional was done via straightforward fixed-volume relaxation for (primitive cell) increasing in 50 Å$^3$ steps from 650 Å$^3$ to 1700 Å$^3$. The phonon frequencies for the 152-atom supercell were calculated using ab initio linear response. As this method converges toward exact second derivatives of the energy, it is more accurate than fitting frozen-phonon results. Due to the large number of degrees of freedom, the phonon calculations are very expensive, and eventually only three calculations were used for the thermodynamics: V = 710 Å$^3$, V = 1200 Å$^3$, and V = 1506 Å$^3$. Linear response was only done using GGA and DFT-D2; the same phonon frequencies $\nu_{\mu}(V)$ were used for each functional in ; only the $U_0$ changed. Because the variation in volume between the $np$ and $lp$ phases is so large, one does not expect the conventional linear Grüneisen approximation for $\nu_{\mu} (V)$ to apply. Instead, we fit the phonon frequencies at intermediate volumes by fitting to the following physically-motivated expression: $$\nu_{\mu}^2 (V) = \nu_{\mu \infty}^2 + C_1/V + C_2/V^2 . \label{eq:phofit}$$ The coefficients in  were determined by fitting the results for the three frequencies calculated. If $\nu_{\mu \infty}^2$ in the fit was less than zero, it was set to zero and the fit recalculated. Due to computational limitations, it is not possible to calculate larger supercells for use in . Instead, the contribution of optical phonons to the thermodynamics was approximated by the phonon spectra calculated for the single 152-atom supercell. The contribution of acoustic phonons to the thermodynamics was approximated by numerical integration of estimated acoustic frequencies over the first Brillouin zone. Further details are given in the Supplementary Information. First, the phonons were calculated for the $np$ and $lp$ structures. All modes were stable for the $np$ structure. For the $lp$ structure, instabilities were found. The most unstable modes, for both the force-constant and dynamical matrices, were hydrogen “flopping" modes in which the H in each hydroxyl group move in the $\pm x$ direction so as to decrease the distance to a pair of carboxylate oxygens (). Fully relaxing this mode maintains orthorhombic symmetry, the 152-atom cell is now a primitive cell. ![Local geometry of MIL-53(Cr) (lp) after DFT relaxation of “H flopping" mode. Each H relaxes to sit approximately 2.4 Å from each of a pair of oxygens (dashed lines); the O are superposed from this vantage point.[]{data-label="fig:flop"}](flop.pdf){width="42mm"} The structure obtained upon relaxation of the flopping instability was taken as the reference $lp$ structure. To obtain the initial structure for the fixed volume relaxations used to determine $U_0(V)$, the ionic coordinates were interpolated (or extrapolated) from the initial $np$ and $lp$ structures. ![Calculated DFT energy for MIL-53(Cr) at 0 K as a function of volume for different density functionals, neglecting zero-point motion. Each curve is scaled so that its minimum is zero.[]{data-label="fig:uo"}](uo.pdf){width="85mm"} The $U_0(V)$ determined for the various density functionals are shown in . The $F(V)$ for T = 293 K are shown in . For every plot in , there are two minima in the free energy, corresponding to $lp$ and $np$ structures. The effect of phonon entropy is to reduce the free energy of the $lp$ structure with respect to the $np$ structure, as expected. Calculations show that the free energies for temperatures up to 500 K and pressures between -30 MPa and 30 MPa maintain two minima for all density functionals tested. ![Calculated Helmholtz free energy for MIL-53(Cr) at 293 K as a function of volume for different density functionals. Each curve is scaled so that its minimum is zero. The effect of atmospheric pressure of about 0.1 MPa is negligible on this scale.[]{data-label="fig:helm"}](helm.pdf){width="85mm"} summarizes and compares the results for the different functionals used. The volumes at which the minima for $U_0$ occur are given by $V_{0np}$ and $V_{0lp}$. The locations of the minima in $F$ at room temperature (RT; 293K) are given by $V_{np}(RT)$ and $V_{lp} (RT)$. The calculated difference in $F$ between the $np$ and $lp$ minima is $\Delta{F}(RT) = F_{lp}(RT) - F_{np}(RT)$. The critical pressure $P_c$ is where the calculated Gibbs free energy of the $np$ and $lp$ phases becomes equal at T = 293 K. $G_{b}(RT;P_c)$ is the calculated free energy barrier between the phases at this pressure. Substantial differences are seen depending on what density functional is used. The general trend is for the GGA functionals and the D2 vdW term to give lower $V_{np}$ and higher $\Delta{F}$ than the metaGGA functionals and D3 or D3(BJ) choices for the vdW interaction. Which functional gives the best agreement with experiment? The experimental unit cell volume of the $lp$ phase of MIL-53(Cr) is 1486 Å$^3$.(Ref. ) The volume of the $np$ phase formed upon sorption of H$_2$O is 1012  Å$^3$,(Ref. ) but this cannot be directly compared with the calculation for the empty cell reported here. As the $np$ phase of MIL-53(Cr) is thermodynamically unstable experimentally, we take the experimental volume[@Liu08; @Nanthamathee15] of MIL-53(Al) $np$, 864 Å$^3$, and estimate that the volume of MIL-53(Cr) should be about 900 Å$^3$ due to the larger ionic radius of Cr$^{3+}$. The best agreement with experiment for the lattice parameters is for the metaGGA-D3(BJ) parameterization, while the second best is for metaGGA-D3. On the other hand, the relative stability of the $lp$ phase found experimentally, $\Delta{F} \approx$ -12.0 kJ mol$^{-1}$ is underestimated by [*all*]{} the functionals chosen. The metaGGA-D3 calculation is best in this regard, as it is the only calculation to yield a negative $\Delta{F}$. All of the metaGGA calculations perform better than GGA in predicting the relative phase stability. As the metaGGA-D3 and metaGGA-D3(BJ) have the best agreement with experiment, their low values of the transition barrier $G_b$, 3.2 to 6.0 kJ mol$^{-1}$ should be considered most reliable. It is interesting to put the comparative results in context of previous studies. In MIL-53, it has previously been found that the D2 vdW overbinds the $np$ phase;[@Haigis14] this work confirms that result. Benchmarking the performance of DFT calculations is currently receiving a great deal of attention[@Kirklin15; @Lejaeghere16; @Tran16]. In Ref. , over sixty different density functionals are compared. Although the RTPSS functional is not tested, the related metaGGA functional TPSS-D3 gives good results for graphite, which suggests that these parameterizations may work well for MIL-53, where the $np$ phase has benzyl rings of carbon approaching each other. Further work is needed to make a full comparison among methods because the current work: (1) includes Hubbard U and J parameters; (2) needs a vdW functional that reproduces the vdW interactions correctly over a wide range of structural distortion, not merely at one equilibrium point. The metaGGA-D3 calculation predicts that the $lp$ phase of MIL-53(Cr) is stable at room temperature, in agreement with experiment. Interestingly, it predicts a transition to the $np$ phase below T = 160 K, similar to what actually occurs for MIL-53(Al). The estimated change in $\Delta{F}$ with temperature is about -0.036 kJ mol$^{-1}$ K$^{-1}$. Applying this to the experimental $\Delta{F} \approx$ -12.0 kJ mol$^{-1}$, the $lp$ phase is expected to remain stable down to T = 0 K, albeit with a free energy advantage of less than 2 kJ mol$^{-1}$. ![Calculated Gibbs free energy for MIL-53(Cr) at 293 K as a function of volume and pressure for the metaGGA-D3 density functional. Each curve is scaled so that its minimum is zero.[]{data-label="fig:gibb"}](gibb.pdf){width="85mm"} The shallowness of the free energy profile suggests that sufficiently large positive or negative pressure would drive the Gibbs free energy G(V, T = 293 K) into a regime where it has only one minimum corresponding to either a $np$ or a $lp$ structure. In , we show G(V, T = 293 K) for various pressures -80 MPa to 80 MPa, using the metaGGA-D3 results. At pressures above about 60 MPa, there is a unique minimum at the $np$ phase; below about -40 MPa, there is one minimum at the $lp$ phase. If the zero in pressure is shifted to correct for the error in the metaGGA-D3 $\Delta{F}$ with respect to experiment, the predicted pressures are shifted to about 80 MPa and -20 MPa, respectively. Of course the prediction of the pressures at which the free energy converts to a single minimum only sets an upper bound on the width of the pressure hysteresis loop; in practice, fluctuations will cause the transitions to occur at less extreme pressures. With this is mind, experimental transition pressures for the hysteresis loop of roughly 50 MPa and 20 MPa for MIL-53(Cr)[@Rodriguez16] are consistent with the DFT results. Note that negative pressures do have physical relevance in microporous materials in the case of sorption- the effective solvation pressure can be either positive or negative depending on the sorbate concentration.[@Ravikovitch06] ![Calculated MIL-53(Cr) lattice parameters and cell angle $\beta$ versus volume.[]{data-label="fig:cell"}](cell.pdf){width="85mm"} In , the crystallographic data for the DFT metaGGA-D2 structural relaxations are shown. The lattice parameters are scaled to the volume of the conventional unit cells. To make the orthorhombic-monoclinic transition clear, the monoclinic cell parameters $a$ and $\beta$ are for an unconventional body-center monoclinic setting. The orthorhombic-monoclinic transition occurs at $V \sim 1500$ Å$^3$, intriguingly close to the experimental cell volume. In addition to the structural transitions, there are three regimes in the behavior of the lattice constants: (1) below about 850 Å$^3$, $a$ $b$ and $c$ all increase with volume; (2) between about 850 Å$^3$ and 1650 Å$^3$, $a$ decreases with volume $b$ increases with volume, and $c$ is nearly flat as the structure flexes; (3) above about 1650 Å$^3$, all lattice parameters increase again. The crossover between regimes (2) and (3) does not occur at the same volume as the monoclinic-orthorhombic transition. To a first approximation, the free energy is nearly flat in regime (2) and increases rapidly above and below this range. The three regimes agree qualitatively with those seen in a recent experiment on the related material MIL-53(Al) under pressure.[@SerraCrespo15] To summarize, we used density functional theory total energy and linear response phonon calculations to compute the free energy profile of MIL-53(Cr) under the quasiharmonic approximation. The density functionals that best match the experimental results give remarkably flat free energy profiles, with a transition barrier of only about a 3 to 6 kJ mol$^{-1}$ between the the narrow pore and large pore phases. I thank Laura Espinal, Kevin F. Garrity, and Winnie Wong-Ng for helpful discussions. This paper was published as J. Phys. Chem. C 2017, [**121**]{}, 4312-4317 (DOI:10.1021/acs.jpcc.6b11692). The Supporting Information is available free of charge on the ACS Publications website at the DOI given immediately above. 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--- abstract: 'We present high performance implementations of the QR and the singular value decomposition of a batch of small matrices hosted on the GPU with applications in the compression of hierarchical matrices. The one-sided Jacobi algorithm is used for its simplicity and inherent parallelism as a building block for the SVD of low rank blocks using randomized methods. We implement multiple kernels based on the level of the GPU memory hierarchy in which the matrices can reside and show substantial speedups against streamed cuSOLVER SVDs. The resulting batched routine is a key component of hierarchical matrix compression, opening up opportunities to perform H-matrix arithmetic efficiently on GPUs.' address: - '$^1$Extreme Computing Research Center (ECRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.' - '$^2$Department of Computer Science, American University of Beirut (AUB), Beirut, Lebanon.' author: - Wajih Halim Boukaram$^1$ - George Turkiyyah$^2$ - Hatem Ltaief$^1$ - 'David E. Keyes$^1$' bibliography: - 'arxiv\_batch\_svd.bib' title: Batched QR and SVD Algorithms on GPUs with Applications in Hierarchical Matrix Compression --- Introduction {#sec:intro} ============ The singular value decomposition (SVD) is a factorization of a general $m \times n$ matrix $A$ of the form $$A = U \Sigma V^*.$$ $U$ is an $m \times m$ orthonormal matrix whose columns $U_i$ are called the left singular vectors. $\Sigma$ is an $m \times n$ diagonal matrix whose diagonal entries $\sigma_i$ are called the singular values and are sorted in decreasing order. $V$ is an $n \times n$ orthonormal matrix whose columns $V_i$ are called the right singular vectors. When $m > n$, we can compute a reduced form $A = \hat{U} \hat{\Sigma} V^*$ where $\hat{U}$ is an $m \times n$ matrix and $\hat{\Sigma}$ is an $n \times n$ diagonal matrix. One can easily obtain the full form from the reduced one by extending $\hat{U}$ with $(m - n)$ orthogonal vectors and $\hat{\Sigma}$ with an $(m - n)$ zero block row. Without any loss of generality, we will focus on the reduced SVD of real matrices in our discussions. The SVD of a matrix is a crucial component in many applications in signal processing and statistics as well as matrix compression, where truncating the $(n - k)$ singular values that are smaller than some threshold gives us a rank-$k$ approximation $\tilde{A}$ of the matrix $A$. This matrix is the unique minimizer of the function $f_k(B) = || A - B ||_F$. In the context of hierarchical matrix operations, effective compression relies on the ability to perform the computation of large batches of independent SVDs of small matrices of low numerical rank. Randomized methods [@halko2011finding] are well suited for computing a truncated SVD of these types of matrices and are built on three computational kernels: the QR factorization, matrix-matrix multiplications and SVDs of smaller $k \times k$ matrices. Motivated by this task, we discuss the implementation of high performance batched QR and SVD kernels on the GPU, focusing on the more challenging SVD tasks. The remainder of this paper is organized as follows. Section \[sec:background\] presents different algorithms used to compute the QR factorization and the SVD as well as some considerations when optimizing for GPUs. Section \[sec:batch\_qr\] discusses the batched QR factorization and compares its performance with existing libraries. Sections \[sec:registers\], \[sec:shared\] and \[sec:block\_global\] discuss the various implementations of the SVD based on the level of the memory hierarchy in which the matrices can reside. Specifically, Section \[sec:registers\] describes the implementation for very small matrix sizes that can fit in registers, Section \[sec:shared\] describes the implementation for matrices that can reside in shared memory, and Section \[sec:block\_global\] describes the block Jacobi implementation for larger matrix sizes that must reside in global memory. Section \[sec:randomized\] details the implementation of the batched randomized SVD routine. We then discuss some details of the application to hierarchical matrix compression in Section \[sec:application\]. We conclude and discuss future work in Section \[sec:conclusion\]. Background {#sec:background} ========== In this section we give a review of the most common algorithms used to compute the QR factorization and the SVD of a matrix as well as discuss some considerations when optimizing on the GPU. QR Factorization ---------------- The QR factorization decomposes an $m \times n$ matrix $A$ into the product of an orthogonal $m \times m$ matrix $Q$ and an upper triangular $m \times n$ matrix $R$ [@golub2013matrix]. We can also compute a reduced form of the decomposition where Q is an $m \times n$ matrix and R is $n \times n$ upper triangular. The most common QR algorithm is based on transforming $A$ into an upper triangular matrix using a series of orthogonal transformations generated using Householder reflectors. Other algorithms such as the Gram-Schmidt or Modified Gram-Schmidt can produce the QR factorization by orthogonalizing a column with all previous columns; however, these methods are less stable than the Householder orthogonalization and the orthogonality of the resulting $Q$ factor suffers with the condition number of the matrix. Another method is based on Givens rotations, where entries in the subdiagonal part of the matrix are zeroed out to form the triangular factor and the rotations are accumulated to form the orthogonal factor. This method is very stable and has more parallelism than the Householder method; however it is more expensive, doing about 50% more work, and it is more challenging to extract the parallelism efficiently on the GPU. For our implementation, we rely on the Householder method due to its numerical stability and simplicity. The method is described in pseudo-code in Algorithm \[alg:qr\]. \[t\] $[Q, R] = [I, A]$ $v = \text{house}(R(i))$ $R = (I - 2vv^T) R$ \[alg:qr:trailing\_update\] $Q = Q (I - 2vv^T)$ SVD Algorithms -------------- Most implementations of the SVD are based on the two-phase approach popularized by Trefethen et al. [@trefethen1997numerical], where the matrix $A$ first undergoes bidiagonalization of the form $A = Q_U B Q_V^T$ where $Q_U$ and $Q_V$ are orthonormal matrices and $B$ is a bidiagonal matrix. The matrix $B$ is then diagonalized using some variant of the QR algorithm, the divide and conquer method or a combination of both to produce a decomposition $B = U_B \Sigma V_B^T$. The complete SVD is then determined as $A = (Q_U U_B) \Sigma (Q_V V_B)^T$ during the backward transformation. These methods require significant algorithmic and programming effort to become robust and efficient while still suffering from a loss of relative accuracy [@demmel1992jacobi]. An alternative is the one-sided Jacobi method where all $n(n-1)/2$ pairs of columns are repeatedly orthogonalized in sweeps using plane rotations until all columns are mutually orthogonal. When the process converges (i.e., all columns are mutually orthogonal up to machine precision), the left singular vectors are the normalized columns of the modified matrix with the singular values as the norms of those columns. The right singular vectors can be computed either by accumulating the rotations or by solving a system of equations. Our application does not need the right vectors, so we omit the details of computing them. Algorithm \[alg:jacobi\] describes the one-sided Jacobi method. Since each pair of columns can be orthogonalized independently, the method is also easily parallelized. The simplicity and inherent parallelism of the method make it an attractive first choice for an implementation on the GPU. \[b\] $G = A_{ij}^T A_{ij}$ \[alg:jacobi:gram\] $R = rot(G)$ $A_{ij} = A_{ij} R$ \[alg:jacobi:rot\] GPU Optimization Considerations ------------------------------- GPU kernels are launched by specifying a grid configuration which lets us organize threads into blocks and blocks into a grid. Launching a GPU kernel causes a short stall (as much as 10 microseconds) as the kernel is prepared for execution. This kernel launch overhead prevents kernels that complete their work faster than the overhead from executing in parallel, essentially serializing them. To overcome this limitation when processing small workloads, the work is batched into a single kernel call when possible [@batchqr_haidar; @batch_haidar]. All operations can then be executed in parallel without incurring the kernel launch overhead, with the grid configuration used to determine thread work assignment. A warp is a group of threads (32 threads in current generation GPUs, such as the NVIDIA K40) within a block that executes a single instruction in lockstep, without requiring any explicit synchronization. The occupancy of a kernel tells us the ratio of active warps to the maximum number of warps that a multiprocessor can host. This metric is dependent on the amount of resources that a kernel uses, such as register and shared memory usage and kernel launch configuration, as well as the compute capability of the card ([@wilt2013cuda] for more details). While not a requirement for good performance [@volkov2010better], it is generally a good idea to aim for high occupancy. Memory on the GPU is organized into a hierarchy of memory spaces as shown in Figure \[fig:memory\_hierarchy\]. At the bottom, we have global memory which is accessible by all threads and is the most plentiful but the slowest memory. The next space of interest is the shared memory which is accessible only by threads within the same block and is configurable with the L1 cache to be at most 48KB per thread block on current generation GPUs. Shared memory is very fast and acts as a programmer controllable cache. Finally, we have the registers which are local to the threads. Registers are the fastest of all memory, but the total number of registers usable by a thread without performance implications is limited. If a kernel needs more registers than the limit, then registers are spilled to “local" memory, which is in the slow but cached global memory. Making good use of the faster memories and avoiding excessive accesses to the slower ones is key to good performance on the GPU. As such, it is common to use blocking techniques in many algorithms, where a block of data is brought in from global memory and processed in one of the faster memories. ![The memory hierarchy of a modern GPU.[]{data-label="fig:memory_hierarchy"}](memory.pdf){width="45.00000%"} Related Work ------------ Batched GPU routines for LU, Cholesky and QR factorizations have been developed in [@batchqr_haidar; @batch_haidar; @charara_batch_tdla] using a block recursive approach which increases data reuse and leads to very good performance for relatively large matrix sizes. GPU routines optimized for computing the QR decomposition of very tall and skinny matrices are presented in [@caqr_anderson] where they develop an efficient transpose matrix-vector computation that is employed with some minor changes in this work. GPU-CPU hybrid algorithms for batched SVD using Jacobi and bidiagonalization methods are introduced in [@kotas_svd] where pair generation for the Jacobi method and the solver phase of the bidiagonalization are handled on the CPU. The work in [@Kang2015] employs the power method to construct a rank 1 approximation for 2D filters in convolutional neural networks. Routines to handle the SVD of many matrices on GPUs is presented in [@badolato_2015] where each thread within a warp computes the SVD of a single matrix. Batched QR Decomposition {#sec:batch_qr} ======================== In this section, we discuss implementation details of our batched QR kernel and compare it with other implementations from the MAGMA 2.2 [@tnld10] and CUBLAS 8 [@nvidia-cublas] libraries. Implementation -------------- One benefit of the Householder algorithm is that the application of reflectors to the trailing matrix (line \[alg:qr:trailing\_update\] of the algorithm) can be blocked together and expressed as a matrix-matrix multiplication (Level 3 BLAS) instead of multiple matrix-vector multiplications (Level 2 BLAS). The increased arithmetic intensity typically allows performance to improve when the trailing matrix is large. However, for small matrix blocks, the overhead of generating the blocked reflectors from their vector form as well as the lower performance of the matrix-matrix multiplication for small matrices hinder performance. We can obtain better performance by applying multiple reflectors in their vector form and performing the transpose matrix-vector multiplication efficiently within a thread block [@caqr_anderson]. First, we perform the regular factorization on a column block $P$ (called a panel). The entire panel is stored in registers, with each thread storing one row of the panel, and the transpose matrix-vector product is computed using a series of reductions using shared memory and warp shuffles [@warp_shfl] which allow threads within a warp to read each other’s registers. Figure \[fig:register\_storage\_reduction\] shows the data layout for a theoretical warp of size 8 with 4 columns in registers and a warp reduction using shuffles. Once we factor the panel, we can apply the reflectors to the trailing sub-matrix in a separate kernel that is optimized for performing the core matrix-vector product in the update. In this second kernel, we load both the factored panel $P$ and a panel $M_i$ of the trailing sub-matrix $M$ to registers and apply the reflectors one at a time, updating the trailing panel in registers. Let us take an example of a $32 \times 8$ trailing panel $M_i$. For each reflector, we compute the matrix-vector product $M_i^Tv$ by flattening the $32 \times 8$ product into a reduction of a 256 vector in shared memory that has been padded to avoid bank conflicts. The reduction can then be serialized until it reaches a size of 32, where a partial reduction to a vector of size 8 can take place in 2 steps. This final vector is the product $M_i^Tv$ which can then be quickly applied to the registers storing $M_i$. This process is repeated for each trailing panel within the same kernel to maximize the use of the reflectors which have been stored in registers. Figure \[fig:qr\_fig\] shows one step of a panel factorization and the application of its reflectors to the trailing submatrix. Since threads are limited to 1024 per block on current architectures, we use the approach developed in [@journals/concurrency/KurzakLDB10] to factorize larger matrices. We first factorize panels up to the thread block limit in a single kernel call. The panels below the first are then factorized by first loading the triangular factor into shared memory and then proceeding with the panel factorization as before, taking the triangular portion into consideration when computing reflectors and updates. To keep occupancy up for the small matrices on devices where the resident block limit could be reached before the thread limit, we assign multiple operations to a single thread block. For a batch of $N$ matrices of dimensions $m \times n$, kernels can be launched using $N/b$ thread blocks of size $m \times b$, where each thread block handles $b$ operations. ![Left: matrix rows allocated to thread registers in a warp. Right: parallel warp reduction using shuffles within registers.[]{data-label="fig:register_storage_reduction"}](reg_svd.pdf){width="0.6\linewidth"} ![One step of the QR factorization where a panel P is factored to produce a triangular factor R and reflectors V which are used to update the trailing sub-matrix M.[]{data-label="fig:qr_fig"}](qr_fig.pdf){width="65.00000%"} Performance ----------- Figures \[fig:batch\_qr\] and \[fig:batch\_qr\_rect\] show the performance of our batched QR for 1000 square and rectangular matrices with a panel width of $16$, tuned for the P100 GPU. We compare against the vendor implementation in CUBLAS as well as the high performance library MAGMA. We can see that our proposed version performs well for rectangular matrices with column size of 32 and starts losing ground against MAGMA for the larger square matrix sizes where the blocked algorithm starts to show its performance benefits. A nested implementation where our kernel can be used to factor relatively large panels in a blocked algorithm will likely show some additional performance improvements for the large square matrices, but we leave that as future work. [0.45]{}  [0.45]{}  Register Memory One-Sided Jacobi {#sec:registers} ================================ In this section we will discuss the first batched SVD kernel where the matrix data is hosted in registers and analyze the performance of the resulting kernel. Implementation -------------- In this implementation, to avoid repeated global memory accesses, we attempt to fit the matrix in register memory using the same layout as the panel in the QR factorization, i.e. one row per thread; however, the number of registers that a thread uses has an impact on occupancy which can potentially lead to lower performance. In addition, once the register count exceeds the limit set by the GPU’s compute capability, the registers spill into “local" memory which resides in cached slow global memory. Since we store an entire matrix row in the registers of one thread, we use the serial one-sided Jacobi algorithm to compute the SVD where column pairs are processed by the threads one at a time. The bulk of the work lies in the computation of the Gram matrix $G = A_{ij}^T A_{ij}$ (line \[alg:jacobi:gram\] of Algorithm \[alg:jacobi\]) and in the update of the columns (line \[alg:jacobi:rot\]). Since the Gram matrix is symmetric, this boils down to three dot products which are executed as parallel reductions within the warp using warp shuffles. The computation of the $2 \times 2$ rotation matrix as well as the convergence test is performed redundantly in each thread. Finally, the column update is done in parallel by each thread on its own register data. As with the QR kernel, we keep occupancy up for the smaller matrix sizes by assigning multiple SVD operations to a single block of threads with each operation assigned to a warp to avoid unnecessary synchronizations. Performance {#subsec:reg_perf} ----------- We generate batches of 1000 test matrices with varying condition numbers using the `latms` LAPACK routine and calculate performance based on the total number of rotations needed for convergence. Figures \[fig:reg\_svd\_perf\] and \[fig:reg\_svd\_occupancy\] show the performance on a P100 GPU of the register-based batched SVD kernel and the effect increased register usage has on occupancy. Profiling the kernel, we see that the Gram matrix computation takes about 500 cycles, column rotations take about 240 cycles, and the redundantly computed convergence test and rotation matrices dominate at 1900 cycles. The fact that the redundant portion of the computation dominates means that it is preferable to assign as few threads as possible when processing column pairs. Due to the low occupancy for the larger matrix sizes and the register spills to local memory for matrices larger than 30, it is obvious that the register approach will not suffice for larger matrix sizes. This leads us to our next implementation based on the slower but more parallel-friendly shared memory. [.45]{}  [.45]{}  Shared Memory One-Sided Jacobi {#sec:shared} ============================== While the register based SVD performs well for very small matrix sizes, we need a kernel that can handle larger sizes and maintain reasonably high occupancy. This leads us to building a kernel based on shared memory, the next level of the GPU memory hierarchy. This section discusses the implementation details of this kernel and analyze its performance when compared with the register kernel. Implementation -------------- In this version, the matrix is stored entirely in shared memory, which is limited to at most 48 KB per thread block on current generation GPUs. Using the same thread assignment as the register based kernel would lead to very poor occupancy due to the high shared memory consumption, where potentially only a few warps will be active in a multiprocessor. Instead, we exploit the inherent parallelism of the one-sided Jacobi to assign a warp to a pair of columns, i.e., there are $n/2$ warps processing an $m \times n$ matrix stored in shared memory. There are a total of $n(n-1)/2$ pairs of columns, so we must generate all pairings in $n-1$ steps, with each step processing $n/2$ pairs in parallel. There are many ways of generating these pairs, including round robin, odd-even, and ring ordering [@parosbj_zhou; @ZHOU19971]. We implement the round robin ordering using shared memory to keep track of the column indexes of the pairs with the first warp in the block responsible for updating the index list after each step. Figure \[fig:round\_robin\] shows this ordering for a matrix with 8 columns. When the number of matrix rows exceeds the size of the warp, the thread-per-row assignment no longer allows us to use fast warp reductions, which would force us to use even more resources, as the reductions would now have to be done in shared memory. Instead, we assign multiple rows to a thread, serializing a portion of the reduction over those rows until warp reductions can be used. This follows our observation in Section \[subsec:reg\_perf\] to assign as few threads as possible to process column pairs, frees up valuable resources and increases the overall performance of the reduction. Row padding is used to keep the rows at multiples of the warp size, and column padding is used to keep the number of columns even. Kernels can then be launched using $32\times n/2$ threads to process each matrix. Figures \[fig:shared\_alloc\] and \[fig:shared\_warp\_reduction\] show examples of the thread allocation and reductions for a $8 \times 8$ matrix using a theoretical warp size of 4. ![Distribution of column pairs to warps at each step of a sweep.[]{data-label="fig:round_robin"}](pair_generation.pdf){width="0.6\linewidth"} [0.3]{} {width="\linewidth"} [0.4]{} {width="\linewidth"} Performance {#performance-1} ----------- Figures \[fig:shared\_svd\_perf\] and \[fig:shared\_svd\_occupancy\] show the performance of the parallel shared SVD kernel compared to the serial register SVD kernel on a P100 GPU. We can see the improved growth in performance in the shared memory kernel due to the greater occupancy as well as the absence of any local memory transactions. Looking at the double precision occupancy, we notice two dips in occupancy at matrix sizes 22 and 32 as the number of resident blocks become limited by the registers/block limits of the device, dropping to 2 and then 1 resident blocks. Performance increases steadily from there as we increase the number of threads assigned to the operation until we reach a matrix size of $64 \times 64$ where we reach the block limit of 1024 threads. To handle larger sizes, we must use a blocked version of the algorithm or the randomized SVD as we see in Sections \[sec:block\_global\] and \[sec:randomized\], respectively. [0.45]{}  [0.45]{}  Global Memory One-Sided Block Jacobi {#sec:block_global} ==================================== When we can no longer store the entire matrix in shared memory, we have to operate on the matrix in the slower global memory. Instead of repeatedly reading and updating the columns one at a time, block algorithms that facilitate cache reuse have been developed [@bevcka1999block1; @bevcka1999block2; @bevcka2015new]. The main benefit of the block Jacobi algorithm is its high degree of parallelism; however, since we implement a batched routine for independent operations, we will use the serial block Jacobi algorithm for individual matrices and rely on the parallelism of the batch processing. The parallel version, where multiple blocks are processed simultaneously, can still be used when the batch size is very small, but we will focus on the serial version. In this section we will discuss the implementation details for two global memory block Jacobi algorithms that differ only in the way block columns are orthogonalized and compare their performance with parallel streamed calls to the cuSOLVER 8 [@nvidia-cusolver] library routines. Gram Matrix Block Jacobi SVD ---------------------------- The block Jacobi algorithm is very similar to the vector Algorithm \[alg:jacobi\], orthogonalizing pairs of blocks columns instead of vectors. The first method of orthogonalizing pairs of block columns is based on the SVD of their Gram matrix. During the $p$-th sweep, each pair of $m \times k$ block columns $A^{(p)}_i$ and $A^{(p)}_j$ is orthogonalized by forming a $2k \times 2k$ Gram matrix $G^{(p)}_{ij} = {[A^{(p)}_i A^{(p)}_j]}^T [A^{(p)}_i A^{(p)}_j] = {A^{(p)}_{ij}}^T A^{(p)}_{ij}$ and generating a block rotation matrix $U^{(p)}_{ij}$, computed as the left singular vectors of $G^{(p)}_{ij}$ (or equivalently its eigenvectors, since it is symmetric positive definite). Updating $A^{p+1}_{ij} = A^p_{ij} U^{(p)}_{ij}$ orthogonalizes the block columns, since we have $${A^{p+1}_{ij}}^T A^{p+1}_{ij} = {U^{(p)}_{ij}}^T {A^p_{ij}}^T A^p_{ij} U^{(p)}_{ij} = {U^{(p)}_{ij}}^T G^{(p)}_{ij} U^{(p)}_{ij} = \Lambda^{p}_{ij},$$ where $\Lambda^{p}_{ij}$ is a diagonal matrix of the singular values of $G^{(p)}_{ij}$. Orthogonalizing all pairs of block columns until the entire matrix is orthogonal will give us the left singular vectors as the normalized columns and the singular values as the corresponding column norms. If the right singular vectors are needed, we can accumulate the action of the block rotation matrices on the identity matrix. For our batched implementation, we use highly optimized batched `syrk` and `gemm` routines from MAGMA to compute $G$ and to apply the block rotations, while the SVD is computed by our shared memory batched kernel. Since different matrices will converge in different numbers of sweeps, we keep track of the convergence of each operation $l$ by computing the norm $e_l$ of the off-diagonal entries of $G$ scaled by its diagonal entries. While this term is an inexact approximation of the off-diagonal terms of the full matrix in each sweep, it is still a good indication of convergence and will cost us at most an extra cheap sweep, since the final sweep will not actually perform any rotations within the SVD of $G$. The entire batched operation will then converge when $e = \max e_l < \epsilon$, where $\epsilon$ is our convergence tolerance. This gives us the Gram matrix path of the batched block Jacobi Algorithm \[alg:block\_jacobi\] to compute the SVD of a batch of matrices in global memory. It is worth noting that the computation of the Gram matrix can be optimized by taking advantage of the special structure of $G$, but since the bulk of the computation is in the SVD of G, it will not result in any significant performance gains. Direct Block Jacobi SVD ----------------------- The Gram matrix method is an indirect way of orthogonalizing block columns and may fail to converge if the matrix is very ill-conditioned. Ill-conditioned matrices can be handled by directly orthogonalizing the columns using their SVD. Since the block columns are rectangular, we first compute their QR decomposition followed by the SVD of the triangular factor $R$. Overwriting the block column $A^p_{ij}$ by the orthogonal factor $Q$ and multiplying it by the left singular vectors of $R$ scaled by the singular values will give us the new block column $A^{p+1}_{ij}$: $$A^p_{ij} = Q^p_{ij} R^p_{ij} = \left( Q^p_{ij} U^p_{ij} \Sigma^p_{ij} \right) {V^p_{ij}}^T = A^{p+1}_{ij} {V^p_{ij}}^T.$$ If the right singular vectors are needed, we can accumulate the action of $V^p_{ij}$ on the identity matrix. For our batched implementation, we use the batch QR routine developed in Section \[sec:batch\_qr\] and `gemm` routines from MAGMA to multiply the orthogonal factor by the left singular vectors, while the SVD is computed by our shared memory batched kernel. The same convergence test used in the Gram matrix method can be used on the triangular factor, since the triangular factor should be close to a diagonal matrix if a pair of block columns are orthogonal. This gives us the direct path of the batched block Jacobi Algorithm \[alg:block\_jacobi\] to compute the SVD of a batch of matrices in global memory. \[t\] $e_l = 0$ $G = \text{batchSyrk}(A_{ij})$ $[A_{ij}, G] = \text{batchQR}(A_{ij})$ $e_l = \text{max}(e_l, \text{scaledOffdiag}(G))$ $U = \text{batchSvd}(G)$ $A_{ij} = \text{batchGemm}(A_{ij}, U)$ $e = \text{max}(e_l)$ Performance {#performance-2} ----------- Figures \[fig:block\_jacobi\_profile1\] and \[fig:block\_jacobi\_profile1\] show the profiling of the different computational kernels involved in the batched block algorithms with a block width of $32$, specifically percentages of total execution time for determining convergence and memory operations, matrix multiplications, QR decompositions and the SVD of the Gram matrix. For the Gram matrix approach, the SVD is the most costly phase, even for the larger operations, while the QR and SVD decompositions take almost the same time for the larger matrices in the direct approach. Figure \[fig:block\_jacobi\_perf\] shows the performance of the batched block Jacobi SVD of 200 matrices using both methods and Figure \[fig:osbjvscustream\] compares the performance of our batched SVD routine with a batched routine that uses the cuSOLVER SVD routine using 20 concurrent streams on a P100 GPU. Increasing the number of streams for cuSOLVER showed little to no performance benefits, highlighting the performance limitations of routines that are bound by kernel launch overhead. The matrices are generated randomly using the `latms` LAPACK routine with a condition number of $10^7$. The Gram matrix approach fails to converge in single precision for these types of matrices, whereas the direct approach always converges; however the Gram matrix approach performs better when it is applicable for the larger matrices due to the strong performance of the matrix-matrix multiplcations. The performance of the block algorithm can be improved by preprocessing the matrix using QR and LQ decompositions to decrease the number of sweeps required for convergence [@Oksa_2006] as well as by adaptively selecting pairs of block columns based on the computed offdiagonal norms of their Gram matrices. These changes are beyond the scope of this paper and will be the focus of future work. [0.45]{} ![Profile of the different phases of the block Jacobi SVD for 200 matrices of varying size on a P100 GPU in double precision. Single precision exhibits similar behavior.[]{data-label="fig:block_jacobi_profile"}](block_perf_1.pdf "fig:") [0.45]{} ![Profile of the different phases of the block Jacobi SVD for 200 matrices of varying size on a P100 GPU in double precision. Single precision exhibits similar behavior.[]{data-label="fig:block_jacobi_profile"}](block_perf_1b.pdf "fig:") [0.45]{}  [0.45]{}  Randomized SVD {#sec:randomized} ============== As mentioned in Section \[sec:intro\], we are often interested in a rank-$k$ approximation of a matrix $A \approx \tilde{U} \tilde{S} \tilde{V}$. We can compute this approximation by first determining the singular value decomposition of the full $m \times n$ matrix $A$ and then truncating the $n-k$ smallest singular values with their corresponding singular vectors; however, when the matrix has low numerical rank $k$, we can obtain the approximation using very fast randomization methods [@halko2011finding]. This section will discuss some details of the algorithm and compare its performance with the full SVD using our one-sided block Jacobi kernel. Implementation -------------- When the singular values of a matrix decay rapidly, we can compute an approximate SVD using a simple two phase randomization method: 1. The first phase determines an approximate orthogonal basis $Q$ for the columns of $A$ ensuring that $A \approx QQ^T A$. When the numerical rank $k$ of $A$ is low, we can be sure that $Q$ has a small number of columns as well. In [@halko2011finding] we see that by drawing $k+p$ sample vectors $y = Aw$ from random input vectors $w$, we can obtain a reliable approximate basis for $A$ which can then be orthogonalized. This boils down to computing a matrix $Y = A \Omega$, where $\Omega$ is a $n \times (k + p)$ random Gaussian sampling matrix, and then computing the QR decomposition of $Y = Q R_y$, where $Q$ is the desired approximate orthogonal basis. 2. The second phase uses the fact that $A \approx QQ^T A$ to compute a matrix $B = Q^T A$ so that we now have $A \approx QB$. Forming the SVD of $B = U_B S V^T$, we finalize our approximation $A \approx QU_B S V^T = U S V^T$. For the wide $(k+p) \times n$ matrix $B$, we can first compute a QR decomposition of its transpose, followed by the SVD of the upper triangular factor. Algorithm \[alg:batch\_rsvd\] shows that the core computations for the randomized method are matrix-matrix multiplications, QR decompositions, and the singular value decompositions of small matrices. Using the batched routines from the previous sections, it is straightforward to form the required randomized batched SVD. More robust randomized SVD algorithms would employ randomized subspace iteration methods to obtain a better basis $Q$ for the columns of $A$ and rely on these same core kernels, but will not be further discussed here. \[t\] $[m, n] = size(A)$ $\Omega = \text{Rand}(n, k + p)$ $Y = \text{batchGemm}(A, \Omega)$ $[Q, R_y] = \text{batchQR}(Y)$ $B = \text{batchGemm}(Q^T, A)$ $[Q_B, R_B] = \text{batchQR}(B^T)$ $[U_R, S, V_R] = \text{batchSvd}(R_B^T)$ $U = \text{batchGemm}(Q, U_R)$ $V = \text{batchGemm}(Q_B, V_R)$ Performance {#performance-3} ----------- Figure \[fig:rsvd\_profile\] shows the profiling of the different kernels used in the randomized batched routine for determining the top 64 singular values and vectors of randomly generated low rank matrices using the `latms` LAPACK routine. The miscellaneous portion includes random number generation using the CURAND library’s default random number generator and a Gaussian distribution, batched transpose operations and memory operations. We can see that the performance of all kernels play almost equally important roles in the performance of the randomized routine as the matrix size grows while keeping the computed rank the same. Figure \[fig:rsvd\_perf\] shows the performance of the batched randomized SVD of 200 operations and Figure \[fig:rsvd\_vs\_osbj\] compares the runtimes of the direct block one-sided Jacobi routine with the randomized SVD on a P100 GPU for the same set of matrices, showing that significant time savings can be achieved even for relatively small blocks. ![Profile of the different phases of the batched randomized SVD for 200 matrices of varying size on a P100 GPU in double precision. Single precision exhibits similar behavior.[]{data-label="fig:rsvd_profile"}](rand_perf_1.pdf) [0.45]{}  [0.45]{}  Application to Hierarchical Matrix Compression {#sec:application} ============================================== As an application of the batched kernels presented, we consider the problem of compressing/recompressing hierarchical matrices. This is a problem of significant importance for building hierarchical matrix algorithms and in fact was our primary motivation for the development of the batched kernels. Hierarchical matrices [@hackbush_2000; @hackbush_h2_2000; @hackbush_1999] have received substantial attention in recent years because of their ability to store and perform algebraic operations in near linear complexity rather than the $O(n^2)$ and $O(n^3)$ that regular dense matrices require. The effectiveness of hierarchical matrices comes from the fact they can approximate a matrix by a (quad)-tree of blocks where many of the blocks in the off-diagonal regions have a rapidly decaying spectrum and can therefore be well-approximated by numerically low rank representations. It is these low rank representations, at different levels of the hierarchical tree, that reduce the memory footprint and operations complexity of the associated matrix algorithms. Hackbush [@hbook] shows that many of the large dense matrices that appear in scientific computing, such as from the discretization of integral operators, Schur complements of discretized PDE operators, and covariance matrices, can be well approximated by these hierarchical representations. Reviewing and analyzing hierarchical matrix algorithms is beyond the scope of this paper. Here we focus on the narrow task of compressing hierarchical matrices. This compression task may be viewed as a generalization of the well-known compression (i.e., low rank approximation) of large dense matrices to the case of hierarchical matrices. For large dense matrices, one way to perform the compression is to generate a single exact or approximate SVD ($U \Sigma V^T$) and truncate the spectrum $\Sigma$ to the desired tolerance, to produce a truncated or “compressed” representation $(\bar{U} \bar{\Sigma} \bar{V}^T)$. For hierarchical matrices, the equivalent operations involve *batched SVDs* on small blocks, with one batched kernel call per level of the tree in the hierarchical representation. The size of the batch in every such call is the number of nodes at the corresponding level in the tree. Compression algorithms with controllable accuracy are important practically, because it is often the case that the hierarchical matrices generated by analytical methods can be compressed with no significant loss of accuracy. Even more importantly, when performing matrix operations such as additiona and multiplication, the apparent ranks of the blocks often grow and have to be recompressed regularly during the operations to prevent superlinear growth in memory requirements. [0.45]{} {width="0.9\linewidth"} [0.45]{} {width="0.8\linewidth"} $\mathcal{H}^2$-matrix representation ------------------------------------- For our application, we use the memory efficient $\mathcal{H}^2$ variant of hierarchical matrices which exhibit linear complexity in time and space for many of its core operations. In the $\mathcal{H}^2$-matrix format, a hierarchical matrix is actually represented by three trees: - Row and column basis column trees $U$ and $V$ that organize the row and column indices of the matrix hierarchically. Each node represents a set of basis vectors for the row and column spaces of the blocks of $A$. Nodes at the leaves of the tree store these vectors explicitly, while inner nodes store only transfer matrices that allow us to implicitly represent basis vectors in terms of their children. A basis tree with this parent-child relationship of the nodes is called a nested basis. For example, in a binary row basis tree $U$ with transfer matrices $E$, we can explicitly compute the basis vectors for a node $i$ with children $i_1$ and $i_2$ at level $l$ as: $$U^{l-1}_i = \begin{bmatrix} U^l_{i_1} & \\ & U^l_{i_2} \end{bmatrix} \begin{bmatrix} E^l_{i_1} \\ E^l_{i_2} \end{bmatrix}.$$ Figure \[fig:basis\_tree\] shows an example of a binary basis tree. - A matrix tree for the hierarchical blocking of $A$ formed by a dual traversal of the nodes of the two basis trees. A leaf is determined when the block is either small enough and stored as an $m \times m$ dense matrix, or when a low rank approximation of the block meets a specified accuracy tolerance. For the latter case, the node is stored as a $k_l \times k_l$ coupling matrix $S$ at each level $l$ of the tree, where $k_l$ is the rank at level $l$. The block $A_{ts}$ of the matrix, where $t$ is the index set of a node in the row basis tree $U$ and $s$ is the index set of a node in the column basis $V$, is then approximated as $A_{ts} \approx U_t S_{ts} V_s^T$. Figure \[fig:hmatrix\] shows the leaves of the matrix quadtree of a simple hierarchical matrix. For the case of symmetric matrices, the $U$ and $V$ trees are identical. Our numerical results below are from a symmetric covariance matrix. Compression ----------- The compression of a symmetric $\mathcal{H}^2$-matrix $A_H$, represented by the two trees $U$ (with its transfer matrices $E$) and $S$, involves generating a new optimal basis tree $\widetilde{U}$ (with its transfer matrices $\widetilde{E}$) in a truncation phase, and a new $\widetilde{S}$ that expresses the contents of the matrix blocks in this new basis in a projection phase. We present a version of the truncation algorithm that generates a memory efficient basis $[\widetilde{U}, \widetilde{E}]$ from a representation of the matrix in a given $[U, E]$ basis. More sophisticated algebraic compression algorithms that involve the use of $S$ in the truncation phase in order to generate a more efficient basis will be the subject of future work. The truncation phase computes the SVD of the nodes of the basis tree $U$ level by level, with all nodes in a level being processed in parallel to produce the new basis $\widetilde{U}$. We have an explicit representation of the basis vectors at the leaves, so we can compute the SVD of all leaf nodes in parallel with our batched kernels and truncate the singular vectors whose singular values are lower than our relative compression threshold $\epsilon$. Truncating the node to the relative threshold using the SVD will give us an approximation of the leaf such that $\frac{||U - \widetilde{U}||_F}{||U||_F} \le \epsilon$. With the new leaf nodes, we can compute projection matrices in a tree $T$, where each node $i$, $T^d_i = \widetilde{U^d_i}^T U^d_i$ and $d$ is the leaf level. Sweeping up the tree, we process the inner nodes while preserving the nested basis property. Using the parent-child relationship of a node $i$ with children $i_1$ and $i_2$ at level $l$, we have: $$U^{l-1}_i = \begin{bmatrix} U^l_{i_1} & \\ & U^l_{i_2} \end{bmatrix}\begin{bmatrix} E^l_{i_1}\\ E^l_{i_2} \end{bmatrix} \approx \begin{bmatrix} \widetilde{U}^l_{i_1} & \\ & \widetilde{U}^l_{i_2} \end{bmatrix}\begin{bmatrix} T^l_{i_1} E^l_{i_1}\\ T^l_{i_2} E^l_{i_2} \end{bmatrix} = \begin{bmatrix} \widetilde{U}^l_{i_1} & \\ & \widetilde{U}^l_{i_2} \end{bmatrix} TE_i$$ After forming the $TE$ matrices using batched matrix-matrix multiplication, we compute their SVD $TE = QSW^T$ using the batched SVD kernel and truncate as we did for the leaves to form the truncated $\widetilde{TE}$ matrices as: $$\widetilde{TE}_i = \widetilde{Q}_i \left( \widetilde{S}_i \widetilde{W}_i^T \right) = \begin{bmatrix} \widetilde{E}^l_{i_1}\\ \widetilde{E}^l_{i_2} \end{bmatrix} T^{l-1}_i$$ where $\widetilde{E}^l$, the block rows of $\widetilde{Q}$, are the new transfer matrices at level $l$ of our compressed nested basis and $T^{l-1}$ are the projection matrices for level $(l-1)$. The key computations involved in this truncation phase consist then of one batched SVD involving the leaves of the tree, followed by a sequence of batched SVDs, one per level of the tree, involving the transfer matrices and data from the lower levels. The projection phase consists of transforming the coupling matrices in the matrix tree using the generated projection matrices of the truncation phase. For each coupling matrix $S_{ts}$, we compute a new coupling matrix $\widetilde{S}_{ts} = T_t S_{ts} T_s^T$ using batched matrix-matrix multiplications. This phase of the operation consumes much less time than the truncation phase on GPUs, because of substantial efficiencies in executing regular arithmetically intensive operations on them. Results ------- As an illustration of the effectiveness of the algebraic compression procedure, we generate covariance matrices of various sizes for a spatial Gaussian process with $n$ observation points placed on a random perturbation of a regular discretization of the unit square $[0,1] \times [0,1]$ and an isotropic exponential kernel with correlation length of $0.1$. Hierarchical representations of the formally dense $n \times n$ covariance matrices are formed analytically by first clustering the points in a KD-tree using a mean split giving us the hierarchical index sets of the basis tree. The basis vectors and transfer nodes are generated using Chebyshev interpolation [@borm2007approximating]. The matrix tree is constructed using a dual traversal of the basis tree [@hackbush_2000; @hackbush_2003], and the coupling matrices are generated by evaluating the kernel at the interpolation points. The approximation error of the constructed matrix is then controlled by varying the number of interpolation points and by varying the leaf admissibility condition during the dual tree traversal. An approximation error of $10^{-7}$ has been used in the following tests and a relative truncation error $\epsilon=\frac{||A_H - \widetilde{A}_H||_F}{||A_H||_F} \le 10^{-7}$ has been used to maintain the accuracy of the compressed matrices. Figure \[fig:compression\] shows the memory consumption before and after compression of hierachical covariance matrices with leaf size $64$ and initial rank $64$ (corresponding to an $8 \times 8$ Chebyshev grid). The dense part remains untouched, while the low rank part of the representation sees a substantial decrease in memory consumption after compression with minimal loss of accuracy. Figure \[fig:compression\_time\] shows the expected asymptotic linear growth in time of the compression algorithm and shows the effect of using the randomized SVD with $32$ samples instead of the full SVD as computed by the shared memory kernel. Figure \[fig:compression\_time2\] shows another example where the admissibility condition is weakened to generate a coarser matrix tree with an increased rank of 121 (corresponding to an $11 \times 11$ Chebyshev grid) and the randomized SVD with $64$ samples also reduces compression time when compared to the full SVD using the direct block Jacobi kernels. [0.45]{}  [0.45]{}  ![Compression time for a coarser matrix tree with initial rank 121 comparing the randomized SVD with 64 samples and the full SVD.[]{data-label="fig:compression_time2"}](compress_perf_3.pdf) Conclusions and Future Work {#sec:conclusion} =========================== In this paper, we described the implementation of efficient batched kernels for the QR decomposition and randomized singular value decomposition of low rank matrices hosted on the GPU. Our batched QR kernel provides significant performance improvements for small matrices over existing state of the art libraries, and our batched SVD routines are the first of their kind on the GPU, with performance exceeding 800/400 GFLOP/s on a batch of 1000 matrices of size $512 \times 512$ in single/double precision. We illustrated the power of these kernels on a problem involving the algebraic compression of hierarchical matrices stored entirely in GPU memory, and demonstrated a high-performance compression algorithm yielding significant memory savings on practical problems. In the future, we plan to investigate alternatives to the one-sided Jacobi algorithm for the SVD of the small blocks in the randomized algorithm and improve the performance of the blocked algorithms using preconditioning and adaptive block column pair selection. We also plan to develop a suite of hierarchical matrix operations suited for execution on modern GPU and manycore architectures. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the NVIDIA Corporation for providing access to the P100 GPU used in this work.

--- abstract: 'In this paper, we apply the statefinder diagnostic to the cosmology with the Abnormally Weighting Energy hypothesis (AWE cosmology), in which dark energy in the observational (ordinary matter) frame results from the violation of weak equivalence principle (WEP) by pressureless matter. It is found that there exist closed loops in the statefinder plane, which is an interesting characteristic of the evolution trajectories of statefinder parameters and can be used to distinguish AWE cosmology from the other cosmological models.' author: - 'Dao-jun Liu' - 'Wei-zhong Liu' title: Statefinder diagnostic for cosmology with the abnormally weighting energy hypothesis --- Understanding the acceleration of the cosmic expansion is one of the deepest problems of modern cosmology and physics. In order to explain the acceleration, an unexpected energy component of the cosmic budget, dark energy, is introduced by many cosmologists. Perhaps the simplest proposal is the Einstein’s cosmological constant $\Lambda$ (vacuum energy), whose energy density remains constant with time. However, due to some conceptual problems associated with the cosmological constant (for a review, see [@ccp]), a large variety of alternative possibilities have been explored. The most popular among them is quintessence scenario which uses a scalar field $\phi$ with a suitably chosen potential $V(\phi)$ so as to make the vacuum energy vary with time. Inclusion of a non-minimal coupling to gravity in quintessence models together with further generalization leads to models of dark energy in a scalar-tensor theory of gravity. Besides, some other models invoke unusual material in the universe such as Chaplygin gas, tachyon, phantom or k-essence (see, for a review, [@dde] and reference therein). The possibility that dark energy comes from the modifications of four-dimensional general theory of relativity (GR) on large scales due to the presence of extra dimensions [@DGP] or other assumptions [@fR] has also been explored. A merit of these models is the absence of matter violating the strong energy condition (SEC). Recently, Füzfa and Alimi propose a completely new interpretation of dark energy that does also not require the violation of strong energy condition [@AWE]. They assume that dark energy does not couple to gravitation as usual matter and weights abnormally, *i.e.*, violates the weak equivalence principle (WEP) on large scales. The abnormally weighting energy (AWE) hypothesis naturally derives from more general effective theories of gravitation motivated by string theory in which the couplings of the different matter fields to the dilaton are not universal in general (see [@AWE07] and the reference therein). In Ref.[@AWE07], Füzfa and Alimi also applied the above AWE hypothesis to a pressureless fluid to explain dark energy effects and further to consider a unified approach to dark energy and dark matter. As so many dark energy models have been proposed, a discrimination between these rivals is needed. A new geometrical diagnostic, dubbed the statefinder pair $\{r, s\}$ is proposed by Sahni *et al* [@statefinder], where $r$ is only determined by the scalar factor $a$ and its derivatives with respect to the cosmic time $t$, just as the Hubble parameter $H$ and the deceleration parameter $q$, and $s$ is a simple combination of $r$ and $q$. The statefinder pair has been used to explor a series of dark energy and cosmological models, including $\Lambda$CDM, quintessence, coupled quintessence, Chaplygin gas, holographic dark energy models, braneworld models, Cardassion models and so on [@SF03; @Pavon; @TJZhang]. In this paper, we apply the statefinder diagnostic to the AWE cosmology. We find that there is a typical characteristic of the evolution of statefinder parameters for the AWE cosmology that can be distinguished from the other cosmological models. As is presented in Ref.[@AWE07], in the AWE cosmology, the energy content of the universe is divided into three parts: a gravitational sector with metric field ($g_{\mu\nu}^{*}$ ) and scalar field ($\phi$) components, a matter sector containing the usual fluids (baryons, photons, normally weighting dark matter if any, etc) and an abnormally weighting energy (AWE) sector. The normally and abnormally weighting matter are assumed to interact only through their gravitational influence without any direct interaction. The corresponding action in the Einstein frame can be written as $$\begin{aligned} \label{action} S&=&\frac{1}{2\kappa_*}\int \sqrt{-g_*}d^4x\{R^*-2g_*^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi\}\nonumber\\ &+&S_m[\psi_m, A^2_m(\phi)g^*_{\mu\nu}]+S_{awe}[\psi_{awe}, A^2_{awe}(\phi)g^*_{\mu\nu}]\end{aligned}$$ where $S_m$ is the action for the matter sector with matter fields $\psi_m$, $S_{awe}$ is the action for AWE sector with fields $\psi_{awe}$, $R^*$ is the curvature scalar, $\kappa_{*}=8\pi G_*$ and $G_*$ is the ’bare’ gravitational coupling constant. $A_{awe}(\phi)$ and $A_{m}(\phi)$ are the constitutive coupling functions to the metric $g_{\mu\nu}^*$ for the AWE and matter sectors respectively. Considering a flat Friedmann-Lemaitre-Robertson-Walker(FLRW) universe with metric $$\label{line1} ds_*^2=-dt_*^2+a_*^2(t_*)dl_*^2,$$ where $a_*(t_*)$ and $dl_*$ are the scale factor and Euclidean line element in the Einstein frame. The Friedmann equation derived from the action (\[action\]) is $$H_*^2=\left(\frac{1}{a_*}\frac{da_*}{dt_*}\right)^2 =\frac{({d\phi}/{dt_*})^2}{3}+\frac{\kappa_*}{3}(\rho_m^*+\rho_{awe}^*)$$ where $\rho_m^*$ and $\rho_{awe}^*$ are energy density of normally and abnormally weighting matter respectively. Assuming further that both the matter sector and AWE sector are constituted by a pressureless fluid, one can obtain the evolution of $\rho_m^*$ and $\rho_{awe}^*$, $$\rho_{m,awe}^*=A_{m,awe}(\phi)\frac{C_{m,awe}}{a_*^3},$$ where $C_{m,awe}$ are two constants to be specified. Introducing a new variable $\lambda=\ln (a_*/a_*^i)$ where $a_*^i$ is a constant, the Klein-Gordon equation ruling the scalar field dynamics is reduced to be $$\label{KG1} \frac{2\phi''}{3-\phi'^2}+\phi'+\frac{R_c\alpha_m(\phi)A_{m}(\phi)+\alpha_{awe}(\phi)A_{awe}(\phi)}{R_cA_{m}(\phi)+A_{awe}(\phi)}=0,$$ where a prime denotes a derivative with respect to $\lambda$, the parameter $R_c= C_m/C_{awe}$ and the functions $\alpha_{m,awe}=d\ln(A_{m,awe}(\phi))/d\phi$. However, Einstein frame, in which the physical degrees of freedom are separated, is not correspond to a physically observable frame. Cosmology and more generally everyday physics are built upon observations based on “normal” matter which couples universally to a unique metric $g_{\mu\nu}$ and according to the AWE action (\[action\]), $g_{\mu\nu}$ defines the observational frame through the following conformal transformation: $$g_{\mu\nu}=A_m^2(\phi)g_{\mu\nu}^*.$$ Therefore, the line element of FLRW metric (\[line1\]) in the observational frame can be written down as $$ds^2=-dt^2+a^2(t)dl^2,$$ where the scale factor $a(t)$ and the element of cosmic time read $$a(t)=A_m(\phi)a_*(t_*)=e^{\lambda}A_m(\phi)a^i_*,\;\;dt=A_m(\phi)dt_*.$$ Therefore, the Friedmann equation in the observational frame reads $$\begin{aligned} \label{H21} H^2\equiv\left(\frac{\dot{a}}{a}\right)^2 =\frac{8\pi G_*}{3}\frac{C_m}{a^3}\frac{A_m^2(\phi)\left(1+\frac{A_{awe}(\phi)}{A_m(\phi)}R_c^{-1}\right)}{\left(1-\alpha_m(\phi)\frac{d\phi}{dN}\right)^2-\frac{1}{3}\left(\frac{d\phi}{dN}\right)^2},\end{aligned}$$ where the overdot denotes the derivation with respect to the time $t$ and $N\equiv \ln(a/a^i)$ ( $a^i$ is value of the scale factor when $t=t_i$). Further, the Friedmann equation (\[H21\]) can be rewritten as $${H^2}={H^2_i}E^2(N)$$ where $$\begin{aligned} \label{E} E^2(N)&=&Fe^{-3N}\frac{A_m^2(\phi)\left(1+\frac{A_{awe}(\phi)}{A_{m}(\phi)}R_c^{-1}\right)} {\left(1-\alpha_m(\phi)\frac{d\phi}{dN}\right)^2-\frac{1}{3}\left(\frac{d\phi}{dN}\right)^2},\end{aligned}$$ where the constant $F$ $$F=\frac{\left(1-\alpha_m(\phi_i)\frac{d\phi}{dN}|_i\right)^2-\frac{1}{3}\left(\frac{d\phi}{dN}|_i\right)^2} {A_m^2(\phi_i)\left(1+\frac{A_{awe}(\phi_i)}{A_{m}(\phi_i)}R_c^{-1}\right)},$$ where $\phi_i$ is the value of field $\phi$ at the moment $t=t_i$ and obviously, at this moment $E=E(0)=1$. We can search for a translation of the AWE cosmology to usual dark energy cosmology. In observational frame, one can search for an effective dark energy density parameter $\Omega_{DE}$ together with its effective equation of state $w$ in a spatially flat universe $$\label{E22} E^2(N)=\Omega_M^{i}e^{-3N}+\Omega_{DE}(N),$$ $$\label{OmegaDE} \Omega_{DE}(N)=(1-\Omega_M^{i})f_X(N),$$ and $$\label{wDE} w(N)=-1-\frac{1}{3}\frac{d\ln f_X(N)}{dN},$$ where $\Omega_M^{i}$ is the total amount of effective dust matter energy density parameter at the time $t=t_i$. Therefore, from Eq.(\[E\]), we have $$f_X(N)=\frac{e^{-3N}}{1-\Omega_M^{i}}\left(F\frac{A_m^2(\phi)\left(1+\frac{A_{awe}(\phi)}{A_{m}(\phi)}R_c^{-1}\right)} {\left(1-\alpha_m(\phi)\frac{d\phi}{dN}\right)^2-\frac{1}{3}\left(\frac{d\phi}{dN}\right)^2}-\Omega_M^{i}\right).$$ Let us choose the coupling functions $A_m{\phi}$ and $A_{awe}(\phi)$ as: $$\label{model} A_m(\phi) = \exp\left(k_m \frac{\phi^2}{2}\right), \;\; A_{awe}(\phi) = \exp\left(k_{awe} \frac{\phi^2}{2}\right),$$ where $k_m$ and $k_{awe}$ are two arbitrarily constant. Therefore, $$\alpha_m(\phi) = k_m \phi, \;\;\; \alpha_{awe}(\phi) = k_{awe} \phi.$$ Note that we assume here that $k_m\neq k_{awe}$, otherwise, it is corresponds to the case of an ordinary scalar-tensor theory. Meanwhile, the Klein-Gordon equation for $\phi$ in the observational frame (\[KG1\]) reduced to be $$\label{KG3} \frac{2\left(\frac{d^2\phi}{dN^2}+k_m \left(\frac{d\phi}{dN}\right)^3\right)} {\left(1-k_m \phi\frac{d\phi}{dN}\right)^3} +\left(3-\frac{\left(\frac{d\phi}{dN}\right)^2}{\left(1-k_m \phi\frac{d\phi}{dN}\right)^2}\right)\left(\frac{\frac{d\phi}{dN}}{1-k_m \phi\frac{d\phi}{dN}} +k_m \phi+\frac{(k_{awe}-k_m) \phi}{1+R_c \exp\left((k_m-k_{awe}) \frac{\phi^2}{2}\right)}\right)=0.$$ It is not difficult to find that if $R_c+k_{awe}/k_m>0$, Eq.(\[KG3\]) have only one limited critical point $(\phi_c,\frac{d\phi}{dN}|_c)=(0,0)$, which is an unstable saddle point. However, in the case of $R_c+k_{awe}/k_m<0$, their exists two critical points $(0,0)$ and $\left(\sqrt{\frac{2}{k_m-k_{awe}}\ln \left(-R_c^{-1}\frac{k_{awe}}{k_m}\right)},0\right)$. It is shown, by performing a linear stability analysis, that the critical point (0,0) is unstable in this case, but $\left(\sqrt{\frac{2}{k_m-k_{awe}}\ln \left(-R_c^{-1}\frac{k_{awe}}{k_m}\right)},0\right)$ is stable. Note that $R_c$ denoting the ratio of energy density of ordinary and AWE matter is certainly positive and we also assume that $k_m >0$ in Eq.(\[model\]). Therefore, if we set a condition that $k_{awe}<-R_c k_m $, an attractor solution of scalar field $\phi$ is expected. From now on, we only deal with the models with this prior condition. The traditional geometrical diagnostics, i.e., the Hubble parameter $H$ and the deceleration parameter $q\equiv {-\ddot{a}a}/{\dot{a}^2}$, are two good choices to describe the expansion state of our universe but they can not characterize the cosmological models uniquely, because a quite number of models may just correspond to the same current value of $H$ and $q$. Fortunately, as is shown in many literatures, the statefinder pair $\{r, s\}$ which is also a geometrical diagnostic, is able to distinguish a series of cosmological models successfully. The statefinder pair $\{r,s\}$ defines two new cosmological parameters in addition to $H$ and $q$: $$r\equiv \frac{\dddot{a}}{aH^3},\;\;\;\;s\equiv \frac{r-1}{3(q-1/2)}.$$ As an important function, the statefinder can allow us to differentiate between a given dark energy model and the simplest of all models, i.e., the cosmological constant $\Lambda$. For the $\Lambda$CDM model, the statefinder diagnostic pair $\{r,s\}$ takes the constant value $\{1,0\}$, and for the SCDM model, $\{1,1\}$. The statefinders $r$ can be easily expressed in terms of the Hubble parameter $H(z)$ and its derivatives as follows: $$r(x)=1-2\frac{H'}{H}x+\left[\frac{H''}{H}+\left(\frac{H'}{H}\right)^2\right]x^2,$$ where the variable $x=1+z$, $q(x)=\frac{H'}{H}x-1$ and $H'$ is the derivative of $H$ with respect to the redshift $z$, and immediately $s$ is also a function of $x$. We can now use this tool to explore the evolutionary trajectories of the universe governed by AWE cosmology. In FIG. \[w-z\], we show the evolutionary trajectories of equation of state of effective dark energy in observational frame for model (\[model\]) with different parameters. For some parameters that is chosen, the equation-of-state parameter $w$ is able to cross the cosmological constant divide $w=-1$ between phantom and quintessence.From FIG. \[q-z\], we find that at high redshifts the standard matter-dominated cosmology is recovered and at low redshifts the universe become accelerating and dark energy dominated as expected. The transition from deceleration to acceleration occurs roughly at $z\approx 1$. The time evolution trajectories of statefinder pairs $\{r,s\}$ and $\{r,q\}$ for model (\[model\]) are shown in FIG. \[r-s\] and FIG. \[r-q\], respectively. The most interesting characteristic of the trajectories is that there is a loop in the plane. Along the time evolution, after passing through the $\Lambda$CDM fixed point $\{r=1,s=0\}$, the statefinder pairs is now going along with a loop in the plane, and at some time in the future they will pass through the $\Lambda$CDM fixed point again. After that, they will go towards the SCDM fixed point $\{r=1,s=1\}$. This character of the trajectories is significantly different from those of the other cosmological models, such as quintessence, Chaplygin gas, brane world ( see, for example, [@SF03]), phantom [@sfphantom], Cardassian [@TJZhang], holographic dark energy [@xzhang05a] agegraphic dark energy models [@caihao], and so on. From FIG. \[r-q\], it is obviously that the acceleration of the universe in this model is a transient phenomenon. In the past, the standard matter-dominated universe is simply mimicked, but in the future, although a SCDM state is an attractor, the universe will go through a series of states, which are different from that of SCDM but decelerating, before the attractor is finally reached. It is worth noting that a class of braneworld models, called “disappearing dark energy” (DDE) [@sahni], in which the current acceleration of the universe is also a transient phase and there exists closed loop in the $\{r,q\}$ plane [@SF03], but there is no closed loop which contains the $\Lambda$CDM fixed point $\{r=1,s=0\}$ in the $\{r,s\}$ plane as that of the model studied in this work. ![The evolution of equation-of-state $w$ for the effective dark energy in the observational frame, where we choose the parameters $k_m=1$,$k_{awe}=-10$ and $R_c=0.1,0.2,0.3$, respectively.[]{data-label="w-z"}](w-z.eps "fig:"){width="5.5cm"}\ ![The evolution of deceleration parameter $q(z)$ in the observational frame, where we choose the parameters $k_m=1$,$k_{awe}=-10$ and $R_c=0.1,0.2,0.3$, respectively.[]{data-label="q-z"}](q-z.eps "fig:"){width="5.5cm"}\ ![Trajectories in the statefinder plane $\{r,s\}$ for the model (\[model\]), where the parameters of the model is chosen as $k_m=1$,$k_{awe}=-10$ and $R_c=0.2,0.3,0.4$, respectively. Two circles at the point $\{1,0\}$ and $\{1,1\}$ in the plane denote the fixed statefinder pairs of LCDM and SCDM model, respectively. The arrows show the direction of the time evolution.[]{data-label="r-s"}](r-s.eps "fig:"){width="5.5cm"}\ ![Trajectories in the statefinder plane $\{r,q\}$ for the model (\[model\]), where the parameters of the model is chosen as $k_m=1$,$k_{awe}=-10$ and $R_c=0.2,0.3,0.4$, respectively. The arrows show the direction of the time evolution.[]{data-label="r-q"}](r-q.eps "fig:"){width="5.5cm"}\ In summary, we investigate the cosmology with the hypothesis of abnormally weighting energy by using the statefinder diagnostic. The statefinder diagnosis provides a useful tool to break the possible degeneracy of different cosmological models by constructing the parameters $\{r, s\}$ or $\{r, q\}$ using the higher derivative of the scale factor. It is found that the trajectories of the statefinder pairs $\{r,s\}$ and $\{r,q\}$ of AWE cosmology in the statfinder plane have a typical characteristic which is distinguished from other cosmological models. We hope that the future high-precision observations offer more accurate data to determine the model parameters more precisely, rule out some models and consequently shed light on the nature of dark energy. This work is supported in part by National Natural Science Foundation of China under Grant No. 10503002 and Shanghai Commission of Science and technology under Grant No. 06QA14039. [99]{} S. Weinberg, Rev. Mod. Phys. **61**, 1 (1989); V. Sahni and A. Starobinsky, Int. J. Mod. Phys. **D9**, 373 (2000); T. Padmanabhan, Phy. Rep. **380**, 235 (2003). E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. **D15**, 1753 (2006). G. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. **B485**, 208 (2000); C. Deffayet, G. Dvali and G. Gabadadze, Phys. Rev. **D65**, 044023 (2002). S. M. Carroll, V. Duvvuri, M. Trodden and M. S. Turner, Phys. Rev. **D70**, 043528 (2004); T. Chiba, Phys. Lett. **B575**, 1 (2003). A. Füzfa and J. -M. Alimi, Phy. Rev. Lett. **97**, 061301 (2006). A. Füzfa and J. -M. Alimi, Phy. Rev. **D75**, 123007 (2007). V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, JETP Lett. **77**, 201 (2003). U. Alam, V. Sahni, T. D. Saini and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. **344**, 1057 (2003). W. Zimdahl, D. Pavon, Gen.Rel.Grav. **36**, 1483 (2004); X. Zhang, Phys.Lett. **B611**, 1 (2005). Z.-L. Yi and T.-J. Zhang, Phys. Rev. **D75**, 083515 (2007). B. Chang, *et al.*, JCAP **0701**, 016 (2007); B. Chang, *et al.*, Chin. Phys. Lett. **24**, 2153 (2007). X. Zhang, Int.J. Mod. Phys. **D14**, 1597 (2005); M. R. Setare, J. Zhang, X. Zhang, JCAP **0703**, 007 (2007); J. Zhang, X. Zhang, H. Liu, arXiv:0705.4145. H. Wei, R.-G. Cai, Phys.Lett. **B655**, 1 (2007). V. Sahni, Y. Shtanov, JCAP **0311**, 014 (2003).

--- abstract: 'The zero-temperature Glauber dynamics of the random-field Ising model describes various ubiquitous phenomena such as avalanches, hysteresis, and related critical phenomena. Here, for a model on a random graph with a special initial condition, we derive exactly an evolution equation for an order parameter. Through a bifurcation analysis of the obtained equation, we reveal a new class of cooperative slow dynamics with the determination of critical exponents.' author: - Hiroki Ohta - 'Shin-ichi Sasa' title: 'A universal form of slow dynamics in zero-temperature random-field Ising model' --- [Department of Pure and Applied Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8902, Japan]{} Slow dynamical behaviors caused by cooperative phenomena are observed in various many-body systems. In addition to well-studied examples such as critical slowing down [@H-H], phase ordering kinetics [@Bray], and slow relaxation in glassy systems [@Cavagna], seemingly different phenomena from these examples have also been discovered successively. In order to judge whether or not an observed phenomenon is qualitatively new, one needs to determine a universality class including the phenomenon. In this context, it is significant to develop a theoretical method for classifying slow dynamics. Here, let us recall a standard procedure for classifying equilibrium critical phenomena. First, for an order parameter $m$ of a mean-field model, a qualitative change in the solutions of a self-consistent equation $m={\cal F}(m)$ is investigated; then, the differences between the results of the mean-field model and finite-dimensional systems are studied by, for example, a renormalization group method. On the basis of this success, an analysis of the dynamics of a typical mean-field model is expected to be a first step toward determining a universality class of slow dynamics. As an example, in the fully connected Ising model with Glauber dynamics, an evolution equation for the magnetization, $\partial_t m={\cal G}(m)$, can be derived exactly. The analysis of this equation reveals that the critical behavior is described by a pitchfork bifurcation in the dynamical system theory [@Gucken]. As another example, an evolution equation for a time-correlation function and a response function was derived exactly for the fully connected spherical $p$-spin glass model [@CHZ; @Kurchan]. The obtained evolution equation represents one universality class related to dynamical glass transitions. The main purpose of this Letter is to present a non-trivial class of slow dynamics by exactly deriving an evolution equation for an order parameter. The model that we consider is the zero-temperature Glauber dynamics of a random-field Ising model, which is a simple model for describing various ubiquitous phenomena such as avalanches, hysteresis, and related critical phenomena [@Inomata; @Sethna; @Vives; @Durin; @Shin1]. As a simple, but still non-trivial case, we analyze the model on a random graph [@fn:global], which is regarded as one type of Bethe lattices [@MP]. Thus far, several interesting results on the quasi-static properties of the model on Bethe lattices have been obtained [@Duxbury1; @Illa; @Dhar1; @Colaiori1; @Alava; @Rosinberg0; @Rosinberg1]. In this Letter, by performing the bifurcation analysis of the derived equation, we determine the critical exponents characterizing singular behaviors of dynamical processes. #### Model: {#model .unnumbered} Let $G(c,N)$ be a regular random graph consisting of $N$ sites, where each site is connected to $c$ sites chosen randomly. For a spin variable $\sigma_i= \pm 1$ and a random field $h_i'$ on the graph $G(c,N)$, the random-field Ising model is defined by the Hamiltonian $$H=-\frac{1}{2}\sum_{i=1}^N\sum_{j\in B_i} \sigma_{i}\sigma_j-\sum_{i=1}^N (h+{h}_i')\sigma_i, \label{model}$$ where $B_i$ represents a set of sites connected to the $i$ site and $h$ is a uniform external field. The random field ${h}_i'$ obeys a Gaussian distribution $D_R({h}_i')$ with variance $R$. We collectively denote $(\sigma_i)_{i=1}^N$ and $(h_i)_{i=1}^N$ by ${{\boldsymbol \sigma}}$ and ${{\boldsymbol h}}$, respectively. Let $u_i$ be the number of upward spins in $B_i$. Then, for a given configuration, we express the energy increment for the spin flip at $i$ site as $-2\sigma_i\Delta_i$, where $$\Delta_i\equiv c-2u_i-(h+{h}_i'). \label{Delta:def}$$ The zero-temperature Glauber dynamics is defined as a stochastic process in the limit that the temperature tends to zero for a standard Glauber dynamics with a finite temperature. Specifically, we study a case in which the initial condition is given by $\sigma_i=-1$ for any $i$. In this case, once $\sigma_i$ becomes positive, it never returns. Thus, the time evolution rule is expressed by the following simple rule: if $\sigma_i=-1$ and $u_i$ satisfies $\Delta_i \le 0 $, the spin flips at the rate of $1/\tau_0$; otherwise, the transition is forbidden. Note that $\sigma_i(t)=-1$ when $\Delta_i(t) >0$, and $\Delta_i(t)$ is a non-increasing function of $t$ in each sample [@Dhar1]. In the argument below, a probability induced by the probability measure for the stochastic time evolution for a given realization ${{\boldsymbol h}}$ is denoted by $P^{{{\boldsymbol h}}}$, and the average of a quantity $X$ over ${{\boldsymbol h}}$ is represented by $\overline{X}$. #### Order parameter equation: {#order-parameter-equation .unnumbered} We first note that the local structure of a random graph is the same as a Cayley tree. In contrast to the case of Cayley trees, a random graph is statistically homogeneous, which simplifies the theoretical analysis. Furthermore, when analyzing the model on a random graph in the limit $N \to \infty$, we may ignore effects of loops. Even with this assumption, the theoretical analysis of dynamical behaviors is not straightforward, because $\sigma_j$ and $\sigma_k$, $j, k \in B_i$, are generally correlated. We overcome this difficulty by the following three-step approach. The first step is to consider a modified system in which $\sigma_i=-1$ is fixed irrespective of the spin configurations. We denote a probability in this modified system by $Q^{{{\boldsymbol h}}}$. We then define $q(t)\equiv\overline{Q^{{{\boldsymbol h}}}(\sigma_j(t)=1)}$ for $j \in B_i$, where $q(t)$ is independent of $i$ and $j$ owing to the statistical homogeneity of the random graph. The second step is to confirm the fact that any configurations with $\Delta_i(t)>0$ in the original system are realized at time $t$ in the modified system as well, provided that the random field and the history of a process are identical for the two systems. This fact leads to a non-trivial claim that $P^{{{\boldsymbol h}}}(\Delta_i(t) > 0)$ is equal to $Q^{{{\boldsymbol h}}}(\Delta_i(t) > 0)$. By utilizing this relation, one may express $P^{{{\boldsymbol h}}}(\Delta_i(t) > 0)$ in terms of $Q^{{{\boldsymbol h}}}(\sigma_j(t)=1)$. The average of this expression over ${{\boldsymbol h}}$, with the definition $\rho(t)\equiv\overline{P^{{{\boldsymbol h}}}(\Delta_i(t) > 0)}$, leads to $$\rho(t) = \sum_{u=0}^{c} \left( \begin{array}{c} c \\ u \end{array} \right) q(t)^{u}(1-q(t))^{c-u} \int_{-\infty}^{c-2u-h} dh' D_R(h'), \label{Gq}$$ where we have employed the statistical independence of $\sigma_j$ and $\sigma_k$ with $j,k \in B_i$ in the modified system. The expression (\[Gq\]) implies that $q(t)$ defined in the modified system has a one-to-one correspondence with the quantity $\rho(t)$ defined in the original system. The third step is to define $p(t)\equiv \overline{Q^{{{\boldsymbol h}}}(\sigma_j=-1, \Delta_j \le 0)}$ and $r(t)\equiv \overline{Q^{{{\boldsymbol h}}}(\Delta_j(t) > 0)}$ for $j \in B_i$. Then, by a procedure similar to the derivation of (\[Gq\]), we find that $dq(t)/dt$ is equal to $p(t)/\tau_0$. $r(t)$ is also expressed as a function of $q(t)$ because $r(t)$ is equal to a probability of $\Delta_j(t) > 0$ in a modified system with $\sigma_i=-1$ and $\sigma_j=-1$ fixed. Concretely, we write $r(t)=1-F(q(t))$, where $$F(q) = \sum_{u=0}^{c-1} \left( \begin{array}{c} c-1 \\ u \end{array} \right) q^{u}(1-q)^{c-1-u} \int_{c-2u-h}^{\infty} dh' D_R(h'). \label{Fq}$$ By combining a trivial relation $p(t)+q(t)+r(t)=1$ with these results, we obtain $$\begin{aligned} \tau_0{\dfrac{d q}{d t}} =F(q)-q, \label{evol:q}\end{aligned}$$ which determines $q(t)$ with the initial condition $q(0)=0$. Here, for a spin configuration at time $t$ under a quenched random field $h_i'$, we define $${{\hat\rho}}(t) \equiv \frac{1}{N}\sum_{i=1}^N\theta(\Delta_i(t)),$$ where $\theta(x)=1$ for $x > 0$ and $\theta(x)=0$ otherwise. Due to the law of large numbers, ${{\hat\rho}}(t)$ is equal to $\rho(t)$ in the limit $N \to \infty$. Since $\rho(t)$ is determined by (\[Gq\]) with (\[evol:q\]), one may numerically compare $\rho(t)$ with ${{\hat\rho}}(t)$ observed in the Monte Calro (MC) simulations. We confirmed that these two results coincided with each other within numerical accuracy. We also numerically checked the validity of (\[evol:q\]) on the basis of the definition of $q$. As a further evidence of the validity of our results, we remark that the stationary solution $q(\infty)$ satisfies $q(\infty)=F(q(\infty))$, which is identical to the fixed-point condition of a recursive equation $Z_{n+1}=F(Z_n)$ in the Cayley tree, where $Z_n$ is a probability at generation $n$. (See Ref. [@Dhar1] for the precise definition of $Z_n$.) In the argument below, we set $\tau_0=1$ without loss of generality and we investigate the case $c=4$ as an example. The obtained results are essentially the same for $ c > 4$; however, the behaviors for $c<4$ are qualitatively different from those for $ c \ge 4$. ![(color online) Phase diagram. Inset: $F(q)-q$ as functions of $q$. $R=1.5$. $h=1.0$ and $h=h_c(R)$.[]{data-label="phase"}](fig1.eps){width="6.5cm"} #### Bifurcation analysis: {#bifurcation-analysis .unnumbered} We start with the analysis of (\[evol:q\]) for $R =1.5$ and $h =1.0$. The qualitative behavior of $q(t)$ is understood from the shape of the graph $F(q)-q$ shown in the inset of Fig. \[phase\]. There are three zeros, $q_1$, $q_2$, and $q_3$, where $0 < q_1 < q_2 <q_3 <1 $. Since $F(q) > q$ in the interval $[0,q_1)$, $q(t) \to q_1$ as $t \to \infty$ with the initial condition $q(0)=0$. This geometrical structure sustains in a region of the parameter space $\alpha\equiv (R,h)$, as shown in Fig. \[phase\]. Let $q_i(\alpha)$ be the parameter dependence of $q_i$. Then, the stable fixed point $q_1(\alpha)$ and the unstable fixed point $q_2(\alpha)$ merge at the solid line, and the stable fixed point $q_3(\alpha)$ and the unstable fixed point $q_2(\alpha)$ merge at the dashed line, both of which correspond to [*saddle-node bifurcations*]{} [@Gucken]. The two lines terminate at a critical point $({R_{\rm sp}}, {h_{\rm sp}})$. Since the trajectories with the initial condition $q(0)=0$ do not exhibit a singularity at the dashed line, only the bifurcations at the solid line are relevant in the present problem. The solid line is called a spinodal line [@Dahmen0]. Now, we investigate the singular behaviors of slow dynamics near the bifurcation points. We first fix the value of $R$ such that $0 < R \ll {R_{\rm sp}}$. Then, a saddle-node bifurcation occurs at $(R,{h_{\rm c}}(R))$ on the solid line. Let ${q_{\rm c}}(R)$ be the saddle-node point such that $q_1=q_2={q_{\rm c}}(R)$. We set $h={h_{\rm c}}(R)+{\epsilon}$ and $q={q_{\rm c}}+u$. From the graph in the inset of Fig. \[phase\], one finds that (\[evol:q\]) becomes $${\dfrac{d u}{d t}}=a_0{\epsilon}+a_2 u^2+O(|u|^3, |{\epsilon}u|), \label{norm-1}$$ when $ |u| \ll 1$ and $|{\epsilon}| \ll 1$. $a_0$ and $a_2$ are numerical constants. Solutions of (\[norm-1\]) are expressed as a scaling form $$u(t)=|{\epsilon}|^{1/2} \bar u_{\pm} (|{\epsilon}|^{1/2} t)$$ when $ |{\epsilon}| \ll 1 $, where $\bar u_{+} $ and $\bar u_{-}$ are ${\epsilon}$-independent functions for ${\epsilon}>0 $ and ${\epsilon}<0$, respectively. The result implies that the characteristic time near $q={q_{\rm c}}$ diverges as $ \tau \simeq |{\epsilon}|^{-1/2}. $ Note that $q(t \to \infty)=q_3$ when ${\epsilon}>0$. Therefore, $q(t \to \infty)$ exhibits a discontinuous change, and the jump width is given by the distance between $q_1(\alpha)=q_2(\alpha)$ and $q_3(\alpha)$ at ${\epsilon}=0$. Next, we focus on the dynamical behaviors near the critical point $({R_{\rm sp}},{h_{\rm sp}})$. By substituting $q={q_{\rm c}}({R_{\rm sp}})+v$ and $(R,h)=({R_{\rm sp}},{h_{\rm sp}})+(\eta,{\epsilon})$ into (\[evol:q\]), we obtain $${\dfrac{d v}{d t}}=c_0 {\epsilon}+ c_1\eta v-c_3 v^3 +O(|v|^4, |{\epsilon}v|, |\eta v^2|), \label{norm-2}$$ when $|v| \ll 1$, $|{\epsilon}| \ll 1$, and $|\eta| \ll 1$. $c_0$, $c_1$, and $c_3$ are numerical constants. System behaviors are classified into two types. First, when $|\eta| \gg |{\epsilon}|^{2/3}$, solutions of (\[norm-2\]) are expressed as $v=|\eta|^{1/2}\bar v_{1\pm}(|\eta| t)$ when $|\eta| \ll 1$. This scaling form is identical to that near a pitchfork bifurcation, which might be related to conjectures presented in Refs. [@Perez; @Colaiori2; @Dahmen1]. Second, when $ |\eta| \ll |{\epsilon}|^{2/3}$, which includes the case in which $R={R_{\rm sp}}$ is fixed, solutions of (\[norm-2\]) are expressed as $$v=|{\epsilon}|^{1/\delta}\bar v_{2 \pm}(|{\epsilon}|^{\zeta} t) \label{scaling:1}$$ when $ |{\epsilon}| \ll 1$, with $\delta=3$ and $\zeta=2/3$. The characteristic time diverges as $\tau \simeq |{\epsilon}|^{-\zeta}$ near the critical point $({R_{\rm sp}},{h_{\rm sp}})$. In addition to the two scaling forms, one can calculate the width of the discontinuous jump of $q$ along the spinodal line, which is proportional to $(-\eta)^{1/2}$ near the critical point [@Colaiori1; @Duxbury1]. #### Finite size fluctuations: {#finite-size-fluctuations .unnumbered} In a system with large but finite $N$, fluctuations of $\hat \rho$ are observed. Their basic characterization is given by the intensity $$\chi_\rho(t)\equiv N {\left\langle}({{\hat\rho}}(t)- {\left\langle}{{\hat\rho}}(t) {\right\rangle})^2 {\right\rangle}.$$ where for a quantity $\hat{X}(t)$ determined by ${{\boldsymbol \sigma}}(t)$ and ${{\boldsymbol h}}$, ${\left\langle}\hat{X}(t){\right\rangle}\equiv\overline{\sum_{{{\boldsymbol \sigma}}} P^{{{\boldsymbol h}}}({{\boldsymbol \sigma}}(t)={{\boldsymbol \sigma}})\hat{X}(t)|_{{{\boldsymbol \sigma}}(t)={{\boldsymbol \sigma}}}}$. The problem here is to determine a singular behavior of $\chi_\rho$ under the condition that $0 < {\epsilon}\ll 1$ and $\eta=0$. Let ${\rho_{\rm c}}$ be defined by (\[Gq\]) with $q={q_{\rm c}}({R_{\rm sp}})$. We then assume $$\begin{aligned} {{\hat\rho}}(t)-{\rho_{\rm c}}=A({\epsilon}, N)\hat{F}(t/\tau({\epsilon}, N))\end{aligned}$$ near ${{\hat\rho}}(t) \simeq {\rho_{\rm c}}$, where $A$ and $\tau$ are typical values of the amplitude and the characteristic time, respectively, and $\hat{F}$ is a time-dependent fluctuating quantity scaled with $A$ and $\tau$. We further conjecture finite size scaling relations $$\begin{aligned} A({\epsilon},N) &=& N^{-1/(\nu \delta)} F_A({\epsilon}N^{1/\nu}), \nonumber \\ \tau({\epsilon},N) &=& N^{\zeta/\nu}F_\tau({\epsilon}N^{1/\nu}), \label{tau:scale}\end{aligned}$$ where $F_A(x) \simeq x^{1/\delta}$ for $x \gg 1$ and $F_\tau(x) \simeq x^{-\zeta}$ for $x \gg 1$; $F_A(x) ={\rm const.}$ and $F_\tau(x) ={\rm const.}$ for $x \ll 1$. Here, the exponent $\nu$ characterizes a cross-over size ${N_{\rm c}}$ between the two regimes as a power-law form ${N_{\rm c}}\simeq {\epsilon}^{-\nu}$. We thus obtain $$\chi_\rho(\tau({\epsilon},N))=N^{\gamma/\nu}F_\chi({\epsilon}N^{1/\nu}), \label{fluc:scale}$$ where $\gamma=\nu-2/\delta$. The values of $\zeta$ and $\delta$ have already been determined. We derive the value of $\nu$ in the following paragraph. It is reasonable to assume that the qualitative behavior of $\hat \rho$ near the critical point $({R_{\rm sp}},{h_{\rm sp}})$ is described by (\[norm-2\]) with small fluctuation effects due to the finite size nature. There are two types of fluctuation effects: one from the stochastic time evolution and the other from the randomness of $h_i'$. The former type is expressed by the addition of a weak Gaussian-white noise with a noise intensity proportional to $1/N$, whereas the latter yields a weak quenched disorder of the coefficients of (\[norm-2\]). In particular, ${\epsilon}$ is replaced with ${\epsilon}+ \hat g/\sqrt{N}$, where $\hat g$ is a time-independent quantity that obeys a Gaussian distribution. Then, two characteristic sizes are defined by a balance between the fluctuation effects and the deterministic driving force. As the influence of the stochastic time evolution rule, the size $N_{\rm s}$ is estimated from a dynamical action of the path-integral expression $\int dt[ N c_4 (\partial_t v-c_{0} {\epsilon}+c_3 v^3)^2+c_5 v^2]$, where the last term is a so-called Jacobian term. $c_4$ and $c_5$ are constants. In fact, the balance among the terms ${\epsilon}^2 \simeq v^6 \simeq v^2/N_{\rm s}$ leads to $N_{\rm s} \simeq {\epsilon}^{-4/3}$. (See Refs. [@Iwata; @Ohta2] for a similar argument.) On the other hand, the size $N_{\rm q}$ associated with the quenched disorder is determined by the balance ${\epsilon}\simeq 1/\sqrt{N_{\rm q}}$, which leads to $N_{\rm q} \simeq {\epsilon}^{-2}$. Since $N_{\rm s} \ll N_{\rm q}$ for ${\epsilon}\ll 1$, the system is dominated by fluctuations when $N \le N_{\rm q}$. With this consideration, we conjecture that ${N_{\rm c}}=N_{\rm q} \simeq {\epsilon}^{-2}$. That is, $\nu=2$. ![(color online) Time evolution of the magnetization for $R=1.781258(\simeq {R_{\rm sp}})$ and $h=1.1$. Note that ${h_{\rm sp}}=1$. The circles are the result obtained by the MC simulations with $N=10^5$, and the solid line corresponds to the solution of $\partial_t m=-(m-1+2\rho)$ with (\[Gq\]) and (\[evol:q\]). One may derive this equation for $m$ in a straightforward manner. Inset: ${\chi_{\rm m}}N^{-2/3}$ as functions of ${\epsilon}N^{1/2}$. The guideline represents ${\chi_{\rm m}}\simeq {\epsilon}^{-4/3}$. []{data-label="scale"}](fig2.eps){width="6.5cm"} In laboratory and numerical experiments, statistical quantities related to the magnetization $\hat m= \sum_{i=1}^N \sigma_i/N$ may be measured more easily than $\chi_\rho$. Since $\hat m$ is not independent of ${{\hat\rho}}$, $\hat m$ also exhibits singular behaviors near the critical point. Concretely, the fluctuation intensity $\chi(t)\equiv N \left({\left\langle}\hat{m}^2(t) {\right\rangle}-{\left\langle}\hat{m}(t){\right\rangle}^2\right)$ is characterized by the above exponents. In order to confirm this claim, we measured $\chi(t)$ by MC simulations. Then, the characteristic time is defined as a time $t_*$ when $\chi(t)$ takes a maximum value ${\chi_{\rm m}}$. Our theory predicts $t_* \simeq {\epsilon}^{-2/3}$ and ${\chi_{\rm m}}=N^{2/3} {\bar\chi_{\rm m}}({\epsilon}N^{1/2})$ with ${\bar\chi_{\rm m}}(x) \simeq x^{-4/3}$ for $x \gg 1$. The numerical results shown in Fig. \[scale\] are consistent with the theoretical predictions. #### Concluding remarks: {#Remark .unnumbered} We have derived the exact evolution equation for the order parameter $\rho$ describing the dynamical behaviors of a random field Ising model on a random graph. From this dynamical system, we have determined the values of the critical exponents: $\zeta=2/3$, $\delta=3$, $\nu=2$, and $\gamma=4/3$. Before ending this Letter, we discuss the critical phenomena in $d$-dimensional random-field Ising models on the basis of our theoretical results. Let $\nu_{\rm m}$ be the critical exponent characterizing the divergence of a correlation length above an upper-critical dimension $d_{\rm u}$. By assuming a standard diffusion coupling for (\[norm-2\]) as an effective description of finite-dimensional systems, we expect $z\equiv\zeta/\nu_{\rm m}=2$. This leads to $\nu_{\rm m}=1/3$. From an application of a hyper-scaling relation to the upper-critical dimension, one also expects the relation $\nu=d_{\rm u}\nu_{\rm m}$ [@Pfeuty]. This leads to $d_{\rm u}=\nu/\nu_{\rm m}=6$, which is consistent with the previous result [@Dahmen0]. We then denote the exponents characterizing the divergences of time and length scales by $\nu_{3}$ and $\zeta_{3}$, respectively, for three-dimensional systems. These values were estimated as $\zeta_3\simeq 1.3$ and $\nu_3\simeq 0.7$ in numerical experiments [@Carpenter; @Perez; @Ohta1]. The theoretical analysis of finite dimensional systems will be attempted as an extension of the present work; this might be complementary to previous studies [@Dahmen0; @Muller]. Finally, we wish to mention that one of the most stimulating studies is to discover such a universality class in laboratory experiments by means of experimental techniques that capture spatial structures directly [@Shin1; @Shin2; @Durin]. We acknowledge M. L. Rosinberg, G. Semerjian, S. N. Majumdar, H. Tasaki, and G. Tarjus for the related discussions. This work was supported by a grant from the Ministry of Education, Science, Sports and Culture of Japan, Nos. 19540394 and 21015005. One of the authors (H.O.) is supported by a Grant-in-Aid for JSPS Fellows. [99]{} P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. [**49**]{}, 435 (1977). A. J. 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--- abstract: 'Short electron pulses are demonstrated to trigger and control magnetic excitations, even at low electron current densities. We show that the tangential magnetic field surrounding a picosecond electron pulse can imprint topologically protected magnetic textures such as skyrmions in a sample with a residual Dzyaloshinskii-Moriya spin-orbital coupling. Characteristics of the created excitations such as the topological charge can be steered via the duration and the strength of the electron pulses. The study points to a possible way for a spatio-temporally controlled generation of skyrmionic excitations.' author: - 'A. F. Schäffer$^1$, H. A. Dürr$^2$, J. Berakdar$^1$' title: Ultrafast imprinting of topologically protected magnetic textures via pulsed electrons --- Tremendous progress has been made towards the realization of spatiotemporally controlled electron sources for probing the materials local structural, electronic and magnetic dynamics [@1; @2; @3; @4]. Working schemes rely on the electron emission from a laser-irradiated nanoscale apex [@5; @6; @7; @8; @9; @10; @11; @12; @13; @14; @15; @16; @17; @18; @19; @20; @21; @22] with the electron pulse duration being controllable with the laser pulse duration. The laser intensity dictates the electron number in the bunch. Electron pulse acceleration and control is achievable by intense THz)fields [@23; @24; @25]. Here we explore the potential of very fast, relativistic electron bunches for a possible control of the magnetic dynamics in a thin film which is traversed by the electrons. Our focus is on the sample spin dynamics triggered by the electric and magnetic fields associated with the electron bunch [@26]. In fact, a pioneering experiment [@27] explored the ultimate speed limit for precessional magnetic dynamics of CoCrPt film driven by the magnetic field ${\mbox{\boldmath$\mathrm{B}$}}({\mbox{\boldmath$\mathrm{r}$}},t)$ of short relativistic electron pulses (with a duration of $\delta = 2.3\,$ps) passing a 14nm thin film of granular CoCrPt ferromagentic samples with grain sizes of $20.6\pm 4$nm. The main experimental results are shown along with our simulations in Fig.\[fig\_comp\]. Prior to the electron-pulse the sample was magnetized homogeneously in $z$ direction. The pulse induced ring pattern of the magnetic domains pointing either up or down (with respect to the easy direction of the magnetic films) is well captured by our micromagnetic simulations and can be interpreted by the analytical model enclosed in the supplementary materials. As pointed out in [@27], the critical precessional angle $\phi\geq\pi/2$ is determined by the local strength of the magnetic field and indicates the achieved angular velocity $\omega$. The pulse duration $\delta$ plays a crucial role [@28]. As discussed in Ref.[@28], an appropriate sequence of ps pulses allows for an optimal control scheme achieving a ballistic magnetic switching, even in the presence of high thermal fluctuations. Longer pulses might drive the system back to the initial state [@28]. So, the critical precessional angle and $\delta$ are the two key parameters [@27] for the established final precessional angle $\phi=\omega\delta$. Note, the demagnetization fields are also relevant, as inferred from Fig. \[fig\_comp\] but they do not change the main picture (further details are in the supplementary materials). ![Comparison between experimental (a)[@27], and numerical results (b), (c). Both numerical simulations and the experimental data cover an area of $150\times 150\,\mu$m$^2$. In contrast to panel (b), in (c) the demagnetizing fields are included in simulations. The grey shading signals the magnetization’s $z$-component with white color meaning $m_z=+\hat{e}_z$ and black $m_z=-\hat{e}_z$. The electrons in the beam impinging normal to the sample have an energy of 28GeV. The pulse’s time-envelope is taken as a Gaussian with a pulse duration of $\sigma_t = 2.3\,$ps, which translates to a number of $n_e\approx 10^{10}$ electrons and an equivalent time-dependence of the generated Oersted field whose radial $\rho$ dependence away from the beam axis derives to $B(\rho)=54.7\,$T$\mu$m$/(\rho+\epsilon)$ (at the peak electron bunch intensity). The cut-off distance $\epsilon=40\,$nm is included in order to avoid a divergent behavior at the origin and can be understood as a rough approximation of the beam width. []{data-label="fig_comp"}](fig1.eps){width=".6\linewidth"} Having substantiated our methods against experiment we turn to the main focus of our study, namely the generation of topologically protected magnetic excitations such as skyrmions via the electron pulses. We consider samples exhibiting Dzyaloshinskii-Moriya (DM) spin-orbital coupling are appropriate. A recent work [@29] evidences that ultrathin nano discs of materials such Co$_{70.5}$Fe$_{4.5}$Si$_{15}$B$_{10} $[@30] sandwiched between Pt and Ru/Ta are well suited for our purpose. The magnetization’s structure may nucleate spontaneously into skyrmionic configurations. We adapted the experimentally verified parameters for this sample and present here the result for the magnetic dynamics triggered by short electron beam pulses. Taking a nano disc of a variable size the ground state with a topological number $|N|=1$ is realized after propagating an initially homogeneous magnetization in $\pm z$ direction according to the Landau-Lifshitz-Gilbert equation (LLG) including DM interactions. The two possible ground states, depending on the initial magnetization’s direction are shown in \[fig\_groundstate\] along with the material’s parameters.\ Our main focus is on how to efficiently and swiftly create skyrmions, an issue of relevance when it comes to practical applications. Previous theoretical predictions (e.g. [@31]) utilize a spin-polarized current for the skyrmion generation. Large currents densities and a finite spin polarization of injected currents are needed, however. Thus, it is of interest to investigate the creation and annihilation of skyrmions with current pulses similar to those discussed above using the surrounding magnetic field. Of interest is the skyrmion generation and modification via a nano-focussed relativistic electron pulse. While currently such pulses can be generated with micron size beam dimensions [@ued_ref] future sources are expected to reach ficus sizes down to the few nm range [@32]. In principle the possibility of beam damage occurring in the beam’s focus as in the case of the experiment in ref.[@27] is present. However, ongoing experiments with relativistic electron beams [@ued_ref] indicate that the use of ultra thin freestanding films may alleviate damage concerns.\ Topologically protected magnetic configurations, like magnetic skyrmions, are well defined quasiparticles. They can be characterized mathematically by the topological or winding number $N=\frac{1}{4\pi}\int {\mbox{\boldmath$\mathrm{m}$}}\cdot\left(\frac{\partial {\mbox{\boldmath$\mathrm{m}$}}}{\partial x}\times\frac{\partial {\mbox{\boldmath$\mathrm{m}$}}}{\partial y}\right){\mathrm{d}}x{\mathrm{d}}y$[@33] which counts simply how often the unit vector of the magnetization wraps the unit sphere when integrated over the two-dimensional sample. Therefore, skyrmions are typically a quasiparticle in thin (mono)layers. The topological number adopts integer values indicating the magnetic configuration to be skyrmionic ($N=\pm 1$) or skyrmion multiplexes ($|N| >1$). If the topological number is not an integer the topological protection is lifted and the magnetic texture is unstable upon small perturbations. The topological stability of skyrmionic states stem from the necessity of flipping at least one single magnetic moment by $180^\circ$, to overcome the barrier and transfer the state into a “trivial” state, like a single domain or vortex state. In the following, we will attempt to overcome this energy barrier with the previous methods so that the magnetization will be converted into a state with a different topological invariant. Advantageous is the spatial structure of the magnetic field curling around the beam’s center, which gives a good point of action in order to manipulate topologically protected configurations.\ ![Magnetic ground states for a nano disc with a diameter of 300nm and a thickness of 1.5nm. The material parameters are $M\ind{sat}=450\times 10^3\,$A/m, $A\ind{ex}=10\,$pJ/m, $\alpha=0.01$, $K_u=1.2\times 10^5\,$J/m$^3$ (out-of plane anisotropy), and the interfacial DMI-constant $D\ind{ind}=0.31\times 10^{-3}\,$mJ/m$^2$. (a) corresponds to $N=1$, whereas (b) possesses $N=-1$, both skyrmions are of the Néel type. Bottom panel illustrates pictorially the influence of the magnetic field associated with the electron bunch. The cones correspond to the initial magnetic configuration as in (a) and (b), whereas the golden arrows show the induced magnetic field. The resulting torque points perpendicular to the magnetization, affecting the magnetic configuration accordingly. []{data-label="fig_groundstate"}](fig2.eps){width="0.6\linewidth"} Using the short electron pulses one may overcome the topological energy barrier with a magnetic “kick” and the magnetization relaxes afterwards, possessing a different winding number. In contrast to the mesoscopic system studied in \[fig\_comp\], in the following not only the far field, but also the near magnetic field of the Gaussian pulses will be treated. To do this the Biot-Savart law is solved numerically and fitted with a model function. For magnetic systems a minimum time of exposure is necessary, whereas the spatial focus of the beam is limited. To overcome this conflict, the pulse duration is fixed at $2.3\,$ps as before, when nothing different is mentioned. Details on the resulting magnetic field can be seen in the supplementary material. Starting from such an electron beam two main parameters can be adjusted to achieve the favored reaction of the nanodiscs. Those are the pulse width and the number of electrons, which will be treated independently. In \[fig\_dur\], the final topological charges after a single Gaussian electron pulse irradiating a nanodisc are plotted as a function of the number of electrons and the width of the Gaussian distributed electrons. The results do not show the transient time evolution of the sample but only the final steady-state values of the winding number. They are obtained by applying an electron pulse, propagating the magnetization during the pulse, and relaxing the magnetic configuration afterwards as to approach a local minimum of the free energy’s hypersurface. ![Varying the number of electrons per pulse or the spatial enlargement of the pulse, the imprinted topological charge can be tuned. The pulse duration is set to $2.3\,$ps. Black and green curves correspond respectively to starting with a magnetic ordering having $+1$ or $-1$ topological charge, as shown in \[fig\_groundstate\] for different pulse widths. Both the blue and red curve start from $N_i=+1$. The sample is a magnetic disc (diameter $d=300\,$nm) which is irradiated with a Gaussian beam pulse with $\sigma_{xy}=30\,$nm (and $90\,$nm) in case of the bottom graph, respectively the upper graphs’ beam has a constant number of $n_e=10^8$ electrons. []{data-label="fig_dur"}](fig3.eps){width="0.6\linewidth"} We note the strong correlation between the change of the topological charge and the number of electrons or accordingly the beam width. Relatively large intervals of both parameters lead to the same final values for $N$. We note that not only the variation of these control parameters, but also of the duration of the pulse is experimentally accessible, particularly in a nanoapex ultrafast transmission electron microscope. Noteworthy, the graphs for opposite initial configurations (see \[fig\_dur\](a)) are axially symmetric with respect to the $x$ axis. This can be explained by the coinciding symmetry centers of the pulse and the skyrmionic structure. This symmetric and robust behavior can be exploited to switch between the accessible different values for the topological charges which are quite close to the ideal integer values that would be realized in an infinitely small discretization.\ Interestingly, the switching between the two stable states occurs repetitively for an increase in the number of electrons, whereas the spatial manipulation of the beam leads to one regime only in which the fields are sufficient to switch the topological number. The first observation can be explained with the schematics shown in fig.\[fig\_groundstate\]c). Depending on the strength of the pulse the magnetic moments are forced to rotate multiple times around the $\hat{e}_\varphi$ vector in a collective manner, as each moment of equal distance to the center experiences the same torque. The final position of the surrounding moments couples backwards to the center and determines the new topological charge. The electron number linearly translates to the peak magnetic field, wheras the beam width has a more complicated influence. When the width is inceased the spatial profile in the $xy$-plane is manipulated, as the maximum magnetic field is shifted towards the disc’s rim and beyond. How the system reacts on this changes depends crucially on the exact profile of the beam, especially on the point of maximum magnetic field strength, as can be seen in \[fig\_dur\](a). This leads to the question of the optimum parameter regime, to manipulate the system reliably, which can not finally be answered as it strongly depends on the experimentally available capabilities. Hence this work focuses on an examplary study on the effect.\ The same switching phenomenon as discussed before can also be observed for different setups. Weaker pulses, as long as they are able to overcome the internal fields to excite the system, can be used as well, but obviously the field’s amplitude translates to the strength of the resulting torque. This implies a longer radiation time needed for pulses of lower intensity to be capable to switch the system. In case of different materials or geometries the accessible topological states have to be investigated, before they can be utilized. Otherwise undesired, interstitial states might be achieved by accident and the switching is not deterministic anymore.\ Aside from the manipulation of the topological charge of a given nano disc system, the creation of skyrmions on extended thin layers is an open challenge. To treat this, we start from a quadratic region with a size of $(800\times 800)\,$nm$^2$ and periodic boundary conditions in $x$ and $y$ direction to avoid finite size effects. To overcome the homogeneously magnetized state, the peak intensity has to be increased further, as well as a focussing down to the scale of the desired skyrmion. The Gaussian profile in the $xy$-plane has a standard deviation of $30\,$nm, whereas the pulse duration is reduced to $500\,$fs and a single pulse includes $10^8$ electrons. Even though this is experimentally challenging, it is necessary to create skyrmions on an extended film. If the beam size is too large, only trivial domain rings are built up. On the other hand the duration has to be long enough to allow the magnetic texture to react to the pulse. A well known characteristics for magnetic skyrmions is a tendency to a blow-up behavior to minimize the exchange energy, when no external stabilization is present. In nano discs the stabilizing factor is the geometric confinement. Several experimental works incorporate an additional external magnetic field perpendicular to the surface to amplify the uniaxial anisotropy in an attempt to block the blow-up effect. As we are mainly interested in the generation of topological defects, we focus on this aspect keeping in mind that a stabilization of the induced skyrmions is necessary to maintain their localized structure. ![Evolution of the topological charge after a single Gaussian beam with $n_e=10^8$ electrons, $\sigma_{xy}=30\,$ nm and $\sigma_t=0.5\,$ps. The initial magnetization points homogeneously in $z$ direction. The system size is $(800\times 800)$nm$^2$ with periodic boundary conditions. On the bottom we present a few snap-shots corresponding to the red curve at time moments depicted on each snapshot. The first and second picture is scaled with a factor of three in order to make the development clearer.[]{data-label="fig_EvHom"}](fig4.eps){width="0.6\linewidth"} As can be seen in \[fig\_EvHom\], the irradiation by the electron beam leads to the injection of a topological charge. The used pulse has the peak intensity at $5\sigma_t=2.5\,$ps to account for its finite rising time. After short oscillations the system relaxes to a skyrmionic state. On the time scale of the electron pulse, the state’s topological nature is perfectly stable but the excitations tend to expand when no further stabilizations are present. Therefore, the diameter of the skyrmion increases and the type changes from Néel to Bloch, the topological charge is still conserved. The situation changes markedly for longer pulses ($\sigma_t \gtrsim 20\,$ps). Just like before the pulse leads to topological excitations. On top of this, a domain-wall ring similar to the results presented above is induced which shields the included skyrmions in two different ways (see \[fig\_ringSk\]). ![Magnetization configuration 1ns after a pulse with a duration of 22ps and correspondingly $10^{11}$ electrons focused on $\sigma_{xy}=10\,$nm. Four skyrmions in the inside of a domain-wall ring are protected against fluctuations outside the ring. The sample covers an area of $800\times 800\,$nm$^2$.“(Multimedia view)”[]{data-label="fig_ringSk"}](fig5.eps){width="0.5\linewidth"} The blow-up behavior is blocked due to the domain wall. The spin waves reflected from the open boundaries are absorbed by the domain ring so that the inside is not affected. In the supplementary [animation](./animation.avi)“(Multimedia view)” the formation of the structure in \[fig\_ringSk\] can be seen. In the first 220ps the time step between the single frames is 4.4ps. Afterwards the movie is accelerated to 22ps/frame which can be easily recognized by the arising spin-waves outside the domain ring. Applying another pulse the ring structure can be opened and the topological charge changed with the same process. Summarizing, we demonstrated the usefulness of ultrashort electron pulses for generating and steering the magnetization dynamics due to the electromagnetic fields associated with the electron pulses. In particular, topologically protected magnetic textures such as skyrmions can be imprinted and manipulated in controllable spatiotemporal way. Supplementary Material {#supplementary-material .unnumbered} ====================== See supplementary material for information on the analytical macrospin approach towards the results shown in \[fig\_comp\], further details on the numerical calculations and the magnetic fields induced by ultrafast electron pulses. Acknowledgements {#acknowledgements .unnumbered} ================ A. F. S. and J. 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--- author: - 'A. Luminari, F. Tombesi, E. Piconcelli, F. Nicastro, K. Fukumura, D. Kazanas, F. Fiore, L. Zappacosta' date: 'Received 27/09/19; accepted 25/11/19' title: 'On the importance of special relativistic effects in modelling ultra-fast outflows' --- Introduction ============ Outflows are ubiquitously observed from a variety of astrophysical sources and their impact on the surrounding environment depends on their energetic. In particular, mildly relativistic and ionised outflows from the innermost regions of Active Galactic Nuclei (AGNs) are often seen in UV and X-ray absorption spectra (e.g., [@Chartas; @T10; @R11; @B19]) and may carry sufficient energy to regulate both the growth of the central super-massive black hole (SMBH) and the evolution of the surrounding host galaxy ([@C14; @F12; @T15; @Z12]). This critically depends on the kinetic power of these outflows, which in turn depends on both their velocity and mass flux ([@Dimatteo; @KP15]). The line-of-sight velocity is typically inferred via the blue-shift of the absorption features imprinted by the outflowing material onto the continuum emission of the central source, compared to the systemic redshift of the host galaxy. The mass outflow rate $\Dot{M}_{out}$, instead, for a given covering factor and distance of the outflow, is estimated by measuring the optical depth of the absorption features. The observed optical depth is considered a proxy of the outflow column density $N_H$ along the line of sight, independently on its outflow velocity $v_{out}$. In this work we show that this assumption no longer holds for outflows escaping the central continuum source of radiation with velocities corresponding to a fraction of the speed of light $c$ (e.g. $v_{out} \buildrel > \over \sim 0.05 c$). For such outflows, the observed (i.e. apparent) optical depth of the spectral features produced by the absorbing material, significantly underestimates the intrinsic $N_H$ and, consequently, the mass transfer rate of the outflows. Therefore, a velocity-dependent correction must be adopted to account for this effect in the estimate of $N_H$. This pure special-relativistic effect is universal (i.e. applies to any fast-moving line-of-sight outflow), and affects not only our estimate of the kinetic power of the outflow but also the ability of the radiative source to effectively accelerate the outflow outwards. For AGN outflows, this may have deep implications on the feedback mechanism and the co-evolution with respect to the host galaxy ([@KH13]). The paper is organised as follows. In Sect. \[physics\] we overview the special relativity treatment for a fast-moving gas embedded in a radiation field. In Section \[prescription\] we show how to incorporate such treatment in modelling outflow spectra. In Section \[conclusions\] we discuss the results and their implications on estimating $\dot{M}_{out}, \dot{E}_{out}$, and we summarise in Sect. \[sect5\]. Special Relativistic Transformation in the Outflow Reference Frame {#physics} ================================================================== According to special relativity, the luminosity $L'$ seen by a clump of gas moving at relativistic speed is reduced of a factor $\Psi$, with respect to a static gas, as follows: $$L'=L\cdot \Psi \label{main}$$ where $L$ is the luminosity seen by an observer at rest and $\Psi$, i.e. the de-boosting factor, is defined as: $$\Psi\equiv\psi^4= \frac{1}{\gamma^4 (1-\beta cos(\theta))^4} \label{main_long}$$ where $\gamma \equiv \frac{1}{\sqrt{1-\beta^2}}$, $\beta=v_{out}/c$, $v_{out}$ is the gas velocity and $\theta$ is the angle between the incident luminosity $L$ and the direction of motion of the gas. Figure \[psi\] shows $\Psi$ as a function of $v_{out}$ for $\theta=180\ deg$, corresponding to a radial outward motion of the gas. The deboosting factor is due to the combination of the space-time dilatation in the gas reference frame, $K'$, and the relativistic Doppler shift of the received radiation ([@RL]). Using Eq. \[main\], the radiative intensity (i.e., the luminosity per solid angle) $\frac{dL'}{d\Omega'}$ received by the outflowing gas in $K'$ can be written as a function of the intensity in the rest frame $K$, as follows: $$\frac{dL'}{d\Omega'}= \Psi \frac{dL}{d\Omega} =\psi dE\cdot \psi^3\frac{1}{dt d\Omega} \label{expl}$$ where $dE, dt, d\Omega$ corresponds to the energy, time and solid angle intervals in $K$. Specifically, in Eq. \[expl\], $\psi dE$ is the energy transformation term, which represents the Doppler shift of the wavelengths in $K'$. The second term, $\psi^3\frac{1}{dt d\Omega}$, indicates a reduction of the intensity due to the space-time dilatation in $K'$. Noteworthy, Eq. \[main\] and \[expl\] also describe the emission from gas moving at relativistic velocity, as usually observed in high velocity systems such as jets in Blazars and GRBs ([@Urry; @G93]). When radiation is emitted along the direction of motion, i.e. $\theta\approx0\ deg$, $\Psi$ increases with increasing $v_{out}$, while $\Psi\leq1$ when it is emitted perpendicularly or backward ($\theta=90\ deg$ and $180\ deg$, respectively). The overall result is to concentrate the emitted radiation into a narrow cone along the direction of motion, an effect known as “relativistic beaming” ([@RL; @EHT]). Another way of describing the reduction of the luminosity seen by the outflowing gas is the following. In $K'$, the luminosity source appears as moving away with velocity $v_{out}$ and $\theta=180\ deg$ (for a pure radial motion), which results into a de-boosting of the received luminosity due to the relativistic beaming, according to Eq. \[main\]. ![Deboosting factor $\Psi$ in the gas reference frame $K'$ as a function of $v_{out}$ assuming $\theta=180$. For speeds lower than 0.1 the speed of light the radiation intercepted by the outflow and the (rest-frame) observer at infinity are virtually the same. For higher speeds, the fraction of intercepted radiation drops dramatically due to special relativistic effects. []{data-label="psi"}](psi_long.pdf){width="9.5cm"} Modelling Outflow Absorption Spectra Including Special Relativistic Effects {#prescription} =========================================================================== We propose to include these special relativistic corrections in modelling spectral absorption features from the outflowing gas, according to the following procedure (see Appendix \[appendix1\] for a detailed description). - [The first step is to transform the incident spectrum $S_I(K)$ from $K$ to $K'$, obtaining $S_I(K')$, according to Eq. \[expl\].]{} - [$S_I(K')$ is then given as input to the radiative transfer code to calculate the transmitted spectrum in the outflowing gas frame $K'$, $S_T(K')$.]{} - [Finally, the relativistic-corrected transmitted spectrum in $K$, i.e. $S_{out}(K)$, is given by: $$S_{out}(K)=S_I(K)\cdot \Delta + S_T(K')\cdot \psi^{-1} \label{sout}$$ where $\Delta\equiv 1-\psi^3$. The term $S_T(K')\cdot \psi^{-1}$ indicates the spectrum $S_T(K')$ in Doppler-shifted (from $K'$ to $K$) frequencies.]{} We note that in the low-velocity limit $v_{out}\ll c$, $\Psi \approx 1, \Delta\approx0$ and the resulting spectrum is $S_{out}(K)=S_T(K')\cdot \psi^{-1}$, as it is usually calculated. For the opposite high-velocity regime $v_{out}\rightarrow c$, $\Psi\approx 0$ and the outflowing gas does not interact with the ionising radiation. In fact, $S_I(K')$ and $S_T(K')$ have null intensity (see Eq. \[expl\]), $\Delta\approx 1$ and $S_{out}(K) \approx S_I(K)$. We use the radiative transfer code *XSTAR*, v2.5 ([@xstar]) to calculate $S_{out}(K)$, which is the spectrum as seen by a rest frame observer in $K$. Figure \[abs\_spectra\] shows the X-ray spectrum in the range $6-16\ keV$ of a power-law continuum source with $\Gamma=2$ and a ionising luminosity $L_{ion}=5\cdot 10^{46}\ erg\ s^{-1}$ in the 1-1000 Ry (1 Ry$= 13.6\ eV$) energy interval, modified by an absorber with $v_{out} = 0.0, 0.3$ and $0.5\ c$. In all cases, we assume an absorbing column density of $N_H=6 \cdot 10^{23} cm^{-2}$ and ionisation parameter $log(\frac{\xi}{erg\ cm\ s^{-1}})=4.5$, which are typical of Ultra Fast Outflows (UFOs) observed in AGNs ([@R09; @T11; @G13]). The middle and right panel of Fig. \[abs\_spectra\] also report the $v_{out}=0$ case for an easier comparison. It can be seen that the absorption features related to the relativistically outflowing gas are both blueshifted and significantly weaker than the $v=0$ case. This effect dramatically increases for increasing velocity, as shown in a more quantitative way in Fig. \[integral\], which displays the column density $N_H$ necessary to reproduce outflow absorption features with a fixed optical depth, as a function of $v_{out}$. The required column density corresponds to $N_H=10^{23} cm^{-2}$ for $v_{out}=0$, and it increases by an order of magnitude for $v_{out}=0.8c$. It is interesting to note that this effect may introduce an observational bias in current X-ray data, which are typically restricted to $E<10 keV$, making outflows at higher velocity more difficult to detect due to the weakening of their spectral features at $E<10 keV$. We also note that @S07 and @S11 presented AGN outflow models including special relativistic effects to provide an estimate of both $N_H$ and $\xi$. However, both studies seem not to account properly for the reduction of the optical depth in the calculation of $S_{out}(K)$. Indeed, in Eq. \[sout\] the relativistic-corrected optical depth of the wind is preserved by transforming the transmitted spectrum back to the source rest frame $K$, while this aspect has not been considered in these studies. {width="19cm"} ![Absorbing gas $N_H$ required to reach a given value of the optical depth as a function of $v_{out}$. Spectral parameters are as in Fig. \[abs\_spectra\].[]{data-label="integral"}](optical_curve.pdf){width="9.5cm"} Discussion {#conclusions} ========== Mass and kinetic energy transfer rates of the outflow (i.e., $\dot{M}_{out}, \dot{E}_{out}$, respectively), linearly depends on $N_H$. Specifically, $\dot{M}_{out}$ can be calculated as follows ([@C12]): $$\dot M_{out} = 4 \pi r N_H \mu m_p C_f v_{out} \label{Mdot}$$ where $r, \mu, m_p, C_f$ are the distance from the source, the mean atomic weight ($\approx1.4$ for solar abundances), the proton mass and the covering factor of the outflow, respectively. $\dot{E}_{out}$ is defined as $\dot{E}_{out}=\frac{1}{2} \dot{M}_{out} v_{out}^2$. Correct estimates of $\dot{M}_{out}$ and $\dot{E}_{out}$ are of fundamental importance to test theoretical models of two-phase expansion of AGN outflows towards galaxy scales, in which kpc-scale galactic outflows are the results of the shock of ultra-fast, accretion disk-scale outflows onto the ISM ([@F12; @Z12; @Menci19]). We find that neglecting special relativistic effects will result into an underestimate of $N_H$ and, in turn, of $\dot{M}_{out},\dot{E}_{out}$. As an example, we correct for these effects the reported values of $\dot{M}_{out},\dot{E}_{out}$ for the UFOs observed in AGNs from @G15 and @F17 (see Figure \[integral\]). Specifically, for the UFOs in @G15 we use the average values between the reported $\dot{M}_{out},\dot{E}_{out}$ calculated using $r_{min}$ and those using $r_{max}$, where $r_{min}$ ($r_{max}$) is the minimum (maximum) inferred launching radius. Values of $\dot{M}_{out},\dot{E}_{out}$ reported in @F17 are calculated in the same way. In Figure \[edot\] we plot the ratio between the relativistic-corrected energy rates, $\dot{E}_{out}^{rel}$, and the original values, $\dot{E}_{out}^0$, as a function of $v_{out}$. $\dot{E}_{out}^{rel}$ is a factor of $>2$ higher than $\dot{E}_{out}^0$ for the fastest observed outflows ($v_{out}\geq 0.3 c$). As shown in Fig. \[integral\], we expect even higher ratios for higher velocity outflows. In this respect, the improved sensitivity and resolution of the new generation X-ray telescopes, such as [*XRISM*]{} and [*Athena*]{}, will be particularly promising and it will allow to partially alleviate the observational bias discussed in Sect. \[prescription\]. Interestingly, evidences for velocities $\geq 0.4-0.5c$ have indeed already been reported for some high luminosity quasars, such as PDS 456 and APM 08279+5255 (see e.g. [@PDS; @APM]). Figure \[ratio\] shows the ratio between the relativistic-corrected mass loss rate, $\dot{M}_{out}^{rel}$, and the mass accretion rate $\dot{M}_{acc}$, as a function of $\lambda_{Edd}\equiv L_{bol}/L_{Edd}$, i.e., the ratio between bolometric and Eddington luminosities. We derive the mass accretion rate as $\dot{M}_{acc}=\frac{L_{Bol}}{\eta c^2}$, assuming $\eta=0.1$ as in a standard @SS73 accretion disk. We note that for almost half of the sources $\frac{\dot{M}_{out}^{rel}}{\dot{M}_{acc}}\geq1$, indicating that $\dot{M}_{out}^{rel}$ is comparable to (or higher than) the mass accretion rate of the disk. This may indicate a limit for the outflow lifetime, after which the accretion disk is depleted and it can no longer sustain the outflow (see e.g. [@B97]). The plot also shows an apparent lack of sources with $\frac{\dot{M}_{out}^{rel}}{\dot{M}_{acc}}>1$ at large $\lambda_{Edd}$. However, the sample is too small to allow us to draw any conclusion. Future observations of high $\lambda_{Edd}$ AGNs are needed to shed light on this aspect. Finally, we compare the outflow momentum rate, defined as $\dot{P}_{out} = \dot{M}_{out} v_{out}$, with the momentum rate of the radiation of the AGN, i.e. $\dot{P}_{rad}= \frac{L_{bol}}{c}$. We obtain a median $\frac{\dot{P}_{out}}{\dot{P}_{rad}}$ value of 0.64 for the original sample, and 0.96 after the relativistic correction. Interestingly, the latter value is consistent with unity, as expected for outflows accelerated through the continuum radiation pressure (the so-called “Eddington winds”, see e.g. [@KP15]). ![Ratio between the relativistic-corrected energy transfer rates, $\dot{E}^{rel}_{out}$, and the original values $\dot{E}^0_{out}$ as a function of $v_{out}$, for a sample of Ultra Fast Outflows observed in AGNs (see Sect. \[conclusions\] for details).[]{data-label="edot"}](erel_avg.pdf){width="9.5cm"} ![Ratio between the relativistic-corrected outflow mass rate $\dot{M}_{out}^{rel}$ and the inflow mass rate $\dot{M}_{acc}$, as a function of $\lambda_{Edd}\equiv L_{bol}/L_{Edd}$. The sample is as in Fig. \[edot\]. Dotted line corresponds to $\frac{\dot{M}_{out}^{rel}}{\dot{M}_{acc}}=1$.[]{data-label="ratio"}](mout_macc_lambdalog_avg.pdf){width="9.5cm"} It is worth noting that in Eq. \[main\_long\] we use the total gas velocity $v_{out}$. We assume that the velocity $v_{los}$ along the line-of-sight (LOS) coincides with $v_{out}$. As a result, the derived relativistic correction must be regarded as a conservative limit. In fact, the correction would increase in presence of an additional velocity component $v_{\perp}$ perpendicular to the LOS, which implies $v_{out}=\sqrt{v_{los}^2+v_{\perp}^2}$. As an example, we consider the MHD model presented by @Fuku10 and @Fuku14, where the outflowing gas is launched from the accretion disk at Keplerian velocity. Close to the launching radius, most of the velocity is in the direction of the disk rotation $\phi$, and it is converted in radial velocity at higher distances (i.e., close to the Alfven point) thanks to MHD effects. For a wind launched at $r_0= 10 r_G$[^1], the rotational speed has a roughly constant value of $v_{\phi}=0.3 c$ until $r\approx 100 r_G$, while the radial velocity (i.e., the component parallel to the LOS) has an average value of $v_{LOS}\approx 0.2c$. In Figure \[psi\] we show that when $v_{out}= 0, \Psi=1$ and the relativistic effects are absent. On the other hand, when $v_{out}\rightarrow c, \Psi \approx 0$ and the effects are the highest. Using $v_{LOS}$ as a proxy for $v_{out}$ in Eq. \[main\_long\] yields $\Psi=0.8$, while using the total velocity (i.e., the composition between $v_r$ and $v_{\phi}$) gives $\Psi=0.6$, a factor 0.25 lower. Noteworthy, @A91 already pointed out that the observed optical depth of the gas depends on the velocity of the outflow relative to the source of radiation (see their Eqs. 2.1, 2.2). Specifically, they concentrated on an outflowing wind which is optically thick with respect to Thompson scattering, and calculated the integrated luminosity of its photosphere. Moreover, @Sumi07 and @S09 considered the impact of special relativistic effects on the emitted radiation from a fast, spherical wind in stars and accreting sources, such as quasars and ULXs. These works further underline the importance of relativistic effects for radiation-matter interaction at high speeds, along with the photo-electric and resonant absorption we investigate in this work. Conclusions {#sect5} =========== In this work we show that special relativistic effects are of fundamental importance for a correct modellisation of the outflow spectral features, even for mildly relativistic velocities ($v_{out} \buildrel > \over \sim 0.05 c$, see Figure \[psi\]). We also provide a simple procedure, that can be implemented in any radiative transfer code, to take into account these effects. We observe a significant reduction of the optical depth of the outflowing gas for fixed $N_H$ and increasing $v_{out}$ (Figs. \[abs\_spectra\] and \[integral\]). This indicates that it is necessary to include a velocity-dependent correction when estimating $N_H$ of the outflow from the optical depth derived by spectral fitting. Such correction is already significant (a factor $\approx 0.5$) for an outflow velocity of $v_{out}=0.1 c$ and reaches values a factor of $\buildrel > \over \sim 10$ for $v_{out} \geq 0.8 c$ (see Fig. \[integral\]). The derived mass and kinetic energy transfer rates linearly depend on $N_H$ (see Eq. \[Mdot\]), and hence must be corrected accordingly. For AGN outflows, this correction can significantly increase both $\dot{M}_{out},\dot{E}_{out}$ and, in turn, the impact of the outflow onto the surrounding environment, and on the feedback mechanism. We plot in Figure \[edot\] and \[ratio\] the relativistic-corrected $\dot{E}_{out},\dot{M}_{out}$ for a sample of Ultra Fast Outflows in AGNs reported in the literature. These pictures further underline the importance of relativistic corrections for a correct assessment of the outflow properties. Furthermore, these corrections are increasingly important in view of the next generation, high-sensitivity X-ray telescopes, which will increase the accuracy of the detection of mildly relativistic outflows, as discussed in Sect. \[conclusions\]. The effects discussed in Sect. \[physics\] have further implications on the radiative driving exerted on the outflowing gas, which will be discussed in a separate work (Luminari et al., *in prep*). Moreover, we also plan to present a new version of the X-ray spectral modelling code WINE [@AL], which includes a relativistic-corrected radiative transfer treatment according to the procedure of Sect. \[prescription\]. *Acknowledgements.* We thank Stefano Ascenzi for useful discussions and Tim Kallman for having provided custom *XSTAR* packages. AL, EP, FT, LZ acknowledge financial support from the Italian Space Agency (ASI) under the contract ASI-INAF n.2017-14-H.0. FT acknowledges support by the Programma per Giovani Ricercatori - anno 2014 “Rita Levi Montalcini”. FF acknowledges support from INAF under the contract PRIN-INAF-2016 FORECAST, and ASI/INAF contract I/037/12/0. 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Following Eq. \[expl\], the incident spectrum in the outflowing gas reference frame, $S_I(K')$, is obtained from $S_I(K)$ by multiplying the frequencies by a factor $\psi$ and the intensity by a factor $\psi^3$. For a given set of outflow parameters ($N_H, log(\xi_0), n_0, v_{out}$), we run radiative transfer simulation by using $S_I(K')$ as incident spectrum. As a result we obtain the transmitted spectrum, $S_T(K')$, displaying the absorption features due to the outflowing gas[^2]. We then calculate the “difference spectrum” as follows: $$S_{T-I}(K')=S_T(K')-S_I(K') \label{diff_spectrum}$$ Accordingly, $S_{T-I}(K')$ represents the absorption features produced by the outflowing gas, with the relativistic-corrected optical depth. As a next step, we calculate the relativistic-corrected, rest-frame absorbed spectrum as follows: $$S_{out}(K)=S_I(K)+S_{T-I}(K')\cdot \psi^{-1} \label{eq_step}$$ where $S_{T-I}(K')\cdot \psi^{-1}$ represents the “difference spectrum” in rest frame ($K$) frequencies, which is obtained by dividing the frequencies by a factor $\psi$. Using Eq. \[diff\_spectrum\], we can thus rewrite Eq. \[eq\_step\] as: $$S_{out}(K)= S_I(K)\cdot \Delta+S_T(K') \cdot \psi^{-1}$$ where $\Delta\equiv 1-\psi^3$ and $S_I(K)\cdot \Delta$ indicates a scaling of the intensity of the spectrum $S_I(K)$ of a factor $\Delta$. In our calculations we assume that the outflow has a net velocity $v_{out}$ and direction $\theta$. From a physical point of view, $v_{out}$ and $\theta$ correspond to the average velocity and direction of the outflow, respectively. Therefore, if a turbulent velocity component is present, the above discussion is still valid, provided that $v_{turb}\ll v_{out}$. Furthermore, if the outflowing velocity is a function of the spatial coordinates, i.e. $v_{out}=v(\overrightarrow{r})$, the above procedure can be implemented by dividing the outflow in small slabs, and assuming $v_{out}$ to be constant in each of them. Finally, the treatment of more complicated scenarios for $v(\overrightarrow{r})$ requiring a first-principle approach are beyond the scope of the present paper. [^1]: The gravitational radius $r_G$ is defined as $r_G=GM/c^2$, where $G$ is the gravitational constant and $M$ is the black hole mass. [^2]: Moreover, in some of the most popular codes, such as *XSTAR*, the emissivity of all the atomic lines listed in the atomic database is saved in a separated file. Emissivities can be used to build relativistic-corrected outflow emission spectra, as we will illustrate in detail in a forthcoming paper.

--- abstract: 'We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat $\Omega$; this problem arises when searching for the optimal shape and location of a shelter zone in order to prevent extinction of the species. On the other hand, we deal with the spectral drop problem, which consists in minimizing a mixed Dirichlet-Neumann eigenvalue in a box $\Omega$. In a previous paper [@mapeve] we proved that the latter one can be obtained as a singular perturbation of the former, when the region outside the refuge is more and more hostile. In this paper we sharpen our analysis in case $\Omega$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.' author: - 'Dario Mazzoleni, Benedetta Pellacci and Gianmaria Verzini' title: Quantitative analysis of a singularly perturbed shape optimization problem in a polygon --- [**AMS-Subject Classification**]{}. [49R05, 49Q10; 92D25, 35P15, 47A75.]{}\ [**Keywords**]{}. [Singular limits, survival threshold, mixed Neumann-Dirichlet boundary conditions, $\alpha$-symmetrization, isoperimetric profile.]{} Introduction {#sec:intro} ============ In this note we investigate some relations between the two following shape optimization problems, settled in a box $\Omega\subset{\mathbb{R}}^N$, that is, a bounded, Lipschitz domain (open and connected). \[def:lambda\] Let $0<\delta<|\Omega|$ and $\beta>\dfrac{\delta}{|\Omega|-\delta}$. For any measurable $D\subset\Omega$ such that $|D| = \delta$, we define the *weighted eigenvalue* $$\label{eq:def_lambda_beta_D} \lambda(\beta,D):=\min \left\{ \dfrac{\int_\Omega |\nabla u|^2\,dx}{\int_D u^2\,dx - \beta \int_{\Omega\setminus D} u^2\,dx} : u\in H^1(\Omega),\ \int_D u^2\,dx>\beta \int_{\Omega\setminus D} u^2\,dx\right\},$$ and the *optimal design problem for the survival threshold as* $$\label{eq:def_od} \operatorname{\Lambda}(\beta,\delta)=\min\Big\{\lambda(\beta,D):D\subset {\Omega},\ |D|=\delta\Big\}.$$ Let $0<\delta<|{\Omega}|$. Introducing the space $H^1_0(D,{\Omega}):=\left\{u\in H^1({\Omega}):u=0\text{ q.e. on }{\Omega}\setminus D\right\}$ (where q.e. stands for quasi-everywhere, i.e. up to sets of zero capacity), we can define, for any quasi-open $D\subset\Omega$ such that $|D| = \delta$, the *mixed Dirichlet-Neumann eigenvalue* as $$\label{eq:def_mu_D} \mu(D,{\Omega}):=\min{\left\{\frac{\int_{{\Omega}}|\nabla u|^2\,dx}{\int_{\Omega}u^2\,dx}:u\in H^1_0(D,{\Omega})\setminus\{0\}\right\}},$$ and the *spectral drop problem* as $$\label{eq:def_sd} \operatorname{M}(\delta)=\min{\Big\{\mu(D,\Omega):D\subset {\Omega},\;\mbox{quasi-open, }|D|=\delta\Big\}}.$$ The two problems above have been the subject of many investigations in the literature. The interest in the study of the eigenvalue $\lambda(\beta,D)$ goes back to the analysis of the optimization of the survival threshold of a species living in a heterogenous habitat $\Omega$, with the boundary $\partial\Omega$ acting as a reflecting barrier. As explained by Cantrell and Cosner in a series of paper [@MR1014659; @MR1105497; @MR2191264] (see also [@ly; @llnp; @mapeve]), the heterogeneity of $\Omega$ makes the intrinsic growth rate of the population, represented by a $L^{\infty}(\Omega)$ function $m(x)$, be positive in favourable sites and negative in the hostile ones. Then, if $m^{+}\not\equiv 0$ and $\int m<0$, it turns out that the positive principal eigenvalue $\lambda=\lambda(m)$ of the problem $$\begin{cases} -\Delta u = \lambda m u &\text{in }\Omega\\ \partial_\nu u = 0 &\text{on }\partial\Omega, \end{cases}$$ i.e. $$\lambda(m)=\left\{\frac{{\int_{\Omega}}|\nabla u|^{2}dx}{{\int_{\Omega}}mu^{2}dx}: u\in H^{1}(\Omega), {\int_{\Omega}}mu^{2}dx>0\right\},$$ acts a survival threshold, namely the smaller $\lambda(m)$ is, the greater the chances of survival become. Moreover, by [@ly], the minimum of $\lambda(m)$ w.r.t. $m$ varying in a suitable class is achieved when $m$ is of bang-bang type, i.e. $m={\mathds{1}_{D}} -\beta {\mathds{1}_{\Omega\setminus D}}$, being $D\subset \Omega$ with fixed measure. As a consequence, one is naturally led to the shape optimization problem introduced in Definition \[def:lambda\]. On the other hand, the spectral drop problem has been introduced in [@buve] as a class of shape optimization problems where one minimizes the first eigenvalue $\mu=\mu(D,{\Omega})$ of the Laplace operator with homogeneous Dirichlet conditions on $\partial D\cap \Omega$ and homogeneous Neumann ones on $\partial D\cap \partial \Omega$: $$\begin{cases} -\Delta u = \mu u &\text{in }D\\ u = 0 &\text{on }\partial D\cap\Omega\\ \partial_\nu u = 0 &\text{on }\partial D\cap\partial\Omega. \end{cases}$$ In our paper [@mapeve], we analyzed the relations between the above problems, showing in particular that $\operatorname{M}(\delta)$ arises from $\operatorname{\Lambda}(\beta,\delta)$ in the singularly perturbed limit $\beta\to+\infty$, as stated in the following result. \[thm:convergence\] If $0<\delta<|\Omega|$, $\beta>\dfrac{\delta}{|\Omega|-\delta}$ and $\dfrac{\delta}{\beta}<{\varepsilon}< |\Omega|-\delta$ then $$\operatorname{M}(\delta+{\varepsilon})\left(1-\sqrt{\frac{\delta}{{\varepsilon}\beta}}\right)^{2}\leq \operatorname{\Lambda}(\beta,\delta)\leq \operatorname{M}(\delta).$$ As a consequence, for every $0<\delta<|\Omega|$, $$\lim_{\beta\to+\infty} \operatorname{\Lambda}(\beta,\delta) = \operatorname{M}(\delta).$$ In respect of this asymptotic result, let us also mention [@derek], where the relation between the above eigenvalue problems has been recently investigated for $D\subset\Omega$ fixed and regular. In [@mapeve], we used the theorem above to transfer information from the spectral drop problem to the optimal design one. In particular, we could give a contribution in the comprehension of the shape of an optimal set $D^{*}$ for $\operatorname{\Lambda}(\beta,\delta)$. This topic includes several open questions starting from the analysis performed in [@MR1105497] (see also [@llnp; @ly]) when $\Omega=(0,1)$: in this case it is shown that any optimal set $D^{*}$ is either $(0,\delta)$ or $(1-\delta,1)$. The knowledge of analogous features in the higher dimensional case is far from being well understood, but it has been recently proved in [@llnp] that when $\Omega$ is an N-dimensional rectangle, then $\partial D^{*}$ does not contain any portion of sphere, contradicting previous conjectures and numerical studies [@MR2214420; @haro; @MR2494032]. This result prevents the existence of optimal *spherical shapes*, namely optimal $D^{*}$ of the form $D^{*}=\Omega\cap B_{r(\delta)}(x_{0})$ for suitable $x_{0}$ and $r(\delta)$ such that $|D^{*}|=\delta$. On the other hand, we have shown that spherical shapes are optimal for $\operatorname{M}(\delta)$, for small $\delta$, when $\Omega$ is an $N$-dimensional polytope. This, together with Theorem \[thm:convergence\], yields the following result. \[thm:orthotope\] Let $\Omega \subset {\mathbb{R}}^N$ be a bounded, convex polytope. There exists $\bar\delta>0$ such that, for any $0<\delta< \bar\delta$: - $D^*$ is a minimizer of the spectral drop problem in $\Omega$, with volume constraint $\delta$, if and only if $D^*=B_{r(\delta)}(x_0)\cap\Omega$, where $x_0$ is a vertex of $\Omega$ with the smallest solid angle; - if $|D|=\delta$ and $D$ is not a spherical shape as above, then, for $\beta$ sufficiently large, $$\lambda(\beta,D)> \lambda (\beta,B_{r(\delta)}(x_0)\cap \Omega).$$ In particular, in case $\Omega = (0,L_1) \times (0,L_2)$, with $L_1\le L_2$, and $0<\delta< L_1^2/\pi$, then any minimizing spectral drop is a quarter of a disk centered at a vertex of $\Omega$. Then, even though the optimal shapes for $\Lambda(\beta,\delta)$ can not be spherical for any fixed $\beta$, they are asymptotically spherical as $\beta\to+\infty$, at least in the qualitative sense described in Theorem \[thm:orthotope\]. The main aim of the present note is to somehow revert the above point of view: we will show that, in case $\operatorname{M}(\delta)$ is explicit as a function of $\delta$, one can use Theorem \[thm:convergence\] in order to obtain quantitative bounds on the ratio $$\frac{\operatorname{\Lambda}(\beta,\delta)}{\operatorname{M}(\delta)}.$$ In particular, we will pursue this program in case $\Omega$ is a planar polygon: indeed, on the one hand, in such case the threshold $\bar\delta$ in Theorem \[thm:orthotope\] can be estimated explicitly; on the other hand, such theorem implies that the optimal shapes for $\operatorname{M}(\delta)$ are spherical, so that $\operatorname{M}(\delta)$ can be explicitly computed. This will lead to quantitative estimates about the convergence of $\operatorname{\Lambda}(\beta,\delta)$ to $\operatorname{M}(\delta)$. As a byproduct of this analysis, we will also obtain some quantitative information on the ratio $$\frac{\lambda(\beta,B_{r(\delta)}(p)\cap \Omega)}{\operatorname{\Lambda}(\beta,\delta)},$$ thus providing a quantitative version of the second part of Theorem \[thm:orthotope\]. These new quantitative estimates are the main results of this note, and they are contained in Theorems \[thm:main1\] and \[thm:main2\], respectively. The next section is devoted to their statements and proofs, together with further details of our analysis. Setting of the problem and main results. {#sec:poli} ======================================== Let $\Omega\subset{\mathbb{R}}^2$ denote a convex $n$-gon, $n\ge 3$. We introduce the following quantities and objects, all depending on $\Omega$: - $\alpha_{\min}$ is the smallest interior angle; - ${{\mathcal{V}}}_{\min}$ is the set of vertices having angle $\alpha_{\min}$; - $e_1,\dots,e_n$ are the (closed) edges; - $d$ denotes the following quantity: $$d=\min\{\operatorname{dist}(e_i\cap e_j,e_k) : i\neq j,\ i\neq k,\ j\neq k\}.$$ Under the above notation, we define the threshold $$\label{eq:deltabar} \bar \delta:=\dfrac{d^2}{2\alpha_{\min}}.$$ Notice that, as far as $n\ge4$, $d$ corresponds to the shortest distance between two non- consecutive edges: $$d=\min\{\operatorname{dist}(x_{i},x_{j}) : x_{i}\in e_{i}, \, x_{j}\in e_{j},\, e_{i} \cap e_{j}=\emptyset\}.$$ Moreover, for any $n$, $$0<\bar\delta <|\Omega|.$$ Indeed, let $e_i\cap e_j\in{{\mathcal{V}}}_{\min}$, with $|e_i|\le|e_j|$. Then $$d\le |e_i| \sin \alpha_{\min}\qquad\text{ and }\qquad |\Omega|\ge \frac12 |e_i||e_j| \sin \alpha_{\min},$$ and the claim follows since $\sin \alpha_{\min} < \alpha_{\min}$. Our main results are the following. \[thm:main1\] Let $\Omega\subset{\mathbb{R}}^2$ denote a convex $n$-gon, let $\bar\delta$ be defined in , and let us assume that $$0 < \delta < \bar\delta.$$ Then $\operatorname{M}(\delta)$ is achieved by $D^{*}$ if and only if $D^*=B_{r(\delta)}(p)\cap \Omega$, where $p\in {{\mathcal{V}}}_{\min}$. Moreover $$\beta > \max\left\{\left(\frac{\delta}{\bar\delta-\delta}\right)^3,1\right\} \qquad\implies\qquad (1+\beta^{-1/3})^{-1}\left(1-\beta^{-1/3}\right)^{2}<\frac{\operatorname{\Lambda}(\beta,\delta)}{\operatorname{M}(\delta)}<1.$$ By taking advantage of the asymptotic information on $\operatorname{\Lambda}(\beta,\delta)/\operatorname{M}(\delta)$, we can deduce the corresponding relation between the eigenvalue of a spherical shape and the minimum $\operatorname{\Lambda}(\beta,\delta)$. \[thm:main2\] Let $\Omega\subset{\mathbb{R}}^2$ denote a convex $n$-gon, $\beta>1$, and let us assume that $$\label{eq:assdelta} \delta <\frac{\beta^{1/3}}{\beta^{1/3}+1}\, \bar \delta,$$ where $\bar\delta$ is defined in . Then, taking $p\in {{\mathcal{V}}}_{\min}$ and $r(\delta)$ such that $|B_{r(\delta)}(p)\cap \Omega|=\delta$, $$1 < \frac{{\lambda}(\beta,B_{r(\delta)}(p)\cap \Omega)}{\Lambda(\beta,\delta)}< \left(1+\beta^{-\frac{1}{3}}\right)\left(1-\beta^{-\frac{1}{3}}\right)^{-2}.$$ To prove our results, we will use the analysis we developed in [@mapeve Section 4] to estimate $\operatorname{M}(\delta)$ by means of $\alpha$-symmetrizations on cones [@MR876139; @MR963504]. To this aim we will first evaluate a suitable isoperimetric constant. For $D\subset\Omega$, we write $${{\mathcal{R}}}(D,\Omega):=\frac{P(D,\Omega)}{2|D\cap\Omega|^{1/2}},$$ where $P$ denotes the relative De Giorgi perimeter. For $0<\delta<|\Omega|$ we consider the isoperimetric problem $$I(\Omega,\delta) := \inf\left\{{{\mathcal{R}}}(D,\Omega) : D\subset\Omega,\ |D|= \delta\right\},$$ and we call $$K(\Omega,\delta) = \inf_{0<\delta'\le\delta} I(\Omega,\delta').$$ Given the unbounded cone with angle $\alpha$, $$\Sigma_\alpha :=\{(r\cos\vartheta,r\sin\vartheta)\in{\mathbb{R}}^2 : 0<\vartheta<\alpha,\; r>0\},$$ it is well known that $$\label{eq:sector} I(\Sigma_\alpha,\alpha r^2/2)={{\mathcal{R}}}(B_r(0)\cap\Sigma_\alpha,\Sigma_\alpha) = \frac{\alpha r}{2|\alpha r^2/2|^{1/2}} = \sqrt{\frac{\alpha}{2}},$$ is independent on $r$, and hence on $\delta=|B_r(0)\cap\Sigma_\alpha|$. As a consequence, also $$K(\Sigma_\alpha,\delta)=\sqrt{\frac{\alpha}{2}},$$ for every $\delta$. \[le:isop\] If $\Omega\subset{\mathbb{R}}^2$ is a convex $n$-gon and $\delta < \bar \delta$, then $I(\Omega,\delta)$ is achieved by $D^*$ if and only if $D^*=B_{r(\delta)}(p)\cap \Omega$, where $p\in {{\mathcal{V}}}_{\min}$. Moreover $K({\Omega},\bar \delta)$ is achieved by the same $D^*$ too. Notice that, by assumption, for any $p\in {{\mathcal{V}}}_{\min}$ the set $D=B_{r(\delta)}(p)\cap \Omega$ is a circular sector of measure $\delta$, with $\partial D \cap \Omega$ a circular arc. Then implies $$\label{eq:lemmino} I(\Omega,\delta) \le I(\Sigma_{\alpha_{\min}},\delta) = \sqrt{\frac{\alpha_{\min}}{2}},$$ and we are left to show the opposite inequality (strict, in case $D$ is not of the above kind). Applying Theorems 4.6 and 5.12 in [@MR3335407], and Theorems 2 and 3 in [@cianchi], we deduce that $I$ is achieved by $D^*_\delta\subset\Omega$, which is an open, connected set, such that $\Gamma:=\partial D^*_\delta\cap {\Omega}$ is either a (connected) arc of circle or a straight line segment. Moreover, $\partial D^*_\delta\cap \partial\Omega$ consists in exactly two points (the endpoints of $\Gamma$), and $\partial D^*_\delta\cap {\Omega}$ reaches the boundary of ${\Omega}$ orthogonally at flat points (i.e. not at a vertex). Hence, there are three possible configurations (see Fig. \[fig:HO\]). 1. The endpoints of $\Gamma$ belong to the interior of two consecutive edges $e_i$ and $e_{i+1}$. In this case $\Gamma$ is orthogonal to both $e_i$ and $e_{i+1}$, and $D^*_\delta$ is a portion of a disk centered at $e_i\cap e_{i+1}$. Recalling , we deduce that $e_i\cap e_{i+1} \in {{\mathcal{V}}}_{\min}$, and the lemma follows. 2. The endpoints of $\Gamma$ belong to the same edge $e_i$. 3. The endpoints of $\Gamma$ belong to two non-consecutive edges. The rest of the proof will be devoted to show that cases B and C can not occur. In case B, assume w.l.o.g. that $e_i\subset\{(x,0)\in{\mathbb{R}}^2\}$ and $\Omega\subset\{(x,y)\in{\mathbb{R}}^2: y\ge0\}=\Sigma_\pi$. Then $D^*_\delta\cap\Omega = D^*_\delta\cap\Sigma_\pi$, $P(D^*_\delta,\Omega) = P(D^*_\delta,\Sigma_\pi)$, and $${{\mathcal{R}}}(D^*_\delta,\Omega) \ge I(D^*_\delta,\Sigma_\pi) = \sqrt\frac{\pi}{2} > \sqrt\frac{\alpha_{\min}}{2},$$ in contradiction with . Finally, in order to rule out configuration C, by definition of $d$ we have $${\mathcal R}(D^{*}_\delta,{\Omega})\geq \frac{d}{2\sqrt{|D^{*}_\delta|}}=\frac{d}{2\sqrt{\delta}}> \sqrt\frac{\alpha_{\min}}{2}$$ whenever $\delta< \bar \delta$, which is fixed as $d/2\alpha_{\min}$. So that we get again a contradiction concluding the proof. Finally, the assertion concerning $K({\Omega},\bar \delta)$ follows by its definition and from the fact that for all $\delta\leq \bar \delta$ (see also [@mapeve Corollary 4.3]), we have just showed that $I({\Omega},\delta)=\sqrt{\frac{\alpha}{2}}$ is a constant independent of $\delta$. (0,0) – (2,0) arc (2:178:0.7) – cycle; (20:1.3) node [$D^*$]{}; (2,0) arc (2:178:0.7); (0,0) – (1.9,1.6) – (3,1.6) – (3.5,1.36) – (3.5,0.5) – (2.5,0) – cycle; (0,0) to \[out=30, in=150\] (2.5,0); at (1,0.2) [$D^{*}$]{}; (0,0) to \[out=30, in=150\] (2.5,0); (0,0) – (1.9,1.6) – (3,1.6) – (3.5,1.36) – (3.5,0.5) – (2.5,0) – cycle; (0,0) – (1.9,1.61)–(1.9,1.61)–(2.2,1.61)– (2.2,1.6) –(2.2,0); at (1,0.2) [$D^{*}$]{}; (2.2,0) – (2.2,1.6); (0,0) – (1.9,1.6) – (3,1.6) – (3.5,1.36) – (3.5,0.5) – (2.5,0) – cycle; (0,0) – (2.14,0) arc (0:17:5.45) – cycle; (20:1.6) node [$D^*$]{}; (2.14,0) arc (0:17:5.45); (0,0) – (1.9,1.6) – (3,1.6) – (3.5,1.36) – (3.5,0.5) – (2.5,0) – cycle; Notice that the threshold $\bar\delta$ in Lemma \[le:isop\] has no reason to be optimal. On the other hand, one can easily check that in the case of a rectangle, as treated in Theorem \[thm:orthotope\] it is actually optimal, since, for $\delta>\bar\delta$, $I(\Omega,\delta)$ is achieved by a rectangle (see e.g. [@mapeve Remark 4.5]). We are now in position to prove our main results. First of all, we take ${\varepsilon}\in (\delta/\beta,\bar \delta-\delta)\not=\emptyset$ by the assumption on $\delta$ and we apply [@mapeve Corollary 4.3] and Lemma \[le:isop\] to deduce that $$\begin{split} \operatorname{M}(\delta)&=K^2({\Omega},\delta)\delta^{-1}\lambda_{1}^{\text{Dir}}=\alpha_{\min}(2\delta)^{-1}\lambda_{1}^{\text{Dir}}\\ \operatorname{M}(\delta+{\varepsilon})&=K^2({\Omega},\delta+{\varepsilon})(\delta+{\varepsilon})^{-1}\lambda_{1}^{\text{Dir}}=\alpha_{\min}[2(\delta+{\varepsilon})]^{-1}\lambda_{1}^{\text{Dir}}, \end{split}$$ where $\lambda_{1}^{\text{Dir}}$ stands for the first eigenvalue of the Dirichlet-Laplacian in the ball of unit radius. By Theorem \[thm:convergence\] we obtain $$1\geq \frac{\operatorname{\Lambda}(\beta,\delta)}{\operatorname{M}(\delta)}\geq \frac{\operatorname{M}(\delta+{\varepsilon})}{\operatorname{M}(\delta)}\left(1-\sqrt{\frac{\delta}{{\varepsilon}\beta}}\right)^{2} =\frac{\delta}{\delta+{\varepsilon}}\left(1-\sqrt{\frac{\delta}{{\varepsilon}\beta}}\right)^{2},$$ for all ${\varepsilon}\in(\delta/\beta,\bar \delta-\delta)$. Then we make the choice of ${\varepsilon}=\delta/\beta^{1/3}$, which is admissible since $\beta>1$ and $\delta<\beta^{1/3}\bar{\delta}/(1+\beta^{1/3})$, and obtain $$\label{eq:fineprimo} 1\geq \frac{\operatorname{\Lambda}(\beta,\delta)}{\operatorname{M}(\delta)}\geq \frac{1}{1+\beta^{-1/3}}\left(1-\beta^{-1/3}\right)^2,$$ yielding the conclusion. Calling $D^*=B_{r(\delta)}(p)\cap \Omega$, for some $p\in {{\mathcal{V}}}_{\min}$ and using conclusion 2 of [@mapeve Lemma 3.1], we infer that ${\lambda}(\beta,D^*)\le \mu(D^*,{\Omega})$. As a consequence we can use Theorem \[thm:main1\] to write $$1\leq \frac{{\lambda}(\beta,D^*)}{\Lambda(\beta,\delta)}\leq \frac{M(\delta)}{\Lambda(\beta,\delta)}\leq (1+\beta^{-1/3})\left(1-\beta^{-1/3}\right)^{-2}.\qedhere$$ The estimate of Theorem \[thm:main2\] can be read as, $$1\leq \frac{{\lambda}(\beta,D^*)}{\Lambda(\beta,\delta)}\leq 1+3\beta^{-1/3}+o(\beta^{-1/3}),\qquad \text{as }\beta\rightarrow\infty.$$ On the other hand, even without using asymptotic expansions, as $\beta$ increases, the estimate becomes more precise. As an example, for all $\beta>8$, one has the explicit estimate$$1\leq \frac{{\lambda}(\beta,D^*)}{\Lambda(\beta,\delta)}\leq 1+15\beta^{-1/3}+14\beta^{-2/3}.$$ Acknowledgments {#acknowledgments .unnumbered} =============== Work partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, by the PRIN-2015KB9WPT Grant: “Variational methods, with applications to problems in mathematical physics and geometry”, and by the INdAM-GNAMPA group. [10]{} H. Berestycki, F. Hamel, and L. Roques. Analysis of the periodically fragmented environment model. [I]{}. [S]{}pecies persistence. , 51(1):75–113, 2005. G. Buttazzo and B. Velichkov. The spectral drop problem. In [*Recent advances in partial differential equations and applications*]{}, volume 666 of [*Contemp. Math.*]{}, pages 111–135. Amer. Math. Soc., Providence, RI, 2016. R. S. Cantrell and C. Cosner. Diffusive logistic equations with indefinite weights: population models in disrupted environments. , 112(3-4):293–318, 1989. R. S. Cantrell and C. Cosner. The effects of spatial heterogeneity in population dynamics. , 29(4):315–338, 1991. R. S. Cantrell and C. Cosner. . Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. A. Cianchi. On relative isoperimetric inequalities in the plane. , 3(2):289–325, 1989. C.-Y. Kao, Y. Lou, and E. Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. , 5(2):315–335, 2008. D. Kielty. Singular limits of sign-changing weighted eigenproblems. , arxiv:1812.03617, 2018. J. Lamboley, A. Laurain, G. Nadin, and Y. Privat. Properties of optimizers of the principal eigenvalue with indefinite weight and [R]{}obin conditions. , 55(6):Paper No. 144, 37, 2016. P.-L. Lions, F. Pacella, and M. Tricarico. Best constants in [S]{}obolev inequalities for functions vanishing on some part of the boundary and related questions. , 37(2):301–324, 1988. Y. Lou and E. Yanagida. Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. , 23(3):275–292, 2006. D. Mazzoleni, B. Pellacci, and G. Verzini. Asymptotic spherical shapes in some spectral optimization problems. , arxiv.org:1811.01623, 2018. F. Pacella and M. Tricarico. Symmetrization for a class of elliptic equations with mixed boundary conditions. , 34(1):75–93, 1985/86. M. Ritoré and E. Vernadakis. Isoperimetric inequalities in [E]{}uclidean convex bodies. , 367(7):4983–5014, 2015. L. Roques and F. Hamel. Mathematical analysis of the optimal habitat configurations for species persistence. , 210(1):34–59, 2007. `dariocesare.mazzoleni@unicatt.it`\ Dipartimento di Matematica e Fisica “N. Tartaglia”, Università Cattolica – Brescia\ via dei Musei 41, 25121 Brescia, Italy\ `benedetta.pellacci@unicampania.it`\ Dipartimento di Matematica e Fisica, Università della Campania “Luigi Vanvitelli”\ viale A. Lincoln 5, Caserta, Italy\ `gianmaria.verzini@polimi.it`\ Dipartimento di Matematica, Politecnico di Milano\ piazza Leonardo da Vinci 32, 20133 Milano, Italy\

--- abstract: 'We analyze the electrostatic interactions between a single graphene layer and a SiO$_2$ susbtrate, and other materials which may exist in its environment. We obtain that the leading effects arise from the polar modes at the SiO$_2$ surface, and water molecules, which may form layers between the graphene sheet and the substrate. The strength of the interactions implies that graphene is pinned to the substrate at distances greater than a few lattice spacings. The implications for graphene nanoelectromechanical systems, and for the interaction between graphene and a STM tip are also considered.' author: - 'J. Sabio$^1$' - 'C. Seoánez$^1$' - 'S. Fratini$^{1,2}$' - 'F. Guinea$^1$' - 'A. H. Castro Neto$^3$' - 'F. Sols$^4$' bibliography: - 'vdW\_sub.bib' title: 'Electrostatic interactions between graphene layers and their environment.' --- Introduction. ============= Graphene is a versatile two dimensional material whose singular electronic and mechanical properties show a great potential for applications in nanoelectronics.[@Netal05b; @GN07; @NGPNG07] Since free floating graphene is subject to crumpling,[@Nelson] the presence of a substrate, and the environment that comes with it, is fundamental for its stabilization. Hence, this environment will have direct impact in the physical properties of graphene. Though the influence of the substrate and other elements of the surroundings has been taken into account in different ways in the literature, the exact part that these are playing is not yet fully understood. On the one hand, the differences observed between samples grown on different substrates constitute an open issue. Most experiments have been carried out in graphene samples deposited over SiO$_2$, or grown over SiC substrates, [@Betal04] and a better understanding of how graphene properties are expected to change would be worthy. On the other hand, there is the question of characterizing all the effects that a particular environment has on graphene electronic and structural properties. Concerning electronic properties, it has been suggested that the low temperature mobility of the carriers is determined by scattering with charged impurities in the SiO$_2$ substrate,[@NM07; @AHGS07] and the effect of these charges can be significantly modified by the presence of water molecules.[@Setal07b] Actually, the very polar modes of SiO$_2$ give a good description of the finite temperature corrections to the mobility.[@PR06; @FG07; @CJXIF07] Supporting this idea, recent experiments show that graphene suspended above the substrate has a higher mobility.[@Betal08; @XuDu08] Experiments also seem to reveal a very important role played by the substrate in the structural properties of graphene. STM measurements suggest that single layer graphene follows the corrugations of the SiO$_2$ substrate,[@ICCFW07; @Setal07] and experiments on graphene nanoelectromechanical systems (NEMS) indicate that the substrate induces significant stresses in few layer graphene samples.[@Betal07] Moreover, the interaction between graphene and the substrate determines the frequency of the out of plane (flexural) vibrations, which can influence the transport properties at finite temperatures.[@KG07; @MNKSEJG07] ![a) Sketch of the system studied in the text. Interaction effects: b) Interaction with water molecules attached to hydroxyl radicals at the substrate. c) Interaction with polar modes at the surface of the substrate. d) van der Waals interaction between the graphene sheet and the metallic gate.[]{data-label="mechanisms"}](Fig1.eps){width="8.5cm"} In order to shed light on the influence of the environment on the graphene properties, we analyze the characteristic energies of interaction with the substrate and other materials present in the experimental setup. This allows us to evaluate the relative importance of the different interactions in the binding and mechanical response of the graphene layer. We also provide estimates of quantities such as equilibrium distances, typical lengthscales of corrugations, and frequencies of vibration, which can be measured in principle in current experimental setups. Throughout the paper we concentrate on SiO$_2$, though results are easily generalized to other substrates. Particularly, we consider: i) the van der Waals forces between graphene and the metallic gate below the SiO$_2$ substrate, ii) the electrostatic forces between the graphene layer and the polar modes of the substrate, iii) the electrostatic forces between graphene and charged impurities which may be present within the substrate and iv) the electrostatic forces between graphene and a water layer which may lay between graphene and the substrate.[@Setal07b] A sketch of the setup studied, and the different interaction mechanisms, is shown in Fig.\[mechanisms\]. We will also mention the possibility of weak chemical bonds between the graphene layer and molecules adjacent to it,[@LPP07; @Wetal07] although they will not be analyzed in detail. We do not consider a possible chemical modification of the graphene layer,[@Eetal07; @Wetal07b] which would change its transport properties. The general features of the electrostatic interactions to be studied are discussed in the next section. Then, we analyze, case by case, the different interactions between the graphene layer and the materials in its environment. Section IV discusses the main implications for the structure and dynamics of graphene, with applications to graphene NEMS and the interaction between graphene and a STM tip. The last section presents the main highlights of our work. Electrostatic interactions between a graphene layer and its environment. ======================================================================== The electrons in the $\pi$ and $\pi^*$ bands of graphene are polarized by electromagnetic potentials arising from charges surrounding it. The van der Waals interactions between metallic systems, and metals and graphene can be expressed as integrals over the dynamic polarizability of both systems. Those, in turn, can be written in terms of the zero point energy of the plasmons.[@TA83; @DWR06] The interaction between the graphene layer and a polarizable dielectric like SiO$_2$ is also given by an integral of the polarizability of the graphene layer times the polarizability of the dielectric. The latter can be approximated by the propagator of the polar modes, which play a similar role to the plasmons in a metal. The interaction between the graphene and static charges of electric dipoles depends only on the static polarizability.[@image] We will calculate these interactions using second order perturbation theory, assuming a perfect graphene sheet so that the momentum parallel to it is conserved. The corresponding diagrams are given in Fig. \[diagrams\]. All interactions depend, to this order, linearly on the polarizability of the graphene layer. In ordinary metallic systems, the Coulomb interaction is changed qualitatively when screening by the graphene electrons is taken into account through an RPA summation of diagrams. This is not the case for undoped graphene. There, the Random Phase Approximation leads to a finite correction $ \pi e^2 / 8 \hbar {v_{\rm F}}\sim 1$ to the dielectric constant, which does not change significantly the estimates obtained using second order perturbation theory. The response function of a graphene layer at half filling is:[@GGV94] $$\chi_G (\vec{q}, i\omega) = \frac{N_v N_s}{16 \hbar} \frac{q^2}{\sqrt{v_F^2 q^2 + \omega^2}}, \label{susc}$$ where $N_s = N_v = 2$ are the valley and spin degeneracy. This expression is obtained assuming a linear dispersion around the $K$ and $K'$ points of the Brillouin Zone. It is valid up to a cutoff in momentum $\Lambda \sim a^{-1}$ and energy $\omega_c \sim {v_{\rm F}}\Lambda$, where $a$ is the lattice spacing. Beyond this scale, the susceptibility has a more complex form, and it is influenced by the trigonal warping of the bands. The component of the electrostatic potential induced by a system at distance $z$ from the graphene layer with momentum $\vec{q}$ is suppressed by a factor $e^{- | \vec{q} | z}$. Hence, the integrations over $\vec{q}$ can be restricted to the region $0 \le q = | \vec{q} | \lesssim q_{max} \sim z^{-1}$. The combination of a term proportional to $e^{- | \vec{q} | z}$ and scale invariant quantities such as the susceptibility in Eq. (\[susc\]) leads to interaction energies which depend as a power law on $z$. In general, we will consider only the leading term, neglecting higher order corrections.[@polar] The calculation described above, which is valid for a single graphene layer at half filling, can be extended to other fillings and to systems with more than one layer. In all cases, the calculations are formally the same, and the interaction energies can be written as integrals over energies and momenta of the susceptibility of the system being considered, which replaces the susceptibility of a single layer, Eq. (\[susc\]). The susceptibility of a doped single layer is well approximated by that of an undoped system, Eq. (\[susc\]), for momenta such that $q \gtrsim {k_{\rm F}}$.[@WSSG06] Analogously, the susceptibilities of a stack of decoupled layers of graphene and multilayered graphene become similar for $q \gtrsim t_\perp / \hbar {v_{\rm F}}$[@G07], where $t_\perp$ is the hopping in the perpendicular direction. The susceptibility of a single undoped plane of graphene, Eq.(\[susc\]) is an increasing function of $q$, so that the integrals are dominated by the region $q \sim q_{max} \sim z^{-1}$. Hence, if $q_{max} \gg {k_{\rm F}}$ or $q_{max} \gg t_\perp / \hbar {v_{\rm F}}$, the interaction energies do not change appreciably from the estimates obtained for a single layer. The corrections can be obtained as an expansion in powers of either ${k_{\rm F}}z$, or $( t_\perp z ) / \hbar {v_{\rm F}}$. The expression Eq. (\[susc\]) can therefore be considered as the lowest order expansion in these parameters. For $z \sim 1$nm, $t_\perp\sim 0.35 eV$ and carrier densities such that $n \sim 10^{10} - 10^{12}$ cm$^{-2}$, we obtain ${k_{\rm F}}z \sim 10^{-2} - 10^{-1}$ and $ t_\perp / \hbar {v_{\rm F}}\sim 10^{-2} - 10^{-1}$. In the following, we will analyze mostly the interaction energies using the expression in Eq. (\[susc\]) for the graphene polarizability. ![Lowest order diagrams which contribute to the interaction between: a) graphene and a metal, b) graphene and a polar dielectric, and c) graphene and a static charge distribution. The thin red bubble stands for the graphene susceptibility. The thick green bubble represents the metallic susceptibility. The wavy green line stands for the propagator of a phonon mode in the dielectric. Crosses stand for static charge distributions, and dashed lines represent the electrostatic potential.[]{data-label="diagrams"}](Fig2.eps){width="6cm"} Interactions with specific environments. ======================================== Metallic gate. -------------- We describe the metallic gate as doped Si, separated from the graphene layer by a 300nm thick slab of SiO$_2$ dielectric. For the Si doping and voltages applied, most of the charge in the Si gate is concentrated on a layer of about $10$ nm thickness,[@S81] much narrower than the distance to the graphene sheet, so that the gate is effectively two dimensional. We describe the susceptibility of the gate as that of a dirty two dimensional electron gas: $$\chi_{gate}(\vec{q},i\omega) = -\frac{dn}{d\mu} \frac{Dq^2}{Dq^2 + |\omega|},$$where $D = {v_{\rm F}}_{gate} l_{gate}$ is the diffusion coefficient of the electrons in the gate, ${v_{\rm F}}_{gate}$ is the Fermi velocity, $l_{gate}$ is the mean free path, and $dn / d \mu$ is the bare compressibility, given by the density of states at the Fermi level (see, for instance, Ref. ). The interaction between the graphene layer and the gate is given by: $$v_q(z) = \frac{2 \pi e^2}{ \epsilon} \frac{e^{-qz}}{q},$$ being $\epsilon$ the static dielectric constant of the SiO$_2$ substrate. The lowest order contribution to the energy in perturbation theory has the following form: $$E_{gate}^{(2)} = - \hbar \sum_q \int_0^{\infty} \frac{d\omega}{2\pi} v_q^2(z) \chi_G (\vec{q},i\omega) \chi_{gate} (\vec{q},i\omega).$$ For future reference, note that we use the symbol $E$ for energies per unit area, and $\mathcal{E}$ for total (integrated) energies. The resulting integrals can be calculated analytically in the limit $z_s\equiv D/4v_F\ll z$: $$E_{gate}^{(2)} = - \frac{1}{12}\frac{dn}{d\mu} \frac{D}{v_F} \frac{e^4}{\epsilon^2} \frac{1}{(2z)^3} \left[ \log \left( \frac{z}{z_s} \right) + \frac{1}{3} \right].$$ The dependence on $z^{-3} \log (z/z_s)$ was obtained in Ref. . We take, as representative parameters for the gate and the graphene layer, $D \approx 10^{-3}$ m$^2$/s, ${v_{\rm F}}= 10^6$ m/s, $z = 300$ nm, $dn/d\mu \simeq g(E_F) = 0.04$ eV$^{-1}$ Å$^{-2}$ and $\epsilon = 4$ for the SiO$_2$ substrate. These parameters lead to interaction energies of order $\sim 10^{-8}$ meV Å$^{-2}$. Polar dielectric. ----------------- The interaction between the graphene layer and the SiO$_2$ substrate can be expressed in terms of the electric fields induced by the surface polar modes of SiO$_2$.[@S72; @WM72; @MA89; @Hetal06] The coupling can be written as: $$H_I = \sum_q M_q \rho_q \left(b_q + b_{-q}^{\dagger} \right),$$ where $\rho_q$ is the electron density operator and $b_q^{\dagger}$,$b_q$ the creation/destruction operators for phonons, and $M_q^2 = (\hbar^2 v_F^2) g e^{-2q z}/(qa)$ is the interaction matrix element, with $g$ a dimensionless coupling constant. In SiO$_2$ we have two dominant phonon modes at $\hbar \Omega_{1}=59$ meV and $\hbar \Omega_{2}=155$ meV, with $g_1 = 5.4\cdot 10^{-3}$ and $g_2 = 3.5 \cdot 10^{-2}$ respectively.[@FG07] The lowest order contribution to the energy is given by: $$E^{(2)}_{subs} = \sum_i \sum_{\vec{q}} \int \frac{d\omega}{2\pi} \chi_G (\vec{q}, i\omega) |M_q(z)|^2 D^{(0)}_i(\vec{q},i\omega),$$ where we have introduced the free phonon propagators: $$D_i^{(0)}(\vec{q},i\omega) = - \frac{2 \Omega_i}{\omega^2 + \Omega_i^2}.$$ The calculation can be again carried out analytically. In the limit $z \ll l_i \equiv {v_{\rm F}}/ \Omega_i$ we obtain: $$E_{subs}^{(2)} = - \sum_i \frac{\hbar v_F}{a} \frac{g_i}{(2z)^2},$$ which gives a $z^{-2}$ dependence on the distance. In the opposite limit $z \gg l_i$, which can be of interest in suspended graphene experiments, one obtains: $$E_{subs}^{(2)} = - \frac{\hbar v_F}{6 a} \frac{1}{(2z)^3}\sum_i l_i g_i \left[\log{\frac{l_i}{4 z}} + \frac{1}{3}\right]$$ Let us give some numerical estimates for both expressions. In the case of graphene deposited over the substrate, we have $z \sim 1$ nm (see Ref. ) and $l_i \ll z$, having interaction energies of order $E^{(2)}_{subs} \sim - 4\times 10^{-1}$meV Å$^{-2}$. For suspended graphene, $z\sim 300$ nm, and the energies are of order $E^{(2)}_{subs} \sim - 10^{-8}$meV Å$^{-2}$, i.e., of the same order than the contribution from the gate. Charges within the substrate. ----------------------------- In this case the calculations are done considering that effectively all the charge is concentrated close to the surface of the SiO$_2$ dielectric. The second order correction to the energy, averaged over the charge distribution, is: $$E^{(2)}_{ch} = - \sum_{\vec{q}} \chi_G(\vec{q},0) v_q^2(z) n_{imp},$$ where we consider a Coulomb interaction $v_q$ between graphene electrons and charges that is statically screened by the effective dielectric constant at the interface, $(\epsilon+1)/2$. Again, this contribution can be carried out analytically: $$E^{(2)}_{ch} = - \left(\frac{2e^2}{\epsilon+1}\right)^2 \frac{\pi n_{imp}}{ 2\hbar {v_{\rm F}}} \frac{1}{2z}. \label{En_charges}$$ This interaction has a $z^{-1}$ dependence, as the image potential in ordinary metals. In this case, however, this behavior arises from the combination of a vanishing density of states and lack of screening in graphene. Reasonable values for the impurity concentration in graphene are in the range $n_{imp} \sim 10^{10} - 10^{12}$ cm$^{-2}$. [@NM07; @AHGS07] Setting $z\sim 1$nm, typical interaction energies are of the order $E_{ch} \sim - 10^{-4} $– $10^{-2}$ meV Å$^{-2}$. The present result is only valid for graphene samples close to the substrate. If the distance to the latter is larger than the typical distance between charges, $d_{imp} \sim \sqrt{n_{imp}} \sim 1 - 10 nm$, the electrons feel the net effect of the effective charge in the substrate. This is zero in average, as there should be a compensated number of positive and negative charges. However, if we consider a finite region of the substrate, fluctuations can locally give rise to a net effective charge. This can be estimated by replacing $N_{imp} = n_{imp} l^2 \rightarrow \sqrt{n_{imp} l^2}$ in the total energy $\mathcal{E}_{ch} = E_{ch}^{(2)} l^2$, where $l$ is the lateral sample size. Layer of water molecules. ------------------------- The properties of the SiO$_2$ surface are dominated, for thermally grown SiO$_2$ layers, by the presence of abundant silanol (SiOH) groups, [@MM90] whose surface density is about $5 \times 10^{14}$ cm$^{-2}$, unless extra steps like thermal annealing in high vacuum are taken during the fabrication process.[@SG95; @DPX98; @N97] Silanol sites are active centers for water absorption, so that the SiO$_2$ surface becomes hydrated under normal conditions,[@BV01] which is probably the case of most of the graphene samples produced by mechanical cleavage.[@Setal07b] Moreover, several layers of water may cover the SiO$_2$ surface, lying between the oxide surface and the graphene samples after the graphene deposition. An analogous situation has been shown to happen in experiments with carbon nanotubes deposited on SiO$_2$. [@Ketal03] The water molecule has an electric dipole, $p_w = 6.2 \times 10^{-30}$ C m $\approx 0.04$e nm. Typical fields applied in present experimental setups are ${\cal E} \sim 0.1$ V nm$^{-1}$. The energy of a water dipole when it is aligned with this field is 4 meV $\sim 50$K, so that, at low temperatures, it will be oriented along the field direction, perpendicular to the substrate and the graphene layer. For this reason, in the following we assume that the water molecules are not charged, and their dipoles are aligned perpendicular to the substrate and the graphene layer. This arrangement can be considered an upper bound to the interaction energy with a neutral water layer, as inhomogeneities and thermal fluctuations will induce deviations in the orientation of the dipoles, and will lower the interaction energy. Note, however, that for high applied electric fields, a charging of water molecules of the order $Q_{{\rm H_2 O}} \sim 0.1 | e |$ has been reported.[@Ketal03] The presence of these extra charges would considerably enhance the interaction between the graphene layer and the water molecules. A water molecule which is located at a distance $z$ from the graphene layer induces an electrostatic potential: $$\Phi ( \vec{q} , z ) = 2 \pi p_w e^{- | \vec{q} | z}.$$ This potential polarizes the graphene layer and gives rise to an interaction energy in a similar way to the static charges discussed in the preceding section. The lowest order contribution to the energy is: $$E^{(2)}_{water} = - \left(e p_w \right)^2 \frac{\pi}{6} \frac{n_w}{\hbar {v_{\rm F}}} \frac{1}{(2z)^3}, \label{int_water}$$ where $n_w$ is the concentration of water molecules and the $z^{-3}$ behavior arises from the dipolar nature of the interactions. For $z = 0.3$nm, which is the approximate thickness of a water monolayer,[@AHM01; @OSAA07] the interaction energy is $E^{(2)}_{water} \sim- 12 n_w$ meV which, for a typical water concentration $n_w=10^{15}$cm$^{-2}$, yields $E^{(2)}_{water}\sim 1$ meV/Å$^2$. The expression in Eq. (\[int\_water\]) can be extended to a semi-infinite stack of water layers. For simplicity we take a distance $z$ between graphene and the uppermost layer of water molecules equal to the interlayer distance. In this case we obtain: $$E^{(2)}_{water} = - \left(e p_w \right)^2 \frac{\pi}{6} \frac{n_w}{\hbar {v_{\rm F}}} \frac{\zeta(3)}{(2z)^3},$$ where $\zeta(3) \approx 1.202$ is Riemann’s zeta function. The present result indicates that the first water layer is the one that mostly contributes to the binding. Van der Waals interaction between graphene layers. -------------------------------------------------- For comparison, in this Section we evaluate the van der Waals interaction between two graphene layers at the equilibrium distance. Using the same approximations as for the other contributions, we recover the result of Ref. , $$E_{G-G}^{(2)} = -\frac{\pi e^4}{16 \hbar v_F} \frac{1}{(2z)^3}.$$ For $z = 0.3$nm, this expression gives an interaction energy of 30 meV Å$^{-2}$. This estimate is similar to other experimental and theoretical values of the graphene-graphene interaction,[@Betal98; @HNI07] and is at least one order of magnitude greater than the other contributions analyzed earlier. Analysis of the results. ======================== Comparison of the different interactions. ----------------------------------------- Numerical estimates for the different interaction energies obtained for reasonable values of the parameters are listed in Table \[table\]. The present results show that the leading interactions are those between graphene and the polar modes of the SiO$_2$ substrate, and between graphene and a possible water layer on top of the substrate. Both effects are of similar order of magnitude in the present approximation, where we have assumed that the water molecules are aligned in the direction normal to the substrate. The interactions for multi-layer graphene samples can be obtained by adding the separate contributions from each layer. The different dependences on distance imply that the relative strength of the interactions in samples with many layers can change compared to the results of Table I. For instance, the effects of the polar substrate $\propto z^{-2}$ and of charged impurities $\propto z^{-1}$, which are of longer range, sum up more effectively than the binding effect of water: the $z^{-3}$ decay of the graphene-water interaction suggests that only the first graphene layer is affected by the presence of water on the substrate. For the same reason, the presence of several layers of aligned water molecules should not increase the binding, as only the closest layer effectively contributes to the interaction energy. On the other hand, the binding effect of water could be enhanced if the molecules were allowed to rotate freely, therefore approaching the high polarizability of liquid water,[@Setal07b] or if they were partly ionized by the applied field,[@Ketal03] leading to additional charges similar to the Coulomb impurities present in the SiO$_2$ substrate. It should be noted that we have considered here only long-range electrostatic interactions, for which reliable expressions can be obtained, in terms of well understood material parameters, like the molecular polarizability, electric dipoles, or surface modes. Still, there is a significant uncertainty in some parameters, like the distance of the relevant charges to the graphene layer and the concentration of charged impurities and water molecules. We have not analyzed the possible formation of chemical bonds between the carbon atoms and the water or silanol groups at the SiO$_2$ surface. Calculations based on the Local Density Functional approximation[@Wetal07; @LPP07] suggest that individual molecules can (weakly) bind to a graphene layer with energies of $10 - 50$ meV, although it is unclear how these estimates are changed when the molecules interact at the same time with the graphene layer and the substrate. --------------------------------- ---------- ----------------------- -------------------- Distance Dependence on Energy (nm) distance (meV Å$^{-2}$) Gate 300 $z^{-3} \log(z/z_s)$ $ 10^{-8}$ Charged impurities 1 $z^{-1}$ $ 10^{-4}-10^{-2}$ SiO$_2$ substrate ($z\ll l_i$) 1 $z^{-2}$ $0.4$ SiO$_2$ substrate ($z \gg l_i$) 300 nm $z^{-3} \log(4z/l_i)$ $10^{-8}$ Water molecules 0.3 $z^{-3}$ $1$ Graphene 0.3 $z^{-3}$ $30$ --------------------------------- ---------- ----------------------- -------------------- : Interaction energy per unit area for the mechanisms studied in the paper. For the numerical estimates we have used typical concentrations of $10^{10}-10^{12}$cm$^{-2}$ charged impurities and $10^{15}$cm$^{-2}$ water molecules[]{data-label="table"} Corrugation of the graphene layer induced by the substrate. ----------------------------------------------------------- The attractive forces calculated in the preceding section imply that graphene is bound to the SiO$_2$ substrate, as observed in experiments. Our previous analysis does not include the short range repulsive forces which determine the equilibrium distance. We assume that the total energy near the surface is the sum of the terms analyzed above, which have a power law dependence on the distance, and a repulsive term, $E_{rep} ( z ) = \epsilon_{rep} ( z_0^n / z^n )$, which also decays as a power law at long distances, with $z_0$ an undetermined length scale. For simplicity, we assume that the leading attractive term is due to the presence of a water layer, which behaves as $E_{water} = - \epsilon_{w} ( z_0^3 / z^3 )$. The total energy per unit area is thus: $$E ( z ) = \epsilon_{rep} \frac{z_0^n}{z^n} - \epsilon_w \frac{z_0^3}{z^3}.$$ At the equilibrium distance, $z_{eq}$, we have: $$\frac{\epsilon_{rep}}{\epsilon_w} = \frac{3}{n} \left( \frac{z_{eq}}{z_{0}} \right)^{n-3},$$ so that: $$\begin{aligned} & &E'' ( z_{eq} ) = \frac{1}{z_{eq}^2} \left[ n ( n+1 ) \epsilon_{rep} \left( \frac{z_0}{z_{eq}} \right)^{n} - 12 \epsilon_w \left( \frac{z_0}{z_{eq}} \right)^{3} \right] \nonumber \\ & &= 3 ( n - 3 ) \frac{\epsilon_w}{z_{eq}^2} \left( \frac{z_0}{z_{eq}} \right)^3 = 3 ( n - 3 ) \frac{E_{water} ( z_{eq} )}{z_{eq}^2}.\end{aligned}$$ Hence, the order of magnitude of the pinning potential induced by the environment on the out of plane modes of graphene is given by $K \propto E_{water} ( z_{eq} ) / z_{eq}^2 \sim 10^{-2}- 10^{-1}$ meV Å$^{-4}$. Defining the out of plane displacement as $h ( \vec{r} )$, the energy stored in a corrugated graphene layer is: $${\cal E} \approx \frac{1}{2} \int d^2 {\vec r} \left[ \kappa (\Delta h)^2 + K h^2 \right],$$ where $\kappa \approx 1$eV is the bending rigidity of graphene.[@KN07; @MO07] For modulations $h ( \vec{r} )$ defined by a length scale $l$, the bending energy dominates if $l \ll l^* = ( \kappa / K )^{1/4}$, while the graphene layer can be considered rigidly pinned to the substrate if $l \gg l^*$. Using our previous estimates, we find $l^* \sim 10$Å, so that the graphene layer should follow closely the corrugations of the substrate. The pinning by the substrate implies that the dispersion of the flexural modes becomes: $$\omega_k = \sqrt{\frac{K}{\rho} + \frac{\kappa k^4}{\rho} }$$ where $\rho$ is the mass density of the graphene layer. At long wavelengths, $\lim_{k \rightarrow 0} \omega_k = \omega_0 \sim 10^{-4} - 10^{-3} {\rm meV} \sim 10^{-3} - 10^{-2}$K. ![Sketch of the deformation of a nanoelectromechanical device studied in the text.[]{data-label="NEMs"}](Fig3.eps){width="7cm"} The estimates obtained above also allow us to analyze the bending of graphene NEMS due to the interaction with the material below, at distance $d$.[@Betal07; @SGN07] We assume that the lateral dimension of the graphene cantilever is $l$, and the maximum displacement of the graphene layer from a flat position is $h$. A sketch of the graphene cantilever is shown in Fig.\[NEMs\]. We consider the force induced by charged impurities in the substrate below the cantilever, as this is the contribution which decays more slowly with the distance to the graphene layer (cf. Table \[table\]). If the distance of the cantilever to the substrate is $d$, and supposing $d \gg h$, the gain in energy due to the deformation of the graphene layer is $\Delta \mathcal{E} \sim \epsilon_{ch} l z_0 h/ \sqrt{n_{imp}} d^2$. We have again defined $z_0$ and $ \epsilon_{ch} $ by rescaling $E_{ch}(d) = \epsilon_{ch} z_0 / d$, with $z_0 \sim 1$ nm and $\epsilon_{ch} \sim 10^{-4}-10^{-2}$ meV Å$^{-2}$ depending on the density of impurities. The factor $l / \sqrt{n_{imp}}$ is included, as already mentioned at the end of Section III.C, to take into account the effect of having an overall neutral distribution of charge in the substrate. This energy should compensate the elastic response to the deformation, $\Delta \mathcal{E}_{el} \sim \kappa h^2 / l^2$, leading to an equilibrium value: $$h \sim \frac{\epsilon_{ch} z_0}{2\kappa} \frac{l^3}{d^2 \sqrt{n_{imp}}} \label{himp}$$ Reminding the dependence of $\epsilon_{ch}$ on the density of impurities, Eq. (\[himp\]) results in an overall behavior $h\propto \sqrt{n_{imp}}$. For structures such that $d\sim 300$ nm, one finds that the suspended graphene sheet deforms significantly (i.e. $h$ becomes comparable with $d$) for lenghts greater than a few $\mu$m. In the case of very pure substrates (i.e. neglecting the presence of charged impurities), the attractive force that bends the graphene sheet would be determined by the next corrections to the energy, which decay as $z^{-3}$. Rewriting $E^{(2)} \sim (\epsilon_w + \epsilon_{ph} + \epsilon_{G}) (z_0/z)^3$ with $\epsilon_{ph} \simeq \epsilon_{G} \sim 0.1 meV / \AA^2$ and $\epsilon_{w} \sim 0.01 meV / \AA^2$, we see that those interactions are dominated by the coupling to the gate and to the polar modes, which are of comparable magnitude. A calculation similar to the one presented above would yield deformations of the order of $h\sim 1 \AA \times (l/d)^4$, resulting even in these cases to unstable graphene sheets for lenghts greater than a few $\mu$m. Interaction with a metallic tip in an STM experiment ---------------------------------------------------- ![Estimate of the threshold voltage as function of graphene-tip separation needed to detach a graphene layer from the substrate. The inset shows a sketch of the geometry considered in the text.[]{data-label="STM"}](Fig4.eps){width="8cm"} It is known that STM tips on graphite surfaces sometimes deform the surface graphene layer,[@Betal87; @Setal91] and the understanding of these deformations can be of interest for current research on graphene.[@ICCFW07; @Setal07; @LA07] The analysis of the electrostatic interactions between a graphene layer and its environment allows us to estimate possible deformations induced by an STM tip. We analyze the setup sketched in the inset of Fig.\[STM\]. The tip has lateral dimension $l$ and it is located at a distance $d$ from a graphene layer. This graphene layer interacts with an underlying substrate, and a voltage $V$ is applied between the graphene layer and the tip. We consider three interactions: i\) An attraction between the tip and the graphene layer, which tends to deform the graphene, in the way shown in Fig.\[STM\]. We assume that this energy is purely electrostatic. A simple estimate can be obtained by describing the setup as a capacitor where the area of the plates is $l^2$, the distance between the plates is $d$, and the applied voltage is $V$. The interaction energy is of the order: $$\mathcal{E}_{G-tip} \approx \frac{V^2 l^2}{8 \pi e^2 d},$$ where we define $V$ in energy units. ii\) The pinning of the graphene layer to the substrate. This contribution opposes the deformation of the layer. We write it as: $$\label{eq:epspin} \mathcal{E}_{pin} \approx \epsilon_{pin} l^2,$$ where $\epsilon_{pin}$ is the pinning energy per unit area. As typical values, we will use 1 meV$/$Å$^2$ for graphene on a water layer, and 30 meV$/$Å$^2$ for graphene interacting with another graphene layer, as in graphite. iii\) The rigidity of the layer against flexural deformations. This term tends to keep the layer flat. The deformed region is likely to be (at least) as large as the size of the STM tip. As a result, an upper bound to the energy stored in a deformation is: $$\mathcal{E}_{el} \approx \kappa \frac{d^2}{l^2}$$ The graphene layer will be deformed when: $$\mathcal{E}_{G-tip} \gtrsim \mathcal{E}_{pin} + \mathcal{E}_{el} \label{threshold_tip}.$$ Note that the approximations involved in obtaining the various terms are valid only if $d \gtrsim a$. We consider a situation where $\epsilon_{pin} , \kappa$ and $l$ are fixed. Eq.(\[threshold\_tip\]) implies that the layer is deformed if the voltage exceeds a threshold: $$V \gtrsim V_{th} ( d ) \approx \sqrt{8 \pi \left(\frac{\kappa e^2 d^3 }{l^4} + \epsilon_{pin}e^2 d\right)}$$ Assuming $l \sim 10 a$ and $d\sim a$, we see that the dominant contribution comes from the pinning term Eq. (\[eq:epspin\]). Hence, in the physically relevant range $a \lesssim d \ll l$, we can write: $$V_{th} ( d ) \approx \sqrt{8 \pi \epsilon_{pin} e^2 d},$$ independent on the tip size. The threshold values for graphene on SiO$_2$ are of about 0.5-2 V for $d \sim 1-10$Å, as schematically shown in Fig.\[STM\]. Conclusions. ============ We have analyzed the electrostatic interactions between a graphene layer and the polarizable materials which may be present in its environment, for samples deposited on SiO$_2$. The strength of these interactions can be obtained in terms of a few well understood microscopic parameters, and they have a simple dependence on the distance between the graphene layer and the system which induces the electrostatic field. The analysis presented here should give reliable estimates of the order of magnitude of the different binding energies, and of their relative strength. We have not considered the possible formation of chemical bonds, which may alter the results when the distances between the carbon atoms in the graphene layer and the surrounding materials is sufficiently small. We find that the leading effects arise from the polar modes of the SiO$_2$ substrate, and water which may form layers on top of it. A summary of the main results is presented in Table \[table\]. The interaction energies with systems with $N$ layers can be obtained, to a first approximation, by adding the contributions from each layer. The estimated magnitude of the interactions suggests that a single graphene layer is pinned to the substrate on length scales greater than a few lattice spacings, $\sim 10$Å. The electrostatic binding modifies the long wavelength, out of plane flexural modes, which acquire a finite frequency, $\omega_0 \sim 10^{-4} - 10^{-3}$ meV. The long range forces considered here can also induce large deformations in graphene NEMS. Besides, we have analyzed the possibility of deformations of the graphene layer by an STM tip. We find that a voltage drop of 0.5 - 2 V between the tip and the sample at distances 1 - 10 Å is sufficient to deform the graphene layer. Acknowledgements. ================= This work was supported by MEC (Spain) through grant FIS2005-05478-C02-01, the Comunidad de Madrid, through the program CITECNOMIK, CM2006-S-0505-ESP-0337, the European Union Contract 12881 (NEST) (J. S., C. S., S. F., and F. G.), MEC through grant FIS2004-05120 and FIS2007-65723, and EU Marie Curie RTN Programme no. MRTN-CT-2003-504574 (F.S. and J.S.). J. S. acknowledges the I3P Program from CSIC, and C.S. the FPU Program from MEC, for funding. We are thankful to A. Bachtold and A. K. Geim for many helpful insights into the relevance of water for current experiments, to S. Vieira for useful information on the interaction between graphene and STM tips, and to A. A. Balandin for a clarifying remark on our results for suspended graphene.

--- abstract: 'We use a Leibnitz rule type inequality for fractional derivatives to prove conditions under which a solution $u(x,t)$ of the k-generalized KdV equation is in the space $L^2(|x|^{2s}\,dx)$ for $s \in \mathbb R_{+}$.' address: | École Polytechnique Fédérale de Lausanne\ MA B1 487\ CH-1015 Lausanne author: - 'J. Nahas' title: 'A decay property of solutions to the k-generalized KdV equation' --- Introduction ============ The the initial value problem for the modified Korteweg-de Vries equation (mKdV), $$\begin{aligned} \partial_tu + \partial_x^3u+\partial_x(u^3)=0, \label{mkdv} \\ u(x,0)=u_0(x), \notag\end{aligned}$$ has applications to fluid dynamics (see [@2009ChPhB..18.4074L], [@1994JNS.....4..355R]), and plasmas (see [@PRUD]). It is also an example of an integrable system (see [@PhysRevLett.19.1095]). Ginibre and Y. Tsutsumi in [@g] proved well-posedness in a weighted $L^2$ space. In [@KPV1], Kenig, Ponce, and Vega proved local well-posedness for $u_0$ in the Sobolev space $H^s$, when $s \ge \frac{1}{4}$ by a contraction mapping argument in mixed $L_x^p$ and $L_T^q$ spaces. Christ, Colliander, and Tao in [@MR2018661] showed that was locally well-posed for $u_0 \in H^s$, when $s \ge \frac{1}{4}$, by using a contraction mapping argument in the Bourgain spaces $X_{s,b}$. Colliander, Keel, Staffilani, Takaoka, and Tao proved global well-posedness for real initial data $u_0 \in H^{s}$, $s > \frac{1}{4}$ in [@CKSTT]. Kishimoto in [@Kish] and Guo in [@MR2531556] proved global well-posedness for real data in the case $s=\frac{1}{4}$. The focus of this work will be , but we will also consider the generalized Korteweg-de Vries equation, $$\left\{ \begin{array}{c l} & \partial_tu + \partial_x^3u + \partial_x (u^{k+1})=0, \label{gkdv} \\ & u(x,0)=u_0(x),\textrm{ } x \in \mathbb R. \end{array} \right.$$ When $k \ge 4$, local well posedness was obtained for initial data $u_0 \in H^s$ with $s \ge \frac{k-4}{2k}$ in [@KPV1] using a contraction mapping argument in mixed $L_x^p$ and $L_T^q$ spaces. When $k=3$, the optimal local well posedness result was proven by Tao in [@MR2286393] for $u_0 \in H^s$ with $s \ge -\frac{1}{6}$ by using Bourgain spaces $X_{s,b}$. Kato in [@Ka] with energy estimates, and the fact that the operator $$\Gamma_K \equiv x+3t\partial_x^2 \notag$$ commutes with $\partial_t+\partial_x^3$, was able to prove the following: if $u_0 \in H^{2k}$ and $|x|^ku_0 \in L^2$ where $k \in \mathbb Z^{+}$, then for any other time $t$ when the solution exists, $|x|^ku(t) \in L_x^2$. Using slightly different techniques, we will prove the following theorem that extends this result slightly to $k \in \mathbb R_+$. \[weak-decay\] Suppose the initial data $u_0$ satisfies $|x|^su_0 \in L^2$, and $u_0 \in H^{2s+\varepsilon}$, for $\varepsilon >0$. Then for any other time $t$, the solution $u(x,t)$ to satisfies $|x|^su(x,t) \in L^2$. When $s \ge \frac{1}{2}$, the result holds for $\varepsilon=0$. Namely, if $|x|^{s}u_0 \in L^2$, and $u_0 \in H^{2s}$, then for any other time $t$, the solution $u(x,t)$ to satisfies $|x|^{s}u(x,t) \in L^2$. Analogous results for the NLS were first proved by Hayashi, Nakamitsu, and M. Tsutsumi in [@MR847012], [@MR880978], and [@MR987792]. They used the vector field $$\Gamma_S = x+2it\nabla, \label{nls-gamma}$$ which commutes with the operator $\partial_t -i\Delta$, and a contraction mapping argument to show that if $u_0 \in L^2(|x|^{2m}\,dx) \cap H^m$, where $m \in \mathbb N$, then the solution $u(x,t)$ at any other time is also in the space $L^2(|x|^{2m}\,dx) \cap H^m$. These results were extended to the case when $m \in \mathbb R_+$ by the author and G. Ponce in [@NP]. The corresponding results for the Benjamin-Ono equation were obtained in [@PonceFons] by G. Ponce and G. Fonseca. Inspired by these persistence results we prove the following as our main result. \[main\] If $u(x,t)$ is a solution of $$\notag \left\{ \begin{array}{c l} & \partial_tu + \partial_x^3u + \partial_x (u^{k+1})=0, \\ & u(x,0)=u_0(x),\textrm{ } x \in \mathbb R, \end{array} \right.$$ such that $u_0 \in H^{s'} \cap L^2(|x|^{s}\,dx)$, where $s \in (0,s']$. If $k=2$, and $s'\ge \frac{1}{4}$, then $u(\cdot,t) \in H^{s'} \cap L^2(|x|^{s}\,dx)$ for all $t$ in the lifespan of $u$. If $k \ge 4$, and $s \ge \frac{k-4}{2k}$, then $u(\cdot,t) \in H^{s'} \cap L^2(|x|^{s}\,dx)$ for all $t$ in the lifespan of $u$. We only prove this property the most interesting case, . Note that the cases in when $k=1$ or $4$ are excluded from Theorem \[main\]. We require our technique to be adapted to Bourgain spaces for these nonlinearities, which is an interesting open question. The difficulty in the case of fractional decay lies in the lack of an operator $\Gamma$ that sufficiently describes the relation between initial decay, and properties of the solution at another time (such as ). In order to solve this problem, we develop a Leibnitz rule type inequality for fractional derivatives. We need some notation to illustrate this idea. If $f$ is a complex valued function on $\mathbb R$, we let $f^{\wedge}$ (or $\hat{f}$) denote the Fourier transform of $f$, and $f^{\vee}$ the inverse Fourier transform. For $\alpha \in \mathbb R$, the operator $D_x^{\alpha}$ is defined as $(D_x^{\alpha}f(x))^{\wedge}(\xi)\equiv |\xi|^{\alpha}f^{\wedge}(\xi)$. Let $U(t)f$ denote the solution $u(x,t)$ to the linear part of $\eqref{mkdv}$, with $u(x,0)=f(x)$. Choose $\eta \in C_0^{\infty}(\mathbb R)$ with $\textrm{supp}(\eta)\subset [\frac{1}{2},2]$ so that $$\sum_{N \in \mathbb Z} (\eta(\frac{x}{2^N})+\eta(-\frac{x}{2^N}))=1 \textrm{ for }x \ne 0. \notag$$ Define the operator $Q_N$ on a function $f$ as $$Q_N(f) \equiv ((\eta(\frac{\xi}{2^N})+\eta(-\frac{\xi}{2^N}))\hat{f}(\xi))^{\vee}. \notag$$ If $\|\cdot\|_Y$ is a norm on some space of functions, we recall that $$\|Q_N(f)\|_{Y l_N^p} \equiv \|(\sum_{N \in \mathbb Z}|Q_N(f)|^p)^{\frac{1}{p}}\|_Y\notag.$$ Using Duhammel’s principle, we can formulate the problem as an integral equation. $$u(x,t) = U(t)u_0 -\int_{0}^{t}U(t-t')\partial_x(u^3(x,t'))\,dt'. \notag$$ Using a Fourier transform, we can see how to commute an $x$ past $U(t)$, $$\begin{aligned} xU(t)f & = (-i\partial_{\xi}(e^{it\xi^3}\hat{f}))^{\vee} \notag \\ & = (3t\xi^2e^{it\xi^3}\hat{f}-ie^{it\xi^3}\partial_{\xi}\hat{f})^{\vee} \notag \\ & = U(t)(3t\partial_x^2f+xf). \notag\end{aligned}$$ We would like to use a similar argument with $|x|^{\frac{1}{8}}$ replacing $x$, but this would require that $D_{\xi}^{\frac{1}{8}}$ obey a product rule. We develop in inequality in Lemma \[my\_prod\_rule\] that is similar enough to the product rule that will allow this argument to work. With Lemma \[my\_prod\_rule\], we will require that $$\left \|D_{\xi}^{\frac{1}{8}}Q_N(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}) \right \|_{L_{\xi}^{\infty}l_N^1} < \infty. \label{main_part}$$ With less sophisticated techniques, we prove Theorem [weak-decay]{} in Section 2. We show in Section 3, then prove our main result in Section 4. The proof of Lemma \[my\_prod\_rule\] is almost identical to the proof of a classical Leibnitz rule inequality. Because this proof requires different techniques than the rest of the paper, we present it in Appendix A. We use the following notation throughout the paper. We let $A \lesssim B$ mean that the quantity $A$ is less than or equal to a fixed constant times the quantity $B$. Let $\langle x \rangle \equiv (1+x^2)^{\frac{1}{2}}$, and similarly, $\langle D_x \rangle$. Weak Persistence Result ======================= Using some standard estimates, we prove Theorem \[weak-decay\] which is a weaker persistence property for IVP for the gKdV equation for low regularity solutions, but holds for more values of $k$ in than our main result. Following an argument by Kato, we multiply by $\phi(x)u(x,t)$ for some function $\phi(x)$, and integrating over $x$ and $t$, we use integration by parts to obtain $$\begin{aligned} & \int_{\mathbb R}\phi(x)u^2(x,T)\,dx-\int_{\mathbb R}\phi(x)u^2(x,0)\,dx -3\int_{[0,T]}\int_{\mathbb R}\phi'(x)(\partial_x u)^2\,dx\,dt \notag \\ & \quad +\int_{[0,T]}\int_{\mathbb R}\phi'''(x)u^2\,dx\,dt + \frac{k+1}{k+2}\int_{[0,T]}\int_{\mathbb R}\phi'(x)u^{k+2}\,dx\,dt=0. \label{weight_eq}\end{aligned}$$ Equation , along with the following two interpolation lemmas are the primary tools for the weak persistence result, Theorem \[weak-decay\]. \[inter-reg-dec\] Let $a,b >0$, and $w(x)> \varepsilon >0$ a locally bounded function. Assume that $\langle D_x \rangle^a f \in L^2(\mathbb R)$ and $ w^b(x) f \in L^2(\mathbb R)$. Then for any $\theta \in (0,1)$ $$\|\langle D_x \rangle^{\theta a}( w^{(1 - \theta)b}(x) f)\|_2 \lesssim \| w^{b}(x) f\|_2^{1-\theta}\|\langle D_x \rangle^af\|_2^{\theta}. \notag$$ This is an easy consequence of the Three Lines Lemma, and the fact that $$\|\langle D_x \rangle^{z a}( w^{(1 - z)b}(x)f)\|_2 \notag$$ is an analytic function in $z$ for $\Re z \in (0,1)$, for a dense set of functions in the space $H^a \cap L^2( w^{2b}(x)\,dx)$. \[23a\] For a solution $u=u(x,t)$ of , $$\|\partial_xu\|_{L^{\frac{1}{s+\frac{1}{2}\varepsilon}}_xL^2_T}\leq c_T \|u_0\|_{H^{2s+\varepsilon}}.$$ Consider the function $$F(z)=\int_{-\infty}^{\infty}\int_0^T D_x^{r(z)}(U(t)u_0)\, \psi(x,z)\,f(t)\,dt dx,$$ where $$r(z)=(1-z)(1+2s+\varepsilon)+z (2s+\varepsilon),\;\;\;\frac{1}{q(z)}=\frac{z}{2}+(1-z),\;\;\;q=\frac{2}{2-2s-\varepsilon},$$ $$\psi(x,z)=|g(x)|^{q/q(z)}\,\frac{g(x)}{|g(x)|},\;\;\;\;\text{with}\;\;\;\;\|g\|_{L^{1/(1-s-\frac{1}{2}\varepsilon)}_x}=\|f\|_{L^2([0,T])}=1,$$ which is analytic for $\Re z \in (0,1)$. Using that $$\|\psi(\cdot,0+iy)\|_2=\|\psi(\cdot,1+iy)\|_1=1,$$ one gets from $H^{2s+\varepsilon}$ persistence and the Kato smoothing effect that $$\label{23} \begin{aligned} & \|\partial_xU(t)u_0\|_{L^{\frac{1}{s+\frac{1}{2}\varepsilon}}_xL^2_T} \leq c \|D_xU(t)u_0\|_{L^{\frac{1}{s+\frac{1}{2}\varepsilon}}_xL^2_T}\\ & \quad \leq c \,\sup_{y\in\R} \| D_x^{1+2s+\varepsilon+iy} U(t)u_0\|_{L^{\infty}_xL^2_T}^{1-2s-\varepsilon}\,\sup_{y\in\R}\|D^{2s+\varepsilon+iy}_xU(t)u_0\|_{L^2_xL^2_T}^{2s+\varepsilon} \notag \\ & \quad \leq c_T \| D_x^{2s+\varepsilon}U(t)u_0\|_{2}. \notag \end{aligned}$$ Inserting the estimate in the proof of the local well posedness for , the result follows. Let $\phi_N$ be a smooth function such that $$\phi_N(x) = \left\{ \begin{array}{c l} \langle x \rangle^{2s} & \textrm{if $|x| \leq N$,} \\ (2N)^{2s} & \textrm{if $|x|>3N$}. \end{array} \right.\notag \notag$$ Then from , $$\begin{aligned} & \int_{\mathbb R}\phi_N(x)u^2(x,T)\,dx-\int_{\mathbb R}\phi_N(x)u^2(x,0)\,dx = \notag \\ & \quad 3\int_{[0,T]}\int_{\mathbb R}\phi'_N(x)(\partial_x u)^2\,dx\,dt -\int_{[0,T]}\int_{\mathbb R}\phi_N'''(x)u^2\,dx\,dt \notag \\ & \quad - \frac{k+1}{k+2}\int_{[0,T]}\int_{\mathbb R}\phi'_N(x)u^{k+2}\,dx\,dt. \label{weight_eq_2}\end{aligned}$$ We only prove the result in the case where $s < 2$ of the KdV equation, when $k=1$. Our main result, Theorem \[main\], is stronger when $k=2$, and $k \ge 4$, and the proof for $s \ge 2$ or $k=3$ is similar. We will use results from [@MR1230283], which state that the smoothing effects and Strichartz estimates that hold for the linearized KdV and mKdV also hold for the KdV. The $\phi_N'''(x)u^{2}$ term in the right hand side of can be bounded by the fact that $\phi_N'''(x) \lesssim 1$ independently of $N$ for $s \leq \frac{1}{2}$, and $L^2$ persistence: $$\left | \int_{[0,T]}\int_{\mathbb R}\phi_N'''(x)u^2\,dx\,dt \right | \lesssim T\|u\|_2^2. \label{missed-term}$$ The bounds on the other terms on the right hand side of depend on whether $s < \frac{1}{2}$ or $s \ge \frac{1}{2}$. We first give the proof of the result in the case that $s < \frac{1}{2}$ Since $|\phi_N'(x)| \lesssim \langle x \rangle^{2s-1}$ independently of $N$, we can bound the first term on the right hand side of by $$\left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)(\partial_xu)^2\,dx \,dt \right | \lesssim \|\langle x \rangle^{s-\frac{1}{2}}\partial_xu\|_{L_x^2L_T^2}^2. \label{1_rhs_weight}$$ Using , Lemma \[23a\], and the Hölder inequality, $$\begin{aligned} & \left |\int_{[0,T]}\int_{\mathbb R}\phi'_N(x)(\partial_x u)^2\,dx\,dt \right | \notag \\ & \quad \lesssim \|\langle x \rangle^{s-\frac{1}{2}}\|_{\frac{2}{1-2(s+\frac{1}{2}\varepsilon)}} \|D_xu(x,t)\|_{L_x^{\frac{1}{s+\frac{1}{2}\varepsilon}}L_T^2} < \infty. \label{weight-eq-first-rhs-term}\end{aligned}$$ For the $\phi_N'(x)u^{k+2}$ term in the right hand side of . We can bound this term with the Hölder inequality, $$\begin{aligned} \left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)u^{3}\,dx\,dt \right | & \lesssim \| \langle x\rangle^{2s-1}|u|^{3} \|_{L_T^1L_x^1} \notag \\ & \leq \| u \|_{L_T^1L_x^{\infty}} \| \langle x\rangle^{s-\frac{1}{2}}u \|_{L_T^{\infty}L_x^2}^2 \notag \\ & \leq T^{\frac{5}{6}} \| u \|_{L_T^6L_x^{\infty}} \|u \|_{L_T^{\infty}L_x^2}^2. \label{weight-eq-sec-rhs-term}\end{aligned}$$ Since $s-\frac{1}{2} < 0$, is finite by the Strichartz estimates in [@MR1230283], and $L^2$ persistence. It follows from that $$\begin{aligned} \left |\int_{\mathbb R}(\phi_N(x)u^2(x,T))\,dx \right | & \leq \int_{\mathbb R}|\phi_N(x)u^2(x,0)|\,dx \notag \\ & \quad + 3\int_{[0,T]}\int_{\mathbb R}|\phi'_N(x)(\partial_x u)^2|\,dx\,dt \notag \\ & \quad + \frac{2}{3}\int_{[0,T]}\int_{\mathbb R}|\phi'_N(x)u^{3}|\,dx\,dt \notag \\ & \quad + \int_{[0,T]}\int_{\mathbb R}|\phi_N'''(x)u^2|\,dx\,dt. \notag\end{aligned}$$ By $|x|^{s}u_0 \in L^2$, , , and , the result follows. We now consider the case that $s \in [\frac{1}{2},1)$. For the first term on the right hand side of , we use Lemma \[inter-reg-dec\], and $H^{2s}$ persistence to obtain $$\begin{aligned} \left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)(\partial_xu)^2\,dx \,dt \right | & \lesssim \|\partial_x u \langle \phi_N'(x) \rangle^{\frac{1}{2}} \|_{L_T^2L_x^2}^2 \notag \\ & \lesssim \|\partial_x (u\langle \phi_N'(x) \rangle^{\frac{1}{2}}) \|_{L_T^2L_x^2}^2 \notag \\ & \quad + \|u(\langle \phi_N'(x) \rangle^{\frac{1}{2}})' \|_{L_T^2L_x^2}^2 \notag \\ & \lesssim \|\frac{\partial_x}{\langle D_x \rangle} \langle D_x \rangle(u\langle \phi_N'(x) \rangle^{\frac{1}{2}}) \|_{L_T^2L_x^2}^2 \notag \\ & \quad + \|u\langle x \rangle ^{s-\frac{3}{2}} \|_{L_T^2L_x^2}^2 \notag \\ & \lesssim \| \langle D_x \rangle(u\langle \phi_N'(x) \rangle^{\frac{1}{2}}) \|_{L_T^2L_x^2}^2 \notag \\ & \quad+ \|u\langle x \rangle ^{s-\frac{3}{2}} \|_{L_T^2L_x^2}^2 \notag \\ & \lesssim \| \langle D_x \rangle^{2s} u \|_{L_T^2L_x^2}^{\frac{1}{s}} \| (\langle \phi_N'(x) \rangle^{\frac{s}{2s-1}})u \|_{L_T^2L_x^2}^{2-\frac{1}{s}} \notag \\ & \quad + \|u\langle x \rangle ^{s-\frac{3}{2}} \|_{L_T^2L_x^2}^2 \notag\end{aligned}$$ Since $\langle \phi_N'(x) \rangle^{\frac{s}{2s-1}} \lesssim \phi_N^{\frac{1}{2}}(x)$, it follows that $$\begin{aligned} \left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)(\partial_xu)^2\,dx \,dt \right | & \lesssim \| \langle D_x \rangle^{2s} u \|_{L_T^2L_x^2}^{\frac{1}{s}} \| \phi_N^{\frac{1}{2}}(x)u \|_{L_T^2L_x^2}^{2-\frac{1}{s}} \notag \\ & \quad + \|u\langle x \rangle ^{s-\frac{3}{2}} \|_{L_T^2L_x^2}^2 \label{case-half}\end{aligned}$$ For the $\phi_N'(x)u^{k+2}$ term, $$\begin{aligned} \left | \int_{[0,T]}\int_{\mathbb R}\phi'_N(x)u^{3}\,dx\,dt \right | & \lesssim \| \langle x\rangle^{2s-1}|u|^{3} \|_{L_T^1L_x^1} \notag \\ & \leq \| u \|_{L_T^1L_x^{\infty}} \| \langle x\rangle^{s-\frac{1}{2}}u \|_{L_T^{\infty}L_x^2}^2 \notag \\ & \leq T^{\frac{5}{6}} \| u \|_{L_T^6L_x^{\infty}} \| \langle x\rangle^{s-\frac{1}{2}}u \|_{L_T^{\infty}L_x^2}^2 \label{big-s-3}\end{aligned}$$ The term in is finite from the first part of the proof since $s-\frac{1}{2} < \frac{1}{2}$. From , , , , the fact that $\phi_N(x) \lesssim \langle x \rangle^{2s}$ and our assumption on $u(x,0)$, $$\begin{aligned} & \|\phi_N^{\frac{1}{2}}(x)u^2(x,T)\|_{L_x^2}^2 \lesssim \|\langle x \rangle^s u^2(x,0)\|_{L_x^2}^2 + \|u\langle x \rangle ^{s-\frac{3}{2}} \|_{L_T^2L_x^2}^2 \notag \\ & \quad +\| \langle x\rangle^{s-\frac{1}{2}}u \|_{L_T^{\infty}L_x^2}^2 + \| \langle D_x \rangle^{2s} u \|_{L_T^2L_x^2}^{\frac{1}{s}} (\int_0^T\| \phi_N^{\frac{1}{2}}(x)u(x,t) \|_{L_x^2}^2\,dt )^{1-1/2s} \notag \\ \label{stay-on-target} & \quad +T\|u\|_{L_x^2}^2 + T^{\frac{5}{6}} \| u \|_{L_T^6L_x^{\infty}} \|u \|_{L_T^{\infty}L_x^2}^2.\end{aligned}$$ The applicaion of Bihari’s inequality (see [@MR0079154]) to yields a bound on $\|\phi_N^{\frac{1}{2}}(x)u(x,T)\|_{2}$ that is independent of $N$. By taking $N$ to infinity, the result follows. Estimating a Derivative ======================= We begin our computation of . We will show that by scaling out the fractional derivative, it will suffice to bound $$\left |Q_N(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}) \right |. \notag$$ Since the operator $Q_N$ is convolution with a function whose Fourier transform is very localized, we require estimates on $$\int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz, \label{project}$$ where $\varphi_{\omega}$ is a function whose Fourier transform has support near $\omega$. We will use a contour integral argument. Because of this, we require estimates on the analytic continuation of $\varphi_{\omega}$. These are contained in the following lemma. \[anal\_cont\] Let $\xi \in \mathbb R$, $z=x+yi$ for $x,y \in \mathbb R$, $\varphi(\xi)$ be a function so that $\hat{\varphi}(x)$ is a smooth function with support in $[\frac{1}{2},2]$, and for $\omega \in \mathbb R \setminus \{0\}$, let $\varphi_{\omega}(\xi)$ be the function with Fourier transform $\hat{\varphi}(\frac{x}{\omega})$. Then $\varphi_{\omega}$ is an entire function that obeys the following estimates. $$|\varphi_{\omega}((\xi-z))| \lesssim \left\{ \begin{array}{c l} \frac{|e^{2\omega y}-e^{\frac{1}{2}\omega y}|}{{\omega}^2y|\xi-z|^2} & \textrm{if $y \ne 0$ and $x \ne \xi$,}\\ \\ \frac{1}{|\omega|(\xi-x)^2} & \textrm{if $y = 0$ and $x \ne \xi$.} \end{array} \right . \notag$$ That $\varphi_{\omega}$ is entire follows from the Paley-Wiener theorem. Let $y \ne 0$. Since $\hat{\varphi}$ is a smooth function with support in $[\frac{1}{2},2]$, we integrate by parts to obtain $$\begin{aligned} \varphi_{\omega}(\xi-z) & = \int_{\mathbb R}\hat{\varphi}(\frac{\zeta}{\omega})\frac{1}{i(\xi-z)}\frac{d}{d\zeta}e^{i\zeta (\xi-z)}\,d\zeta \notag \\ & = -\int_{[\frac{\omega}{2},2\omega]}\frac{1}{\omega}\hat{\varphi}^{'}(\frac{\zeta}{\omega})\frac{1}{i(\xi-z)}e^{i\zeta (\xi-z)}\,d\zeta \notag\\ & = \int_{[\frac{\omega}{2},2\omega]}\frac{1}{\omega}\hat{\varphi}^{'}(\frac{\zeta}{\omega})\frac{1}{(\xi-z)^2}\frac{d}{d\zeta}e^{i\zeta (\xi-z)}\,d\zeta \notag\\ & = -\int_{[\frac{\omega}{2},2\omega]}\frac{1}{{\omega}^2}\hat{\varphi}^{''}(\frac{\zeta}{\omega})\frac{1}{(\xi-z)^2}e^{i\zeta (\xi-z)}\,d\zeta. \label{phi_parts}\end{aligned}$$ From we conclude that $$\begin{aligned} |\varphi_{\omega}(\xi-z)| & \leq \left |\int_{[\frac{\omega}{2},2\omega]}\frac{1}{{\omega}^2}\hat{\varphi}^{''}(\frac{\zeta}{\omega})\frac{1}{(\xi-z)^2}e^{i\zeta (\xi-z)}\,d\zeta \right | \notag \\ & \leq \int_{[\frac{\omega}{2},2\omega]}\frac{1}{{\omega}^2}|\hat{\varphi}^{''}(\frac{\zeta}{\omega})|\frac{1}{|\xi-z|^2}e^{\zeta y}\,d\zeta \notag \\ & \leq c_{\varphi}\frac{|e^{2\omega y}-e^{\frac{1}{2}\omega y}|}{{\omega}^2y|\xi-z|^2}. \notag\end{aligned}$$ The case $y=0$ follows from taking the limit as $y \rightarrow 0$ of the first estimate. From Lemma \[anal\_cont\], we can infer the following about the analyticity of the integrand in . \[anal\_cont\_cor\] For $\xi \in \mathbb R$, the function $$\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}$$ is analytic on $\mathbb C \setminus \{z: |\Im{z}| \ge 1, \Re{z}=0\}$. The estimate in Lemma \[anal\_cont\] has good $x$ dependence away from $\xi$. To estimate near $z=\xi$, we use an analytic continuation of the integrand and the Cauchy integral theorem, which we now describe. The function $\varphi_{\omega}$ oscillates with frequency near $\omega$. For a fixed $z_0 \in \mathbb R$, we think of the function $\exp(itz^3)$ as oscillating with frequency $tz^2_0$ near the value $z_0$. For $z=\xi$ where $t\xi^2 \ll \omega$, the function $\varphi_{\omega}$ oscillates much faster than $\exp(itz^3)$, so Lemma \[anal\_cont\] shows that analytic continuation of $$\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}} \notag$$ changes this rapid oscillation into decay, which yields good $\omega$ dependence for . To formalize this, we make the following definition. Given $t>0$, and $\omega>0$, we say that $\xi \in \mathbb R$ is **near** if $$|\xi| \leq \frac{1}{10}\sqrt{\frac{\omega}{t}}. \notag$$ Where the oscillation of $\exp(itz^3)$ is much larger than $\omega$, an analytic continuation of $\exp(itz^3)$ has a similar property. We say that $\xi \in \mathbb R$ is **far** if $$|\xi| > 10\sqrt{\frac{\omega}{t}}. \notag$$ In the intermediate case where the oscillation of $\exp(itz^3)$ is comparable to $\omega$, analytic continuation does not help. This is where the worst behavior of the estimate occurs. We say that $\xi \in \mathbb R$ is **intermediate** if $$\frac{1}{10}\sqrt{\frac{\omega}{t}} < |\xi| \leq 10\sqrt{\frac{\omega}{t}}. \notag$$ These heuristics are formalized in Lemma \[convol\], then used to estimate in Lemma \[finite\]. We require an elementary integral estimate for Lemma \[convol\]. One expects that since $\sin t \approx t$, then $$\begin{aligned} \int_{[0,\pi]}\frac{e^{a\sin{s}}-e^{b\sin{s}}}{\sin{s}}\,ds & \approx \int_{[0,\pi]}\frac{e^{a s}-e^{b s}}{s}\,ds \notag \\ & = \int_{[0,\pi]}\frac{e^{a s}-e^{b s}}{as}a\,ds \notag \\ & = \int_{[0,\pi]}\frac{e^{t}-e^{\frac{b}{a} t}}{t}\,dt. \notag $$ This is what the next lemma proves. \[e\_sin\] Let $a<b<0$. Then $$\left | \int_{[0,\pi]}\frac{e^{a\sin{s}}-e^{b\sin{s}}}{\sin{s}}\,ds\right | \lesssim (\pi\frac{a}{b}-1)+1+\frac{b}{\pi a}e^{-\frac{\pi a}{b}}. \notag$$ By making the change of variable $r=-(s-\frac{\pi}{2})$, we have for an arbitrary function $f$, $$\int_{[\frac{\pi}{2},\pi]}f(\sin{s})\,ds=\int_{[0,\frac{\pi}{2}]}f(\sin{r})\,dr. \notag$$ Therefore, $$\begin{aligned} \left | \int_{[0,\pi]}\frac{e^{b\sin{s}}-e^{a\sin{s}}}{\sin{s}}\,ds\right | & = 2\int_{[0,\frac{\pi}{2}]}\frac{e^{b\sin{s}}-e^{a\sin{s}}}{\sin{s}}\,ds. \label{e_sin_1}\end{aligned}$$ Notice that for $s \in [0,\frac{\pi}{2}]$, $\frac{2s}{\pi} \leq \sin{s} \leq 2s$. We use this to bound . $$\begin{aligned} \int_{[0,\frac{\pi}{2}]}\frac{e^{b\sin{s}}-e^{a\sin{s}}}{\sin{s}}\,ds & \lesssim \int_{[0,\frac{\pi}{2}]}\frac{e^{\frac{2b}{\pi}s}-e^{2as}}{s}\,ds \notag \\ & = \int_{[0,\frac{\pi}{2}]}\frac{e^{\frac{2b}{\pi}s}-e^{2as}}{\frac{2b}{\pi}s}\frac{2b}{\pi}\,ds \notag \\ & = \int_{[b,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr. \label{e_sin_2}\end{aligned}$$ Because $a<b<0$, it follows that $\frac{a}{b} > 1$, and $\frac{\pi a}{b} > 1$. For $r < 0$, $\frac{\pi a}{b}r < r$, so that $e^{\frac{\pi a}{b}r}-e^{r} < 0$, and therefore $$\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r} > 0. \label{e_sin_2_pos}$$ By \[e\_sin\_2\_pos\], the integrand in is positive, so we can bound it with $$\begin{aligned} \int_{[b,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr & \leq \int_{[-\infty,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr \notag \\ & = \int_{[-\infty,-1]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr + \int_{[-1,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr \notag \\ & \leq \frac{b}{\pi a}e^{-\frac{\pi a}{b}}+e^{-1} + \int_{[-1,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr. \label{e_sin_3}\end{aligned}$$ By Taylor expansion and an error estimate for alternating sums, $$\begin{aligned} \int_{[-1,0]}\frac{e^{\frac{\pi a}{b}r}-e^{r}}{r}\,dr & = \int_{[-1,0]}\sum_{n=1}^{\infty}(\frac{(\frac{\pi a}{b})^n-1}{n!}r^{n-1})\,dr \notag \\ & = -\sum_{n=1}^{\infty}(\frac{(\frac{\pi a}{b})^n-1}{n!}\frac{(-1)^{n}}{n}) \notag \\ & \leq (\frac{\pi a}{b})-1. \label{e_sin_4}\end{aligned}$$ Combining and , the result follows. \[convol\] Let $\varphi(\xi)$ be a function so that the Fourier transform $\hat{\varphi}(x)$ is a smooth function with support in $[\frac{1}{2},2]$, and for $\omega \in \mathbb R\setminus \{0\}$, let $\varphi_{\omega}(\xi)$ be the function such that $\hat{\varphi}_{\omega}=\varphi(\frac{x}{\omega})$. Then $$\left |\int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \lesssim \left\{ \begin{array}{c l} (1+t){\omega}^{-\frac{1}{8}} & \textrm{if $\omega >0$,} \\ & \textrm{and }|\xi| \textrm{ intermediate,}\\ (1+t)|\omega|^{-1} & \textrm{else}. \end{array} \right.\notag$$ We consider separately the four different cases, ${\omega}<0$, ${\omega}>0$ and $|\xi|$ near, ${\omega}>0$ and $|\xi|$ intermediate, and ${\omega}>0$ and $|\xi|$ far. *Case ${\omega}<0$:* (-3,0) – (3,0) node\[right\] [$x$]{}; (0,-1.2) – (0,2.5) node\[above\] [$yi$]{}; (-2,0) –(1,0) node\[below\] [$\gamma_1$]{}; (1,0) –(2,0) node\[below\] [$R$]{}; (2,0) – (2,0.5) node\[right\] [$\gamma_2$]{}; (2,0.5) – (2,1.2) node\[right\] [$R+\frac{i}{2}$]{}; (2,1.2) – (-1,1.2) node\[above\] [$\gamma_3$]{}; (-1,1.2) – (-2,1.2) node\[left\] [$-R+\frac{i}{2}$]{}; (-2,1.2) – (-2,0.5) node\[left\] [$\gamma_4$]{}; (-2,0.5) – (-2,0) node\[below\] [$-R$]{}; Instead of integrating over $\mathbb R$ in , we will compute the integral over the contours $\gamma_1$ through $\gamma_4$ in Figure 1, taking the limit as $R$ approaches infinity. By Corollary \[anal\_cont\_cor\] and the Cauchy integral theorem, $$\begin{aligned} \int_{\gamma_1}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz & = -\int_{\gamma_2}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz -\int_{\gamma_3}\ldots \notag \\ & \quad -\int_{\gamma_4}\ldots \notag\end{aligned}$$ We will use estimates on the integrals over $\gamma_2$, $\gamma_3$, and $\gamma_4$ to estimate . Along $\gamma_2$, $$\begin{aligned} & \left |\int_{[0,\frac{1}{2}]} \varphi_{\omega}(\xi-R-yi)\frac{e^{it(R+yi)^3}}{(1+(R+yi)^2)^{\frac{1}{8}}}i\,dy \right | \lesssim \notag \\ & \int_{[0,\frac{1}{2}]} \left | \varphi_{\omega}(\xi-R-yi)\frac{e^{it(R+yi)^3}}{(1+(R+yi)^2)^{\frac{1}{8}}}i \right | \,dy \lesssim \notag \\ & \int_{[0,\frac{1}{2}]} \frac{|e^{2{\omega}y}-e^{\frac{1}{2}{\omega}y}|}{{\omega}^2y|\xi-R-yi|^2}\frac{e^{-t(3R^2-y^2)y}}{(1+R^2)^{\frac{1}{8}}} \,dy. \label{gam_2}\end{aligned}$$ For fixed ${\omega}$, approaches 0 as $R \rightarrow \infty$. A similar estimate applies for $\gamma_4$. We can estimate the integral along $\gamma_3$ using Lemma \[anal\_cont\], $$\begin{aligned} & \left |\int_{[-R,R]} \varphi_{\omega}(\xi-x-\frac{i}{2})\frac{e^{it(x+\frac{i}{2})^3}}{(1+(x+\frac{i}{2})^2)^{\frac{1}{8}}}\,dx \right | \lesssim \notag \\ & \int_{[-R,R]}\left | \varphi_{\omega}(\xi-x-\frac{i}{2})\frac{e^{it(x+\frac{i}{2})^3}}{(1+(x+\frac{i}{2})^2)^{\frac{1}{8}}}\right | \,dx \lesssim \notag \\ & \int_{[-R,R]} \frac{|e^{{\omega}}-e^{\frac{1}{4}{\omega}}|}{{\omega}^2((\xi-x)^2+1)}\frac{e^{-t(\frac{3}{2}x^2-\frac{1}{8})}}{(1+x^2)^{\frac{1}{8}}}\,dx \lesssim \notag \\ & \frac{|e^{{\omega}}-e^{\frac{1}{4}{\omega}}|}{{\omega}^2}\int_{\mathbb R} \frac{1}{((\xi-x)^2+1)}\frac{1}{(1+x^2)^{\frac{1}{8}}}\,dx \lesssim \notag \\ & \frac{|e^{{\omega}}-e^{\frac{1}{4}{\omega}}|}{{\omega}^2}. \label{gam_3}\end{aligned}$$ From and we estimate , $$\left | \int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \lesssim \frac{|e^{{\omega}}-e^{\frac{1}{4}{\omega}}|}{{\omega}^2} \lesssim (1+t){|\omega|}^{-1}\notag.$$ *End of Case ${\omega}<0$.* Let $\varepsilon$ be some positive number that will be specified later. For the remaining three cases, we split up the integral in the following manner. $$\begin{aligned} \int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz & = \int_{\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \notag \\ & \quad + \int_{ B_{\frac{1}{10}\varepsilon}(\xi)}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz. \notag\end{aligned}$$ We estimate the integral over $\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)$ using the decay of $\varphi_{\omega}$, from Lemma \[anal\_cont\]. $$\begin{aligned} \left | \int_{\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | & \leq \int_{\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)}\left | \varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}} \right |\,dz \notag \\ & \leq \int_{\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)}\left | \varphi_{\omega}(\xi-z) \right |\,dz \notag \\ & \leq \int_{\mathbb R \setminus B_{\frac{1}{10}\varepsilon}(\xi)} \frac{1}{{\omega}(\xi-x)^2} \,dx \lesssim \frac{1}{{\omega}\varepsilon}. \label{r-ep}\end{aligned}$$ In the next three cases we estimate $$\int_{ B_{\frac{1}{10}\varepsilon}(\xi)}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz. \label{b+ep}$$ *Case $\omega > 0$, near:* (-2,0) –(-0.5,0); (-0.5,0) –(0,0) node\[above\] [$\xi$]{}; (0,0) –(2,0) node\[above\] [$\xi+\frac{1}{10}\varepsilon$]{}; +(2,0) arc (0:-45:2) node\[right\] [$\Gamma_1$]{}; +(1.414,-1.414) arc (-45:-90:2) node\[below\] [$\xi-\frac{i}{10}\varepsilon$]{}; +(0,-2) arc (-90:-180:2) node\[above\] [$\xi-\frac{1}{10}\varepsilon$]{}; By Corollary \[anal\_cont\_cor\] and the Cauchy integral theorem, we can estimate by approximating the integral along the semicircle arc $\Gamma_1$ in Figure 2, as long as we avoid the rays where the integrand is not analytic. If $\frac{1}{10}{\omega}^{\frac{1}{2}}t^{-\frac{1}{2}}<1$, then let $\varepsilon={\omega}^{\frac{1}{2}}t^{-\frac{1}{2}}$. Otherwise, let $\varepsilon=1$. We illustrate the estimate only for the case $\varepsilon=1$, as the other case follows by a similar argument. $$\begin{aligned} & \left | \int_{[2\pi,\pi]} \varphi_{\omega}(-\frac{\varepsilon}{10} e^{is}) \frac{1}{(1+(\xi+\frac{\varepsilon}{10} e^{is})^2)^{\frac{1}{8}}} e^{it(\xi+\frac{\varepsilon}{10} e^{is})^3} i\frac{\varepsilon}{10} e^{is}\,ds \right | \lesssim \notag \\ & \int_{[2\pi, \pi]} \left | \varphi_{\omega}(-\frac{\varepsilon}{10} e^{is}) \frac{1}{(1+(\xi+\frac{\varepsilon}{10} e^{is})^2)^{\frac{1}{8}}} e^{it(\xi+\frac{\varepsilon}{10} e^{is})^3} \right | \varepsilon\,ds \lesssim \notag \\ & \int_{[2\pi,\pi]}t\frac{|e^{\frac{1}{5}{\omega}\varepsilon \sin{s}}-e^{\frac{1}{20}{\omega}\varepsilon\sin{s}}|}{{\omega}^3\varepsilon\sin{s}} e^{-\frac{t}{10}(3(\xi+\frac{1}{10}\varepsilon\cos{s})^2-\frac{1}{100}\varepsilon^2\sin^2{s})\varepsilon\sin{s}} \varepsilon\,ds \label{near_1}\end{aligned}$$ Since $|\xi| \leq \frac{1}{10}\sqrt{\frac{{\omega}}{t}}$ and $\varepsilon = 1 \leq \sqrt{\frac{{\omega}}{t}}$, it follows that $$\begin{aligned} |\frac{t}{10}(3(\xi+\frac{1}{10}\varepsilon\cos{s})^2-\frac{1}{100}\varepsilon^2)| & \leq \frac{t}{10}(3(|\xi|+\frac{1}{10}\varepsilon)^2+\frac{1}{100}\varepsilon^2) \notag \\ & \leq \frac{13}{1000}{\omega}. \notag\end{aligned}$$ Using this and Lemma \[e\_sin\], we bound with $$\begin{aligned} & \int_{[2\pi,\pi]}t\frac{|e^{\frac{1}{5}{\omega}\varepsilon\sin{s}}-e^{\frac{1}{20}{\omega}\varepsilon\sin{s}}|}{{\omega}^3\sin{s}} e^{-\frac{13}{1000}{\omega}\varepsilon\sin{s}} \,ds \notag \\ & \quad \lesssim \frac{t}{{\omega}^3}\int_{[2\pi,\pi]}\frac{|e^{0.187{\omega}\varepsilon\sin{s}}-e^{0.037{\omega}\varepsilon\sin{s}}|}{\sin{s}} \,ds \lesssim \frac{t}{\omega}. \label{near}\end{aligned}$$ From and , we have the estimate $$\left |\int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \lesssim \frac{1}{\omega}+\frac{t}{\omega} \lesssim (1+t){\omega}^{-1}. \notag$$ *End of Case $\omega > 0$, near.* *Case $\omega > 0$, intermediate:* To estimate , we use the Young inequality, and the fact that $\|\varphi_{\omega}\|_1$ is uniformly bounded in ${\omega}$. Let $\varepsilon=\frac{1}{10}\sqrt{\frac{{\omega}}{t}}$. $$\begin{aligned} & \left | \int_{B_{\frac{1}{10}\varepsilon}(\xi)} \varphi(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \notag \\ & \lesssim \left | \varphi_{\omega} \ast (\chi_{B_{\frac{1}{10}\varepsilon}(\xi)}(z) \frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}) \right | \notag \\ & \lesssim \|\varphi_{\omega}\|_1 \left \|\chi_{B_{\frac{1}{10}\varepsilon}(\xi)}(z) \frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}} \right \|_{\infty} \label{inter_1}\end{aligned}$$ Since $\xi$ is intermediate and $\varepsilon=\frac{1}{10}\sqrt{\frac{{\omega}}{t}}$, any $z \in B_{\frac{1}{10}\varepsilon}(\xi)$ will obey the estimate $z \approx \sqrt{\frac{\omega}{t}}$. This estimate on $z$ allows us to bound the $\|\cdot\|_{\infty}$ term in by $$\left \|\chi_{B_{\frac{1}{10}\varepsilon}(\xi)}(z) \frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}} \right \|_{\infty} \lesssim t^{\frac{1}{8}}|{\omega}|^{-\frac{1}{8}}. \label{inter}$$ From and , we have the estimate $$\left |\int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \lesssim \frac{\sqrt{t}}{{\omega}^{\frac{3}{2}}}+t^{\frac{1}{8}}{\omega}^{-\frac{1}{8}} \lesssim (1+t){\omega}^{-\frac{1}{8}}. \notag$$ *End of Case $\omega > 0$, intermediate.* *Case $\omega > 0$, far:* (-2,0) –(-0.5,0) ; (-0.5,0) –(0,0) node\[below\] [$\xi$]{}; (0,0) –(2,0) node\[below\] [$\xi+\frac{1}{10}\varepsilon$]{}; +(2,0) arc (0:45:2) node\[above\] [$\Gamma_2$]{}; +(1.414,1.414) arc (45:90:2) node\[above\] [$\xi+\frac{i}{10}\varepsilon$]{}; +(0,2) arc (90:180:2) node\[below\] [$\xi-\frac{1}{10}\varepsilon$]{}; Let $\varepsilon=\sqrt{\frac{{\omega}}{t}}$. We use an argument similar the near case, integrating along the the semicircle arc $\Gamma_2$ in Figure 3, $$\begin{aligned} & \left | \int_{[0,\pi]} \varphi_{\omega}(-\frac{1}{10}\varepsilon e^{is}) \frac{1}{(1+(\xi+\frac{1}{10}\varepsilon e^{is})^2)^{\frac{1}{8}}} e^{it(\xi+\frac{1}{10}\varepsilon e^{is})^3} \frac{i}{10}\varepsilon e^{is}\,ds \right | \lesssim \notag \\ & \int_{[0,\pi]} \left | \varphi_{\omega}(-\frac{1}{10}\varepsilon e^{is}) \frac{1}{(1+(\xi+\frac{1}{10}\varepsilon e^{is})^2)^{\frac{1}{8}}} e^{it(\xi+\frac{1}{10}\varepsilon e^{is})^3} \right | \varepsilon\,ds \lesssim \notag \\ & \int_{[0,\pi]}t\frac{|e^{\frac{1}{5}{\omega}\varepsilon \sin{s}}-e^{\frac{1}{20}{\omega}\varepsilon \sin{s}}|}{{\omega}^3\varepsilon \sin{s}} e^{-\frac{t}{10}(3(\xi+\frac{1}{10}\varepsilon \cos{s})^2-\frac{1}{100}\varepsilon^2 \sin^2{s})\varepsilon \sin{s}} \varepsilon\,ds. \label{far_eq}\end{aligned}$$ Since $\xi > 10\sqrt{\frac{{\omega}}{t}}$, $$\begin{aligned} & \quad -29.402{\omega} \notag \\ & \leq -\frac{t}{10}(3(10\sqrt{\frac{{\omega}}{t}}-\frac{1}{10}\sqrt{\frac{{\omega}}{t}})^2-\frac{1}{100}\sqrt{\frac{{\omega}}{t}}) \notag \\ & \leq -\frac{t}{10}(3(\xi+\frac{1}{10}\varepsilon \cos{s})^2-\frac{1}{100}\varepsilon^2 \sin^2{s}). \notag\end{aligned}$$ We use this with Lemma \[e\_sin\] to bound by $$\begin{aligned} & \quad \int_{[0,\pi]}t\frac{|e^{\frac{1}{5}{\omega}\varepsilon \sin{s}}-e^{\frac{1}{20}{\omega}\varepsilon \sin{s}}|}{{\omega}^3\sin{s}} e^{-29.402{\omega}\varepsilon \sin{s}} \,ds \notag \\ & \lesssim \frac{t}{{\omega}^3}\int_{[0,\pi]}\frac{|e^{-29.202{\omega}\varepsilon \sin{s}}-e^{-29.352{\omega}\varepsilon \sin{s}}|}{\sin{s}} \,ds \lesssim \frac{t}{{\omega}^3}. \label{far}\end{aligned}$$ From and , we have the estimate $$\left |\int_{\mathbb R}\varphi_{\omega}(\xi-z)\frac{e^{itz^3}}{(1+z^2)^{\frac{1}{8}}}\,dz \right | \lesssim \frac{\sqrt{t}}{{\omega}^{\frac{3}{2}}}+\frac{t}{{\omega}^3} \lesssim (1+t){\omega}^{-\frac{3}{2}} \lesssim (1+t){\omega}^{-1}. \notag$$ *End of Case $\omega > 0$, far.* \[finite\] $$\left \|D_{\xi}^{\frac{1}{8}} \left (\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}} \right ) \right \|_{L_{\xi}^{\infty}l_{N}^1} \lesssim 1+t. \notag$$ The operator $Q^5_N$ (see also Appendix A) is defined by $$Q^5_Nf \equiv (|\frac{x}{2^N}|^{\frac{1}{8}}(\eta(\frac{x}{2^N})+\eta(\frac{-x}{2^N}))\hat{f}(x))^{\vee}. \notag$$ Since $Q_N$ is just convolution against the Fourier transform of a scaled smooth function, by rescaling we obtain $$\left \|Q_ND_{\xi}^{\frac{1}{8}}\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}} \right \|_{L_{\xi}^{\infty}l_N^1} = \left \|2^{\frac{N}{8}}Q^5_N \left (\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}} \right ) \right \|_{L_{\xi}^{\infty}l_N^1}. \notag$$ We can estimate the low frequency part using the Young inequality in the following manner, $$\begin{aligned} \left \|2^{\frac{N}{8}} Q^5_N \left (\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}} \right ) \right \|_{L_{\xi}^{\infty}l_{N \leq 0}^1} & \leq \sum_{N \leq 0}2^{\frac{N}{8}} \left \|Q^5_N(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}) \right \|_{L_{\xi}^{\infty}} \notag \\ & \lesssim \sum_{N \leq 0}2^{\frac{N}{8}} \left \|\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}} \right \|_{L_{\xi}^{\infty}} \notag \\ & \lesssim \sum_{N \leq 0}2^{\frac{N}{8}} \lesssim 1. \notag\end{aligned}$$ We use Lemma \[convol\], noting that if $t$ is fixed, for each $|\xi|$, there is a unique dyadic $2^N$ so that $\xi$ is intermediate. We use this to bound the remaining frequencies. $$\begin{aligned} \left \|2^{\frac{N}{8}}Q^5_N(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}) \right \|_{L_{\xi}^{\infty}l_{N > 1}^1} & = \left \|\sum_{N=1}^{\infty}2^{\frac{N}{8}}|Q^5_N(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}})| \right \|_{L_{\xi}^{\infty}} \notag \\ & \lesssim \left \|\sum_{2^N \vert \textrm{ $\xi$ not intermediate}}2^{\frac{N}{8}} \left |Q^5_{N}(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}) \right | \right \|_{L_{\xi}^{\infty}} \notag \\ & \quad + \left \|\sum_{2^N \vert \textrm{ $\xi$ intermediate}}2^{\frac{N}{8}}|Q^5_{N}(\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}})| \right \|_{L_{\xi}^{\infty}} \notag \\ & \lesssim (\sum_{N=1}^{\infty}2^{\frac{N}{8}} 2^{-N} + 1)(1+t). \notag\end{aligned}$$ Decay Estimates for mKdV Solutions ================================== With our bound from Lemma \[finite\], we will show that our main result follows. This will come from the fact that for $\alpha \in (0,1)$, $$\|D_{x}^{\alpha}(fg)-gD_{x}^{\alpha}f\|_2 \lesssim \|Q_ND_{x}^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2. \label{prod_rule}$$ A classical Leibnitz type inequality for fractional derivatives is the following (see [@KPV1]). \[stand\_prod\] Let $0 < \alpha, \alpha_1, \alpha_2 < 1$, $\alpha=\alpha_1+\alpha_2$, $1 <p,p_1,p_2 < \infty$, and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$. In addition, the $\alpha_1=\alpha$, $p=p_2$, and $p_1=\infty$ is allowed. Then the following holds for functions $f,g$ on $\mathbb R^n$. $$\|D_x^{\alpha}(fg)-D^{\alpha}(f)g-fD_x^{\alpha}(g)\|_p \lesssim \|D_x^{\alpha_1}g\|_{p_1}\|D_x^{\alpha_2}f\|_{p_2} \notag$$ The proof uses the Littlewood-Paley Theorem (see [@S]), which states that for any function $f$, if $1 < p < \infty$, then $$\label{little_pale} \|Q_N(f)\|_{L_x^pl_N^2} \lesssim \|f\|_p \lesssim \|Q_N(f)\|_{L_x^pl_N^2}.$$ Lemma \[stand\_prod\] is not sufficient for our argument in the previous section, since we need to put the *derivative* term in the infinity norm. A product rule like this can be obtained by following the proof of Lemma \[stand\_prod\] line for line. The only difference is that since fails for $p=\infty$, $\|Q_N(D_x^{\alpha}g)\|_{L_{x}^{\infty}l_N^2}$ is not equivalent to $\|D_x^{\alpha}g\|_{\infty}$. This idea was inspired by [@KT], where the authors use $\|Q_N\cdot\|_{l_N^2L_x^{4}L_T^{\infty}}$ in an estimate where the $\|\cdot\|_{L_x^{4}L_T^{\infty}}$ norm may fail. \[my\_prod\_rule\] Let $0 < \alpha < 1$ and $1 < p < \infty$. For functions $f$ and $g$, $$\|D_x^{\alpha}(fg)-gD_x^{\alpha}f-fD_x^{\alpha}g\|_p \lesssim (\|Q_ND_x^{\alpha}g\|_{L_x^{\infty}l_N^2} +\|D_x^{\alpha}g\|_{L_x^{\infty}})\|f\|_p. \notag$$ In particular, $$\|D_x^{\alpha}(fg)-gD_x^{\alpha}f\|_2 \lesssim \|Q_ND_x^{\alpha}g\|_{L_x^{\infty}l_N^1}\|f\|_2. \notag$$ The proof is in Appendix A. For a number $1 \leq p \leq \infty$, let $p'$ denote the conjugate exponent. We recall the following properties of the operator $U(t)$, $$\begin{aligned} \|\partial_x\int_0^tU(t-t')f(x,t')\,dt'\|_{L_x^2} \lesssim \|f\|_{L_x^1L_T^2}, \label{kato_smooth_dual} \\ \|\int_0^tU(t-t')f(t')\,dt'\|_{L_x^{2}} \lesssim \|f\|_{L_T^{q'}L_x^{p'}}, \label{strichartz} \\ \text{where $p \ge 2$, and $q$ satisfy } \frac{1}{q}=\frac{1}{6}-\frac{1}{3p}. \notag\end{aligned}$$ The proof of can be found in [@B], or [@KPV1]. Inequality follows from the fact that $U(t)$ is an $L_x^2$ isometry, along with the dual of the homogenous Strichartz estimate for $U(t)$ (see [@g], page 1392). The existence theorem for solutions to is proved by a contraction mapping argument, which can also be found in [@B]. \[exist\] Let $\|\cdot\|_{Y_T}$ denote the norm such that $$\begin{aligned} \|f\|_{Y_T} & \equiv \|f\|_{L_x^4L_T^{\infty}}+\|D_x^{\frac{1}{4}}\partial_xf\|_{L_x^{\infty}L_T^{2}} \notag \\ & \quad +\|f\|_{L_T^{\infty}H^{\frac{1}{4}}}+\|\partial_xf\|_{L_x^{20}L_T^{\frac{5}{2}}}+\|D_x^{\frac{1}{4}}f\|_{L_x^{5}L_T^{10}}, \notag \\ Y_T & \equiv \{f \vert \textrm{ } \|f\|_{Y_T} < \infty\}, \notag\end{aligned}$$ and let $u_0 \in L^2$, and $\Phi$ be the map from $Y_{T}$ to $Y_{T}$ such that $$\Phi(u) \equiv U(t)u_0-\int_{0}^{t}U(t-t')\partial_x(u^3(t'))\,dt'. \notag$$ Then $$\|\Phi(u)\|_{Y_T} \lesssim \|u_0\|_{H^{\frac{1}{4}}}+T^{\frac{1}{2}}\|u\|_{Y_T}^3. \label{fixed}$$ This implies by contraction mapping that there exist $T=c\|D_x^{\frac{1}{4}}u\|_2^{-4}$ and a unique strong solution $u(t)$ of the IVP . The proof requires a Leibnitz rule type inequality for $L_{x}^pL_{T}^q$ norms, which we need as well. Let $\alpha \in (0,1)$, $\alpha_1,\alpha_2 \in [0,\alpha]$ with $\alpha=\alpha_1+\alpha_2$. Let $p,q,p_1,p_2,q_2\in (1,\infty)$, $q_1 \in (1,\infty]$ be such that $$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2} \textrm{ and } \frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}. \notag$$ Then $$\begin{aligned} \|D_x^{\alpha}(fg)-fD_x^{\alpha}g-gD_x^{\alpha}f\|_{L_x^pL_T^q} \lesssim \|D_x^{\alpha_1}f\|_{L_x^{p_1}L_T^{q_1}}\|D_x^{\alpha_2}f\|_{L_x^{p_2}L_T^{q_2}} \notag\end{aligned}$$ Moreover, for $\alpha_1=0$, the value $q_1=\infty$ is allowed. We will need an estimate on the Fourier transform $k(x)$ of $(1+\xi^2)^{-\frac{1}{8}}$. We expect $k$ to have good decay properties since it is the inverse Fourier transform of a smooth function. Since $$|\hat{\xi}|^{-\frac{1}{4}}=c_0|x|^{-\frac{3}{4}}, \label{basic_ft}$$ we expect that $k(x) \approx |x|^{-\frac{3}{4}}$ for small $x$. This is formalized in the following lemma. \[k\_est\_lem\] Let $k(x)$ denote the Fourier transform of the function $(1+\xi^2)^{-\frac{1}{8}}$. Then for any $n \in \mathbb N$, $$|k(x)| \lesssim \frac{1}{|x|^{\frac{3}{4}}(1+x^{2n})}. \label{k_est}$$ In particular, $$\int_{\mathbb R}|x|^{\frac{1}{8}}|k(x)| < \infty. \notag$$ For $x>1$, we can repeatedly integrate by parts as follows: $$\begin{aligned} \int_{\mathbb R}(1+\xi^2)^{-\frac{1}{8}}e^{-ix\xi}\,d\xi & = \int_{\mathbb R}(1+\xi^2)^{-\frac{1}{8}}\frac{1}{-ix}\frac{d}{\,d\xi}e^{-ix\xi}\,d\xi \notag \\ & = \frac{1}{ix}\int_{\mathbb R}\frac{1}{4}\xi(1+\xi^2)^{-\frac{9}{8}}e^{-ix\xi}\,d\xi \notag \\ & = \frac{1}{ix}\int_{\mathbb R}\frac{1}{4}\xi(1+\xi^2)^{-\frac{9}{8}}\frac{1}{-ix}\frac{d}{\,d\xi}e^{-ix\xi}\,d\xi \notag \\ & = \ldots \notag\end{aligned}$$ This argument gives us the decay in . When $x < 1$, we split up the integral over the region $S=[-|x|^{-1},|x|^{-1}]$. $$\begin{aligned} \int_{\mathbb R}(1+\xi^2)^{-\frac{1}{8}}e^{-ix\xi}\,d\xi & = \int_{S}(1+\xi^2)^{-\frac{1}{8}}e^{-ix\xi}\,d\xi \notag \\ & \quad +\int_{\mathbb R \setminus S}(1+\xi^2)^{-\frac{1}{8}}e^{-ix\xi}\,d\xi \notag \\ & = \mathcal{A}+\mathcal{B}. \notag\end{aligned}$$ Since $(1+\xi^{-2})^{-\frac{1}{8}}$ is bounded, $$\begin{aligned} |\mathcal{A}| & \lesssim \int_{S}(1+\xi^2)^{-\frac{1}{8}}\,d\xi \notag \\ & = \int_{S}|\xi|^{-\frac{1}{4}}(1+\xi^{-2})^{-\frac{1}{8}}\,d\xi \notag \\ & \lesssim \int_{S}|\xi|^{-\frac{1}{4}}\,d\xi \lesssim |x|^{-\frac{3}{4}}. \notag \notag\end{aligned}$$ By integration by parts, $$\begin{aligned} \mathcal{B} & = \int_{\mathbb R \setminus S}(1+\xi^2)^{-\frac{1}{8}}\frac{1}{-ix}\frac{d}{\,d\xi}e^{-ix\xi}\,d\xi \notag \\ & = (1+x^{-2})^{-\frac{1}{8}}\frac{e^{i}}{-ix}+(1+x^{-2})^{-\frac{1}{8}}\frac{e^{-i}}{ix} \notag \\ & \quad +\frac{1}{ix}\int_{\mathbb R \setminus S}\frac{1}{4}\xi(1+\xi^2)^{-\frac{9}{8}}e^{-ix\xi}\,d\xi. \notag\end{aligned}$$ Therefore, $$\begin{aligned} |\mathcal{B}| & \lesssim |x|^{-\frac{3}{4}} +\frac{1}{|x|}\int_{\mathbb R \setminus S}|\xi|(1+\xi^2)^{-\frac{9}{8}}\,d\xi \notag \\ & = |x|^{-\frac{3}{4}} +\frac{1}{|x|}\int_{\mathbb R \setminus S}|\xi|^{-\frac{5}{4}}(1+\xi^{-2})^{-\frac{9}{8}}\,d\xi \notag \\ & \lesssim |x|^{-\frac{3}{4}} +\frac{1}{|x|}\int_{\mathbb R \setminus S}|\xi|^{-\frac{5}{4}}\,d\xi \lesssim |x|^{-\frac{3}{4}}. \notag\end{aligned}$$ Combining our estimates for $|\mathcal{A}|$ and $|\mathcal{B}|$, the result follows. Before proving Theorem \[main\], we prove the corresponding decay result for solutions to the linear part of . This is necessary for the proof of Theorem \[main\], and it is also a simpler case that illustrates the main idea of our proof of Theorem \[main\]. We note that it is also possible to prove this result using an argument like Lemma 2 in [@NP], but this proof does not generalize to solutions of as discussed in the introduction. \[main\_lin\] For $u_0 \in C_0^{\infty}(\mathbb R)$, $$\||x|^{s}U(t)u_0(x)\|_2 \lesssim (1+|t|+|t|^{s})\|u_0\|_{H^{2s}}+ \||x|^{s}u_0\|_2. \notag$$ For concreteness, it will suffice to prove the result in the case $s=\frac{1}{8}$. By the definition of $U(t)$ and the triangle inequality, $$\begin{aligned} & \||x|^{\frac{1}{8}} U(t)u_0\|_{2} = \|D_{\xi}^{\frac{1}{8}}\left ( e^{it\xi^3}\hat{u}_0\right )\|_{L_{\xi}^2} \notag \\ & =\left \| D_{\xi}^{\frac{1}{8}}\left ( \frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}(1+\xi^2)^{\frac{1}{8}}\hat{u}_0\right ) \right \|_{2} \notag \\ & \lesssim \left \|D_{\xi}^{\frac{1}{8}}\left ( \frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}(1+\xi^2)^{\frac{1}{8}}\hat{u}_0\right ) -\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}D_{\xi}^{\frac{1}{8}}((1+\xi^2)^{\frac{1}{8}}\hat{u}_0) \right \|_{2} \notag \\ & \quad +\|\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}D_{\xi}^{\frac{1}{8}}((1+\xi^2)^{\frac{1}{8}}\hat{u}_0)\|_{2} \notag \\ & \equiv \mathcal{I} + \mathcal{II}. \notag\end{aligned}$$ We can write term $\mathcal{II}$ as $$\begin{aligned} \mathcal{II} & = \|(1+\xi^2)^{-\frac{1}{8}}D_{\xi}^{\frac{1}{8}}((1+\xi^2)^{\frac{1}{8}}\hat{u}_0)\|_{2} \notag \\ & = \|[(1+\xi^2)^{-\frac{1}{8}},D_{\xi}^{\frac{1}{8}}]((1+\xi^2)^{\frac{1}{8}}\hat{u}_0) +D_{\xi}^{\frac{1}{8}}\hat{u}_0\|_{2} \notag \\ & \leq \|[(1+\xi^2)^{-\frac{1}{8}},D_{\xi}^{\frac{1}{8}}]((1+\xi^2)^{\frac{1}{8}}\hat{u}_0)\|_2 +\|D_{\xi}^{\frac{1}{8}}\hat{u}_0\|_{2} \label{term_2}\end{aligned}$$ We need to bound the commutator term in $\mathcal{II}$. For any function $h$, we use the Plancherel theorem, the Young inequality, and Lemma \[k\_est\_lem\] to obtain $$\begin{aligned} \|[(1+\xi^2)^{-\frac{1}{8}},D_{\xi}^{\frac{1}{8}}]h\|_{L_{\xi}^2} & = \|\int_{\mathbb R}(|x|^{\frac{1}{8}}-|y|^{\frac{1}{8}})k(x-y)\hat{h}(y)\,dy\|_{L_{x}^2} \notag \\ & \lesssim \|\int_{\mathbb R}|x-y|^{\frac{1}{8}}|k(x-y)||\hat{h}(y)|\,dy\|_{L_{x}^2} \notag \\ & \lesssim c_{k}\|\hat{h}\|_{L_{x}^2} = c_{k}\|h\|_{L_{\xi}^2}. \notag\end{aligned}$$ We apply this to , $$|\mathcal{II}| \lesssim c_k\|u_0\|_{H^{\frac{1}{4}}}+\||x|^{\frac{1}{8}}u_0\|_2. \notag$$ For term $\mathcal{I}$, we use Lemma \[my\_prod\_rule\] with Lemma \[finite\]. $$\begin{aligned} |\mathcal{I}| & \lesssim \|Q_N\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}\|_{L_{x}^{\infty}l_N^1} \|u_0\|_2. \notag \\ & \lesssim (1+t)\|u_0\|_2. \notag\end{aligned}$$ Combining our estimates for $\mathcal{I}$ and $\mathcal{II}$, the result follows. For concreteness, we prove the result in the most interesting case when $k=2$, $s=s'=\frac{1}{8}$, and $t>0$. We use a contraction mapping argument to prove our decay estimate. The resolution space is $$\begin{aligned} \|f\|_{Z_T} & \equiv \||x|^{\frac{1}{8}}f\|_{L_T^{\infty}L_x^2} + \|f\|_{Y_T}. \notag \\ Z_T & \equiv \{f \vert \textrm{ } \|f\|_{Z_T} < \infty\}. \notag\end{aligned}$$ Let $f(t)\equiv \partial_x(u^3(t))$ for convenience, and consider $$\Phi(u)(x,t) = U(t)u(x,0)-\int_0^tU(t-t')f(t')dt'. \label{sol_map}$$ Multiply by $|x|^{\frac{1}{8}}$. The $|x|^{\frac{1}{8}}U(t)u(x,0)$ term is bounded by Lemma \[main\_lin\] along with a density argument. We concentrate on the nonlinear term: $$\begin{aligned} & \left \||x|^{\frac{1}{8}}\int_0^t U(t-t')f(t')\,dt' \right \|_{L_T^{\infty}L_x^2} = \left \|D_{\xi}^{\frac{1}{8}}\left ( \int_0^te^{i(t-t')\xi^3}f^{\wedge}(t')\,dt' \right ) \right \|_{L_T^{\infty}L_{\xi}^2} \notag \\ & =\left \|D_{\xi}^{\frac{1}{8}}\left (\int_0^t\frac{e^{i(t-t')\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}(1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t') \right )\,dt' \right \|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \lesssim \|D_{\xi}^{\frac{1}{8}}\left ( \int_0^t\frac{e^{i(t-t')\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}(1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t')\,dt'\right ) \notag \\ & \quad -\int_0^t\frac{e^{i(t-t')\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}D_{\xi}^{\frac{1}{8}}((1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t'))\,dt' \|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \quad +\|\int_0^t\frac{e^{i(t-t')\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}D_{\xi}^{\frac{1}{8}}((1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t'))\,dt'\|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \equiv I + II. \label{def-I}\end{aligned}$$ We bound term $II$ in a similar fashion to term $\mathcal{II}$ in Lemma \[main\_lin\]: $$\begin{aligned} II & \lesssim \|\int_0^t {e^{i(t-t')\xi^3}}[\frac{1}{(1+\xi^2)^{\frac{1}{8}}},D_{\xi}^{\frac{1}{8}}]((1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t'))\,dt'\|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \quad + \|\int_0^t {e^{i(t-t')\xi^3}}D_{\xi}^{\frac{1}{8}}(f^{\wedge}(t'))\,dt'\|_{L_T^{\infty}L^2_{\xi}} \notag \\ & \lesssim \|[\frac{1}{(1+\xi^2)^{\frac{1}{8}}},D_{\xi}^{\frac{1}{8}}]((1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t'))\|_{L_T^1L^2_{\xi}} \notag \\ & \quad + \|\int_0^t {e^{i(t-t')\xi^3}}D_{\xi}^{\frac{1}{8}}(f^{\wedge}(t'))\,dt'\|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \lesssim \|((1+\xi^2)^{\frac{1}{8}}f^{\wedge}(t'))\|_{L_T^1L^2_{\xi}} + \|\int_0^t {e^{i(t-t')\xi^3}}D_{\xi}^{\frac{1}{8}}(f^{\wedge}(t'))\,dt'\|_{{L_T^{\infty}L_{\xi}^2}} \notag \\ & \lesssim \|((1+D_x^2)^{\frac{1}{8}}f(t'))\|_{L_T^1L^2_{x}} + \|\int_0^{t}U(t-t')|x|^{\frac{1}{8}}f(t')\,dt'\|_{{L_T^{\infty}L_x^2}} \notag \\ & \equiv II.1+II.2. \notag\end{aligned}$$ Specializing to the case of the mKdV, $f(t')=\partial_{x}(u^3(t'))$, we bound $II.1$ using Theorem \[exist\]: $$\begin{aligned} II.1 & \lesssim\|\partial_x(u^3)\|_{L_T^1L_x^2} +\|D_x^{\frac{1}{4}}\partial_x(u^3)\|_{L_T^1L_x^2} \notag \\ & \lesssim T^{\frac{1}{2}}\|\partial_x(u^3)\|_{L_T^2L_x^2} +T^{\frac{1}{2}}\|D_x^{\frac{1}{4}}\partial_x(u^3)\|_{L_T^2L_x^2} \notag \\ &\lesssim T^{\frac{1}{2}}\|u\|_{L_x^{4}L_T^{\infty}}^2\|\partial_xu\|_{L_x^{\infty}L_T^{2}}+T^{\frac{1}{2}}\|u^2\|_{L_x^2L_T^{\infty}}\|D_x^{\frac{1}{4}}\partial_xu\|_{L_x^{\infty}L_T^2} \notag \\ & \quad +T^{\frac{1}{2}}\|D_x^{\frac{1}{4}}(u^2)\|_{L_x^{\frac{20}{9}}L_T^{10}}\|\partial_xu\|_{L_x^{20}L_T^{\frac{5}{2}}} \notag \\ & \lesssim T^{\frac{1}{2}}\| u\|_{L_x^{4}L_T^{\infty}}^2\|\partial_xu\|_{L_x^{\infty}L_T^{2}}+ T^{\frac{1}{2}}\|u\|_{L_x^4L_T^{\infty}}^2\|D_x^{\frac{1}{4}}\partial_xu\|_{L_x^{\infty}L_T^2} \notag \\ & \quad +T^{\frac{1}{2}}\|u\|_{L_x^{4}L_T^{\infty}}\|D_x^{\frac{1}{4}}u\|_{L_x^{5}L_T^{10}}\|\partial_xu\|_{L_x^{20}L_T^{\frac{5}{2}}} \notag \\ & \lesssim T^{\frac{1}{2}}\|u\|_{Z_T}^3. \notag\end{aligned}$$ Let $\phi(x) \in C_0^{\infty}(\mathbb R)$ have the property that $\phi(x)=1$ for $x \in (-1,1)$. We handle $II.2$ with the following argument: $$\begin{aligned} & \|\int_0^{t}U(t-t')|x|^{\frac{1}{8}}f(t')\,dt'\|_{L_T^{\infty}L_x^2} \lesssim \|\int_0^{t}U(t-t')|x|^{\frac{1}{8}}\phi(x)f(t')\,dt'\|_{L_T^{\infty}L_x^2} \notag \\ & \quad + \|\int_0^{t}U(t-t')|x|^{\frac{1}{8}}(1-\phi(x))f(t')\,dt'\|_{L_T^{\infty}L_x^2} \notag \\ & \lesssim \|\int_0^{t}U(t-t')[|x|^{\frac{1}{8}}(1-\phi(x)),\partial_x]u^3(t')\,dt'\|_{L_T^{\infty}L_x^2} \notag \\ & \quad +\|\int_0^{t}U(t-t')\partial_x((|x|^{\frac{1}{8}}(1-\phi(x)))u^3(t'))\,dt'\|_{L_T^{\infty}L_x^2}\notag \\ & \quad +\|\int_0^{t}U(t-t')|x|^{\frac{1}{8}}\phi(x)\partial_x(u^3(t'))\,dt'\|_{L_T^{\infty}L_x^2}\notag \\ & \equiv II.2.a+II.2.b+II.2.c. \notag\end{aligned}$$ For $II.2.a$, we use , and that for any function $h$, and $p \ge 1$, $$\|[|x|^{\frac{1}{8}}(1-\phi(x)),\partial_x]h\|_p \lesssim \|\frac{\partial}{\partial x}(|x|^{\frac{1}{8}}(1-\phi(x)))\|_{\infty}\|h\|_p, \notag$$ along with the Sobolev inequality to obtain the bound $$\begin{aligned} II.2.a & \lesssim \|u^3\|_{L_T^{\frac{12}{11}}L_x^{\frac{4}{3}}}=\|u\|_{L_T^{\frac{36}{11}}L_x^{4}}^3 \notag \\ & \lesssim T^{\frac{11}{12}}\|u\|_{L_{T}^{\infty}H^{\frac{1}{4}}}^3 \lesssim T^{\frac{11}{12}}\|u\|_{Z_T}^3. \notag\end{aligned}$$ We use to estimate $II.2.b$: $$\begin{aligned} II.2.b & \lesssim \||x|^{\frac{1}{8}}(1-\phi(x))u^3\|_{L_x^1L_T^2} \notag \\ & \lesssim \|u^2\|_{L_x^2L_T^{\infty}}\||x|^{\frac{1}{8}}(1-\phi(x))u\|_{L_x^2L_T^{2}} \notag \\ & \lesssim \|u\|_{L_x^4L_T^{\infty}}^2(\||x|^{\frac{1}{8}}u\|_{L_T^{2}L_x^2} +\|u\|_{L_T^{2}L_x^2}) \notag \\ & \lesssim T^{\frac{1}{2}}\|u\|_{L_x^4L_T^{\infty}}^2(\||x|^{\frac{1}{8}}u\|_{L_T^{\infty}L_x^2} +\|u\|_{L_T^{\infty}L_x^2}) \notag \\ & \lesssim T^{\frac{1}{2}}\|u\|_{Z_T}^3. \notag\end{aligned}$$ We use Theorem \[exist\] and the fact that $\phi$ has compact support to control $II.2.c$: $$\begin{aligned} II.2.c & \lesssim \||x|^{\frac{1}{8}}\phi(x)\partial_x(u^3)\|_{L_T^1L_x^2} \notag \\ & \lesssim \|\partial_x(u^3)\|_{L_T^1L_x^2} \notag \\ & \lesssim T^{\frac{1}{2}}\|u\|_{L_x^4L_T^{\infty}}^2\|\partial_xu\|_{L_x^{\infty}L_T^{2}} \notag \\ & \lesssim T^{\frac{1}{2}}\|u\|_{Z_T}^3. \notag\end{aligned}$$ Term $I$ from can be controlled using Theorem \[exist\], and the same argument as the bound for $II.1$: $$\begin{aligned} I & \lesssim \|Q_ND_{\xi}^{\frac{1}{8}}\frac{e^{it\xi^3}}{(1+\xi^2)^{\frac{1}{8}}}\|_{L_T^{2}L_{\xi}^{\infty}l_N^1} \|(1+D_x^2)^{\frac{1}{8}}(u^2\partial_x u)\|_{L_T^2L_{x}^2} \notag \\ & \lesssim (1+T^{\frac{3}{2}}) (\|u^2\partial_x u\|_{L_T^2L_{x}^2}+\|D_x^{\frac{1}{4}}(u^2\partial_x u)\|_{L_T^2L_{x}^2}) \notag \\ & \lesssim T^{\frac{1}{2}}(1+T) (\|u\|_{L_{x}^{4}L_{T}^{\infty}}^2\|\partial_xu\|_{L_{x}^{\infty}L_{T}^{2}} + \|u\|_{L_x^4L_T^{\infty}}^2\|D_x^{\frac{1}{4}}\partial_xu\|_{L_x^{\infty}L_T^2} \notag \\ & \quad + \|u\|_{L_x^{4}L_T^{\infty}}\|D_x^{\frac{1}{4}}u\|_{L_x^{5}L_T^{10}}\|\partial_xu\|_{L_x^{20}L_T^{\frac{5}{2}}}) \notag \\ & \lesssim T^{\frac{1}{2}}(1+T)\|u\|_{Z_T}^3 \lesssim T^{\frac{1}{2}}(1+T)\|u\|_{Z_T}^3. \notag\end{aligned}$$ Putting these estimates together, $$\begin{aligned} \||x|^{\frac{1}{8}}u\|_{L_T^{\infty}L_x^2} & \lesssim \||x|^{\frac{1}{8}}U(t)u_0\|_{L_T^{\infty}L_x^2}+I+II.1+II.2.a+II.2.b+II.2.c \notag \\ & \lesssim \||x|^{\frac{1}{8}}u_0\|_{L_x^2}+(1+T)\|u_0\|_{H^{\frac{1}{4}}} +T^{\frac{1}{2}}(1+T^{\frac{5}{12}}+T)\|u\|_{Z_T}^3. \label{contract}\end{aligned}$$ In order to get a contraction, we need to bound $\|u\|_{Y_T}$ in terms of $\|u\|_{Z_T}$. This follows from estimate in Theorem \[exist\]. By combining this with , we obtain a contraction by taking $T$ small enough, $$\|u\|_{Z_T} \lesssim \||x|^{\frac{1}{8}}u_0\|_{L_x^2}+(1+T)\|u_0\|_{H^{\frac{1}{4}}} +T^{\frac{1}{2}}(1+T^{\frac{5}{12}}+T)\|u\|_{Z_T}^3. \notag$$ In order to show that $\||x|^{\frac{1}{8}}u(t)\|_{L_x^2}$ is finite, for $t \in [0,T)$, apply $\||x|^{\frac{1}{8}}\cdot\|_2$ to instead of $\||x|^{\frac{1}{8}}\cdot\|_{L_T^{\infty}L_x^2}$, keeping in mind that $\||x|^{\frac{1}{8}}u\|_{L_T^{\infty}L_x^2}$ is finite. Appendix A ========== For our proof of Lemma \[prod\_rule\], we closely follow the proof of Theorem A.8 in [@KPV1]. This requires more notation. Let $\alpha_1=0$, $\alpha_2=\alpha \in [0,1]$. For a function $f$, let $$P_Nf \equiv \sum_{j \leq N-3}Q_jf. \notag$$ Define $p(x)$ to be the function so that $$(P_Nf)^{\wedge} = p(2^{-N}x)\hat{f}. \notag$$ Let $\tilde{p} \in C_{0}^{\infty}(\mathbb R)$, with $\tilde{p}(x) =1$ for $x \in [-100,100]$, and let $$(\tilde{P}_Nf)^{\wedge}(x)=\tilde{p}(2^{-N}x)\hat{f}. \notag$$ Let $\tilde{\eta}\in C_0^{\infty}(\mathbb R)$ with $\tilde{\eta}(x)=1$ for $x \in [\frac{1}{4},4]$, and supp$\tilde{\eta} \in [\frac{1}{8},8]$. Then define $(\tilde{Q}_kf)^{\wedge}(x)=\tilde{\eta}(2^{-k}x)\hat{f}$. Let $$\begin{aligned} \Psi^{i}(x)=|x|^{\alpha_j}p(x) & \textrm{, } \eta^{j}(x)=\frac{\eta(x)}{|x|^{\alpha_j}}, \notag \\ (\Psi_j^k)^{\wedge}(x)=\Psi_j(2^{-k})\hat{f}(x) & \textrm{, and } (Q_k^jf)^{\wedge}(x)=\eta^j(2^{-k}x)\hat{f}(x). \notag\end{aligned}$$ Similarly, with $\eta^3(x)=|x|^{\alpha}\tilde{p}(x)$, $\eta^4(x)=|x|^{\alpha_1}\eta(x)$, and $\eta^5(x)=|x|^{\alpha_2}\eta(x)$ we define $Q_k^3,Q_k^4,Q_k^5$. Let $$\begin{aligned} \eta^{\nu,j}(x) & =\exp(i\nu x)\eta^j(x), \notag \\ \eta^{\mu,j}(x) & =\exp(i\mu x)x|x|^{-\alpha_j}p(x), \notag\end{aligned}$$ with $j=1,2$ and $Q_k^{\nu,j}, Q_k^{\mu,j}$ the corresponding operators. The following is Proposition A.2 from [@KPV1]. \[prop\_a2\] $$\begin{aligned} D_x^{\alpha}(fg)-&fD_x^{\alpha}g-gD_x^{\alpha}f \notag \\ &= \sum_{|j|<2}2^{j\alpha_2}\sum_{k}Q_k^3(Q_k^1(D^{\alpha_1}f)Q_{k-j}^2(D^{\alpha_2}g)) \notag \\ & \quad + \sum_{k}\tilde{Q}_k(\Psi_k^1(D^{\alpha_2}g)Q_k^1(D^{\alpha_1}f)) \notag \\ & \quad + \sum_{k}\tilde{Q}_k(Q_k^2(D^{\alpha_2}g)\Psi_k^2(D^{\alpha_1}f)) \notag \\ & \quad + \sum_{|j| \leq 2}2^{j\alpha_2}\sum_{k}Q_k^1(D^{\alpha_1}f)Q_{k-j}^4(D^{\alpha_2}g) \notag \\ & \quad + \sum_{|j| \leq 2}2^{j\alpha_2}\sum_{k}Q_{k-j}^2(D^{\alpha_2}g)Q_{k}^5(D^{\alpha_1}f) \notag \\ & \quad +\int_{\mathbb R}\int_{\mathbb R} \left [ \sum_{k} \tilde{Q}_k(Q_k^{\nu,1}(D^{\alpha_1}f)Q_k^{\mu,2}(D^{\alpha_2}g)) \right ]r_1(\mu,\nu)\,d\nu\,d\mu \notag \\ & \quad +\int_{\mathbb R}\int_{\mathbb R} \left [ \sum_{k} \tilde{Q}_k(Q_k^{\nu,2}(D^{\alpha_2}g)Q_k^{\mu,1}(D^{\alpha_1}f)) \right ]r_2(\mu,\nu)\,d\nu\,d\mu, \notag\end{aligned}$$ where $r_1,r_2 \in \mathcal{S}(\mathbb R^2)$. From Lemma \[prop\_a2\], we need to bound four types of terms: 1. $\sum_{-\infty}^{\infty} Q_k(Q_k(f)Q_k(D_x^{\alpha}g))$ 2. $\sum_{-\infty}^{\infty} Q_k(\Psi_k(f)Q_k(D_x^{\alpha}g))$ 3. $\sum_{-\infty}^{\infty} Q_k(Q_k(f)\Psi_k(D_x^{\alpha}g))$ 4. $\sum_{-\infty}^{\infty} Q_k(f)Q_k(D_x^{\alpha}g)$ Let $\mathcal{M}h$ denote the Hardy Maximal operator applied to the function $h$. We control the first term using duality, $$\begin{aligned} & |\int_{\mathbb R} \sum_{-\infty}^{\infty} Q_k(Q_k(f)Q_k(D_x^{\alpha}g))h\,dx| = |\int_{\mathbb R} \sum_{-\infty}^{\infty} Q_k(f)Q_k(D_x^{\alpha}g)Q_k(h)\,dx| \notag \\ & \lesssim \int_{\mathbb R} \sqrt{\sum_{-\infty}^{\infty} |Q_k(f)|^2|Q_k(D_x^{\alpha}g)|^2}\sqrt{\sum_{-\infty}^{\infty}|Q_n(h)|^2}\,dx \notag \\ & \lesssim \|Q_k(f)Q_k(D_x^{\alpha}g)\|_{L_x^pl_k^2}\|Q_n(h)\|_{L_x^{p'}l_n^2} \notag \\ & \lesssim \|\mathcal{M}(f)\|_{L_x^p}\|Q_k(D_x^{\alpha}g)\|_{L_x^{\infty}l_k^2}\|Q_n(h)\|_{L_x^{p'}l_n^2} \notag \\ & \lesssim \|f\|_{p'}\|Q_k(D_x^{\alpha}g)\|_{L_x^{\infty}l_k^2}\|Q_n(h)\|_{L_x^{p'}l_n^2} \notag \\ & \lesssim \|f\|_p\|Q_k(D_x^{\alpha}g)\|_{L_x^{\infty}l_k^2}\|h\|_{L_x^{p'}}. \notag\end{aligned}$$ The second item is treated as the first, with $\Psi_k(f)$ replacing $Q_k(f)$. A similar argument is used on the third term, with $\Psi_k(D_x^{\alpha}g)$ replacing $\Psi_k(f)$, and the fact that $$\|\mathcal{M}(D_x^{\alpha}g)\|_{L_x^{\infty}} \lesssim \|g\|_{L_x^{\infty}}, \notag$$ because $\mathcal{M}$ is a bounded operator from $L^{\infty}$ to $L^{\infty}$. The Last term is treated with Cauchy-Schwartz, $$\begin{aligned} \|\sum_{-\infty}^{\infty}Q_k(f)Q_k(D_x^{\alpha}g)\|_{p} & \lesssim \|\|Q_n(f)\|_{l_n^2}\|Q_k(D_x^{\alpha}g)\|_{l_k^2}\|_p \notag \\ & \lesssim \|Q_n(f)\|_{L_x^{p}l_k^2}\|Q_k(D_x^{\alpha}g)\|_{L_x^{\infty}l_k^2}. \notag\end{aligned}$$ This proves the the first part of the lemma. The second part follows from $$\|D_x^{\alpha}(fg)-fD_x^{\alpha}g-gD_x^{\alpha}f\|_p \ge \|D_x^{\alpha}(fg)-gD_x^{\alpha}f\|_p-\|fD_x^{\alpha}g\|_p, \notag$$ the observation that $$|D^{\alpha}g| \leq \sum_{N}|Q_N(D^{\alpha}g)|, \notag$$ and for arbitrary functions $\varphi_N$, $$\|\varphi_N\|_{l_N^2} \leq \|\varphi_N\|_{l_N^{\infty}}^{\frac{1}{2}}\|\varphi_N\|_{l_N^1}^{\frac{1}{2}} \leq \|\varphi_N\|_{l_N^1}. \notag$$ **Acknowledgments:** I would like to thank Gustavo Ponce for many fruitful discussions, and Luiz Farah for reading earlier drafts. [10]{} I. Bihari. A generalization of a lemma of [B]{}ellman and its application to uniqueness problems of differential equations. , 7:81–94, 1956. Michael Christ, James Colliander, and Terrence Tao. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. , 125(6):1235–1293, 2003. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. 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--- abstract: 'Let $F$ be a non-archimedean local field with residue field $\bbF_q$ and let $\mathbf{G}=GL_{2/F}$. Let $\bfq$ be an indeterminate and let $\cH^{(1)}(\bfq)$ be the generic pro-$p$ Iwahori-Hecke algebra of the $p$-adic group $\mathbf{G}(F)$. Let $V_{\mathbf{\whG}}$ be the Vinberg monoid of the dual group $\mathbf{\whG}$. We establish a generic version for $\cH^{(1)}(\bfq)$ of the Kazhdan-Lusztig-Ginzburg antispherical representation, the Bernstein map and the Satake isomorphism. We define the flag variety for the monoid $V_{\mathbf{\whG}}$ and establish the characteristic map in its equivariant $K$-theory. These generic constructions recover the classical ones after the specialization $\bfq=q\in\bbC$. At $\bfq=q=0\in \overline{\bbF}_q$, the antispherical map provides a dual parametrization of all the irreducible $\cH^{(1)}_{\overline{\bbF}_q}(0)$-modules.' author: - Cédric PEPIN and Tobias SCHMIDT title: '****' --- Introduction ============ Let $F$ be a non-archimedean local field with ring of integers $o_F$ and residue field $\bbF_q$. Let $\bfG$ be a connected split reductive group over $F$. Let $\cH_k=(k[I\setminus\bfG(F)/I],\star) $ be the Iwahori-Hecke algebra, i.e. the convolution algebra associated to an Iwahori subgroup $I\subset \bfG(F)$, with coefficients in an algebraically closed field $k$. On the other hand, let $\widehat{\bfG}$ be the Langlands dual group of $\bfG$ over $k$, with maximal torus and Borel subgroup $\widehat{\bfT}\subset \widehat{\bfB}$ respectively. Let $W_0$ be the finite Weyl group. When $k=\bbC$, the irreducible $\cH_{\bbC}$-modules appear as subquotients of the Grothendieck group $K^{\widehat{\bfG}}( \widehat{\bfG}/ \widehat{\bfB})_{\bbC}$ of $\widehat{\bfG}$-equivariant coherent sheaves on the dual flag variety $\widehat{\bfG}/ \widehat{\bfB}$. As such they can be parametrized by the isomorphism classes of irreducible tame $\widehat{\bfG}(\bbC)$-representations of the Weil group $\cW_F$ of $F$, thereby realizing the tame local Langlands correspondence (in this setting also called the Deligne-Lusztig conjecture for Hecke modules): Kazhdan-Lusztig [@KL87], Ginzburg [@CG97]. Their approach to the Deligne-Lusztig conjecture is based on two steps: the first step develops the theory of the so-called [*antispherical representation*]{} leading to a certain dual parametrization of Hecke modules. The second step links these dual data to representations of the group $\cW_F$. The antispherical representation is a distinguished faithful action of the Hecke algebra $\cH_{\bbC}$ on its maximal commutative subring $\cA_{\bbC}\subset\cH_{\bbC}$ via $\cA_{\bbC}^{W_0}$-linear operators: elements of the subring $\cA_{\bbC}$ act by multiplication, whereas the standard Hecke operators $T_s\in\cH_{\bbC}$, supported on double cosets indexed by simple reflections $s\in W_0$, act via the classical Demazure operators [@D73; @D74]. The link with the geometry of the dual group comes then in two steps. First, the classical Bernstein map $\tilde{\theta}$ identifies the ring of functions $\bbC[\widehat{\bfT}]$ with $\cA_{\bbC}$, such that the invariants $\bbC[\widehat{\bfT}]^{W_0}$ become the center $Z(\cH_{\bbC})=\cA_{\bbC}^{W_0}$. Second, the characteristic homomorphism $c_{\mathbf{\whG}}$ of equivariant $K$-theory identifies the rings $\bbC[\widehat{\bfT}]$ and $K^{\widehat{\bfG}}( \widehat{\bfG}/ \widehat{\bfB})_{\bbC}$ as algebras over the representation ring $\bbC[\widehat{\bfT}]^{W_0}=R(\widehat{\bfG})_{\bbC}$. When $k=\overline{\bbF}_q$ any irreducible $\widehat{\bfG}(\overline{\bbF}_q)$-representation of $\cW_F$ is tame and the Iwahori-Hecke algebra needs to be replaced by the bigger pro-$p$-Iwahori-Hecke algebra $$\cH_{\overline{\bbF}_q}^{(1)}=(\overline{\bbF}_q[I^{(1)}\setminus \bfG(F)/I^{(1)}],\star).$$ Here, $I^{(1)}\subset I$ is the unique pro-$p$ Sylow subgroup of $I$. The algebra $\cH_{\overline{\bbF}_q}^{(1)}$ was introduced by Vignéras and its structure theory developed in a series of papers [@V04; @V05; @V06; @V14; @V15; @V16; @V17]. More generally, Vignéras introduces and studies a generic version $\cH^{(1)}(\bfq_{*})$ of this algebra which is defined over a polynomial ring $\bbZ[\bfq_{*}]$ in finitely many indeterminates $\bfq_s$. The mod $p$ ring $\cH_{\overline{\bbF}_q}^{(1)}$ is obtained by specialization $\bfq_s=q$ followed by extension of scalars from $\bbZ$ to $\overline{\bbF}_q$, in short $\bfq_s=q=0$. Let now $\bfG=\mathbf{GL_n}$ be the general linear group, so that $\bfq_s$ is independent of $s$. Our aim is to show that there is a Kazhdan-Lusztig theory for the generic pro-$p$ Iwahori-Hecke algebra $\cH^{(1)}(\bfq)$. On the one hand, it gives back (and actually improves!) the classical theory after passing to the direct summand $\cH(\bfq)\subset \cH^{(1)}(\bfq)$ and then specializing $\bfq=q\in\bbC$. On the other hand, it gives a genuine mod $p$ theory after specializing to $\bfq=q=0\in \overline{\bbF}_q$. In the generic situation, the role of the Langlands dual group is taken by its Vinberg monoid $V_{\widehat{\bfG}}$ and its flag variety. The monoid comes with a fibration $\bfq : V_{\widehat{\bfG}}\rightarrow\bbA^1$ and the dual parametrisation of $\cH_{\overline{\bbF}_q}^{(1)}$-modules is achieved by working over the $0$-fiber $V_{\widehat{\bfG},0}$. In this article, we only explain the case of the group $\bfG=\mathbf{GL_2}$, and we are currently generalizing this material to the general linear group $\mathbf{GL_n}$. Moreover, we will treat the link with two-dimensional mod $p$ representations of the Weil group $\cW_F$ and the mod $p$ local Langlands program for $\mathbf{GL_2}$ in a forthcoming sequel to this paper. From now on, let $k=\overline{\bbF}_q$ and $\bfG=\mathbf{GL_2}$ and let $\bfq$ be an indeterminate. We let $\bfT\subset\bfG$ be the torus of diagonal matrices. Although our primary motivation is the extreme case $\bfq=q=0$, we will prove all our results in the far more stronger generic situation. It also allows us to find the correct normalizations in the extreme case and to recover and improve the classical theory over $\bbC$ (typically, the formulas become cleaner, e.g. in the Bernstein and in the Satake isomorphism). Let $\cA^{(1)}(\bfq) \subset \cH^{(1)}(\bfq)$ be the maximal commutative subring and $\cA^{(1)}(\bfq)^{W_0} = Z(\cH^{(1)}(\bfq))$ be its ring of invariants. We let $\tilde{\bbZ}:=\bbZ[\frac{1}{q-1},\mu_{q-1}]$ and denote by $\tilde{\bullet}$ the base change from $\bbZ$ to $\tilde{\bbZ}$. The algebra $\tilde{\cH}^{(1)}(\bfq)$ splits as a direct product of subalgebras $\tilde{\cH}^{\gamma}(\bfq)$ indexed by the orbits $\gamma$ of $W_0$ in the set of characters of the finite torus $\bbT:=\bfT(\bbF_q)$. There are regular resp. non-regular components corresponding to $|\gamma|=2$ resp. $|\gamma|=1$ and the algebra structure of $\tilde{\cH}^{\gamma}(\bfq)$ in these two cases is fundamentally different. We define an analogue of the Demazure operator for the regular components and call it the [*Vignéras operator*]{}. Passing to the product over all $\gamma$, this allows us to single out a distinguished $Z(\tilde{\cH}^{(1)}(\bfq))$-linear operator on $\tilde{\cA}^{(1)}(\bfq)$. Our first main result is the existence of the [*generic pro-$p$ antispherical representation*]{}: [**Theorem A.**]{} (cf. \[sA2q\], \[sA1q\]) *There is a (essentially unique) faithful representation* $$\xymatrix{ \tilde{\sA}^{(1)}(\bfq):\tilde{\cH}^{(1)}(\bfq) \ar[r] & \operatorname{End}_{Z(\tilde{\cH}^{(1)}(\bfq))}(\tilde{\cA}^{(1)}(\bfq)) }$$ such that - $$\tilde{\sA}^{(1)}(\bfq)|_{\tilde{\cA}^{(1)}(\bfq)}=\textrm{ the natural inclusion $\tilde{\cA}^{(1)}(\bfq)\subset\operatorname{End}_{Z(\tilde{\cH}^{(1)}(\bfq))}(\tilde{\cA}^{(1)}(\bfq))$}$$ - $$\tilde{\sA}^{(1)}(\bfq)(T_s)=\textrm{ the Demazure-Vign\'eras operator on $\tilde{\cA}^{(1)}(\bfq)$}.\;\;\;\;\;\;\;\;\;\;$$ Restricting the representation $\tilde{\sA}^{(1)}(\bfq)$ to the Iwahori component, its base change $\bbZ[\bfq]\rightarrow \bbZ[\bfq^{\pm \frac{1}{2}} ]$ coincides with the classical antispherical representation of Kazhdan-Lusztig and Ginzburg. We call the left $\tilde{\cH}^{(1)}(\bfq)$-module defined by $\tilde{\sA}^{(1)}(\bfq)$ the *generic antispherical module* $\tilde{\cM}^{(1)}$. Let $\operatorname{Mat}_{2\times 2}$ be the $\bbZ$-monoid scheme of $2\times 2$-matrices. The Vinberg monoid $V_{\widehat{\bfG}}$, as introduced in [@V95], in the particular case of $\mathbf{GL_2}$ is the $\bbZ$-monoid scheme $$V_{\mathbf{GL_2}}:=\operatorname{Mat}_{2\times 2}\times\bbG_m.$$ It implies the striking interpretation of the formal indeterminate $\bfq$ as a regular function. Indeed, denote by $z_2$ the canonical coordinate on $\bbG_m$. Let $\bfq$ be the homomorphism from $V_{\mathbf{GL_2}}$ to the multiplicative monoid $(\bbA^1,\cdot )$ defined by $(f,z_2)\mapsto \det(f)z_2^{-1}$: $$\xymatrix{ V_{\mathbf{GL_2}} \ar[d]_{\bfq} \\ \bbA^1. }$$ The fibration $\bfq$ is trivial over $\bbA^{1}\setminus \{ 0 \}$ with fibre $\mathbf{GL_2}$. The special fiber at $\bfq=0$ is the $\bbZ$-semigroup scheme $$V_{\mathbf{GL_2},0} := \bfq^{-1}(0)= \operatorname{Sing}_{2\times 2}\times\bbG_m,$$ where $\operatorname{Sing}_{2\times 2}$ represents the singular $2\times 2$-matrices. Let $\operatorname{Diag}_{2\times 2}\subset \operatorname{Mat}_{2\times 2}$ be the submonoid scheme of diagonal $2\times 2$-matrices, and set $$V_{\mathbf{\whT}}:=\operatorname{Diag}_{2\times 2}\times\bbG_m\subset V_{\mathbf{GL_2}}= \operatorname{Mat}_{2\times 2}\times\bbG_m.$$ This is a diagonalizable $\bbZ$-monoid scheme. Restricting the above $\bbA^1$-fibration to $V_{\mathbf{\whT}}$ we obtain a fibration, trivial over $\bbA^{1}\setminus \{ 0 \}$ with fibre $\mathbf{\whT}$. Its special fibre at $\bfq=0$ is the $\bbZ$-semigroup scheme $$V_{\mathbf{\whT},0} := \bfq|_{V_{\mathbf{\whT}}} ^{-1}(0)= \operatorname{SingDiag}_{2\times2}\times\bbG_m,$$ where $\operatorname{SingDiag}_{2\times2}$ represents the singular diagonal $2\times 2$-matrices. To ease notion, we denote the base change to $\overline{\bbF}_q$ of these $\bbZ$-schemes by the same symbols. Let $\bbT^{\vee}$ be the finite abelian dual group of $\bbT$. As $\bbT^{\vee}$ has order prime to $p$, it defines a constant finite diagonalizable $\overline{\bbF}_q$-group scheme. We let $R(V^{(1)}_{\mathbf{\whT}})$ be the representation ring of the extended monoid $$V^{(1)}_{\mathbf{\whT}}:=\bbT^{\vee}\times V_{\mathbf{\whT}}.$$ Our second main result is the existence of the [*generic pro-$p$ Bernstein isomorphism*]{}. [**Theorem B.**]{} (cf. \[genB1\]) *There exists a ring isomomorphism $$\xymatrix{ \sB^{(1)}(\bfq):\cA^{(1)}(\bfq)\ar[r]^>>>>>{\sim} & R(V^{(1)}_{\mathbf{\whT}}) }$$ with the property: Restricting the isomorphism $\sB^{(1)}(\bfq)$ to the Iwahori component, its base change $\bbZ[\bfq]\rightarrow \bbZ[\bfq^{\pm \frac{1}{2}} ]$ recovers[^1] the classical Bernstein isomorphism $\tilde{\theta}$.* The extended monoid $V^{(1)}_{\mathbf{\whT}}$ has a natural $W_0$-action and the isomorphism $\sB^{(1)}(\bfq)$ is equivariant. We call the resulting ring isomorphism $$\xymatrix{ \sS^{(1)}(\bfq):=\sB^{(1)}(\bfq)^{W_0}: \cA^{(1)}(\bfq)^{W_0}\ar[r]^<<<<<{\sim} & R(V^{(1)}_{\mathbf{\whT}})^{W_0} }$$ the *generic pro-$p$-Iwahori Satake isomorphism*. Our terminology is justified by the following. Let $K=\bfG(o_F)$. Recall that the spherical Hecke algebra of $\mathbf{G}(F)$ with coefficients in any commutative ring $R$ is defined to be the convolution algebra $$\cH_{R}^{\operatorname{sph}}:=(R[K\backslash \mathbf{G}(F)/K],\star)$$ generated by the $K$-double cosets in $\mathbf{G}(F)$. We define a [*generic spherical Hecke algebra*]{} $\cH^{\operatorname{sph}}(\bfq)$ over the ring $\bbZ[\bfq]$. Its base change $\bbZ[\bfq]\rightarrow R$, $\bfq\mapsto q$ coincides with $\cH_{R}^{\operatorname{sph}}$. Our third main result is the existence of the [*generic Satake isomorphism*]{}. (cf. \[ThgenSat\]) [*There exists a ring isomorphism $$\xymatrix{ \sS(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r]^<<<<<{\sim} & R(V_{\mathbf{\whT}})^{W_0} }$$ with the propery: Base change $\bbZ[\bfq]\rightarrow \bbZ[\bfq^{\pm \frac{1}{2}} ]$ and specialization $\bfq\mapsto q\in\bbC$ recoversthe classical Satake isomorphism between $\cH^{\operatorname{sph}}_{\bbC}$ and $R(\mathbf{\whT})_{\bbC}^{W_0}$.* ]{} We emphasize that the possibility of having a generic Satake isomorphism is conceptually new and of independent interest. Its definition relies on the deep Kazhdan-Lusztig theory for the intersection cohomology on the affine flag manifold. Its proof follows from the classical case by specialization (to an infinite number of points $q$). The special fibre $\sS(0)$ recovers Herzig’s mod $p$ Satake isomorphism [@H11], by choosing certain ‘Steinberg coordinates’ on $V_{\mathbf{\whT},0}$. As a corollary we obtain the *generic central elements morphism* as the unique ring homomorphism $$\xymatrix{ \sZ(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r] & \cA(\bfq)\subset\cH(\bfq) }$$ making the diagram $$\xymatrix{ \cA(\bfq) \ar[rr]_{\sim}^{\sB^{(1)}(\bfq)|_{\cA(\bfq)}} && R(V_{\mathbf{\whT}}) \\ \cH^{\operatorname{sph}}(\bfq) \ar[u]^{\sZ(\bfq)} \ar[rr]_{\sim}^{\sS(\bfq)} && R(V_{\mathbf{\whT}})^{W_0} \ar@{^{(}->}[u] }$$ commutative. The morphism $\sZ(\bfq)$ is injective and has image $Z(\cH(\bfq))$. Base change $\bbZ[\bfq]\rightarrow \bbZ[\bfq^{\pm \frac{1}{2}} ]$ and specialization $\bfq\mapsto q\in\bbC$ recoversBernstein’s classical central elements morphism. Its specialization $\bfq\mapsto q=0\in\overline{\bbF}_q$ coincides with Ollivier’s construction from [@O14]. Our fourth main result is the [*characteristic homomorphism*]{} in the equivariant $K$-theory over the Vinberg monoid $V_{\mathbf{\whG}}$. The monoid $V_{\mathbf{\whG}}$ carries an action by multiplication on the right from the $\bbZ$-submonoid scheme $$V_{\mathbf{\whB}}:=\operatorname{UpTriang}_{2\times 2}\times\bbG_m\subset \operatorname{Mat}_{2\times 2}\times\bbG_m=V_{\mathbf{\whG}}$$ where $\operatorname{UpTriang}_{2\times 2}$ represents the upper triangular $2\times 2$-matrices. We explain in an appendix how to construct (virtual) quotients in the context of semigroups and how to construct categories of equivariant vector bundles and their $K$-theory on such quotients. Although maybe well-known, we could not find this material in the literature. The usual induction functor for vector bundles gives a characteristic homomorphism, which is an isomorphism in the case of monoids. Applying this general formalism, the *flag variety* $V_{\mathbf{\whG}}/V_{\mathbf{\whB}}$ resp. its extended version $V^{(1)}_{\mathbf{\whG}}/V^{(1)}_{\mathbf{\whB}}$ is defined as a $\bbZ$-monoidoid (instead of a groupoid). (cf. \[cVGL2\]) *Induction of equivariant vector bundles defines a characteristic isomorphism* $$\xymatrix{ c_{V^{(1)}_{\mathbf{\whG}}}: R(V^{(1)}_{\mathbf{\whT}}) \ar[r]^<<<<<{\sim} & K^{V^{(1)}_{\mathbf{\whG}}}(V^{(1)}_{\mathbf{\whG}}/V^{(1)}_{\mathbf{\whB}}). }$$ The ring isomorphism is $R(V^{(1)}_{\mathbf{\whT}})^{W_0}=R(V^{(1)}_\mathbf{\whG})$-linear and compatible with passage to $\bfq$-fibres. Over the open complement $\bfq\neq 0$, its Iwahori-component coincides with the classical characteristic homomorphism $c_{\mathbf{\whG}}$ between $R(\mathbf{\whT})$ and $K^{\mathbf{\whG}}(\mathbf{\whG}/\mathbf{\whB})$. We define the [*category of Bernstein resp. Satake parameters*]{} $\operatorname{BP}_{\mathbf{\whG}}$ resp. $\operatorname{SP}_{\mathbf{\whG}}$ to be the category of quasi-coherent modules on the $\tilde{\bbZ}$-scheme $V^{(1)}_{\mathbf{\whT}}$ resp. $V^{(1)}_{\mathbf{\whT}}/W_0$. By Theorem B, restriction of scalars to the subring $\tilde{\cA}^{(1)}(\bfq)$ or $Z(\tilde{\cH}^{(1)}(\bfq))$ defines a functor $B$ resp. $P$ from the category of $\tilde{\cH}^{(1)}(\bfq)$-modules to the categories $\operatorname{BP}_{\mathbf{\whG}}$ resp. $\operatorname{SP}_{\mathbf{\whG}}$. For example, the Bernstein resp. Satake parameter of the antispherical module $\tilde{\cM}^{(1)}$ equals the structure sheaf $\cO_{V^{(1)}_{\mathbf{\whT}}}$ resp. the quasi-coherent sheaf corresponding to the $R(V^{(1)}_{\mathbf{\whT}})^{W_0}$-module $K^{V^{(1)}_{\mathbf{\whG}}}(V^{(1)}_{\mathbf{\whG}}/V^{(1)}_{\mathbf{\whB}})$. We call $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq))\ar[d]^P & \\ \operatorname{SP}_{\mathbf{\whG}} }$$ the *generic parametrization functor*. In the other direction, we define the *generic antispherical functor* $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)) & \\ \operatorname{SP}_{\mathbf{\whG}} \ar[u]_{\operatorname{ASph}} }$$ to be the functor $\operatorname{ASph}:= (\tilde{\cM}^{(1)}\otimes_{Z(\tilde{\cH}^{(1)}(\bfq))}\bullet)\circ S^{-1}$ where $S$ is the Satake equivalence between $Z(\tilde{\cH}^{(1)}(\bfq))$-modules and $\operatorname{SP}_{\mathbf{\whG}}$. The relation between these functors is expressed by the commutative diagram: $$\xymatrix{ &\operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq))\ar[d]^{B} \ar[dr]^{P} & \\ \operatorname{SP}_{\mathbf{\whG}} \ar[ur]^{\operatorname{ASph}} \ar[r]_{\pi^*} & \operatorname{BP}_{\mathbf{\whG}}\ar[r]_{\pi_*}& \operatorname{SP}_{\mathbf{\whG}}, }$$ where $\pi: V^{(1)}_{\mathbf{\whT}}\rightarrow V^{(1)}_{\mathbf{\whT}}/W_0$ is the canonical projection. In the final section, we pass to the special fibre, i.e. we perform the base change $\bbZ[\bfq]\rightarrow k=\overline{\bbF}_q$, $\bfq\mapsto q=0$. Identifying the $k$-points of the $k$-scheme $V^{(1)}_{\mathbf{\whT}}/W_0$ with the skyscraper sheaves on it, the antispherical functor $\operatorname{ASph}$ induces a map $$\xymatrix{ \operatorname{ASph}:\big(V^{(1)}_{\mathbf{\whT}}/W_0 \big)(k)\ar[r] & \{\textrm{left $\cH_{\overline{\bbF}_q}^{(1)}$-modules}\}. }$$ Considering the decomposition of $V^{(1)}_{\mathbf{\whT}}/W_0$ into its connected components indexed by $\gamma\in \bbT^{\vee}/W_0$, the antispherical map decomposes as a disjoint union of maps $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big( V^{(1)}_{\mathbf{\whT}}/W_0\big)_{\gamma}(k) \ar[r] & \{\textrm{left $\cH_{\overline{\bbF}_q}^{\gamma}$-modules}\}.& }$$ We come to our last main result, the mod $p$ dual parametrization of [*all*]{} irreducible $\cH_{\overline{\bbF}_q}^{(1)}$-modules via the antispherical map. [**Theorem E.**]{} (cf. \[ASphreg\], \[ASphnonreg\]) ** - Let $\gamma\in \bbT^{\vee}/W_0$ regular. The antispherical map induces a bijection $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big( V^{(1)}_{\mathbf{\whT}}/W_0\big)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{simple finite dimensional left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}/\sim. }$$ The singular locus of the parametrizing $k$-scheme $$\big(V^{(1)}_{\mathbf{\whT},0}/W_0\big)_{\gamma}\simeq V_{\mathbf{\whT},0}=\operatorname{SingDiag}_{2\times2}\times\bbG_m$$ is given by $(0,0)\times\bbG_m\subset V_{\mathbf{\whT},0}$ in the standard coordinates, and its $k$-points correspond to the supersingular Hecke modules through the correspondence $\operatorname{ASph}^{\gamma}$. - Let $\gamma\in \bbT^{\vee}/W_0$ be non-regular. Consider the decomposition $$\big(V^{(1)}_{\mathbf{\whT},0}/W_0\big)_{\gamma}=V_{\mathbf{\whT},0}/W_0\simeq \bbA^1\times\bbG_m= D(2)_{\gamma}\cup D(1)_{\gamma}$$ where $D(1)_{\gamma}$ is the closed subscheme defined by the parabola $z_2=z_1^2$ in the Steinberg coordinates $z_1,z_2$ and $D(2)_{\gamma}$ is the open complement. The antispherical map induces bijections $$\xymatrix{ \operatorname{ASph}^{\gamma}(2):D(2)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{simple $2$-dimensional left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}/\sim }\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $$\xymatrix{ \operatorname{ASph}^{\gamma}(1):D(1)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{antispherical pairs of characters of $\cH^{\gamma}_{\overline{\bbF}_q}$}\}/\sim. }$$ The branch locus of the covering $$V_{\mathbf{\whT},0}\lra V_{\mathbf{\whT},0}/W_0\simeq \big(V^{(1)}_{\mathbf{\whT},0}/W_0\big)_{\gamma}$$ is contained in $D(2)_{\gamma}$, with equation $z_1=0$ in Steinberg coordinates, and its $k$-points correspond to the supersingular Hecke modules through the correspondence $\operatorname{ASph}^{\gamma}(2)$. We refer to the main body of the text for the unexplained notions. Also, let us mention that our methods of proofs are elementary. Once the Vinberg monoid is introduced, the generic Satake isomorphism is formulated and the generic antispherical module is constructed, everything else follows from Vignéras’ structure theory of the generic pro-$p$-Iwahori Hecke algebra and her classification of the irreducible representations. We finish by relating the present article to our prior preprint [@PS1]. In [@PS1] we tried to work with Hecke actions on mod $p$ reductions of rings of the type $\bbZ[\mathbf{\whT}]$ in order to define the antispherical module. From the point of view of the Vinberg monoid, this is unnatural since it mixes the $0$-fiber of the generic Hecke algebras with the $1$-fiber $\mathbf{\whT}=V_{\mathbf{\whT},1}$. This implies, as we believe now, misleading definitions (for example the use of intersection theory on regular components) and, most importantly, prevents the theory to pass to a generic level. In the present article, we take up this theory again, in a completely general way. [*Notation:*]{} In general, the letter $F$ denotes a locally compact complete non-archimedean field with ring of integers $o_F$. Let $\bbF_q$ be its residue field, of characteristic $p$ and cardinality $q$. We denote by $\bfG$ the algebraic group $\mathbf{GL_2}$ over $F$ and by $G:=\bfG(F)$ its group of $F$-rational points. Let $\bfT\subset\bfG$ be the torus of diagonal matrices. Finally, $I\subset G$ denotes the upper triangular standard Iwahori subgroup and $I^{(1)}\subset I$ denotes the unique pro-$p$ Sylow subgroup of $I$. The pro-$p$-Iwahori-Hecke algebra ================================= The generic pro-$p$-Iwahori Hecke algebra ----------------------------------------- \[extendedWeylgroup\] We let $$W_0=\{1,s\}\hskip10pt \text{resp.}\hskip10pt \Lambda=\bbZ\times\bbZ$$ be the *finite Weyl group* of $\bfG$ resp. the *lattice of cocharacters* of $\bfT$. If $$\bbT=k^{\times}\times k^{\times}$$ denotes the *finite torus* $\bfT(\bbF_q)$, then $W_0$ acts naturally on $\bbT\times\Lambda$. The *extended Weyl group* of $\bfG$ is $$W^{(1)}=\bbT\times\Lambda\rtimes W_0.$$ It contains the *affine Weyl group* and the *Iwahori-Weyl group* $$W_{\operatorname{aff}}=\bbZ(1,-1)\rtimes W_0\subseteq W=\Lambda\rtimes W_0.$$ The affine Weyl group $W_{\operatorname{aff}}$ is a Coxeter group with set of simple reflexions $S_{\operatorname{aff}}=\{s_0,s\}$, where $s_0=(1,-1)s$. Moreover, setting $u=(1,0)s\in W$ and $\Omega=u^{\bbZ}$, we have $W=W_{\operatorname{aff}}\rtimes \Omega.$ The length function $\ell$ on $W_{\operatorname{aff}}$ can then be inflated to $W$ and $W^{(1)}$. \[defgenericprop\] Let $\bfq$ be an indeterminate. The *generic pro-$p$ Iwahori Hecke algebra* is the $\bbZ[\bfq]$-algebra $\cH^{(1)}(\bfq)$ defined by generators $$\cH^{(1)}(\bfq):=\bigoplus_{w\in W^{(1)}} \bbZ[\bfq] T_w$$ and relations: - braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W^{(1)}$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$T_s^2=\bfq + c_sT_s\quad \textrm{if $s\in S_{\operatorname{aff}}$},$$ where $c_s:=\sum_{t\in (1,-1)(k^{\times})}T_t$. \[distinguishedelements\] The identity element is $1=T_1$. Moreover we set $$S:=T_s,\quad U:=T_u\quad\textrm{and}\quad S_0:=T_{s_0}=USU^{-1}.$$ \[defprop\] Let $R$ be any commutative ring. The *pro-$p$ Iwahori Hecke algebra of $G$ with coefficients in $R$* is defined to be the convolution algebra $$\cH_{R}^{(1)}:=(R[I^{(1)}\backslash G/I^{(1)}],\star)$$ generated by the $I^{(1)}$-double cosets in $G$. ***(Vignéras, [@V16 Thm. 2.2])*** \[presprop\] Let $\bbZ[\bfq]\ra R$ be the ring homomorphism mapping $\bfq$ to $q$. Then the $R$-linear map $$\xymatrix{ \cH^{(1)}(\bfq)\otimes_{\bbZ[\bfq]}R \ar[r] & \cH_{R}^{(1)} }$$ sending $T_w$, $w\in W^{(1)}$, to the characteristic function of the double coset $I^{(1)}\backslash w/I^{(1)}$, is an isomorphism of $R$-algebras. Idempotents and component algebras ---------------------------------- \[finiteT\] Recall the finite torus $\bbT=\bfT(\bbF_q)$. Let us consider its group algebra $\tilde{\bbZ}[\bbT]$ over the ring $$\tilde{\bbZ}:=\bbZ[\frac{1}{q-1},\mu_{q-1}].$$ As $q-1$ is invertible in $\tilde{\bbZ}$, so is $|\bbT|=(q-1)^2$. We denote by $\bbT^{\vee}$ the set of characters $$\lambda: \bbT\rightarrow \mu_{q-1}\subset\widetilde{\bbZ}$$ of $\bbT$, with its natural $W_0$-action given by $^s\lambda(t_1,t_2)=\lambda(t_2,t_1)$ for $(t_1,t_2)\in\bbT$. The set of $W_0$-orbits in $\bbT^{\vee}/W_0$ has cardinality $\frac{q^2-q}{2}$. Also $W^{(1)}$ acts on $\bbT^{\vee}$ through the canonical quotient map $W^{(1)}\rightarrow W_0$. Because of the braid relations in $\cH^{(1)}(\bfq)$, the rule $t\mapsto T_t$ induces an embedding of $\tilde{\bbZ}$-*algebras* $$\tilde{\bbZ}[\bbT]\subset \cH_{\tilde{\bbZ}}^{(1)}(\bfq):=\cH^{(1)}(\bfq)\otimes_{\bbZ}\tilde{\bbZ}.$$ For all $\lambda\in \bbT^{\vee}$ and for $\gamma\in \bbT^{\vee}/W_0$, we define $$\varepsilon_{\lambda}:=|\bbT|^{-1}\sum_{t\in\bbT}\lambda^{-1}(t)T_t\hskip10pt \text{and} \hskip10pt \varepsilon_{\gamma}:=\sum_{\lambda\in\gamma}\varepsilon_{\lambda}.$$ \[decompprop\] The elements $\varepsilon_{\lambda}$, $\lambda\in \bbT^{\vee}$, are idempotent, pairwise orthogonal and their sum is equal to $1$. The elements $\varepsilon_{\gamma}$, $\gamma\in \bbT^{\vee}/W_0$, are idempotent, pairwise orthogonal, their sum is equal to $1$ and they are central in $\cH^{(1)}_{\widetilde{\bbZ}}(\bfq)$. The $\tilde{\bbZ}[\bfq]$-algebra $\cH^{(1)}_{\tilde{\bbZ}}(\bfq)$ is the direct product of its sub-$\tilde{\bbZ}[\bfq]$-algebras $\cH^{\gamma}_{\tilde{\bbZ}}(\bfq):=\cH^{(1)}_{\tilde{\bbZ}}(\bfq)\varepsilon_{\gamma}$: $$\cH^{(1)}_{\tilde{\bbZ}}(\bfq)=\prod_{\gamma\in\bbT^{\vee}/W_0}\cH^{\gamma}_{\tilde{\bbZ}}(\bfq).$$ In particular, the category of $\cH_{\tilde{\bbZ}}^{(1)}(\bfq)$-modules decomposes into a finite product of the module categories for the individual component rings $\cH^{(1)}_{\tilde{\bbZ}}(\bfq)\varepsilon_{\gamma}$. The elements $\varepsilon_{\gamma}$ are central because of the relations $T_sT_t=T_{s(t)}T_s$, $T_{s_0}T_t=T_{s_0(t)}T_{s_0}$ and $T_uT_t=T_{s(t)}T_u$ for all $t\in(1,-1)k^{\times}$. Following the terminology of [@V04], we call $|\gamma|=2$ a [*regular*]{} case and $|\gamma|=1$ a [*non-regular*]{} (or [*Iwahori*]{}) case. The Bernstein presentation {#The Bernstein presentation} -------------------------- The inverse image in $W^{(1)}$ of any subset of $W$ along the canonical projection $W^{(1)}\ra W$ will be denoted with a superscript ${}^{(1)}$. ***(Vignéras [@V16 Th. 2.10, Cor 5.47])*** \[Bernsteinpresprop\] The $\bbZ[\bfq]$-algebra $\cH^{(1)}(\bfq)$ admits the following *Bernstein presentation*: $$\cH^{(1)}(\bfq)=\bigoplus_{w\in W^{(1)}}\bbZ[\bfq] E(w)$$ satisfying - braid relations $$E(w)E(w')=E(ww')\quad\textrm{for $w,w'\in W_0^{(1)}$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$E(s)^2=\bfq E(s^2)+c_sE(s)\quad \textrm{if $s\in S_0^{(1)}$},$$ where $c_{ts}:=T_{s(t)}c_s$ for all $t\in\bbT$ and $s\in S_0$ - product formula $$E(\lambda)E(w)=\bfq^{\frac{\ell(\lambda)+\ell(w)-\ell(\lambda w)}{2}}E(\lambda w)\quad \textrm{for $\lambda\in\Lambda^{(1)}$ and $w\in W^{(1)}$}$$ - Bernstein relations for $s\in s_{\beta}^{(1)}\subset S_0^{(1)}$ and $\lambda\in\Lambda^{(1)}$ : set $V:=\bbR\Phi^{\vee}$ and let $$\nu:\Lambda^{(1)}\ra V$$ be the homomorphism such that $\lambda\in\Lambda^{(1)}$ acts on $V$ by translation by $\nu(\lambda)$ ; then the Bernstein element $$B(\lambda,s):=E(s \lambda s^{-1})E(s)-E(s)E(\lambda)$$ $$\begin{aligned} =& 0 & \textrm{if $\lambda\in(\Lambda^s)^{(1)}$} \\ = &\operatorname{sign}(\beta\circ\nu(\lambda))\sum_{k=0}^{|\beta\circ\nu(\lambda)|-1}\mathbf{q}(k,\lambda)c(k,\lambda)E(\mu(k,\lambda)) & \textrm{if $\lambda\in\Lambda^{(1)}\setminus(\Lambda^s)^{(1)}$}\end{aligned}$$ where $\bfq(k,\lambda)c(k,\lambda)\in\bbZ[\bfq][\bbT]$ and $\mu(k,\lambda)\in\Lambda^{(1)}$ are explicit, cf. [@V16 Th. 5.46] and references therein. \[AIH\] Let $$\cA(\bfq):=\bigoplus_{\lambda\in\Lambda}\bbZ[\bfq]E(\lambda)\subset \cA^{(1)}(\bfq):=\bigoplus_{\lambda\in\Lambda^{(1)}}\bbZ[\bfq]E(\lambda)\subset\cH^{(1)}(\bfq).$$ It follows from the product formula that these are *commutative sub-$\bbZ[\bfq]$-algebras of $\cH^{(1)}(\bfq)$*. Moreover, by definition [@V16 5.22-5.25], we have $E(t)=T_t$ for all $t\in\bbT$, so that $\bbZ[\bbT]\subset \cA^{(1)}(\bfq)$. Then, again by the product formula, the commutative algebra $\cA^{(1)}(\bfq)$ decomposes as the tensor product of the subalgebras $$\cA^{(1)}(\bfq) =\bbZ[\bbT]\otimes_{\bbZ} \cA(\bfq).$$ Also, after base extension $\bbZ\ra\tilde{\bbZ}$, we can set $\cA_{\tilde{\bbZ}}^{\gamma}(\bfq):=\cA_{\tilde{\bbZ}}^{(1)}(\bfq)\varepsilon_{\gamma}$, and obtain the decomposition $$\cA_{\tilde{\bbZ}}^{(1)}(\bfq)=\prod_{\gamma\in\bbT^{\vee}/W_0}\cA^{\gamma}_{\tilde{\bbZ}}(\bfq)\subset\prod_{\gamma\in\bbT^{\vee}/W_0}\cH^{\gamma}_{\tilde{\bbZ}}(\bfq)=\cH^{(1)}_{\tilde{\bbZ}}(\bfq).$$ \[presAq\] Let $X$,Y$,z_2$ be indeterminates. There exists a unique ring homomorphism $$\xymatrix{ \bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2)\ar[r] & \cA(\bfq) }$$ such that $$X\lmapsto E(1,0),\quad Y\lmapsto E(0,1)\quad\textrm{and}\quad z_2\lmapsto E(1,1).$$ It is an isomorphism. Moreover, for all $\gamma\in\bbT^{\vee}/W_0$, $$\cA_{\tilde{\bbZ}}^{\gamma}(\bfq)= \left\{ \begin{array}{ll} (\tilde{\bbZ}\varepsilon_{\lambda}\times\tilde{\bbZ}\varepsilon_{\mu})\otimes_{\bbZ}\cA(\bfq) & \textrm{ if $\gamma=\{\lambda,\mu\}$ is regular} \\ \tilde{\bbZ}\varepsilon_{\lambda}\otimes_{\bbZ}\cA(\bfq)& \textrm{ if $\gamma=\{\lambda\}$ is non-regular}. \end{array} \right.$$ For any $(n_1,n_2)\in\bbZ^2=\Lambda$, we have $\ell(n_1,n_2)=|n_1-n_2|$. Hence it follows from product formula that $z_2$ is invertible and $XY=\bfq z_2$, so that we get a $\bbZ[\bfq]$-algebra homomorphism $$\xymatrix{ \bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2)\ar[r] & \cA(\bfq). }$$ Moreover it maps the $\bbZ[\bfq][z_2^{\pm1}]$-basis $\{X^n\}_{n>1}\coprod\{1\}\coprod \{Y^n\}_{n>1}$ to the $\bbZ[\bfq][z_2^{\pm1}]$-basis $$\{E(n,0)\}_{n>1}\coprod\{1\}\coprod \{E(0,n)\}_{n>1},$$ and hence is an isomorphism. The rest of the lemma is clear since $\cA_{\tilde{\bbZ}}^{(1)}(\bfq)=\tilde{\bbZ}[\bbT]\otimes_{\bbZ}\cA(\bfq)$ and $\tilde{\bbZ}[\bbT]=\prod_{\lambda\in\bbT^{\vee}}\tilde{\bbZ}\varepsilon_{\lambda}$. In the following, we will sometimes view the isomorphism of the lemma as an identification and write $X=E(1,0), Y=E(0,1)$ and $z_2=E(1,1)$. \[centerHI1\] The rule $E(\lambda)\mapsto E(w(\lambda))$ defines an action of the finite Weyl group $W_0=\{1,s\}$ on $\cA^{(1)}(\bfq)$ by $\bbZ[\bfq]$-algebra homomorphisms. By [@V05 Th. 4] (see also [@V14 Th. 1.3]), the subring of $W_0$-invariants is equal to the center of $\cH^{(1)}(\bfq)$, and the same is true after the scalar extension $\bbZ\ra\tilde{\bbZ}$. Now the action on $\cA_{\tilde{\bbZ}}^{(1)}(\bfq)$ stabilizes each component $\cA^{\gamma}_{\tilde{\bbZ}}(\bfq)$ and then the resulting subring of $W_0$-invariants is the center of $\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)$. In terms of the description of $\cA^{\gamma}_{\tilde{\bbZ}}(\bfq)$ given in Lemma \[presAq\], this translates into : \[centergamma\] Let $\gamma\in\bbT^{\vee}/W_0$. - If $\gamma=\{\lambda,\mu\}$ is regular, then the map $$\begin{aligned} \cA_{\tilde{\bbZ}}(\bfq) & \lra & \cA_{\tilde{\bbZ}}^{\gamma}(\bfq)^{W_0}=Z(\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)) \\ a & \lmapsto & a\varepsilon_{\lambda}+s(a)\varepsilon_{\mu}\end{aligned}$$ is an isomorphism of $\tilde{\bbZ}[\bfq]$-algebras. It depends on the choice of order $(\lambda,\mu)$ on the set $\gamma$. - If $\gamma=\{\lambda\}$ is non-regular, then $$Z(\cH_{\tilde{\bbZ}}^{\gamma}(\bfq))=\cA_{\tilde{\bbZ}}^{\gamma}(\bfq)^{W_0}=\tilde{\bbZ}[\bfq][z_2^{\pm1},z_1]\varepsilon_{\lambda}$$ with $z_1:=X+Y$. \[BernsteinVSIM\] One can express $X,Y,z_2\in\cA^{(1)}(\bfq)\subset\cH^{(1)}(\bfq)$ in terms of the distinguished elements \[distinguishedelements\]. This is an application of [@V16 Ex. 5.30]. We find: $$(1,0)=s_0u=us\in\Lambda\Rightarrow X:=E(1,0)=(S_0-c_{s_0})U=U(S-c_s),$$ $$(0,1)=su\in\Lambda \Rightarrow Y:=E(0,1)=SU,$$ $$(1,1)=u^2 \in\Lambda\Rightarrow z_2:=E(1,1)=U^2.$$ Also $$z_1:=X+Y=U(S-c_s)+SU.$$ The generic regular antispherical representation {#regcomponents} ================================================ The generic regular Iwahori-Hecke algebras {#genericgammalagebrareg} ------------------------------------------ Let $\gamma=\{\lambda,\mu\}\in\bbT^{\vee}/W_0$ be a regular orbit. We define a model $\cH_2(\bfq)$ over $\bbZ$ for the component algebra $\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)\subset \cH_{\tilde{\bbZ}}^{(1)}(\bfq)$. The algebra $\cH_2(\bfq)$ itself will not depend on $\gamma$. By construction, the $\tilde{\bbZ}[\bfq]$-algebra $\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)$ admits the following presentation: $$\cH_{\tilde{\bbZ}}^{\gamma}(\bfq) =(\tilde{\bbZ}\varepsilon_{\lambda}\times\tilde{\bbZ}\varepsilon_{\mu})\otimes_{\bbZ}'\bigoplus_{w\in W} \bbZ[\bfq] T_w,$$ where $\otimes_{\bbZ}'$ is the tensor product *twisted* by the $W$-action on $\{\lambda,\mu\}$ through the quotient map $W\ra W_0$, together with the - braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$T_s^2=\bfq \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$ Let $\bfq$ be an indeterminate. The *generic second Iwahori-Hecke algebra* is the $\bbZ[\bfq]$-algebra $\cH_2(\bfq)$ defined by generators $$\cH_2(\bfq):=(\bbZ\varepsilon_1\times\bbZ\varepsilon_2)\otimes_{\bbZ}'\bigoplus_{w\in W} \bbZ[\bfq] T_w,$$ where $\otimes'$ is the tensor product twisted by the $W$-action on $\{1,2\}$ through the quotient map $W\ra W_0=\mathfrak{S}_2$, and relations: - braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$T_s^2=\bfq \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$ \[presH2q\] The identity element of $\cH_2(\bfq)$ is $1=T_1$. Moreover we set in $\cH_2(\bfq)$ $$S:=T_s,\quad U:=T_u\quad\textrm{and}\quad S_0:=T_{s_0}=USU^{-1}.$$ Then one checks that $$\cH_2(\bfq)=(\bbZ\varepsilon_1\times\bbZ\varepsilon_2)\otimes_{\bbZ}'\bbZ[\bfq][S,U^{\pm 1}],\quad S^2=\bfq,\quad U^2S=SU^2$$ is a presentation of $\cH_2(\bfq)$. Note that the element $U^2$ is invertible in $\cH_2(\bfq)$. \[H2VSHgamma\] Choosing the ordering $(\lambda,\mu)$ on the set $\gamma=\{\lambda,\mu\}$ and mapping $\varepsilon_1\mapsto\varepsilon_{\lambda}, \varepsilon_2\mapsto \varepsilon_{\mu}$ defines an isomorphism of $\tilde{\bbZ}[\bfq]$-algebras $$\xymatrix{ \cH_2(\bfq)\otimes_{\bbZ}\tilde{\bbZ}\ar[r]^<<<<<{\sim} & \cH_{\tilde{\bbZ}}^{\gamma}(\bfq), }$$ such that $S\otimes 1\mapsto S\varepsilon_{\gamma}$, $U\otimes 1\mapsto U\varepsilon_{\gamma}$ and $S_0\otimes 1\mapsto S_0\varepsilon_{\gamma}$. \[presA2q\] We identify two important commutative subrings of $\cH_2(\bfq)$. We define $\cA_2(\bfq)\subset\cH_2(\bfq)$ to be the $\bbZ[\bfq]$-subalgebra generated by the elements $\varepsilon_1$, $\varepsilon_2$, $US$, $SU$ and $U^{\pm 2}$. Let $X,Y$ and $z_2$ be indeterminates. Then there is a unique $(\bbZ\varepsilon_1\times\bbZ\varepsilon_2)\otimes_{\bbZ}\bbZ[\bfq]$-algebra homomorphism $$(\bbZ\varepsilon_1\times\bbZ\varepsilon_2)\otimes_{\bbZ}\bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2) \lra \cA_2(\bfq)$$ such that $X\mapsto US, Y\mapsto SU, z_2\mapsto U^2$, and it is an isomorphism. In particular, $\cA_2(\bfq)$ is a [*commutative*]{} subalgebra of $\cH_2(\bfq)$. The isomorphism \[H2VSHgamma\] identifies $\cA_2(\bfq)\otimes_{\bbZ}\tilde{\bbZ}$ with $\cA_{\tilde{\bbZ}}^{\gamma}(\bfq)$. Moreover, permuting $\varepsilon_1$ and $\varepsilon_2$, and $X$ and $Y$, extends to an action of $W_0=\mathfrak{S}_2$ on $\cA_2(\bfq)$ by homomorphisms of $\bbZ[\bfq]$-algebras, whose invariants is the center $Z(\cH_2(\bfq))$ of $\cH_2(\bfq)$, and the map $$\begin{aligned} \bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2) & \lra & \cA_2(\bfq)^{W_0}=Z(\cH_2(\bfq)) \\ a & \lmapsto & a\varepsilon_1+s(a)\varepsilon_2\end{aligned}$$ is an isomorphism of $\bbZ[\bfq]$-algebras. This is a consequence of \[H2VSHgamma\], \[BernsteinVSIM\], \[presAq\] and \[centergamma\]. In the following, we will sometimes view the above isomorphisms as identifications. In particular, we will write $X=US, Y=SU$ and $z_2=U^2$. The Vignéras operator --------------------- In this subsection and the following, we will investigate the structure of the $Z(\cH_2(\bfq))$-algebra $\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$ of $Z(\cH_2(\bfq))$-linear endomorphisms of $\cA_2(\bfq)$. Recall from the preceding subsection that $Z(\cH_2(\bfq))= \cA_2(\bfq)^s$ is the subring of invariants of the commutative ring $\cA_2(\bfq)$. We have $$\cA_2(\bfq)=\cA_2(\bfq)^s\varepsilon_1\oplus\cA_2(\bfq)^s\varepsilon_2$$ as $\cA_2(\bfq)^{s}$-modules. This is immediate from the two isomorphisms in \[presA2q\]. According to the lemma, we may use the $\cA_2(\bfq)^{s}$-basis $\varepsilon_1,\varepsilon_2$ to identify $\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$ with the algebra of $2\times2$-matrices over $\cA_2(\bfq)^{s}=\bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2)$. The endomorphism of $\cA_2(\bfq)$ corresponding to the matrix $$V_s(\bfq):=\left (\begin{array}{cc} 0 & Y\varepsilon_1+X\varepsilon_2 \\ z_2^{-1}(X\varepsilon_1+Y\varepsilon_2) & 0 \end{array} \right)$$ will be called *the Vignéras operator on $\cA_2(\bfq)$*. \[relationVsq\] We have $$V_s(\bfq)^2=\bfq.$$ This is a short calculation. The generic regular antispherical representation {#HnilqGL2} ------------------------------------------------ In the following theorem we define the generic regular antispherical representation of the algebra $\cH_2(\bfq)$ on the $Z(\cH_2(\bfq))$-module $\cA_2(\bfq)$. Note that the commutative ring $\cA_2(\bfq)$ is naturally a subring $$\cA_2(\bfq)\subset \operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq)),$$ an element $a\in \cA_2(\bfq)$ acting by multiplication $b\mapsto ab$ on $ \cA_2(\bfq)$. \[sA2q\] There exists a unique $\bbZ[\bfq]$-algebra homomorphism $$\xymatrix{ \sA_2(\bfq):\cH_2(\bfq) \ar[r] & \operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq)) }$$ such that - $$\sA_2(\bfq)|_{\cA_2(\bfq)}=\textrm{ the natural inclusion $\cA_2(\bfq)\subset\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$}$$ - $$\sA_2(\bfq)(S)=V_s(\bfq).$$ Recall that $\cH_2(\bfq)=(\bbZ\varepsilon_1\times\bbZ\varepsilon_2)\otimes_{\bbZ}'\bbZ[\bfq][S,U^{\pm1}]$ with the relations $S^2=\bfq$ and $U^2S=SU^2$. In particular $\sA_2(\bfq)(S):=V_s(\bfq)$ is well-defined thanks to \[relationVsq\]. Now let us consider the question of finding the restriction of $\sA_2(\bfq)$ to the subalgebra $\bbZ[\bfq][S,U^{\pm1}]$. As the $\bbZ[\bfq]$-algebra $\cA_2(\bfq)\cap\bbZ[\bfq][S,U^{\pm1}]$ is generated by $$z_2=U^2,\quad X=US\quad\textrm{and}\quad Y=SU,$$ such a $\bbZ[\bfq]$-algebra homomorphism exists if and only if there exists $$\sA_2(\bfq)(U)\in\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$$ satisfying 1. $\sA_2(\bfq)(U)$ is invertible ; 2. $\sA_2(\bfq)(U)^2=\sA_2(\bfq)(U^2)=\sA_2(\bfq)(z_2)=z_2\operatorname{Id}$ ; 3. $\sA_2(\bfq)(U)V_s(\bfq)=\textrm{ multiplication by $X$}$ 4. $V_s(\bfq)\sA_2(\bfq)(U)=\textrm{ multiplication by $Y$}$. As before we use the $Z(\cH_2(\bfq))$-basis $\varepsilon_1,\varepsilon_2$ of $\cA_2(\bfq)$ to identify $\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$ with the algebra of $2\times 2$-matrices over the ring $Z(\cH_2(\bfq))=\cA_2(\bfq)^s$. Then, by definition, $$V_s(\bfq)= \left (\begin{array}{cc} 0 & Y\varepsilon_1+X\varepsilon_2 \\ z_2^{-1}(X\varepsilon_1+Y\varepsilon_2) & 0 \end{array} \right).$$ Moreover, the multiplications by $X$ and by $Y$ on $\cA_2(\bfq)$ correspond then to the matrices $$\left (\begin{array}{cc} X\varepsilon_1+Y\varepsilon_2 & 0 \\ 0 & Y\varepsilon_1+X\varepsilon_2 \end{array} \right) \quad\textrm{and}\quad \left (\begin{array}{cc} Y\varepsilon_1+X\varepsilon_2 & 0 \\ 0 & X\varepsilon_1+Y\varepsilon_2 \end{array} \right).$$ Now, writing $$\sA_2(\bfq)(U)= \left (\begin{array}{cc} a& c \\ b& d \end{array} \right)$$ we have: $$\sA_2(\bfq)(U)^2=z_2\operatorname{Id}\Longleftrightarrow \left (\begin{array}{cc} a^2+bc& c(a+d) \\ b(a+d)& d^2+bc \end{array} \right) = \left (\begin{array}{cc} z_2& 0 \\ 0& z_2 \end{array} \right),$$ $$\sA_2(\bfq)(U)V_s(\bfq)=\textrm{ multiplication by $X$}$$ $$\Longleftrightarrow \left (\begin{array}{cc} cz_2^{-1}(X\varepsilon_1+Y\varepsilon_2)& a(Y\varepsilon_1+X\varepsilon_2) \\ dz_2^{-1}(X\varepsilon_1+Y\varepsilon_2)& b(Y\varepsilon_1+X\varepsilon_2) \end{array} \right) = \left (\begin{array}{cc} X\varepsilon_1+Y\varepsilon_2 & 0 \\ 0 & Y\varepsilon_1+X\varepsilon_2 \end{array} \right)$$ and $$V_s(\bfq)\sA_2(\bfq)(U)=\textrm{ multiplication by $Y$}$$ $$\Longleftrightarrow \left (\begin{array}{cc} b(Y\varepsilon_1+X\varepsilon_2)& d(Y\varepsilon_1+X\varepsilon_2) \\ az_2^{-1}(X\varepsilon_1+Y\varepsilon_2)& cz_2^{-1}(X\varepsilon_1+Y\varepsilon_2) \end{array} \right) = \left (\begin{array}{cc} Y\varepsilon_1+X\varepsilon_2 & 0 \\ 0 & X\varepsilon_1+Y\varepsilon_2 \end{array} \right).$$ Each of the two last systems admits a unique solution, namely $$\sA_2(\bfq)(U) = \left (\begin{array}{cc} a& c \\ b& d \end{array} \right) = \left (\begin{array}{cc} 0& z_2 \\ 1& 0 \end{array} \right),$$ which is also a solution of the first one. Moreover, the determinant $$ad-bc=-z_2$$ is invertible. Finally, $\cA_2(\bfq)$ is generated by $\cA_2(\bfq)\cap\bbZ[\bfq][S,U^{\pm1}]$ together with $\varepsilon_1$ and $\varepsilon_2$. The latter are assigned to map to the projectors $$\textrm{multiplication by $\varepsilon_1$}=\left (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \quad\textrm{and}\quad \textrm{multiplication by $\varepsilon_2$}=\left (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right).$$ Thus it only remains to check that $$\left (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \sA_2(\bfq)(S) = \sA_2(\bfq)(S) \left (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)$$ and $$\left (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) \sA_2(\bfq)(S) = \sA_2(\bfq)(S) \left (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right),$$ and similarly with $\sA_2(\bfq)(U)$ in place of $\sA_2(\bfq)(U)$, which is straightforward. The map $\sA_2(\bfq)$, together with the fact that it is an isomorphism (see below), is a rewriting of a theorem of Vignéras, namely [@V04 Cor. 2.3]. In loc. cit., the algebra $\cH_2(\bfq)$ is identified with the algebra of $2\times 2$-matrices over the ring $\bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2)$. In our approach, we have replaced the *abstract rank $2$ module* underlying the standard representation of this matrix algebra, by the *subring $\cA_2(\bfq)$ of $\cH_2(\bfq)$* with $\{\varepsilon_1,\varepsilon_2\}$ for the canonical basis. In this way, we are able to formulate the property that the restriction of $\sA_2(\bfq)$ to the subring $\cA_2(\bfq)\subset\cH_2(\bfq)$ is the action by multiplication. This observation will be crucial to find the analogue of the representation $\sA_2(\bfq)$ in the *non*-regular case. \[sA2sf\] The homomorphism $\sA_2(\bfq)$ is an isomorphism. It follows from \[presH2q\] and \[presA2q\] that the $\bbZ[\bfq]$-algebra $\cH_2(\bfq)$ is generated by the elements $$\varepsilon_1,\ \varepsilon_2,\ S,\ U,\ SU$$ as a module over its center $Z(\cH_2(\bfq))$. Moreover, as $SU^2=U^2S=:z_2S$ and $SU=:Y$, we have $$S=z_2^{-1}YU=z_2^{-1}Y(\varepsilon_1U+\varepsilon_2U)=z_2^{-1}(Y\varepsilon_1+X\varepsilon_2)\varepsilon_1U+z_2^{-1}(X\varepsilon_1+Y\varepsilon_2)\varepsilon_2U,$$ $$U=\varepsilon_1U+\varepsilon_2U\quad\textrm{and}\quad SU=(Y\varepsilon_1+X\varepsilon_2)\varepsilon_1+(X\varepsilon_1+Y\varepsilon_2)\varepsilon_2.$$ Consequently $\cH_2(\bfq)$ is generated as a $Z(\cH_2(\bfq))$-module by the elements $$\varepsilon_1,\ \varepsilon_2,\ z_2^{-1}\varepsilon_1U,\ \varepsilon_2U.$$ Since $$\sA_2(\bfq)(U) := \left (\begin{array}{cc} 0& z_2 \\ 1& 0 \end{array} \right),$$ these four elements are mapped by $\sA_2(\bfq)$ to $$\left (\begin{array}{cc} 1& 0 \\ 0& 0 \end{array} \right),\ \left (\begin{array}{cc} 0& 0 \\ 0& 1 \end{array} \right),\ \left (\begin{array}{cc} 0& 1 \\ 0& 0 \end{array} \right),\ \left (\begin{array}{cc} 0& 0 \\ 1& 0 \end{array} \right).$$ As $\sA_2(\bfq)$ indentifies $Z(\cH_2(\bfq))\subset\cH_2(\bfq)$ with the center of the matrix algebra $$\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))=\operatorname{End}_{Z(\cH_2(\bfq))}(Z(\cH_2(\bfq))\varepsilon_1\oplus Z(\cH_2(\bfq))\varepsilon_2),$$ it follows that the elements $\varepsilon_1$, $\varepsilon_2$, $z_2^{-1}\varepsilon_1U$, $\varepsilon_2U$ are linearly independent over $Z(\cH_2(\bfq))$ and that $\sA_2(\bfq)$ is an isomorphism. We record the following corollary of the proof. \[dimOverCenter\] The ring $\cH_2(\bfq)$ is a free $Z(\cH_2(\bfq))$-module on the basis $ \varepsilon_1,\varepsilon_2,z_2^{-1}\varepsilon_1U,\varepsilon_2U. $ We end this section by noting an equivariance property of $\sA_2(\bfq)$. As already noticed, the finite Weyl group $W_0$ acts on $\cA_2(\bfq)$ by $\bbZ[\bfq]$-algebra automorphisms, and the action is clearly faithful. Moreover $\cA_2(\bfq)^{W_0}=Z(\cH_2(q))$. Hence $W_0$ can be viewed as a subgroup of $\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$, and we can let it act on $\operatorname{End}_{Z(\cH_2(\bfq))}(\cA_2(\bfq))$ by conjugation. The embedding $\sA_2(\bfq)|_{\cA_2(\bfq)}$ is $W_0$-equivariant. Indeed, for all $a,b\in\cA_2(\bfq)$ and $w\in W_0$, we have $$\sA_2(\bfq)(w(a))(b)=w(a)b=w(aw^{-1}(b))=(waw^{-1})(b)=(w\sA_2(\bfq)(a)w^{-1})(b).$$ The generic non-regular antispherical representation {#IwahoriKtheory} ==================================================== The generic non-regular Iwahori-Hecke algebras {#genericgammalagebranonreg} ---------------------------------------------- Let $\gamma=\{\lambda\}\in\bbT^{\vee}/W_0$ be a non-regular orbit. As in the regular case, we define a model $\cH_1(\bfq)$ over $\bbZ$ for the component algebra $\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)\subset \cH_{\tilde{\bbZ}}^{(1)}(\bfq)$. The algebra $\cH_1(\bfq)$ itself will not depend on $\gamma$. By construction, the $\tilde{\bbZ}[\bfq]$-algebra $\cH_{\tilde{\bbZ}}^{\gamma}(\bfq)$ admits the following presentation: $$\cH_{\tilde{\bbZ}}^{\gamma}(\bfq) =\bigoplus_{w\in W} \bbZ[\bfq] T_w\varepsilon_{\lambda},$$ with the - braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$T_s^2=\bfq +(q-1)T_s\quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$ \[defgeneric1\] Let $\bfq$ be an indeterminate. The *generic Iwahori-Hecke algebra* is the $\bbZ[\bfq]$-algebra $\cH_1(\bfq)$ defined by generators $$\cH_1(\bfq):=\bigoplus_{w\in W} \bbZ[\bfq] T_w$$ and relations: - braid relations $$T_wT_{w'}=T_{ww'}\quad\textrm{for $w,w'\in W$ if $\ell(w)+\ell(w')=\ell(ww')$}$$ - quadratic relations $$T_s^2=\bfq +(\bfq-1)T_s \quad \textrm{if $s\in S_{\operatorname{aff}}$}.$$ \[presH1q\] The identity element of $\cH_1(\bfq)$ is $1=T_1$. Moreover we set in $\cH_1(\bfq)$ $$S:=T_s,\quad U:=T_u\quad\textrm{and}\quad S_0:=T_{s_0}=USU^{-1}.$$ Then one checks that $$\cH_1(\bfq)=\bbZ[\bfq][S,U^{\pm 1}],\quad S^2=\bfq+(\bfq-1)S,\quad U^2S=SU^2$$ is a presentation of $\cH_1(\bfq)$. Note that the element $U^2$ is invertible in $\cH_1(\bfq)$. \[H1VSHgamma\] Sending $1$ to $\varepsilon_{\gamma}$ defines an isomorphism of $\tilde{\bbZ}[\bfq]$-algebras $$\xymatrix{ \cH_1(\bfq)\otimes_{\bbZ}\tilde{\bbZ}\ar[r]^<<<<<{\sim} & \cH_{\tilde{\bbZ}}^{\gamma}(\bfq), }$$ such that $S\otimes 1\mapsto S\varepsilon_{\gamma}$, $U\otimes 1\mapsto U\varepsilon_{\gamma}$ and $S_0\otimes 1\mapsto S_0\varepsilon_{\gamma}$. \[presA1q\] We define $\cA_1(\bfq)\subset\cH_1(\bfq)$ to be the $\bbZ[\bfq]$-subalgebra generated by the elements $(S_0-(\bfq-1))U$, $SU$ and $ U^{\pm 2}$. Let $X,Y$ and $z_2$ be indeterminates. Then there is a unique $\bbZ[\bfq]$-algebra homomorphism $$\bbZ[\bfq][z_2^{\pm1}][X,Y]/(XY-\bfq z_2) \lra \cA_1(\bfq)$$ such that $X\mapsto (S_0-(\bfq-1))U$, $Y\mapsto SU$, $z_2\mapsto U^2$, and it is an isomorphism. In particular, $\cA_1(\bfq)$ is a [*commutative*]{} subalgebra of $\cH_1(\bfq)$. The isomorphism \[H1VSHgamma\] identifies $\cA_1(\bfq)\otimes_{\bbZ}\tilde{\bbZ}$ with $\cA_{\tilde{\bbZ}}^{\gamma}(\bfq)$. Moreover, permuting $X$ and $Y$ extends to an action of $W_0=\mathfrak{S}_2$ on $\cA_1(\bfq)$ by homomorphisms of $\bbZ[\bfq]$-algebras, whose invariants is the center $Z(\cH_1(\bfq))$ of $\cH_1(\bfq)$ and $$\bbZ[\bfq][z_2^{\pm1}][z_1]\xrightarrow{\sim}\cA_1(\bfq)^{W_0}=Z(\cH_1(\bfq))$$ with $z_1:=X+Y$. This is a consequence of \[H1VSHgamma\], \[BernsteinVSIM\], \[presAq\] and \[centergamma\]. In the following, we will sometimes view the above isomorphisms as identifications. In particular, we will write $$X=(S_0-(\bfq-1))U=U(S-(\bfq-1)),\quad Y=SU\quad\textrm{and}\quad z_2=U^2\quad\textrm{in}\quad\cH_1(\bfq).$$ It is well-known that the generic Iwahori-Hecke algebra $\cH_1(\bfq)$ is a $\bfq$-deformation of the group ring $\bbZ[W]$ of the Iwahori-Weyl group $W=\Lambda\rtimes W_0$. More precisely, specializing the chain of inclusions $\cA_1(\bfq)^{W_0}\subset \cA_1(\bfq) \subset \cH_1(\bfq)$ at $\bfq=1$, yields the chain of inclusions $ \bbZ[\Lambda]^{W_0}\subset \bbZ[\Lambda] \subset \bbZ[W].$ The Kazhdan-Lusztig-Ginzburg operator ------------------------------------- As in the regular case, we will study the $Z(\cH_1(\bfq))$-algebra $\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$ of $Z(\cH_1(\bfq))$-linear endomorphisms of $\cA_1(\bfq)$. Recall that $Z(\cH_1(\bfq))= \cA_1(\bfq)^s$ is the subring of invariants of the commutative ring $\cA_1(\bfq)$. \[1Ybasis\] We have $$\cA_1(\bfq)=\cA_1(\bfq)^{s}X\oplus\cA_1(\bfq)^s=\cA_1(\bfq)^{s}\oplus\cA_1(\bfq)^sY$$ as $\cA_1(\bfq)^{s}$-modules. Applying $s$, the two decompositions are equivalent; so it suffices to check that $\bbZ[z_2^{\pm1}][X,Y]$ is free of rank 2 with basis $1,Y$ over the subring of symmetric polynomials $\bbZ[z_2^{\pm1}][X+Y,XY]$. First if $P=QY$ with $P$ and $Q$ symmetric, then applying $s$ we get $P=QX$ and hence $Q(X-Y)=0$ which implies $P=Q=0$. It remains to check that any monomial $X^iY^j$, $i,j\in\bbN$, belongs to $$\bbZ[z_2^{\pm1}][X+Y,XY]+\bbZ[z_2^{\pm1}][X+Y,XY]Y.$$ As $X=(X+Y)-Y$ and $Y^2=-XY+(X+Y)Y$, the later is stable under multiplication by $X$ and $Y$; as it contains $1$, the result follows. The basis $\{1,Y\}$ specializes at $\bfq=1$ to the so-called [*Pittie-Steinberg basis*]{} [@St75] of $\bbZ[\Lambda]$ over $\bbZ[\Lambda]^{W_0}$. $$D_s:=\textrm{ projector on $\cA_1(\bfq)^sY$ along $\cA_1(\bfq)^{s}$ }$$ $$D_s':=\textrm{ projector on $\cA_1(\bfq)^s$ along $\cA_1(\bfq)^{s}X$ }$$ $$D_s(\bfq):=D_s-\bfq D_s'.$$ The operators $D_s$ and $D_s'$ specialize at $\bfq=1$ to the *Demazure operators* on $\bbZ[\Lambda]$, as introduced in [@D73; @D74]. \[relationDsq\] We have $$D_s(\bfq)^2=(1-\bfq)D_s(\bfq)+\bfq.$$ Noting that $Y=z_1-X$, we have $$D_s(\bfq)^2(1)=D_s(\bfq)(-\bfq)=\bfq^2=(1-\bfq)(-\bfq)+\bfq=((1-\bfq)D_s(\bfq)+\bfq)(1)$$ and $$\begin{aligned} D_s(\bfq)^2(Y)&=&D_s(\bfq)(Y-\bfq z_1)\\ &=&Y-\bfq z_1-\bfq z_1(-\bfq)\\ &=&(1-\bfq)(Y-\bfq z_1)+\bfq Y\\ &=&((1-\bfq)D_s(\bfq)+\bfq)(Y).\end{aligned}$$ The generic non-regular antispherical representation {#HIqGL2} ---------------------------------------------------- We define the generic non-regular antispherical representation of the algebra $\cH_1(\bfq)$ on the $Z(\cH_1(\bfq))$-module $\cA_1(\bfq)$. The commutative ring $\cA_1(\bfq)$ is naturally a subring $$\cA_1(\bfq)\subset \operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq)),$$ an element $a\in \cA_1(\bfq)$ acting by multiplication $b\mapsto ab$ on $ \cA_1(\bfq)$. \[sA1q\] There exists a unique $\bbZ[\bfq]$-algebra homomorphism $$\xymatrix{ \sA_1(\bfq):\cH_1(\bfq) \ar[r] & \operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq)) }$$ such that - $$\sA_1(\bfq)|_{\cA_1(\bfq)}=\textrm{ the natural inclusion $\cA_1(\bfq)\subset\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$}$$ - $$\sA_1(\bfq)(S)=-D_s(\bfq).$$ Recall that $\cH_1(\bfq)=\bbZ[\bfq][S,U^{\pm1}]$ with the relations $S^2=(\bfq-1)S+\bfq$ and $U^2S=SU^2$. In particular $\sA_1(\bfq)(S):=-D_s(\bfq)$ is well-defined thanks to \[relationDsq\]. On the other hand, the $\bbZ[\bfq]$-algebra $\cA_1(\bfq)$ is generated by $$z_2=U^2,\quad X=US+(1-\bfq)U\quad\textrm{and}\quad Y=SU.$$ Consequently, there exists a $\bbZ[\bfq]$-algebra homomorphism $\sA_1(\bfq)$ as in the statement of the theorem if and only if there exists $$\sA_1(\bfq)(U)\in\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$$ satisfying 1. $\sA_1(\bfq)(U)$ is invertible ; 2. $\sA_1(\bfq)(U)^2=\sA_1(\bfq)(U^2)=\sA_1(\bfq)(z_2)=z_2\operatorname{Id}$ ; 3. $\sA_1(\bfq)(U)(-D_s(\bfq))+(1-\bfq)\sA_1(\bfq)(U)=\textrm{ multiplication by $X$}$ 4. $-D_s(\bfq)\sA_1(\bfq)(U)=\textrm{ multiplication by $Y$}$. Let us use the $Z(\cH_1(\bfq))$-basis $1,Y$ of $\cA_1(\bfq)$ to identify $\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$ with the algebra of $2\times 2$-matrices over the ring $Z(\cH_1(\bfq))=\cA_1(\bfq)^s$. Then, by definition, $$-D_s(\bfq)= \left (\begin{array}{cc} 0& 0 \\ 0& -1 \end{array} \right) + \bfq \left (\begin{array}{cc} 1& z_1 \\ 0& 0 \end{array} \right) = \left (\begin{array}{cc} \bfq& \bfq z_1 \\ 0& -1 \end{array} \right).$$ Moreover, as $X=z_1-Y$, $XY=\bfq z_2$ and $Y^2=-XY+(X+Y)Y=-\bfq z_2+z_1Y$, the multiplications by $X$ and by $Y$ on $\cA_1(\bfq)$ get identified with the matrices $$\left (\begin{array}{cc} z_1& \bfq z_2 \\ -1& 0 \end{array} \right) \quad\textrm{and}\quad \left (\begin{array}{cc} 0& -\bfq z_2 \\ 1& z_1 \end{array} \right).$$ Now, writing $$\sA_1(\bfq)(U)= \left (\begin{array}{cc} a& c \\ b& d \end{array} \right)$$ we have: $$\sA_1(\bfq)(U)^2=z_2\operatorname{Id}\Longleftrightarrow \left (\begin{array}{cc} a^2+bc& c(a+d) \\ b(a+d)& d^2+bc \end{array} \right) = \left (\begin{array}{cc} z_2& 0 \\ 0& z_2 \end{array} \right),$$ $$\sA_1(\bfq)(U)(-D_s(\bfq))+(1-\bfq)\sA_1(\bfq)(U)=\textrm{ multiplication by $X$}$$ $$\Longleftrightarrow \left (\begin{array}{cc} a& \bfq(a z_1-c) \\ b& \bfq(b z_1-d) \end{array} \right) = \left (\begin{array}{cc} z_1& \bfq z_2 \\ -1& 0 \end{array} \right)$$ and $$-D_s(\bfq)\sA_1(\bfq)(U)=\textrm{ multiplication by $Y$}$$ $$\Longleftrightarrow \left (\begin{array}{cc} \bfq(a+z_1b)& \bfq(c+z_1d) \\ -b& -d \end{array} \right) = \left (\begin{array}{cc} 0& -\bfq z_2 \\ 1& z_1 \end{array} \right).$$ Each of the two last systems admits a unique solution, namely $$\sA_1(\bfq)(U)= \left (\begin{array}{cc} a& c \\ b& d \end{array} \right) = \left (\begin{array}{cc} z_1& z_1^2-z_2 \\ -1& -z_1 \end{array} \right),$$ which is also a solution of the first one. Moreover, the determinant $$ad-bc=-z_1^2+(z_1^2-z_2)=-z_2$$ is invertible. The relation between our generic non-regular representation $\sA_1(\bfq)$ and the theory of Kazhdan-Lusztig [@KL87], and Ginzburg [@CG97], is the following. Introducing a square root $\bfq^{\frac{1}{2}}$ of $\bfq$ and extending scalars along $\bbZ[\bfq]\subset\bbZ[\bfq^{\pm\frac{1}{2}}]$, we obtain the Hecke algebra $\cH_1(\bfq^{\pm\frac{1}{2}})$ together with its commutative subalgebra $\cA_1(\bfq^{\pm\frac{1}{2}})$. The latter contains the elements $\tilde{\theta}_{\lambda}$, $\lambda\in\Lambda$, introduced by Bernstein and Lusztig, which are defined as follows: writing $\lambda=\lambda_1-\lambda_2$ with $\lambda_1,\lambda_2$ antidominant, one has $$\tilde{\theta}_{\lambda}:=\tilde{T}_{e^{\lambda_1}}\tilde{T}_{e^{\lambda_2}}^{-1}:=\bfq^{-\frac{\ell(\lambda_1)}{2}}\bfq^{\frac{\ell(\lambda_2)}{2}} T_{e^{\lambda_1}}T_{e^{\lambda_2}}^{-1}.$$ They are related to the Bernstein basis $\{E(w),\ w\in W\}$ of $\cH_1(\bfq)$ introduced by Vignéras (which is analogous to the Bernstein basis of $\cH^{(1)}(\bfq)$ which we have recalled in \[Bernsteinpresprop\]) by the formula: $$\forall \lambda\in\Lambda,\ \forall w\in W_0,\quad E(e^{\lambda}w)=\bfq^{\frac{\ell(e^{\lambda}w)-\ell(w)}{2}}\tilde{\theta}_{\lambda}T_w\quad\in \cH_1(\bfq)\subset\cH_1(\bfq^{\pm\frac{1}{2}}).$$ In particular $E(e^{\lambda})=\bfq^{\frac{\ell(e^{\lambda})}{2}}\tilde{\theta}_{\lambda}$, and by the product formula (analogous to the product formula for $\cH^{(1)}(\bfq)$, cf. \[Bernsteinpresprop\]), the $\bbZ[\bfq^{\pm\frac{1}{2}}]$-linear isomorphism $$\begin{aligned} \tilde{\theta}: \bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda] & {\stackrel{\sim}{\longrightarrow}}& \cA_1(\bfq^{\pm\frac{1}{2}})\\ e^{\lambda} & \lmapsto & \tilde{\theta}_{\lambda}\end{aligned}$$ is in fact multiplicative, i.e. it is an isomorphism of $\bbZ[\bfq^{\pm\frac{1}{2}}]$-algebras. Consequently, if we base change our action map $\sA_1(\bfq)$ to $\bbZ[\bfq^{\pm\frac{1}{2}}]$, we get a representation $$\xymatrix{ \sA_1(\bfq^{\pm\frac{1}{2}}):\cH_1(\bfq^{\pm\frac{1}{2}}) \ar[r] & \operatorname{End}_{Z(\cH_1(\bfq^{\pm{\frac{1}{2}}}))}(\cA_1(\bfq^{\pm\frac{1}{2}})) \simeq \operatorname{End}_{\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda]^{W_0}}(\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda]), }$$ which coincides with the natural inclusion $\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda]\subset\operatorname{End}_{\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda]^{W_0}}(\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda])$ when restricted to $\cA_1(\bfq^{\pm\frac{1}{2}})\simeq\bbZ[\bfq^{\pm\frac{1}{2}}][\Lambda]$, and which sends $S$ to the opposite $-D_s(\bfq)$ of the $\bfq$-deformed Demazure operator. Hence, this is the antispherical representation defined by Kazhdan-Lusztig and Ginzburg. In particular, $\sA_1(1)$ is the usual action of the Iwahori-Weyl group $W=\Lambda\rtimes W_0$ on $\Lambda$, and $\sA_1(0)$ can be thought of as a degeneration of the latter. \[sAqinj\] \[injIGL2\] \[sA1sf\] The homomorphism $\sA_1(\bfq)$ is injective. It follows from \[presH1q\] and \[presA1q\] that the ring $\cH_1(\bfq)$ is generated by the elements $$1,\ S,\ U,\ SU$$ as a module over its center $Z(\cH_1(\bfq))=\bbZ[\bfq][z_1,z_2^{\pm1}]$. As the latter is mapped isomorphically to the center of the matrix algebra $\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$ by $\sA_1(\bfq)$, it suffices to check that the images $$1,\ \sA_1(\bfq)(S),\ \sA_1(\bfq)(U),\ \sA_1(\bfq)(SU)$$ of $1, S, U, SU$ by $\sA_1(\bfq)$ are free over $Z(\cH_1(\bfq))$. So let $\alpha,\beta,\gamma,\delta\in Z(\cH_1(\bfq))$ (which is an integral domain) be such that $$\alpha \left (\begin{array}{cc} 1& 0 \\ 0& 1 \end{array} \right) + \beta \left (\begin{array}{cc} \bfq& \bfq z_1 \\ 0& -1 \end{array} \right) + \gamma \left (\begin{array}{cc} z_1& z_1^2-z_2 \\ -1& -z_1 \end{array} \right) + \delta \left (\begin{array}{cc} 0& -\bfq z_2 \\ 1& z_1 \end{array} \right) =0.$$ Then $$\left\{ \begin{array}{lll} \alpha+\beta\bfq+\gamma z_1&=&0\\ -\gamma+\delta&=&0\\ \beta\bfq z_1+\gamma(z_1^2-z_2)-\delta\bfq z_2&=&0 \\ \alpha-\beta+(\delta-\gamma)z_1&=&0. \end{array} \right.$$ We obtain $\delta=\gamma$, $\alpha=\beta$ and $$\left\{ \begin{array}{lll} \alpha(1+\bfq)+\gamma z_1&=&0\\ \alpha\bfq z_1+\gamma(z_1^2-z_2-\bfq z_2)&=&0. \\ \end{array} \right.$$ The latter system has determinant $$(1+\bfq)(z_1^2-z_2-\bfq z_2)-\bfq z_1^2=z_1^2-z_2-2\bfq z_2-\bfq^2z_2$$ which is nonzero (its specialisation at $\bfq=0$ is equal to $z_1^2-z_2\neq 0$), whence $\alpha=\gamma=0=\beta=\delta$. We record the following two corollaries of the proof. \[dimOverCenter\] The ring $\cH_1(\bfq)$ is a free $Z(\cH_1(\bfq))$-module on the basis $1, S, U, SU$. \[faithfulatzero\] The homomorphism $\sA_1(0)$ is injective. We end this section by noting an equivariance property of $\sA_1(\bfq)$. As already noticed, the finite Weyl group $W_0$ acts on $\cA_1(\bfq)$ by $\bbZ[\bfq]$-algebra automorphisms, and the action is clearly faithful. Moreover $\cA_1(\bfq)^{W_0}=Z(\cH_1(q))$. Hence $W_0$ can be viewed as a subgroup of $\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$, and we can let it act on $\operatorname{End}_{Z(\cH_1(\bfq))}(\cA_1(\bfq))$ by conjugation. The embedding $\sA_1(\bfq)|_{\cA_1(\bfq)}$ is $W_0$-equivariant. Indeed, for all $a,b\in\cA_1(\bfq)$ and $w\in W_0$, we have $$\sA_1(\bfq)(w(a))(b)=w(a)b=w(aw^{-1}(b))=(waw^{-1})(b)=(w\sA_1(\bfq)(a)w^{-1})(b).$$ Geometric representation theory =============================== The Vinberg monoid of the dual group $\mathbf{\whG}=\mathbf{GL_2}$ {#Vinsubsection} ------------------------------------------------------------------ The Langlands dual group over $k:=\overline{\bbF}_q$ of the connected reductive algebraic group $GL_2$ over $F$ is $\mathbf{\whG}=\mathbf{GL_2}$. We recall the $k$-monoid scheme introduced by Vinberg in [@V95], in the particular case of $\mathbf{GL_2}$. It is in fact defined over $\bbZ$, as the group $\mathbf{GL_2}$. In the following, all the fiber products are taken over the base ring $\bbZ$. Let $\operatorname{Mat}_{2\times 2}$ be the $\bbZ$-monoid scheme of $2\times 2$-matrices (with usual matrix multiplication as operation). The *Vinberg monoid for $\mathbf{GL_2}$* is the $\bbZ$-monoid scheme $$V_{\mathbf{GL_2}}:=\operatorname{Mat}_{2\times 2}\times\bbG_m.$$ The group $\mathbf{GL_2}\times\bbG_m$ is recovered from the monoid $V_{\mathbf{GL_2}}$ as its group of units. The group $\mathbf{GL_2}$ itself is recovered as follows. Denote by $z_2$ the canonical coordinate on $\bbG_m$. Then *let $\bfq$ be the homomorphism from $V_{\mathbf{GL_2}}$ to the multiplicative monoid $(\bbA^1,\cdot )$ defined by $(f,z_2)\mapsto \det(f)z_2^{-1}$*: $$\xymatrix{ V_{\mathbf{GL_2}} \ar[d]_{\bfq} \\ \bbA^1. }$$ Then $\mathbf{GL_2}$ is recovered as the fiber at $\bfq=1$, canonically: $$\bfq^{-1}(1)= \{ (f,z_2) : \det(f)=z_2 \} {\stackrel{\sim}{\longrightarrow}}\mathbf{GL_2},\quad (f,z_2)\mapsto f.$$ The fiber at $\bfq=0$ is the $\bbZ$-semigroup scheme $$V_{\mathbf{GL_2},0} := \bfq^{-1}(0)= \operatorname{Sing}_{2\times 2}\times\bbG_m,$$ where $\operatorname{Sing}_{2\times 2}$ represents the singular $2\times 2$-matrices. Note that it has no identity element, i.e. it is a semigroup which is not a monoid. \[VT\] Let $\operatorname{Diag}_{2\times 2}\subset \operatorname{Mat}_{2\times 2}$ be the submonoid scheme of diagonal $2\times 2$-matrices, and set $$V_{\mathbf{\whT}}:=\operatorname{Diag}_{2\times 2}\times\bbG_m\subset V_{\mathbf{GL_2}}= \operatorname{Mat}_{2\times 2}\times\bbG_m.$$ This is a diagonalizable $\bbZ$-monoid scheme with character monoid $$\bbX^{\bullet}(V_{\mathbf{\whT}})=\bbN(1,0)\oplus\bbN(0,1)\oplus\bbZ(1,1)\subset\bbZ(1,0)\oplus\bbZ(0,1)=\Lambda=\bbX^{\bullet}(\mathbf{\whT}).$$ In particular, setting $X:=e^{(1,0)}$ and $Y:=e^{(0,1)}$ in the group ring $\bbZ[\Lambda]$, we have $$\mathbf{\whT}=\operatorname{Spec}(\bbZ[X^{\pm1},Y^{\pm1}])\subset \operatorname{Spec}(\bbZ[z_2^{\pm1}][X,Y])=V_{\mathbf{\whT}}.$$ Again, this closed subgroup is recovered as the fiber at $\bfq=1$ of the fibration $\bfq|_{V_{\mathbf{\whT}}} : V_{\mathbf{\whT}}\ra \bbA^1$, and the fiber at $\bfq=0$ is the $\bbZ$-semigroup scheme $\operatorname{SingDiag}_{2\times2}\times\bbG_m$ where $\operatorname{SingDiag}_{2\times2}$ represents the singular diagonal $2\times 2$-matrices: $$\xymatrix{ \mathbf{\whT}\ar@{^{(}->}[r] \ar[d] & V_{\mathbf{\whT}} \ar[d]_{\bfq} & \ar@{_{(}->}[l]\operatorname{SingDiag}_{2\times2}\times\bbG_m \ar[d]\\ \operatorname{Spec}(\bbZ)\ar@{^{(}->}[r]^<<<<<1 &\bbA^1 & \ar@{_{(}->}[l]_>>>>>>>>>0 \operatorname{Spec}(\bbZ). }$$ In terms of equations, the $\bbA^1$-family $$\xymatrix{ \bfq:V_{\mathbf{\whT}}=\operatorname{Diag}_{2\times 2}\times\bbG_m=\operatorname{Spec}(\bbZ[z_2^{\pm1}][X,Y])\ar[r] &\bbA^1 }$$ is given by the formula $\bfq(\operatorname{diag}(x,y),z_2)=\det(\operatorname{diag}(x,y))z_2^{-1}=xyz_2^{-1}$. Hence, after fixing $z_2\in\bbG_m$, the fiber over a point $\bfq\in\bbA^1$ is the hyperbola $xy=\bfq z_2$, which is non-degenerate if $\bfq \neq 0$, and is the union of the two coordinate axis if $\bfq =0$. The associated flag variety and its equivariant $K$-theory ---------------------------------------------------------- Let $\mathbf{\whB}\subset\mathbf{GL_2}$ be the Borel subgroup of upper triangular matrices, let $\operatorname{UpTriang}_{2\times 2}$ be the $\bbZ$-monoid scheme representing the upper triangular $2\times 2$-matrices, and set $$V_{\mathbf{\whB}}:=\operatorname{UpTriang}_{2\times 2}\times\bbG_m\subset \operatorname{Mat}_{2\times 2}\times\bbG_m=:V_{\mathbf{GL_2}}.$$ Then we can apply to this inclusion of $\bbZ$-monoid schemes the general formalism developed in the Appendix \[Appendix\]. In particular, the *flag variety $V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}$* is defined as a $\bbZ$-monoidoid. Moreover, after base changing along $\bbZ\ra k$, we have defined a ring $K^{V_{\mathbf{GL_2}}}(V_{\mathbf{GL_2}}/V_{\mathbf{\whB}})$ of $V_{\mathbf{GL_2}}$-equivariant $K$-theory on the flag variety, together with an induction isomorphism $$\xymatrix{ \cI nd_{V_{\mathbf{\whB}}}^{V_{\mathbf{GL_2}}}:R(V_{\mathbf{\whB}}) \ar[r]^<<<<<{\sim} & K^{V_{\mathbf{GL_2}}}(V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}) }$$ from the ring $R(V_{\mathbf{\whB}})$ of right representations of the $k$-monoid scheme $V_{\mathbf{\whB}}$ on finite dimensional $k$-vector spaces. \[retraction\] Now, we have the inclusion of monoids $V_{\mathbf{\whT}}=\operatorname{Diag}_{2\times 2}\times\bbG_m\subset V_{\mathbf{\whB}}=\operatorname{UpTriang}_{2\times 2}\times\bbG_m$, which admits the retraction $$\begin{aligned} V_{\mathbf{\whB}} & \lra & V_{\mathbf{\whT}} \\ \bigg(\left (\begin{array}{cc} x & c\\ 0 & y \end{array} \right),\ z_2\bigg) & \lmapsto & \bigg(\left (\begin{array}{cc} x & 0\\ 0 & y \end{array} \right),\ z_2\bigg).\end{aligned}$$ Let $\operatorname{Rep}(V_{\mathbf{\whT}})$ be the category of representations of the commutative $k$-monoid scheme $V_{\mathbf{\whT}}$ on finite dimensional $k$-vector spaces. The above preceding inclusion and retraction define a *restriction functor* and an *inflation functor* $$\xymatrix{ \operatorname{Res}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}}:\operatorname{Rep}(V_{\mathbf{\whB}}) \ar@<1ex>[r] & \operatorname{Rep}(V_{\mathbf{\whT}}) :\operatorname{Infl}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}}. \ar@<1ex>[l] }$$ These functors are exact and compatible with the tensors products and units. \[ResInfl\] The ring homomorphisms $$\xymatrix{ \operatorname{Res}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}}:R(V_{\mathbf{\whB}}) \ar@<1ex>[r] & R(V_{\mathbf{\whT}}) :\operatorname{Infl}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}} \ar@<1ex>[l] }$$ are isomorphisms, which are inverse one to the other. We have $\operatorname{Res}_{V_{\mathbf{\whT}}}\circ \operatorname{Infl}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}}=\operatorname{Id}$ by construction. Conversely, let $M$ be an object of $\operatorname{Rep}(V_{\mathbf{\whB}})$. The solvable subgroup $\mathbf{\whB}\times\bbG_m\subset V_{\mathbf{\whB}}$ stabilizes a line $L\subseteq M$. As $\mathbf{\whB}\times\bbG_m$ is dense in $V_{\mathbf{\whB}}$, the line $L$ is automatically $V_{\mathbf{\whB}}$-stable. Moreover the unipotent radical $\mathbf{\whU}\subset \mathbf{\whB}$ acts trivially on $L$, so that $\mathbf{\whB}\times\bbG_m$ acts on $L$ through the quotient $\mathbf{\whT}\times\bbG_m$. Hence, by density again, $V_{\mathbf{\whB}}$ acts on $L$ through the retraction $V_{\mathbf{\whB}}\ra V_{\mathbf{\whT}}$. This shows that any irreducible $M$ is a character inflated from a character of $V_{\mathbf{\whT}}$. In particular, the map $R(V_{\mathbf{\whT}}) \rightarrow R(V_{\mathbf{\whB}})$ is surjective and hence bijective. \[cVGL2\] We have a ring isomorphism $$\xymatrix{ c_{V_{\mathbf{GL_2}}}:=\cI nd_{V_{\mathbf{\whB}}}^{V_{\mathbf{GL_2}}}\circ\operatorname{Infl}_{V_{\mathbf{\whT}}}^{V_{\mathbf{\whB}}}:\bbZ[X,Y,z_2^{\pm1}]\cong R(V_{\mathbf{\whT}}) \ar[r]^<<<<<{\sim} & K^{V_{\mathbf{GL_2}}}(V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}), }$$ that we call *the characteristic isomorphism in the equivariant $K$-theory of the flag variety $V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}$*. We have a commutative diagram *specialization at $\bfq=1$* $$\xymatrix{ \bbZ[X,Y,z_2^{\pm1}] \ar[rr]^{c_{V_{\mathbf{GL_2}}}}_{\sim} \ar@{->>}[d] && K^{V_{\mathbf{GL_2}}}(V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}) \ar@{->>}[d] \\ \bbZ[X^{\pm1},Y^{\pm1}]\ar[rr]^{c_{\mathbf{GL_2}}}_{\sim} && K^{\mathbf{GL_2}}(\mathbf{GL_2}/\mathbf{\whB}). }$$ The vertical map on the left-hand side is given by specialization $\bfq = 1$, i.e. by the surjection $$\bbZ[X,Y,z_2^{\pm 1 }] = \bbZ[\bfq][X,Y,z_2^{\pm 1 }] / (XY-\bfq z_2) \longrightarrow \bbZ[X,Y,z_2^{\pm 1 }] / (XY-z_2) = \bbZ[X^{\pm 1},Y^{\pm 1}].$$ The vertical map on the right-hand side is given by restricting equivariant vector bundles to the $1$-fiber of $\bfq: V_{\mathbf{GL_2}}\ra\bbA^1$, thereby recovering the classical theory. Let $\operatorname{Rep}(V_{\mathbf{GL_2}})$ be the category of right representations of the $k$-monoid scheme $V_{\mathbf{GL_2}}$ on finite dimensional $k$-vector spaces. The inclusion $V_{\mathbf{\whB}}\subset V_{\mathbf{GL_2}}$ defines a restriction functor $$\xymatrix{ \operatorname{Res}^{V_{\mathbf{GL_2}}}_{V_{\mathbf{\whB}}}:\operatorname{Rep}(V_{\mathbf{GL_2}}) \ar[r] & \operatorname{Rep}(V_{\mathbf{\whB}}), }$$ whose composition with $\operatorname{Res}^{V_{\mathbf{\whB}}}_{V_{\mathbf{\whT}}}$ is the restriction from $V_{\mathbf{GL_2}}$ to $V_{\mathbf{\whT}}$: $$\xymatrix{ \operatorname{Res}^{V_{\mathbf{GL_2}}}_{V_{\mathbf{\whT}}}=\operatorname{Res}^{V_{\mathbf{\whB}}}_{V_{\mathbf{\whT}}}\circ \operatorname{Res}^{V_{\mathbf{GL_2}}}_{V_{\mathbf{\whB}}}:\operatorname{Rep}(V_{\mathbf{GL_2}}) \ar[r] & \operatorname{Rep}(V_{\mathbf{\whT}}). }$$ These restriction functors are exact and compatible with the tensors products and units. The action of the Weyl group $W_0$ on $\Lambda=\bbX^{\bullet}(\mathbf{\whT})$ stabilizes $\bbX^{\bullet}(V_{\mathbf{\whT}})\subset \bbX^{\bullet}(\mathbf{\whT})$, consequently $W_0$ acts on $V_{\mathbf{\whT}}$ and the inclusion $\mathbf{\whT}\subset V_{\mathbf{\whT}}$ is $W_0$-equivariant. Explicitly, $W_0=\{1,s\}$ and $s$ acts on $V_{\mathbf{\whT}}=\operatorname{Diag}_{2\times 2}\times\bbG_m$ by permuting the two diagonal entries and trivially on the $\bbG_m$-factor. \[charVin\] The ring homomorphism $$\xymatrix{ \operatorname{Res}^{V_{\mathbf{GL_2}}}_{V_{\mathbf{\whT}}}:R(V_{\mathbf{GL_2}}) \ar[r] & R(V_{\mathbf{\whT}}) }$$ is injective, with image the subring $R(V_{\mathbf{\whT}})^{W_0}\subset R(V_{\mathbf{\whT}})$ of $W_0$-invariants. The resulting ring isomorphism $$\xymatrix{ \chi_{V_{\mathbf{GL_2}}}:R(V_{\mathbf{GL_2}}) \ar[r]^<<<<<{\sim} & R(V_{\mathbf{\whT}})^{W_0} }$$ is the *character isomorphism of $V_{\mathbf{GL_2}}$*. This is a general result on the representation theory of $V_{\mathbf{\whG}}$. Note that in the case of $\mathbf{\whG}=\mathbf{GL_2}$, we have $$R(V_{\mathbf{\whT}})^{W_0}=\bbZ[X+Y,XYz_2^{-1}=:\bfq,z_2^{\pm1}]\subset \bbZ[X,Y,z_2^{\pm1}]=R(V_{\mathbf{\whT}}).$$ Dual parametrization of generic Hecke modules ============================================= We keep all the notations introduced in the preceding section. In particular, $k=\overline{\bbF}_q$. The generic Bernstein isomorphism --------------------------------- Recall from \[AIH\] the subring $\cA(\bfq)\subset\cH^{(1)}(\bfq)$ and the remarkable Bernstein basis elements $E(1,0)$, $E(0,1)$ and $E(1,1)$. Also recall from \[VT\] the representation ring $R(V_{\mathbf{\whT}})=\bbZ[X,Y,z_2^{\pm1}]$ of the diagonalizable $k$-submonoid scheme $V_{\mathbf{\whT}}\subset V_{\mathbf{\whG}}$ of the Vinberg $k$-monoid scheme of the Langlands dual $k$-group $\mathbf{\whG}=\mathbf{GL_2}$ of $GL_{2,F}$. \[genB\] There exists a unique ring homomorphism $$\xymatrix{ \sB(\bfq):\cA(\bfq) \ar[r]& R(V_{\mathbf{\whT}}) }$$ such that $$\sB(\bfq)(E(1,0))=X,\quad \sB(\bfq)(E(0,1))=Y,\quad \sB(\bfq)(E(1,1))=z_2\quad\textrm{and}\quad\sB(\bfq)(\bfq)=XYz_2^{-1}.$$ It is an isomorphism. This is a reformulation of the first part of \[presAq\]. Then recall from \[AIH\] the subring $\cA^{(1)}(\bfq)=\bbZ[\bbT]\otimes_{\bbZ}\cA(\bfq)\subset\cH^{(1)}(\bfq)$ where $\bbT$ is the finite abelian group $\bfT(\bbF_q)$. Let $\bbT^{\vee}$ be the finite abelian dual group of $\bbT$. As $\bbT^{\vee}$ has order prime to $p$, it defines a constant finite diagonalizable $k$-group scheme, whose group of characters is $\bbT$, and hence whose representation ring $R(\bbT^{\vee})$ identifies with $\bbZ[\bbT]$: $t\in\bbT\subset \bbZ[\bbT]$ corresponds to the character $\operatorname{ev}_t$ of $\bbT^{\vee}$ given by evaluation at $t$. \[genB1\] There exists a unique ring homomorphism $$\xymatrix{ \sB^{(1)}(\bfq):\cA^{(1)}(\bfq)\ar[r] & R(\bbT^{\vee}\times V_{\mathbf{\whT}}) }$$ such that $$\sB^{(1)}(\bfq)(E(1,0))=X,\quad \sB^{(1)}(\bfq)(E(0,1))=Y,\quad \sB^{(1)}(\bfq)(E(1,1))=z_2,\quad\sB^{(1)}(\bfq)(\bfq)=XYz_2^{-1}$$ $$\textrm{and}\quad\forall t\in\bbT,\ \sB^{(1)}(\bfq)(T_t)=\operatorname{ev}_t.$$ It is an isomorphism, that we call the *generic (pro-$p$) Bernstein isomorphism*. \[B1qchar\] Also we have from \[ResInfl\] the ring isomorphism $$\xymatrix{ \operatorname{Infl}^{\bbT^{\vee}\times V_{\mathbf{\whB}}}_{\bbT^{\vee} \times V_{\mathbf{\whT}}}=\operatorname{Id}_{\bbZ[\bbT]}\otimes_{\bbZ}\operatorname{Res}^{V_{\mathbf{\whB}}}_{V_{\mathbf{\whT}}}: R(\bbT^{\vee} \times V_{\mathbf{\whT}})=\bbZ[\bbT]\otimes_{\bbZ}R(V_{\mathbf{\whT}})\ar[r]^<<<<<{\sim} & R(\bbT^{\vee}\times V_{\mathbf{\whB}})=\bbZ[\bbT]\otimes_{\bbZ}R(V_{\mathbf{\whB}}) }$$ and from \[ringmon\] the ring isomorphism $$\xymatrix{ \cI nd_{\bbT^{\vee}\times V_{\mathbf{\whB}}}^{\bbT^{\vee}\times V_{\mathbf{\whG}}}:R(\bbT^{\vee}\times V_{\mathbf{\whB}}) \ar[r]^<<<<<{\sim} & K^{\bbT^{\vee}\times V_{\mathbf{\whG}}}(\bbT^{\vee}\times V_{\mathbf{\whG}}/\bbT^{\vee}\times V_{\mathbf{\whB}}), }$$ and hence by composition the *characteristic isomorphism* $$\xymatrix{ c_{\bbT^{\vee}\times V_{\mathbf{\whG}}}:R(\bbT^{\vee} \times V_{\mathbf{\whT}}) \ar[r]^<<<<<{\sim} & K^{\bbT^{\vee}\times V_{\mathbf{\whG}}}(\bbT^{\vee}\times V_{\mathbf{\whG}}/\bbT^{\vee}\times V_{\mathbf{\whB}}). }$$ Whence a ring isomorphism $$\xymatrix{ c_{\bbT^{\vee}\times V_{\mathbf{\whG}}}\circ \sB^{(1)}(\bfq):\cA^{(1)}(\bfq) \ar[r]^<<<<<{\sim} & K^{\bbT^{\vee}\times V_{\mathbf{\whG}}}(\bbT^{\vee}\times V_{\mathbf{\whG}}/\bbT^{\vee}\times V_{\mathbf{\whB}}). }$$ \[SpecB1q\] The representation ring $R(V_{\mathbf{\whT}})$ is canonically isomorphic to the ring $\bbZ[V_{\mathbf{\whT}}]$ of regular functions of $V_{\mathbf{\whT}}$ considered now as a diagonalizable monoid scheme over $\bbZ$. Also recall from \[finiteT\] the ring extension $\bbZ\subset\tilde{\bbZ}$, and denote by $\tilde{\bullet}$ the base change functor from $\bbZ$ to $\tilde{\bbZ}$. For example, we will from now on write $\tilde{\cA}^{(1)}(\bfq)$ instead of $\cA_{\tilde{\bbZ}}^{(1)}(\bfq)$. We have the constant finite diagonalizable $\tilde{\bbZ}$-group scheme $\bbT^{\vee}$, whose group of characters is $\bbT$, and whose ring of regular functions is $$\tilde{\bbZ}[\bbT]=\prod_{\lambda\in \bbT^{\vee}}\tilde{\bbZ}\varepsilon_{\lambda}.$$ Hence applying the functor $\operatorname{Spec}$ to $\tilde{\sB}^{(1)}(\bfq)$, we obtain the commutative diagram of $\tilde{\bbZ}$-schemes $$\xymatrix{ \operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq)) \ar[dr]_{\pi_0\times\bfq} & & \bbT^{\vee}\times V_{\mathbf{\whT}} \ar[dl]^{\operatorname{Id}\times\bfq} \ar[ll]_>>>>>>>>>>>>>>{\operatorname{Spec}(\tilde{\sB}^{(1)}(\bfq))}^>>>>>>>>>>>>>>{\sim} \\ &\bbT^{\vee}\times\bbA^1 & }$$ where $\pi_0:\operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq))\ra \bbT^{\vee}$ is the decomposition of $\operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq))$ into its connected components. In particular, for each $\lambda\in \bbT^{\vee}$, we have the subring $\tilde{\cA}^{\lambda}(\bfq)= \tilde{\cA}^{(1)}(\bfq)\varepsilon_\lambda$ of $\tilde{\cA}^{(1)}(\bfq)$ and the isomorphism $$\xymatrix{ \operatorname{Spec}(\tilde{\cA}^{\lambda}(\bfq)) & &\{\lambda\} \times V_{\mathbf{\whT}} \ar[ll]_>>>>>>>>>>>{\operatorname{Spec}(\tilde{\sB}^{\lambda}(\bfq))}^>>>>>>>>>>>{\sim} }$$ of $\tilde{\bbZ}$-schemes over $\{\lambda\}\times\bbA^1$. In turn, each of these isomorphisms admits a model over $\bbZ$, obtained by applying $\operatorname{Spec}$ to the ring isomorphism in \[presA1q\] $$\xymatrix{ \sB_1(\bfq):\cA_1(\bfq)\ar[r]^<<<<<{\sim} & R(V_{\mathbf{\whT}}). }$$ The generic Satake isomorphism ------------------------------ Recall part of our notation: $\mathbf{G}$ is the algebraic group $\mathbf{GL_2}$ (which is defined over $\bbZ$), $F$ is a local field and $G:=\mathbf{G}(F)$. Also we have denoted by $o_F$ the ring of integers of $F$. Now we set $K:=\mathbf{G}(o_F)$. \[defsph\] Let $R$ be any commutative ring. The *spherical Hecke algebra of $G$ with coefficients in $R$* is defined to be the convolution algebra $$\cH_{R}^{\operatorname{sph}}:=(R[K\backslash G/K],\star)$$ generated by the $K$-double cosets in $G$. By the work of Kazhdan and Lusztig, the $R$-algebra $\cH_{R}^{\operatorname{sph}}$ depends on $F$ only through the cardinality $q$ of its residue field. Indeed, choose a uniformizer $\varpi\in o_F$. For a dominant cocharacter $\lambda\in\Lambda^+$ of $\mathbf{T}$, let $\mathbbm{1}_{\lambda}$ be the characteristic function of the double coset $K\lambda(\varpi)K$. Then $(\mathbbm{1}_{\lambda})_{\lambda\in \Lambda^+}$ is an $R$-basis of $\cH_{R}^{\operatorname{sph}}$. Moreover, for all $\lambda,\mu,\nu\in\Lambda^+$, there exist polynomials $$N_{\lambda,\mu;\nu}(\bfq)\in\bbZ[\bfq]$$ depending only on the triple $(\lambda,\mu,\nu)$, such that $$\mathbbm{1}_{\lambda}\star\mathbbm{1}_{\mu}=\sum_{\nu\in\Lambda^+}N_{\lambda,\mu;\nu}(q)\mathbbm{1}_{\nu}$$ where $N_{\lambda,\mu;\nu}(q)\in\bbZ\subset R$ is the value of $N_{\lambda,\mu;\nu}(\bfq)$ at $\bfq=q$. These polynomials are uniquely determined by this property since when $F$ vary, the corresponding integers $q$ form an infinite set. Their existence can be deduced from the theory of the spherical algebra with coefficients in $\bbC$, as $\cH_{R}^{\operatorname{sph}}=R\otimes_{\bbZ}\cH_{\bbZ}^{\operatorname{sph}}$ and $\cH_{\bbZ}^{\operatorname{sph}}\subset \cH_{\bbC}^{\operatorname{sph}}$ (e.g. using arguments similar to those in the proof of \[ThgenSat\] below). Let $\bfq$ be an indeterminate. The *generic spherical Hecke algebra* is the $\bbZ[\bfq]$-algebra $\cH^{\operatorname{sph}}(\bfq)$ defined by generators $$\cH^{\operatorname{sph}}(\bfq):=\oplus_{\lambda\in\Lambda^+}\bbZ[\bfq]T_{\lambda}$$ and relations: $$T_{\lambda}T_{\mu}=\sum_{\nu\in\Lambda^+}N_{\lambda,\mu;\nu}(\bfq)T_{\nu}\quad\textrm{for all $\lambda,\mu\in\Lambda^+$}.$$ \[ThgenSat\] There exists a unique ring homomorphism $$\xymatrix{ \sS(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r] & R(V_{\mathbf{\whT}}) }$$ such that $$\sS(\bfq)(T_{(1,0)})=X+Y,\quad \sS(\bfq)(T_{(1,1)})=z_2\quad\textrm{and}\quad\sS(\bfq)(\bfq)=XYz_2^{-1}.$$ It is an isomorphism onto the subring $R(V_{\mathbf{\whT}})^{W_0}$ of $W_0$-invariants $$\xymatrix{ \sS(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r]^>>>>>{\sim} & R(V_{\mathbf{\whT}})^{W_0} \subset R(V_{\mathbf{\whT}}). }$$ In particular, the algebra $\cH^{\operatorname{sph}}(\bfq)$ is commutative. Let $$\xymatrix{ \sS_{\operatorname{cl}}:\cH_{\bbC}^{\operatorname{sph}}\ar[r]^{\sim} & \bbC[\bbX^{\bullet}(\mathbf{\whT})]^{W_0} }$$ be the ‘classical’ isomorphism constructed by Satake [@S63]. We use [@Gr98] as a reference. For $\lambda\in\Lambda^+$, let $\chi_{\lambda}\in \bbZ[\bbX^{\bullet}(\mathbf{\whT})]^{W_0}$ be the character of the irreducible representation of $\mathbf{\whG}$ of highest weight $\lambda$. Then $(\chi_{\lambda})_{\lambda\in\Lambda^+}$ is a $\bbZ$-basis of $\bbZ[\bbX^{\bullet}(\mathbf{\whT})]^{W_0}$. Set $f_{\lambda}:=\sS_{\operatorname{cl}}^{-1}(q^{\lan\rho,\lambda\ran}\chi_{\lambda})$, where $2\rho=\alpha:=(1,-1)$. Then for each $\lambda,\mu\in\Lambda^+$, there exist polynomials $d_{\lambda,\mu}(\bfq)\in\bbZ[\bfq]$ such that $$f_{\lambda}=\mathbbm{1}_{\lambda}+\sum_{\mu<\lambda}d_{\lambda,\mu}(q)\mathbbm{1}_{\mu}\in \cH_{\bbC}^{\operatorname{sph}},$$ where $d_{\lambda,\mu}(q)\in\bbZ$ is the value of $d_{\lambda,\mu}(\bfq)$ at $\bfq=q$; the polynomial $d_{\lambda,\mu}(\bfq)$ depends only on the couple $(\lambda,\mu)$, in particular it is uniquely determined by this property. As $(\mathbbm{1}_{\lambda})_{\lambda\in \Lambda^+}$ is a $\bbZ$-basis of $\cH_{\bbZ}^{\operatorname{sph}}$, so is $(f_{\lambda})_{\lambda\in\Lambda^+}$. Then let us set $$f_{\lambda}(\bfq):=T_{\lambda}+\sum_{\mu<\lambda}d_{\lambda,\mu}(\bfq)T_{\mu}\in \cH^{\operatorname{sph}}(\bfq).$$ As $(T_{\lambda})_{\lambda\in \Lambda^+}$ is a $\bbZ[\bfq]$-basis of $\cH^{\operatorname{sph}}(\bfq)$, so is $(f_{\lambda}(\bfq))_{\lambda\in\Lambda^+}$. Next consider the following $\bbZ[\bfq^{\frac{1}{2}}]$-linear map: $$\begin{aligned} \sS_{\operatorname{cl}}(\bfq):\bbZ[\bfq^{\frac{1}{2}}]\otimes_{\bbZ[\bfq]}\cH^{\operatorname{sph}}(\bfq)&\lra& \bbZ[\bfq^{\frac{1}{2}}]\otimes_{\bbZ}\bbZ[\bbX^{\bullet}(\mathbf{\whT})]=\bbZ[\bfq^{\frac{1}{2}}][\bbX^{\bullet}(\mathbf{\whT})] \\ 1\otimes f_{\lambda}(\bfq) & \lmapsto & \bfq^{\lan\rho,\lambda\ran}\chi_{\lambda}.\end{aligned}$$ We claim that it is a ring homomorphism. Indeed, for $h_1(\bfq),h_2(\bfq)\in \bbZ[\bfq^{\frac{1}{2}}]\otimes_{\bbZ[\bfq]}\cH^{\operatorname{sph}}(\bfq)$, we need to check the identity $$\sS_{\operatorname{cl}}(\bfq)(h_1(\bfq)h_2(\bfq))=\sS_{\operatorname{cl}}(\bfq)(h_1(\bfq))\sS_{\operatorname{cl}}(\bfq)(h_2(\bfq))\in \bbZ[\bfq^{\frac{1}{2}}][\bbX^{\bullet}(\mathbf{\whT})].$$ Projecting in the $\bbZ[\bfq^{\frac{1}{2}}]$-basis $\bbX^{\bullet}(\mathbf{\whT})$, the latter corresponds to (a finite number of) identities in the ring $\bbZ[\bfq^{\frac{1}{2}}]$ of polynomials in the variable $\bfq^{\frac{1}{2}}$. Now, by construction and because $\sS_{\operatorname{cl}}$ is a ring homomorphism, the desired identities hold after specialyzing $\bfq$ to any power of a prime number; hence they hold in $\bbZ[\bfq^{\frac{1}{2}}]$. Also note that $\sS_{\operatorname{cl}}(\bfq)$ maps $1=T_{(0,0)}$ to $1=\chi_{(0,0)}$ by definition. It can also be seen that $\sS_{\operatorname{cl}}(\bfq)$ is injective using a specialization argument: if $h(\bfq)\in\bbZ[\bfq^{\frac{1}{2}}]\otimes_{\bbZ[\bfq]}\cH^{\operatorname{sph}}(\bfq)$ satisfies $\sS_{\operatorname{cl}}(\bfq)(h(\bfq))=0$, then the coordinates of $h(\bfq)$ (in the basis $(1\otimes f_{\lambda}(\bfq))_{\lambda\in\Lambda^+}$ say, one can also use the basis $(1\otimes T_{\lambda})_{\lambda\in\Lambda^+}$) are polynomials in the variable $\bfq^{\frac{1}{2}}$ which must vanish for an infinite number of values of $\bfq$, and hence they are identically zero. Let us describe the image of $\cH^{\operatorname{sph}}(\bfq)\subset \bbZ[\bfq^{\frac{1}{2}}]\otimes_{\bbZ[\bfq]}\cH^{\operatorname{sph}}(\bfq)$ under the ring embedding $\sS_{\operatorname{cl}}(\bfq)$. By construction, we have $$\sS_{\operatorname{cl}}(\bfq)(\cH^{\operatorname{sph}}(\bfq))=\bigoplus_{\lambda\in\Lambda^+}\bbZ[\bfq]\bfq^{\lan\rho,\lambda\ran}\chi_{\lambda}.$$ Explicitly, $$\Lambda^+=\bbN(1,0)\oplus\bbZ(1,1)\subset\bbZ(1,0)\oplus\bbZ(0,1)=\Lambda,$$ so that $$\sS_{\operatorname{cl}}(\bfq)(\cH^{\operatorname{sph}}(\bfq))=\bigg(\bigoplus_{n\in\bbN}\bbZ[\bfq]\bfq^{\frac{n}{2}}\chi_{(n,0)}\bigg)\otimes_{\bbZ}\bbZ[\chi_{(1,1)}^{\pm1}].$$ On the other hand, recall that the ring of symmetric polynomials in the two variables $e^{(1,0)}$ and $e^{(0,1)}$ is a graded ring generated the two characters $\chi_{(1,0)}=e^{(1,0)}+e^{(0,1)}$ and $\chi_{(1,1)}=e^{(1,0)}e^{(0,1)}$: $$\bbZ[e^{(1,0)},e^{(0,1)}]^{s}=\bigoplus_{n\in\bbN}\bbZ[e^{(1,0)},e^{(0,1)}]_n^s=\bbZ[\chi_{(1,0)},\chi_{(1,1)}].$$ As $\chi_{(1,0)}$ is homogeneous of degree 1 and $\chi_{(1,1)}$ is homogeneous of degree 2, this implies that $$\bbZ[e^{(1,0)},e^{(0,1)}]_n^s=\bigoplus_{\substack{(a,b)\in\bbN^2\\a+2b=n}}\bbZ\chi_{(1,0)}^a\chi_{(1,1)}^b.$$ Now if $a+2b=n$, then $\bfq^{\frac{n}{2}}\chi_{(1,0)}^a\chi_{(1,1)}^b=(\bfq^{\frac{1}{2}}\chi_{(1,0)})^a(\bfq\chi_{(1,1)})^b$. As the symmetric polynomial $\chi_{(n,0)}$ is homogeneous of degree $n$, we get the inclusion $$\sS_{\operatorname{cl}}(\bfq)(\cH^{\operatorname{sph}}(\bfq))\subset\bbZ[\bfq][\bfq^{\frac{1}{2}}\chi_{(1,0)},\bfq\chi_{(1,1)}]\otimes_{\bbZ}\bbZ[\chi_{(1,1)}^{\pm1}] =\bbZ[\bfq][\bfq^{\frac{1}{2}}\chi_{(1,0)},\chi_{(1,1)}^{\pm1}].$$ Since by definition of $\sS_{\operatorname{cl}}(\bfq)$ we have $\sS_{\operatorname{cl}}(\bfq)(f_{(1,0)}(\bfq))=\bfq^{\frac{1}{2}}\chi_{(1,0)}$, $\sS_{\operatorname{cl}}(\bfq)(f_{(1,1)}(\bfq))=\chi_{(1,1)}$ and $\sS_{\operatorname{cl}}(\bfq)(f_{(-1,-1)}(\bfq))=\chi_{(-1,-1)}=\chi_{(1,1)}^{-1}$, this inclusion is an equality. We have thus obtained the $\bbZ[\bfq]$-algebra isomorphism: $$\xymatrix{ \sS_{\operatorname{cl}}(\bfq)|_{\cH^{\operatorname{sph}}(\bfq)}:\cH^{\operatorname{sph}}(\bfq) \ar[r]^<<<<<{\sim} & \bbZ[\bfq][\bfq^{\frac{1}{2}}\chi_{(1,0)},\chi_{(1,1)}^{\pm1}]. }$$ Also note that $T_{(1,0)}\mapsto \bfq^{\frac{1}{2}}\chi_{(1,0)}$ and $T_{(1,1)}\mapsto \chi_{(1,1)}$ since $T_{(1,0)}=f_{(1,0)}(\bfq)$ and $T_{(1,1)}=f_{(1,1)}(\bfq)$. Finally, recall that $V_{\widehat{\bfT}}$ being the diagonalizable $k$-monoid scheme $\operatorname{Spec}(k[X,Y,z_2^{\pm1}])$, we have $$R(V_{\widehat{\bfT}})^{W_0}=\bbZ[X,Y,z_2^{\pm1}]^{W_0}=\bbZ[X+Y,XY,z_2^{\pm1}]=\bbZ[X+Y,XYz_2^{-1},z_2^{\pm1}].$$ Hence we can define a ring isomorphism $$\xymatrix{ \iota:\bbZ[\bfq][\bfq^{\frac{1}{2}}\chi_{(1,0)},\chi_{(1,1)}^{\pm1}]\ar[r]^<<<<<{\sim} & R(V_{\widehat{\bfT}})^{W_0} }$$ by $\iota(\bfq):=XYz_2^{-1}$, $\iota(\bfq^{\frac{1}{2}}\chi_{(1,0)})=X+Y$ and $\iota(\chi_{(1,1)})=z_2$. Composing, we get the desired isomorphism $$\xymatrix{ \sS(\bfq):=\iota\circ\sS_{\operatorname{cl}}(\bfq)|_{\cH^{\operatorname{sph}}(\bfq)}:\cH^{\operatorname{sph}}(\bfq) \ar[r]^<<<<<{\sim} & R(V_{\widehat{\bfT}})^{W_0}. }$$ Note that $\sS(\bfq)(T_{(1,0)})=X+Y$, $\sS(\bfq)(T_{(1,1)})=z_2$, $\sS(\bfq)(\bfq)=XYz_2^{-1}$, and that $\sS(\bfq)$ is uniquely determined by these assignments since the ring $\cH^{\operatorname{sph}}(\bfq)$ is the polynomial ring in the variables $\bfq$, $T_{(1,0)}$ and $T_{(1,1)}^{\pm 1}$, thanks to the isomorphism $\sS_{\operatorname{cl}}(\bfq)|_{\cH^{\operatorname{sph}}(\bfq)}$. \[Remiota\] The choice of the isomorphism $\iota$ in the preceding proof may seem *ad hoc*. However, it is natural from the point of view of the Vinberg fibration $\bfq:V_{\mathbf{\whT}}\ra \bbA^1$. First, as pointed out by Herzig in [@H11 §1.2], one can make the classical complex Satake transform $\sS_{\operatorname{cl}}$ integral, by removing the factor $\delta^{\frac{1}{2}}$ from its definition, where $\delta$ is the modulus character of the Borel subgroup. Doing so produces a ring embedding $$\xymatrix{ \cS':\cH_{\bbZ}^{\operatorname{sph}}\ar@{^{(}->}[r] & \bbZ[\bbX^{\bullet}(\mathbf{\whT})]. }$$ The image of $\cS'$ is not contained in the subring $\bbZ[\bbX^{\bullet}(\mathbf{\whT})]^{W_0}$ of $W_0$-invariants. In fact, $$\cS'(T_{(1,0)})=qe^{(1,0)}+e^{(0,1)}\quad\textrm{and}\quad\cS'(T_{(1,1)})=e^{(1,1)},$$ so that $$\xymatrix{ \cS':\cH_{\bbZ}^{\operatorname{sph}}\ar[r]^<<<<<{\sim} & \bbZ[(qe^{(1,0)}+e^{(0,1)}),e^{\pm (1,1)}]\subset \bbZ[\bbX^{\bullet}(\mathbf{\whT})]. }$$ Now, $$\bbZ[\bbX^{\bullet}(\mathbf{\whT})]=\bbZ[\mathbf{\whT}]=\bbZ[V_{\mathbf{\whT},1}],$$ where $\mathbf{\whT}\cong V_{\mathbf{\whT},1}$ is the fiber *at $1$* of the fibration $\bfq:V_{\mathbf{\whT}}\ra \bbA^1$ considered over $\bbZ$. But the algebra $\cH_{\bbZ}^{\operatorname{sph}}$ is the specialisation *at $q$* of the generic algebra $\cH^{\operatorname{sph}}(\bfq)$. From this perspective, the morphism $\cS'$ is unnatural, since it mixes a $1$-fiber with a $q$-fiber. To restore the $\bfq$-compatibility, one must consider the composition of $\bbQ\otimes_{\bbZ}\cS'$ with the isomorphism $$\begin{aligned} \bbQ[V_{\mathbf{\whT},1}]=\bbQ[X,Y,z_2^{\pm1}]/(XY-z_2)&\xrightarrow{\sim} &\bbQ[V_{\mathbf{\whT},q}]=\bbQ[X,Y,z_2^{\pm1}]/(XY-qz_2)\\ X & \mapsto & q^{-1}X \\ Y & \mapsto & Y \\ z_2 & \mapsto & z_2.\end{aligned}$$ But then one obtains the formulas $$\begin{aligned} \cH_{\bbQ}^{\operatorname{sph}}&\xrightarrow{\sim} &\bbQ[V_{\mathbf{\whT},q}]=\bbQ[X,Y,z_2^{\pm1}]/(XY-qz_2)\\ T_{(1,0)} & \mapsto & X+Y \\ T_{(1,1)} & \mapsto & z_2 \\ q & \mapsto & XYz_2^{\pm1}.\end{aligned}$$ This composed map is defined over $\bbZ$, its image is the subring $\bbZ[V_{\mathbf{\whT},q}]^{W_0}$ of $W_0$-invariants, and it is precisely the specialisation $\bfq=q$ of the isomorphism $\sS(\bfq)$ from \[ThgenSat\]. \[DefgenSat\] We call $$\xymatrix{ \sS(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r]^<<<<<{\sim} & R(V_{\mathbf{\whT}})^{W_0} }$$ the *generic Satake isomorphism*. Composing with the inverse of the character isomorphism $\chi_{V_{\mathbf{\whG}}}^{-1}: R(V_{\mathbf{\whT}})^{W_0}{\stackrel{\sim}{\longrightarrow}}R(V_{\mathbf{\whG}})$ from \[charVin\], we arrive at an isomorphism $$\xymatrix{ \chi_{V_{\mathbf{\whG}}}^{-1}\circ\sS(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r]^<<<<<{\sim} & R(V_{\mathbf{\whG}}). }$$ Next, recall the generic Iwahori-Hecke algebra $\cH_1(\bfq)$ \[defgeneric1\], and the commutative subring $\cA_1(\bfq)\subset \cH_1(\bfq)$ \[presA1q\] together with the isomorphism $\sB_1(\bfq)$ in \[SpecB1q\]. \[sZ1def\] The *generic central elements morphism* is the unique ring homomrphism $$\xymatrix{ \sZ_1(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r] & \cA_1(\bfq)\subset\cH_1(\bfq) }$$ making the diagram $$\xymatrix{ \cA_1(\bfq) \ar[rr]_{\sim}^{\sB_1(\bfq)} && R(V_{\mathbf{\whT}}) \\ \cH^{\operatorname{sph}}(\bfq) \ar[u]^{\sZ_1(\bfq)} \ar[rr]_{\sim}^{\sS(\bfq)} && R(V_{\mathbf{\whT}})^{W_0} \ar@{^{(}->}[u] }$$ commutative. \[sZ1iso\] By construction, the morphism $\sZ_1(\bfq)$ is injective, and is uniquely determined by the following equalities in $\cA_1(\bfq)$: $$\sZ_1(\bfq)(T_{(1,0)})=z_1,\quad \sZ_1(\bfq)(T_{(1,1)})=z_2\quad\textrm{and}\quad\sZ_1(\bfq)(\bfq)=\bfq.$$ Moreover the group $W_0$ acts on the ring $\cA_1(\bfq)$ and the invariant subring $\cA_1(\bfq)^{W_0}$ is equal to the center $Z(\cH_1(\bfq))\subset \cH_1(\bfq)$. As the isomorphism $\sB_1(\bfq)$ is $W_0$-equivariant by construction, we obtain that the image of $\sZ_1(\bfq)$ indeed is equal to the *center of the generic Iwahori-Hecke algebra $\cH_1(\bfq)$*: $$\xymatrix{ \sZ_1(\bfq):\cH^{\operatorname{sph}}(\bfq)\ar[r]^>>>>>{\sim} & Z(\cH_1(\bfq))\subset \cA_1(\bfq)\subset \cH_1(\bfq). }$$ \[compgenBSiso\] Under the identification $R(V_{\mathbf{\whT}})=\bbZ[V_{\mathbf{\whT}}]$ of \[SpecB1q\], the elements $\sS(\bfq)(T_{(1,0)})=X+Y$, $\sS(\bfq)(\bfq)=\bfq$, $\sS(\bfq)(T_{(1,1)})=z_2$, correspond to the *Steinberg choice of coordinates* $z_1$, $\bfq$, $z_2$ on the affine $\bbZ$-scheme $V_{\mathbf{\whT}}/W_0=\operatorname{Spec}(\bbZ[V_{\mathbf{\whT}}]^{W_0})$. On the other hand, the *Trace of representations morphism* $\operatorname{Tr}:R(V_{\mathbf{\whG}})\ra \bbZ[V_{\mathbf{\whG}}]^{\mathbf{\whG}}$ fits into the commutative diagram $$\xymatrix{ R(V_{\mathbf{\whT}})^{W_0} \ar@{=}[d] &\ar[l]^{\sim}_{\chi_{V_{\mathbf{\whG}}}} R(V_{\mathbf{\whG}}) \ar[d]^{\operatorname{Tr}} \\ \bbZ[V_{\mathbf{\whT}}]^{W_0} &\ar[l]^{\sim}_{\operatorname{Ch}} \bbZ[V_{\mathbf{\whG}}]^{\mathbf{\whG}} }$$ where $\chi_{V_{\mathbf{\whG}}}$ is the character isomorphism of \[charVin\], and $\operatorname{Ch}$ is the *Chevalley isomorphism* which is constructed for the Vinberg monoid $V_{\mathbf{\whG}}$ by Bouthier in [@Bo15 Prop. 1.7]. So we have the following commutative diagram of $\bbZ$-schemes $$\xymatrix{ \operatorname{Spec}(\cA_1(\bfq)) \ar@{->>}[d]_{\operatorname{Spec}(\sZ_1(\bfq))} & & V_{\mathbf{\whT}} \ar@{->>}[d] \ar[ll]_>>>>>>>>>>>>>>>>>>>{\operatorname{Spec}(\sB_1(\bfq))}^>>>>>>>>>>>>>>>>>>>{\sim} \ar@{^{(}->}[rr] && V_{\mathbf{\whG}}\ar@{->>}[d] \\ \operatorname{Spec}(\cH^{\operatorname{sph}}(\bfq)) \ar[dr]^{\sim}_{(T_{(1,0)},\bfq,T_{(1,1)})} & & V_{\mathbf{\whT}}/W_0 \ar[dl]^{(z_1,\bfq,z_2)}_{\sim} \ar[ll]^>>>>>>>>>>>>>>>>>>>{\sim}_>>>>>>>>>>>>>>>>>>>{\operatorname{Spec}(\sS(\bfq))} \ar[rr]_{\sim}^{\operatorname{Spec}(\operatorname{Ch})} && V_{\mathbf{\whG}}//\mathbf{\whG}\\ &\bbA^2\times\bbG_m. & }$$ Note that for $\mathbf{\whG}=\mathbf{GL_2}$, the composed *Chevalley-Steinberg map* $V_{\mathbf{\whG}}\ra \bbA^2\times\bbG_m$ is given explicitly by attaching to a $2\times 2$ matrix its characteristic polynomial (when $z_2=1$). We have recalled that for the generic pro-$p$-Iwahori-Hecke algebra $\cH^{(1)}(\bfq)$ too, the center can be described in terms of $W_0$-invariants, namely $Z(\cH^{(1)}(\bfq))=\cA^{(1)}(\bfq)^{W_0}$, cf. \[centerHI1\]. As the generic Bernstein isomorphism $\sB^{(1)}(\bfq)$ is $W_0$-equivariant by construction, cf. \[genB1\], we can make the following definition. \[DefgenSat1\] We call $$\xymatrix{ \sS^{(1)}(\bfq):=\sB^{(1)}(\bfq)^{W_0}:\cA^{(1)}(\bfq)^{W_0}\ar[r]^<<<<<{\sim} & R(\bbT^{\vee}\times V_{\mathbf{\whT}})^{W_0} }$$ the *generic pro-$p$-Iwahori Satake isomorphism*. \[SpecS1q\] Note that we have $\bbT^{\vee}\times V_{\mathbf{\whT}}=\coprod_{\gamma\in \bbT^{\vee}/W_0} \coprod_{\lambda\in\gamma} V_{\mathbf{\whT}}$ and the $W_0$-action on this scheme respects the $\gamma$-components. We obtain $$(\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0 = \coprod_{\gamma\in \bbT^{\vee}/W_0} (\coprod_{\lambda\in\gamma} V_{\mathbf{\whT}})/W_0.$$ If $\gamma$ is regular, then $(\coprod_{\lambda\in\gamma} V_{\mathbf{\whT}})/W_0 \simeq V_{\mathbf{\whT}}$, the isomorphism depending on a choice of order on the set $\gamma$, cf. \[centergamma\]. Hence, passing to $\tilde{\bbZ}$ as in \[SpecB1q\], with $\tilde{\cH}^{(1)}(\bfq):=\cH_{\tilde{\bbZ}}^{(1)}(\bfq)$, we obtain the following commutative diagram of $\tilde{\bbZ}$-schemes. $$\xymatrix{ \operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq)) \ar@{->>}[d] && \bbT^{\vee}\times V_{\mathbf{\whT}} \ar@{->>}[d] \ar[ll]_{\operatorname{Spec}(\tilde{\sB}^{(1)}(\bfq))}^>>>>>>>>>>>>>>>>>>>{\sim} \\ \operatorname{Spec}(Z(\tilde{\cH}^{(1)}(\bfq))) \ar[d]^{\rotatebox{90}{$\sim$}} && (\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0 \ar[ll]^{\sim}_{\operatorname{Spec}(\tilde{\sS}^{(1)}(\bfq))} \ar[d]_{\rotatebox{90}{$\sim$}}^{\ref{centergamma}} \\ (\bbA^2\times\bbG_m)^{\bbT^{\vee}/W_0} &&\coprod_{(\bbT^{\vee}/W_0)_{\operatorname{reg}}}V_{\mathbf{\whT}}\coprod_{(\bbT^{\vee}/W_0)_{\operatorname{non-reg}}} V_{\mathbf{\whT}}/W_0, \ar[ll]^>>>>>>>>>>>{\sim} }$$ where the bottom isomorphism of the diagram is given by the standard coordinates $(x,y,z_2)$ on the regular components and by the Steinberg coordinates $(z_1,\bfq,z_2)$ on the non-regular components. The generic parametrization {#genericpara} --------------------------- We keep the notation $\bbZ\subset \tilde{\bbZ}$ for the ring extension of \[finiteT\]. Then we have defined the $\tilde{\bbZ}$-scheme $\bbT^{\vee}\times V_{\mathbf{\whT}}$ in \[SpecB1q\], and we have considered in \[SpecS1q\] its quotient by the natural $W_0$-action. Also recall that $\mathbf{\whG}=\mathbf{GL_2}$ is the Langlands dual $k$-group of $GL_{2,F}$. The category of quasi-coherent modules on the $\tilde{\bbZ}$-scheme $(\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0$ will be called the *category of Satake parameters*, and denoted by $\operatorname{SP}_{\mathbf{\whG}}$: $$\operatorname{SP}_{\mathbf{\whG}}:=\operatorname{QCoh}\bigg((\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0\bigg).$$ For $\gamma \in \bbT^{\vee}/W_0$, we also define $\operatorname{SP}^{\gamma}_{\mathbf{\whG}}:=\operatorname{QCoh}\bigg(( \coprod_{\lambda\in\gamma} V_{\mathbf{\whT}})/W_0 \bigg)$. Now, over $\tilde{\bbZ}$, we have the isomorphism $$\xymatrix{ i_{\tilde{\sS}^{(1)}(\bfq)}:=\operatorname{Spec}(\tilde{\sS}^{(1)}(\bfq)):(\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0\ar[r]^<<<<<{\sim} & \operatorname{Spec}(Z(\tilde{\cH}^{(1)}(\bfq))) }$$ from the scheme $(\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0$ to the spectrum of the center $Z(\tilde{\cH}^{(1)}(\bfq))$ of the generic pro-$p$-Iwahori Hecke algebra $\tilde{\cH}^{(1)}(\bfq)$, cf. \[SpecS1q\]. The category of modules over $Z(\tilde{\cH}^{(1)}(\bfq))$ is equivalent to the category of Satake parameters: $$\xymatrix{ S:=(i_{\tilde{\sS}^{(1)}(\bfq)})^*:\operatorname{Mod}(Z(\tilde{\cH}^{(1)}(\bfq))) \ar@<1ex>[r]_<<<<<{\sim} & \operatorname{SP}_{\mathbf{\whG}}:(i_{\tilde{\sS}^{(1)}(\bfq)})_*. \ar@<1ex>[l] }$$ The equivalence $S$ will be referred to as the *functor of Satake parameters*. The quasi-inverse $(i_{\tilde{\sS}^{(1)}(\bfq)})_*$ will be denoted by $S^{-1}$. Still from \[SpecS1q\], these categories decompose as products over $\bbT^{\vee}/W_0$ (considered as a finite set), compatibly with the equivalences: for all $\gamma\in \bbT^{\vee}/W_0$, $$\xymatrix{ S^{\gamma}:=(i_{\tilde{\sS}^{\gamma}(\bfq)})^*:\operatorname{Mod}(Z(\tilde{\cH}^{\gamma}(\bfq))) \ar@<1ex>[r]_<<<<<{\sim} & \operatorname{SP}_{{\mathbf{\whG}}}^{\gamma}:(i_{\tilde{\sS}^{\gamma}(\bfq)})_*, \ar@<1ex>[l] }$$ where $$\operatorname{SP}_{{\mathbf{\whG}}}^{\gamma}\simeq \left\{ \begin{array}{ll} \operatorname{QCoh}(V_{\mathbf{\whT}}) & \textrm{ if $\gamma$ is regular} \\ \operatorname{QCoh}(V_{\mathbf{\whT}}/W_0)& \textrm{ if $\gamma$ is non-regular}. \end{array} \right.$$ In the regular case, the latter isomorphism depends on a choice of order on the set $\gamma$. \[Striv\] In particular, we have the trivial orbit $\gamma:=\{1\}$. The corresponding component $\tilde{\cH}^{\{1\}}(\bfq)$ of $\tilde{\cH}^{(1)}(\bfq)$ is canonically isomorphic to the $\tilde{\bbZ}$-base change of the generic non-regular Iwahori-Hecke algebra $\cH_1(\bfq)$. Hence from \[sZ1iso\] we have an isomorphism $$\xymatrix{ \tilde{\sZ}_1(\bfq):\tilde{\cH}^{\operatorname{sph}}(\bfq)\ar[r]^>>>>>{\sim} & Z(\tilde{\cH}^{\{1\}}(\bfq))\subset \tilde{\cA}^{\{1\}}(\bfq)\subset \tilde{\cH}^{\{1\}}(\bfq)\subset \tilde{\cH}^{(1)}(\bfq). }$$ Using these identifications, the equivalence $S^{\gamma}$ for $\gamma:=\{1\}$ can be rewritten as $$\xymatrix{ S^{\{1\}}:\operatorname{Mod}(\tilde{\cH}^{\operatorname{sph}}(\bfq)) \ar[r]^<<<<<{\sim} & \operatorname{SP}_{{\mathbf{\whG}}}^{\{1\}}. }$$ The category of quasi-coherent modules on the $\tilde{\bbZ}$-scheme $\bbT^{\vee}\times V_{\mathbf{\whT}}$ will be called the *category of Bernstein parameters*, and denoted by $\operatorname{BP}_{\mathbf{\whG}}$: $$\operatorname{BP}_{\mathbf{\whG}}:=\operatorname{QCoh}(\bbT^{\vee}\times V_{\mathbf{\whT}}).$$ Over $\tilde{\bbZ}$, we have the isomorphism $$\xymatrix{ i_{\tilde{\sB}^{(1)}(\bfq)}:=\operatorname{Spec}(\tilde{\sB}^{(1)}(\bfq)):\bbT^{\vee}\times V_{\mathbf{\whT}}\ar[r]^<<<<<{\sim} & \operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq))) }$$ from the scheme $\bbT^{\vee}\times V_{\mathbf{\whT}}$ to the spectrum of the commutative subring $\tilde{\cA}^{(1)}(\bfq)$ of the generic pro-$p$-Iwahori Hecke algebra $\tilde{\cH}^{(1)}(\bfq)$, cf. \[SpecB1q\]. Also we have the *restriction functor* $$\xymatrix{ \operatorname{Res}^{\tilde{\cH}^{(1)}(\bfq)}_{\tilde{\cA}^{(1)}(\bfq)}:\operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)) \ar[r] & \operatorname{Mod}(\tilde{\cA}^{(1)}(\bfq))\cong\operatorname{QCoh}(\operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq))) }$$ from the category of left $\tilde{\cH}^{(1)}(\bfq)$-modules to the one of $\tilde{\cA}^{(1)}(\bfq)$-modules, equivalently of quasi-coherent modules on $\operatorname{Spec}(\tilde{\cA}^{(1)}(\bfq))$. The *functor of Bernstein parameters* is the composed functor $$\xymatrix{ B:=(i_{\tilde{\sB}^{(1)}(\bfq)})^*\circ\operatorname{Res}^{\tilde{\cH}^{(1)}(\bfq)}_{\tilde{\cA}^{(1)}(\bfq)}:\operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)) \ar[r] & \operatorname{BP}_{\mathbf{\whG}}. }$$ Still from \[SpecB1q\], the category $\operatorname{BP}_{{\mathbf{\whG}}}$ decomposes as a product over the finite group $\bbT^{\vee}$: $$\operatorname{BP}_{{\mathbf{\whG}}}\cong\prod_{\lambda\in\bbT^{\vee}} \operatorname{BP}_{{\mathbf{\whG}}}^{\lambda}, \quad\textrm{where}\quad \forall \lambda\in\bbT^{\vee},\ \operatorname{BP}_{{\mathbf{\whG}}}^{\lambda}\simeq\operatorname{QCoh}(V_{\mathbf{\whT}}).$$ \[pi\] Denoting by $$\xymatrix{ \bbT^{\vee}\times V_{\mathbf{\whT}}\ar@{->>}[d]_{\pi} \\ (\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0 }$$ the canonical projection, the compatibilty between the functors $S$ and $B$ of Satake and Bernstein parameters is expressed by the commutativity of the diagram $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)) \ar[r]^<<<<<<B \ar[d]_{\operatorname{Res}_{Z(\tilde{\cH}^{(1)}(\bfq))}^{\tilde{\cH}^{(1)}(\bfq)}} & \operatorname{BP}_{\mathbf{\whG}}\ar[d]^{\pi_*} \\ \operatorname{Mod}(Z(\tilde{\cH}^{(1)}(\bfq))) \ar[r]^<<<<<S_>>>>>{\sim} & \operatorname{SP}_{\mathbf{\whG}}. }$$ The *generic parametrization functor* is the functor $$P:=S\circ \operatorname{Res}_{Z(\tilde{\cH}^{(1)}(\bfq))}^{\tilde{\cH}^{(1)}(\bfq)} = \pi_*\circ B:$$ $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq))\ar[d] & \\ \operatorname{SP}_{\mathbf{\whG}}. }$$ It follows from the definitions that for all $\gamma\in \bbT^{\vee}/W_0$, the fiber of $P$ over the direct factor $\operatorname{SP}_{\mathbf{\whG}}^{\gamma}\subset \operatorname{SP}_{\mathbf{\whG}}$ is the direct factor $\operatorname{Mod}(\tilde{\cH}^{\gamma}(\bfq))\subset \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq))$: $$P^{-1}(\operatorname{SP}_{\mathbf{\whG}}^{\gamma})=\operatorname{Mod}(\tilde{\cH}^{\gamma}(\bfq))\subset \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)).$$ Accordingly the parametrization functor $P$ decomposes as the product over the finite set $\bbT^{\vee}/W_0$ of functors $$\xymatrix{ P^{\gamma}:\operatorname{Mod}(\tilde{\cH}^{\gamma}(\bfq))\ar[r] & \operatorname{SP}_{\mathbf{\whG}}^{\gamma}. }$$ \[Strivv\] In the case of the trivial orbit $\gamma:=\{1\}$, it follows from \[Striv\] that $P^{\{1\}}$ factors as $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{\{1\}}(\bfq)) \ar[d]_{\operatorname{Res}^{\tilde{\cH}^{\{1\}}(\bfq)}_{\tilde{\cH}^{\operatorname{sph}}(\bfq)}} \ar[dr]^{P^{\{1\}}}& \\ \operatorname{Mod}(\tilde{\cH}^{\operatorname{sph}}(\bfq)) \ar[r]_<<<<<<{\sim}^>>>>{S^{\{1\}}} & \operatorname{SP}_{\mathbf{\whG}}^{\{1\}}. }$$ The generic antispherical module -------------------------------- Recall the generic regular and non-regular antispherical representations $\sA_2(\bfq)$ \[sA2q\] and $\sA_1(\bfq)$ \[sA1q\] of $\cH_2(\bfq)$ and $\cH_1(\bfq)$. Thanks to \[H2VSHgamma\] and \[H1VSHgamma\], they are models over $\bbZ$ of representations $\tilde{\sA}^{\gamma}(\bfq)$ of the regular and non-regular components $\tilde{\sA}^{\gamma}(\bfq)$, $\gamma\in\bbT^{\vee}/W_0$, of the generic pro-$p$-Iwahori Hecke algebra $\tilde{\cH}^{(1)}(\bfq)$ over $\tilde{\bbZ}$, cf. \[decompprop\] and \[AIH\]. Taking the product over $\bbT^{\vee}/W_0$ of these representations, we obtain a representation $$\xymatrix{ \tilde{\sA}^{(1)}(\bfq):\tilde{\cH}^{(1)}(\bfq) \ar[r] & \operatorname{End}_{Z(\tilde{\cH}^{(1)}(\bfq))}(\tilde{\cA}^{(1)}(\bfq)). }$$ By construction, the representation $\tilde{\sA}^{(1)}(\bfq)$ depends on a choice of order on each regular orbit $\gamma$. We call $\tilde{\sA}^{(1)}(\bfq)$ the *generic antispherical representation*, and the corresponding left $\tilde{\cH}^{(1)}(\bfq)$-module $\tilde{\cM}^{(1)}$ the *generic antispherical module*. \[propantisph\] 1. The generic antispherical representation is faithful. 2. The Bernstein parameter of the antispherical module is the structural sheaf: $$B(\cM^{(1)})=\cO_{\bbT^{\vee}\times V_{\mathbf{\whT}}}.$$ 3. The Satake parameter of the antispherical module is the $\tilde{R}(\bbT^{\vee}\times V_{\mathbf{\whG}})$-module of $\bbT^{\vee}\times V_{\mathbf{\whG}}$-equivariant $K$-theory of the flag variety of $\bbT^{\vee}\times V_{\mathbf{\whG}}$: $$\tilde{c}_{\bbT^{\vee}\times V_{\mathbf{\whG}}}:S(\cM^{(1)})\xrightarrow{\sim} \tilde{K}^{\bbT^{\vee}\times V_{\mathbf{\whG}}}(\bbT^{\vee}\times V_{\mathbf{\whG}}/\bbT^{\vee}\times V_{\mathbf{\whB}}).$$ Part 1. follows from \[sA2sf\] and \[sA1sf\], part 2. from the property *(i)* in \[sA2q\] and \[sA1q\], and part 3. from the characteristic isomorphism in \[B1qchar\]. Now, being a left $\tilde{\cH}^{(1)}(\bfq)$-module, the antispherical module $\tilde{\cM}^{(1)}$ defines a functor $$\xymatrix{ \tilde{\cM}^{(1)}\otimes_{Z(\tilde{\cH}^{(1)}(\bfq))}\bullet:\operatorname{Mod}(Z(\tilde{\cH}^{(1)}(\bfq)))\ar[r] & \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)). }$$ On the other hand, recall the canonical projection $\pi:\bbT^{\vee}\times V_{\mathbf{\whT}}\ra (\bbT^{\vee}\times V_{\mathbf{\whT}})/W_0$ from \[pi\]. Then point 2. of \[propantisph\] has the following consequence. The diagram $$\xymatrix{ \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)) \ar[r]^<<<<<<{B} & \operatorname{BP}_{\mathbf{\whG}} \\ \operatorname{Mod}(Z(\tilde{\cH}^{(1)}(\bfq))) \ar[r]^<<<<{S}_>>>>>>{\sim} \ar[u]^{ \tilde{\cM}^{(1)}\otimes_{Z(\tilde{\cH}^{(1)}(\bfq))}\bullet} & \operatorname{SP}_{\mathbf{\whG}} \ar[u]_{\pi^*} }$$ is commutative. The *generic antispherical functor* is the functor $$\operatorname{ASph}:= (\tilde{\cM}^{(1)}\otimes_{Z(\tilde{\cH}^{(1)}(\bfq))}\bullet)\circ S^{-1}:$$ $$\xymatrix{ \operatorname{SP}_{\mathbf{\whG}}\ar[r] & \operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq)). }$$ The diagram $$\xymatrix{ &&\operatorname{Mod}(\tilde{\cH}^{(1)}(\bfq))\ar[d]^{P} \\ \operatorname{SP}_{\mathbf{\whG}} \ar[urr]^{\operatorname{ASph}} \ar[r]_{\pi^*} & \operatorname{BP}_{\mathbf{\whG}}\ar[r]_{\pi_*}& \operatorname{SP}_{\mathbf{\whG}} }$$ is commutative. One has $P\circ \operatorname{ASph}= \pi_* \circ (B \circ \operatorname{ASph}) = \pi_* \circ \pi^*$ by the preceding corollary. By construction, the antispherical functor $\operatorname{ASph}$ decomposes as a product of functors $\operatorname{ASph}^{\gamma}$ for $\gamma\in \bbT^{\vee}/W_0$, and accordingly the previous diagram decomposes over $\bbT^{\vee}/W_0$. In particular for $\gamma=\{1\}$ we have the commutative diagram $$\xymatrix{ &&\operatorname{Mod}(\tilde{\cH}^{\{1\}}(\bfq))\ar[d]^{P^{\{1\}}} \ar[dr]^{\operatorname{Res}^{\tilde{\cH}^{\{1\}}(\bfq)}_{\tilde{\cH}^{\operatorname{sph}}(\bfq)}} \\ \operatorname{SP}_{\mathbf{\whG}}^{\{1\}} \ar[urr]^{\operatorname{ASph}^{\{1\}}} \ar[r]_{\pi^*} & \operatorname{BP}_{\mathbf{\whG}}^{\{1\}}\ar[r]_{\pi_*}& \operatorname{SP}_{\mathbf{\whG}}^{\{1\}} & \operatorname{Mod}(\tilde{\cH}^{\operatorname{sph}}(\bfq)). \ar[l]^>>>>>>>>{S^{\{1\}}}_>>>>>>>>{\sim} }$$ The theory at $\bfq=q=0$ ======================== We keep all the notations introduced in the preceding section. In particular, $k=\overline{\bbF}_q$. Geometric representation theory at $\bfq=0$ ------------------------------------------- Recall from \[Vinsubsection\] the $k$-semigroup scheme $$V_{\mathbf{GL_2},0}=\operatorname{Sing}_ {2\times 2}\times\bbG_m,$$ which can even be defined over $\bbZ$, and which is obtained as the $0$-fiber of $$\xymatrix{ V_{\mathbf{GL_2}}\ar[d]^{\bfq} \\ \bbA^1. }$$ It admits $$V_{\mathbf{\whT},0}=\operatorname{SingDiag}_ {2\times 2}\times\bbG_m$$ as a commutative subsemigroup scheme. The latter has the following structure: it is the pinching of the monoids $$\bbA_X^1\times\bbG_m:=\operatorname{Spec}(k[X,z_2^{\pm1}]) \quad \textrm{and} \quad \bbA_Y^1\times\bbG_m:=\operatorname{Spec}(k[Y,z_2^{\pm1}])$$ along the sections $X=0$ and $Y=0$. These monoids are semisimple, with representation rings $$R(\bbA_X^1\times\bbG_m)=\bbZ[X,z_2^{\pm1}] \quad \textrm{and} \quad R(\bbA_Y^1\times\bbG_m)=\bbZ[Y,z_2^{\pm1}].$$ There are three remarkable elements in $V_{\mathbf{\whT},0}$, namely $$\varepsilon_X:=(\operatorname{diag}(1,0),1),\quad \varepsilon_Y:=(\operatorname{diag}(0,1),1)\quad\textrm{and}\quad\varepsilon_0:=(\operatorname{diag}(0,0),1).$$ They are idempotents. Now let $M$ be a finite dimensional $k$-representation of $V_{\mathbf{\whT},0}$. The idempotents act on $M$ as projectors, and as the semigroup $V_{\mathbf{\whT},0}$ is commutative, the $k$-vector space $M$ decomposes as a direct sum $$M=\bigoplus_{(\lambda_X,\lambda_Y,\lambda_0)\in\{0,1\}^3} M(\lambda_X,\lambda_Y,\lambda_0)$$ where $$M(\lambda_X,\lambda_Y,\lambda_0)=\{m\in M\ |\ m\varepsilon_X=\lambda_Xm,\ m\varepsilon_Y=\lambda_Ym,\ m\varepsilon_0=\lambda_0m\}.$$ Moreover, since $V_{\mathbf{\whT},0}$ is commutative, each of these subspaces is in fact a subrepresentation of $M$. As $\varepsilon_X\varepsilon_Y=\varepsilon_0\in V_{\mathbf{\whT},0}$, we have $M(1,1,0)=0$. Next, as $\varepsilon_X$ is the unit of the monoid $\bbA_X^1$, if $\lambda_X=0$ then $\operatorname{Res}^{V_{\mathbf{\whT},0}}_{\bbA_X^1}M(\lambda_X,\lambda_Y,\lambda_0)$ must be the null representation, in particular we must have $\lambda_0=0$; hence $M(0,0,1)=M(0,1,1)=0$. Considering $\varepsilon_Y$ instead of $\varepsilon_X$, we get similarly that $M(0,0,1)=M(1,0,1)=0$. Consequently $$M=M(1,0,0)\bigoplus M(0,1,0)\bigoplus M(1,1,1)\bigoplus M(0,0,0).$$ The restriction $\operatorname{Res}^{V_{\mathbf{\whT},0}}_{\bbA_X^1}M(1,0,0)$ is a representation of the monoid $\bbA_X^1$ where $0$ acts by $0$, and $\operatorname{Res}^{V_{\mathbf{\whT},0}}_{\bbA_Y^1}M(1,0,0)$ is the null representation. Hence, if for $n>0$ we still denote by $X^n$ the character of $V_{\mathbf{\whT},0}$ which restricts to the character $X^n$ of $\bbA_X^1\times\bbG_m$ and the null map of $\bbA_Y^1\times\bbG_m$, then $M(1,0,0)$ decomposes as a sum of weight spaces $$M(1,0,0)=\oplus_{n>0}M(X^n):=\oplus_{n>0,m\in\bbZ}M(X^nz_2^m).$$ Similarly $$M(0,1,0)=\oplus_{n>0}M(Y^n):=\oplus_{n>0,m\in\bbZ}M(Y^nz_2^m).$$ Finally, $V_{\mathbf{\whT},0}$ acts through the projection $V_{\mathbf{\whT},0}\ra\bbG_m$ on $$M(1,1,1)=:M(1)=\oplus_{m\in\bbZ}M(z_2^m),$$ and by $0$ on $$M(0,0,0)=:M(0).$$ Thus we have obtained the following \[RVT0\] The category $\operatorname{Rep}(V_{\mathbf{\whT},0})$ is semisimple, and there is a ring isomorphism $$R(V_{\mathbf{\whT},0})\cong\big(\bbZ[X,Y,z_2^{\pm1}]/(XY)\big)\times\bbZ.$$ Next let $$V_{\mathbf{\whB},0}=\operatorname{SingUpTriang}_ {2\times 2}\times\bbG_m\subset V_{\mathbf{GL_2},0}=\operatorname{Sing}_{2\times 2}\bbG_m$$ be the subsemigroup scheme of singular upper triangular $2\times 2$-matrices. It contains $V_{\mathbf{\whT},0}$, and the inclusion $V_{\mathbf{\whT},0}\subset V_{\mathbf{\whB},0}$ admits a retraction $V_{\mathbf{\whB},0}\ra V_{\mathbf{\whT},0}$, namely the specialisation at $\bfq=0$ of the retraction \[retraction\]. Let $M$ be an object of $\operatorname{Rep}(V_{\mathbf{\whB},0})$. Write $$\operatorname{Res}^{V_{\mathbf{\whB},0}}_{V_{\mathbf{\whT},0}}M=M(1,0,0)\oplus M(0,1,0)\oplus M(1)\oplus M(0).$$ For a subspace $N\subset M$, consider the following property: (P${}_N$) *the subspace $N\subset M$ is a subrepresentation, and $V_{\mathbf{\whB},0}$ acts on $N$ through the retraction of $k$-semigroup schemes $V_{\mathbf{\whB},0}\ra V_{\mathbf{\whT},0}$*. Let us show that (P${}_{M(0,1,0)}$) is true. Indeed for $m\in M(0,1,0)=\oplus_{n>0}M(Y^n)$, we have $$m\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) = (m\varepsilon_Y)\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) =m\varepsilon_0=0 =m\left (\begin{array}{cc} x & 0\\ 0 & 0 \end{array} \right)$$ and $$m\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) = (m\varepsilon_Y)\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) =m\left (\begin{array}{cc} 0 & 0\\ 0 & y \end{array} \right).$$ Next assume $M(0,1,0)=0$, and let us show that in this case (P${}_{M(0)}$) is true. Indeed for $m\in M(0)$, we have $$m\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) = m\bigg(\varepsilon_X\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right)\bigg) =(m\varepsilon_X)\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) =0,$$ and if we decompose $$m':=m\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) =m_{(1,0,0)}'+m_1'+m_0'\in M(1,0,0)\oplus M(1)\oplus M(0),$$ then by applying $\varepsilon_X$ on the right we see that $0=m_{(1,0,0)}'+m_1'$ so that $m'\in M(0)$ and hence $$m\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) = m\bigg(\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right)\varepsilon_Y\bigg) =m'\varepsilon_Y=0.$$ Next assume $M(0,1,0)=M(0)=0$, and let us show that in this case (P${}_{M(1,0,0)}$) is true. Indeed, let $m\in M(1,0,0)=\oplus_{n>0}M(X^n)$. Then for any $c\in k$, $$m':=m\left (\begin{array}{cc} 0 & c\\ 0 & 0 \end{array} \right)$$ satisfies $m'\varepsilon_X=0$, $m'\varepsilon_Y=m'$, $m'\varepsilon_0=0$, i.e. $m'\in M(0,1,0)$, and hence is equal to $0$ by our assumption. It follows that $$m\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) = (m\varepsilon_X)\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) = m\bigg(\varepsilon_X\left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right)\bigg) =m\left (\begin{array}{cc} 0 & c\\ 0 & 0 \end{array} \right) =0 =m\left (\begin{array}{cc} 0 & 0\\ 0 & y \end{array} \right).$$ On the other hand, if we decompose $$m':=m \left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) =m_{(1,0,0)}'+m_1'\in M(1,0,0)\oplus M(1),$$ then by applying $\varepsilon_0$ on the right we find $0=m_1'$, i.e. $m'\in M(1,0,0)$ and hence $$m \left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) =m'=m'\varepsilon_X=m\bigg(\left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right)\varepsilon_X\bigg) = m\left (\begin{array}{cc} x & 0\\ 0 & 0 \end{array} \right).$$ Finally assume $M(0,1,0)=M(0)=M(1,0,0)=0$, and let us show that in this case (P${}_{M(1)}$) is true, i.e. that $V_{\mathbf{\whB},0}$ acts through the projection $V_{\mathbf{\whB},0}\ra\bbG_m$ on $M=M(1)$. Indeed for any $m$ we have $$m \left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right) = \bigg(m \left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right)\bigg)\varepsilon_0 = m\bigg( \left (\begin{array}{cc} x & c\\ 0 & 0 \end{array} \right)\varepsilon_0\bigg) =m\varepsilon_0=m$$ and $$m \left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right) = \bigg(m \left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right)\bigg)\varepsilon_0 = m\bigg( \left (\begin{array}{cc} 0 & c\\ 0 & y \end{array} \right)\varepsilon_0\bigg) =m\varepsilon_0=m.$$ It follows from the preceding discussion that the irreducible representations of $V_{\mathbf{\whB},0}$ are the characters, which are inflated from those of $V_{\mathbf{\whT},0}$ through the retraction $V_{\mathbf{\whB},0}\ra V_{\mathbf{\whT},0}$. As a consequence, considering the *restriction* and *inflation* functors $$\xymatrix{ \operatorname{Res}_{V_{\mathbf{\whT},0}}^{V_{\mathbf{\whB},0}}:\operatorname{Rep}(V_{\mathbf{\whB},0}) \ar@<1ex>[r] & \operatorname{Rep}(V_{\mathbf{\whT},0}) :\operatorname{Infl}_{V_{\mathbf{\whT},0}}^{V_{\mathbf{\whB},0}}, \ar@<1ex>[l] }$$ which are exact and compatible with tensor products and units, we get: \[ResInfl0\] The ring homomorphisms $$\xymatrix{ \operatorname{Res}_{V_{\mathbf{\whT},0}}^{V_{\mathbf{\whB},0}}:R(V_{\mathbf{\whB},0}) \ar@<1ex>[r] & R(V_{\mathbf{\whT},0}) :\operatorname{Infl}_{V_{\mathbf{\whT},0}}^{V_{\mathbf{\whB},0}}, \ar@<1ex>[l] }$$ are isomorphisms, which are inverse one to the other. Finally, note that $\varepsilon_0\in V_{\mathbf{GL_2}}(k)$ belongs to all the left $V_{\mathbf{GL_2}}(k)$-cosets in $V_{\mathbf{GL_2}}(k)$. Hence, by \[Indsemi\], the catgory $\operatorname{Rep}(V_{\mathbf{\whB},0})$ is equivalent to the one of induced vector bundles on the semigroupoid flag variety $V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}$: $$\xymatrix{ \cI nd_{V_{\mathbf{\whB},0}}^{V_{\mathbf{GL_2},0}}:\operatorname{Rep}(V_{\mathbf{\whB},0}) \ar[r]^<<<<<{\sim} & \cC_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}) \subset \cC^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}). }$$ We have a ring isomorphism $$\xymatrix{ \cI nd_{V_{\mathbf{\whB},0}}^{V_{\mathbf{GL_2},0}}\circ \operatorname{Infl}_{V_{\mathbf{\whT},0}}^{V_{\mathbf{\whB},0}} : R(V_{\mathbf{\whT},0}) \ar[r]^<<<<<{\sim} & K_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}). }$$ \[defrel\] We call *relevant* the full subcategory $$\operatorname{Rep}(V_{\mathbf{\whT},0})^{\operatorname{rel}}\subset \operatorname{Rep}(V_{\mathbf{\whT},0})$$ whose objects $M$ satisfy $M(0)=0$. Correspondingly, we have relevant full subcategories $$\operatorname{Rep}(V_{\mathbf{\whB},0})^{\operatorname{rel}}\subset \operatorname{Rep}(V_{\mathbf{\whB},0})\quad\textrm{and}\quad \cC_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0})^{\operatorname{rel}}\subset\cC_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}).$$ We have a ring isomorphism $$\xymatrix{ c_{V_{\mathbf{GL_2},0}}:=\bbZ[X,Y,z_2^{\pm1}]/(XY)\cong R(V_{\mathbf{\whT},0})^{\operatorname{rel}} \ar[r]^<<<<<{\sim} & K_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0})^{\operatorname{rel}}, }$$ that we call *the characteristic isomorphism in the equivariant $K$-theory of the flag variety $V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0}$*. We have a commutative diagram *specialization at $\bfq=0$* $$\xymatrix{ \bbZ[X,Y,z_2^{\pm1}] \ar[rr]^{c_{V_{\mathbf{GL_2}}}}_{\sim} \ar@{->>}[d] && K^{V_{\mathbf{GL_2}}}(V_{\mathbf{GL_2}}/V_{\mathbf{\whB}}) \ar@{->>}[d]\\ \bbZ[X,Y,z_2^{\pm1}]/(XY)\ar[rr]^{c_{V_{\mathbf{GL_2},0}}}_{\sim} && K_{\cI nd}^{V_{\mathbf{GL_2},0}}(V_{\mathbf{GL_2},0}/V_{\mathbf{\whB},0})^{\operatorname{rel}}, }$$ where the vertical right-hand side map is given by restricting equivariant vector bundles to the $0$-fiber of $\bfq: V_{\mathbf{GL_2}}\ra\bbA^1$. The mod $p$ Satake and Bernstein isomorphisms --------------------------------------------- In the sequel, we will denote by $(\bullet)_{\overline{\bbF}_q}$ the *specialization at $\bfq=q=0$*, i.e. the base change functor along the ring morphism $$\begin{aligned} \bbZ[\bfq] & \lra & \overline{\bbF}_q=:k\\ \bfq & \lmapsto & 0.\end{aligned}$$ Also we fix an embedding $\mu_{q-1}\subset\overline{\bbF}_q^{\times}$, so that the above morphism factors through the inclusion $\bbZ[\bfq] \subset \tilde{\bbZ}[\bfq]$, where $\bbZ\subset\tilde{\bbZ}$ is the ring extension considered in \[finiteT\]. **The mod $p$ Satake and pro-$p$-Iwahori Satake isomorphisms.** Specializing \[DefgenSat\], we get an isomorphism of $\overline{\bbF}_q$-algebras $$\xymatrix{ \sS_{\overline{\bbF}_q}:\cH_{\overline{\bbF}_q}^{\operatorname{sph}}\ar[r]^<<<<<{\sim} & \overline{\bbF}_q[V_{\mathbf{\whT},0}]^{W_0}=\big(\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY)\big)^{W_0}. }$$ In [@H11], Herzig constructed an isomorphism $$\xymatrix{ \sS_{\operatorname{Her}}:\cH_{\overline{\bbF}_q}^{\operatorname{sph}}\ar[r]^<<<<<{\sim} & \overline{\bbF}_q[\bbX^{\bullet}(\mathbf{\whT})_-]=\overline{\bbF}_q[e^{(0,1)},e^{\pm(1,1)}] }$$ (this is $\overline{\bbF}_q\otimes_{\bbZ}\cS'$, with the notation $\cS'$ from \[Remiota\]). They are related by the Steinberg choice of coordinates $z_1:=X+Y$ and $z_2$ on the quotient $V_{\mathbf{\whT},0}/W_0$, cf. \[compgenBSiso\], i.e. by the following commutative diagram $$\xymatrix{ \cH_{\overline{\bbF}_q}^{\operatorname{sph}}\ar[rr]_<<<<<<<<<<<<<<<<<<<<<<{\sim}^<<<<<<<<<<<<<<<<<<<<<<{\sS_{\overline{\bbF}_q}} \ar[dr]_{\sS_{\operatorname{Her}}}^{\sim} && \big(\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY)\big)^{W_0} \\ &\overline{\bbF}_q[e^{(0,1)},e^{\pm(1,1)}] \ar[ur]_{\quad \quad e^{(0,1)}\mapsto z_1,\ e^{(1,1)}\mapsto z_2}^{\sim}. & }$$ Specializing \[DefgenSat1\], we get an isomorphism of $\overline{\bbF}_q$-algebras $$\xymatrix{ \sS_{\overline{\bbF}_q}^{(1)}:(\cA_{\overline{\bbF}_q}^{(1)})^{W_0}\ar[r]^<<<<<{\sim} & \overline{\bbF}_q[\bbT^{\vee}\times V_{\mathbf{\whT},0}]^{W_0}=\big(\overline{\bbF}_q[\bbT][X,Y,z_2^{\pm1}]/(XY)\big)^{W_0}. }$$ \[B1qcharmodp\] **The mod $p$ Bernstein isomorphism.** Specializing \[genB1\], we get an isomorphism of $\overline{\bbF}_q$-algebras $$\xymatrix{ \sB_{\overline{\bbF}_q}^{(1)}:\cA_{\overline{\bbF}_q}^{(1)}\ar[r]^<<<<<{\sim} & \overline{\bbF}_q[\bbT^{\vee}\times V_{\mathbf{\whT},0}]=\overline{\bbF}_q[\bbT][X,Y,z_2^{\pm1}]/(XY). }$$ Moreover, similarly as in \[B1qchar\] but here using \[ResInfl0\] and \[ringsemigr\], we get the *characteristic isomorphism* $$\xymatrix{ c_{\bbT^{\vee}\times V_{\mathbf{\whG},0}}:R(\bbT^{\vee}\times V_{\mathbf{\whT},0}) \ar[r]^<<<<<{\sim} & K_{\cI nd}^{\bbT^{\vee}\times V_{\mathbf{\whG},0}}(\bbT^{\vee}\times V_{\mathbf{\whG},0}/\bbT^{\vee}\times V_{\mathbf{\whB},0}). }$$ Whence by \[RVT0\] (and recalling \[defrel\]) an isomorphism $$\xymatrix{ c_{\bbT^{\vee}\times V_{\mathbf{\whG},0}\overline{\bbF}_q}^{\operatorname{rel}}\circ \sB_{\overline{\bbF}_q}^{(1)}:\cA_{\overline{\bbF}_q}^{(1)} \ar[r]^<<<<<{\sim} & K_{\cI nd,\overline{\bbF}_q}^{\bbT^{\vee}\times V_{\mathbf{\whG},0}}(\bbT^{\vee}\times V_{\mathbf{\whG},0}/\bbT^{\vee}\times V_{\mathbf{\whB},0})^{\operatorname{rel}}. }$$ Also, specializing \[SpecB1q\], $\sB_{\overline{\bbF}_q}^{(1)}$ splits as a product over $\bbT^{\vee}$ of $\overline{\bbF}_q$-algebras isomorphisms $\sB_{\overline{\bbF}_q}^{\lambda}$, each of them being of the form $$\xymatrix{ \sB_{1,\overline{\bbF}_q}:\cA_{1,\overline{\bbF}_q}\ar[r]^<<<<<{\sim} & \overline{\bbF}_q[V_{\mathbf{\whT},0}]=\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY). }$$ \[sZ1isomodp\] **The mod $p$ central elements embedding.** Specializing \[sZ1def\], we get an embedding of $\overline{\bbF}_q$-algebras $$\xymatrix{ \sZ_{1,\overline{\bbF}_q}:\cH_{\overline{\bbF}_q}^{\operatorname{sph}}\ar[r]^<<<<<{\sim} & Z(\cH_{1,\overline{\bbF}_q}) \subset \cA_{1,\overline{\bbF}_q} \subset \cH_{1,\overline{\bbF}_q} }$$ making the diagram $$\xymatrix{ \cA_{1,\overline{\bbF}_q} \ar[rr]_<<<<<<<<<<<{\sim}^<<<<<<<<<<<{\sB_{1,\overline{\bbF}_q}} && \overline{\bbF}_q[V_{\mathbf{\whT},0}]=\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY) \\ \cH^{\operatorname{sph}}_{\overline{\bbF}_q} \ar@{^{(}->}[u]^{\sZ_{1,\overline{\bbF}_q}} \ar[rr]_<<<<<<<<<{\sim}^<<<<<<<<<{\sS_{\overline{\bbF}_q} } && \overline{\bbF}_q[V_{\mathbf{\whT},0}]^{W_0}=\big(\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY)\big)^{W_0}\ar@{^{(}->}[u] }$$ commutative. Then $\sZ_{1,\overline{\bbF}_q}$ coincides with the central elements construction of Ollivier [@O14 Th. 4.3] for the case of $\mathbf{GL_2}$. This follows from the explicit formulas for the values of $\sZ_{1}(\bfq)$ on $T_{ (1,0) }$ and $T_{(1,1) }$, cf. \[sZ1iso\]. The mod $p$ parametrization --------------------------- The category of quasi-coherent modules on the $k$-scheme $(\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0$ will be called the *category of mod $p$ Satake parameters*, and denoted by $\operatorname{SP}_{\mathbf{\whG},0}$: $$\operatorname{SP}_{\mathbf{\whG},0}:=\operatorname{QCoh}\bigg((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\bigg).$$ For $\gamma \in \bbT^{\vee}/W_0$, we also define $\operatorname{SP}^{\gamma}_{\mathbf{\whG},0}:=\operatorname{QCoh}\bigg(( \coprod_{\lambda\in\gamma} V_{\mathbf{\whT},0})/W_0 \bigg)$. Similarly to the generic case \[genericpara\], the mod $p$ pro-$p$-Iwahori Satake isomorphism induces an equivalence of categories $$\xymatrix{ S:\operatorname{Mod}(Z(\cH_{\overline{\bbF}_q}^{(1)})) \ar[r]^<<<<<{\sim} & \operatorname{SP}_{\mathbf{\whG},0}, }$$ that will be referred to as the *functor of mod $p$ Satake parameters*, and which decomposes as a product over the finite set $\bbT^{\vee}/W_0$: $$\xymatrix{ S=\prod_{\gamma}S^{\gamma}:\prod_{\gamma}\operatorname{Mod}(Z(\cH_{\overline{\bbF}_q}^{\gamma}))\ar[r]^<<<<<{\sim} & \prod_{\gamma}\operatorname{SP}_{\mathbf{\whG},0}^{\gamma}\simeq \prod_{\gamma\ \operatorname{reg}}\operatorname{QCoh}(V_{\mathbf{\whT},0})\prod_{\gamma\ \operatorname{non-reg}}\operatorname{QCoh}(V_{\mathbf{\whT},0}/W_0). }$$ For $\gamma=\{1\}$ and using \[sZ1isomodp\] we get an equivalence $$\xymatrix{ S^{\{1\}}:\operatorname{Mod}(\cH_{\overline{\bbF}_q}^{\operatorname{sph}}) \ar[r]^<<<<<{\sim} & \operatorname{SP}_{\mathbf{\whG},0}^{\{1\}}=\operatorname{QCoh}(V_{\mathbf{\whT},0}/W_0). }$$ Note that under this equivalence, the characters $\cH_{\overline{\bbF}_q}^{\operatorname{sph}}\ra \overline{\bbF}_q$ correspond to the skyscraper sheaves on $V_{\mathbf{\whT},0}/W_0$, and hence to its $k$-points. Choosing the Steinberg coordinates $(z_1,z_2)$ on the $k$-scheme $V_{\mathbf{\whT},0}/W_0$, they may also be regarded as the $k$-points of $\operatorname{Spec}(k[\bbX^{\bullet}(\mathbf{\whT})_-])$, which are precisely the mod $p$ Satake parameters defined by Herzig in [@H11]. The category of quasi-coherent modules on the $k$-scheme $\bbT^{\vee}\times V_{\mathbf{\whT},0}$ will be called the *category of mod $p$ Bernstein parameters*, and denoted by $\operatorname{BP}_{\mathbf{\whG},0}$: $$\operatorname{BP}_{\mathbf{\whG},0}:=\operatorname{QCoh}(\bbT^{\vee}\times V_{\mathbf{\whT},0}).$$ Similarly to the generic case \[genericpara\], the inclusion $\cH^{(1)}_{\overline{\bbF}_q}\supset \cA^{(1)}_{\overline{\bbF}_q}$ together with the mod $p$ Bernstein isomorphism define a *functor of mod $p$ Bernstein parameters* $$\xymatrix{ B:\operatorname{Mod}(\cH^{(1)}_{\overline{\bbF}_q}) \ar[r] & \operatorname{BP}_{\mathbf{\whG},0}. }$$ Moreover the category $\operatorname{BP}_{\mathbf{\whG},0}$ decomposes as a product over the finite group $\bbT^{\vee}$: $$\operatorname{BP}_{{\mathbf{\whG}},0}=\prod_{\lambda} \operatorname{BP}_{{\mathbf{\whG}},0}^{\lambda}=\prod_{\lambda}\operatorname{QCoh}(V_{\mathbf{\whT},0}).$$ Let $\pi:\bbT^{\vee}\times V_{\mathbf{\whT},0}\ra (\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0$ be the canonical projection. The *mod $p$ parametrization functor* is the functor $$P:=S\circ \operatorname{Res}_{Z(\cH^{(1)}_{\overline{\bbF}_q})}^{\cH^{(1)}_{\overline{\bbF}_q}} = \pi_*\circ B:$$ $$\xymatrix{ \operatorname{Mod}(\cH^{(1)}_{\overline{\bbF}_q})\ar[d] & \\ \operatorname{SP}_{\mathbf{\whG},0}. }$$ The functor $P$ decomposes as a product over the finite set $\bbT^{\vee}/W_0$: $$\xymatrix{ P=\prod_{\gamma}P^{\gamma}:\prod_{\gamma}\operatorname{Mod}(\cH_{\overline{\bbF}_q}^{\gamma})\ar[r]^<<<<<{\sim} & \prod_{\gamma}\operatorname{SP}_{\mathbf{\whG},0}^{\gamma}. }$$ In the case of the trivial orbit $\gamma:=\{1\}$, $P^{\{1\}}$ factors as $$\xymatrix{ \operatorname{Mod}(\cH^{\{1\}}_{\overline{\bbF}_q}) \ar[d]_{\operatorname{Res}^{\cH^{\{1\}}_{\overline{\bbF}_q}}_{\cH^{\operatorname{sph}}_{\overline{\bbF}_q}}} \ar[dr]^{P^{\{1\}}}& \\ \operatorname{Mod}(\cH^{\operatorname{sph}}_{\overline{\bbF}_q}) \ar[r]_<<<<<{\sim}^>>>>>{S^{\{1\}}} & \operatorname{SP}_{\mathbf{\whG},0}^{\{1\}}. }$$ The mod $p$ antispherical module -------------------------------- We call $$\xymatrix{ \sA_{\overline{\bbF}_q}^{(1)}:\cH^{(1)}_{\overline{\bbF}_q} \ar[r] & \operatorname{End}_{Z(\cH^{(1)}_{\overline{\bbF}_q})}(\cA^{(1)}_{\overline{\bbF}_q}) }$$ the *mod $p$ antispherical representation*, and the corresponding left $\cH^{(1)}_{\overline{\bbF}_q}$-module $\cM^{(1)}_{\overline{\bbF}_q}$ the *mod $p$ antispherical module*. \[propantisphmodp\] 1. The mod $p$ antispherical representation is faithful. 2. The mod $p$ Bernstein parameter of the antispherical module is the structural sheaf: $$B(\cM^{(1)}_{\overline{\bbF}_q})=\cO_{\bbT^{\vee}\times V_{\mathbf{\whT},0}}.$$ 3. The mod $p$ Satake parameter of the antispherical module is the $R_{\overline{\bbF}_q}(\bbT^{\vee}\times V_{\mathbf{\whT},0})^{\operatorname{rel},W_0}$-module of the relevant induced $\bbT^{\vee}\times V_{\mathbf{\whG,0}}$-equivariant $K_{\overline{\bbF}_q}$-theory of the flag variety of $\bbT^{\vee}\times V_{\mathbf{\whG},0}$: $$c_{\bbT^{\vee}\times V_{\mathbf{\whG},0}\overline{\bbF}_q}^{\operatorname{rel}}:S(\cM^{(1)}_{\overline{\bbF}_q})\xrightarrow{\sim} K^{\bbT^{\vee}\times V_{\mathbf{\whG},0}}_{\cI nd,\overline{\bbF}_q}(\bbT^{\vee}\times V_{\mathbf{\whG},0}/\bbT^{\vee}\times V_{\mathbf{\whB},0})^{\operatorname{rel}}.$$ Part 1. follows from \[sA2sf\] and \[faithfulatzero\], part 2. from the property *(i)* in \[sA2q\] and \[sA1q\], and part 3. from the characteristic isomorphism in \[B1qcharmodp\]. The diagram $$\xymatrix{ \operatorname{Mod}(\cH^{(1)}_{\overline{\bbF}_q}) \ar[r]^<<<<<<{B} & \operatorname{BP}_{\mathbf{\whG},0} \\ \operatorname{Mod}(Z(\cH^{(1)}_{\overline{\bbF}_q})) \ar[r]^<<<<{S}_>>>>>>{\sim} \ar[u]^{ \cM^{(1)}_{\overline{\bbF}_q}\otimes_{Z(\cH^{(1)}_{\overline{\bbF}_q})}\bullet} & \operatorname{SP}_{\mathbf{\whG},0} \ar[u]_{\pi^*} }$$ is commutative. The *mod $p$ antispherical functor* is the functor $$\operatorname{ASph}:= (\cM^{(1)}_{\overline{\bbF}_q}\otimes_{Z(\cH^{(1)}_{\overline{\bbF}_q})}\bullet)\circ S^{-1}:$$ $$\xymatrix{ \operatorname{SP}_{\mathbf{\whG},0}\ar[r] & \operatorname{Mod}(\cH^{(1)}_{\overline{\bbF}_q}). }$$ \[PASph\] The diagram $$\xymatrix{ &&\operatorname{Mod}(\cH^{(1)}_{\overline{\bbF}_q})\ar[d]^{P} \\ \operatorname{SP}_{\mathbf{\whG},0} \ar[urr]^{\operatorname{ASph}} \ar[r]_{\pi^*} & \operatorname{BP}_{\mathbf{\whG},0}\ar[r]_{\pi_*}& \operatorname{SP}_{\mathbf{\whG},0} }$$ is commutative. The antispherical functor $\operatorname{ASph}$ decomposes as a product of functors $\operatorname{ASph}^{\gamma}$ for $\gamma\in \bbT^{\vee}/W_0$, and accordingly the previous diagram decomposes over $\bbT^{\vee}/W_0$. In particular for $\gamma=\{1\}$ we have the commutative diagram $$\xymatrix{ &&\operatorname{Mod}(\cH^{\{1\}}_{\overline{\bbF}_q})\ar[d]^{P^{\{1\}}} \ar[dr]^{\operatorname{Res}^{\cH^{\{1\}}_{\overline{\bbF}_q}}_{\cH^{\operatorname{sph}}_{\overline{\bbF}_q}}} \\ \operatorname{SP}_{\mathbf{\whG},0}^{\{1\}} \ar[urr]^{\operatorname{ASph}^{\{1\}}} \ar[r]_{\pi^*} & \operatorname{BP}_{\mathbf{\whG},0}^{\{1\}}\ar[r]_{\pi_*}& \operatorname{SP}_{\mathbf{\whG},0}^{\{1\}} & \operatorname{Mod}(\cH^{\operatorname{sph}}_{\overline{\bbF}_q}). \ar[l]^>>>>>>>>{S^{\{1\}}}_>>>>>>>>{\sim} }$$ Now, identifying the $k$-points of the $k$-scheme $(\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0$ with the skyscraper sheaves on it, the antispherical functor $\operatorname{ASph}$ induces a map $$\xymatrix{ \operatorname{ASph}:\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)(k)\ar[r] & \{\textrm{left $\cH^{(1)}_{\overline{\bbF}_q}$-modules}\}. }$$ Considering the decomposition of $(\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0$ into its connected components: $$(\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0=\coprod_{\gamma\in (\bbT^{\vee}/W_0)}\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}\simeq\coprod_{\gamma\in (\bbT^{\vee}/W_0)_{\operatorname{reg}}}V_{\mathbf{\whT},0}\coprod_{\gamma\in (\bbT^{\vee}/W_0)_{\operatorname{non-reg}}}V_{\mathbf{\whT},0}/W_0,$$ the antispherical map decomposes as a disjoint union of maps $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}(k)\simeq V_{\mathbf{\whT},0}(k)\ar[r] & \{\textrm{left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}& \textrm{for $\gamma$ regular}, }$$ $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}(k)\simeq(V_{\mathbf{\whT},0}/W_0)(k)\ar[r] & \{\textrm{left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}& \textrm{for $\gamma$ non-regular}. }$$ In the regular case, we make the standard choice of coordinates $$V_{\mathbf{\whT},0}(k)=\bigg(\{(x,0)\ |\ x\in k\}\coprod_{(0,0)}\{(0,y)\ |\ y\in k\}\bigg)\times \{z_2\in k^{\times}\}$$ and we identify $\cH^{\gamma}_{\overline{\bbF}_q}$ with $\cH_{2,\overline{\bbF}_q}$ using \[H2VSHgamma\]. A point $v\in V_{\mathbf{\whT},0}(k)$ corresponds to a character $$\theta_v:Z(\cH_{2,\overline{\bbF}_q})\simeq\overline{\bbF}_q[X,Y,z_2^{\pm1}]/(XY)\lra\overline{\bbF}_q,$$ and then $\operatorname{ASph}^{\gamma}(v)$ identifies with the central reduction $$\cA_{2,\theta_v}:=\cA_{2,\overline{\bbF}_q}\otimes_{Z(\cH_{2,\overline{\bbF}_q}),\theta_v}\overline{\bbF}_q$$ of the mod $p$ regular antispherical representation $\sA_{2,\overline{\bbF}_q}$ specializing \[sA2q\]. The latter being an isomorphism by \[sA2sf\], so is $$\xymatrix{ \sA_{2,\theta_v}:\cH_{2,\theta_v} \ar[r]^<<<<{\sim} & \operatorname{End}_{\overline{\bbF}_q}(\cA_{2,\theta_v}). }$$ Consequently $\cH_{2,\theta_v}$ is a matrix algebra and $\cA_{2,\theta_v}$ is the unique simple finite dimensional left $\cH_{2, \overline{\bbF_q}}$-module with central character $\theta_v$, up to isomorphism. It is the *standard module with character $\theta_v$*, with *standard basis* $\{\varepsilon_1,\varepsilon_2\}$ (in particular its $\overline{\bbF}_q$-dimension is $2$). Conversely, any simple finite dimensional $\cH_{2, \overline{\bbF}_q}$-module has a central character, by Schur’s lemma. Following [@V04], a central character $\theta$ is called [*supersingular*]{} if $\theta(X+Y)=0$, and the standard module with character $\theta$ is called [*supersingular*]{} if $\theta$ is. Since $XY=0$, one has $\theta(X+Y)=0$ if and only if $\theta(X)=\theta(Y)=0$. \[ASphreg\] Let $\gamma\in \bbT^{\vee}/W_0$ regular. Then the antispherical map induces a bijection $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{simple finite dimensional left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}/\sim. }$$ The singular locus of the parametrizing $k$-scheme $$\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}$$ is given by $(0,0)\times\bbG_m\subset V_{\mathbf{\whT},0}$ in the standard coordinates, and its $k$-points correspond to the supersingular Hecke modules through the correspondence $\operatorname{ASph}^{\gamma}$. In the non-regular case, we make the Steinberg choice of coordinates $$(V_{\mathbf{\whT},0}/W_0)(k)=\{z_1\in k\}\times \{z_2\in k^{\times}\}$$ and we identify $\cH^{\gamma}_{\overline{\bbF}_q}$ with $\cH_{1,\overline{\bbF}_q}$ using \[H1VSHgamma\]. A point $v\in (V_{\mathbf{\whT},0}/W_0)(k)$ corresponds to a character $$\theta_v:Z(\cH_{1,\overline{\bbF}_q})\simeq\overline{\bbF}_q[z_1,z_2^{\pm1}]\lra\overline{\bbF}_q,$$ and then $\operatorname{ASph}^{\gamma}(v)$ identifies with the central reduction $$\cA_{1,\theta_v}:=\cA_{1,\overline{\bbF}_q}\otimes_{Z(\cH_{1,\overline{\bbF}_q}),\theta_v}\overline{\bbF}_q$$ of the mod $p$ non-regular antispherical representation $\sA_{1,\overline{\bbF}_q}$ specializing \[sA1q\]. Now recall from [@V04 1.4] the classification of the simple finite dimensional $\cH_{1,\overline{\bbF}_q}$-modules: they are the characters and the simple standard modules. The characters $$\cH_{1,\overline{\bbF}_q}= \overline{\bbF}_q [S,U^{\pm 1}]\lra \overline{\bbF}^\times_q$$ are parametrized by the set $\{0,-1\}\times \overline{\bbF}^\times_q$ via evaluation on the elements $S$ and $U$. On the other hand, given $v=(z_1,z_2)\in k\times k^{\times}=\overline{\bbF}_q \times \overline{\bbF}_q^{\times}$, a [*standard module with character $\theta_v$*]{} over $\cH_{1,\overline{\bbF}_q}$ is defined to be a module of type $$M_2(z_1,z_2):=\overline{\bbF}_q m \oplus \overline{\bbF}_q Um,\hskip15pt Sm=-m, \hskip15pt SUm=z_1 m, \hskip15pt U^2m=z_2 m$$ (in particular its $\overline{\bbF}_q$-dimension is $2$). The center $Z(\cH_{1,\overline{\bbF}_q})$ acts on $M_2(z_1,z_2)$ by the character $\theta_v$. In particular such a module is uniquely determined by its central character. It is simple if and only if $z_2\neq z_1^2$. It is called [*supersingular*]{} if $z_1=0$. Set $$\xymatrix{ \sA_{1,\theta_v}:=\sA_{1,\overline{\bbF}_q}\otimes_{Z(\cH_{1,\overline{\bbF}_q}),\theta_v}\overline{\bbF}_q: \cH_{1,\theta_v}\ar[r] & \operatorname{End}_{\overline{\bbF}_q}(\cA_{1,\theta_v}). }$$ - Assume $z_2\neq z_1^2$. Then $\sA_{1,\theta_v}$ is an isomorphism, and the $\cH_{1,\overline{\bbF}_q}$-module $\cA_{1,\theta_v}$ is isomorphic to the simple standard module $M_2(z_1,z_2)$. - Assume $z_2=z_1^2$. Then $\sA_{1,\theta_v}$ has a $1$-dimensional kernel, and the $\cH_{1,\overline{\bbF}_q}$-module $\cA_{1,\theta_v}$ is a non-split extension of the character $(0,z_1)$ by the character $(-1,-z_1)$. The proof of Proposition \[sAqinj\] shows that $\cH_{1,\theta_v}$ has an $\overline{\bbF}_q$-basis given by the elements $1, S, U, SU$, and that their images $$1,\ \sA_{1,\theta_v} (S),\ \sA_{1,\theta_v} (U),\ \sA_{1,\theta_v}(S)\sA_{1,\theta_v}(U)$$ by $\sA_{1,\theta_v}$ are linearly independent over $\overline{\bbF}_q$ if and only if $z_1^2-z_2\neq 0$. If $z_2\neq z_1^2$, then $\sA_{1,\theta_v}$ is injective, and hence bijective since $\dim_{\overline{\bbF}_q} \cA_{1,\theta_v}=2$ from \[1Ybasis\]. Moreover $S\cdot Y=-Y$ and $U\cdot Y=(z_1^2-z_2)-z_1Y$ and so $SUY=S ((z_1^2-z_2 ) -z_1Y) = S(-z_1Y)=z_1Y$, so that $$\cA_{1,\theta_v}=\overline{\bbF}_qY\oplus \overline{\bbF}_qU\cdot Y=M_2(z_1,z_2).$$ If $z_2=z_1^2$, then the proof of Proposition \[sAqinj\] shows that $\sA_{1,\theta_v}$ has a $1$-dimensional kernel which is the $\overline{\bbF}_q$-line generated by $-z_1(1+S)+U+SU$. Moreover $\overline{\bbF}_qY\subset\cA_{1,\theta_v}$ realizes the character $(-1,-z_1)$ of $\cH_{1,\overline{\bbF}_q}$, and $\cA_{1,\theta_v}/\overline{\bbF}_qY\simeq\overline{\bbF}_q1$ realizes the character $(0,z_1)$. Finally the $0$-eigenspace of $S$ in $\cA_{1,\theta_v}$ is $\overline{\bbF}_q1$, which is not $U$-stable, so that the character $(0,z_1)$ does not lift in $\cA_{1,\theta_v}$. Geometrically, the function $z_2-z_1^2$ on $V_{\mathbf{\whT},0}/W_0$ defines a family of parabolas $$\xymatrix{ V_{\mathbf{\whT},0}/W_0\ar[d]^{z_2-z_1^2},\\ \bbA^1 }$$ whose parameter is $4\Delta$, where $\Delta$ is the discriminant of the parabola. Then the locus of $V_{\mathbf{\whT},0}/W_0$ where $z_2=z_1^2$ corresponds to the parabola at $0$, having vanishing discriminant (at least if $p\neq 2$). We will say that a pair of characters of $\cH_{1,\overline{\bbF}_q}=\overline{\bbF}_q [S,U^{\pm 1}]\ra \overline{\bbF}_q^{\times}$ is *antispherical* if there exists $z_1\in \overline{\bbF}^\times_q$ such that, after evaluating on $(S,U)$, it is equal to $$\{(0,z_1),(-1,-z_1)\}.$$ Note that the set of characters $\cH_{1,\overline{\bbF}_q}\ra \overline{\bbF}_q^{\times}$ is the disjoint union of the antispherical pairs, by the very definition. \[ASphnonreg\] Let $\gamma\in \bbT^{\vee}/W_0$ non-regular. Consider the decomposition $$\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}=D(2)_{\gamma}\cup D(1)_{\gamma}$$ where $D(1)_{\gamma}$ is the closed subscheme defined by the parabola $z_2=z_1^2$ in the Steinberg coordinates $z_1,z_2$ and $D(2)_{\gamma}$ is the open complement. Then the antispherical map induces bijections $$\xymatrix{ \operatorname{ASph}^{\gamma}(2):D(2)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{simple $2$-dimensional left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}/\sim }$$ $$\xymatrix{ \operatorname{ASph}^{\gamma}(1):D(1)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{antispherical pairs of characters of $\cH^{\gamma}_{\overline{\bbF}_q}$}\}/\sim. }$$ The branch locus of the covering $$V_{\mathbf{\whT},0}\lra V_{\mathbf{\whT},0}/W_0\simeq \big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}$$ is contained in $D(2)_{\gamma}$, with equation $z_1=0$ in Steinberg coordinates, and its $k$-points correspond to the supersingular Hecke modules through the correspondence $\operatorname{ASph}^{\gamma}(2)$. The matrices of $S$, $U$ and $S_0=USU^{-1}$ in the $\overline{\bbF}_q$-basis $\{1,Y\}$ of the supersingular module $\cA_{1,\theta_v}\cong M_2(0,z_2)$ are $$S= \left (\begin{array}{cc} 0 & 0\\ 0 & -1 \end{array} \right),\hskip15pt U= \left (\begin{array}{cc} 0 & -z_2\\ -1 & 0 \end{array} \right), \hskip15pt S_0= \left (\begin{array}{cc} -1 & 0\\ 0 & 0 \end{array} \right).$$ The two characters of the *finite subalgebra* $\overline{\bbF}_q[S]$ corresponding to $S\mapsto 0$ and $S\mapsto -1$ are realized by $1$ and $Y$. From the matrix of $S_0$, we see in fact that the whole *affine subalgebra* $\overline{\bbF}_q[S_0,S]$ acts on $1$ and $Y$ via the two *supersingular affine characters*, which by definition are the characters different from the trivial character $(S_0,S)\mapsto(0,0)$ and the sign character $(S_0,S)\mapsto(-1,-1)$. Finally, let $v$ be any $k$-point of the parametrizing space $(\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0$. As a particular case of \[PASph\], the Bernstein parameter of the antispherical module $\operatorname{ASph}(v)$ is the structure sheaf of the fiber of the quotient map $\pi$ at $v$, and its Satake parameter is the underlying $k$-vector space: $$B(\operatorname{ASph}(v))=\cO_{\pi^{-1}(v)}\quad\textrm{and}\quad S(\operatorname{ASph}(v))=\pi_*\cO_{\pi^{-1}(v)}.$$ Appendix: Virtual quotients for actions of semigroups {#Appendix} ===================================================== A *semigroup* is a set equipped with an internal law which is *associative*. If the law admits a (necessary unique) identity element then the semigroup is a *monoid*, and if furthermore every element is invertible then it is a group. These set theoretic notions induce corresponding notions for set-valued functors on a given category, in particular on the category of schemes. Using the Yoneda embedding, we get the notions of a semigroup scheme, monoid scheme and group scheme (over a fixed base scheme). In this appendix, we consider the following setup. We fix a base scheme $S$ and let $(Sch/S)$ be the category of schemes over $S$. We fix a semigroup scheme $G$ over $S$ and a subsemigroup scheme $B\subset G$ (i.e. a subsemigroup functor which is representable by a scheme). We denote by $\alpha_{G,G}:G\times G\ra G$ the law of $G$ (resp. $\alpha_{B,B}:B\times B\ra B$ the law of $B$). If $G$ is a monoid we denote by $e_G$ its identity section and then we suppose that $B\subset G$ is a submonoid: $e_B:=e_G\in B$. If $G$ is a group then we suppose that $B\subset G$ is a subgroup, and denote by $i_G:G\ra G$ the inverse map of $G$ (resp. $i_B:B\ra B$ the inverse map of $B$). Virtual quotients ----------------- Recall that an [*$S$-space in groupoids*]{} is a pair of sheaves of sets $(R,U)$ on $(Sch/S)$ with five morphisms $s,t,e,c,i$ (source, target, identity, composition, inversion) $$\xymatrix{ R \ar@<1ex>[r]^{s} \ar@<-1ex>[r]_{t} & U\ar[r]^{e} & R & \quad R \times_{s,U,t}R \ar[r]^>>>>>{c} & R & \quad R \ar[r]^>>>>>{i} & R }$$ satifying certain natural compatibilities. Given a groupoid space, one defines the fibered groupoid over $(Sch/S)$ to be the category $[R,U]'$ over $(Sch/S)$ whose objects resp. morphisms over a scheme $T$ are the elements of the set $U(T)$ resp. $R(T)$. Given a morphisms $f: T'\rightarrow T$ in $(Sch/S)$ one defines the pull-back functor $f^* : [R,U]'(T)\rightarrow [R,U]'(T')$ using the maps $U(T)\rightarrow U(T')$ and $R(T)\rightarrow R(T')$. An equivalent terminology for ‘fibered groupoid over $(Sch/S)$’ is ‘prestack over $S$’, and given a Grothendieck topology on $(Sch/S)$, one can associate a stack to a prestack; in the case of the prestack $[R,U]'$, the associated stack is denoted by $[R,U]$. If $X$ is a scheme equipped with a (right) action of a group scheme $B$, one takes $U=X$, $R=X\times B$, and let $s$ be the action of the group and $t=p_1$ be the first projection. Then $c$ is the product in the group and $e,i$ are defined by means of the identity and the inverse of $B$. By definition, the quotient stack $[X/B]$ is the stack $[X\times B,X]$. For all of this, we refer to [@LM00 (2.4.3)]. In the context of semigroups, we adopt the same point of view, however, the maps $e$ and $i$ are missing. This leads to the following definition. \[defG/B\] The *virtual quotient* associated to the inclusion of semigroups $B\subset G$ is the semigroupoid consisting of the source and target maps $\alpha_{G,B}:=\alpha_{G,G}|_{G\times B}$ and first projection $p_1$ $$\xymatrix{ G\times B \ar@<1ex>[r]^{\alpha_{G,B}} \ar@<-1ex>[r]_{p_1} & G }$$ together with the composition $$\begin{aligned} c:(G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) & \lra & G\times B \\ \big((g,b),(gb,b')\big) & \longmapsto & (g,bb'). \end{aligned}$$ We denote it by $G/B$. Writing that these data define a semigroupoid means that they satisfy the following axioms: - $\alpha_{G,B}\circ c =\alpha_{G,B}\circ p_2$ and $p_1\circ c =p_1\circ p_1$ where we have denoted the two projections $(G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ra G\times B$ by $p_1,p_2$ ; - (associativity) the two composed maps $$\xymatrix{ (G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) {}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ar@<1ex>[r]^<<<<<<{c\times\operatorname{id}_{G\times B}} \ar@<-1ex>[r]_>>>>>>{\operatorname{id}_{G\times B}\times c} & (G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ar[r]^<<<<c & (G\times B) }$$ are equal. If $B\subset G$ is an inclusion of monoids, then $G/B$ becomes a monoidoid thanks to the additional datum of the identity map $$\xymatrix{ \varepsilon: G \ar[rr]^{\operatorname{id}_G\times e_B} && G\times B. }$$ This means that the following additional axioms are satisfied: - $\alpha_{G,B}\circ (\operatorname{id}_G\times e_B) = p_1 \circ (\operatorname{id}_G\times e_B) = \operatorname{id}_G$ ; - (identity element) the two composed maps $$\xymatrix{ G\times B=(G\times B){}_{\alpha_{G,B}}\times_G G=G\times_G {}_{p_1}(G\times B) \ar@<1ex>[r]^<<<<<<{\varepsilon\times\operatorname{id}_{G\times B}} \ar@<-1ex>[r]_>>>>>>{\operatorname{id}_{G\times B}\times\varepsilon} & (G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ar[r]^<<<<<c & (G\times B) }$$ are equal. If $B\subset G$ is an inclusion of groups, then $G/B$ becomes a groupoid thanks to the additional datum of the inverse map $$\xymatrix{ i: G\times B \ar[rr]^<<<<<<<<<<{\alpha_{G,B}\times i_B} && G\times B. }$$ This means that the following additional axioms are satisfied: - $\alpha_{G,B}\circ (\alpha_{G,B} \times i_B)=p_1$ and $p_1\circ (\alpha_{G,B}\times i_B)=\alpha_{G,B}$ ; - (inverse) the two diagrams $$\xymatrix{ G\times B \ar[d]_{\alpha_{G,B}} \ar[rrr]^<<<<<<<<<<<<<<{(\alpha_{G,B}\times i_B)\times\operatorname{id}_{G\times B}} &&& (G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ar[d]^c \\ G \ar[rrr]^{\operatorname{id}_G\times e_B} &&& G\times B }$$ $$\xymatrix{ G\times B \ar[d]_{p_1} \ar[rrr]^<<<<<<<<<<<<<<{\operatorname{id}_{G\times B}\times(\alpha_{G,B}\times i_B)} &&& (G\times B){}_{\alpha_{G,B}}\times_{G} {}_{p_1} (G\times B) \ar[d]^c \\ G \ar[rrr]^{\operatorname{id}_G\times e_B} &&& G\times B }$$ are commutative. Categories on the virtual quotient ---------------------------------- Let $\cC$ be a category fibered over $(Sch/S)$. \[defCG/B\] The *(fiber of the) category $\cC$ over $G/B$* is the category $\cC(G/B)$ defined by: - an object of $\cC(G/B)$ is a couple $(\cF,\phi_B)$ where $\cF$ is an object of $\cC(G)$ and $$\phi_B:p_1^*\cF \lra \alpha_{G,B}^*\cF$$ is a morphism in $\cC(G\times B)$ satisfying the following cocycle condition: considering the maps $$G\times B\times B \lra G$$ $$p_1=p_1\circ(\operatorname{id}_G\times\alpha_{B,B})=p_1\circ p_{12}$$ $$q:=\alpha_{G,B}\circ(\operatorname{id}_G\times\alpha_{B,B})=\alpha_{G,B}\circ(\alpha_{G,B}\times\operatorname{id}_B)$$ $$r:=p_1\circ(\alpha_{G,B}\times\operatorname{id}_B)=\alpha_{G,B}\circ p_{12},$$ the diagram in $\cC(G\times B\times B)$ $$\xymatrix{ p_1^*\cF\ar[dr]_{p_{12}^*\phi_B} \ar[rr]^{(\operatorname{id}_G\times\alpha_{B,B})^*\phi_B} && q^*(\cF,\phi_B) \\ &r^*\cF \ar[ur]_{(\alpha_{G,B}\times\operatorname{id}_B)^*\phi_B}& }$$ is commutative ; - a morphism $(\cF^1,\phi_B^1)\ra(\cF^2,\phi_B^2)$ in $\cC(G/B)$ is a morphism $\varphi:\cF^1\ra\cF^2$ in $\cC(G)$ such that the diagram in $\cC(G\times B)$ $$\xymatrix{ p_1^*\cF^1 \ar[r]^{p_1^*\varphi} \ar[d]_{\phi_B^1} &p_1^*\cF^2 \ar[d]^{\phi_B^2} \\ \alpha_{G,B}^*\cF^1 \ar[r]^{\alpha_{G,B}^*\varphi} & \alpha_{G,B}^*\cF^2 }$$ is commutative. \[defCG/Bmon\] If $B\subset G$ is an inclusion of monoids, then an object of $\cC(G/B)$ is a couple $(\cF,\phi_B)$ as in \[defCG/B\] which is required to satisfy the additional condition that the morphism in $\cC(G)$ $$\xymatrix{ \varepsilon^*(\phi_B):=(\operatorname{id}_G\times e_B)^*\phi_B:\cF\ar[r] & \cF }$$ is equal to the identity. Homomorphisms in $\cC(G/B)$ remain the same as in the case of semigroups. \[defCG/Bgroup\] If $B\subset G$ is an inclusion of groups, then given an object $(\cF,\phi_B)$ of $\cC(G/B)$ as in \[defCG/Bmon\], the morphism $\phi_B$ in $\cC(G\times B)$ is automatically an isomorphism, whose inverse is equal to $i^*(\phi_B):=(\alpha_{G,B}\times i_B)^*(\phi_B)$. The category $\cC(G/B)$ coincides therefore with the category attached to the underlying inclusion of monoids. Equivariant categories on the virtual quotient ---------------------------------------------- By taking the direct product $\operatorname{id}_G\times\bullet$ of all the maps appearing in the definition \[defG/B\] of the semigroupoid $G/B$, we get a semigroupoid $G\times G/B$, whose source and target maps are $$\xymatrix{ (G\times G)\times B \ar@<1ex>[r]^<<<<<{\alpha_{G\times G,B}} \ar@<-1ex>[r]_<<<<{p_1} & G\times G. }$$ Then given $\cC$ we define the category $\cC(G\times G/B)$ exactly as we defined the category $\cC(G/B)$, but now using the semigroupoid $G\times G/B$ instead of $G/B$. Applying once more $\operatorname{id}_G\times\bullet$, we also get the semigroupoid $G\times G\times G/B$ with source and target maps $$\xymatrix{ (G\times G\times G)\times B \ar@<1ex>[r]^<<<<<{\alpha_{G\times G\times G,B}} \ar@<-1ex>[r]_<<<<{p_1} & G\times G\times G, }$$ and then the category $\cC(G\times G\times G/B)$. A morphism $f:G\times G\ra G$ is *$B$-equivariant* if the diagram $$\xymatrix{ (G\times G)\times B \ar[r]^<<<<<{f\times\operatorname{id}_B} \ar[d]_{\alpha_{G\times G,B}} & G\times B \ar[d]^{\alpha_{G,B}} \\ G\times G \ar[r]^f & G }$$ commutes. Then there is a well-defined *pull-back functor* $$\xymatrix{ f^*:\cC(G/B)\ar[r] & \cC(G\times G/B), }$$ given by the rules $(\cF,\phi_B)\mapsto (f^*\cF,(f\times\operatorname{id}_B)^*\phi_B)$ and $\varphi\mapsto f^*\varphi$. One defines similarly the $B$-equivariant morphisms $f:G\times G\times G \ra G\times G$ and the associated pull-back functors $f^*:\cC(G\times G/B)\ra \cC(G\times G\times G/B)$. With this preparation, we will now be able to define the $G$-equivariant version of the category $\cC(G/B)$. It relies on the semigroupoid $G\backslash G$ consisting of the source and target maps $$\xymatrix{ G\times G \ar@<1ex>[r]^<<<<<{\alpha_{G,G}} \ar@<-1ex>[r]_<<<<<{p_2} & G }$$ together with the composition $$\begin{aligned} (G\times G){}_{\alpha_{G,G}}\times_{G} {}_{p_2} (G\times G) & \lra & G\times G \\ \big((g_1,g_0),(g_2,g_1g_0)\big) & \longmapsto & (g_2g_1,g_0). \end{aligned}$$ Note that the source and target maps $\alpha_{G,G}$ and $p_2$ are $B$-equivariant. \[defCGG/B\] The *($G$-)equivariant (fiber of the) category $\cC$ over $G/B$* is the category $\cC^G(G/B)$ defined by: - an object of $\cC^G(G/B)$ is a triple $(\cF,\phi_B,{}_G\phi)$ where $(\cF,\phi_B)$ is an object of $\cC(G/B)$ and $${}_G\phi:p_2^*(\cF,\phi_B) \lra \alpha_{G,G}^*(\cF,\phi_B)$$ is an *isomorphism* in $\cC(G\times G/B)$ satisfying the following cocycle condition: considering the $B$-equivariant maps $$G\times G\times G \lra G$$ $$p_3$$ $$q:=\alpha_{G,G}\circ(\alpha_{G,G}\times\operatorname{id}_G)=\alpha_{G,G}\circ(\operatorname{id}_G\times\alpha_{G,G})$$ $$r:=p_2\circ(\operatorname{id}_G\times\alpha_{G,G})=\alpha_{G,G}\circ p_{23},$$ and the $B$-equivariant maps $\alpha_{G,G}\times\operatorname{id}_G$, $p_{23}$, $\operatorname{id}_G\times\alpha_{G,G}$ from $G\times G\times G$ to $G\times G$, the diagram in $\cC(G\times G\times G/B)$ $$\xymatrix{ p_3^*(\cF,\phi_B)\ar[dr]_{p_{23}^*{}_G\phi} \ar[rr]^{(\alpha_{G,G}\times\operatorname{id}_G)^*{}_G\phi} && q^*(\cF,\phi_B) \\ &r^*(\cF,\phi_B) \ar[ur]_{(\operatorname{id}_G\times\alpha_{G,G})^*{}_G\phi}& }$$ is commutative ; - a morphism $(\cF^1,\phi_B^1,{}_G\phi^1)\ra(\cF^2,\phi_B^2,{}_G\phi^2)$ in $\cC^G(G/B)$ is a morphism $\varphi:(\cF^1,\phi_B^1)\ra(\cF^2,\phi_B^2)$ in $\cC(G/B)$ such that the diagram in $\cC(G\times G/B)$ $$\xymatrix{ p_2^*(\cF^1,\phi_B^1) \ar[r]^{p_2^*\varphi} \ar[d]_{{}_G\phi^1} &p_2^*(\cF^2,\phi_B^2) \ar[d]^{{}_G\phi^2} \\ \alpha_{G,G}^*(\cF^1,\phi_B^1) \ar[r]^{\alpha_{G,G}^*\varphi} & \alpha_{G,G}^*(\cF^2,\phi_B^2) }$$ is commutative (which by definition means that the diagram in $\cC(G\times G)$ $$\xymatrix{ p_2^*\cF^1 \ar[r]^{p_2^*\varphi} \ar[d]_{{}_G\phi^1} &p_2^*\cF^2 \ar[d]^{{}_G\phi^2} \\ \alpha_{G,G}^*\cF^1 \ar[r]^{\alpha_{G,G}^*\varphi} & \alpha_{G,G}^*\cF^2 }$$ is commutative). \[defCGG/Bmon\] If $B\subset G$ is an inclusion of monoids, then an object of $\cC^G(G/B)$ is a triple $(\cF,\phi_B,{}_G\phi)$ as in \[defCGG/B\], where now the object $(\cF,\phi_B)$ of $\cC(G/B)$ is as in \[defCG/Bmon\], which is required to satisfy the additional condition that the morphism in $\cC(G)$ $$\xymatrix{ (e_G\times \operatorname{id}_G)^*{}_G\phi:\cF\ar[r] & \cF }$$ is equal to the identity. Homomorphisms in $\cC^G(G/B)$ remain the same as in the case of semigroups. \[defCGG/Bgroup\] As in the non-equivariant setting, cf. \[defCG/Bgroup\], if $B\subset G$ is an inclusion of groups, then the category $\cC^G(G/B)$ coincides with the category attached to the underlying inclusion of monoids. Induction of representations ---------------------------- From now on, the fixed base scheme is a field $k$ and $\cC$ is the fibered category of vector bundles. \[defRepB\] The category $\operatorname{Rep}(B)$ of *right representations of the $k$-semigroup scheme $B$ on finite dimensional $k$-vector spaces* is defined as follows: - an object of $\operatorname{Rep}(B)$ is a couple $(M,\alpha_{M,B})$ where $M$ is a finite dimensional $k$-vector space and $$\alpha_{M,B}:M\times B \lra M$$ is a morphism of $k$-schemes such that $$\forall (m,b_1,b_2)\in M\times B\times B,\quad \alpha_{M,B}(\alpha_{M,B}(m,b_1),b_2)=\alpha_{M,B}(m,\alpha_{B,B}(b_1,b_2)).$$ - a morphism $(M^1,\alpha_{M^1,B})\ra (M^2,\alpha_{M^2,B})$ in $\operatorname{Rep}(B)$ is a $k$-linear map $f:M^1\ra M^2$ such that $$\forall (m,b)\in M^1\times B,\quad f(\alpha_{M_1,B}(m,b))=\alpha_{M_2,B}(f(m),b).$$ We define an *induction functor* $$\xymatrix{ \cI nd_B^G:\operatorname{Rep}(B) \ar[r] & \cC^G(G/B) }$$ as follows. Let $(M,\alpha_{M,B})$ be an object of $\operatorname{Rep}(B)$. Set $\cF:=G\times M\in\cC(G)$. There are canonical identifications $p_1^*\cF=G\times M\times B$ and $\alpha_{G,B}^*\cF=G\times B\times M$ in $\cC(G\times B)$. Set $$\begin{aligned} \phi_B:G\times M\times B &\lra & G\times B\times M \\ (g,m,b) & \mapsto & (g,b,\alpha_{M,B}(m,b)).\end{aligned}$$ Then $(\cF,\phi_B)$ is an object of $\cC(G/B)$. Next, there are canonical identifications $p_2^*\cF=G\times G\times M$ and $\alpha_{G,G}^*\cF=G\times G\times M$ in $\cC(G\times G)$. Set $${}_G\phi:=\operatorname{id}_{G\times G\times M}.$$ Then ${}_G\phi$ is an isomorphism $p_2^*(\cF,\phi_B)\ra\alpha_{G,G}^*(\cF,\phi_B)$ in $\cC(G\times G/B)$, and $((\cF,\phi_B),{}_G\phi)$ is an object of $\cC^G(G/B)$. Let $f:(M^1,\alpha_{M^1,B})\ra (M^2,\alpha_{M^2,B})$ be a morphism in $\operatorname{Rep}(B)$. Then $$\operatorname{id}_G\times f:\cF^1=G\times M^1\lra \cF^2=G\times M^2$$ defines a morphism $\varphi:((\cF^1,\phi_B^1),{}_G\phi^1)\ra ((\cF^2,\phi_B^2),{}_G\phi^2)$ in $\cC^G(G/B)$. These assignments are functorial. \[Indsemi\] The functor $\cI nd_B^G$ is faithful. Suppose moreover that the $k$-semigroup scheme $G$ has the following property: *There exists a $k$-point of $G$ which belongs to all the $G(\ok)$-left cosets in $G(\ok)$, and the underlying $k$-scheme of $G$ is locally of finite type.* Then the functor $\cI nd_B^G$ is fully faithful. Faithfulness is obvious. Now let $\varphi:\cI nd_B^G(M^1)=G\times M^1\ra \cI nd_B^G(M^2)=G\times M^2$. The compatibilty of $\varphi$ with ${}_G\phi^i$, $i=1,2$, reads as $$\operatorname{id}_G\times\varphi=\alpha_{G,G}^*\varphi:G\times G\times M^1\lra G\times G\times M^2.$$ For $g\in G(\ok)$, denote by $\phi_g:M_{\ok}^1\ra M_{\ok}^2$ the fiber of $\varphi$ over $g$. Taking the fiber at $(g',g)$ in the above equality implies that $\varphi_g=\varphi_{g'g}$ for all $g,g'\in G(\ok)$, i.e. $\varphi_g$ depends only on the left coset $G(\ok)g$, hence is independent of $g$ if all the left cosets share a common point. Assuming that such a point exists and is defined over $k$, let $f:M^1\ra M^2$ be the corresponding $k$-linear endomorphism. Then $\varphi-\operatorname{id}_G\times f$ is a linear morphism between two vector bundles on $G$, which vanishes on each geometric fiber. Then it follows from Nakayama’s Lemma that $\varphi-\operatorname{id}_G\times f=0$ on $G$, at least if the latter is locally of finite type over $k$. \[Indcat\] When the functor $\cI nd_B^G$ is fully faithful, we call its essential image the *category of induced vector bundles on $G/B$*, and denote it by $\cC_{\cI nd}^G(G/B)$: $$\xymatrix{ \cI nd_B^G:\operatorname{Rep}(B) \ar[r]^<<<<<<{\sim} & \cC_{\cI nd}^G(G/B) \subset \cC^G(G/B). }$$ If $B\subset G$ is an inclusion of monoids, then an object of $\operatorname{Rep}(B)$ is a couple $(M,\alpha_{M,B})$ as in \[defRepB\] which is required to satisfy the additional condition that the $k$-morphism $$\xymatrix{ \alpha_{M,B}\circ (\operatorname{id}_M\times e_B):M\ar[r] & M }$$ is equal to the identity. Homomorphisms in $\operatorname{Rep}(B)$ remain the same as in the case of semigroups. In particular, comparing with \[defCGG/Bmon\], the same assignments as in the case of semigroups define an induction functor $$\xymatrix{ \cI nd_B^G: \operatorname{Rep}(B) \ar[r] & \cC^G(G/B). }$$ Now set $e:=e_B=e_G\in B(k)\subset G(k)$, the identity element. We define a functor *fiber at $e$* $$\xymatrix{ \operatorname{Fib}_e: \cC^G(G/B) \ar[r] & \operatorname{Rep}(B) }$$ as follows. Let $(\cF,\phi_B,{}_G\phi)$ be an object of $\cC^G(G/B)$. Set $M:=\cF|_e$, a finite dimensional $k$-vector space. There are canonical identifications $(p_1^*\cF)|_{e\times B}=M\times B$, $(\alpha_{G,B}^*\cF)|_{e\times B}=(\alpha_{G,G}^*\cF)|_{B\times e}=\cF|_B$ and $(p_2^*\cF)|_{B\times e}=B\times M$. Set $$\xymatrix{ \alpha_{M,B}:M\times B\ar[r]^<<<<<<{\phi_B|_{e\times B}} & \cF|_B & B\times M \ar[l]_{{}_G\phi|_{B\times e}}^{\sim} \ar[r]^<<<<<<{p_2} & M. }$$ Then $(M,\alpha_{M,B})$ is an object of $\operatorname{Rep}(B)$. Let $\varphi:(\cF^1,\phi_B^1,{}_G\phi^1)\ra (\cF^2,\phi_B^2,{}_G\phi^2)$ be a morphism in $\cC^G(G/B)$. Then $$f=\varphi_e:\cF^1|_e=M^1\lra \cF^2|_e=M^2$$ defines a morphism $(M^1,\alpha_{M^1,B})\ra (M^2,\alpha_{M^2,B})$ in $\operatorname{Rep}(B)$. These assignments are functorial. \[IndFib\] For an inclusion of $k$-monoid schemes $B\subset G$ with unit $e$, the functors $\cI nd_B^G$ and $\operatorname{Fib}_e$ are equivalences of categories, which are quasi-inverse one to the other. Left to the reader. Analogous to the property \[defCGG/Bgroup\] for equivariant vector bundles, we have that if $B\subset G$ is an inclusion of groups, then given an object $(M,\alpha_{M,B})$ of $\operatorname{Rep}(B)$, the right $B$-action on $M$ defined by $\alpha_{M,B}$ factors automatically through the $k$-group scheme opposite to the one of $k$-linear automorphisms of $M$, the inverse of $\alpha_{M,B}(\bullet,b)$ being equal to $\alpha_{M,B}(\bullet,i_B(b))$ for all $b\in B$. The category $\operatorname{Rep}(B)$ coincides therefore with the category attached to the underlying monoid of $B$. In particular, we have the functors $\cI nd_B^G$ and $\operatorname{Fib}_e$ attached to the underlying inclusion of monoids $B\subset G$, for which Proposition \[IndFib\] holds. Grothendieck rings of equivariant vector bundles ------------------------------------------------ \[ringsemigr\] For a $k$-semigroup scheme $B$, the category $\operatorname{Rep}(B)$ is abelian $k$-linear symmetric monoidal with unit. Hence, for an inclusion of $k$-semigroup schemes $B\subset G$ such that the functor $\cI nd_B^G$ is fully faithful, the essential image $\cC_{\cI nd}^G(G/B)$ has the same structure. In particular, it is an abelian category whose Grothendieck group $K_{\cI nd}^G(G/B)$ is a commutative ring, which is isomorphic to the ring $R(B)$ of right representations of the $k$-semigroup scheme $B$ on finite dimensional $k$-vector spaces: $$\xymatrix{ \cI nd_B^G:R(B) \ar[r]^<<<<<{\sim} & K_{\cI nd}^G(G/B). }$$ \[ringmon\] If $B\subset G$ is an inclusion of monoids, then it follows from \[IndFib\] that the category $\cC^G(G/B)$ is abelian $k$-linear symmetric monoidal with unit. In particular, it is an abelian category whose Grothendieck group $K^G(G/B)$ is a commutative ring, which is isomorphic to the ring $R(B)$ of right representations of the $k$-monoid scheme $B$ on finite dimensional $k$-vector spaces: $$\xymatrix{ \cI nd_B^G:R(B) \ar[r]^<<<<<{\sim} & K^G(G/B). }$$ If $B\subset G$ is an inclusion of groups, then \[ringmon\] applies to the underlying inclusion of monoids. [99]{} , [*Dimension des fibres de Springer affines pour les groupes*]{}, Transform. 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--- abstract: | Recent researches show that the fluctuations of the dielectric mirrors coating thickness can introduce a substantial part of the future laser gravitational-wave antennae total noise budget. These fluctuations are especially large in the high-reflectivity end mirrors of the Fabry-Perot cavities which are being used in the laser gravitational-wave antennae. We show here that the influence of these fluctuations can be substantially decreased by using additional short Fabry-Perot cavities, tuned in anti-resonance instead of the end mirrors. author: - 'F.Ya.Khalili' title: 'Reducing the mirrors coating noise in laser gravitational-wave antennae by means of double mirrors' --- Introduction ============ One of the basis components of laser gravitational-wave antennae [@Abramovici1992; @Abramovici1996; @WhitePaper1999] are high-reflectivity mirrors with multilayer dielectric coating. Recent researches [@Levin1998; @Crooks2002; @Harry2002; @Nakagava2002; @Penn2003; @03a1BrVy; @03a1BrSa; @Cagnoli2003; @Fejer2004; @Harry2004] have shown that fluctuations of the coating thickness produced by, in particular, Brownian and thermoelastic noise in a coating, can introduce substantial part of the total noise budget of the future laser gravitational-wave antennae. For example, estimates, done in [@03a1BrVy] show that the thermoelastic noise value can be close to the Standard Quantum Limit (SQL) [@03a1BrGoKhMaThVy] which corresponds to the sensitivity level of the Advanced LIGO project [@WhitePaper1999] or even can exceed it in some frequency range. For this reason it was proposed in [@04a1BrVy] to replace end mirrors by coatingless corner reflectors. It was shown in this article that by using these reflectors, it is possible, in principle, to obtain sensitivity much better than the SQL. However, the corner reflectors require substantial redesign of the gravitational-wave antennae core optics and suspension system. At the same time, the value of the mirror surface fluctuations depends on the number of dielectric layers which form the coating. It can be explained in the following way. The most of the light is reflected from the first couple of the layers. At the same time, fluctuations of the mirror surface are created by the thickness fluctuations of all underlying layers, and the larger is the layers number, the larger is the surface noise. Therefore, the surface fluctuations are relatively small for the input mirrors ([ITM]{}) of the Fabry-Perot cavities of the laser gravitational-wave antennae with only a few coating layers and $1-{\cal R}\sim 10^{-2}$ (${\cal R}$ is the mirror power reflectivity), and is considerably larger for the end mirrors ([ETM]{}) with coating layers number $\sim 40$ and $1-{\cal R}\lesssim 10^{-5}$. \[ct\]\[lb\][$L=4\,{\rm Km}$]{} \[ct\]\[lb\][$l\lesssim 10\,{\rm m}$]{} \[cb\]\[lb\][[ITM]{}]{} \[cb\]\[lb\][[IETM]{}]{} \[cb\]\[lb\][[EETM]{}]{} ![Schematic layout of a Fabry-Perot cavity with double mirror system instead of the end mirror: [ITM]{} and [IETM]{} are similar moderate reflective mirrors; [EETM]{} is a high-reflective one.[]{data-label="fig:fabry_dbl_mirror"}](fabry_dbl_mirror.eps){width="5in"} In this paper another, less radical way of reducing the coating noise, exploiting this feature, is proposed. It is based on the use of an additional short Fabry-Perot cavity instead of the end mirror (see Fig.\[fig:fabry\_dbl\_mirror\]). It should be tuned in anti-resonance, [*i.e*]{} its optical length $l$ should be close to $l=(N+1/4)\lambda$, where $\lambda$ is a wavelength. The back side of the first mirror have to have a few layers of an antireflection coating. It can be shown that in this case reflectivity of this cavity will be defined by the following equation: $$\label{R_simple} 1-{\cal R} \approx \frac{(1-{\cal R}_1)(1-{\cal R}_2)}{4} \,,$$ where ${\cal R}_{1,2}$ are the reflectivities of the first ([EETM]{} on Fig.\[fig:fabry\_dbl\_mirror\]) and the second ([IETM]{}) mirrors. Phase shift in the reflected beam produced by small variations $y$ in position of the second mirror reflecting surface relative to the first one will be equal to $$\label{phi_simple} \phi \approx \frac{1-{\cal R}_1}{4}\times 2ky \,,$$ where $k=2\pi/\lambda$ is a wave number. It is supposed for simplicity that there is no absorption in the first mirror material; more general formulae are presented below. It follows from these formulae that the first mirror can have a moderate value of reflectivity and, therefore, a small number of coating layers. In particular, it can be identical to the input mirror of the main Fabry-Perot cavity ([ITM]{}). At the same time, influence of the coating noise of the second (very-high-reflective) mirror will be suppressed by a factor of $(1-{\cal R}_1)/4$, which can be as small as $\sim 10^{-2}\div 10^{-3}$. \[cb\]\[lb\][[ETM]{}]{} ![The double reflector based on a single mirror.[]{data-label="fig:single_mirror"}](single_mirror.eps){width="2in"} In principle, another design of the double reflector is possible, which consists of one mirror only, see Fig.\[fig:single\_mirror\]. Both surfaces of this mirror have to have reflective coatings: the thin one on the face side and the thick one on the back side. In this case the additional Fabry-Perot cavity is created [*inside*]{} this mirror. However, in this case thermoelastic fluctuations of the the back surface coating will bend the mirror and thus will create unacceptable large mechanical fluctuations of the face surface. Estimates show that using this design, it possible to reduce the face surface fluctuations by factor $\sim 3$ only [@vyat_priv]. So the design with two [*mechanically isolated*]{} reflectors only will be considered here. In the next section more detail analysis of this system is presented. Analysis of the double-mirror reflector ======================================= \[rc\]\[lb\][$a$]{} \[rc\]\[lb\][$b$]{} \[lc\]\[lb\][$a_0$]{} \[lc\]\[lb\][$b_0$]{} \[lc\]\[lb\][$a_1$]{} \[lc\]\[lb\][$b_1$]{} \[rc\]\[lb\][$a_2$]{} \[rc\]\[lb\][$b_2$]{} \[cb\]\[lb\][[IETM]{}]{} \[cb\]\[lb\][[EETM]{}]{} ![The double mirror reflector.[]{data-label="fig:dbl_mirror"}](dbl_mirror.eps){width="3.5in"} The rightmost part of Fig.\[fig:fabry\_dbl\_mirror\] is presented in Fig.\[fig:dbl\_mirror\], where the following notation is used: $a, b$ are the amplitudes of the incident and reflected waves for the first mirror, respectively; $a_0, b_0$ are the amplitudes of the waves traveling in the left and right directions, respectively, just behind the first mirror coating; $a_1, b_1$ are the same for the waves just behind the first mirror itself; $a_2, b_2$ are the amplitudes of the incident and reflected waves for the second mirror, respectively. These amplitudes satisfy the following equations: \[main\_eqs\] $$\begin{aligned} a_0 &= -R_1b_0 + iT_1a \,, \\ a_1 &= T_0a_0 + A_1n_a \,, \\ a_2 &= \theta a_1 \,, \\ b &= -R_1a + iT_1b_0 \,, \\ b_0 &= T_0b_1 + A_0n_b \,, \\ b_1 &= \theta b_2 \,, \\ b_2 &= -R_2a_2 + A_2n_2 \,, \end{aligned}$$ where: $n_a,n_b,n_2$ are independent zero-point oscillations generated in the first ($n_a,n_b$) and the second ($n_2$) mirrors; $\theta = e^{ikl_1}$, where $l_1$ is the distance between the first mirror back surface and the second mirror; $-R_1$ and $iT_1$ are the amplitude reflectivity and transmittance of the first mirror coating, respectively, $R_1^2+T_1^2=1$; $T_0$ and $A_0$ are the amplitude transmittance and absorption of the first mirror bulk, respectively, $|T_0|^2+A_0^2=1$; $-R_2$ and $A_2$ are the amplitude reflectivity and absorption of the second mirror, respectively, $R_2^2+A_2^2=1$. $R_1,T_1,A_0,R_2,A_2$ are real values; $T_0$ is a complex one, its argument corresponds to the phase shift in the first mirror bulk. Here we do not consider absorption in the first mirror coating for two reasons: (i) it is relatively small and (ii) it exists both in traditional one-mirror reflectors and in the one considered here, and the main goal of this short article is to emphasize the [*differences*]{} between these two types of reflectors. We also suppose that the mirrors move rather slowly: $$\frac{dl}{dt} \ll \frac{l}{c} \,.$$ In the case of the gravitational-wave signal characteristic frequencies $\Omega\lesssim 10^3\,{\rm s}^{-1}$ and relatively short length $l\lesssim 1\,{\rm m}$ this inequality is fulfilled pretty well. It follows from equations (\[main\_eqs\]) that the reflected beam amplitude is equal to $$\label{dbl_soln_b} b = \frac{(R_2T_0^2\theta^2-R_1)a - iR_2A_0T_0T_1\theta^2n_a + iA_2T_0T_1\theta n_2 + iA_0T_1n_b}{1-R_1R_2T_0^2\theta^2} \,.$$ This solution can be presented in the following form: $$b = \tilde R a + A n \,,$$ where $$\tilde R = \frac{R_2T_0^2\theta^2 - R_1}{1 - R_1R_2T_0^2\theta^2}$$ is the equivalent complex reflection factor for the scheme considered, $$A = \frac{T_1\sqrt{1 - R_2^2|T_0|^4}}{|1-R_1R_2T_0^2\theta^2|}$$ is its equivalent absorption factor, and $$n = \frac{1}{A}\,\frac{ - iR_2A_0T_0T_1\theta^2n_a + iA_2T_0T_1\theta n_2 + iA_0T_1n_b}{1-R_1R_2T_0^2\theta^2}$$ is the sum noise normalized as zero-point fluctuations. As mentioned above, this system should be tuned in anti-resonance: $$l \equiv \frac{1}{k}\arg{T_0\theta} = \frac{\pi}{k}\left(N + \frac{1}{2}\right) + y \,, \\$$ where $N$ is an integer and $y \ll \lambda$. In this case $$T_0\theta = i(-1)^N|T_0|e^{iky} \approx i(-1)^N(1+iky) \,,$$ and $$\tilde R \approx -Re^{i\phi} \,,$$ where $$R = 1-\frac{(1-R_1)(1-R_2|T_0|^2)}{1 + R_1R_2|T_0|^2}\,,$$ and $$\phi \approx \frac{2R_2|T_0|^2T_1^2}{(R_2|T_0|^2 + R_1)(1 + R_1R_2|T_0|^2)}\,ky$$ is the phase shift produced by the deviation $y$ in the distance $l$. Suppose that factors $T_1, A_0, A_2$ are small. In this case $$1-R \approx \frac{(1-R_1)(1-R_2+A_0^2)}{2} \label{R_amp} \,, \\$$ $$\phi \approx (1-R_1)ky \,. \label{phi_amp}$$ Using power reflection and absorption factors instead of the amplitude ones: $$\begin{gathered} {\cal R} = R^2 \,, \\ {\cal R}_{1,2} = R_{1,2}^2 \,, \\ {\cal A}_0 = A_0^2 \,,\end{gathered}$$ equations (\[R\_amp\]), (\[phi\_amp\]) can be rewritten as follows: $$1- {\cal R} \approx \frac{(1-{\cal R}_1)(1-{\cal R}_2 + 2{\cal A}_0)}{4} \,, \label{R_pow} \\$$ $$\phi \approx \frac{1-{\cal R}_1}{2}\,ky \,.$$ Conclusion ========== The main goal of this short article is just to claim the idea, so the detailed design of the additional cavity is not presented here. However, the following important topics have to be discussed in brief. The first one concerns the optimal value of the [IETM]{} mirror reflectivity. The smaller is $1-{\cal R}_1$, the larger is suppression factor for the [EETM]{} mirror surface noises; at the same time, the larger is the [IETM]{} mirror coating noise. The rigorous optimization requires exact knowledge of the coating noise dependence on the coating layers number. A crude estimate based in the exponential dependence of the [IETM]{} mirror transmittance ${\cal T}_1 \approx 1-{\cal R}_1$ on the coating layers number gives that the optimal transmittance value is relatively large, ${\cal T}_1 \sim 10^{-1}$. On the other hand, smaller values of the [IETM]{} mirror transmittance, down to the input ([ITM]{}) mirror transmittance ${\cal T}_{\sf ITM}$ are also acceptable. Therefore, identical [ITM]{} and [IETM]{} mirrors can be used. In the Advanced LIGO interferometer, the input mirrors transmittance will be equal to ${\cal T}_{\sf ITM}\approx 5\times 10^{-3}$, and its bulk absorption will be equal to ${\cal A}_{\rm ITM} \approx 10^{-5}$ [@RefDesign]. Using such mirror as an [IETM]{} mirror in the scheme proposed in this article, and mirror with commercially available value of $1-{\cal R}_2 \approx 10^{-5}$ as an [EETM]{} mirror, it is possible to create a double-mirror reflector with $1-{\cal R} < 10^{-6}$ and suppression factor for the [EETM]{} surface fluctuations $\dfrac{1-{\cal R}_1}{4} \approx 10^{-3}$. The second issue concerns the optical power circulating through the [IETM]{} mirror. It is easy to show using equations (\[main\_eqs\]), that it is $\dfrac{4}{1-{\cal R}_1}\sim 10^3$ times smaller that the power circulating in the main cavities. In the Advanced LIGO topology, it will be approximately equal to the power circulating through the [ITM]{} mirrors and the beamsplitter (about 1KW). It is necessary to note also that $y$ in the calculations presented above includes not only coating noise of the [EETM]{} mirror but all possible kinds of its surface fluctuations, including ones caused by Brownian and thermoelastic fluctuations in this mirror bulk, Brownian fluctuation in its suspension, seismic noise as well as the mirror quantum fluctuations. This feature simplifies greatly the [EETM]{} mirror design because the requirements for all these noise sources can be reduced by a factor of $(1-{\cal R}_1)/4$. In particular, the SQL value $\sqrt{\dfrac{\hbar}{m\Omega^2}}$ for this mirror ($m$ is its mass and $\Omega$ is the observation frequency) can be larger by a factor of $\left(\dfrac{1-{\cal R}_1}{4}\right)^{-1}$. Therefore, its mass can be, in principle, $\left(\dfrac{1-{\cal R}_1}{4}\right)^{-2} \sim 10^6$ times smaller than for the main ([ITM]{} and [IETM]{}) mirrors. Of course, such a small mirror hardly can be used in the real interferometer. This estimates shows only that the quantum noise does not impose any practical limitation on the ${\sf EETM}$ mirror mass. Acknowledgments {#acknowledgments .unnumbered} =============== The author is grateful to V.B.Braginsky, S.L.Danilishin, G.Harry, D.Ottaway, D.Shoemaker and S.P.Vyatchanin for useful remarks. This work was supported by NSF grant PHY0098715, by Russian Ministry of Industry and Sciences contracts 40.02.1.1.1.1137 and 40.700.12.0086, and by Russian Foundation of Basic Researches Grant 03.02.16975-a. [10]{} , Science [**256**]{}, 325 (1992). , Physics Letters A [**218**]{}, 157 (1996). , , 1999, . , Physical Review D [**57**]{}, 659 (1998). , Classical and Quantum Gravity [**19**]{}, 883 (2002). , Classical and Quantum Gravity [**19**]{}, 897 (2002). , Physical Review D [**65**]{}, 102001 (2002). , Classical and Quantum Gravity [**20**]{}, 2917 (2003). , Physics Letters A [**312**]{}, 169 (2003). , Physics Letters A [**315**]{}, 175 (2003). , (2003). , arXiv:gr-qc/0402034 (2004). , (2004). , Physical Review D [**67**]{}, 082001 (2003). , arXive:cond-mat/0402650 (2004). S.P.Vyatchanin, [*private communication.*]{} www.ligo.caltech.edu/AdvLIGO

--- abstract: 'We describe the TreePM method for carrying out large N-Body simulations to study formation and evolution of the large scale structure in the Universe. This method is a combination of Barnes and Hut tree code and Particle-Mesh code. It combines the automatic inclusion of periodic boundary conditions of PM simulations with the high resolution of tree codes. This is done by splitting the gravitational force into a short range and a long range component. We describe the splitting of force between these two parts. We outline the key differences between TreePM and some other N-Body methods.' author: - | J.S.Bagla\ Harish-Chandra Research Institute, Chhatnag Road, Jhunsi,\ Allahabad 211019, INDIA\ e-mail:jasjeet@mri.ernet.in date: 'Received 2002 June 13; accepted 2002 November 14' title: 'TreePM: A code for Cosmological N-Body Simulations' --- \[firstpage\] gravitation, methods: numerical, cosmology: large scale structure of the universe Introduction ============ Observations suggest that the present universe is populated by very large structures like galaxies, clusters of galaxies etc. Current models for formation of these structures are based on the assumption that gravitational amplification of density perturbations resulted in the formation of large scale structures. In absence of analytical methods for computing quantities of interest, numerical simulations are the only tool available for study of clustering in the non-linear regime. Last two decades have seen a rapid development of techniques and computing power for cosmological simulations and the results of these simulations have provided valuable insight into the study of structure formation. The simplest N-Body method that has been used for studying clustering of large scale structure is the Particle Mesh method (PM hereafter). The genesis of this method is in the realisation that the Poisson equation is an algebraic equation in Fourier space, hence if we have a tool for switching to Fourier space and back, we can calculate the gravitational potential and the force with very little effort. It has two elegant features in that it provides periodic boundary conditions by default, and the force is softened naturally so as to ensure collisionless evolution of the particle distribution. However, softening of force done at grid scale implies that the force resolution is very poor. This limits the dynamic range over which we can trust the results of the code between a few grid cells and about a quarter of the simulation box (Bouchet and Kandrup, 1985; Bagla and Padmanabhan, 1997. Many efforts have been made to get around this problem, mainly in the form of P$^3$M (Particle-Particle Particle Mesh) codes (Efstathiou et al, 1985; Couchman 1991). In these codes, the force computed by the particle mesh part of the code is supplemented by adding the short range contribution of nearby particles, to improve force resolution. The main problem with this approach is that the particle-particle summation of the short range force takes a lot of time in highly clustered situations. Another, more subtle problem is that the force computed using the PM method has anisotropies and errors in force at grid scale – these errors are still present in the force calculated by combining the PM force with short range corrections (Bouchet and Kandrup, 1985). A completely different approach to the problem of computing force are codes based on the tree method. In this approach we consider groups of particles at a large distance to be a single entity and compute the force due to the group rather than sum over individual particles. There are different ways of defining a group, but by far the most popular method is that due to Barnes and Hut (1986). Applications of this method to Cosmological simulations require including periodic boundary conditions. This has been done using Ewald’s method (Ewald, 1921; Rybicki, 1986; Hernquist, Bouchet and Suto, 1991; Springel, Yoshida and White, 2001). Ewald’s method is used to tabulate the correction to the force due to periodic boundary conditions. This correction term is stored on a grid (in relative separation of a pair of particles) and the interpolated value is added to the pairwise force. Some attempts have been made to combine the high resolution of a tree code with the natural inclusion of periodic boundary conditions in a PM code by simply extending the P$^3$M method and replacing the particle-particle part for short range correction with a local tree (Xu, 1995). In this paper we present a hybrid N-Body method that attempts to combine the good features of the PM and the tree method, while avoiding the problems of the P$^3$M and the TPM methods. Our approach is to divide force into long and short range components using partitioning of unity, instead of taking the PM force as given. This allows us greater control over errors, as we shall see below. The plan of the paper is as follows: §[2]{} introduces the basic formalism of both the tree and PM codes. §[2.3]{} gives the mathematical model for the TreePM code. We analyse errors in force for the TreePM code in §[3]{}. Computational requirements of our implementation of the TreePM code are discussed in §[4]{}. A discussion of the relative merits of the TreePM method with respect to other N-Body methods follows in §[5]{}. The TreePM Method ================= Tree Code --------- We use the approach followed by Barnes and Hut (1986). In this, the simulation volume is taken to be a cube. The tree structure is built out of cells and particles. Cells may contain smaller cells (subcells) within them. Subcells can have even smaller cells within them, or they can contain a particle. We start with the simulation volume and add particles to it. If two particles end up in the same subcell, the subcell is geometrically divided into smaller subcells until each subcell contains either subcells or at most one particle. The cubic simulation volume is the root cell. In three dimensions, each cubic cell is divided into eight cubic subcells. Cells, as structures, have attributes like total mass, location of centre of mass and pointers to subcells. Particles, on the other hand have the traditional attributes like position, velocity and mass. More details can be found in the original paper (Barnes and Hut, 1986). Force on a particle is computed by adding contribution of other particles or of cells. A cell that is sufficiently far away can be considered as a single entity and we can just add the force due to the total mass contained in the cell from its centre of mass. If the cell is not sufficiently far away then we must consider its constituents, subcells and particles. Whether a cell can be accepted as a single entity for force calculation is decided by the cell acceptance criterion (CAC). We compute the ratio of the size of the cell $d$ and the distance $r$ from the particle in question to its centre of mass and compare it with a threshold value $$\theta = \frac{d}{r} \leq \theta_c \label{trwalk}$$ The error in force increases with $\theta_c$. There are some potentially serious problems associated with using $\theta_c \geq 1/\sqrt{3}$, a discussion of these is given in Salmon and Warren (1994). One can also work with completely different definitions of the CAC (Salmon and Warren, 1994; Springel, Yoshida and White, 2001). Irrespective of the criterion used, the number of terms that contribute to the force on a particle is much smaller than the total number of particles, and this is where a tree code gains in terms of speed over direct summation. We will use the Barnes and Hut tree code and we include periodic boundary conditions for computing the short range force of particles near the boundaries of the simulation cube. Another change to the standard tree walk is that we do not consider cells that do not have any spatial overlap with the region within which the short range force is calculated. We also use an optimisation technique to speed up force calculation (Barnes, 1990). Particle Mesh Code ------------------ A PM code is the obvious choice for computing long range interactions. Much has been written about the use of these in cosmological simulations (e.g., see Hockney and Eastwood, 1988) so we will not go into details here. PM codes solve for the gravitational potential in the Fourier space. These use Fast Fourier Transforms (FFT) to compute Fourier transforms, and as FFT requires data to be defined on a regular grid the concept of mesh is introduced. The density field represented by particles is interpolated onto the mesh. Poisson equation is solved in Fourier space and an inverse transform gives the potential (or force) on the grid. This is then differentiated and interpolated to the position of each particle in order to calculate the displacements. Use of a grid implies that forces are not accurate at the scale smaller than the grid cells. A discussion of errors in force in a PM code can be found in Efstathiou et al (1985) and elsewhere (Bouchet and Kandrup, 1985; Bagla and Padmanabhan, 1997). The error in force can be very large at small scales but it drops to an acceptable number beyond a few grid cells, and is negligible at large scales. We use the Cloud-in-Cell weight function for interpolation. We solve the Poisson equation using the natural kernel, $-1/k^2$; this is called the poor man’s Poisson solver (Hockney and Eastwood, 1988). We compute the gradient of the potential in Fourier space. TreePM Code ----------- We now turn to the question of combining the tree and the PM code. We wish to split the inverse square force into a long range force and a short range force. The gravitational potential can be split into two parts in Fourier space (Ewald, 1921). $$\begin{aligned} \varphi_k &=& - \frac{4 \pi G \varrho_k}{k^2} \label{pm_std}\\ &=& - \frac{4 \pi G \varrho_k}{k^2} \exp\left(-k^2 r_s^2\right) - \frac{4 \pi G \varrho_k}{k^2} \left(1 - \exp\left(-k^2 r_s^2\right)\right)\nonumber\\ &=& \varphi_k^l + \varphi_k^s \nonumber \\ \varphi_k^l &=& - \frac{4 \pi G \varrho_k}{k^2} \exp\left(-k^2 r_s^2\right) \label{longr}\\ \varphi_k^s &=& - \frac{4 \pi G \varrho_k}{k^2} \left(1 - \exp\left(-k^2 r_s^2\right)\right) \label{shortr}\end{aligned}$$ where $\varphi^l$ and $\varphi^s$ are the long range and the short range potentials, respectively. The splitting is done at the scale $r_s$. $G$ is the gravitational coupling constant and $\varrho$ is density. The expression for the short range force in real space is: $${\bf f}^s({\bf r}) = - \frac{G m {\bf r}}{r^3} \left({\rm erfc}\left(\frac{r}{2 r_s}\right) + \frac{r}{r_s \sqrt{\pi}} \exp\left(-\frac{r^2}{4 r_s^2}\right)\right) \label{fshort}$$ Here, ${\rm erfc}$ is the complementary error function. These equations describe the mathematical model for force in the TreePM code. The long range potential is computed in the Fourier space, just as in a PM code, but using eqn.(\[longr\]) instead of eqn.(\[pm\_std\]). This potential is then used to compute the long range force. The short range force is computed directly in real space using eqn.(\[fshort\]). In the TreePM method this is computed using the tree approximation. The short range force falls rapidly at scales $r \gg r_s$, and hence we need to take this into account only in a small region around each particle. =4truein We have plotted the long range and the short range force (eqn.(\[fshort\])) as a function of $r$ in fig.1 to show their dependence on scale. We have chosen $r_s=1$ here. The short range force closely follows the total force up to about $2 r_s$ and then falls rapidly, its magnitude falls below $1\%$ of the total force by $5 r_s$. The long range force reaches a peak around $2 r_s$. It makes up most of the total force beyond $3.5 r_s$. It falls with scale below $2 r_s$, becoming negligible below $r_s / 2$. Evaluation of special functions for calculating the short range force can be time consuming. To save time, we compute an array containing the magnitude of the short range force. The force between any two objects, particle-cell or particle-particle, is computed by linearly interpolating between the nearby array elements multiplied by the unit vector ${\bf r}$. It is necessary for the array to sample the force at sufficiently closely spaced values of $r$ in order to keep error in interpolation small. Error Estimation ================ In this section we will study errors in force introduced by various components of the TreePM code. We will only list salient points here and the reader is referred to a more comprehensive study for details (Bagla and Ray, 2002). We start by estimating the error in force due to one particle. The long range force of a particle is calculated using the PM method, but using eqn.(\[longr\]) instead of eqn.(\[pm\_std\]). The cutoff at high wave numbers largely removes the effect of the grid and we find that the dispersion in the long range force is very small, e.g. for $r_s \geq 1$ grid length the dispersion is smaller than $1\%$ of the total force at all scales. There is a systematic offset in the long range force that is larger than the dispersion. This offset is induced by the interpolating function, and can be corrected (White, 2000; Bagla and Ray, 2002) by de-convolving the square of the interpolating function (we need to interpolate twice). This deconvolution does not affect the dispersion in any significant manner. There are no errors in computing the short range force for one particle, hence the only source of errors is in the calculation of the long range force in this case. All the errors arise due to anisotropies in the long range force. The errors in the long range force increase as we approach small scales, but the contribution of the long range force to the total force falls sharply below $2r_s$ and hence the errors also drop rapidly. There is a peak in errors around $2r_s$–$3r_s$, and for $r_s=1$ maximum rms error in force of one particle is $1\%$ of the total force. In calculating the total force, we added the short range force to the long range force at all scales. However, this is not necessary as beyond some scale, the contribution of small scale force to the total force drops to a negligible fraction of the total force. We will call the scale upto which we add the small scale force as $r_{cut}$. The short range force is just below $1\%$ of the total force at $r_{cut}=5r_s$. We choose this value of $r_{cut}$ for the TreePM code. =4truein The other source of error is the tree approximation that we use for computing the short range force. The first correction term is due to the quadrapole moment of the particle distribution in the cell, however the magnitude of this error is larger than in the inverse square force due to a more rapid variation in force with distance. In the worst case, this error can be more than twice the error in the corresponding case of inverse square force (Bagla and Ray, 2002). In more generic cases, errors due to this effect tend to cancel out and the net error is small. Apart from this effect, there is also a dispersion introduced by the tree approximation. The magnitude of this dispersion varies monotonically with $\theta_c$. One factor that we have to weigh in is that the execution time is small for large $\theta_c$ and small $r_{cut}$. Given these considerations, the obvious solution is to choose the smallest $r_s$ and the largest $\theta_c$ that gives us a sufficiently accurate force field. It is important to estimate the errors in a realistic situation, even though we do not expect errors to add up coherently in most situations. We test errors for two distributions of particles: a homogeneous distribution and a clumpy distribution. For the homogeneous distribution, we use randomly distributed particles in a box. We use $262144$ particles in a $64^3$ box for this distribution. We compute the force using a reference setup ($r_s=4$, $r_{cut}=6 r_s$, $\theta_c=0$) and the setup we wish to test ($r_s=1$, $r_{cut}=5 r_s$, $\theta_c=0.5$). It can be shown that the errors in the reference setup are well below $0.5\%$ for the entire range of scales (Bagla and Ray, 2002). We compute the fractional error in force acting on each particle, this is defined as, $$\epsilon = \frac{\left\vert {\bf f} - {\bf f}_{ref} \right\vert}{\left\vert {\bf f}_{ref} \right\vert} .$$ Fig.2 shows the cumulative distribution of fractional errors. The curves show the fraction of particles with error greater than $\epsilon$. The thick line shows this for the homogeneous distribution. Error $\epsilon$ for $99\%$ of particles is less than $3.5\%$. Results for the clumpy distribution of particles are shown by the dashed line. We used the output of a CDM simulation (fig.3a) run with the TreePM code. Errors in this case are much smaller, as compared to the homogeneous distribution, as in the case of tree code (Hernquist, Bouchet and Suto, 1991). Error $\epsilon$ for $99\%$ of particles is around $2\%$, as compared to $3.5\%$ for the homogeneous distribution. =3.2truein =3.2truein There are two noteworthy features of this figure. One is that the error for the homogeneous distribution is higher. The main reason for this is similar to that in tree codes, though the effect is much smaller here. When we are dealing with a homogeneous distribution, the total force on each particle is very small because forces due to nearly identical mass distributions on opposite sides cancel out. This near cancellation of large numbers gives rise to errors that decrease as the net result of these cancellations grows. In a tree code, we calculate the force due to all the particles in the simulation box whereas in the TreePM method we add up the contribution of only those within a sphere of radius $r_{cut}$. This is the reason for the difference in these two curves being much less pronounced than the corresponding curves for the tree code (Hernquist, Bouchet and Suto, 1991). The other feature is that the shape of the curves for the homogeneous distribution and the clumpy distribution is different. This is because we begin to see the effect of the error due to tree approximation in case of clumpy distribution. In case of the homogeneous distribution, the distribution of particles is close to isotropic around any given particle and hence the error cancels out. This error can be controlled by reducing $\theta_c$. We end this section with a brief comparison of the TreePM code with a PM code. We ran a simulation of the sCDM model ($262144$ particles, $64$h$^{-1}$Mpc box) with a PM code (Bagla and Padmanabhan, 1997) and with the TreePM code discussed here. Fig.3 shows a slice from these simulations; fig.3a shows the simulation with the TreePM code and fig.3b shows the same for a PM code. The large scale structures are the same in the two but there are significant differences at small scales. The halos are much more compact in the TreePM simulation, and large halos show more substructure. These differences are also clear in the two point correlation function $\bar\xi(r)$ plotted in fig.4. The thick line shows the correlation from the TreePM simulation and the dashed line shows the same for the PM simulation. As expected from fig.3 and from general considerations, the correlation function in the TreePM simulation matches with that from the PM simulation at large scales, but at small scales, the TreePM simulation has a higher correlation function. =4truein We have checked the accuracy of evolution by checking the rate of growth for the correlation function in the linear regime and also by looking for scale invariance of the correlation function for power law models. For more details please see (Bagla and Ray, 2002). Computational Resources ======================= In this section, we describe the computational resources required for the present implementation of the TreePM code. Given that we have combined the tree and the PM code, the memory requirement is obviously greater than that for either one code. We need four arrays for the PM part, the potential and the force. The rest is exactly the same as a standard Barnes and Hut tree code. With efficient memory management, we need less than $160$MB of RAM for a simulation with $128^3$ particles in a $128^3$ mesh for most part. In absence of memory management, this requirement can go up to 250MB. These are the numbers for floating point numbers, if we use double precision variables then this requirement goes up by a factor of two. ---------------- ------------- ------------- ------------- ----------- ------------- $N_{particle}$ time time time time time (ms) (ms) (ms) (ms) TreePM TreePM TreePM TreePM tree unclustered unclustered unclustered clustered unclustered P-4 PIII Alpha Alpha Alpha $32768$ 0.57 0.59 2.94 $262144$ 0.78 0.80 3.75 $2097152$ 0.34 0.89 1.22 1.28 6.03 ---------------- ------------- ------------- ------------- ----------- ------------- : Time taken by the code, per time step per particle. Column 1 lists the number of particles. Column 2, 3, 4 and 5 list the time taken (per time step per particle) by the TreePM code for an unclustered and a clustered particle distribution. Column 6 lists the same number for a tree code for an unclustered distribution of particles. All the times are in milli seconds. Table 1 lists the time required per time step per particle for three values of the number of particles. These were run on a 533MHz Alpha workstation (EV5) and compiled with the native F90 compiler, a $1$GHz Pentium III desktop or a $1.6$GHz P-4 and compiled with the Intel F90 compiler. Column 1 lists the number of particles and col.2, 3 and 4 list the time per step per particle for an unclustered distribution. This number increases much slower than the total number of particles, as expected from the theoretical scaling of $O(N\ln{N})$. Column 5 of table gives the same number for a highly clustered particle distribution, similar in clustering strength to that shown in fig.3. Column 6 lists the time per step per particle taken by the tree code for the particle distribution used in col.4. It is clear that the TreePM code is faster than the tree code by a factor of about $4.5$. It is also clear that this code performs well even on inexpensive hardware. The performance of this code can be improved further by including features like individual time steps for particles. It is expected that adding individual time steps will improve the performance by a factor of two or more. Comparison with other Methods ============================= Amongst other codes that try to augment the performance of PM codes are the P$^3$M (Efstathiou et al, 1985; Couchman, 1991) codes and the TPM code (Xu, 1995). Following subsections compare TreePM with these codes. P$^3$M and AP$^3$M ------------------ There are two main differences between P$^3$M codes (Efstathiou et al, 1985; Couchman, 1991) and the TreePM code presented here. One is that most P$^3$M codes use the natural cutoff provided by the grid for the long range force, i.e. these take the PM force to be the long range force. Hence errors in the PM force are present in the P$^3$M force. In contrast, the TreePM code uses an explicit cutoff that allows us to limit errors near the grid scale. The second difference is in terms of the time taken for the adding the short range correction as a function of clustering. In both instances, the short range force is added for particles within a fixed radius $r_{cut}$. This process is of order $O(N n r_{cut}^3 (1 + \bar\xi(r_{cut})) )$ for the P$^3$M method, where $N$ is the number of particles in the simulation, $n$ is the number density of particles and $\bar\xi(r_{cut})$ is the average number of excess particles around a particle, here excess is measured compared to a homogeneous distribution of particles with the same number density. At early times this reduces to $O(N n r_{cut}^3)$, but at late times, when the density field has become highly non-linear ($\bar\xi(r_{cut}) \gg 1$), it becomes $O(N n r_{cut}^3 \bar\xi(r_{cut}))$. As the density field becomes more and more clumpy, the number of operations required for computing the short range force increase rapidly. This is to be compared with the number of operations required for adding the short range correction in the TreePM code: $O(N \log(n r_{cut}^3 (1 + \bar\xi(r_{cut}))) )$. The linear and the non-linear limits of this expression are $O(N \log(n r_{cut}^3))$ and $O(N \log(n r_{cut}^3 \bar\xi(r_{cut})))$, respectively. Thus the variation in the number of operations with increase in clustering is much less for TreePM code than a P$^3$M code. The problem is not as severe as outlined for the Adaptive P$^3$M code (Couchman, 1991) but it still persists. Therefore the TreePM code has a clear advantage over the P$^3$M and AP$^3$M code for simulations of models where $\bar\xi(r_{cut})$ is very large. In turn, P$^3$M codes have one significant advantage over TreePM, these require much less memory. This gives P$^3$M codes an advantage on small machines and for simulations of models where $\bar\xi(r_{cut})$ is not much larger than unity. TPM --- Before we go into the differences between the TreePM and TPM methods, we would like to summarise the TPM method (Xu, 1995) here. The TPM method is an extension of the P$^3$M method in that the PM force is taken to be the long range force and a short range force is added to it. Tree method is used for adding the short range correction instead of the particle-particle method. There are some further differences, e.g. correction is added only for particles in high density regions implying that the resolution is non-uniform. At each time step, high density regions are identified and a local tree is constructed in each of these regions for computing the short range correction. Thus, there are two clear differences between the TreePM and the TPM method: - The TPM code uses the usual PM force to describe the long range component. In contrast, the TreePM code uses an explicit cutoff ($r_s$). - TreePM treats all the particles on an equal footing, we compute the short range (eqn(\[fshort\])) and the long range force for each particle. In the TPM code, the short range force is computed only for particles in the high density regions. Discussion ========== Preceeding sections show that we have developed a new method for doing cosmological N-Body simulations with a clean mathematical model. The model splits force into long and short range forces using a parameter $r_s$. By choosing this parameter judiciously, in conjunction with two other parameters that arise in the implementation of this model ($r_{cut}$ and $\theta_c$) we can obtain a configuration that matches our requirements for the error budget. It is possible to devise a more complex scheme for splitting the force into two parts but the one we have chosen seems to be the optimal scheme from the point of view of errors in force calculation as well as CPU time (Bagla and Ray, 2002). Apart from improving control over errors, the TreePM code also leads to a significant gain in speed over the traditional tree code. TreePM code is also amenable to parallelisation along the lines of (Dubinski, 1996), and is likely to scale well because the communication overhead is much more limited. Work in this direction is in progress and will be reported elsewhere (Bagla, 2002). Acknowledgement {#acknowledgement .unnumbered} =============== I would like to thank Rupert Croft, Lars Hernquist, Suryadeep Ray, Volker Springel and Martin White for insightful comments and discussions. Part of the work reported in this paper was done while the author was at the Harvard-Smithsonian Center for Astrophysics. [10]{} Bagla J.S. and Padmanabhan T. 1997, Pramana – Journal of Physics 49, 161 Bagla J.S. and Ray S. 2002, Manuscript in Preparation. Bagla J.S. 2002, To appear in proceedings of [*Numerical Simulations in Astrophysics $2002$*]{}. Barnes, J.E. 1990, J.Comp.Phys. 87, 161 Barnes J. and Hut P. 1986, Nature 324, 446 Bouchet F.R. and Kandrup H.E. 1985, ApJ 299, 1 Couchman H.M.P. 1991, ApJL 368, L23 Dubinski J. 1996, New Astronomy 1, 133 Efstathiou G., Davis M., Frenk C.S. and White S.D.M. 1985, ApJS 57, 241 Ewald P.P. 1921, Ann.Physik 64, 253 Hernquist L. 1987, ApJS 64, 715 Hernquist L., Bouchet F.R. and Suto Y. 1991, ApJS 75, 231 Hockney R.W. and Eastwood J.W. 1988, [*Computer Simulation using Particles*]{}, (New York: McGraw Hill) Rybicki G.B. 1986, in [*The Use of Supercomputers in Stellar Dynamics*]{}, ed. P.Hut and S.McMillan (Berlin: Springer), p.181 Salmon J.K. and Warren M.S. 1994, J.Comp.Phys. 111, 136 Springel V., Yoshida N. and White S.D.M. 2001, New Astronomy 6, 79 White M. 2000, Private communication. Xu G. 1995, ApJS 98, 355 \[lastpage\]

--- abstract: 'In this paper we propose a novel Bayesian kernel based solution for regression in complex fields. We develop the formulation of the Gaussian process for regression (GPR) to deal with complex-valued outputs. Previous solutions for kernels methods usually assume a *complexification* approach, where the real-valued kernel is replaced by a complex-valued one. However, based on the results in complex-valued linear theory, we prove that both a kernel and a *pseudo-kernel* are to be included in the solution. This is the starting point to develop the new formulation for the complex-valued GPR. The obtained formulation resembles the one of the *widely* linear minimum mean-squared (WLMMSE) approach. Just in the particular case where the outputs are proper, the pseudo-kernel cancels and the solution simplifies to a real-valued GPR structure, as the WLMMSE does into a *strictly* linear solution. We include some numerical experiments to show that the novel solution, denoted as widely non-linear complex GPR (WCGPR), outperforms a *strictly* complex GPR where a pseudo-kernel is not included.' bibliography: - 'CGPR.bib' - 'murilloGP.bib' - 'biblio.bib' --- Introduction ============ Complex-valued signals are present in the modeling of many systems in a wide range of fields such as optics, electromagnetics, acoustics and telecommunications, among others. The study of linear solutions for complex-valued signals has been addressed in detail in the literature. These solutions can be roughly classified into those that assume properness and those that do not. A proper complex random signal is uncorrelated with its complex conjugate [@Neeser93]. In the proper scenario, solutions for the real-valued case can be usually rewritten for the complex-valued scenario by just replacing transpose by Hermitian. However, in the improper case, the solutions are more involved and the concept of *widely* linear is introduced. Accordingly, the linear minimum mean-squared error (LMMSE) can be simply rewritten by taking into account the covariance between two random vectors. However, if the outputs are improper, an additional term must be added to include the pseudo-covariance [@Tulay11; @Schreier06]. Hence, both covariance and pseudo-covariance must be taken into account. Many non-linear tools for complex fields have been developed within the artificial neural network research community [@Mandic09; @hirose13]. In kernel methods, we may find a few results for kernel principal analysis [@Papaioannou14], classification [@Steinwart06] or regression [@OgunfunmiP11; @Bouboulis12; @Tobar12; @Boloix14]. These solutions are usually introduced as a *complexificacion* of the kernel [@Bouboulis12]. In the complexification approach, real-valued kernel tools are adapted to the complex-valued scenario by just rewriting the kernel to deal with complex-valued outputs, and inputs. However, as discussed above for linear solutions, this may suffice for the proper case, but not for the general one. Bearing this in mind, we investigate in this paper how pseudo-covariance matrices should be included in the solutions. In particular, we focus in Gaussian process for regression (GPR). Gaussian processes (GPs) are kernel Bayesian tools for discriminative machine learning [@OHagan78; @Rasmussen06; @PerezCruz13gp]. They have been successfully applied to regression, classification and dimensionality reduction. GPs can be interpreted as a family of kernel methods with the additional advantage of providing a full conditional statistical description for the predicted variable. Also, hyperparameters can be learned by maximizing the marginal likelihood, avoiding cross-validation. For real fields, GPs applied to regression can be casted as a non-linear MMSE [@PerezCruz13gp]: they present a similar structure as the LMMSE, where we replace the linear covariances by kernels, and the regularization term also depends on the prior of the weights of the generalized regression [@Rasmussen06]. In the following, we propose to develop a new formulation of GPR for complex-valued signals. We start analyzing the prediction for the real and imaginary part separately. Then we merge the results into a complex-valued formulation. In the general improper case, we show that the solution depends on both a kernel and a pseudo-kernel, to propose a *widely* complex GPR (WCGPR). Widely linear MMSE (WLMMSE) estimation ====================================== In this section we review the *widely* concept for complex-valued signals by describing the widely linear minimum mean-squared error (WLMMSE) estimation. The WLMMSE estimation of a zero-mean signal $\fv\newd: \Omega \rightarrow \CC^\d$ from the zero-mean measurement $\yv: \Omega \rightarrow \CC^\n$ is [@Picinbono95; @Schreier06] $$\begin{aligned} {\hat{\fv}}_{\newd}&=\matr{W}_{1}\yv+\matr{W}_{2}\yv^{*},\end{aligned}$$ or by making use of the augmented notation, where the complex signals are stacked on their conjugates: $$\begin{aligned} \aug{\hat{\fv}}_{\newd}=\left[ \begin{array}{c} {\hat{\fv}}_{\newd}\\ {\hat{\fv}}_{\newd}^{*}\\ \end{array}\right]=\aug{\matr{W}}\,\aug{\yv}=\left[ \begin{array}{c c} \matr{W}_{1} & \matr{W}_{2}\\ \matr{W}_{2}^{*} & \matr{W}_{1}^{*}\\ \end{array}\right]\left[ \begin{array}{c} \yv\\ \yv^{*}\\ \end{array}\right].\end{aligned}$$ The widely linear estimator is determined such that the mean-squared error is minimized, i.e., the error between the augmented estimator and the augmented signal, $\aug{\vect{e}}=\aug{\hat{\fv}}_{\newd}-\aug{{\fv}}_{\newd}$, must be orthogonal to the augmented measurement, $\aug{\yv}$, [@Picinbono95; @Schreier06]: $$\begin{aligned} \LABEQ{W} \aug{\matr{W}}=\aug{\matr{R}}_{\fv\newd\yv}\aug{\matr{R}}_{\yv\yv}\inv=\left[ \begin{array}{cc} {\matr{R}}_{\fv\newd\yv} & {\matr{\tilde{R}}}_{\fv\newd\yv}\\ {\matr{\tilde{R}}}_{\fv\newd\yv}^* &{\matr{R}}_{\fv\newd\yv}^*\\ \end{array}\right]\left[ \begin{array}{cc} {\matr{R}}_{\yv\yv} & {\matr{\tilde{R}}}_{\yv\yv}\\ {\matr{\tilde{R}}}_{\yv\yv}^* &{\matr{R}}_{\yv\yv}^*\\ \end{array}\right]\inv,\end{aligned}$$ where $\aug{\matr{R}}_{\yv\yv}$ is the augmented covariance matrix of the measurements, with covariance matrix $\matr{R}_{\yv\yv}=\mathbb{E}\left[\yv\yv\her\right]$ and pseudo-covariance or complementary covariance matrix $\matr{\tilde{R}}_{\yv\yv}=\mathbb{E}\left[\yv\yv^\top\right]$. Similarly, $\aug{\matr{R}}_{\fv\newd\yv}$ is composed by $\matr{R}_{\fv\newd\yv}=\mathbb{E}\left[\fv\newd\yv\her\right]$ and $\matr{\tilde{R}}_{\fv\newd\yv}=\mathbb{E}\left[\fv\newd\yv^\top\right]$. Now, by using the matrix-inversion lemma in , the WLMMSE estimation yields $$\begin{aligned} \LABEQ{fWLMMSE} {\hat{\fv}}_{\newd}&=\left[\matr{R}_{\fv\newd\yv}-\matr{\tilde{R}}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}^{-*}\matr{\tilde{R}}^*_{\yv\yv}\right]\matr{P}_{\yv\yv}\inv\yv\nonumber\\&+\left[\matr{\tilde{R}}_{\fv\newd\yv}-\matr{{R}}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}\inv\matr{\tilde{R}}_{\yv\yv}\right]\matr{P}_{\yv\yv}^{-*}\yv^{*},\end{aligned}$$ where $\matr{P}_{\yv\yv}=\matr{{R}}_{\yv\yv}-\matr{\tilde{R}}_{\yv\yv}\matr{{R}}_{\yv\yv}^{-*}\matr{\tilde{R}}^*_{\yv\yv}$ is the error covariance matrix for linearly estimating $\yv$ from $\yv^*$. Finally, the error covariance matrix $\matr{Q}=\mathbb{E}\left[\vect{e}\vect{e}\her\right]$ of the error vector $\vect{e}={\hat{\fv}}_{\newd}-\vect{{f}}\newd$ is [@Schreier06] $$\begin{aligned} \LABEQ{QWLMMSE} \matr{Q}&=\matr{R}_{\fv\newd\fv\newd}-\left[\matr{R}_{\fv\newd\yv}-\matr{\tilde{R}}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}^{-*}\matr{\tilde{R}}^*_{\yv\yv}\right]\matr{P}_{\yv\yv}\inv\matr{R}\her_{\fv\newd\yv}\nonumber\\&-\left[\matr{\tilde{R}}_{\fv\newd\yv}-\matr{{R}}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}\inv\matr{\tilde{R}}_{\yv\yv}\right]\matr{P}_{\yv\yv}^{-*}\matr{\tilde{R}}\her_{\fv\newd\yv}.\end{aligned}$$ It is important to note that the WLMMSE compared to the strictly linear MMSE commonly used fully exploits the dimensions of the problem, including the real and imaginary parts of every signal involved. Just in the case where the error of the LMMSE estimate is orthogonal to $\yv^*$ [@Schreier06], $$\label{eq:WLMMSEisLMMSE} \matr{\tilde{R}}_{\fv\newd\yv}-\matr{{R}}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}\inv\matr{\tilde{R}}_{\yv\yv}=0,$$ and both estimators provide the same solution $\hat{\fv}_{\newd}=\matr{R}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}\inv\yv$. Composite Gaussian Processes for Regression =========================================== Once we have defined the WLMMSE we next aim at developing the formulation for the GPR, to later relate both results. We first face the case where real and imaginary parts are estimated separately, to later merge the solutions into one complex-valued expression in the next section. GP for regression can be presented as a nonlinear regressor that expresses the input-output relation through function $f(\x)$, known as latent function, that follows a GP and underlies the regression problem $$\LABEQ{regression} {\y}=\f(\x)+\epsilon,$$ where the input vector is $\x\in\CC^\d$, and the error $\epsilon$ is modeled as additive zero-mean Gaussian noise. Given a training set $\tset=\{(\x(i),y({i})) | i = 1, . . . , \n\}=\{\X_\n,\yv_\n\}$, we aggregate the input vectors as columns in matrix $\X_\n$ and the outputs are stacked in the complex column vector $\yv=\left[{\y}({1}), . . . , {\y}(\n)\right]^{\top}=\fv(\X_\n)+\noiseOutv=\fv+\noiseOutv$. The latent function provides the multidimensional Gaussian complex-valued random vector $\fv=\left[{f}(\x({1})), . . . , {f}(\x(\n))\right]^{\top}$, where $f(\x(i))\in \CN$. The goal of the regression is to predict the value of $\fv\newd\triangleq\left[{f}(\x\newd({1})), . . . , {f}(\x\newd(m))\right]^{\top}$ for new inputs ${\X\newd }_{m}=\left[\x\newd({1}), . . . , \x\newd({m})\right]$. The straightforward way of applying GPR to complex signals is to process a composite vector where we append the imaginary values to the real ones. Then two GPs can be learned, one for the real part and another for the imaginary part of the output, either independently or using a multi-output learning or vector scheme [@Micchelli05; @Boyle05; @Alvarez12]. The model in can be rewritten in *composite* form as $$\begin{aligned} \label{eq:lrmreal} \yv\com&=\left[ \begin{array}{c} {\yv\rr}\\ {\yv\jj}\\ \end{array}\right]=\left[ \begin{array}{c}\fv\rr(\X_\n)\\ \fv\jj(\X_\n)\\ \end{array}\right]+\left[ \begin{array}{c} {\noiseOutv}\rr\\ {\noiseOutv}\jj\\ \end{array}\right]={\fv\com}(\X_\n)+\noiseOutv\com,\end{aligned}$$ where ${\yv\com}$, ${\fv\com}$ and $\noiseOutv\com$ are the composite (real) vectors for the outputs, the latent function and the noise, respectively. We assume that the real additive noise $\noiseOutv\com$ is i.i.d. Gaussian with zero mean and variance $\matr{\Sigma}_{\noiseOutv\com}$. If we assume a zero mean process and specify the covariance function of the process $\k\com(\x_{i},\x_{l})$, we can write out the corresponding $2n\times 2n$ covariance matrix $\K\com(\X_\n,\X_\n)$ elementwise from $\X_\n$, and generate the Gaussian prior $\fv\com\sim \calg{N}\left(\vect{0},\K\com(\X_\n,\X_\n)\right)$. Therefore, the observations are also Gaussian distributed, $\yv\com\sim \calg{N}\left(\vect{0},\K\com(\X_\n,\X_\n)+\matr{\Sigma}_{\noiseOutv\com}\right)=\calg{N}\left(\vect{0},\matr{C}\com\right)$, and the joint distribution of the training outputs, $\yv\com$, and the test predictions ${{\fv\com}\newd}={{\fv\com}\newd}(\X\newd)$ according to the prior yield $$\begin{aligned} \left[ \begin{array}{c} \yv\com\\ {\fv\com}\newd\\ \end{array}\right]\sim\calg{N}\hspace{-1pt}\left( \matr{0}, \left[ \begin{array}{cc} \matr{C}\com & \K\com(\X_\n,\X\newd)\\ \K\com(\X\newd,\X_\n) & \K\com(\X\newd,\X\newd)\\ \end{array}\hspace{-1pt}\right]\right)\hspace{-2pt}.\end{aligned}$$ The conditional distribution for the predictions ${{\fv\com}\newd}$ given the observations yields the predictive distribution $$\begin{aligned} {{\fv\com}\newd}| \X_\star,\X,\yv\com \sim \calg{N}\left(\vect{\mu}_{{\fv\com}\newd},\matr{\Sigma}_{{\fv\com}\newd}\right),\end{aligned}$$ and we arrive at the key predictive equations for GPR, the mean and variance given by: $$\begin{aligned} \label{eq:meanreal} &\vect{\mu}_{{\fv\com}\newd}=\K\com(\X\newd,\X_\n)\matr{C}\com\inv\yv\com, \\ &\matr{\Sigma}_{{\fv\com}\newd}=\K\com(\X\newd,\X\newd)-\K\com(\X\newd,\X_\n)\matr{C}\com\inv\K\com(\X_\n,\X\newd).\label{eq:varreal}\end{aligned}$$ Note that in the predictions and we have matrices $\K\rrrr$, $\K\rrjj$, $\K\jjrr$ and $\K\jjjj$, that are block matrices in the vector kernel matrix $$\begin{aligned} \LABEQ{Cmo} \K_{\Rext}(\X_\n,\X_\n)= \left[\begin{array}{c c} \K\rrrr(\X_\n,\X_\n) & \K\rrjj(\X_\n,\X_\n) \\ \K\jjrr(\X_\n,\X_\n) & \K\jjjj(\X_\n,\X_\n) \end{array}\right].\end{aligned}$$ Widely Complex Gaussian Process Regression ========================================== The model in can also be rewritten in the augmented vector notation by stacking the complex signals on their conjugates: $$\begin{aligned} \aug{\yv}=\left[ \begin{array}{c} \yv\\ \yv^{*}\\ \end{array}\right]=\left[ \begin{array}{c} {\fv(\X_\n)}\\ {\fv}^{*}(\X_\n)\\ \end{array}\right]+\left[ \begin{array}{c} {\noiseOutv}\\ {\noiseOutv}^{*}\\ \end{array}\right] &=\aug{\fv}(\X_\n)+\aug{\noiseOutv}\nonumber\\&=\aug{\fv}+\aug{\noiseOutv},\label{eq:modelaugcomplexlinear}\end{aligned}$$ where ${\aug{\yv}}$, $\aug{\fv}$ and $\aug{\noiseOutv}$ are the augmented vectors for the outputs, the latent function vector and the noise, respectively. There exists a simple relation between the composite vector and the augmented vector : $\aug{\yv}=\T\yv\com$, where $$\T=\left[ \begin{array}{c c} \matr{I}_{\n} & \j\matr{I}_{\n} \\ \matr{I}_{\n} & -\j\matr{I}_{\n} \\ \end{array}\right]\in \mathbb{C}^{2n\times 2n},$$ and $\T\T\her=\T\her\T=2\matr{I}_{2\n}$. Also, $\aug{\noiseOutv}=\T\noiseOutv\com$ and $\aug{\fv}=\T\fv\com$. This simple transformation allows us to calculate the augmented mean vector and the augmented covariance matrix of the prediction $\fv\newd$ from and , which are $\aug{\vect{\mu}}_{\fv\newd}=\T\vect{\mu}_{{\fv\com}\newd}$ and $\aug{\matr{\Sigma}}_{\fv\newd}=\T\matr{\Sigma}_{{\fv\com}\newd}\T\her$, respectively: $$\begin{aligned} \aug{\vect{\mu}}_{\fv\newd}&=\left[ \begin{array}{c} {\vect{\mu}}_{\fv\newd}\\ {\vect{\mu}}_{\fv\newd}^{*}\\ \end{array}\right]=\aug{\K}(\X\newd,\X_\n)\aug{\matr{C}}\inv\aug{\yv},\label{eq:meancomplex}\\ \aug{\matr{\Sigma}}_{\fv\newd}&=\aug{\K}(\X\newd,\X\newd)-\aug{\K}(\X\newd,\X_\n)\aug{\matr{C}}\inv\aug{\K}(\X_\n,\X\newd),\label{eq:varcomplex}\end{aligned}$$ where the augmented covariance matrix of the augmented observations, $\aug{\matr{C}}=\mathbb{E}\left[\aug{\yv}\aug{\yv}\her\right]=\T\matr{C}\com\T\her$, is defined as $$\begin{aligned} \label{eq:augC} \aug{\matr{C}}&=\left[ \begin{array}{cc} \matr{C} & \matr{\tilde{C}}\\ \matr{\tilde{C}}^{*}&\matr{C}^{*}\\ \end{array}\right]=\aug{\K}(\X_\n,\X_\n)+\aug{\matr{\Sigma}}_{\noiseOutv}\nonumber\\&=\left[ \begin{array}{cc} \K(\X_\n,\X_\n)& \matr{\tilde{K}}(\X_\n,\X_\n)\\ \matr{\tilde{K}}^{*}(\X_\n,\X_\n)^* &\K^{*}(\X_\n,\X_\n)\\ \end{array}\right]+\left[ \begin{array}{cc} \matr{\Sigma}_{\noiseOutv} & \matr{\tilde\Sigma}_{\noiseOutv}\\ \matr{\tilde\Sigma}_{\noiseOutv}^* &\matr{\Sigma}_{\noiseOutv}^*\\ \end{array}\right].\end{aligned}$$ Matrix $\aug{\matr{\Sigma}}_{\noiseOutv}=\T\Sigma_{\noiseOutv\com}\T\her$ is the augmented covariance matrix of the noise, and $\aug{\K}(\X_\n,\X_\n)=\T\K\com(\X_\n,\X_\n)\T\her$ is the augmented covariance matrix of $\aug{\fv}=\aug{\fv}(\X_\n)$, composed by the covariance matrix $\K(\X_\n,\X_\n)= \mathbb{E}\left[\fv(\X_\n)\fv\her(\X_\n)\right]$ and the pseudo-covariance or complementary covariance matrix $\matr{\tilde{K}}(\X_\n,\X_\n)=\mathbb{E}\left[\fv(\X_\n)\fv^\top(\X_\n)\right]$. Notice that in the general complex case, two functions must be defined to calculate matrices $\K(\X_\n,\X_\n)$ and $\matr{\tilde{K}}(\X_\n,\X_\n)$, respectively, i.e., we need a covariance function or kernel $\k(\x_{i},\x_{l})$, and a pseudo-covariance function or *pseudo-kernel*, $\tilde{\k}(\x_{i},\x_{l})$. Using the matrix-inversion lemma to find $\aug{\matr{C}}\inv$ in (\[eq:meancomplex\]) yields the mean of the prediction $$\begin{aligned} \label{eq:meanfstar} \vect{\mu}_{{\fv}\newd}&=\left[\K(\X\newd,\X_\n)- \matr{\tilde{K}}(\X\newd,\X_\n)\matr{C}^{-*}\matr{\tilde{C}}^{*}\right]\matr{P}\inv\yv\nonumber\\&+\left[ \matr{\tilde{K}}(\X\newd,\X_\n)-\K(\X\newd,\X_\n)\matr{{C}}\inv\matr{\tilde{C}}\right]\matr{P}^{-*}\yv^{*},\end{aligned}$$ where $\matr{P}=\matr{C}-\matr{\tilde{C}}\matr{C}^{-*}\matr{\tilde{C}}^{*}$. Also, the covariance matrix yields $$\begin{aligned} \label{eq:varfstar} &{\matr{\Sigma}}_{{\fv}\newd}=\K(\X\newd,\X\newd)\nonumber\\&-\left[\K(\X\newd,\X)- \matr{\tilde{K}}(\X\newd,\X)\matr{C}^{-*}\matr{\tilde{C}}^{*}\right]\matr{P}\inv\K(\X,\X\newd)\nonumber\\ &-\left[ \matr{\tilde{K}}(\X\newd,\X)-\K(\X\newd,\X)\matr{{C}}\inv\matr{\tilde{C}}\right]\matr{P}^{-*}\matr{\tilde{K}}^{*}(\X,\X\newd).\end{aligned}$$ Relation to the widely linear MMSE (WLMMSE) estimation ------------------------------------------------------ At this point it is important to remark the similarity of the widely linear MMSE estimation (WLMMSE) with the complex GPR developed above. Notice the similarity between the WLMMSE estimation in and the mean of the complex GPR prediction in (\[eq:meanfstar\]). The role of matrices $\matr{R}_{\yv\yv}$, $\matr{\tilde{R}}_{\yv\yv}$, $\matr{R}_{\fv\newd\yv}$ and $\matr{\tilde{R}}_{\fv\newd\yv}$ in the WLMMSE estimation in is played in the complex GPR prediction in (\[eq:meanfstar\]) by $\matr{C}$, $\matr{\tilde{C}}$, $\K$ and $\matr{\tilde{K}}$, respectively, i.e., covariances and pseudo-covariances are replaced by kernels and pseudo-kernels. Therefore, the mean of the proposed complex GPR prediction can be cast as a nonlinear extension to the widely linear MMSE estimation, and we may denote it as widely nonlinear complex GPR (WCGPR). The same kind of similarity is found between the error covariance matrix $\matr{Q}$ in and the WCGPR predictive covariance matrix in (\[eq:varfstar\]). In we stated that the WLMMSE estimate and the strictly linear MMSE estimate are identical, and equal to $\hat{\fv}_{\newd}=\matr{R}_{\fv\newd\yv}\matr{{R}}_{\yv\yv}\inv\yv$, if and only if (\[eq:WLMMSEisLMMSE\]) holds. Similarly, in the context of WCGPR the prediction mean (\[eq:meanfstar\]) simplifies to $$\begin{aligned} \label{eq:meanfstarred} \vect{\mu}_{{\fv}\newd}&=\left[\K(\X\newd,\X_\n)- \matr{\tilde{K}}(\X\newd,\X_\n)\matr{C}^{-*}\matr{\tilde{C}}^{*}\right]\matr{P}\inv\yv\nonumber\\&=\K(\X\newd,\X_\n)\matr{C}\inv\yv,\end{aligned}$$ if $$\label{eq:GPTotalsimp} \left[ \matr{\tilde{K}}(\X\newd,\X_\n)-\K(\X\newd,\X_\n)\matr{{C}}\inv\matr{\tilde{C}}\right]=0.$$ This takes place when, e.g. both ${\fv}$ and ${\noiseOutv}$ are proper. In this scenario, since both $\matr{\tilde{K}}$ and $\matr{\tilde{C}}$ cancel the second term in vanishes. This case is analogous to the strictly linear MMSE and this solution for proper complex GPR, that assumes a null pseudo-covariance, could be denoted as a strictly nonlinear complex GPR. This is the case studied in [@Boloix14]. Note that, in the same way that the WLMMSE compared to the strictly linear MMSE fully exploits the dimensions of the problem, the WCGPR presented in this paper also fully exploits the dimensions of the problems, while the complex GPR for the proper case in [@Boloix14] does not. This advantage is highlighted in the next section devoted to experiments. Numerical Experiments ===================== {width="7.5cm"} We propose the following example where we generated a sample function of a complex Gaussian process, then added a complex Gaussian noise to it, randomly chose training samples and tried to learn the sample function of the process by using (\[eq:meanfstar\]). In order to generate a complex Gaussian process we followed the procedure in [@Picinbono97], and the sample function of the process $f(\x)$ can be written as the output of a *widely* linear filter driven by complex proper white, zero-mean, unit-variance noise, $S(\x)=S\rr(\x)+\j S\jj(\x)$: $$\begin{aligned} \LABEQ{f} f(\x)&=\left(h_{r1}(\x)+\j h_{j1}(\x)\right) \star S(\x)\nonumber\\&+\left(h_{r2}(\x)+\j h_{j2}(\x)\right) \star S^{*}(\x).\end{aligned}$$ This procedure, sketched in , allows for the generation of both proper or improper Gaussian processes with the desired second order statistics. In this example, the filters used were parameterized exponentials: $$\begin{aligned} h(\x)=v\exp{\left(-\frac{\x\her\x}{\gamma}\right)},\end{aligned}$$ where $\gamma=0.6$ and $v=4$ for $h_{r1}(\x)$, $v=5$ for $h_{j1}(\x)$, $v=1$ for $h_{r2}(\x)$, and $v=-3$ for $h_{j2}(\x)$. We generated $100$ samples in $[-5,5]$ for both the real and the imaginary parts of the inputs to get a set of $10000$ complex-valued inputs, and the filters were normalized to have unit norm. The real part of sample function $f(\x)$ obtained is shown in (top). Complex Gaussian noise with variance $\sigma_{\epsilon}^2$ and complementary variance $\rho\sigma_{\epsilon}^2$ was added to represent measurement uncertainty. In this example we set $\sigma_{\epsilon}=0.0165$ and $\rho=0.8\exp(\j 3\pi/2)$. A set of $\n=500$ training noisy samples were randomly chosen. These samples have been depicted as circles in (top). ![Real part of the sample function of a complex Gaussian process $f(\x)$ (top) and real part of the mean WCGPR estimation (\[eq:meanfstar\]) (bottom) versus the real and imaginary parts of the input, $x$. The training samples are depicted as blue circles.](fig13){width="8.5cm"} We calculated the mean (\[eq:meanfstar\]) and variance (\[eq:varfstar\]) of the predictive distribution using the training samples. The real part of the predictive mean (\[eq:meanfstar\]) is depicted in (bottom). The mean squared error of the estimation was $10\log_{10}MSE =-12.6$ dB, computed for 10000 inputs. In a slice of the surface in is shown. The real part of the sample function of the process is plotted (black line) versus the real part of the input, the imaginary part of the input $(x)$ was fixed to the value $0.4545$. The real part of the prediction in (\[eq:meanfstar\]) is depicted in red line, along with the grey shaded area that represents the pointwise mean plus and minus two times the standard deviation. The blue circles mark the training samples. We have also compared the predictive capabilities of the proposed *widely* complex GPR in (\[eq:meanfstar\]) with that of the prediction for the proper case in (\[eq:meanfstarred\]). In the mean of the prediction in (\[eq:meanfstarred\]) is plotted as a blue line. It is shown that the proposed WGPR prediction is always closer to the actual value of $f(\x)$ than the prediction for the proper case, as expected. Finally, in we compare the mean square error of the estimation of the same $f(x)$ as before for the proposed WCGPR (\[eq:meanfstar\]) and the proper case estimation (\[eq:meanfstarred\]) versus the number of training samples. The noise variance was increased to $\sigma_{\epsilon}=0.165$ in order to check the good behavior of the proposed complex-valued regressor under a ten-fold higher noise level. All other parameters were set to the same values used in the previous experiments. It can be seen in that the proposed *widely* complex GPR performs better that the proper case estimation, with a reduction in the $MSE$ close to $2$ dB at its best. ![Real part of the sample function of a complex Gaussian process $f(\x)$ (black line) and real part of the predictive WCGPR mean (\[eq:meanfstar\]) (red line) versus the real part of the input $x$. The imaginary part of the inputs is fixed to $0.4545$. Training samples are depicted as blue circles. The blue line depicts the predictive mean for the proper CGPR case (\[eq:meanfstarred\]).](figure14.pdf){width="8.5cm"} ![Averaged $10\log_{10}(MSE)$ versus the number of training samples for the predictive WCGPR mean (\[eq:meanfstar\]) and the proper CGPR case (\[eq:meanfstarred\]).](figure1){width="8.5cm"} Conclusion ========== We have shown that developing complex-valued non-linear kernel based solutions does not suffice to replace kernels by its complex versions. In the general case, another kernel matrix, the so-called pseudo-kernel matrix must be included. We have focused on GPR to develop a novel formulation, denoted as *widely* non-linear complex-valued GPR (WCGPR), after the *widely* linear MMSE, as it exhibits a quite similar structure. The pseudo-kernel or pseudo-covariance matrix in this formulation models the covariance between the outputs and their conjugates. If this pseudo-covariance cancels, i.e. the outputs are proper, WCGPR yields a strict non-linear complex formulation, as the WLMMSE yields a strict LMMSE. Other special cases can be also derived from this general solution. Through numerical experiments we show that the proposed formulation outperforms the strictly non-linear complex-valued GPR when learning a complex Gaussian process generated using widely linear filters.

--- abstract: | We investigate the classical and quantum dynamics of an electron confined to a circular quantum dot in the presence of homogeneous $B_{dc}+B_{ac}$ magnetic fields. The classical motion shows a transition to chaotic behavior depending on the ratio $\epsilon=B_{ac}/B_{dc}$ of field magnitudes and the cyclotron frequency ${\tilde\omega_c}$ in units of the drive frequency. We determine a phase boundary between regular and chaotic classical behavior in the $\epsilon$ vs ${\tilde\omega_c}$ plane. In the quantum regime we evaluate the quasi-energy spectrum of the time-evolution operator. We show that the nearest neighbor quasi-energy eigenvalues show a transition from level clustering to level repulsion as one moves from the regular to chaotic regime in the $(\epsilon,{\tilde\omega_c})$ plane. The $\Delta_3$ statistic confirms this transition. In the chaotic regime, the eigenfunction statistics coincides with the Porter-Thomas prediction. Finally, we explicitly establish the phase space correspondence between the classical and quantum solutions via the Husimi phase space distributions of the model. Possible experimentally feasible conditions to see these effects are discussed. Pacs: 05.45.+b address: | [*Department of Physics and Center for Interdisciplinary Research on Complex Systems,\ Northeastern University, Boston Massachusetts 02115, USA*]{} author: - 'R. Badrinarayanan and Jorge V. José' title: | Classical and Quantum Chaos in a quantum dot\ in time-periodic magnetic fields --- Introduction {#sec:intro} ============ In this paper, we present results of a study of the behavior of an electron confined to a disk of finite radius, subjected to spatially uniform, constant ($B_{dc}$) plus time-varying ($B_{ac}$) perpendicular magnetic fields. This allows us to analyze an old problem which exhibits some very novel behavior because of the time-dependent field. Without this time varying component of the field, the electronic states form the oscillator-like Landau levels[@fock]. With the addition of confinement, this constant field problem was studied in great detail by Dingle[@dingle]. He obtained perturbative solutions and subsequently others obtained numerical and exact[@robnik] solutions. The solutions depend on the ratio of the cyclotron radius $\rho_c$ to the confinement radius $R_0$. One of the most important consequences of confinement is the presence of ‘skipping orbits’, which play an important role, for example, in the Quantum Hall Effect[@prange]. This problem is of significant interest as a consequence of two independent developments over the past few years. One, the important advances in our knowledge of classical chaos[@ll], and to a lesser extent, it’s quantum and semiclassical counterparts[@casati1]; and two, the spectacular advances in the fabrication of very clean mesoscopic quantum devices[@beenakker], where a high-mobility two-dimensional electron gas is trapped within a boundary of controlled shape. We attempt to begin to bring the two fields together by asking how this model system behaves from the classical dynamical point of view and what it’s quantum signatures are. We predict ranges of fields and frequencies where some novel effects may be experimentally observable. In this paper, we consider the single-electron case and leave for a future publication the many electron problem. This paper is organized as follows: In section II we define the model with its classical and quantum-mechanical properties, elucidate the important parameters in the problem and describe the general method of solution. In section III, we investigate the properties of the classical model. Based on a combination of analytic and numerical analysis, we obtain a ‘phase diagram’ in the parameter space of the system, which separates the quasi-integrable from the chaotic regions. This phase diagram is shown in Fig.1. The vertical axis is the ratio $\epsilon=B_{ac}/B_{dc}$ of the magnitudes of the fields, and the horizontal axis is the Larmor frequency normalized to the [*a.c.*]{} drive frequency, ${\tilde\omega_c}=\omega_c/\omega_0$. This phase diagram is of paramount importance in making the connection between the classical and quantum solutions. The values of the d.c. field $B_{dc}$ and drive frequency $\omega_0$ depend on the radius of the dot $R_0$ and certain other parameters. However, to give an idea of the magnitudes of the physical parameters involved, let us pick two representative points on the diagram: $({\tilde{\omega_c}},\epsilon)$ = (0.1, 0.1) corresponds to $\omega_0$ = 20 GHz and $B_{dc}$ = 20 gauss when $R_0 = 1\mu m$, while $\omega_0$ = 800 MHz and $B_{dc}$ = 0.08 gauss for $R_0 = 5\mu m$. Similarly, $({\tilde{\omega_c}},\epsilon)$ = (2.0, 2.0) corresponds to $\omega_0$ = 20 GHz and $B_{dc}$ = 800 gauss for $R_0 = 1\mu m$, while $\omega_0$ = 20 GHz and $B_{dc}$ = 32 gauss for $R_0 = 5\mu m$. The details of the these estimates are presented in Section V. We analytically obtain conditions and look at various kinds of fixed points of the classical solutions. In section IV we study the spectral statistics of the quantum evolution operator, which shows clear signatures of the classical transition from quasi-integrabality to chaos. We also discuss the eigenfunctions properties in different regimes using the $\chi ^2$ distribution of $\nu$ freedoms as a convenient parameterization of the results. Then, we turn to semiclassical correspondences, where we use a phase-space approach to the quantum eigenfunctions, and make direct connections with various types of classical phase space periodic orbits. In section V we discuss possible experimental scenarios where the predicted effects may be observable. Finally, in section VI we summarize our results and present our conclusions. The Model {#sec:model} ========= The model of a quantum dot we consider here is that of an electron confined to a disk of radius $R_0$ subject to steady ([*d.c.*]{}) and time-periodic ([*a.c.*]{}) magnetic fields. Choosing the cylindrical gauge, where the vector potential ${\bf A}(\vec \rho,t) = {1\over 2}B(t)\, \rho\, \hat e_\phi$, $B(t)$ being the time-dependent magnetic field, the quantum mechanical single-particle Hamiltonian in the coordinate representation is given by $$\label{eq:a} H = -\frac{{\hbar^2}}{2m^*}\left( \frac{d^2}{d\rho^2} + \frac{1}{\rho}\frac{d}{d\rho} + \frac{1}{\rho^2}\frac{d^2}{d\phi^2} \right) + \frac{1}{8} m^* \Omega^2(t) \rho^2 + \frac{1}{2} \Omega(t) L_z, \quad 0 \leq \rho \leq R_0,$$ where $m^*$ is the effective mass of the electron (roughly 0.067$m_e$ in GaAs-AlGaAs semiconductor quantum dots) [@beenakker], $L_z$ is the operator of the conserved angular momentum, and $\Omega(t) = e^{*}B(t)/m^*c$, $e^{*}$ and $c$ being the effective electronic charge ($e^{*}\sim 0.3e$) and the speed of light, respectively. Let the magnetic field be of the form, $B(t) = B_{dc} + B_{ac} f(t)$, where $f(t)=f(t+T_0)$ is some periodically time varying function. We can separate the Hamiltonian $H=H_{dc} + H_1(t)$, where $$\label{eq:b} H_{dc} = -\frac{{\hbar^2}}{2m^*}\left( \frac{d^2}{d\rho^2} + \frac{1}{\rho}\frac{d}{d\rho}\right) + \frac{{\hbar^2\ell ^2}}{2m^*}\frac{1}{\rho^2} + \frac{1}{8} m^* \omega_{c}^2(t) \rho^2 + \frac{1}{2} \frac{\ell \hbar \omega_c}{2},$$ and $H_1(t)=\frac{1}{8}m^*\left(2B_{dc}B_{ac}f(t) + B_{ac}^2f^2(t)\right)\rho^2$. Here $H_{dc}$ is the standard static Hamiltonian for a charge in a homogeneous constant perpendicular magnetic field, that includes the para- and dia-magnetic contributions, with $\omega_c = \frac{e B_{dc}}{m^* c}$. With the additional dropping of a term of the form $L_z B_{ac}f(t)$ which can trivially be removed by a unitary transformation, $H_1(t)$ gives the time-dependent contribution to $H$. Note that $H_1(t)=H_1(t+T_0)$. In the limit in which $H_1(t)$ is much smaller than $H_{dc}$ one can study the modification to the solutions associated to $H_{dc}$ by standard time-dependent perturbation theory. As can be seen from the classical phase diagram given in Fig. 1 the boundary between regular and chaotic behavior in fact occurs for $\epsilon=B_{ac}/B_{dc}>1$ and ${\tilde{\omega_c}}>1$. We are then led to approximate $H_1(t)$ by, $$\label{eq:bb} H_1(t)=\frac{1}{8}m^*\left({(\epsilon\omega_{c})}^2 \sum _{n=-\infty}^{\infty}\delta(t-nT_0)\right)\rho^2.$$ With this simplification, the Hamiltonian (\[eq:a\]) is then approximated by the sum of Eqs. (\[eq:b\]) and (\[eq:bb\]). This choice is also motivated by the following factors: 1. Calculational ease: the delta function is the paradigm for time-dependent systems because one can proceed further in the analysis without recourse to drastic approximations; 2. Effects of chaos: since our primary objective is to explore the quantum manifestations of classical chaos, we are more interested in the general issues of chaos, rather than specific functional forms. Even for a more ‘physical’ choice of $f(t)=A\cos(\omega t)$, one can easily show that the resulting functional form of $\Omega^2(t)$ can be approximated sensibly as above; and 3. Classical considerations: as shown in the Appendix, starting from the Lorenz force plus Maxwell’s equations, one can write the classical equations of motion [*exactly*]{} including the self-induced fields, even for the magnetic field given by $B(t) = B_{dc} + B_{ac} T_0\sum_{n=-\infty}^{\infty} \delta (t-nT_0)$. Classically, the associated Lagrangian is linear in the vector potential. There are regularization problems, however, when using this form in the quantum Hamiltonian, since in this case there is an ill defined $A_{ac}^2(t)$ term present. However, the model $H=H_{dc}+H_1$ is well defined. In order to more clearly see what the relevant parameters in the problem are, we go over to dimensionless units, defined by rescaling all lengths to the disk radius $R_0$, all masses by the effective mass $m^*$ and all times by the period of the $a.c.$ field, $T_0$. Thus, we define \[allf\] $$\label{eq:f1} r = \rho/R_0; \qquad 0\le r \le 1,$$ $$\label{eq:f2} \tau = t/T_0 \equiv \frac{\omega_0}{2\pi}t, \qquad {\tilde\omega_c} = \omega_c/\omega_0, \qquad {\tilde\hbar} = \frac{\hbar}{m^*\omega_0 R_0^2}.$$ In these units, equations (\[eq:b\]) and (\[eq:bb\]) become \[allg\] $$\label{eq:g0} \tilde H = \tilde H_{dc} + \tilde H_1(\tau)$$ $$\label{eq:g1} \tilde H_{dc} = -\frac{{\tilde\hbar}^2}{2}\left( \frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} \right) + \frac{\ell^2 {\tilde\hbar}^2}{2 r^2} + \frac{1}{2} \left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + \ell\, {\tilde\hbar} \frac{{\tilde\omega_c}}{2} ,$$ $$\label{eq:g2} \tilde H_1(\tau) = \frac{1}{2}\ \eta\ r^2 \sum_{n=-\infty}^{\infty} \delta (\tau-n), \quad {\rm where}$$ $$\eta = \left(\frac{\epsilon\,{\tilde\omega_c}}{2}\right)^2,$$ and the corresponding solutions to the time-independent part, along with the boundary and normalization conditions, are given by \[allh\] $$\label{eq:h1} \tilde H_{dc}\, \tilde\Psi_{n\ell}(r,\phi) = \tilde E_{n\ell}\, \tilde\psi_{n\ell}(r) \frac{e^{i \ell \phi}}{\sqrt{2 \pi}},$$ $$\label{eq:h2} \tilde\Psi(r,\phi) = \sum_{n=1}^{\infty} \sum_{\ell=-\infty}^{\infty} \tilde\psi_{n\ell}(r) \frac{e^{i \ell \phi}}{\sqrt{2 \pi}},$$ $$\label{eq:h3} \tilde\psi_{n\ell}(r = 1) = 0, \qquad {\rm and} \qquad \int_{0}^{1} \tilde\psi_{n\ell}^{2}(r) \, r \,dr = 1.$$ As was first pointed out by Dingle[@dingle], the physically acceptable solutions to equations (\[allh\]) are the Whittaker functions of the first kind[@abramowitz], $$\label{eq:i} \tilde\psi_{n\ell}(r) = \sqrt{\frac{2}{N_{n\ell}}}\; \frac{1}{r}\;M_{\chi_{n\ell},{\mid \ell\mid}/2}(2\pi F\, r^2),$$ where the frustration parameter $F = \frac{\Phi}{\Phi_{0}}$ is the ratio of the flux threading the dot to the quantum of flux $\Phi_0=h/2e$. The quantities $\chi_{n\ell}$ are related to the eigenvalues via $$\label{eq:k} \chi_{n\ell} = \frac{1}{2}( \tilde E_{n\ell} - \ell ),$$ and are determined precisely by the requirement that the wavefunction vanishes at the boundary, equation (\[eq:h3\]), $M_{\chi_{n\ell},{\mid \ell\mid}/2}(F) = 0$. In the limit of no confinement, $R_0 \rightarrow \infty$, we recover the usual Laguerre polynomial solutions for the $\tilde\psi_{n\ell}$’s. The frustration parameter $F$ can also be written as $$\label{eq:m} F = \frac{1}{4\pi}\left(\frac{R_0}{\ell_H}\right)^2, \qquad {\rm where} \qquad \ell_H=\left(\frac{\hbar c}{eB_{dc}}\right)^{1/2},$$ that is, it’s proportional to the square of the ratio of the confinement radius to the magnetic length. When $2\pi F \ll 1$, the problem is equivalent to that of a nearly free electron, bound by a very weak magnetic field, and so is amenable to a perturbative treatment. In the opposite limit, the boundary can essentially be neglected, and we recover the results of Dingle mentioned previously. It is in the intermediate regime, when the two lengths are comparable, that we expect the effects of confinement to be nontrivial, especially in the presence of strong time-dependent fields. In principle, we are able to cover the entire range of parameter values within the same framework by means of a numerical evaluation of the Whittaker functions. However, the Whittaker functions are not very well suited to large scale computations, because of the time required to evaluate each individual function. We choose instead to perform most of our calculations in a Fourier sine basis, which is numerically much faster, and use the Whittaker basis as a check on our results. Choosing the (orthonormalized) basis functions as, $$\label{eq:eq1} \chi_{n\ell} = \sqrt{\frac{2}{r}} \sin(n\pi r) \frac{e^{i\ell\phi}}{\sqrt{2\pi}},$$ one can show, after a straightforward calculation, that the matrix elements of $\tilde H_{dc}$ are given by, $$\begin{aligned} \label{eq:eq2} (\tilde H_{dc})_{mn} &=& \Bigg\{ \frac{{\tilde\hbar}^2}{2}(n\pi)^2 + n\pi (\ell^2-\frac{1}{4}) {\tilde\hbar}^2{\rm Si}(2n\pi) + \frac{1}{2}\left (\frac{{\tilde\omega_c}}{2}\right)^2 \left(\frac{1}{3}-\frac{1}{2n^2\pi^2}\right) + \ell{\tilde\hbar} \frac{{\tilde\omega_c}}{2} \Bigg\} ~ \delta_{mn} \nonumber \\ &+& \Bigg\{ \frac{\pi}{2}(\ell^2-\frac{1}{4}){\tilde\hbar}^2\left \{ (m+n){\rm Si} \left[(m+n)\pi\right] - (m-n){\rm Si}\left[(m-n)\pi\right]\right\} \Bigg . \nonumber \\ &+& \Bigg . \frac{1}{2}\left(\frac{{\tilde\omega_c}}{2}\right)^2 \frac{(-)^{m+n}}{\pi^2} \frac{8mn}{(m^2-n^2)^2} \Bigg\}~ (1-\delta_{mn})\end{aligned}$$ where ${\rm Si}(x)$ is the Sine integral. One can similarly compute matrix elements of other needed operators. Having worked out a suitable set of basis functions, we now proceed to tackle the full time-dependent problem. The Schrödinger equation for the time evolution operator is, $$\label{eq:n} i {\tilde\hbar}\,\frac{\partial}{\partial\tau}\,U(\tau,\tau_0) = ( \tilde H_{dc} + \tilde H_1(\tau) )\, U(\tau,\tau_0).$$ Since we have a periodic system, $\tilde H(\tau+1) = \tilde H(\tau)$, from the Floquet theorem[@casati2], it is sufficient to determine the one-period time evolution operator $U(\tau_0 + 1,\tau_0)$, from \[allo\] $$\label{eq:o1} i {\tilde\hbar}\,\frac{\partial}{\partial\tau}\,U(\tau) = ( \tilde H_{dc} + \tilde V_1(\tau) )\,U(\tau) \, , \quad 0 < \tau \le 1,$$ $$\label{eq:o2} \tilde V_1(\tau) = \tilde V \, \delta(\tau-1), \qquad {\rm where} \qquad \tilde V = \frac{1}{2}\ \eta\ r^2,$$ where the parameter $\eta$ has been defined previously. All the information about the dynamics of the system is contained within this Floquet operator, since $\Phi(r,\phi,\tau+1) = U\, \Phi(r,\phi,\tau)$, where $\Phi$ is the total wave function. Because of the periodic $\delta$-kicked dynamics, we can immediately integrate equation (\[eq:o1\]) to get $$\label{eq:p} U_\ell(1,0) = \exp\left(-\frac{i}{{\tilde\hbar}}\,\tilde V\right)\, \exp\left(-\frac{i}{{\tilde\hbar}}\,\tilde H_{dc}\right).$$ The subscript $\ell$ has been attached to $U$ to emphasize that the evolution operator has been restricted to that single $\ell$ value. In other words, states with different values of $\ell$ evolve independently, an immediate consequence of the conservation of angular momentum in this system. The rightmost exponential operator in equation (\[eq:p\]) evolves the wave function from just after the ‘kick’ at $\tau=0$ to just before the kick at one period under the influence of $\tilde H_{dc}$, while the operator to it’s left propagates it from just before to just after the kick at a period. Since $U$ is an Unitary operator, the spectrum of it’s eigenvalues can be represented as $$\label{eq:q} U_\ell\,\phi_{n\ell} = e^{i\,\varepsilon_{n\ell}}\,\phi_{n\ell}.$$ The set of eigenvalues $\{\varepsilon_{n\ell} \in (0,2\pi]\}$, are collectively known as the Quasi-energy Eigenvalues (QEE), and the eigenfunctions $\{\phi_{n\ell}\}$ as the Quasi-energy Eigenfunctions (QEF) of $U$. The investigation of the quantum dynamics of the system is completely equivalent to determining the nature of the QEE and QEF. The fundamental task is thus to obtain the Quasi-energy Spectrum (QES) of the evolution operator given by equation (\[eq:p\]). Classical Dynamics ================== We begin the discussion of the behavior of the model by looking at it’s classical dynamics. The classical Hamiltonian corresponding to the quantum one given by equations (\[allg\]) is, \[alls\] $$\label{eq:s0} \tilde H = \tilde H_{dc} + \tilde H_1(\tau)$$ $$\label{eq:s1} \tilde H_{dc} = \frac{1}{2} p_r^2 + \frac{J^2}{2 r^2} + \frac{1}{2} \left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + J \frac{{\tilde\omega_c}}{2} ,$$ $$\label{eq:s2} \tilde H_1(\tau) = \frac{1}{2}\ \eta\ r^2 \sum_{n=-\infty}^{\infty} \delta (\tau-n),$$ where $p_r$ is the radial momentum and $J$ is the [*conserved*]{} angular momentum. To make quantitative correspondences between the classical and quantum results, we always set the numerical values of the angular momenta in the two cases to be equal, [*i.e.*]{}, we set $J=\ell\, {\tilde\hbar}$. In between the ‘kicks’ at a period, and as long as it does not hit the boundary at $r=1$, the electron’s motion is governed by the static Hamiltonian $\tilde H_{dc}$. The equation of motion in this case is $$\ddot r = -\left(\frac{{\tilde\omega_c}}{2}\right)^2 r + \frac{J^2}{r^3},$$ whose solution, in terms of the energy $E$, $$\label{eq:t} E = \frac{1}{2} p_r^2 + \frac{J^2}{2 r^2} + \frac{1}{2} \left(\frac{{\tilde\omega_c}}{2}\right)^2 r^2 + J \frac{{\tilde\omega_c}}{2},$$ is given by $$\label{eq:u} \left( \begin{array}{c} r(\tau) \\ p_r(\tau) \end{array} \right) = \left( \begin{array}{c} \sqrt{\frac{2}{{\tilde\omega_c}}\left[ b + a \sin\left\{ {\tilde\omega_c}(\tau-\tau_0) + \sin^{-1}\left(\frac{\frac{1}{2}{\tilde\omega_c} r_0^2 - b}{a} \right)\right\}\right]} \\ \frac{a}{r(\tau)}\cos\left\{ {\tilde\omega_c}(\tau-\tau_0) + \sin^{-1} \left(\frac{\frac{1}{2}{\tilde\omega_c} r_0^2 - b}{a}\right)\right\} \end{array} \right),$$ where $$\label{eq:v} b = 2E/{\tilde\omega_c} - J \quad {\rm and} \quad a = \sqrt{b^2 - J^2}.$$ Here, $r_0$ and $\tau_0$ are initial conditions. For a given energy $E$, the motion is constrained by the centrifugal barrier on one side, and the smaller of the wall radius (equal to 1) and the constraint imposed by the attractive quadratic potential on the other: \[allw\] $$\label{eq:w1} r_{min} \le r(\tau) \le {\rm Min}\{r_{max},1\}, \qquad {\rm where}$$ $$\label{eq:w2} r_{min} = \sqrt{\frac{2}{{\tilde\omega_c}}(b-a)},\qquad {\rm and} \qquad r_{max} = \sqrt{\frac{2}{{\tilde\omega_c}}(b+a)}.$$ Note that the equations of motion are nonlinear here, even in the walls’ absence. The effect of collision with the wall (or centrifugal barrier) is simply to reverse the direction of motion: $$\label{eq:x} \left( \begin{array}{c} r(\tau_c^+) \\ p_r(\tau_c^+) \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \left( \begin{array}{c} r(\tau_c^-) \\ p_r(\tau_c^-) \end{array} \right),$$ where $\tau_c$ is the time of collision with the wall (or barrier). Finally, the effects of the kicks at $\tau=n$ are obtained by integrating the equations of motion over an infinitesimal duration around $n$: $$\label{eq:y} \left( \begin{array}{c} r(n^+) \\ p_r(n^+) \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ \eta r & 1 \end{array} \right) \left( \begin{array}{c} r(n^-) \\ p_r(n^-) \end{array} \right).$$ If we denote the mapping due to the ‘free’ evolution of the particle under the influence of $H_{dc}$ by $M_0$ (equations (\[eq:u\])), that due to the walls by equation by $M_{wall}$ (equations (\[eq:x\]), and the mapping due to the kick by $M_{kick}$ (equations (\[eq:y\])), then the complete one-period map is typically given by the product of several $M$’s for a given energy, [*i.e.*]{}, $$M_{T} = \left( M_0\cdot M_{wall}\right)^N \cdot M_{kick}.$$ In general, the map is very complicated, and very sensitive to initial conditions. By recording the values at each successive period, we obtain a surface-of-section of the trajectory of the particle in phase space. There are three independent parameters in the problem: ${\tilde\omega_c}$, $\epsilon$ and ${\tilde\hbar}$. However, for quantitative correspondences to be made later with the quantum results, as mentioned earlier, we keep the angular momentum $J=\ell{\tilde\hbar}$ fixed, which reduces the number of parameters to the first two. The transition to chaos is manifested in the parameter space spanned by $(\epsilon,{\tilde\omega_c})$ (see Fig. 1). All of our subsequent results refer to this space. We did investigate the effects of varying $J$ by varying ${\tilde\hbar}$ for fixed $\ell$, and the results are even quantitatively very similar. The first (and most obvious) evidence of chaotic behavior is seen in the Poincaré surface of section in $(r,p_r)$. In Figures 2(a)–(d) we show the sections corresponding to $\epsilon$ values of 0.5, 1.5, 1.95 and 2.5, respectively, while ${\tilde\hbar}=0.01$, ${\tilde\omega_c}=2.0$ and $\ell=5$ are held fixed. (The reason for this particular choice has to do with the $(\epsilon,{\tilde\omega_c})$ ‘phase diagram’ for this system, which is explained in more detail shortly.) In the quasi-integrable regime (Figs. 2(a),(b)), the phase space is dominated by invariant tori, which are close to those of the unperturbed problem. As the value of ${\tilde\omega_c}$ is increased, the tori begin to break up, and isolated chaotic islands begin to appear (Fig. 2(c)), until finally, all evidence of invariant curves disappears and all we see is the uniform chaotic sea (Fig. 2(d)). These values of $(\epsilon,{\tilde\omega_c})$ corresponding to the integrable, intermediate and chaotic regions will be retained throughout what follows to make comparisons between the classical and quantum results. Corresponding to the transition from regular to chaotic behavior, we begin to see the appearance of diffusive growth in the averaged energy (or squared momentum) of a localized ensemble of initial conditions. Figure 3(a) shows the average energy as a function of time for the parameters corresponding to the quasi-integrable regime, while Figure 3(b) corresponds to parameter values in the chaotic regime. In contrast to the behavior in the quasi-integrable regime, where the energy $E$ is regular, oscillatory quasiperiodic functioning of time around a constant value, in the chaotic regime $E$ grows linearly (or $p_r$ grows quadratically) with time. (Here and subsequently, ‘time’ refers to stroboscopic time, just after every kick ). A quantitative measure of the degree of chaos in the system is to calculate the largest Lyapunov exponent. (In our reduced two-dimensional phase space since the flow is Hamiltonian, the Lyapunov exponents come in pairs of opposite sign.) Because our phase-space is bounded, we use a slightly modified approach from that used for an unbounded system to the calculation of the exponent, as outlined in Reichl[@reichl]. The (largest) Lyapunov exponent is defined by, $$\label{eq:eq17} \lambda_n(\tau,{\rm{\bf X}}_{0,0},{\rm{\bf Y}}_{0,0}) = \frac{1}{n\tau} \sum_{j=1}^{n} \ln\left(\frac{d_j}{d_0}\right),$$ where $d_0 = |{\rm{\bf Y}}_{0,0}~-~{\rm{\bf X}}_{0,0}|$ is the Euclidean distance between the position of neighboring trajectories labelled by [**X**]{}$_{0,0}$ and [**Y**]{}$_{0,0}$, and $\{d_j\}, j=1,\dots,n$ are the sequence of distances generated between the trajectories at $n$ successive time steps. If $d_0$ is not too big, then the limit, $lim_{n\uparrow\infty} \lambda_n(\tau,{\rm{\bf X}}_{0,0},{\rm{\bf Y}}_{0,0}) = \lambda({\rm{\bf X}}_{0,0})$ exists, and is independent of both $d_0$ and $\tau$. Furthermore, $\lambda({\rm{\bf X}}_{0,0})$ is zero if [**X**]{}$_{0,0}$ is chosen in a regular region, while it is positive if [**X**]{}$_{0,0}$ is chosen to lie in a chaotic region. With the help of the Lyapunov exponent we constructed the ‘phase diagram’ shown in Fig.1 for this system in the $(\epsilon,{\tilde\omega_c})$ parameter space in the following fashion. For a given set of parameters $(\epsilon,{\tilde\omega_c})$, we choose a very large number (typically $10^6$) initial conditions ${\rm{\bf X}}_{0,0}$ spread uniformly in $(r,p_r)$ phase space. Next we randomly choose a nearby phase space point ${\bf Y}_{0,0}$ within a circle of radious $d_0$, centered about ${\bf X}_{0,0}$. We calculate the Lyapunov exponent, using formula (25), from the successive evaluation of the distances $d_j$ for each $j$ iteration of the mapping. This process is repeated for several nearby ${\bf Y}_{0,0}$ trajectories. When the Lyapunov exponent reaches saturation we average the resulting value over the set of initial conditions to find $\lambda$. If this asymptotic value is positive, the system is defined as chaotic. To put a stricter criterion on the degree of chaos, we choose a threshold value of the exponent $\lambda_c$ beyond which the system is in the regime of hard chaos. We set $\lambda_c$ arbitrarily to the value 1, but as a check we generated Poincaré phase portraits to confirm chaos by looking for featureless ([*i.e.,*]{} no invariant tori) phase portraits. In this way, by varying the parameters $(\epsilon,{\tilde\omega_c})$ in a continuous fashion over the whole plane, running the map repeatedly and obtaining the resulting $\lambda$’s, we obtained the ‘phase diagram’ for this system, including a distinct ‘phase boundary’ separating the quasi-integrable and hard chaos regions. Of course, this phase boundary depends on the precise value of the cutoff $\lambda_c$ we choose. Nevertheless, we checked that on varying the cutoff $\lambda_c$, the phase boundary shifts only slightly and furthermore, the shape of the boundary remains qualitatively the same. Indeed, to a high degree of precision, the phase boundary can be fitted by $$\label{eq:eq18} {\tilde\omega_c} = C(\lambda_c)/\epsilon,$$ where $C(\lambda_c)$ is a constant which depends on the value of the cutoff. Figure 1 shows the phase diagram for a cutoff $\lambda_c=1$. We observe from the classical Poincaré sections that there is a symmetry line in the $(r,p_r)$ plane. This arises from the time-reversal invariance present in the problem as follows. Consider a particle kicked at $\tau=0$. The position $r_0$ remains unchanged, while the momentum changes : $p_r^{(+)} = p_r^{(-)} + \eta\ r_0$. Denoting $p_r^{+}$ by $p_0$, then at time $0^{(-)}$ the particle had momentum $p_r^{(-)} = p_0 -\eta\ r_0$. Taking into account the fact that the angular momentum is conserved, we see that propagating a particle [*forward*]{} in time from $(r_0,p_0)$ is the same as propagating it [*backward*]{} from $(r_0,\eta\ r_0-p_0)$. Thus, the motion is symmetric about the line $p_r = -\frac{1}{2}\eta\ r$. This symmetry is, of course, present in the quantum problem also, where it will be exploited when calculating the Husimi distributions of the QEF’s. In the classical case, we exploit its existence to plot the stable manifolds around hyperbolic fixed points, which are otherwise very difficult to do because of their extreme sensitivity to perturbations. Although the map is very complicated, there are a few periodic orbit cases that one can analytically study. By following the trajectory of the periodic orbit in phase space, and given the mapping equations, we can reconstruct the initial conditions giving rise to the orbit. For example, the fixed point shown in the Fig.5 (for $\ell=5$, ${\tilde\hbar}=0.008$, ${\tilde\omega_c} =2\sqrt{2}$ and $\epsilon=1.0$), labeled F, is given by $r_0 = 0.75528003154206\ldots$, $p_0 = 2.43838534012017\ldots$. Quantum to Classical correspondence {#subsubsec:sqfad} =================================== As mentioned in the introduction, one of the clear quantum manifestations of classical chaos (QMCC) emerges when one compares the spectral properties of specific model systems as appropriate parameters are tuned to classically produce a transition from integrable to completely chaotic regimes. In this section we follow the general thinking developed in Random Matrix Theories (RMT) to implement different tests to quantify the spectral properties of the model. These properties are obtained from a direct diagonalization of the one-period time evolution matrix. For the results presented here we vary the value of $\epsilon$ while keeping $J$, ${\tilde\hbar}$ and ${\tilde\omega_c}$ fixed, so as to go from the integrable to the chaotic regime in the phase diagram that coincide with the values considered in the classical case. We note that the appropriate RMT statistical ensemble is a COE rather than a CUE, because this model has a false-T breaking symmetry. Next we discuss the RMT tests and their application to the results obtained for the QEE of our quantum dot model. Nearest neighbor QEE distributions ---------------------------------- A local measure often used in RMT is the distribution of nearest-neighbor energy level separations, $P(s)$, where $s=\varepsilon_{n+1}-\varepsilon_n$. In the extreme integrable and chaotic regimes it has been established[@bohigas2; @jose] that $P(s)$ takes the Poisson or Wigner distribution forms, $$\label{eq:qee} P_P(s) = e^{-s} \qquad {\rm and} \qquad P_W(s) = {\pi\over 2}\,s\,e^{-{\pi\over 4}s^2},$$ respectively. A convenient and often successful parameterization of the $P(s)$ obtained in the transition between $P_P$ to $P_W$ is provided by the Brody interpolation formula[@brody]: $$P_{\nu}(s) = \gamma(\nu+1)\,s^{\nu}\,\exp(-\gamma s^{\nu+1}),$$ where $\gamma = \left[\Gamma\left({\nu+2\over\nu+1}\right)\right]^ {\nu+1}$, and $\Gamma(x)$ is the Gamma function. This distribution is normalized and, by construction, has mean spacing $\langle{s}\rangle=1$. We recover the Poisson case taking $\nu =0$ and Wigner for $\nu =1$. A criticism to the Brody distribution is, however, that there is no first principles justification for its validity. The fact remains that it does fit the specific results found when considering explicit model systems. Results of the transition, as parameterized by $\nu$, are shown in Figure 5. We also calculated higher-order eigenvalue spectral correlations[@bohigas1]. The average number of levels in an interval of length L is $<n(L)> = {1\over L}\sum_{\alpha< }n(\alpha ,L),$ where the $< >$ stands for spectral average, and $n(\alpha ,L)$ is the number of levels in an interval of length $L$ starting at $\alpha$ and ending at $\alpha + L$. Also important are the various moments of the level distribution. The one considered here is the second moment of the average number of levels in a given stretch of length $L$ of the spectrum, the $\Sigma^2(L)$ statistic $$<\Sigma ^{2}(L)> = \left <(n(\alpha ,L) - <n(\alpha ,L)>)^2 \right >.$$ Another often calculated statistic is the Dyson-Mehta $\Delta _3(L)$ which measures the stiffness of the spectrum. This is defined by $${\Delta _3(L,\alpha )}={1\over L}{min_{A,B}} {\int _{\alpha }^{\alpha +L}} {[{\tilde N}(x)-Ax-B]^2}\,dx,$$ where $\tilde N(x)$ is the unfolded number density. In our case there is no need to unfold the spectrum since it is fully contained between 0 and 2$\pi$; $\Delta _3$ is just the least mean square deviation of ${\tilde N}(x)$ from the mean straight line behavior. This statistic is directly proportional to the $<{\Sigma ^2}>$ by $\Delta _3 (L)={2\over {L^4}} \int _0^L(L^3-sL^2x+x^3) \Sigma ^2(x)dx$, and thus can be calculated for the Circular Orthogonal Ensembles (COE) as well[@mehta]. The specific theoretical predictions for the averaged ${<{\Delta_3(L)}>}={1\over L}{\sum _{\alpha } \Delta_3(L,\alpha )}$, are ${{\Delta_3^{(COE)}(L)}}\>={{1\over \pi ^2}}\ell n\{L\} -0.007,\> \>$ and ${<{\Delta_3^{(Poisson)}(L)}>}\>={L\over 15}\> $. These results are correct in the asymptotic limit valid for $15\leq L$. In Fig.6 we present our results for $<\Delta_3>$ and $<\Sigma ^2>$. In these figures one clearly sees the transition from Poisson-like (dashes) to COE-like (solid line) behavior as $\epsilon/{{\tilde\hbar}}$ is varied. We note that the $\Delta_3$ statistic does not saturate in the COE limit, even for the maximum interval $L$ that we looked at, as would be expected from semiclassical arguments originally proposed by Berry[@berry]. Furthermore, note that for the largest $L$ considered the Poisson limit does not present the knee seen in other completely integrable systems as was found before[@jose]. All in all the results shown in Fig.6 are consistent with what we have come to expect for the transition between regular and chaotic regions. Quasi-energy eigenfunction statistics -------------------------------------- Here we consider the statistical properties of the eigenfunction overlaps with the natural basis vectors. it has been conjectured [@alhassid] that as the classical motion changes from chaotic to regular, this distribution of overlaps can be represented by a $\chi^2$-distribution in $\nu$ degrees of freedom, with $\nu$ varying from 1 in the chaotic regime (the Porter-Thomas limit) to 0 in the regular region (the Poisson limit): $$P_{\nu}(y) = {(\nu /2)^{\nu /2}\over \Gamma (\nu /2)} \enskip y^{\nu /2 -1}\enskip \exp(-\nu y/2).$$ Here $y \equiv \mid \langle\lambda |nl\rangle\mid ^2$, where $\mid{\lambda}\rangle$ label the QEF and $|{nl}\rangle$ label a set of $N$ orthogonal basis vectors. (The $y$’s have been rescaled so that $\langle{y}\rangle=1$.) We have tested this hypothesis for the overlap strengths for the same parameter values as for the quasi-energy eigenvalue statistics. The results are shown in Fig. 7, plotted on a logarithmic scale. These results show the general trend of decreasing $\nu$ as we cross the phase boundary from regular to chaotic classical motion. However, we note that as we go from the COE to the Poisson limits, the fits to the $\chi^2$ get worse. Note especially the shift of the maxima away from zero. This discrepancy is connected to the fact that the results are strongly basis dependent when not in the universal COE limit. Semiclassical correspondences ----------------------------- We can now make a direct comparison between the classical and quantum results by employing a phase space approach. To do this, we use the Husimi representation of the QEF. The Husimi distribution, interpreted as a probability density, is a coarse-grained version of the Wigner function which goes smoothly to the semiclassical limit[@chang]. In practice, the most often used technique of coarse-graining is to take the overlap of the QEF with coherent oscillator states. For the radial coordinate the coherent state is $$\Psi_{r_0,p_0}^G(r) = ({\sigma\over\pi{\tilde\hbar}})^{1\over 4}\, \exp\left\{-{\sigma\over 2{\tilde\hbar}}(r - r_0)^2 + i{p_0\over {\tilde\hbar}}(r - {r_0\over 2}) \right\},$$ which is a minimum-uncertainty Gaussian wavepacket centered at $(r_0,p_0)$, with root mean-squared deviations given by $\Delta\rho = \sqrt{{\tilde\hbar}/2\sigma}$, $\Delta p = \sqrt{{\tilde\hbar}\sigma/2}$, and $\sigma\,$ is the ‘squeezing’ parameter. This parameter is adjusted when making comparisons to the classical phase-space plots. The Husimi distribution of a single QEF $\phi_{\varepsilon}(r)$, is then defined by $${\cal F}_{\phi_{\varepsilon}}(r_0,p_0) = \left| \>\int_0^1 \Psi_{r_0,p_0}^G(r)\,\phi_{\varepsilon}(r)\,dr\> \right|^2.$$ The Husimi distribution is obtained by scanning through the values of $(r_0,p_0)$ in the region of interest in phase space, and the result is compared with the classical surface-of-section. We begin the comparison by noting the symmetry about the line $p=-\eta r$ in the Husimi contour plots in Fig.8. As mentioned earlier, this feature carries over from the classical results for the same reasons as there, and it is in fact used to effectively halve the numerical effort. All calculations reported here were carried out for relative cyclotron frequency ${\tilde\omega_c}=2\sqrt{2}$, angular momentum quantum number $\ell=5$, relative [*a.c.*]{} to [d.c.]{} field strength $\epsilon=1$ and scaled ${\tilde\hbar}=0.008$. In this case, all terms in the Hamiltonian are comparable in magnitude, which means that we are in a non-perturbative regime. Furthermore, we can clearly see both from the phase diagram and the surface-of-section that this places the system on the order-chaos border, where the dynamics is quite ‘mixed’. A few calculations were done for different values of the parameters, but no new qualitative features emerged. In choosing the value of ${\tilde\hbar}$, we were guided by the following considerations. The value of ${\tilde\hbar}$ has to be small enough so that the system is well into the semiclassical regime, yet large enough so that the dimension of the truncated Hilbert space $N$ (which grows as the inverse square of ${\tilde\hbar}$) is large enough to preserve unitarity. Moreover, $N$ has to be such that the largest eigenenergy of $H_{dc}$ has to be larger than the maximum energy of the classical particle in the region of interest in phase space. All the interesting features seen in this model are manifested in this regime. Finally, the classical conserved angular momentum $J$ was kept identical to the quantum value, $\ell{\tilde\hbar}$. The classical analysis was carried out for different values of the angular momentum $J$[@badri1]. First, we iterated a single (arbitrarily chosen) initial condition several thousand times, which typically leads to the chaotic background as shown in the figures. Embedded in this background are KAM tori centered around elliptic fixed points, defined by choosing appropriate initial conditions. In Figure 8, we show several such tori, and in particular, a fixed point of period 4 which was determined earlier analytically. Also shown in each of the figures is a hyperbolic fixed point of order 6, marked by its stable and unstable manifolds. The fixed points were determined by using a modified Powell method of determining zeros of coupled nonlinear sets of equations[@recipes]. This method, like all multidimensional root-finding techniques, requires a good initial guess to converge to the fixed point, but once given it determines the root and the Monodromy matrix (the Jacobian or the determinant of the linearized version of the map equations reliably and accurately. The fixed points are elliptic, parabolic or hyperbolic if the discriminant obtained from the eigenvalues ([*i.e.*]{}, $(Trace)^2-4\cdot (Determinant)$) is negative, zero or positive, respectively. In all cases, it was verified, within numerical error, that the map was area-preserving, [*i.e.*]{}, the determinant was equal to one. The unstable manifold was obtained by iterating the map along the direction given by the eigenvector corresponding to the eigenvalue larger than one. The [*stable*]{} manifold is given by the time reversed version of the unstable one. Comparison of the Husimi distributions ${\cal F}_{\phi{\varepsilon}}(r_0,p_0)$ with the classical phase space plots show some striking similarities. There are, for many QEF, many structures which unmistakably correspond to elliptic, parabolic and hyperbolic periodic orbits, as seen in Fig. 8. For example, the Husimi representation of one of the QEF sits on top of the analytic period-two fixed point marked as F. Also, seen in the figure are Husimis which peak [*exactly*]{} on top of the unstable hyperbolic period-6 fixed point, referred to in the literature as ‘scars’[@jensen]. This correspondence is so robust, in fact, that often when a good guess to the [*classical*]{} hyperbolic fixed points are unavailable, the Husimis are used as a guide to the location of the fixed point (being unstable, hyperbolic fixed points cannot be located without a very good initial guess). These enhanced probability densities are conjectured to play as important a role in quantum mechanics as the hyperbolic orbits play in classical chaos. Finally, a rare but persistent occurrence in all the cases considered is that of a single Husimi distributions peaked simultaneously over [*both*]{} elliptic and hyperbolic fixed points, reflecting a purely quantum-mechanical tunneling across the KAM tori. Here we have only shown representative results of the correspondence between Husimi distributions and classical solutions. Experimental Feasibility {#sec:exp} ======================== Before concluding, we present some experimental scenarios where the predicted effects may be observable. A ‘typical’ GaAs-AlGaAs semiconductor quantum dot device [@marcus1],[@levy] has a radius $R_0$ of between 0.1 and 10$\mu$m, a sheet density $n\sim 10^{11}$ cm$^{-2}$, and a mobility $\mu\sim 2.65 \times 10^5$ cm$^2$/V$\cdot$s. The typical level spacing $\Delta\epsilon \sim 0.05$ meV or $\sim 500$ mK. The operating temperatures is generally of the order of 0.1 K, so $kT\sim 0.01$ meV is smaller than $\Delta\epsilon$, and thus the discrete spectrum can be accessed. A typical elastic mean free path $\l_\phi\sim 10\mu$m, and the phase coherence length varies between 15 and 50 $\mu$m. The power injected is typically $<$ 1 nW, which avoids the problem of electron heating. Given these parameters, we can estimate in physical units the field strengths and frequencies required to observe the effects predicted by our model. Let us first calculate these assuming a dot radius $R_0\sim 1\mu$m. The fundamental kick frequency $\omega_0$ in our problem can be deduced from Eqs. \[eq:f2\] as $\omega_0 = \hbar/(m^*R_0^2\tilde\hbar) \simeq [1/\tilde\hbar]\,2\times 10^{9} {\rm s}^{-1} .$ From this, we can deduce the required $d.c.$ and $a.c.$ magnetic field magnitudes: $$\begin{aligned} \label{allexp-2} B_{dc} &=& \frac{\omega_0 m^* c}{e^*}\,{{\tilde\omega_c}} \nonumber \\ &\simeq& 20 \frac{{\tilde\omega_c}}{{\tilde\hbar}} \,{\rm Gauss} \\ B_{ac} &=& \epsilon B_{dc} \simeq 20 \frac{\epsilon{\tilde\omega_c}}{{\tilde\hbar}} \,{\rm Gauss} .\end{aligned}$$ Finally, the Larmor frequency associated with the $a.c.$ field is given by $\omega_{ac} = \epsilon {\tilde\omega_c} \simeq \epsilon{\tilde\omega_c}/(\tilde\hbar)\, 2\times 10^7 \,{\rm s}^{-1}.$ The dot radius $R_0$ in Ref. [@levy] is about 5$\mu$m. For this radius, the frequency and $d.c.$ magnetic field magnitudes are, $\omega_0 \simeq \,8\times 10^{7} {\rm s}^{-1}{{\tilde\hbar}}^{-1}$ and $B_{dc} \simeq 0.8 \tilde\omega_c/\tilde\hbar\,{\rm Gauss}. $ With these values, we can see what physical parameters correspond to the integrable and chaotic regimes. We fix ${\tilde\hbar}=0.1$, and choose as representative parameters $(\epsilon,{\tilde\omega_c})^{(reg)} = (0.1,0.1)$ where the motion is regular, and the parameters $(\epsilon,{\tilde\omega_c})^{(chaos)} = (2.0,2.0)$ where the motion is chaotic. Then, for $R_0\sim 1\mu$m, the frequency and $a.c.$ fields corresponding to the regular regime are, $$\omega_0^{(reg)} \simeq 20 \,{\rm GHz}, \,\,\,\, B_{ac}^{(reg)} \simeq 20 \,{\rm Gauss} ,$$ while those corresponding to the chaotic regime are, $$\label{allexp-6} \omega_0^{(chaos)} \simeq 20 \,{\rm GHz} ,\,\,\,\, B_{ac}^{(chaos)} \simeq 800 \,{\rm Gauss} .$$ For the case $R_0 \sim 5\mu$m case, the frequencies and fields are, for the regular regime, $$\label{allexp-7} \omega_0^{(reg)} \simeq 800 \,{\rm MHz}, \,\,\,\, B_{ac}^{(reg)} \simeq 0.08 \,{\rm Gauss} ,$$ and for the chaotic regime, $$\label{allexp-8} \omega_0^{(chaos)} \simeq 800 \,{\rm MHz}, \,\,\,\, B_{ac}^{(chaos)} \simeq 32 \,{\rm Gauss}.$$ With the appropriate techniques of measurement, for example by using an array of $\sim 10^5$ [*isolated*]{} quantum dots to increase the magnitude of the signal, and using a highly sensitive electromagnetic superconducting microresonator to measure the response, as was done by Reulet, [*et. al.*]{} in Ref. [@reulet] to measure the dynamic conductance of mesoscopic rings threaded by Aharonov-Bohm fluxes. We believe that an experimental realization of this system is feasible. Conclusions {#sec:conc} =========== We have shown that the model of an electron in a rigid quantum dot structure subject to constant and periodically kicked magnetic fields shows a transition to chaos, depending on the relationship between the strengths of the fields and the cyclotron frequency of the steady field. This relationship is characterized by a ‘phase diagram’ in parameter space shown in Fig. 1. The nature of various periodic orbits were investigated. The quantum signatures of this transition are evidenced in two measures. First, as the classical system goes from integrable to chaotic, the statics of the quasienergy spectrum follow the route from Poisson-like to COE-like. Second, the contour plots of the Husimi distribution of the quasienergy eigenfunctions clearly exhibit the phenomenon of ‘scarring’ over unstable periodic orbits. Finally, we have presented some experimental ranges of the parameters where the effects of chaos in the system may be observable. To sum up, all tests applied to the classical quantum correspondence are in full agreement with the established quantum manifestations of classical chaos. The many electron problem will be treated elsewhere [@badri2]. We thank G. Chu for useful discussions. This work has been supported by Office of Naval Research grant number ONR-N00014-92-1666 and by NSF grant DMR-95-21845. {#section .unnumbered} In this appendix, we show that the classical particle and field equations of motion can be written exactly for a periodically kicked magnetic field. Starting from the Lorenz force equation, \[allapa\] $$\label{eq:apa1} m^*\frac{d^2{\bf r}}{dt^2} = m^*\frac{d{\bf v}}{dt} = e^*\left\{\frac{{\bf v}} {c}\times {\bf B}(t) + {\bf E}(t) \right\},$$ $$\label{eq:apa2} {\rm where}\qquad{\bf B}(t) = \left( B_{dc} + B_{ac} T_0\sum_{n=-\infty}^ {\infty}\delta(t-nT_0) \right){\hat e_z} \equiv \left\{ B_{dc} + B_{ac}\Delta(t) \right\}{\hat e_z},$$ $$\label{eq:apa3} {\rm and}\qquad {\bf E}(t) = -\frac{1}{c}\frac{\partial} {\partial t}{\bf A}(t) = \frac{B_{ac}}{2c} {\dot\Delta}(t)\,\left({\bf r} \times{\hat e_z}\right).$$ Then, on substituting Eqs.(\[eq:apa2\]) and (\[eq:apa3\]) in Eq.(\[eq:apa1\]), and using the definition of $\omega_c$, we get, $$\label{eq:apb} \frac{d{\bf v}}{dt} = \omega_c \left({\bf v}\times{\hat e_z}\right) + \epsilon\omega_c \left({\bf v}\times{\hat e_z}\right) \Delta(t) + \frac{\epsilon\omega_c}{2}\left({\bf r}\times{\hat e_z}\right) {\dot\Delta}(t).$$ Using the standard property of the delta function, $\int{f(x)\delta^{\prime} (x-a)\,dx} = -f^{\prime}(a)$, the last term becomes ($\epsilon=\frac{B_{ac}}{B_{dc}}$), $$\label{eq:apc} \frac{\epsilon\omega_c}{2}\left({\bf v}\times{\hat e_z}\right) \Delta(t).$$ Thus, the [*exact*]{} equations of motion can be written as, $$\label{eq:apd} \frac{d{\bf v}}{dt} = \omega_c \left({\bf v} \times{\hat e_z}\right)\left\{ 1 + \frac{\epsilon}{2} T_0\sum_{n=-\infty}^{\infty} \delta(t-nT_0)\right\}.$$ Note that the only difference we have from including the induced [**E**]{} field is a factor of 1/2 in the kicked component of the [**B**]{} field. The reason the same analysis cannot be done the same way in the quantum problem is that there it is the vector potential that is the relevant dynamical variable. Thus if we use an [**A**]{} = [**A**]{}$_{dc}$ + [**A**]{}$_{ac}(t)$ with [**A**]{}$_{ac}(t) \simeq \sum_{n=-\infty}^{\infty}B_{ac}(\rho)\delta(t-nT_0)$, we see that we have a mathematical ambiguity in the definition of [**A**]{}$_{ac}^2$. Nonetheless, one can carry out the nonrelativistic analysis with our model Hamiltonian that contains, we believe, the essential physics of the problem and yet is mathematically tractable. V. Fock, Z. Phys. [**47**]{}, 446 (1928); L. Landau, Z. Phys. [**64**]{}, 629 (1930). R. B. Dingle, Proc. Roy. Soc. [**A211**]{}, 500 (1952); R. B. Dingle, Proc. Roy. Soc. [**A212**]{}, 47 (1952). M. Robnik, J. Phys. A: Math. Gen. [**19**]{}, 3619 (1986). R. Prange and S. Girvin, eds., [*The Quantum Hall Effect*]{}, 2nd ed., Springer-Verlag (1990). For a review, see A. J. Lichtenberg and A. M. Lieberman, [*Regular and chaotic dynamics*]{}, 2nd ed., Springer-Verlag (1992). For a recent comprehensive overview, see G. Casati and B. Chirikov, eds., [*Quantum Chaos: Between order and disorder*]{}, Cambridge University Press (1995). C. W. J. Beenakker and H. van Houten, [Solid State Physics]{} [**44**]{}, 1, eds. H. Ehrenreich and D. Turnbull, Academic Press (1991). L. J. Slater, [*Handbook of Mathematical Functions*]{}, ed. Milton Abramowitz and Irene A. Stegun, National Bureau of Standards App. Math. Series [**55**]{}, 503 (1968). G. Casati and L. Molinari, Prog. Th. Phys. Suppl. [**98**]{} 287 (1989). L. E. Reichl, [*The transition to chaos in conservative classical systems: quantum manifestations*]{}, Springer-Verlag, New York (1992). O. Bohigas, M. -J. Giannoni and C. Schmit, Phys. Rev. Lett. [**52**]{}, 1 (1984). J. V. José and R. Cordery, Phys. Rev. Lett. [**56**]{}, 290 (1986). T. A. Brody, Lett. Nuovo Cim. [**7**]{}, 482 (1973). O. Bohigas and M. -J. Giannoni in [ *Mathematical and Computational Methods in Nuclear Physics*]{}, Granada, Spain, ed. J. S. Dehesa [*et. al.*]{}, Springer-Verlag, Berlin (1984). M. L. Mehta, [*Random Matrices: an enlarged and revised second edition*]{}, Academic Press (1991). M. V. Berry, Proc. Roy. Soc. [**A400**]{}, 299 (1985). Y. Alhassid and R. D. Levine, Phys. Rev. Lett. [**57**]{}, 2879 (1986); K. Zyczkowski, J. Phys. [**A23**]{}, 4427 (1991). S. -J. Chang and K.-J. Shi, Phys. Rev. A [**34**]{}, 7 (1986). The classical analysis parallels a similar calculation for a different problem, see R. Badrinarayanan, J. V. José and G. Chu, Physica [**D83**]{}, 1 (1995); see also, G. Chu and J. V. José, J. Stat. Phys. [**68**]{}, 153 (1992). W. H. Press [*et.al.*]{}, [*Numerical recipes in Fortran*]{}, 2nd ed., Cambridge University Press, 406 (1992). R. V. Jensen, M. M. Sanders and M. Saraceno, Phys. Rev. Lett. [**63**]{}, 2771 (1989). C. M. Marcus [*et. al.*]{}, Phys. Rev. Lett. [**69**]{}, 506 (1992). L. P. 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--- abstract: 'We prove that for a chainable continuum $X$ and every $x\in X$ with only finitely many coordinates contained in a zigzag there exists a planar embedding $\phi:X\to \phi(X)\subset\R^2$ such that $\phi(x)$ is accessible, partially answering a question of Nadler and Quinn from 1972. Two embeddings $\phi,\psi:X \to \R^2$ are called strongly equivalent if $\phi \circ \psi^{-1}: \psi(X) \to \phi(X)$ can be extended to a homeomorphism of $\R^2$. We also prove that every nondegenerate indecomposable chainable continuum can be embedded in the plane in uncountably many ways that are not strongly equivalent.' address: - 'Departamento de Matemática Aplicada, IME-USP, Rua de Matão 1010, Cidade Universitária, 05508-090 São Paulo SP, Brazil' - 'Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria' - 'AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland. – and – National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic' author: - 'Ana Anušić, Henk Bruin, Jernej Činč' title: Planar embeddings of chainable continua --- [^1] Introduction ============ It is well-known that every chainable continuum can be embedded in the plane, see [@Bing]. In this paper we develop methods to study nonequivalent planar embeddings, similar to methods used by Lewis in [@Lew] and Smith in [@Sm] for the study of planar embeddings of the pseudo-arc. Following Bing’s approach from [@Bing] (see Lemma \[lem:patterns\]), we construct nested intersections of discs in the plane which are small tubular neighborhoods of polygonal lines obtained from the bonding maps. Later we show that this approach produces all possible planar embeddings of chainable continua which can be covered with planar chains with *connected* links, see Theorem \[thm:allemb\]. From that we can produce uncountably many nonequivalent planar embeddings of the same chainable continuum. \[def:equivembed\] Let $X$ be a chainable continuum. Two embeddings $\phi,\psi:X \to \R^2$ are called [*equivalent*]{} if there is a homeomorphism $h$ of $\R^2$ such that $h(\phi(X)) = \psi(X)$. They are [*strongly equivalent*]{} if $\psi \circ \phi^{-1}: \phi(X)\to \psi(X)$ can be extended to a homeomorphism of $\R^2$. That is, equivalence requires some homeomorphism between $\phi(X)$ and $\psi(X)$ to be extended to $\R^2$ whereas strong equivalence requires the homeomorphism $\psi \circ \phi^{-1}$ between $\phi(X)$ and $\psi(X)$ to be extended to $\R^2$. Clearly, strong equivalence implies equivalence, but in general not the other way around, see for instance Remark \[rem:n\_emb\]. We say a nondegenerate continuum is [*indecomposable*]{}, if it is not the union of two proper subcontinua. \[q:uncountably\] Are there uncountably many nonequivalent planar embeddings of every chainable indecomposable continuum? This question is listed as Problem 141 in a collection of continuum theory problems from 1983 by Lewis [@LewP] and was also posed by Mayer in his dissertation in 1983 [@MayThesis] (see also [@May]) using the standard definition of equivalent embeddings. We give a positive answer to the adaptation of the above question using strong equivalence, see Theorem \[thm:Mayer\]. If the continuum is the inverse limit space of a unimodal map and not hereditarily decomposable, then the result holds for both definitions of equivalent, see Remark \[rem:otherdef\]. In terms of equivalence, this generalizes the result in [@embed], where we prove that every unimodal inverse limit space with bonding map of positive topological entropy can be embedded in the plane in uncountably many nonequivalent ways. The special construction in [@embed] uses symbolic techniques which enable direct computation of accessible sets and prime ends (see [@AC]). Here we utilize a more direct geometric approach. One of the main motivations for the study of planar embeddings of tree-like continua is the question of whether the *plane fixed point property* holds. The problem is considered to be one of the most important open problems in continuum theory. Is it true that every continuum $X \subset \R^2$ not separating the plane has the fixed point property, every continuous $f: X\to X$ has a fixed point? There are examples of tree-like continua without the fixed point property, see Bellamy’s example in [@Bell]. It is not known whether Bellamy’s example can be embedded in the plane. Although chainable continua are known to have the fixed point property (see [@Ha]), insight in their planar embeddings may be of use to the general setting of tree-like continua. Another motivation for this study is the following long-standing open problem. For this we use the following definition. Let $X\subset\R^2$. We say that $x\in X$ is [*accessible*]{} (from the complement of $X$) if there exists an arc $A\subset\R^2$ such that $A\cap X=\{x\}$. \[Nadler and Quinn 1972, pg. 229 in [@Nadler]\] \[q:NaQu\] Let $X$ be a chainable continuum and $x\in X$. Can $X$ be embedded in the plane such that $x$ is accessible? We will introduce the notion of a *zigzag* related to the admissible permutations of graphs of bonding maps and answer Nadler and Quinn’s question in the affirmative for the class of *non-zigzag* chainable continua (see Corollary \[cor:nonzigzag\]). From the other direction, a promising possible counterexample to Question \[q:NaQu\] is the one suggested by Minc (see Figure \[fig:Minc\] and the description in [@Minc]). However, the currently available techniques are insufficient to determine whether the point $p\in X_M$ can be made accessible or not, even with the use of thin embeddings, see Definition \[def:thin\]. Section \[sec:notation\] gives basic notation, and we review the construction of natural chains in Section \[sec:chains\]. Section \[sec:permuting\] describes the main technique of permuting branches of graphs of linear interval maps. In Section \[sec:stretching\] we connect the techniques developed in Section \[sec:permuting\] to chains. Section \[sec:emb\] applies the techniques developed so far to accessibility of points of chainable planar continua; this is the content of Theorem \[thm:algorithm\] which is used as a technical tool afterwards. Section \[sec:zigzags\] introduces the concept of zigzags of a graph of an interval map. Moreover, it gives a partial answer to Question \[q:NaQu\] and provides some interesting examples by applying the results from this section. Section \[sec:thin\] gives a proof that the permutation technique yields all possible thin planar embeddings of chainable continua. Furthermore, we pose some related open problems at the end of this section. Finally, Section \[sec:nonequivalent\] completes the construction of uncountably many planar embeddings that are not equivalent in the strong sense, of every chainable continuum which contains a nondegenerate indecomposable subcontinuum and thus answers Question \[q:uncountably\] for strong equivalence. We conclude the paper with some remarks and open questions emerging from the study in the final section. Notation {#sec:notation} ======== Let $\N = \{ 1,2,3,\dots\}$ and $\N_0=\{0,1,2,3,\dots\}$ be the positive and nonnegative integers. Let $f_i: I=[0,1]\to I$ be continuous surjections for $i\in\N$ and let [ *inverse limit space*]{} $$X_{\infty}=\underleftarrow{\lim}\{I, f_i\}=\{(x_0, x_1, x_2, \dots): f_i(x_i)=x_{i-1}, i\in\N\} %\subset I^{\infty}$$ be equipped with the subspace topology endowed from the product topology of $I^{\infty}$. Let $\pi_i: X_{\infty}\to I$ be the [*coordinate projections*]{} for $i\in\N_0$. Let $X$ be a metric space. A *chain in $X$* is a set $\chain=\{\ell_1\ldots, \ell_n\}$ of open subsets of $X$ called *links*, such that $\ell_i\cap\ell_j\neq\emptyset$ if and only if $|i-j|\leq 1$. If also $\cup_{i=1}^n \ell_i = X$, then we speak of a [*chain cover*]{} of $X$. We say that a chain $\chain$ is *nice* if additionally all links are open discs (in $X$). The *mesh of a chain $\chain$* is $\mesh(\chain)=\max\{\diam{\ell_i}: i=1, \ldots, n\}$. A continuum $X$ is [*chainable*]{} if there exist chain covers of $X$ of arbitrarily small mesh. We say that $\chain'=\{\ell'_1, \ldots, \ell'_m\}$ *refines* $\chain$ and write $\chain'\preceq\chain$ if for every $j\in\{1, \ldots, m\}$ there exists $i\in\{1, \ldots, n\}$ such that $\ell'_j\subset\ell_i$. We say that $\chain'$ *properly refines* $\chain$ and write $\chain'\prec\chain$ if additionally $\ell'_j\subset\ell_i$ implies that the closure $\overline{\ell'_j}\subset\ell_i$. Let $\chain'\preceq\chain$ be as above. The *pattern of $\chain'$ in $\chain$*, denoted by $Pat(\chain', \chain)$, is the ordered $m$-tuple $(a_1, \ldots, a_m)$ such that $\ell'_{j}\subset\ell_{a(j)}$ for every $j\in\{1, \ldots, m\}$ where $a(j)\in \{1,\ldots,n \}$. If $\ell'_{j}\subset \ell_{i}\cap\ell_{i+1}$, we take $a(j)=i$, but that choice is just by convention. For chain $\chain=\{\ell_1, \ldots, \ell_n\}$, write $\chain^*=\cup_{i=1}^n\ell_i.$ Construction of natural chains, patterns and nested intersections {#sec:chains} ================================================================= First we construct *natural chains* $\chain_n$ for every $n\in\N$ (the terminology originates from [@Br2]). Take some nice chain cover $C_0=\{l_1^0, \ldots, l_{k(0)}^0\}$ of $I$ and define $\chain_0:=\pi_0^{-1}(C_0)=\{\ell_1^0, \ldots, \ell_{k(0)}^0\}$, where $\ell_i^0=\pi_0^{-1}(l_i^0)$. Note that $\chain_0$ is a chain cover of $X_{\infty}$ (but the links are not necessarily connected sets in $X_{\infty}$). Now take a nice chain cover $C_1=\{l_1^1, \ldots, l_{k(1)}^1\}$ of $I$ such that for every $j\in\{1, \ldots, k(1)\}$ there exists $j'\in\{1, \ldots, k(0)\}$ such that $f_1(\overline{l_j^1})\subset l_{j'}^0$ and define $\chain_1:=\pi_1^{-1}(C_1)$. Note that $\chain_1$ is a chain cover of $X_{\infty}$. Also note that $\chain_1\prec\chain_0$ and $Pat(\chain_1, \chain_0)=\{a^1_1, \ldots, a^1_{k(1)}\}$ where $f_1(\pi_1(\ell_j^1))\subset\pi_0(\ell_{a^1_j}^0)$ for each $j\in\{1, \ldots, k(1)\}$. So the pattern $Pat(\chain_1, \chain_0)$ can easily be calculated by just following the graph of $f_1$. Inductively we construct $\chain_{n+1}=\{\ell^{n+1}_1, \ldots, \ell^{n+1}_{k(n+1)}\}:=\pi_{n+1}^{-1}(C_{n+1})$, where $C_{n+1}=\{l^{n+1}_1, \ldots, l^{n+1}_{k(n+1)}\}$ is some nice chain cover of $I$ such that for every $j\in\{1, \ldots, k(n+1)\}$ there exists $j'\in\{1, \ldots, k(n)\}$ such that $f_{n+1}(\overline{l_j^{n+1}})\subset l_{j'}^n$. Note that $\chain_{n+1}\prec\chain_n$ and $Pat(\chain_{n+1}, \chain_n)=(a^{n+1}_1, \ldots, a^{n+1}_{k(n+1)})$, where $f_{n+1}(\pi_{n+1}(\ell_j^{n+1}))\subset\pi_n(\ell_{a^{n+1}_j}^{n})$ for each $j\in\{1, \ldots, k(n+1)\}$. Throughout the paper we use the straight letter $C$ for chain covers of the interval $I$ and the script letter $\chain$ for chain covers of the inverse limits space. Note that the links of $C_n$ can be chosen small enough to ensure that $\mesh(\chain_n)\to 0$ as $n\to\infty$ and note that $X_{\infty}=\cap_{n\in\N_0}\chain_n^*$. \[lem:patterns\] Let $X$ and $Y$ be compact metric spaces and let $\{\chain_n\}_{n \in \N_0}$ and $\{\D_n\}_{n \in \N_0}$ be sequences of chains in $X$ and $Y$ respectively such that $\chain_{n+1}\prec\chain_n$, $\D_{n+1}\prec\D_n$ and $Pat(\chain_{n+1}, \chain_n)=Pat(\D_{n+1}, \D_n)$ for each $n\in\N_0$. Assume also that $\mesh(\chain_n)\to 0$ and $\mesh(\D_n)\to 0$ as $n\to\infty$. Then $X'=\cap_{n\in\N_0}\chain_n^*$ and $Y'=\cap_{n\in\N_0}\D_n^*$ are nonempty and homeomorphic. To see that $X'$ and $Y'$ are nonempty note that they are nested intersections of nonempty closed sets. Define $\chain_k = \{\ell_1^k, \ldots, \ell_{n(k)}^k\}$ and $\D_k = \{L_1^k, \ldots, L_{n(k)}^k\}$ for each $k\in\N_0$. Let $x\in X'$. Then $x=\cap_{k\in\N_0}\ell_{i(k)}^k$ for some $\ell_{i(k)}^k\in\chain_k$ such that $\overline{\ell_{i(k)}^k}\subset\ell_{i(k-1)}^{k-1}$ for each $k\in\N$. Define $h: X'\to Y'$ as $h(x):=\cap_{k\in\N_0}L_{i(k)}^k$. Since the patterns agree and diameters tend to zero, $h$ is a well-defined bijection. We show that it is continuous. First note that $h(\ell_{i(m)}^m\cap X')=L_{i(m)}^m\cap Y'$ for every $m\in\N_0$ and every $i(m)\in\{1, \ldots, n(m)\}$, since if $x=\cap_{k\in\N_0}\ell_{i(k)}^k\subset\ell_{i(m)}^m$, then there is $k'\in\N_0$ such that $\ell_{i(k)}^k\subset\ell_{i(m)}^m$ for each $k\geq k'$. But then $L_{i(k)}^k\subset L_{i(m)}^m$ for each $k\geq k'$, thus $h(x)=\cap_{k\in\N_0}L_{i(k)}^k\subset L_{i(m)}^m$. The other direction follows analogously. Now let $U\subset Y'$ be an open set and $x\in h^{-1}(U)$. Since diameters tend to zero, there is $m\in\N_0$ and $i(m)\in\{1, \ldots, n(m)\}$ such that $h(x)\in L_{i(m)}^m\cap Y'\subset U$ and thus $x\in\ell_{i(m)}^m\cap X'\subset h^{-1}(U)$. So $h^{-1}(U)\subset X'$ is open and that concludes the proof. In the following sections we will construct nested intersections of nice planar chains such that their patterns are the same as the patterns of refinements $\chain_n\prec\chain_{n-1}$ of natural chains of $X_{\infty}$ (as constructed at the beginning of this section) and such that the diameters of links tend to zero. By the previous lemma, this gives embeddings of $X_{\infty}$ in the plane. We note that the previous lemma holds in a more general setting (with an appropriately generalized definition of patterns), for graph-like continua and graph chains, see [@Medd]. Permuting the graph {#sec:permuting} =================== Let $C=\{l_1, \ldots, l_n\}$ be a chain cover of $I$ and let $f: I\to I$ be a continuous surjection which is piecewise linear with finitely many critical points $0=t_0<t_1< \ldots< t_m<t_{m+1}=1$ (so we include the endpoints of $I=[0,1]$ in the set of critical points). In the rest of the paper we work with continuous surjections which are piecewise linear (so with finitely many critical points); we call them [*piecewise linear surjections*]{}. Without loss of generality we assume that for every $i\in\{0, \ldots, m\}$ and $l \in C$, $f([t_i, t_{i+1}]) \not\subset l$. Define $H_{j}=f([t_j, t_{j+1}])\times \{j\}$ for each $j\in\{0, \ldots, m\}$ and $V_j=\{f(t_j)\}\times[j-1, j]$ for each $j\in\{1, \ldots, m\}$. Note that $H_{j-1}$ and $H_{j}$ are joined at their left endpoints by $V_j$ if there is a local minimum of $f$ in $t_{j}$ and they are joined at their right endpoints if there is a local maximum of $f$ in $t_j$ (see Figure \[fig:1\]). The line $H_0\cup V_1\cup H_1\cup\ldots\cup V_m\cup H_m=:G_f$ is called the *flattened graph of $f$ in $\R^2$*. \[def:flat\] A permutation $p:\{0, 1, \ldots, m\}\to\{0, 1, \ldots, m\}$ is called a *$C$-admissible permutation of $G_f$* if for every $i\in\{0, \ldots, m-1\}$ and $k\in\{0, \ldots, m\}$ such that $p(i)<p(k)<p(i+1)$ or $p(i+1)<p(k)<p(i)$ it holds that: 1. $f(t_{i+1})\not\in f([t_k, t_{k+1}])$, or 2. $f(t_{i+1})\in f([t_k, t_{k+1}])$ but $f(t_k)$ or $f(t_{k+1})$ is contained in the same link of $C$ as $f(t_{i+1})$. Denote a $C$-admissible permutation of $G_f$ by $$p^{C}(G_f)=p(H_0)\cup p(V_1)\cup\ldots\cup p(V_m)\cup p(H_m),$$ for $p(H_j)=f([ \tilde{t}_j, \tilde{t}_{j+1}])\times\{p(j)\}$ and $p(V_j)=\{f(\tilde{t}_j)\}\times[ p(j-1), p(j)]$, where $\tilde{t}_j$ is chosen such that $f(t_j)$ and $f(\tilde{t}_j)$ are contained in the same link of $C$, and such that $p^C(G_f)$ has no self intersections for every $j\in\N$. A line $p^{C}(G_f)$ will be called a *permuted graph of $f$ with respect to $C$*. Let $E(p^C(G_f))$ be the endpoint of $p(H_0)$ corresponding to $(f(\tilde{t}_0),p(0))$. Note that $p(V_j)$ from Definition \[def:flat\] is a vertical line in the plane which joins the endpoints of $p(H_{j-1})$ and $p(H_{j})$ at $f(\tilde{t}_{j})$, see Figure \[fig:1\]. If $p(j)=m$, we say that $H_j$ is *at the top of $p^C(G_f)$*. (0,0)–(0,1)–(1,1)–(1,0)–(0,0); (0,0)–(1/4,1)–(0.5,0.3)–(3/4,1)–(1,0.3); (1/4,-0.01)–(1/4,0.01); at (0.27,-0.07) [$t_{1}$]{}; (0.5,-0.01)–(0.5,0.01); at (0.52,-0.07) [$t_{2}$]{}; (3/4,-0.01)–(3/4,0.01); at (0.77,-0.07) [$t_{3}$]{}; at (0.5,-0.25) [$(a)$]{}; (-0.01,1/4)–(0.01,1/4); (-0.01,1/2)–(0.01,1/2); (-0.01,3/4)–(0.01,3/4); at (-0.05,1/8) [$l_1$]{}; at (-0.05,3/8) [$l_2$]{}; at (-0.05,5/8) [$l_3$]{}; at (-0.05,7/8) [$l_4$]{}; (1.5,0)–(2.5,0); (1.5, -0.01)–(1.5,0.01); (1.75, -0.01)–(1.75,0.01); (2, -0.01)–(2,0.01); (2.25, -0.01)–(2.25,0.01); (2.5, -0.01)–(2.5,0.01); at (1.625,-0.07) [$l_1$]{}; at (1.875,-0.07) [$l_2$]{}; at (2.125,-0.07) [$l_3$]{}; at (2.375,-0.07) [$l_4$]{}; (1.5,1/4)–(2.5,1/4)–(2.5,1/2)–(1.85,1/2)–(1.85,3/4)–(2.5,3/4)–(2.5, 1)–(1.85,1); at (1.45,1/4) [$0$]{}; at (1.45,1/2) [$1$]{}; at (1.45,3/4) [$2$]{}; at (1.45,1) [$3$]{}; at (2,0.3) [$H_0$]{}; at (2.25,0.55) [$H_1$]{}; at (2.25,0.8) [$H_2$]{}; at (2.25,1.05) [$H_3$]{}; at (2.58,0.375) [$V_0$]{}; at (1.77,0.625) [$V_1$]{}; at (2.58,0.875) [$V_2$]{}; at (2,-0.25) [$(b)$]{}; (3.5,0)–(4.5,0); (3.5, -0.01)–(3.5,0.01); (3.75, -0.01)–(3.75,0.01); (4, -0.01)–(4,0.01); (4.25, -0.01)–(4.25,0.01); (4.5, -0.01)–(4.5,0.01); at (3.625,-0.07) [$l_1$]{}; at (3.875,-0.07) [$l_2$]{}; at (4.125,-0.07) [$l_3$]{}; at (4.375,-0.07) [$l_4$]{}; (3.5,1)–(4.5,1)–(4.5,1/4)–(3.85,1/4)–(3.85,0.5)–(4.4,0.5)–(4.4,3/4)–(3.85 ,3/4); at (3.2,1/4) [$0=p(1)$]{}; at (3.2,1/2) [$1=p(2)$]{}; at (3.2,3/4) [$2=p(3)$]{}; at (3.2,1) [$3=p(0)$]{}; at (4.25,0.3) [$p(H_1)$]{}; at (4.25,0.55) [$p(H_2)$]{}; at (4.25,0.8) [$p(H_3)$]{}; at (4,1.05) [$p(H_0)$]{}; at (4.65,0.625) [$p(V_0)$]{}; at (3.7,0.375) [$p(V_1)$]{}; (4.4,0.625)–(4.5,0.625); (4.5,0.625)–(4.4,0.625); (3.5,1) circle (0.01); at (3.45,1) [$E$]{}; at (4,-0.25) [$(c)$]{}; Note that a flattened graph $G_f$ is just a graph of $f$ for which its critical points have been extended to vertical intervals. These vertical intervals were introduced for the definition of a permuted graph. After permuting the flattened graph, we can quotient out the vertical intervals in the following way. For every $i=1, \ldots, m$, pick a point $q_i\in p(V_i)$. There exists a homotopy $F:I\times\R^2\to\R^2$ such that $F(1,y)=q_i$ for every $y\in p(V_i)$ and every $i=1, \ldots, m$, and for every $t \in I$, $F(t, \cdot): p^C(G_f)) \to \R^2$ is injective, and such that $\pi_x(F(t,(x,y)))=x$ for every $(x,y)\in\R^2$ and $t\in I$. Here $\pi_x(x,y)$ denotes a projection on the first coordinate. From now on $p^C(G_f)$ will always stand for the quotient $F(1,p^C(G_f))$, but for clarity in the figures of Sections \[sec:stretching\] and \[sec:emb\] we will continue to draw it with long vertical intervals. Chain refinements, their composition and stretching {#sec:stretching} =================================================== Let $f: I\to I$ be a piecewise linear surjection, $p$ an admissible $C$-permutation of $G_f$ and $\eps>0$. We call a nice planar chain $\chain=\{\ell_1, \ldots, \ell_n\}$ a *tubular $\eps$-chain with nerve $p^{C}(G_f)$* if - $\chain^*$ is an $\eps$-neighborhood of $p^{C}(G_f)$, and - there exists $n\in\N$ and arcs $A_1\cup \ldots \cup A_n=p^{C}(G_f)$ such that $\ell_i$ is the $\eps$-neighborhood of $A_i$ for every $i\in\N$. Denote a nerve $p^{C}(G_f)$ of $\chain$ by $\cN_{\chain}$. When there is no need to specify $\varepsilon$ and $\cN_{\chain}$ we just say that $\chain$ is *tubular*. \[def:horizontal\] A planar chain $\chain=\{\ell_1, \ldots, \ell_n\}$ will be called [ *horizontal*]{} if there are $\delta>0$ and a chain of open intervals $\{l_1, \ldots, l_n\}$ in $\R$ such that $\ell_i=l_i\times(-\delta, \delta)$ for every $i\in\{1, \ldots, n\}$. \[rem:stretch\] Let $\chain$ be a tubular chain. There exists a homeomorphism $\widetilde{H}:\R^2\to\R^2$ such that $\widetilde{H}(\chain)$ is a horizontal chain and $\widetilde{H}^{-1}(\chain')$ is tubular for every tubular $\chain'\prec \widetilde{H}(\chain)$. Moreover, for $\chain=\{\ell_1, \ldots, \ell_n\}$ denote by $\cN_{\widetilde{H}(\chain)}=I\times\{0\}$. Note that $\chain\setminus(\ell_1\cup\ell_n\cup \cN_{\chain})$ has two components and thus it makes sense to call the components upper and lower. Let $S$ be the upper component of $\chain\setminus(\ell_1\cup\ell_n\cup \cN_{\chain})$. There exists a homeomorphism $H:\R^2\to\R^2$ which has all the properties of a homeomorphism $\widetilde{H}$ above and in addition satisfies: - the endpoint $H(E(p^{C}(G_f)))=(0,0)$ (recall $E$ from Definition \[def:flat\]) and - $H(S)$ is the upper component of $H(\chain^*)\setminus(H(\ell_1)\cup H(\ell_n)\cup H(A))$. Applying $H$ to a chain $\chain$ is called the *stretching of $\chain$* (see Figure \[fig:stretch2\]). (3,1)–(1,1)–(1,0.5)–(4,0.5)–(4,0)–(2,0); (3.1,1.1)–(0.9,1.1)–(0.9,0.4)–(3.9,0.4)–(3.9,0.1)–(1.9,0.1)–(1.9, -0.1)–(4.1,-0.1)–(4.1,0.6)–(1.1,0.6)–(1.1,0.9)–(3.1,0.9)–(3.1,1.1); (2.5,0.9)–(2.5,1.1); (1.5,0.9)–(1.5,1.1); (0.9,0.75)–(1.1,0.75); (1.5,0.4)–(1.5,0.6); (2.5,0.4)–(2.5,0.6); (3.5,0.4)–(3.5,0.6); (3.9,0.25)–(4.1,0.25); (3.5,-0.1)–(3.5,0.1); (2.5,-0.1)–(2.5,0.1); at (3.2,1.2) [$\chain$]{}; (3,1) circle (0.03); (3.1,1.1)–(0.9,1.1)–(0.9,0.4)–(3.9,0.4)–(3.9,0.1)–(1.9,0.1)– (1.9,0)–(2,0)–(4,0)–(4,0.5)–(1,0.5)–(1,1)–(3.1,1)–(3.1,1.1); (2.5,-0.2)–(2.5,-0.5); at (2.65,-0.35) [$H$]{}; at (1,0.5) ; at (2,0.5) ; at (3,0.5) ; at (4,0.5) ; at (5,0.5) ; at (6,0.5) ; at (7,0.5) ; at (8,0.5) ; at (9,0.5) ; at (10,0.5) ; (1,0.5)–(10,0.5); (1,0.5) circle (0.05); at (10,1.5) [$H(\chain)$]{}; (10.5,0.5)–(10.5,2)–(0.5,2)–(0.5,0.5)–cycle; \[rem:ref\] Let $X_{\infty}, \{C_i\}_{i\in\N_0}, \{\chain_i\}_{i\in\N_0}$ be as defined in Section \[sec:chains\]. For $i\in\N_0$, let $\D_i$ be a horizontal chain with the same number of links as $\chain_i$ and such that $p^{C_i}(G_{f_{i+1}})\subset\D_i^*$ for some $C_i$-admissible permutation $p$. Fix $\eps>0$ and note that, after possibly dividing links of $\chain_{i+1}$ into smaller links (refining the chain $C_{i+1}$ of $I$), there exists a tubular chain $\D_{i+1}\prec\D_i$ with nerve $p^{C_i}(G_{f_{i+1}})$ such that $Pat(\D_{i+1}, \D_i)=Pat(\chain_{i+1}, \chain_i)$ and $\mesh(\D_{i+1})<\eps$, see Figure \[fig:reftube\]. at (1,0.5) ; at (2,0.5) ; at (3,0.5) ; at (4,0.5) ; at (5,0.5) ; (2,1)–(4,1)–(4,0.5)–(1,0.5)–(1,0.25)–(5,0.25); (1.95,1.05)–(4.05,1.05)–(4.05,0.45)–(1.05,0.45)–(1.05,0.3)–(5.05, 0.3)–(5.05,0.2)–(0.95,0.2)–(0.95,0.55)–(3.95,0.55)–(3.95,0.95)–(1.95, 0.95)–(1.95,1.05); (2.4,0.95)–(2.4,1.05); (2.5,0.95)–(2.5,1.05); (2.9,0.95)–(2.9,1.05); (3,0.95)–(3,1.05); (3.2,0.95)–(3.2,1.05); (3.5,0.95)–(3.5,1.05); (3.5,0.45)–(3.5,0.55); (3,0.45)–(3,0.55); (2,0.45)–(2,0.55); (2.5,0.45)–(2.5,0.55); (1.5,0.45)–(1.5,0.55); (0.95,0.35)–(1.05,0.35); (1.5,0.2)–(1.5,0.3); (1.9,0.2)–(1.9,0.3); (2.1,0.2)–(2.1,0.3); (2.5,0.2)–(2.5,0.3); (3.1,0.2)–(3.1,0.3); (3.5,0.2)–(3.5,0.3); (4.5,0.2)–(4.5,0.3); (4.6,0.2)–(4.6,0.3); at (5.3,1.3) [$\D_i$]{}; at (4.1,1.15) [$\D_{i+1}$]{}; Let $H:\R^2\to\R^2$ be a stretching of some tubular chain $\chain$. If $\chain'$ is a nice chain in $\R^2$ refining $\chain$ and there is an interval map $g: I\to I$ such that $p^{C}(G_g)$ is a nerve of $H(\chain')$, then we say that *$\chain'$ follows $p^{C}(G_g)$ in $\chain$*. Now we discuss compositions of chain refinements. Let $f, g: I\to I$ be piecewise linear surjections. Let $0=t_0<t_1< \ldots< t_{m}<t_{m+1}=1$ be the critical points of $f$ and let $0=s_0<s_1< \ldots< s_{n}<s_{n+1}=1$ be the critical points of $g$. Let $C_1$ and $C_2$ be nice chain covers of $I$, let $p_1:\{0, 1, \ldots, m\}\to\{0, 1, \ldots, m\}$ be an admissible $C_1$-permutation of $G_f$ and let $p_2:\{0, 1, \ldots, n\}\to\{0, 1, \ldots, n\}$ be an admissible $C_2$-permutation of $G_g$. Assume $\chain''\prec\chain'\prec\chain$ are nice chains in $\R^2$ such that $\chain$ is horizontal and $p_1^{C_1}(G_f)\subset\chain^*$ (recall that $\chain^*$ denotes the union of the links of $\chain$), $\chain'$ is a tubular chain with $\cN_{\chain'}=p_1^{C_1}(G_f)$, and $\chain''$ follows $p_2^{C_2}(G_g)$ in $\chain'$. Then $\chain''$ follows $f\circ g$ in $\chain$ with respect to a $C_1$-admissible permutation of $G_{f\circ g}$ which we will denote by $p_1*p_2$ (see Figures \[fig:3\] and \[fig:4\]). at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (1,0)–(2,0)–(2,0.5)–(1,0.5)–(1,1)–(4,1)–(4,1.5)–(1,1.5); at (1.5,0.1) [$H_0$]{}; at (1.5,0.65) [$H_1$]{}; at (2.5,1.15) [$H_2$]{}; at (2.5,1.65) [$H_3$]{}; at (-0.6,2) [$\chain$]{}; at (4.3,1.7) [$N_{\chain'}$]{}; (6,-0.5)–(6,2.5)–(9,2.5)–(9,-0.5)–(6,-0.5); (6,-0.5)–(6.75,0.75)–(7.5,-0.5)–(8.25,2.5)–(9,-0.5); (5.95,0)–(6.05,0); (5.95,1)–(6.05,1); (5.95,2)–(6.05,2); (6.75,-0.45)–(6.75,-0.55); (7.5,-0.45)–(7.5,-0.55); (8.25,-0.5)–(9,-0.5); (8.25,-0.45)–(8.25,-0.55); at (6.8,-0.7) [$t_1$]{}; at (8.3,-0.7) [$t_3$]{}; at (7.55,-0.7) [$t_2$]{}; at (9.5,2) [$\Gamma_f$]{}; at (5,-1) [$(a)$]{}; at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (2,0.5)–(4,0.5)–(4,1.5)–(1,1.5); at (3,0.65) [$H_0$]{}; at (2.5,1.65) [$H_1$]{}; at (-0.3,2) [$H(\chain')$]{}; at (4.1,1.7) [$N_{H(\chain'')}$]{}; (6,-0.5)–(6,2.5)–(9,2.5)–(9,-0.5)–(6,-0.5); (6,0.5)–(7.5,2.5)–(9,-0.5); (5.95,0)–(6.05,0); (5.95,1)–(6.05,1); (5.95,2)–(6.05,2); (6,2)–(6,2.5); (7.5,-0.5)–(7.75,-0.5); (7.5,2.5)–(7.75,2); (7.5,-0.45)–(7.5,-0.55); at (5.75,0) [$t_1$]{}; at (5.75,1) [$t_2$]{}; at (5.75,2) [$t_3$]{}; at (9.5,2) [$\Gamma_g$]{}; (3.5,1.5)–(4,1.5); at (5,-1) [$(b)$]{}; at (1.5,-0.7) [$t_1$]{}; at (2.5,-0.7) [$t_2$]{}; at (3.5,-0.7) [$t_3$]{}; at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (1,0)–(2,0)–(2,0.5)–(1,0.5)–(1,1)–(4,1)–(4,1.65)–(1,1.65)–(1,1.5)–(3.85 ,1.5)–(3.85,1.15)–(0.85,1.15)–(0.85,0.35)–(1.6,0.35); (4,1.65)–(1,1.65); at (2.5,1.79) [$H_{13}$]{}; at (2.5,1.35) [$H_{03}$]{}; at (3.2,1.25) [$H_{02}$]{}; at (3.2,0.8) [$H_{12}$]{}; at (1.5,0.65) [$H_{11}$]{}; at (1.5,0.2) [$H_{01}$]{}; at (1.5,-0.15) [$H_{10}$]{}; (6,-0.5)–(6,2.5)–(9,2.5)–(9,-0.5)–(6,-0.5); (6,0.2)–(6.3,-0.5)–(28.5/4,2.5)–(7.5,-0.5)–(7.7,2.5)–(8.25,-0.5)–(8.7, 0.75)–(9,-0.5); (5.95,0)–(6.05,0); (5.95,1)–(6.05,1); (5.95,2)–(6.05,2); (7.5,-0.45)–(7.5,-0.55); (7.5,-0.5)–(7.7,2.5); at (9.5,2) [$\Gamma_{f\circ g}$]{}; at (0,2) [$\chain$]{}; at (4.2,1.9) [$N_{\chain''}$]{}; at (5.5,-1) [$(c)$]{}; Define $$A_{ij}=\{x\in I: x\in[s_i, s_{i+1}], g(x)\in[t_j, t_{j+1}]\},$$ for $i\in\{0, 1, \ldots, n\}, j\in\{0, 1, \ldots, m\}$, $A_{ij}$ are maximal intervals on which $f\circ g$ is injective and possibly $A_{ij} = \emptyset$. Let $H_{ij}$ be the horizontal branch of $G_{f\circ g}$ corresponding to the interval $A_{ij}$. We want to see which branch $H_{ij}$ corresponds to the top of $(p_1*p_2)^{C_1}(G_{f\circ g})$. Denote the top of $p_1^{C_1}(G_f)$ by $p_1(H_{T_1})$, $p_1(T_1)=m$. Denote the top of $p_2^{C_2}(G_g)$ by $p_2(H_{T_2})$, $p_2(T_2)=n$. By the choice of orientation of $H$, the top of $(p_1*p_2)^{C_1}(G_{f\circ g})$ is $H_{T_2T_1}$ (see Figures \[fig:3\] and \[fig:4\]). at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (1,1)–(2,1)–(2,0.5)–(1,0.5)–(1,0)–(4,0)–(4,1.5)–(1,1.5); at (2.5,0.1) [$p_1(H_2)$]{}; at (1.5,0.65) [$p_1(H_1)$]{}; at (1.5,1.15) [$p_1(H_0)$]{}; at (2.5,1.65) [$p_1(H_3)$]{}; at (0,2) [$\chain$]{}; at (4.3,1.7) [$p_1(N_{\chain'})$]{}; at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (1,0.5)–(4,0.5)–(4,1.5)–(2,1.5); at (2.5,0.65) [$p_2(H_1)$]{}; at (3,1.65) [$p_2(H_0)$]{}; at (5.3,2.3) [$H(\chain')$]{}; at (4.5,1.7) [$p_2(N_{H(\chain'')})$]{}; (3.5,1.5)–(4,1.5); at (1.5,-0.7) [$t_1$]{}; at (2.5,-0.7) [$t_2$]{}; at (3.5,-0.7) [$t_3$]{}; at (1,1) ; at (2,1) ; at (3,1) ; at (4,1) ; (1,1)–(2,1)–(2,0.5)–(1,0.5)–(1,0)–(4,0)–(4,1.5)–(1,1.5)–(1,1.65)–(4.15, 1.65)–(4.15,-0.15)–(0.85,-0.15)–(0.85, 0.65)–(1.6,0.65); (1,1.65)–(4.15,1.65); at (0,2) [$\chain$]{}; at (4.2,1.9) [$p_1*p_2(G_{f\circ g})$]{}; at (2,1.8) [$p_1*p_2(H_{03})$]{}; Construction of the embeddings {#sec:emb} ============================== Let $X_{\infty}=\underleftarrow{\lim}\{I, f_i\}$ where for every $i\in\N$ the map $f_i$ is a continuous piecewise linear surjection with critical points $0=t_0^i<t_1^i< \ldots< t_{m(i)}^i<t_{m(i)+1}^i=1$. Let $I_k^i=[t_k^i, t_{k+1}^i]$ for every $i\in\N$ and every $k\in\{0, \ldots, m(i)\}$. We construct chains $(C_n)_{n\in\N_0}$ and $(\chain_n)_{n\in\N_0}$ as before, such that for each $i \in \N_0$, $k\in\{0, \ldots, m(i+1)\}$ and $l \in C_i$, $f_{i+1}(I_{k}^{i+1}) \not\subset l$. The flattened graph of $f_{i}$ will be denoted by $G_{f_{i}}=H^i_0\cup V^i_1\cup\ldots\cup V^i_{m(i)}\cup H^i_{m(i)}$ for each $i\in\N_0$. \[thm:algorithm\] Let $x=(x_0, x_1, x_2, \dots)\in X_{\infty}$ be such that for each $i\in\N_0$, $x_i\in I_{k(i)}^i$ and there exists an admissible permutation (with respect to $C_{i-1}$) $p_i:\{0, \ldots, m(i)\}\to\{0, \ldots, m(i)\}$ of $G_{f_{i}}$ such that $p_i(k(i))=m(i)$. Then there exists a planar embedding of $X_{\infty}$ such that $x$ is accessible. Fix a strictly decreasing sequence $(\eps_i)_{i\in\N}$ such that $\eps_i\to 0$ as $i\to\infty$. Let $\D_0$ be a nice horizontal chain in $\R^2$ with the same number of links as $\chain_0$. By Remark \[rem:ref\] we can find a tubular chain $\D_1\prec\D_0$ with nerve $p_1^{C_0}(G_{f_1})$, such that $Pat(\D_1, \D_0)=Pat(\chain_1, \chain_0)$ and $\mesh(\D_1)<\eps_1$. Note that $p_1(k(1))=m(1)$. Let $F:\R^2\to\R^2$ be a stretching of $\D_1$ (see Remark \[rem:stretch\]). Again using Remark \[rem:ref\] we can define $F(\D_2)\prec F(\D_1)$ such that $\mesh(\D_2)<\eps_2$ ($F$ is uniformly continuous), $Pat(F(\D_2), F(\D_1))=Pat(\chain_2, \chain_1)$ and nerve of $F(\D_2)$ is $p_2^{C_1}(G_{f_2})$. Thus $H^2_{k(2)}$ is the top of $N_{F(\D_2)}$. By the arguments in the previous section, the top of $N_{\D_2}$ is $H_{k(2)k(1)}$. As in the previous section, denote the maximal intervals of monotonicity of $f_1\circ\ldots\circ f_i$ by $$A_{n(i)\ldots n(1)}:=\{x\in I: x\in I_{n(i)}^i, f_i(x)\in I_{n(i-1)}^{i-1}, \ldots, f_1\circ\ldots\circ f_{i-1}(x)\in I_{n(1)}^1\},$$ and denote the corresponding horizontal intervals of $G_{f_1\circ\ldots\circ f_i}$ by $H_{n(i)\ldots n(1)}$. Assume that we have constructed a sequence of chains $\D_i\prec\D_{i-1}\prec\ldots\prec\D_1\prec\D_0$. Take a stretching $F:\R^2\to\R^2$ of $\D_i$ and define $F(\D_{i+1})\prec F(\D_i)$ such that $\mesh(\D_{i+1})<\eps_{i+1}$, $Pat(F(\D_{i+1}), F(\D_i))=Pat(\chain_{i+1}, \chain_i)$ and such that a nerve of $F(\D_{i+1})$ is $p_{i+1}^{C_i}(G_{f_{i+1}})$, which is possible by Remark \[rem:ref\]. Note that the top of ${\mathcal N}_{\D_{i+1}}$ is $H_{k(i+1)\ldots k(1)}$. Since $Pat(F(\D_{i+1}), F(\D_i))=Pat(\D_{i+1}, \D_i)$ for every $i\in\N_0$ and by the choice of the sequence $(\eps_i)$, Lemma \[lem:patterns\] yields that $\cap_{n\in\N_0}\D_n^*$ is homeomorphic to $X_{\infty}$. Let $\phi(X_{\infty})=\cap_{n\in\N_0}\D_n^*$. To see that $x$ is accessible, note that $H=\lim_{i\to\infty}H_{k(i)\ldots k(1)}$ is a well-defined horizontal arc in $\phi(X_{\infty})$ (possibly degenerate). Let $H=[a, b]\times\{h\}$ for some $h\in\R$. Note that for every $y=(y_1, y_2)\in\phi(X_{\infty})$ it holds that $y_2\leq h$. Thus every point $p=(p_1, h)\in H$ is accessible by the vertical planar arc $\{p_1\}\times[ h, h+1]$. Since $x\in H$, the construction is complete. Zigzags {#sec:zigzags} ======= \[def:zigzag\] Let $f: I\to I$ be a continuous piecewise monotone surjection with critical points $0=t_0<t_1< \ldots< t_{m}<t_{m+1}=1$. Let $I_k=[t_k, t_{k+1}]$ for every $k\in\{0, \ldots, m\}$. We say that $I_k$ is *inside a zigzag of $f$* if there exist critical points $a$ and $e$ of $f$ such that $a<t_k<t_{k+1}<e\in I$ and either 1. $f(t_k)>f(t_{k+1})$ and $f|_{[a,e]}$ assumes its global maximum at $a$ and its global minimum at $e$, or 2. $f(t_k)<f(t_{k+1})$ and $f|_{[a,e]}$ assumes its global minimum at $a$ and its global maximum at $e$. Then we say that $x\in \mathring{I}_k=I_k\setminus\{t_k,t_{k+1}\}$ is [*inside a zigzag of $f$*]{} (see Figure \[fig:zigzag\]). We also say that *$f$ contains a zigzag* if there is a point inside a zigzag of $f$. (0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(1/5,1)–(2/5,1/4)–(3/5,3/4)–(4/5,1/2)–(1,1); (2/5,-0.02)–(2/5,0.02); at (0.5, 0.8) [ $f$]{}; at (2/5, -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [$x$]{}; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [$t_4$]{}; (1,-0.02)–(1,0.02); at (1, -0.07) [$e$]{}; (3/5,3/4)–(4/5,1/2); (0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(0.2,0.85)–(0.4,0.5)–(0.6,0.75)–(0.8,0.25)–(1,1); (0,-0.02)–(0,0.02); at (0.5, 0.8) [ $g$]{}; at (0, -0.07) [$a$]{}; (3/5,-0.02)–(3/5,0.02); at (3/5, -0.07) [$t_3$]{}; (7/10,-0.02)–(7/10,0.02); at (7/10, -0.07) [$x$]{}; (4/5,-0.02)–(4/5,0.02); at (4/5, -0.07) [$t_4$]{}; (1,-0.02)–(1,0.02); at (1, -0.07) [$e$]{}; (0.6,0.75)–(0.8,0.25); \[lem:zigzag\] Let $f: I\to I$ be a continuous piecewise linear surjection with critical points $0=t_0<t_1<\ldots <t_m<t_{m+1}=1$. If there is $k\in\{0, \ldots, m\}$ such that $I_k=[t_k, t_{k+1}]$ is not inside a zigzag of $f$, then there exists an admissible permutation $p$ of $G_f$ (with respect to any nice chain $C$) such that $p(k)=m$. Assume that $I_k$ is not inside a zigzag of $f$. Without loss of generality assume that $f(t_k)>f(t_{k+1})$. If $f(a)\geq f(t_{k+1})$ for each $a\in[0, t_k]$ (or if $f(e)\leq f(t_k)$ for each $e\in[t_{k+1}, 1]$) we are done (see Figure \[fig:refl1\]) by simply reflecting all $H_i$, $i<k$ over $H_k$ (or reflecting all $H_i$, $i>k$ over $H_k$ in the second case). (0,1)–(0,2.5)–(0.2,2.5)–(0.2,2)–(0.4,2)–(0.4,3)–(0.6,3)–(0.6,1.5)–(0.8, 1.5)–(0.8,2.3)–(1,2.3)–(1,0.5)–(1.2,0.5)–(1.2,3.5); (1,2.3)–(1,0.5); (1.5,2)–(2,2); (2.5,0.5)–(3.7,0.5)–(3.7,3.5); (2.5,2.3)–(2.5,0.5); (3.5,1)–(3.5,2.5)–(3.3,2.5)–(3.3,2)–(3.1,2)–(3.1,3)–(2.9,3)–(2.9, 1.5)–(2.7,1.5)–(2.7,2.3)–(2.5,2.3); at (0.95,2.4) [$f(t_k)$]{}; at (1.2,0.38) [$f(t_{k+1})$]{}; Therefore, assume that there exists $a\in[0, t_k]$ such that $f(a)<f(t_{k+1})$ and there exists $e\in[t_{k+1},1]$ such that $f(e)>f(t_k)$. Denote the largest such $a$ by $a_1$ and the smallest such $e$ by $e_1$. Since $I_k$ is not inside a zigzag, there exists $e'\in[t_{k+1}, e_1]$ such that $f(e')\leq f(a_1)$ or there exists $a'\in[a_1, t_k]$ such that $f(a')\geq f(e_1)$. Assume the first case and take $e'$ for which $f|_{[t_{k+1}, e_1]}$ attains its global minimum (in the second case we take $a'$ for which $f|_{[a_1, t_k]}$ attains its global maximum). Reflect $f|_{[a_1, t_k]}$ over $f|_{[t_k, e']}$ (in the second case we reflect $f|_{[t_{k+1}, e_1]}$ over $f|_{[a', t_{k+1}]}$). Then, $H_k$ becomes the top of $G_{f|_{[a_1, e_1]}}$ (see Figure \[fig:refl2\]). (0,0.5)–(0,2)–(0.2,2)–(0.2,1)–(0.4,1)–(0.4,1.5)–(0.6,1.5)–(0.6,0)–(0.8, 0)–(0.8,2.5); (0.2,2)–(0.2,1); at (0.02,0.4) [$a_1$]{}; at (0.1,2.1) [$t_k$]{}; at (0.3,0.9) [$t_{k+1}$]{}; at (0.7,-0.1) [$e'$]{}; at (0.8,2.55) [$e_1$]{}; (1.3,1.25)–(1.8,1.25); (2.5,2)–(2.5,1)– (2.7,1)–(2.7,1.5)–(2.9,1.5)–(2.9,0)–(3.3,0)–(3.3,2.5); (2.5,2)–(2.5,1); (2.5,2)–(3.1,2)–(3.1,0.5); If $f(a)\geq f(e')$ for each $a\in[0, a_1]$ (or if $f(e)\leq f(a')$ for all $e\in[e_1, 1]$ in the second case), we are done. So assume that there is $a_2\in[0, a_1]$ such that $f(a_2)<f(e')$ and take the largest such $a_2$. Then there exists $a''\in[a_2, a_1]$ such that $f(a'')\geq f(e_1)$, and take $a''$ for which $f|_{[a_2, a_1]}$ attains its global maximum. If $f(e)\leq f(a'')$ for each $e\in[e_1, 1]$, we reflect $f|_{[a_2, a'']}$ over $f|_{[e_1, 1]}$ and are done. If there is (minimal) $e_2>e_1$ such that $f(e_2)>f(a'')$, then there exists $e''\in[e_1, e_2]$ such that $f(e'')\leq f(a_2)$ and for which $f|_{[e_1, e_2]}$ attains a global minimum. In that case we reflect $f|_{[a'', t_k]}$ over $f|_{[t_k, e']}$ and $f|_{[a_2, a'']}$ over $f|_{[t_k,e'']}$ (see Figure \[fig:refl3\]). (-0.4,-0.2)–(-0.4,2.7)–(-0.2,2.7)–(-0.2,0.5)– (0,0.5)–(0,2)–(0.2,2)–(0.2,1)–(0.4,1)–(0.4,1.5)–(0.6,1.5)–(0.6,0)–(0.8, 0)–(0.8,2.5)–(1,2.5)–(1,-0.5)–(1.2,-0.5)–(1.2,3); (0.2,2)–(0.2,1); at (-0.25,2.8) [$a''$]{}; at (-0.4,-0.3) [$a_2$]{}; at (-0.05,0.4) [$a_1$]{}; at (0.1,2.1) [$t_k$]{}; at (0.3,0.9) [$t_{k+1}$]{}; at (0.7,-0.1) [$e'$]{}; at (0.95,2.55) [$e_1$]{}; at (1.1,-0.6) [$e''$]{}; at (1.2,3.05) [$e_2$]{}; (1.3,1.25)–(1.8,1.25); (2.5,2)–(2.5,1)– (2.7,1)–(2.7,1.5)–(2.9,1.5)–(2.9,0)–(3.5,0)–(3.5,2.5)–(3.7,2.5)–(3.7, -0.5)–(4.1,-0.5)–(4.1,3); (2.5,2)–(2.5,1); (2.5,2)–(3.1,2)–(3.1,0.5)–(3.3,0.5)–(3.3,2.7); (3.3,2.7)–(3.9,2.7)–(3.9,-0.2); Thus we have constructed a permutation such that $H_k$ becomes the top of $G_{f|_{{[a_2, e_2]}}}$. We proceed by induction. \[thm:zigzag\] Let $X_{\infty}=\underleftarrow{\lim}\{I, f_i\}$ where each $f_i: I\to I$ is a continuous piecewise linear surjection. If $x=(x_0, x_1, x_2, \dots)\in X_{\infty}$ is such that for each $i\in\N$, $x_i$ is not inside a zigzag of $f_i$, then there exists an embedding of $X_{\infty}$ in the plane such that $x$ is accessible. The proof follows by Lemma \[lem:zigzag\] and Theorem \[thm:algorithm\]. \[cor:nonzigzag\] Let $X_{\infty}=\underleftarrow{\lim}\{I, f_i\}$ where each $f_i: I\to I$ is a continuous piecewise linear surjection which does not have zigzags. Then, for every $x\in X_{\infty}$ there exists an embedding of $X_{\infty}$ in the plane such that $x$ is accessible. Note that if $T:I\to I$ is a unimodal map and $x\in\UIL$, then $\UIL$ can be embedded in the plane such that $x$ is accessible by the previous corollary. This is Theorem 1 of [@embed]. This easily generalizes to an inverse limit of open interval maps (generalized Knaster continua). The following lemma shows that given arbitrary chains $\{C_i\}$, the zigzag condition from Lemma \[lem:zigzag\] cannot be improved. Let $f: I\to I$ be a continuous piecewise linear surjection with critical points $0=t_0<t_1<\ldots <t_m<t_{m+1}=1$. If $I_k=[t_k, t_{k+1}]$ is inside a zigzag for some $k\in\{0, \ldots, m\}$, then there exists a nice chain $C$ covering $I$ such that $p(k)\neq m$ for every admissible permutation $p$ of $G_f$ with respect to $C$. Take a nice chain cover $C$ of $I$ such that $\mesh\, (C)<\min\{|f(t_i)-f(t_j)|: i,j\in\{0, \ldots, m+1\}, f(t_i)\neq f(t_j)\}$. Assume without loss of generality that $f(t_k)>f(t_{k+1})$ and let $t_i<t_k<t_{k+1}<t_{j}$ be such that minimum and maximum of $f|_{[t_i, t_j]}$ are attained at $t_i$ and $t_j$ respectively. Assume $t_i$ is the largest and $t_{j}$ is the smallest index with such properties. Let $p$ be some permutation. If $p(i)<p(j)<p(k)$, then by the choice of $C$, $p(H_j)$ intersects $p(V_{i'})$ for some $i'\in\{i, \ldots, k\}$. We proceed similarly if $p(j)<p(i)<p(k)$. Let $X_{\infty}=\underleftarrow{\lim}\{I, f_i\}$ and $x=(x_0, x_1, x_2, \ldots)\in X_{\infty}$. If there exist piecewise linear continuous surjections $g_i: I\to I$ and a homeomorphism $h: X_{\infty}\to \underleftarrow{\lim}\{I, g_i\}$ such that every projection of $h(x)$ is not in a zigzag of $g_i$, then $X_{\infty}$ can be embedded in the plane such that $x$ is accessible. We have the following two corollaries. See also Examples \[ex:2sin1x\]-\[ex:Nadler\]. Let $X_{\infty}=\underleftarrow{\lim}\{I, f_i\}$ where each $f_i: I\to I$ is a continuous piecewise linear surjection. If $x=(x_0, x_1, x_2, \dots)\in X_{\infty}$ is such that $x_i$ is inside a zigzag of $f_i$ for at most finitely many $i\in\N$, then there exists an embedding of $X_{\infty}$ in the plane such that $x$ is accessible. Since $\underleftarrow{\lim}\{I, f_i\}$ and $\underleftarrow{\lim}\{I, f_{i+n}\}$ are homeomorphic for every $n\in\N$, the proof follows using Theorem \[thm:zigzag\]. \[cor:iterations\] Let $f$ be a continuous piecewise linear surjection with finitely many critical points and $x=(x_0, x_1, x_2, \dots)\in X_{\infty}=\underleftarrow{\lim}\{I, f\}$. If there exists $k\in\N$ such that $x_i$ is not inside a zigzag of $f^k$ for all but finitely many $i$ , then there exists a planar embedding of $X_{\infty}$ such that $x$ is accessible. Note that $\underleftarrow{\lim}\{I, f^k\}$ and $X_{\infty}$ are homeomorphic. We give applications of Corollary \[cor:iterations\] in the following examples. \[ex:2sin1x\] Let $f$ be a piecewise linear map such that $f(0)=0$, $f(1)=1$ and with critical points $\frac{1}{4}, \frac{3}{4}$, where $f(\frac{1}{4})=\frac{3}{4}$ and $f(\frac{3}{4})=\frac{1}{4}$ (see Figure \[fig:spiral\]). (0,0)–(1,1); (0,0) – (1/4, 3/4) – (3/4, 1/4) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/4, 0) – (1/4, 1); (1/2, 0) – (1/2, 1); (3/4, 0) – (3/4, 1); (0, 1/4) – (1, 1/4); (0, 1/2) – (1, 1/2); (0, 3/4) – (1, 3/4); at (1/4,-0.1) [$\frac{1}{4}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (3/4,-0.1) [$\frac{3}{4}$]{}; at (-0.1,1/4) [$\frac{1}{4}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,3/4) [$\frac{3}{4}$]{}; (1/2,0.5) circle (0.02); Note that $X=\underleftarrow{\lim}\{I,f\}$ consists of two rays compactifying on an arc and therefore, for every $x\in X$, there exists a planar embedding making $x$ accessible. However, the point $\frac{1}{2}$ is inside a zigzag of $f$. Figure \[fig:squared\] shows the graph of $f^2$. Note that the point $\frac{1}{2}$ is not inside a zigzag of $f^2$ and that gives an embedding of $X$ such that $(\frac{1}{2}, \frac{1}{2}, \ldots)$ is accessible. (0,0)–(1,1); (0,0) – (1/12, 3/4) – (1/4,1/4) – (3/4, 3/4) – (11/12,1/4) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/4, 0) – (1/4, 1); (1/2, 0) – (1/2, 1); (3/4, 0) – (3/4, 1); (0, 1/4) – (1, 1/4); (0, 1/2) – (1, 1/2); (0, 3/4) – (1, 3/4); at (1/4,-0.1) [$\frac{1}{4}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (3/4,-0.1) [$\frac{3}{4}$]{}; at (-0.1,1/4) [$\frac{1}{4}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,3/4) [$\frac{3}{4}$]{}; (1/2,0.5) circle (0.02); (2,0)–(2,3/4)–(2+1/12,3/4)–(2+1/12,1/4)–(2+3/4,1/4)–(2+3/4,3/4)–(2+1/2, 3/4)–(2+1/2,0.3)–(2+0.3,0.3)–(2+0.3,1); (2.75,0.5) circle (0.02); Let $x=(x_0, x_1, x_2, \ldots)\in X$ be such that $x_i\in[1/4,3/4]$ for all but finitely many $i\in\N_0$. Then, the embedding in Figure \[fig:squared\] will make $x$ accessible. For other points $x=(x_0, x_1, x_2, \ldots)\in X$ there exists $N\in\N$ such that $x_i\in[0,1/4]$ for each $i>N$ or $x_i\in[3/4,1]$ for each $i>N$ so the standard embedding makes them accessible. In fact, the embedding in Figure \[fig:squared\] will make every $x\in X$ accessible. \[ex:2knsater\] Assume that $f$ is a piecewise linear map with $f(0)=0$, $f(1)=1$ and critical points $f(\frac{3}{8})=\frac{3}{4}$ and $f(\frac{5}{8})=\frac{1}{4}$ (see Figure \[fig:doubleKnaster\]). (0,0)–(1,1); (0,0) – (3/8, 3/4) – (5/8, 1/4) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/4, 0) – (1/4, 1); (1/2, 0) – (1/2, 1); (3/4, 0) – (3/4, 1); (0, 1/4) – (1, 1/4); (0, 1/2) – (1, 1/2); (0, 3/4) – (1, 3/4); at (1/4,-0.1) [$\frac{1}{4}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (3/4,-0.1) [$\frac{3}{4}$]{}; at (-0.1,1/4) [$\frac{1}{4}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,3/4) [$\frac{3}{4}$]{}; (0.5,0.5) circle (0.02); (0,0)–(1,1); (0,0) – (3/16, 3/4) – (1/4+1/16, 1/4) – (3/8,1/2) – (1/2-1/16,1/4) – (1/2+1/16,3/4) – (1/2+1/8,1/2) – (3/4-1/16,3/4) – (3/4,1/2) – (3/4+1/16,1/4) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/4, 0) – (1/4, 1); (1/2, 0) – (1/2, 1); (3/4, 0) – (3/4, 1); (0, 1/4) – (1, 1/4); (0, 1/2) – (1, 1/2); (0, 3/4) – (1, 3/4); at (1/4,-0.1) [$\frac{1}{4}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (3/4,-0.1) [$\frac{3}{4}$]{}; at (-0.1,1/4) [$\frac{1}{4}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,3/4) [$\frac{3}{4}$]{}; (0.5,0.5) circle (0.02); Note that $X=\underleftarrow{\lim}\{I,f\}$ consists of two Knaster continua joined at their endpoints together with two rays both converging to these two Knaster continua. Note that $(\frac{1}{2}, \frac{1}{2}, \ldots)$ can be embedded accessibly with the use of $f^2$, see Figure \[fig:doubleKnaster\]. However, as opposed to the previous example, $X$ cannot be embedded such that every point is accessible (this follows already by a result of Mazurkiewicz [@Maz] which says that there exist nonaccessible points in any planar embedding of a nondegenerate indecomposable continuum). It is proven by Minc and Transue in [@MincTrans] that such an embedding of a chainable continuum exists if and only if it is [*Suslinean*]{}, contains at most countably many mutually disjoint nondegenerate subcontinua. \[ex:Nadler\] Let $f: I\to I$ be as in Figure \[fig:iter\]. This is Nadler’s candidate from [@Nadler] for a negative answer to Question \[q:NaQu\]. However, in what follows we show that every point can be made accessible via some planar embedding of $\underleftarrow{\lim}(I,f)$. Let $n\in\N$. If $J\subset I$ is a maximal interval such that $f^n|_J$ is increasing, then $J$ is not inside a zigzag of $f^n$, see Figure \[fig:iter\]. (0,0)–(1,1); (0,0) – (1/5, 1/5) – (2/5, 4/5) – (3/5,1/5) – (4/5, 4/5) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/5, 0) – (1/5, 1); (2/5, 0) – (2/5, 1); (3/5, 0) – (3/5, 1); (4/5, 0) – (4/5, 1); (0, 1/5) – (1, 1/5); (0, 2/5) – (1, 2/5); (0, 3/5) – (1, 3/5); (0, 4/5) – (1, 4/5); at (1/5,-0.1) [$\frac{1}{5}$]{}; at (2/5,-0.1) [$\frac{2}{5}$]{}; at (3/5,-0.1) [$\frac{3}{5}$]{}; at (4/5,-0.1) [$\frac{4}{5}$]{}; at (-0.1,1/5) [$\frac{1}{5}$]{}; at (-0.1,2/5) [$\frac{2}{5}$]{}; at (-0.1,3/5) [$\frac{3}{5}$]{}; at (-0.1,4/5) [$\frac{4}{5}$]{}; (1/5,1/5)–(2/5,4/5); (3/5,1/5)–(4/5,4/5); (0,0)–(1,1); (0,0) – (1/5, 1/5) – (4/15, 4/5) – (5/15,1/5) – (6/15, 4/5) – (7/15, 1/5) – (8/15,4/5) – (9/15, 1/5) – (10/15, 4/5) – (11/15,1/5) – (12/15, 4/5) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/5, 0) – (1/5, 1); (2/5, 0) – (2/5, 1); (3/5, 0) – (3/5, 1); (4/5, 0) – (4/5, 1); (0, 1/5) – (1, 1/5); (0, 2/5) – (1, 2/5); (0, 3/5) – (1, 3/5); (0, 4/5) – (1, 4/5); at (1/5,-0.1) [$\frac{1}{5}$]{}; at (2/5,-0.1) [$\frac{2}{5}$]{}; at (3/5,-0.1) [$\frac{3}{5}$]{}; at (4/5,-0.1) [$\frac{4}{5}$]{}; at (-0.1,1/5) [$\frac{1}{5}$]{}; at (-0.1,2/5) [$\frac{2}{5}$]{}; at (-0.1,3/5) [$\frac{3}{5}$]{}; at (-0.1,4/5) [$\frac{4}{5}$]{}; (1/5,1/5)–(4/15,4/5); (5/15,1/5)–(6/15,4/5); (7/15,1/5)–(8/15,4/5); (9/15,1/5)–(10/15,4/5); (11/15,1/5)–(12/15,4/5); We will code the orbit of points in the invariant interval $[ 1/5, 4/5]$ in the following way. For $y\in [ 1/5, 4/5]$ let $i(y)=(y_n)_{n\in\N_0}\subset\{0, 1, 2\}^{\infty}$, where $$y_n=\begin{cases} 0,& f^n(y)\in[ 1/5, 2/5],\\ 1,& f^n(y)\in[ 2/5, 3/5],\\ 2,& f^n(y)\in[ 3/5, 4/5]. \end{cases}$$ The definition is somewhat ambiguous with a problem occurring at points $2/5$ and $3/5$. Note, however, that $f^n(2/5)=4/5$ and $f^n(3/5)=1/5$ for all $n\in\N$. So every point in $[1/5, 4/5]$ will have a unique itinerary, except the preimages of $2/5$ (to which we can assign two itineraries $a_1\ldots a_n\frac 01 2222\ldots$) and preimages of $3/5$, (to which we can assign two itineraries $a_1\ldots a_n\frac 12 0000\ldots$), where $\frac 01$ means “$0$ or $1$” and $a_1, \ldots, a_n\in\{0,1,2\}$. Note that if $i(y)=1y_2\ldots y_n1$, where $y_i\in\{0, 2\}$ for every $i\in\{2, \ldots, n\}$, then $y$ is contained in an increasing branch of $f^{n+1}$. This holds also if $n=1$, $y_2\ldots y_n=\emptyset$. Also, if $i(y)=0\ldots$ or $i(y)=2\ldots$, then $y$ is contained in an increasing branch of $f$. See Figure \[fig:coding\]. (0,0)–(1,1); (0,0) – (1/5, 1/5) – (2/5, 4/5) – (3/5,1/5) – (4/5, 4/5) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/5, 0) – (1/5, 1); (2/5, 0) – (2/5, 1); (3/5, 0) – (3/5, 1); (4/5, 0) – (4/5, 1); (0, 1/5) – (1, 1/5); (0, 2/5) – (1, 2/5); (0, 3/5) – (1, 3/5); (0, 4/5) – (1, 4/5); at (3/10,-0.1) [$0$]{}; at (5/10,-0.1) [$1$]{}; at (7/10,-0.1) [$2$]{}; (1/5,1/5)–(2/5,4/5); (3/5,1/5)–(4/5,4/5); (0,0)–(1,1); (0,0) – (1/5, 1/5) – (4/15, 4/5) – (5/15,1/5) – (6/15, 4/5) – (7/15, 1/5) – (8/15,4/5) – (9/15, 1/5) – (10/15, 4/5) – (11/15,1/5) – (12/15, 4/5) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/5, 0) – (1/5, 1); (2/5, 0) – (2/5, 1); (3/5, 0) – (3/5, 1); (4/5, 0) – (4/5, 1); (0, 1/5) – (1, 1/5); (0, 2/5) – (1, 2/5); (0, 3/5) – (1, 3/5); (0, 4/5) – (1, 4/5); at (7/30,-0.1) ; at (9/30,-0.1) ; at (11/30,-0.1) ; at (13/30,-0.1) ; at (15/30,-0.1) ; at (17/30,-0.1) ; at (19/30,-0.1) ; at (21/30,-0.1) ; at (23/30,-0.1) ; (1/5,1/5)–(4/15,4/5); (5/15,1/5)–(6/15,4/5); (7/15,1/5)–(8/15,4/5); (9/15,1/5)–(10/15,4/5); (11/15,1/5)–(12/15,4/5); We extend the symbolic coding to $X$. For $x=(x_0, x_1, x_2, \ldots)\in X$ with itinerary $i(x)$ let $(y_k)_{k\in\Z}$ be defined by $$y_k=\begin{cases} i(x)_k,& k \geq 0, \text{ and for } k < 0,\\ 0,& x_{-k}\in[ 1/5, 2/5],\\ 1,& x_{-k}\in[ 2/5, 3/5],\\ 2,& x_{-k}\in[ 3/5, 4/5]. \end{cases}$$ Again, the assignment is injective everywhere except at preimages of critical points $2/5$ or $3/5$. Now fix $x=(x_0, x_1, x_2, \ldots)\in X$ with its backward itinerary $\ovl x=\ldots y_{-2}y_{-1}y_0$ (assume the itinerary is unique, otherwise choose one of the two possible backward itineraries). Assume first that $y_k\in\{0, 2\}$ for every $k\leq 0$. Then, for every $k\in\N_0$ it holds that $i(x_k)=0\ldots$ or $i(x_k)=2\ldots$ so $x_k$ is in an increasing branch of $f$ and thus not inside a zigzag of $f$. By Theorem \[thm:zigzag\] it follows that there is an embedding making $x$ accessible. Similarly, if there exists $n\in\N$ such that $y_k\neq 1$ for $k<-n$, we use that $X$ is homeomorphic to $\underleftarrow{\lim}\{I, f_j\}$ where $f_1=f^n$, $f_j=f$ for $j\geq 2$. Assume that $\ovl x=\ldots 1(\frac02)^{n_3}1(\frac02)^{n_2}1(\frac02)^{n_1}$ where $\frac02$ means “$0$ or $2$” and $n_i\geq 0$ for $i\in\N$. We will assume that $n_1>0$; the general case follows similarly. Note that $i(x_{n_1-1})=(\frac02)^{n_1}\ldots$ and so it is contained in an increasing branch of $f^{n_1-1}$. Note further that $i(x_{n_1+1+n_2})=1(\frac02)^{n_2}1(\frac02)^{n_1}\ldots$ and so it is contained in an increasing branch of $f^{n_2+2}$. Also $f^{n_2+2}(x_{n_1+1+n_2})=x_{n_1-1}$. Further we note that $i(x_{n_1+1+n_2+1+n_3-1})=(\frac02)^{n_3}1(\frac02)^{n_2}1(\frac02)^{n_1}$ and so it is contained in an increasing branch of $f^{n_3}$. Furthermore, $f^{n_3}(x_{n_1+1+\\n_2+1+n_3-1})=x_{n_1+1+n_2}$. In this way, we see that for every even $k\geq 4$ it holds that $$i(x_{n_1+1+n_2+1+\ldots+1+n_k})=1\left(\frac02\right)^{n_k} 1\ldots1\left(\frac02\right)^{n_1}\ldots$$ and so it is contained in an increasing branch of $f^{n_k+2}$. Also, $f^{n_k+2}(x_{n_1+1+n_2+1+\ldots+1+n_k})=x_{n_1+1+n_2+1+\ldots+1+n_{k-1}-1}$. Similarly, $$i(x_{n_1+1+n_2+1+\ldots+1+n_k+1+n_{k+1}-1})=\left(\frac02\right)^{n_{k+1}} 1\ldots1\left(\frac02\right)^{n_1}\ldots$$ so $x_{n_1+1+n_2+1+\ldots+1+n_k+1+n_{k+1}-1}$ is in an increasing branch of $f^{n_{k+1}}$. Note also that $f^{n_{k+1}}(x_{n_1+1+n_2+1+\ldots+1+n_k+1+n_{k+1}-1})=x_{ n_1+1+n_2+1+\ldots+1+n_k}$. So we have the following sequence $$\ldots \overset{f^{n_5}}{\longrightarrow} x_{n_1+1+\ldots +1+n_4}\overset{f^{n_4+2}}{\longrightarrow} x_{n_1+1+n_2+1+n_3-1}\overset{f^{n_3}}{\longrightarrow} x_{n_1+1+n_2}\overset{f^{n_2+2}}{\longrightarrow}x_{n_1-1}\overset{f^{n_1-1}}{\longrightarrow}x_0,$$ where the chosen points in the sequence are not contained in zigzags of the corresponding bonding maps. Let $$f_i=\begin{cases} f^{n_1-1},& i=1,\\ f^{n_i+2},& i \text{ even,}\\ f^{n_i},& i>1 \text{ odd}. \end{cases}$$ Then, $\underleftarrow{\lim}\{I, f_i\}$ is homeomorphic to $X$ and by Theorem \[thm:zigzag\] it can be embedded in the plane such that every $x\in\underleftarrow{\lim}\{I, f_i\}$ is accessible. Thin embeddings {#sec:thin} =============== We have proven that if a chainable continuum $X$ has an inverse limit representation such that $x\in X$ is not contained in zigzags of bonding maps, then there is a planar embedding of $X$ making $x$ accessible. Note that the converse is not true. The pseudo-arc is a counter-example, because it is homogeneous, so each of its points can be embedded accessibly. However, the crookedness of the bonding maps producing the pseudo-arc implies the occurrence of zigzags in every representation. Since the pseudo-arc is hereditarily indecomposable, no point is contained in an arc. To the contrary, in Minc’s continuum $X_M$ (see Figure \[fig:Minc\]), every point is contained in an arc of length at least $\frac{1}{3}$. (0,0) – (1/3, 1) – (5/12, 1/3) – (7/12,2/3) – (2/3, 0) – (1,1); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/3, 0) – (1/3, 1); (1/2, 0) – (1/2, 1); (2/3, 0) – (2/3, 1); (0, 1/3) – (1, 1/3); (0, 1/2) – (1, 1/2); (0, 2/3) – (1, 2/3); at (1/3,-0.1) [$\frac{1}{3}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (2/3,-0.1) [$\frac{2}{3}$]{}; at (-0.1,1/3) [$\frac{1}{3}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,2/3) [$\frac{2}{3}$]{}; (1/2,0.5) circle (0.015); at (0.9,0.9) [$f$]{}; at (0.53,0.45) [$p$]{}; (0,0) – (1/9, 1) – (5/36, 1/3) – (7/36,2/3) – (2/9, 0) – (1/3,1)–(3/8,0)–(3/8+1/96,2/3)–(10/24-1/96,1/3)–(10/24,1)–(5/12+1/24, 1/3)–(1/2,1/2); (0,0) – (1/9, 1) – (5/36, 1/3) – (7/36,2/3) – (2/9, 0) – (1/3,1)–(3/8,0)–(3/8+1/96,2/3)–(10/24-1/96,1/3)–(10/24,1)–(5/12+1/24, 1/3)–(1/2,1/2); (0,0) – (0,1) – (1,1) – (1,0) – (0,0); (1/3, 0) – (1/3, 1); (1/2, 0) – (1/2, 1); (2/3, 0) – (2/3, 1); (0, 1/3) – (1, 1/3); (0, 1/2) – (1, 1/2); (0, 2/3) – (1, 2/3); at (1/3,-0.1) [$\frac{1}{3}$]{}; at (1/2,-0.1) [$\frac{1}{2}$]{}; at (2/3,-0.1) [$\frac{2}{3}$]{}; at (-0.1,1/3) [$\frac{1}{3}$]{}; at (-0.1,1/2) [$\frac{1}{2}$]{}; at (-0.1,2/3) [$\frac{2}{3}$]{}; (1/2,0.5) circle (0.015); at (0.53,0.45) [$p$]{}; at (0.9,0.9) [$f^2$]{}; In the next two definitions we introduce the notion of thin embedding, used under this name in [@DT]. In [@ABC-q] the notion of thin embedding was referred to as [*$C$-embedding*]{}. Let $Y\subset\R^2$ be a continuum. We say that $Y$ is [*thin chainable*]{} if there exists a sequence $({\chain_n})_{n\in\N}$ of chains in $\R^2$ such that $Y=\cap_{n\in\N}\chain^*_n$, where $\chain_{n+1}\prec\chain_n$ for every $n\in\N$, $\textrm{mesh}(\chain_n)\to 0$ as $n\to\infty$, and the links of $\chain_n$ are connected sets in $\R^2$ (note that links are open in the topology of $\R^2$). \[def:thin\] Let $X$ be a chainable continuum. We say that an embedding $\phi: X\to\R^2$ is a [*thin embedding*]{} if $\phi(X)$ is thin chainable. Otherwise $\phi$ is called a [*thick embedding*]{}. Note that in [@Bing] Bing shows that every chainable continuum has a thin embedding in the plane. Is there a planar embedding of Minc’s chainable continuum $X_M$ which makes $p$ accessible? Or as a special case, is there a *thin embedding* of $X_M$ which makes $p$ accessible? An [*Elsa continuum*]{} (see [@Na-Elsa]) is a continuum consisting of a ray compactifying on an arc (in [@BrBr] this was called an arc+ray continuum). An example of a thick embedding of an Elsa continuum was constructed by Bing (see Figure \[fig:Bing\]). (0,-1)–(0,1); plot ([3+0.5\*cos()]{}, [0.5\*sin()]{}); plot ([3+0.3\*cos()]{}, [0.3\*sin()]{}); plot ([3+0.1\*cos()]{}, [0.1\*sin()]{}); plot ([2.1+0.4\*cos()]{}, [0.5\*sin()]{}); plot ([2.1+0.25\*cos()]{}, [0.3\*sin()]{}); plot ([2.1+0.1\*cos()]{}, [0.1\*sin()]{}); plot ([1.6+0.1\*cos()]{}, [0.1\*sin()]{}); plot ([1.6+0.25\*cos()]{}, [0.3\*sin()]{}); plot ([1.6+0.4\*cos()]{}, [0.5\*sin()]{}); (2.2,0)–(2.2,1); (2.35,0)–(2.35,1); plot ([2.275+0.075\*cos()]{}, [1+0.075\*sin()]{}); (1.5,0)–(1.5,-1); (1.35,0)–(1.35,-1); plot ([1.425+0.075\*cos()]{}, [-1+0.075\*sin()]{}); plot ([3.6+0.1\*cos()]{}, [0.1\*sin()]{}); plot ([3.6+0.3\*cos()]{}, [0.3\*sin()]{}); plot ([3.6+0.5\*cos()]{}, [0.5\*sin()]{}); (3.7,0)–(3.7,1); (3.9,0)–(3.9,1); plot ([3.8+0.1\*cos()]{}, [1+0.1\*sin()]{}); (2.9,0)–(2.9,-1); (2.7,0)–(2.7,-1); plot ([2.8+0.1\*cos()]{}, [-1+0.1\*sin()]{}); (0.3,0) circle (0.02); (0.6,0) circle (0.02); (0.9,0) circle (0.02); An example of a thick embedding of the $3$-Knaster continuum was given by Dbski and Tymchatyn in [@DT]. An arc has a unique planar embedding (up to equivalence), so all of its planar embeddings are thin. Therefore, it is natural to ask the following question. \[Question 1 in [@ABC-q]\] Which chainable continua have a thick embedding in the plane? Given a chainable continuum $X$, let $\E_C(X)$ denote the set of all planar embeddings of $X$ obtained by performing admissible permutations of $G_{f_i}$ for every representation $X$ as $\underleftarrow{\lim}\{I, f_i\}$. The next theorem says that the class of all planar embeddings of chainable continuum $X$ obtained by performing admissible permutations of graphs $G_{f_i}$ is the class of all thin planar embeddings of $X$ up to the equivalence relation between embeddings. \[thm:allemb\] Let $X$ be a chainable continuum and $\phi: X\to\R^2$ a thin embedding of $X$. Then there exists an embedding $\psi\in\E_C(X)$ which is equivalent to $\phi$. Recall that $\chain^*_n = \bigcup_{\ell \in \chain_n} \ell$. Let $\phi(X)=\cap_{n\in\N_0}\chain^*_n$, where the links of $\chain_n$ are open, connected sets in $\R^2$. Furthermore, $\chain_{n+1}\prec\chain_n$ for every $n\in\N_0$. Note that we assume that links of $\chain_n$ have a polygonal curve for a boundary, using a brick decomposition of the plane. We argue that we can also assume that every $\chain^*_n$ is simply connected. This goes in a few steps. In every step we first state the claim we can obtain and then argue in the rest of the step how to obtain it. 1. Without loss of generality we can assume that the separate links of $\chain_n$ are simply connected, by filling in the holes. That is, if a link $\ell\in\chain_n$ is such that $\R^2\setminus\ell$ separates the plane, instead of $\ell$, we take $\ell \cup \bigcup_{i}V_i$, where the $V_i$ are the bounded components of $\R^2\setminus\ell$. Filling in the holes thus merges all the links contained in $\ell \cup \bigcup_i V_i$ into a single link. This does not change the mesh nor the pattern of the chain. 2. We can assume without loss of generality that $\ell_{i+1}$ doesn’t separate $\overline{\ell_i}$, i.e., $\overline{\ell_i} \setminus \ell_{i+1}$ is connected. Indeed, let $K_j$ be the components of $\overline{\ell_i} \cap \partial \ell_{i+1}$ that separate $\ell_i$, and take pairwise disjoint neighbourhoods $U_j$ of $K_j$ so small that $\ell_{i+1}$ doesn’t separate $\overline{U_j}$ and such that $\overline{U_j}\cap\overline{\ell_k}=\emptyset$ for all $k\neq i, i+1$ (recall that the chain is taut). Pick a component $L$ of $\ell_i \setminus \ell_{i+1}$ that intersects $\ell_{i-1}$ (or just any component if $i = 1$). Replace $\ell_i$ with $L \cup U_j$, where $K_j \subset \partial L$ and replace $\ell_{i+1}$ with $\ell_{i+1}\cup(\ell_i\setminus\overline{L})$. The new cover is a chain, and $\ell_{i+1}$ no longer separates $\overline{\ell_i}$. 3. We can assume that for every hole $H$ between the links $\ell_i$ and $\ell_{i+1}$ (a connected bounded component of $\R^2\setminus (\ell_i\cup\ell_{i+1})$) it holds that if $H \cap \ell_j \neq \emptyset$ for some $j$, then $\ell_j \subset \ell_i \cup \ell_{i+1} \cup H$. Denote by $U_i$ the union of bounded component of $\R^2\setminus (\ell_i\cup\ell_{i+1})$ and note that $\{\ell_1\cup U_1\cup\ell_2, \ell_3\cup U_3\cup\ell_4, \ldots, \ell_{k(n)-1}\cup U_{k(n)-1}\cup\ell_{k(n)}\}$ is again a chain. (It can happen that the first or last few links are merged into one link. Also, we can assume that $n$ is even by merging the last two links if necessary.) Denote for simplicity $\tilde{\ell}_i=\ell_{2i-1}\cup U_{2i-1}\cup\ell_{2i}$ for every $i\in \{1,2,\ldots, n/2\}$. We claim that if $\tilde{\ell}_j\cap H\neq\emptyset$ for some hole between $\tilde{\ell}_i$ and $\tilde{\ell}_{i+1}$, then $\tilde{\ell}_j\subset \tilde{\ell}_i\cup\tilde{\ell}_{i+1}\cup H$. Assume the contrary, and take without loss of generality that $j>i+1$. Then necessarily $j=i+2$ and $\tilde{\ell}_{i+2}$ separates $\tilde{\ell}_{i+1}$ so that at least two components of $\tilde \ell_{i+1} \setminus \tilde \ell_{i+2}$ intersect $\tilde{\ell}_i$. That is, $\tilde{\ell}_{i+2}$ separates $\ell_{2i+1}$. But this is a contradiction since $\tilde{\ell}_{i+2}=\ell_{2i+3}\cup U_{2i+3}\cup\ell_{2i+4}$ can only intersect $\ell_{2i+1}$ if $\ell_{2i+1}\subset U_{2i+3}$ in which case $\tilde{\ell}_{i+2}$ does not separate $\ell_{2i+1}$. 4. If there is a hole between links $\tilde \ell_i$ and $\tilde \ell_{i+1}$, then we can fill it in a similar way as in Step (1). That is, letting $\tilde U_i$ be the union of bounded components of $\R^2 \setminus (\tilde \ell_i \cup \tilde \ell_{i+1})$, the links of the modified chain are $\{\tilde \ell_1, \ldots, \tilde \ell_{i - 1}, \tilde \ell_i\cup \tilde U_i\cup\tilde \ell_{i+1}, \tilde \ell_{i+2}, \ldots, \tilde \ell_{k(n)/2}\}$. (It can happen that $\tilde \ell_j\subset \tilde U_i$ for all $j>N\geq i+1$ or $j<N\leq i$, but then we merge all these links.) We do this for each $i \in \{ 1, \dots, k(n)/2 \}$ where there is a hole between $\tilde \ell_i$ and $\tilde \ell_{i+1}$, so not just the odd values of $i$ as in Step (2). Due to the claim in Step (2), the result is again a chain. These modified chains can have a larger mesh (up to four times the original mesh), but still satisfy $\chain_{n+1}\prec\chain_n$ for every $n\in\N_0$ and $\mesh(\chain_n)\to 0$ as $n\to\infty$. In the rest of the proof we construct homeomorphisms $F_j$, $0 \leq j \leq n\in\N_0$ and $G_n:=F_n\circ\ldots \circ F_1\circ F_0$, which straighten the chains $\chain_n$ to horizontal chains. The existence of such homeomorphisms follows from the generalization of the piecewise linear Schoenflies’ theorem given in [@Moise Section 3]. Take a homeomorphism $F_0:\R^2\to\R^2$ which maps $\chain_0$ to a horizontal chain. Then $F_0(\chain_1)\prec F_0(\chain_0)$ and there is a homeomorphism $F_1:\R^2\to\R^2$ which is the identity on $\R^2\setminus F_0(\chain_0)^*$ (recall that $\chain_n^*$ denotes the union of links of $\chain_n$), and which maps $F_0(\chain_1)^*$ to a tubular neighborhood of some permuted flattened graph with $\mesh(F_1(F_0(\chain_1)))<\mesh(\chain_1)$.\ Note that $G_n(\chain_{n+1})\prec G_n(\chain_{n})$ and there is a homeomorphism $F_{n+1}:\R^2\to\R^2$ which is the identity on $\R^2\setminus G_n(\chain_{n})^*$ and which maps $G_n(\chain_{n+1})^*$ to a tubular neighborhood of some flattened permuted graph with $\mesh(F_{n+1}(G_n(\chain_{n+1})))<\mesh(\chain_{n+1})$.\ Note that the sequence $(G_n)_{n\in\N_0}$ is uniformly Cauchy and $G := \lim_{n\to\infty}G_n$ is well-defined. By construction, $G:\R^2\to\R^2$ is a homeomorphism and $G\circ\phi\in\E_C(X)$. Is there a chainable continuum $X$ and a thick embedding $\psi$ of $X$ such that the set of accessible points of $\psi(X)$ is different from the set of accessible points of $\phi(X)$ for any thin embedding $\phi$ of $X$? Uncountably many nonequivalent embeddings {#sec:nonequivalent} ========================================= In this section we construct, for every chainable continuum containing a nondegenerate indecomposable subcontinuum, uncountably many embeddings which are pairwise not strongly equivalent. Recall that $\phi, \psi\colon X\to \R^2$ are strongly equivalent if $\phi\circ\psi^{-1}$ can be extended to a homeomorphism of $\R^2$. The idea of the construction is to find uncountably many composants in some indecomposable planar continuum which can be embedded accessibly in more than a point. The conclusion then follows easily with the use of the following theorem. \[thm:Maz\] Let $X\subset\R^2$ be an indecomposable planar continuum. There are at most countably many composants of $X$ which are accessible in at least two points. Let $X=\underleftarrow{\lim}\{I,f_i\}$, where $f_i: I\to I$ are continuous piecewise linear surjections. Let $f: I\to I$ be a continuous surjection. An interval $I'\subset I$ is called a *surjective interval* if $f(I')=I$ and $f(J)\neq I$ for every $J\subsetneq I'$. Let $A_1, \ldots, A_n$, $n \geq 1$, be the surjective intervals of $f$ ordered from left to right. For every $i\in\{1, \ldots, n\}$ define the *right accessible set* by $R(A_i) = \{ x\in A_i : f(y)\neq f(x) \text{ for all } x<y\in A_i\}$ (see Figure \[fig:LR\]). (0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(1/9,0.6)–(2/9,0.4)–(1/3,1)–(0.4,0.8)–(0.45,1)–(2/3,0)–(7/9, 0.6)–(8/9,0.4)–(1,1); \[shift=[(1,1)]{},rotate=180\] (0,0.6)–(1/9,0.6); (0,5.4\*0.02)–(0.02,5.4\*0.02); (0,5.4\*0.04)–(0.04,5.4\*0.04); (0,5.4\*0.06)–(0.06,5.4\*0.06); (0,5.4\*0.08)–(0.08,5.4\*0.08); (0,5.4\*0.1)–(0.1,5.4\*0.1); (0,0.676)–(0.27,0.676); (0,0.78)–(0.29,0.78); (0,0.9)–(0.315,0.9); (0,0.97)–(0.325,0.97); at (1.1,1) [$f$]{}; (0,-0.1)–(1/3,-0.1); at (1/6,-0.2) [$A_1$]{}; (0.45,-0.1)–(2/3,-0.1); at (0.55,-0.2) [$A_2$]{}; (2/3,-0.1)–(1,-0.1); at (5/6,-0.2) [$A_3$]{}; (0,0)–(1/9-0.035,0); (2/9,0.4)–(2/9,0); (1/9-0.035,0.4)–(1/9-0.035,0); (1/3,1)–(1/3,0); (2/9,0)–(1/3,0); at (-0.2,0.24) [$R(A_1)$]{}; (-0.2,0.2)–(0.05,0); (-0.2,0.2)–(0.3,0); (2/3,0)–(0.74,0); (0.74,0.4)–(0.74,0); (8/9,0.4)–(8/9,0); (8/9,0)–(1,0); at (1.2,0.24) [$R(A_3)$]{}; (1.2,0.2)–(0.7,0); (1.2,0.2)–(8/9+1/18,0); We will first assume that the map $f_i$ contains at least three surjective intervals for every $i\in\N$. We will later see that this assumption can be made without loss of generality. \[rem:image\] Assume that $f:I\to I$ has $n\geq 3$ surjective intervals. Then $A_1\cap A_n=\emptyset$ and $f([l,r])=I$ for every $l\in A_1$ and $r\in A_n$. Also $f([l,r])=I$ for every $l\in A_i$ and $r\in A_j$ where $j-i\geq 2$. \[lem:preimages\] Let $J\subset I$ be a closed interval and $f:I\to I$ a map with surjective intervals $A_1, \ldots A_n$, $n\geq 1$. For every $i\in\{1, \ldots, n\}$ there exists an interval $J^i\subset A_i$ such that $f(J^i)=J$, $f(\partial J^i)=\partial J$ and $J^i\cap R(A_i)\neq\emptyset$. Consider the interval $J=[a,b]$ and fix $i\in\{1, \ldots, n\}$. Let $a_i,b_i\in R(A_i)$ be such that $f(a_i)=a$ and $f(b_i)=b$. Assume first that $b_i<a_i$ (see Figure \[fig:JL\]). Find the smallest $\tilde a_i>b_i$ such that $f(\tilde a_i)=a$. Then $J^i:=[b_i, \tilde a_i]$ has the desired properties. If $a_i<b_i$, then take $J^i=[a_i, \tilde b_i]$, where $\tilde b_i>a_i$ is the smallest such that $f(\tilde b_i)=b$. \[yscale=1,xscale=-1\] (0,0)–(1,0)–(1,1)–(0,1)–(0,0); (0,0)–(1/9,0.6)–(2/9,0.4)–(1/3,1)–(2/3,0)–(7/9,0.6)–(8/9,0.4)–(1,1); (0,0.5)–(0,0.7); at (-0.1,0.6) [$J$]{}; (0,0.5)–(0.5/5.4,0.5); (0,0.7)–(0.7/5.4+0.15,0.7); (0.5/5.4+0.15,0.5)–(0.7/5.4+0.15,0.7); (0.5/5.4+0.15,0.5)–(0.5/5.4+0.15,0); (0.7/5.4+0.15,0.7)–(0.7/5.4+0.15,0); (0.5/5.4+0.15,0)–(0.7/5.4+0.15,0); at (0.28,-0.15) [$J^3$]{}; (0.5/5.4,0.5)–(0.5/5.4,0); at (0.5/5.4,-0.05) [$a_i$]{}; at (0.7/5.4+0.15,-0.05) [$b_i$]{}; at (-0.03,0.5) [$a$]{}; at (-0.03,0.7) [$b$]{}; at (1.1,1) [$f$]{}; The following definition is a slight generalization of the notion of the “top” of a permutation $p(G_f)$ of the graph $\Gamma_f$. Let $f: I\to I$ be a piecewise linear surjection and for a chain $C$ of $I$, let $p$ be a admissible $C$-permutation of $G_f$. For $x\in I$ denote the point in $p(G_f)$ corresponding to $f(x)$ by $p(f(x))$. We say that $x$ is *topmost in $p(G_f)$* if there exists a vertical ray $\{f(x)\}\times[h, \infty)$, where $h\in\R$, which intersects $p(G_f)$ only in $p(f(x))$. If $A_1, \ldots, A_n$ are surjective intervals of $f: I\to I$, then every point in $R(A_n)$ is topmost. Also, for every $i=1, \ldots, n$ there exists a permutation of $G_f$ such that every point in $R(A_i)$ is topmost. \[lem:twopoints\] Let $f: I\to I$ be a map with surjective intervals $A_1, \ldots A_n$, $n\geq 1$. For $[a, b]=J\subset I$ and $i\in\{1, \ldots, n\}$ denote by $J^i$ an interval from Lemma \[lem:preimages\]. There exists an admissible permutation $p_i$ of $G_f$ such that both endpoints of $J^i$ are topmost in $p_i(G_f)$. Let $A_i=[l_i, r_i]$. Assume first that $f(l_i)=0$ and $f(r_i)=1$, thus $a_i<b_i$ (recall the notation $a_i, \tilde a_i$ and $b_i, \tilde b_i$ from the proof of Lemma \[lem:preimages\]). Find the smallest critical point $m$ of $f$ such that $m\geq \tilde b_i$ and note that $f(x)> f(a)$ for all $x\in A_i$, $x>m$. So we can reflect $f|_{[m,r_i]}$ over $f|_{[a_i,m]}$ and $f|_{[r_i,1]}$ over $f|_{[0,l_{i}]}$. This makes $a_i$ and $\tilde b_i$ topmost, see Figure \[fig:topmost\]. In the case when $f(l_i)=1, f(r_i)=0$, thus $a_i>b_i$, we have that $f(x)<f(b)$ for all $x\in A_i$, $x>m$ so we can again reflect $f|_{[m,r_i]}$ over $f|_{[a_i,m]}$ making $\tilde a_i$ and $b_i$ topmost. (0,0)–(0.25,0.5)–(0.4,0.3)–(0.6,0.7)–(0.8,0.1)–(1,1); (1,0.07)–(0.035,0.07); (1,0.6)–(0.55,0.6); (0.6,0.7) circle (0.015); at (0.7,0.75) [$(m,f(m))$]{}; (0.55,0.6)–(0.4,0.3)–(0.25,0.5)–(0.035,0.07); at (1.05,0.6) [$b$]{}; at (1.05,0.07) [$a$]{}; (1.2,0.5)–(1.5,0.5); (2,0)–(2.25,0.5)–(2.4,0.3)–(2.6,0.7); (2.035,0.07)–(3,0.07); (2.55,0.6)–(3,0.6); (2.6,0.7)–(2,0.7); (2,0.7)–(1.8,0.1)–(1.6,1); (2.55,0.6)–(2.4,0.3)–(2.25,0.5)–(2.035,0.07); \[lem:Mayerprep\] Let $X=\underleftarrow{\lim}\{I,f_i\}$, where each $f_i: I\to I$ is a continuous piecewise linear surjection and assume that $X$ is indecomposable. If $f_i$ contains at least three surjective intervals for every $i\in\N$, then there exist uncountably many planar embeddings of $X$ that are not strongly equivalent. For every $i\in\N$ let $k_i\geq 3$ be the number of surjective branches of $f_i$ and fix $L_i, R_i\in\{1, \ldots, k_i\}$ such that $|L_i-R_i|\geq 2$. Let $J\subset I$ and $(n_i)_{i\in\N}\in\prod_{i\in\N}\{L_i,R_i\}$. Then $$J^{(n_i)}:=J\stackrel{\text{$f_1$}}{\leftarrow} J^{n_1}\stackrel{\text{$f_2$}}{\leftarrow} J^{n_1n_2}\stackrel{\text{$f_3$}}{\leftarrow} J^{n_1n_2n_3}\stackrel{\text{$f_4$}}{\leftarrow}\ldots$$ is a well-defined subcontinuum of $X$. Here we used the notation $J^{nm}=(J^n)^m$. Moreover, Lemma \[lem:twopoints\] and Theorem \[thm:algorithm\] imply that $X$ can be embedded in the plane such that both points in $\partial J\leftarrow \partial J^{n_1}\leftarrow \partial J^{n_1n_2}\leftarrow \partial J^{n_1n_2n_3}\leftarrow\ldots$ are accessible. Remark \[rem:image\] implies that for every $f: I\to I$ with surjective intervals $A_1, \ldots, A_n$, every $|i-j|\geq 2$ and every $J\subset I$ it holds that $f([J^i, J^j])=I$, where $[J^i, J^j]$ denotes the convex hull of $J^i$ and $J^j$. So if $(n_i), (m_i)\in\prod_{i\in\N}\{L_i,R_i\}$ differ at infinitely many places, then there is no proper subcontinuum of $X$ which contains both $J^{(n_i)}$ and $J^{(m_i)}$, they are contained in different composants of $X$. Now Theorem \[thm:Maz\] implies that there are uncountably many planar embeddings of $X$ that are not strongly equivalent. Next we prove that the assumption of at least three surjective intervals can be made without loss of generality for every nondegenerate indecomposable chainable continuum. For $X=\underleftarrow{\lim}\{I,f_i\}$, where each $f_i: I\to I$ is a continuous piecewise linear surjection, we show that there is $X'=\underleftarrow{\lim}\{I,g_i\}$ homeomorphic to $X$ such that $g_i$ has at least three surjective intervals for every $i\in\N$. We will build on the following remark. \[rem:May\] Assume that $f, g: I\to I$ each have at least two surjective intervals. Note that then $f\circ g$ has at least three surjective intervals. So if $f_i$ has two surjective intervals for every $i\in\N$, then $X$ can be embedded in the plane in uncountably many nonequivalent ways. Let $\eps>0$ and let $f: I\to I$ be a continuous surjection. We say that $f$ is $P_{\eps}$ if for every two segments $A,B\subset I$ such that $A\cup B= I$ it holds that $d_{H}(f(A), I)<\eps$ or $d_{H}(f(B), I)<\eps$, where $d_{H}$ denotes the Hausdorff distance. \[rem:threepts\] Let $f: I\to I$ and $\eps>0$. Note that $f$ is $P_{\eps}$ if and only if there exist $0\leq x_1<x_2<x_3\leq 1$ such that one of the following holds - $|f(x_1)-0|,|f(x_3)-0|<\eps$, $|f(x_2)-1|<\eps$, or - $|f(x_1)-1|,|f(x_3)-1|<\eps$, $|f(x_2)-0|<\eps$. For $n<m$ denote by $f_n^m=f_n\circ f_{n+1}\circ\ldots\circ f_{m-1}$. \[thm:kuykendall\] The inverse limit $X=\underleftarrow{\lim}\{I,f_i\}$ is indecomposable if and only if for every $\eps>0$ and every $n\in\N$ there exists $m>n$ such that $f_n^m$ is $P_{\eps}$. Furthermore, we will need the following strong theorem. \[thm:Miod\] Two continua $\underleftarrow{\lim}\{I, f_i\}$ and $\underleftarrow{\lim}\{I, g_i\}$ are homeomorphic if and only if for every sequence of positive integers $\eps_i\to 0$ there exists an infinite diagram as in Figure \[fig:Miod\], \(1) at (0,1) [$I$]{}; (2) at (0.5,0) [$I$]{}; (3) at (1,1) [$I$]{}; (4) at (1.5,0) [$I$]{}; (5) at (2,1) [$I$]{}; (6) at (2.5,0) [$I$]{}; (7) at (3,1) [$I$]{}; (8) at (3.5,0) [$I$]{}; (9) at (4,1) [$\ldots$]{}; (10) at (4.5,0) [$\ldots$]{}; (3) to (1); (5) to (3); (7) to (5); (9) to (7); (4) to (2); (6) to (4); (8) to (6); (10) to (8); (2) to (1); (3) to (2); (4) to (3); (5) to (4); (6) to (5); (7) to (6); (8) to (7); at (0.55,1.15) [$f_{n_1}^{n_2}$]{}; at (1.55,1.15) [$f_{n_2}^{n_3}$]{}; at (2.55,1.15) [$f_{n_3}^{n_4}$]{}; at (3.55,1.15) [$f_{n_4}^{n_5}$]{}; at (1.05,-0.2) [$g_{m_1}^{m_2}$]{}; at (2.05,-0.2) [$g_{m_2}^{m_3}$]{}; at (3.05,-0.2) [$g_{m_3}^{m_4}$]{}; at (4.05,-0.2) [$g_{m_4}^{m_5}$]{}; where $(n_i)$ and $(m_i)$ are sequences of strictly increasing integers, $f_{n_i}^{n_{i+1}}=f_{n_i+1}\circ\ldots\circ f_{n_{i+1}}$, $g_{m_i}^{m_{i+1}}=g_{m_i+1}\circ\ldots\circ g_{m_{i+1}}$ for every $i\in\N$ and every subdiagram as in Figure \[fig:Miod2\] is $\eps_i$-commutative. \(1) at (0,1) [$I$]{}; (2) at (0.5,0) [$I$]{}; (3) at (1,1) [$I$]{}; (4) at (1.5,0) [$\ldots$]{}; (5) at (2,1) [$\ldots$]{}; (6) at (2.5,0) [$I$]{}; (7) at (3,1) [$I$]{}; (8) at (3.5,0) [$I$]{}; (3) to (1); (5) to (3); (4) to (2); (6) to (4); (2) to (1); (3) to (2); (7) to (5); (7) to (6); (8) to (6); (8) to (7); at (0.55,1.15) [$f_{n_i}^{n_{i+1}}$]{}; at (1.55,1.15) [$f_{n_{i+1}}^{n_{i+2}}$]{}; at (2.55,1.15) [$f_{n_k}^{n_{k+1}}$]{}; at (1.05,-0.2) [$g_{m_i}^{m_{i+1}}$]{}; at (2.05,-0.2) [$g_{m_{k-1}}^{m_k}$]{}; at (3.05,-0.2) [$g_{m_{k}}^{m_{k+1}}$]{}; \(2) at (0.5,0) [$I$]{}; (3) at (1,1) [$I$]{}; (4) at (1.5,0) [$I$]{}; (5) at (2,1) [$\ldots$]{}; (6) at (2.5,0) [$\ldots$]{}; (7) at (3,1) [$I$]{}; (8) at (3.5,0) [$I$]{}; (9) at (4,1) [$I$]{}; (5) to (3); (4) to (2); (6) to (4); (3) to (2); (7) to (5); (8) to (6); (8) to (7); (4) to (3); (9) to (8); (9) to (7); at (1.55,1.15) [$f_{n_{i+1}}^{n_{i+2}}$]{}; at (2.55,1.15) [$f_{n_k}^{n_{k+1}}$]{}; at (3.55,1.15) [$f_{n_{k+1}}^{n_{k+2}}$]{}; at (1.05,-0.2) [$g_{m_i}^{m_{i+1}}$]{}; at (2.05,-0.2) [$g_{m_{i+1}}^{m_{i+2}}$]{}; at (3.05,-0.2) [$g_{m_{k}}^{m_{k+1}}$]{}; \[thm:Mayer\] Every nondegenerate indecomposable chainable continuum $X$ can be embedded in the plane in uncountably many ways that are not strongly equivalent. Let $X=\underleftarrow{\lim}\{I,f_i\}$, where each $f_i: I\to I$ is a continuous piecewise linear surjection. If all but finitely many $f_i$ have at least three surjective intervals, we are done by Lemma \[lem:Mayerprep\]. If for all but finitely many $i$ the map $f_i$ has two surjective intervals, we are done by Remark \[rem:May\]. Now fix a sequence $(\eps_i)$ such that $\eps_i>0$ for every $i\in\N$ and $\eps_i\to 0$ as $i\to\infty$. Fix $n_1=1$ and find $n_2>n_1$ such that $f_{n_1}^{n_2}$ is $P_{\eps_1}$. Such $n_2$ exists by Theorem \[thm:kuykendall\]. For every $i\in\N$ find $n_{i+1}>n_i$ such that $f_{n_i}^{n_{i+1}}$ is $P_{\eps_i}$. The continuum $X$ is homeomorphic to $\underleftarrow{\lim}\{I, f_{n_i}^{n_{i+1}}\}$. Every $f_{n_i}^{n_{i+1}}$ is piecewise linear and there exist $x_1^i<x_2^i<x_3^i$ as in Remark \[rem:threepts\]. Take them to be critical points and assume without loss of generality that they satisfy condition $(a)$ of Remark \[rem:threepts\]. Define a piecewise linear surjection $g_i: I\to I$ with the same set of critical points as $f_{n_i}^{n_{i+1}}$ such that $g_i(c)=f_{n_i}^{n_{i+1}}(c)$ for all critical points $c\not\in\{x_1, x_2, x_3\}$ and $g_i(x_1)=g_i(x_3)=0$, $g_i(x_2)=1$. Then $g_i$ is $\eps_i$-close to $f_{n_i}^{n_{i+1}}$. By Theorem \[thm:Miod\], $\underleftarrow{\lim}\{I, f_{n_i}^{n_{i+1}}\}$ is homeomorphic to $\underleftarrow{\lim}\{I, g_i\}$. Since every $g_i$ has at least two surjective intervals, this finishes the proof by Remark \[rem:May\]. Specifically, Theorem \[thm:Mayer\] proves that the pseudo-arc has uncountably many embeddings that are not strongly equivalent. Lewis [@Lew] has already proven this with respect to the standard version of equivalence, by carefully constructing embeddings with different prime end structures. In the next theorem we expand the techniques from this section to construct uncountably many strongly nonequivalent embeddings of every chainable continuum that contains a nondegenerate indecomposable subcontinuum. First we give a generalisation of Lemma \[lem:twopoints\]. \[lem:indecsubc\] Let $f: I\to I$ be a surjective map and let $K\subset I$ be a closed interval. Let $A_1, \ldots, A_n$ be the surjective intervals of $f|_K: K\to f(K)$, and let $J^i$, $i \in \{1,\dots, n\}$, be intervals from Lemma \[lem:preimages\] applied to the map $f|_K$.\ Assume $n\geq 4$. Then there exist $\alpha, \beta\in\{1, \ldots, n\}$ such that $|\alpha-\beta|\geq 2$ and such that there exist admissible permutations $p_{\alpha}, p_{\beta}$ of $G_f$ such that both endpoints of $J^{\alpha}$ are topmost in $p_{\alpha}(G_{f|_K})$ and such that both endpoints of $J^{\beta}$ are topmost in $p_{\beta}(G_{f|_K})$. Let $K=[k_l, k_r]$ and $f(K)=[K_l, K_r]$. Let $x>k_r$ be the smallest local extremum of $f$ such that $f(x)>K_r$ or $f(x)<K_l$. A surjective interval $A_i=[l_i, r_i]$ will be called increasing (decreasing) if $f(l_i)=K_l$ ($f(r_i)=K_l$). [**Case 1.**]{} Assume $f(x)>K_r$ (see Figure \[fig:extperm\]). If $A_i=[l_i, r_i]$ is increasing, since $f(x)>K_r$, there exists an admissible permutation which reflects $f|_{[m, x]}$ over $f|_{[a_i, m]}$ and leaves $f|_{[x, 1]}$ fixed. Here $m$ is chosen as in the proof of Lemma \[lem:twopoints\]. Since there are at least four surjective intervals, at least two are increasing. This finishes the proof. [**Case 2.**]{} If $f(x)<K_l$ we proceed as in the first case but for decreasing $A_i$. (0,0)–(0,1)–(1,1)–(1,0)–(0,0); (0,0)–(0,-0.2); (1,0)–(1,-0.2); (0,0)–(-0.2,0); (0,1)–(-0.2,1); at (0,-0.3) [$k_l$]{}; at (1,-0.3) [$k_r$]{}; at (-0.3,0) [$K_l$]{}; at (-0.3,1) [$K_r$]{}; at (0.4,0.5) [$\ldots$]{}; (0.5,0.5)–(0.6,0)–(0.7,0.6)–(0.8,0.4)–(0.9,1)–(1,0.7)–(1.1,1.1)–(1.2, -0.1); (0.633,0.2)–(0.69,0.55); at (0.7,0.65) [$f(m)$]{}; at (1.1,1.15) [$f(x)$]{}; (1.3,0.5)–(1.45,0.5); (1.6,0.5)–(1.6,0)–(2.1,0)–(2.1,0.6)–(2,0.6)–(2,0.4)–(1.9,0.4)–(1.9, 1)–(1.8,1)–(1.8,0.7)–(1.7,0.7)–(1.7,1.1)–(2.3,1.1)–(2.3,-0.1)–(2.4,-0.1); at (2.05,0.65) [$f(m)$]{}; at (2,1.15) [$f(x)$]{}; (2.1,0.2)–(2.1,0.55); \[thm:Mayersubc\] Let $X$ be a chainable continuum that contains a nondegenerate indecomposable subcontinuum $Y$. Then $X$ can be embedded in the plane in uncountably many ways that are not strongly equivalent. Let $$Y:=Y_0\stackrel{\text{$f_1$}}{\leftarrow} Y_1\stackrel{\text{$f_2$}}{\leftarrow} Y_2\stackrel{\text{$f_3$}}{\leftarrow} Y_3\stackrel{\text{$f_4$}}{\leftarrow}\ldots.$$ If $\phi, \psi: X\to \R^2$ are strongly equivalent planar embeddings of $X$, then $\phi|_Y, \psi|_Y$ are strongly equivalent planar embeddings of $Y$. We will construct uncountably many strongly nonequivalent planar embeddings of $Y$ extending to planar embeddings of $X$, which will complete the proof. According to Theorem \[thm:kuykendall\] and Theorem \[thm:Miod\] we can assume that $f_i|_{Y_i}: Y_i\to Y_{i-1}$ has at least four surjective intervals for every $i\in\N$. For a closed interval $J\subset Y_{j-1}$, let $\alpha_j, \beta_j$ be integers from Lemma \[lem:indecsubc\] applied to $f_j: Y_j\to Y_{j-1}$, and denote the appropriate subintervals of $Y_j$ by $J^{\alpha_j}$, $J^{\beta_j}$. For every sequence $(n_i)_{i\in\N}\in\prod_{i\in\N}\{\alpha_i, \beta_i\}$ we obtain a subcontinuum of $Y$: $$J^{(n_i)}:=J\stackrel{\text{$f_1$}}{\leftarrow} J^{n_1}\stackrel{\text{$f_2$}}{\leftarrow} J^{n_1n_2}\stackrel{\text{$f_3$}}{\leftarrow} J^{n_1n_2n_3}\stackrel{\text{$f_4$}}{\leftarrow}\ldots$$ We use the notation of the proof of Lemma \[lem:Mayerprep\]. Lemma \[lem:indecsubc\] implies that for every sequence $(n_i)$ there exists an embedding of $Y$ such that both points of $\partial J\leftarrow \partial J^{n_1}\leftarrow \partial J^{n_1n_2}\leftarrow \partial J^{n_1n_2n_3}\leftarrow\ldots$ are accessible and which can be extended to an embedding of $X$. This completes the proof. We have proven that every chainable continuum containing a nondegenerate indecomposable subcontinuum has uncountably many embeddings that are not strongly equivalent. Thus we pose the following question. \[q:above\] Which hereditarily decomposable chainable continua have uncountably many planar embeddings that are not equivalent and/or strongly equivalent? Mayer has constructed uncountably many nonequivalent planar embeddings (in both senses) in [@May] of the $\sin\frac 1x$ continuum by varying the rate of convergence of the ray. This approach readily generalizes to any Elsa continuum. We do not know whether the approach can be generalized to all chainable continua which contain a dense ray. Specifically, it would be interesting to see if $\underleftarrow{\lim}\{I, f_{Feig}\}$ (where $f_{Feig}$ denotes the logistic interval map at the Feigenbaum parameter) can be embedded in uncountably many nonequivalent ways. However, this approach would not generalize to the remaining hereditarily decomposable chainable continua since there exist hereditarily decomposable chainable continua which do not contain a dense ray, see [@Jan]. \[rem:n\_emb\] In Figure \[fig:n\_embed\] we give examples of planar continua which have exactly $n\in\N$ or countably many nonequivalent planar embeddings. However, except for the arc, all the examples we know are not chainable. (1,1)–(2,1); (1,1)–(0,1); (1,1)–(1,2); (1,1)–(1,0); (2,1)–(1,0)–(0,1)–(1,2)–(2,1); (4, 0)–(4,2); (4, 0)–(6,0); (4, 1.8)–(6,1.8); (4, 1)–(6,1); (4, 0.5)–(6,0.5); (4, 0.25)–(6,0.25); (4, 0.125)–(6,0.125); (4, 0.0625)–(6,0.0625); Is there a non-arc chainable continuum for which there exist at most countably many nonequivalent planar embeddings? \[rem:otherdef\] For inverse limit spaces $X$ with a single [**unimodal**]{} bonding map that are not hereditarily decomposable, Theorems \[thm:Mayer\] and \[thm:Mayersubc\] hold with the standard notion of equivalence as well, for details see [@embed]. This is because every self-homeomorphism of $X$ is known to be pseudo-isotopic (two self-homeomorphisms $f, g$ of $X$ are called [*pseudo-isotopic*]{} if $f(C)=g(C)$ for every composant $C$ of $X$) to a power of the shift homeomorphism (see [@BBS]), and so every composant can only be mapped to one in a countable collection of composants. Hence, if uncountably many composants can be made accessible in at least two points, then there are uncountably many nonequivalent embeddings. In general there are no such rigidity results on the group of self-homeomorphisms of chainable continua. For example, there are uncountably many self-homeomorphisms of the pseudo-arc up to pseudo-isotopy, since it is homogeneous and all arc-components are degenerate. Thus we ask the following question. For which indecomposable chainable continua is the group of all self-homeomorphisms up to pseudo-isotopy at most countable? [ABCD]{} A. Anušić, H. Bruin, J. Činč, [ *Problems on planar embeddings of chainable continua and accessibility*]{}, In: Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali, Topology Proc. [**52**]{} (2018), 283–285. A. Anušić, H. Bruin, J. Činč, [ *Uncountably many planar embeddings of unimodal inverse limit spaces*]{}, DCDS - Series A [**37**]{} (2017), 2285–2300. A. Anušić, J. Činč, [*Accessible points of planar embeddings of tent inverse limit spaces*]{}, Diss. Math. DOI: 10.4064/dm776-1-2019, 2019. M. Barge, H. Bruin, S. Štimac, [*The Ingram Conjecture,*]{} Geom. Topol. [**16**]{} (2012), 2481–2516. D. P. Bellamy, [*A tree-like continuum without the fixed point property*]{}, Houston J. Math. [**6**]{} (1979), 1–13. R. H. Bing, [*Snake-like continua*]{}, Duke Math J. [**18**]{} (1951), 653–663. B. Brechner, [*On stable homeomorphisms and imbeddings of the pseudo-arc*]{}, Illinois J. Math. [**22**]{} Issue 4 (1978), 630–661. K. Brucks, H. Bruin, [*Subcontinua of inverse limit spaces of unimodal maps*]{}, Fund. Math. [**160**]{} (1999), 219–246. H. Bruin, [*Inverse limit spaces of post-critically finite tent maps*]{}, Fund. Math., [**165**]{} (2000), 125–138. W. Dbski, E. Tymchatyn, [*A note on accessible composants in Knaster continua*]{}, Houston J. Math. [**19**]{} (1993), no. 3, 435–442. O. H. Hamilton, [*A fixed point theorem for pseudo-arcs and certain other metric continua*]{}, Proc. Amer. Math. Soc. [ **2**]{} (1951), 173–174. Z. Janiszewski, [*Über die Begriffe “Linie” und “Fläche”,*]{} International Congress of Math., Cambridge, 1912. D. P. Kuykendall, [*Irreducibility and indecomposability in inverse limits*]{}, Fund. Math. [**80**]{} (1973), 265–270. W. Lewis, [*Embeddings of the pseudo-arc in $E^2$*]{}, Pacific J. Math., [**93**]{}, no. 1, (1981), 115–120. W. Lewis, [*Continuum theory problems*]{}, Proceedings of the 1983 topology conference (Houston, Tex., 1983). Topology Proc. [**8**]{} no. 2 (1983), 361–394. J. C. Mayer, [*Embeddings of plane continua and the fixed point property,*]{} Ph. D. dissertation, University of Florida, 1982, 1-247. J. C. Mayer, [*Inequivalent embeddings and prime ends*]{}, Topology Proc. [**8**]{} (1983), 99–159. S. Mazurkiewicz, [*Un théorème sur l’accessibilité des continus indécomposables*]{}, Fund. Math. [**14**]{} (1929), 271–276. J. Meddaugh, [*Embedding one-dimensional continua into $T\times I$*]{}, Topology and its Appl. [**153**]{} (2006) 3519–3527. P. Minc [*Embedding tree-like continua in the plane*]{}, In: Problems in Continuum Theory in Memory of Sam B. Nadler, Jr. Ed. Logan Hoehn, Piotr Minc, Murat Tuncali, Topology Proc. [**52**]{} (2018), 296–300. P. Minc, W. R. R. Transue, [*Accessible points of hereditarily decomposable chainable continua*]{}, Trans. Amer. Math.  Soc. [**2**]{} (1992), 711–727. J. Mioduszewski, [*Mappings of inverse limits*]{}, Colloquium Mathematicum [**10**]{} (1963), 39–44. E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, New York (1977). S. B. Nadler, Jr., [*Continua whose cone and hyperspace are homeomorphic*]{}, Trans. Amer. Math. Soc. [**230**]{} (1977), 321–345. S. B. Nadler, Jr., *Some results and problems about embedding certain compactifications*, Proceedings of the University of Oklahoma Topology Conference (1972) 222–233. M. Smith, [*Plane indecomposable continua no composant of which is accessible at more than one point*]{}, Fund. Math. [**111**]{} (1981), 61–69. [^1]: AA was supported in part by Croatian Science Foundation under the project IP-2014-09-2285. HB and JČ were supported by the FWF stand-alone project P25975-N25. JČ was partially supported by the FWF Schrödinger Fellowship stand-alone project J-4276 and by University of Ostrava grant IRP201824 “Complex topological structures”. We gratefully acknowledge the support of the bilateral grant *Strange Attractors and Inverse Limit Spaces*, Österreichische Austauschdienst (OeAD) - Ministry of Science, Education and Sport of the Republic of Croatia (MZOS), project number HR 03/2014.

--- abstract: '[We study experimentally and theoretically a cold trapped Bose gas under critical rotation, *i.e.* with a rotation frequency close to the frequency of the radial confinement. We identify two regimes: the regime of explosion where the cloud expands to infinity in one direction, and the regime where the condensate spirals out of the trap as a rigid body. The former is realized for a dilute cloud, and the latter for a Bose-Einstein condensate with the interparticle interaction exceeding a critical value. This constitutes a novel system in which repulsive interactions help in maintaining particles together.]{}' author: - 'P. Rosenbusch$^1$, D.S. Petrov$^{2,3}$, S. Sinha$^1$, F. Chevy$^1$, V. Bretin$^1$, Y. Castin$^1$, G. Shlyapnikov$^{1,2,3}$, and J. Dalibard$^1$' date: Received title: Critical rotation of a harmonically trapped Bose gas --- The rotation of a macroscopic quantum object is a source of spectacular and counter-intuitive phenomena. In superfluid liquid helium contained in a cylindrical bucket rotating around its axis $z$, one observes the nucleation of quantized vortices for a sufficiently large rotation frequency $\Omega$ [@Donnelly]. A similar phenomenon occurs in a Bose-Einstein condensate confined in a rotating harmonic trap [@ENS; @MIT; @Boulder; @Oxford]. In particular, vortices are nucleated when the rotation resonantly excites surface modes of the condensate. This occurs for particular rotation frequencies in the interval $0<\Omega \leq \omega_\bot/\sqrt{2}$, where $\omega_\bot$ is the trap frequency in the $xy$ plane perpendicular to the rotation axis $z$. Several theoretical studies have recently considered the critical rotation of the gas, *i.e.* $\Omega\sim \omega_\bot$, which presents remarkable features [@Rokhsar; @Mottelson; @Gunn; @Stringari; @Zoller; @Ho; @Fetter; @Baym; @Sinova]. From a classical point of view, for $\Omega=\omega_\bot$ the centrifugal force compensates the harmonic trapping force in the $xy$ plane. Hence the motion of a single particle of mass $m$ in the frame rotating at frequency $\Omega$ is simply due to the Coriolis force $2 m {\bf{\dot r}}\times {\bf \Omega} $. This force is identical to the Lorentz force acting on a particle of charge $q$ in the magnetic field ${\bf B}= 2 (m/q)\, {\bf \Omega}$. The analogy between the motion of charged particles in a magnetic field and neutral particles in a rotating frame also holds in quantum mechanics. In this respect, a quantum gas of atoms confined in a harmonic trap rotating at the critical frequency is analogous to an electron gas in a uniform magnetic field. One can then expect [@Gunn; @Zoller] to observe phenomena related to the Quantum Hall Effect. This paper presents an experimental and theoretical study of the dynamics of a magnetically trapped rubidium ($^{87}$Rb) gas stirred at a frequency close to $\omega_\bot$. We show that the single particle motion is dynamically unstable for a window of frequencies $\Omega$ centered around $\omega_\bot$. This result entails that the center-of-mass of the atom cloud (without or with interatomic interactions) is destabilized, since its motion is decoupled from any other degree of freedom for a harmonic confinement. This also implies that a gas of non-interacting particles “explodes”, which we indeed check experimentally. When one takes into account the repulsive interactions between particles, which play an important role in a $^{87}$Rb condensate, one would expect naively that this explosion is enhanced. However, we show experimentally that this is not the case, and repulsive interactions can “maintain the atoms together". This has been predicted for a Bose-Einstein condensate in the strongly interacting – Thomas-Fermi (TF)– regime [@Stringari]. Here we derive the minimal interaction strength which is necessary to prevent the explosion. This should help studies of the Quantum Hall related physics in the region of critical rotation. Consider a gas of particles confined in an axisymmetric harmonic potential $V_0({\bf r})$, with frequency $\omega_z$ along the trap axis $z$, and $\omega_\bot$ in the $xy$ plane. To set this gas into rotation, one superimposes a rotating asymmetric potential in the $xy$ plane. In the reference frame rotating at an angular frequency $\Omega$ around the $z$ axis, this potential reads $V_1({\bf r})=\epsilon m\omega_\bot^2 (Y^2 -X^2)/2$, where $\epsilon>0$. The rotating frame coordinates $X,Y$ are deduced from the lab frame coordinates $x,y$ by a rotation at an angle $\Omega t$. For a non-interacting gas, the equation of motion for each particle reads: $$\begin{aligned} & & \ddot X -2 \Omega \dot Y+\left(\omega_\bot^2(1-\epsilon)-\Omega^2 \right)X=0 \label{eq:Xmotion}\\ & & \ddot Y +2 \Omega \dot X+\left(\omega_\bot^2(1+\epsilon)-\Omega^2 \right)Y=0, \label{eq:Ymotion}\end{aligned}$$ while the motion along $z$ is not affected by the rotation. One deduces from this set of equations that the motion in the $xy$ plane is dynamically unstable if the stirring frequency $\Omega$ is in the interval $[\omega_\bot \;\sqrt{1-\epsilon},\omega_\bot\; \sqrt{1+\epsilon}]$. In particular, for $\Omega=\omega_\bot$ and $\epsilon \ll 1$, one finds that the quantity $X+Y$ diverges as $\exp{(\epsilon \omega_\bot t/2)}$, whereas $X-Y$ remains finite. To test this prediction we use a $^{87}$Rb cold gas in a Ioffe-Pritchard magnetic trap, with frequencies $\omega_x= \omega_y=2\pi\times 180$ Hz, and $\omega_z=2\pi \times 11.7$ Hz. The initial temperature of the cloud pre-cooled using optical molasses is 100 $\mu$K. The gas is further cooled by radio-frequency evaporation. For the first set of experiments we stop the evaporation before the Bose-Einstein condensation is reached. The resulting sample contains $10^7$ atoms at a temperature $T\sim 5\,\mu$K. It is dilute, with a central density $\sim 10^{12}$ cm$^{-3}$, and atomic interactions can be neglected (mean-field energy $\ll k_B T$). The second set of experiments corresponds to a much colder sample ($T<50$ nK), *i.e.* to a quasi-pure condensate with $10^5$ atoms. After evaporative cooling, the atomic cloud is stirred during an adjustable period $t$ by a focused laser beam of wavelength $852$ nm and waist $w_0=20 \mu$m, whose position is controlled using acousto-optic modulators [@ENS]. The beam is switched on abruptly and it creates a rotating optical-dipole potential which is nearly harmonic over the extension of the cloud. We measure the transverse density profile of the condensate after a period of free expansion. In this pursuit, we suddenly switch off the magnetic field and the stirrer, allow for a 25 ms free-fall, and image the absorption of a resonant laser by the expanded cloud. The imaging beam propagates along the $z$ axis. We fit the density profile of the sample assuming a Gaussian shape for the non-condensed cloud, and a parabolic TF shape for the quasi-pure condensate. We extract from the fit the long and short diameters in the plane $z=0$, and the average position of the cloud. The latter gives access to the velocity of the center-of-mass of the atom cloud before time of flight. ![Center-of-mass displacement after free expansion (log-scale) *vs.* stirring time for $\Omega\!=\!\omega_\bot$ and $\epsilon\!=\!0.09$. (a) Non-condensed cloud with $10^7$ atoms, $T\!=\!5\;\mu$K; (b) Condensate with $10^5$ atoms. Solid line: exponential fit to the data.[]{data-label="fig:CMinstability"}](fig1.eps) The center-of-mass displacement as a function of the stirring time $t$ is shown in Fig. \[fig:CMinstability\]. We choose here $\epsilon=0.09$ and $\Omega=\omega_\bot$, so that the motion predicted by Eqs.(\[eq:Xmotion\]-\[eq:Ymotion\]) is dynamically unstable. To ensure reliable initial conditions, we deliberately offset the center of the rotating potential by a few micrometers with respect to the atom cloud. We find the instability for the center-of-mass motion both for the non-condensed cloud (Fig. \[fig:CMinstability\]a) and for the quasi-pure condensate (Fig. \[fig:CMinstability\]b). The center-of-mass displacement increases exponentially, with an exponent consistent with the measured $\epsilon$. We consider now the evolution of the size of the atom cloud as a function of $t$ (Fig. \[fig:sizeincrease\]). The non-condensed cloud exhibits the behavior expected from the single particle dynamics, *i.e.* the “explosion” in the $X=Y$ direction. The cloud becomes more and more elliptical in the $xy$ plane. The long radius increases with time, while the short one remains approximately constant (Fig. \[fig:sizeincrease\]a). On the opposite, we find that the condensate remains circular (Fig. \[fig:sizeincrease\]b), with no systematic increase in size. We then obtain the following counter-intuitive result: for a significant repulsive interaction the atoms remain in a compact cloud, while they fly apart if the interaction is negligible. We observe this stability of the shape of the condensate rotating at the critical velocity for $\epsilon \leq 0.2$. Above this value of $\epsilon$ we find that the atomic cloud rapidly disintegrates. For $\epsilon\approx 0.3$, after a stirring time of $50$ ms, we observe several fragments in the time-of-flight picture. ![Long [($\blacksquare$)]{} and short ($\circ$) diameters of the atom cloud *vs.* stirring time, for $\Omega=\omega_\bot$. (a) Non-condensed cloud ; (b) Quasi-pure condensate (same parameters as in Fig. \[fig:CMinstability\]).[]{data-label="fig:sizeincrease"}](fig2.eps) We now perform a theoretical analysis of how the interparticle interaction stabilizes a rotating condensate. To this end we use the 2D ($x,y$) time-dependent Gross-Pitaevskii (GP) equation for an idealized cylindrical trap ($\omega_z=0$). In the rotating frame the GP equation reads: $$i\partial_{t}\psi\!=\!\frac{1}{2}\left[\!-\Delta\! +\!(\!1\!-\!\epsilon)X^{2}\!\!+\!(\!1\!+\!\epsilon)Y^{2}\!\! +\!2g|\psi|^{2}\!\!-\!2\Omega\hat{L} \right]\psi,\!\! \label{eq:gpe}$$ where $\hat L$ is the $z$ component of the angular momentum operator. In Eq.(\[eq:gpe\]) the coordinates are given in units of the initial harmonic oscillator length $\sqrt{\hbar/m\omega_\bot}$, and the frequencies in units of $\omega_\bot$. The condensate wave function $\psi(X,Y,t)$ is normalized to unity, and the effective coupling constant is $g\!=\!4\pi a\tilde N $, with $a$ being the positive scattering length, and $\tilde N$ the number of particles per unit axial length. The effective coupling $g$ depends on density and characterizes the ratio of the mean-field interparticle interaction to the radial frequency $\omega_\bot$. Since the trapping potential is harmonic, the average center of mass motion of the condensate is described by the classical equations (\[eq:Xmotion\]-\[eq:Ymotion\]) and is decoupled from the evolution of the condensate wave function in the center of mass reference frame. We shall therefore restrict to wave functions $\psi$ centered at $x=y=0$ for all times. We start with a variational analysis of the steady state of the condensate in the rotating frame, using a Gaussian ansatz for the condensate wave function [@cirac]: $$\psi(X,Y) \propto \exp(i\alpha XY-\beta X^2/2-\gamma Y^2/2)\ .$$ We use the symmetry properties of the Hamiltonian and assume that the condensate wave function remains invariant under the combination of a time reversal ($\psi \rightarrow \psi^{*}$) and a reflection with respect to the $XZ$ plane. This implies that the parameters $\alpha$, $\beta$, $\gamma$ are real numbers. We extremize the GP energy functional with respect to these parameters. Extremizing with respect to the phase parameter $\alpha$ gives $\alpha=\Omega(\gamma-\beta)/(\gamma+\beta)$. As $\beta$ and $\gamma$ should be finite and positive this sets the constraint $\alpha^2<\Omega^2$. Extremizing over $\beta$ and $\gamma$ and expressing $\beta/\gamma$ in terms of $\alpha$, we obtain a closed equation for $\alpha$: $$\begin{aligned} \!\!\!& & (\epsilon/\Omega)\,[\alpha^2+2\alpha\Omega(\Omega^2-1) /\epsilon+\Omega^2] \nonumber\\ \!\!\!& & -(g/2 \pi)\sqrt{1-\alpha^2/\Omega^2}\,[\alpha^3+(1-2\Omega^2) \alpha-\Omega\epsilon]=0.\,\,\,\,\,\,\,\,\,\, \label{eq:great}\end{aligned}$$ In the non-interacting case ($g=0$) the Gaussian ansatz is exact, and $\alpha$ is the root of a quadratic equation. This ansatz also captures the scaling properties of the rotating condensate in the regime of strong interactions. In the TF limit ($g\rightarrow \infty$) the first line of (\[eq:great\]) can be neglected and we recover the cubic equation for $\alpha$ derived in [@Stringari]. For $g=0$ the parameter $\alpha$ is complex in the interval of rotation frequencies $\sqrt{1 - \epsilon}<\Omega<\sqrt{1 + \epsilon}$, and there is no steady state solution for the condensate wave function at these $\Omega$ [@Stringari]. For a finite $g$ the lower border $\Omega_-$ of this frequency interval remains equal to $\sqrt{1-\epsilon}$ irrespective of the value of $g$. The upper border, dashed curves in Fig.3, decreases with increasing $g$ at a given $\epsilon$. For small anisotropy $\epsilon<1/5$ it reaches the lower border at a critical coupling strength. For larger $g$ the steady states exist at any $\Omega$. If $\epsilon>1/5$, the upper border never reaches $\Omega_-=\sqrt{1-\epsilon}$ and for any $g$ one has an interval of $\Omega$ where steady state solutions are absent. The $\epsilon=1/5$ threshold was derived for the TF limit in [@Stringari]. We now analyze the time evolution of the condensate after the stirring potential has been switched on. For this purpose we use an approximate scaling approach to the solution of Eq.(\[eq:gpe\]). We assume (and later on check) that the evolution of the condensate shape is well described by dilations with factors $b_{u}(t)$ and $b_{v}(t)$ along the axes $\tilde X$ and $\tilde Y$, rotating at an angular frequency $\dot\phi(t)$ with respect to the laboratory frame $x,y$. To determine $b_{u}(t), b_{v}(t)$, and $\phi(t)$, we write the wave function as $$\psi(\tilde X,\tilde Y,t)=(b_{u}b_{v})^{-1/2} \chi(u,v,t)\exp\{i\Phi(\tilde X,\tilde Y,t)\} \ , \label{eq:scaling}$$ where we have set $u=\tilde X/b_{u}$, $v=\tilde Y/b_{v}$, and $$\!\Phi(\tilde X,\tilde Y,t)={\tilde \alpha}(t)\tilde X\tilde Y\!+(\dot b_{u}/2b_{u})\tilde X^2\!+(\dot b_{v}/2b_{v})\tilde Y^2\!,$$ with ${\tilde \alpha}=-\dot \phi \tanh\xi$ and $\xi(t)=\ln(b_{v}/b_{u})$. Then the GP equation reduces to the following equation for $\chi(u,v,t)$: $$\begin{aligned} & & i\left(\partial_t -\frac{\dot \phi}{\cosh\xi}(u\partial_v-v\partial_u) \right)\chi = \Big[-\frac{\partial^2_u}{2b_{u}^2}-\frac{\partial^2_v} {2b_{v}^2} \nonumber \\ & & \qquad \quad + \frac{1}{2}\left(\nu_{u}^2b_{u}^2u^2 +\nu_{v}^2b_{v}^2v^2 \right) +\frac{g|\chi|^2}{b_{u}b_{v}}\Big]\;\chi. \label{rescaledGP}\end{aligned}$$ The “frequencies" $\nu_{u}$ and $\nu_{v}$ are given by $$\label{omegas} \nu_{u,v}^2=1\mp\epsilon\cos(2\Omega t-2\phi)+{\tilde \alpha}^2\mp 2\dot \phi {\tilde \alpha}+\ddot b_{u,v}/b_{u,v}.$$ In Eq.(\[rescaledGP\]) we took into account that $u,v$ are eigenaxes of the condensate, which requires the absence of terms proportional to $uv$ and is provided by the relation $$\label{momentum} \epsilon\sin(2\Omega t-2\phi)-\dot{\tilde \alpha}-{\tilde \alpha} (\dot b_{u}/b_{u}+\dot b_{v}/b_{v})+ \dot\phi\dot\xi=0.$$ We now replace the lhs of Eq.(\[rescaledGP\]) by $\tilde \mu \chi$, where $\tilde\mu$ follows from the normalization condition for $\chi$. The solution of the resulting equation is a function of $b_{u,v},\phi$ and $\nu_{u,v}$. We then require that (\[eq:scaling\]) is a relevant scaling transform, *i.e.*, that the function $\chi(u,v,t)$ is most similar to the initial function $\chi(u,v,0)$. More precisely we set the averages $\langle u^2\rangle_t$, $\langle v^2\rangle_t$ equal to their values at $t=0$. This fixes $\nu_{u}$ and $\nu_{v}$ in terms of $b_{u},b_{v},\phi$. The solution of Eqs.(\[omegas\]) and (\[momentum\]) then gives the desired scaling parameters. The omitted lhs of Eq.(\[rescaledGP\]) only insignificantly changes $\nu_{u}$ and $\nu_{v}$. It vanishes in both the TF regime and for an ideal-gas condensate. For the TF limit our procedure gives the same results as the scaling approach of [@Olshanii]. =8.5cm -0.5cm For an abrupt switching of the rotating potential we use the initial conditions $b_{u,v}(0)=1$, $\dot b_{u,v}(0)=\phi(0)=\dot \phi(0)=0$. We find two types of solution: (i) oscillating functions $b_{u,v}(t)$, (ii) one of the scaling parameters eventually grows exponentially. Case (ii) describes an infinite expansion of the condensate in one direction, similarly to the expansion of the ideal gas under rotation. For a given $\epsilon$ we obtain the upper ($\Omega_+$) and lower ($\Omega_-$) instability borders in the $\Omega-g$ space (see Fig. \[fig:delta1\]). The lower border is always equal to $\sqrt{1-\epsilon}$. The upper border $\Omega_+(g)$ decreases with increasing $g$ and for $\epsilon < 0.17$ it reaches $\Omega_-$ at a critical value of the coupling strength. For $\epsilon> 0.17$ we have $\Omega_+>\Omega_-$ at any $g$. The obtained results can be understood by comparing the frequency $\Omega_q$ of the rotating quadrupole mode of the condensate with the rotation frequency $\Omega\sim 1$ of the perturbation $V_1({\bf r})$. In the absence of interaction one has $\Omega_{q}=1$, and the corresponding resonance leads to the condensate explosion. The interactions reduce the frequency of the rotating quadrupole mode ($\Omega_q=1/\sqrt{2}$ in the TF limit), suppressing the resonance at $\Omega\approx 1$: the deformation of the condensate induced by $V_1$ remains small, at least for $\epsilon$ smaller than the detuning from the resonance. For larger $\epsilon$ the condensate explodes. In the presence of interactions ($1/\sqrt{2}<\Omega_q<1$), one could expect naively that the explosion occurs for a resonant drive with $\Omega \sim \Omega_q$, even for small $\epsilon$. This is not the case because of a non-linear character of the dynamics. As the system starts to elongate under the action of the resonant excitation, it becomes closer to an ideal gas, for which $\Omega_q=1$. The gas is then driven away from the resonance and its deformation stops. This explains why the lower instability border $\Omega_-$ is independent of $g$. The scaling method can also be used to identify stationary solutions, by setting $\dot \phi=\Omega $ and constant $b_{u,v}$ in Eqs.(\[omegas\]-\[momentum\]). The results nearly coincide with the ones from the Gaussian ansatz. The existence of these solutions can also be explored using an adiabatic switching of the rotating potential. As shown in Fig. \[fig:delta1\] the domain of instability for an abrupt switching of the rotation includes the domain for the absence of stationary solutions. In Fig.\[PSh\] we present the minimum coupling strength $g_c(\epsilon)$ required for the stability of the shape of the condensate, for both abrupt and adiabatic switching of the potential rotating at $\Omega=1$. =8.5cm -0.5cm To summarize, our analysis shows that the condensate can preserve its shape and size for rotation frequencies in the instability window for the center of mass motion, $\sqrt{1-\epsilon}<\Omega<\sqrt{1+\epsilon}$. This means that the condensate will spiral out of the trap as a rigid body after the rotation is switched on. For TF condensates this is the case if $\epsilon\alt 0.28$, which explains why the repulsive interparticle interaction maintains particles together in our experiment. On the other hand, for larger $\epsilon$ there are always frequencies at which even TF condensates are unstable. This gives an account for the destruction of the condensate in our experiment at $\epsilon \approx 0.3$. One can think of observing related effects in rotating ion clouds in electromagnetic traps (see [@Bollinger] and refs. therein). Even though our picture properly describes the experiment, we are likely not to deal with the ground state of the system. On a very long time scale, the rotating gas can evolve to a more complex state, for instance to a multi-vortex state (with possible quantum melting) discussed in [@Rokhsar; @Ho; @Gunn; @Fetter; @Baym; @Sinova], or to a Quantum-Hall-like state [@Gunn; @Zoller]. However these states have been discussed for the axially symmetric case and a natural development will be to include a finite rotating anisotropy $\epsilon$, which is a necessary ingredient for most present experiments. In particular, for $\Omega=\omega_\bot \sqrt{1- \epsilon}$, one reaches a 1-body Hamiltonian corresponding to an unbound motion (with a gauge field) in the $X$ direction, similar to a Quantum Hall fluid in a narrow channel. 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--- author: - 'R. Liseau' - 'C. Risacher' - 'A. Brandeker' - 'C. Eiroa' - 'M. Fridlund' - 'R. Nilsson' - 'G. Olofsson' - 'G.L. Pilbratt' - | \ P. Thébault date: 'Received ; accepted ' title: 'q$^{1}$Eri: a solar-type star with a planet and a dust belt[^1]' --- [Far-infrared excess emission from main-sequence stars is due to dust produced by orbiting minor bodies. In these disks, larger bodies, such as planets, may also be present and the understanding of their incidence and influence currently presents a challenge.]{} [Only very few solar-type stars exhibiting an infrared excess and harbouring planets are known to date. Indeed, merely a single case of a star-planet-disk system has previously been detected at submillimeter (submm) wavelengths. Consequently, one of our aims is to understand the reasons for these poor statistics, i.e., whether these results reflected the composition and/or the physics of the planetary disks or were simply due to observational bias and selection effects. Finding more examples would be very significant.]{} [The selected target, [q$^{1}$Eri]{}, is a solar-type star, which was known to possess a planet, [q$^{1}$Eri]{}b, and to exhibit excess emission at IRAS wavelengths, but had remained undetected in the millimeter regime. Therefore, submm flux densities would be needed to better constrain the physical characteristics of the planetary disk. Consequently, we performed submm imaging observations of [q$^{1}$Eri]{}.]{} [The detected dust toward [q$^{1}$Eri]{} at 870[$\mu$m]{} exhibits the remarkable fact that the entire SED, from the IR to mm-wavelengths, is fit by a single-temperature blackbody function (60K). This would imply that the emitting regions are confined to a narrow region (ring) at radial distances much larger than the orbital distance of [q$^{1}$Eri]{}b, and that the emitting particles are considerably larger than some hundred micron. However, the 870[$\mu$m]{} source is extended, with a full-width-half-maximum of roughly 600AU. Therefore, a physically more compelling model also invokes a belt of cold dust (17K), located at 300AU from the star and about 60AU wide.]{} [The minimum mass of 0.04[$M_{\oplus}$]{} (3[$M_{\rm Moon}$]{}) of 1mm-size icy ring-particles is considerable, given the stellar age of [$\stackrel {>}{_{\sim}}$]{}1Gyr. These big grains form an inner edge at about 25AU, which may suggest the presence of an unseen outer planet ([q$^{1}$Eri]{}c). ]{} Introduction ============ During the end stages of early stellar evolution, dusty debris disks are believed to be descendents of gas-rich protoplanetary disks. These had been successful to varying degrees in building a planetary system. What exactly determines the upper cut-off mass of the bodies in individual systems, and on what time scales, is not precisely known. However, the presence of debris around matured stars is testimony to the action of orbiting bodies, where a large number of smaller ones are producing the dust through collisional processes and where a small number of bigger bodies, if any, are determining the topology (disks, rings and belts, clumps) through gravitational interaction. The time evolution of the finer debris is believed to be largely controlled by non-gravitational forces, though. By analogy, many debris disks are qualitatively not very different from the asteroid and Kuiper belts and the zodiacal dust cloud in the solar system [@mann2006]. For solar-type stars on the main-sequence, which are known to exhibit infrared excess due to dust disks, one might expect, therefore, a relatively high incidence of planetary systems around them. Surveying nearly 50 FGK stars with known planets for excess emission at 24[$\mu$m]{} and 70[$\mu$m]{}, @trilling2008 detected about 10-20% at 70[$\mu$m]{}, but essentially none at 24[$\mu$m]{}, implying that these planetary disks are cool ($<100$K) and large ($> 10$AU). However, in general, the conjecture that the infrared excess arises from disks lacks as yet observational confirmation due to insufficient spatial resolution. In fact, until very recently, there was only one main-sequence system known that has an extended, resolved disk/belt structure and (at least) one giant planet, viz. [$\epsilon \, {\rm Eri}$]{}, a solar-type star at the distance of only three parsec [@greaves1998; @greaves2005]. Its planetary companion, [$\epsilon \, {\rm Eri}$]{}b, has been detected indirectly by astrometric and radial velocity (RV) methods applied to the star [@hatzes2000; @benedict2006], whereas attempts to directly detect the planet have so far been unsuccessful [@itoh2006; @janson2007]. As its name indicates, the object of the present study, [q$^{1}$Eri]{}, happens to belong to the same celestial constellation of Eridanus, albeit at a larger distance ($D=17.35 \pm 0.2$pc) and is, as such, unrelated to [$\epsilon \, {\rm Eri}$]{}. The planet was discovered with the RV technique [for a recent overview, see @butler2006]. These RV data suggest that the semimajor axis of the Jupiter-mass planet [q$^{1}$Eri]{}b is about 2AU (Table\[star\]). It seems likely that regions inside this orbital distance have been largely cleared by the planet, whereas outside the planetary orbit, substantial amounts of material might still be present. In fact, IRAS and ISO data were suggestive of significant excess radiation above the photospheric emission at wavelengths longward of about 20[$\mu$m]{}. @zuckerman2004 interpreted these data in terms of dust in a disk at the orbital distance of 30AU and at a temperature of about 55K. @chen2006 fitted the far-infrared emission with the corresponding values of 20AU and 70K, respectively. @trilling2008 derived 20AU and 60K. In their entire sample of more than 200 stars, [q$^{1}$Eri]{} (=HD10647) has by far the highest 70[$\mu$m]{} excess. At mm-wavelengths, @schutz2005 failed to detect the disk and assigned an upper limit to the dust mass of 6[$M_{\rm Moon}$]{}. This is unsatisfactory, as the proper characterization of the dust around [q$^{1}$Eri]{} would require valid long wavelength data. In the following, observations of [q$^{1}$Eri]{} at 870[$\mu$m]{} are described and their implications discussed. Observations and Data Reductions ================================ APEX, the Atacama Pathfinder EXperiment, is a 12m diameter submillimeter telescope situated at an altitude of 5100m on the Llano Chajnantor in northern Chile. The telescope is operated by the Onsala Space Observatory, the Max-Planck-Institut für Radioastronomie, and the European Southern Observatory. The Large Apex BOlometer CAmera [LABOCA, @siringo2007] is a multi-channel bolometer array for continuum observations with 60GHz band width and centered on the wavelength of 870[$\mu$m]{}. The array, having a total field of view of 11[$^{\prime}$]{}, is spatially undersampled and we therefore adopted spiral pattern observing as the appropriate technique [@siringo2007]. This procedure results in fully-sampled maps with a uniform noise distribution over an area of about 8[$^{\prime}$]{}. During the nights of August1-4, 2007, we obtained 32 such individual maps, for about 7.5min each with central coordinates RA= and Dec= (J2000). The LABOCA beam width at half power (HPBW) is $\pm$ . We focussed LABOCA on the planet Jupiter and the rms-pointing accuracy of the telescope was 3[$^{\prime \prime}$]{} to 4[$^{\prime \prime}$]{}. We reduced the data with the BoA software [@siringo2007], which included flat fielding, baseline removal, despiking and iteratively removing the sky noise, and filtering out the low frequencies of the $1/f$-noise, with the cut-off frequency corresponding to several arcminutes. The software also accounts for the map reconstruction and the absolute calibration, using the opacities determined from numerous skydips (zenith opacities were in the range 0.1 to 0.3) and observations of the planets Uranus and Mars. The final result is an rms-noise-weighted average map (Fig.\[obs\]). [ll]{} Parameter & Value\ \ [**The star [q$^{1}$Eri]{}**]{} &\ Distance, $D$ & 17.35pc\ Spectral type and luminosity class & F8-9V\ Effective temperature, $T_{\rm eff}$ & 6100K\ Luminosity, $L_{\rm star}$ & 1.2[$L_{\odot}$]{}\ Surface gravity, $\log {g}$ & 4.4 (in cm s$^{-2}$)\ Radius, $R_{\rm star}$ & 1.1[$R_{\odot}$]{}\ Mass $M_{\rm star}$ & 1.1[$M_{\odot}$]{}\ Metallicity, \[Fe/H\] & $-0.08$\ Age & $(>1-2)$Gyr\ &\ Period, $P$ & $2.75 \pm 0.15$yr\ Semimajor axis, $a_{\rm orbit}$ & $2.0 \pm 0.2$AU\ Eccentricity, $e$ & $0.2 \pm 0.2$\ Mass, $M \sin{i}$ & $0.9 \pm 0.2$[$M_{\rm Jupiter}$]{}\ $^{\star}$ See @butler2006 and references cited in the text. Results ======= The final product of the reduction process is the 870[$\mu$m]{} image presented in Fig.\[obs\], which shows the central $5^{\prime} \times 5^{\prime}$ of the LABOCA map. The peak flux in the map is found at the position of [q$^{1}$Eri]{} and a few other pointlike features of low intensity are also present, one of which is close to the star. If not merely noise, these low signals could be due to extragalactic background sources, as the displayed number density is consistent with that observed elsewhere [e.g., @lagache2005; @bertoldi2007; @ivison2007]. Other, complementary observations (e.g., optical, IR, X-rays, radio interferometry) would be required for their identification. The derived flux densities of [q$^{1}$Eri]{} are provided in Table\[parameters\], which presents the results from fitting the data to a two-dimensional Gaussian function. The indicated errors are formal fit errors only, based on $1 \sigma$ rms values. In individual cases, e.g., [*pa*]{}, realistic errors could be twice as large. The error on the integrated flux density in Table\[parameters\] also includes an uncertainty of 10% in the absolute calibration. The 870[$\mu$m]{} source is at best only marginally resolved in the North-South direction (formal fit result is 23[$^{\prime \prime}$]{} $\pm$ 1[$^{\prime \prime}$]{}), whereas it is clearly elongated in approximately the East-West direction (37[$^{\prime \prime}$]{} $\pm$ 2[$^{\prime \prime}$]{}). At the distance of [q$^{1}$Eri]{}, this corresponds to a disk diameter of 640AU and assuming a circular shape, these disk dimensions yield an inclination with respect to the line of sight of $i \ge 52$[$^{\circ}$]{}, not excluding the possibility that the disk is seen essentially edge-on ($i$ close to 90[$^{\circ}$]{}). The vertical disk scale height is undetermined. Discussion ========== Physical conditions and the age of the system --------------------------------------------- The spectral type of [q$^{1}$Eri]{} is slightly earlier than that of the Sun [F8-9 V, @nordstrom2004; @decin2000; @decin2003; @zuckerman2004; @chen2006], with the effective temperature being bracketed by the extremes 6040K [@nordstrom2004] and 6260K [@chen2006] , with the mean of 6150K, i.e., essententially the value given by @butler2006 [6105 K] (see Table\[star\]). Literature estimations of likely ages for [q$^{1}$Eri]{} span the range 0.3 to 4.8Gyr [with an entire range of 0.0 to 7.0Gyr, @decin2000; @zuckerman2004; @decin2003; @nordstrom2004; @chen2006]. However, the level of chromospheric activity ($\log{R^{\prime}_{\rm HK}}=-4.7$) suggests an age of 1.9Gyr [see Eq.15 of @wright2004]. The star has also been detected in X-rays with ROSAT ($\log{L_{\rm X}}=28.3$, J.Sanz, private communication), yielding 1.2Gyr [@ribas2005; @guinan2007]. This value is also consistent with the stellar rotation period of about 10days [uncorrected for $\sin i$; @ecuvillon2007]. It is clear that the star is definitely on the main-sequence and that the age of the system likely exceeds [10$^{9}$]{}yr. [ll]{} Parameter & Value\ \ Peak offset$^a$, ($\Delta \alpha,\,\Delta \delta$) & (+4[$^{\prime \prime}$]{}, +3[$^{\prime \prime}$]{}), (error: $\pm 4$[$^{\prime \prime}$]{})\ Peak flux density$^a$, $F_{\nu}$(0, 0), $\lambda=870$[$\mu$m]{}& $(16.2 \pm 0.8)$mJy/beam\ Integrated flux density, $\int\!F_{\nu} d\alpha d\delta$& $(39.4 \pm 4.1)$mJy, $F_{\nu} \ge 2 \sigma$\ Major axis$^a$ (FWHM) & 37[$^{\prime \prime}$]{}$\pm$ 2[$^{\prime \prime}$]{} (640 $\pm$ 35)AU\ Position angle$^a$, $pa$ & 55[$^{\circ}$]{} $\pm$ 4[$^{\circ}$]{} (north over east)\ Minor axis$^a$ (FWHM) & 23[$^{\prime \prime}$]{} $\pm$ 1[$^{\prime \prime}$]{}\ Inclination angle, $i$ & $\ge 52$[$^{\circ}$]{} (90[$^{\circ}$]{}=edge-on)\ Fractional luminosity, $L_{\rm bb}/L_{\rm star}$ & $1.1 \times 10^{-4}$\ Inner (outer)$^b$ Temperature, $T_{\rm bb}$ & 60K (17K)\ Inner (outer)$^b$ Radius, $r_{\rm bb}$ & 25AU (300AU)\ Inner (outer)$^b$ Width, $\Delta r_{\rm bb}$ & 0.02AU (60AU)\ Inner (outer)$^b$ Minimum mass$^{c}$, $M_{\rm dust}$ & 0.04[$M_{\oplus}$]{} (0.15[$M_{\oplus}$]{})\ $^{a}$ Two-dimensional Gaussian fits with $1 \sigma$ [*formal*]{} fitting uncertainties.\ $^{b}$ An outer dust belt is implied by the extent of [q$^{1}$Eri]{} at 870[$\mu$m]{}.\ $^{c}$ $\kappa_{10^{11.5}\,{\rm Hz}}=2$cm$^2$g$^{-1}$ ($\rho=1.18$g[cm$^{-3}$]{}, $a_{\rm max}=1$mm, $n(a) \propto a^{-3.5}$). The nature of the emitting particles ------------------------------------ The absence of spectral features in the 10 to 30[$\mu$m]{} region suggests that the dust grains are considerably larger than 10[$\mu$m]{} [@chen2006; @schutz2005]. Remarkably, the spectral energy distribution (SED) of the excess emission can be fit by a single-temperature blackbody of 60K, from the infrared to the submm/mm regime (see Fig.\[SED\]). The blackbody character is determined by the LABOCA flux and independent of the relative weights assigned to the mid- and far-infrared data. The radial distance from the central star, at which a grain has attained thermal equilibrium, is approximately given by $[(1-A)/(16 \pi\,\epsilon\,\sigma)\,(L_{\rm star}/T^4)]^{1/2}$, where $A$ and $\epsilon$ are the integrated reflectivity and emissivity, respectively. For a blackbody this reduces to $r_{\rm bb} = (R_{\rm star}/2)\,(T_{\rm eff}/T_{\rm bb})^2$, which for $T_{\rm bb}=60$K, yields a minimum distance of 25AU for the [q$^{1}$Eri]{} dust (Table\[parameters\]). Taken at face value, this would mean that the range in dust temperatures is very limited: single values lower than 50K or as high as 100K can be excluded (without giving higher weight to the Spitzer data, $T_{\rm bb}$ becomes closer to 70K). Therefore, $r_{\rm bb}$, is determined to better than within a factor of two. For unit filling factor, the blackbody emitting regions would appear to be confined to a very narrow ring-like structure (see Table\[parameters\]). The fractional luminosity is $1 \times 10^{-4}$ and the emission is optically thin. The blackbody fit also implies that the emitting particles have sizes largely in excess of 100[$\mu$m]{} ($2 \pi a > \lambda$) and that these grains have grey opacities in the infrared to submm, i.e., $\kappa \neq \kappa(\lambda)$. Given the available evidence, it is not possible, however, to tell the actual sizes of the particles or their absolute opacities. Some insight may be gained from the work of @miyake1993, who explored the optical properties of dust that produces small values of the opacity index, and presented opacities over a broad range in frequency and particle size. Maximum opacity, max$\kappa \sim 2$cm$^2$g$^{-1}$, was found for the size $a_{\rm max}=1$mm at $\nu = 10^{11.5}$Hz ($\lambda \sim 1$mm) and for larger particles, $\kappa$ decreases rapidly (as $\sim 1/\sqrt{a_{\rm max}}$). This assumes compact spheres of density $\rho=1.18$g[cm$^{-3}$]{} (well-mixed silicates and water ice) and being distributed in size according to $n(a) \propto a^p$, with $p=-3.5$. The adopted density is consistent with values determined for Kuiper Belt objects [@grundy2007 and references therein]. The value of $\kappa_{1\,{\rm mm}}$ is not strongly dependent on $p$, as long as $-4 \le p \le -2$ [@miyake1993]. In general, these results are in agreement with other work [e.g., @krugel1994; @stognienko1995]. For this maximum value of $\kappa$, the 870[$\mu$m]{} flux density yields a minimum mass $M_{\rm dust} = F_{\nu}\,D^2/\kappa_{\nu}\,B_{\nu}(T_{\rm dust}) \ge 3$[$M_{\rm Moon}$]{} (0.04[$M_{\oplus}$]{}, see Table\[parameters\]). This minimum mass is larger than the ’blackbody mass’ one would infer from the effective area of the blackbody ($6.85 \times 10^{26}$cm$^2$) and for the same $a$ and $\rho$, but consistent with the (re-scaled) result of @schutz2005. The fact that the 870[$\mu$m]{} source appears linearly resolved, speaks against the narrow ring scenario, and the existence of a more massive and colder belt ($T=17$K at $r=300$AU, say) can at present not be excluded (see Fig.\[SED\]). The relative width of such a cold belt would seem less implausible, viz. $\Delta r/r \ge 0.2$ and, hence, its physical width would be at least 60AU. Also, the equilibrium temperature of the dust would be esssentially constant. With the same parameters as before, the mass would scale simply as the ratio of the temperatures, yielding 13[$M_{\rm Moon}$]{}. Of course, at shorter wavelengths, the spectrum would have to be steeper than the blackbody SED, implying values of $\beta > 0$, where $\beta$ parameterizes the frequency dependence of the opacity, i.e., $\kappa_{\nu} \propto \nu^{\,\beta}$. Together with the 17K blackbody, the SED can be fit with a modified 60K blackbody with $\beta=1.0$ longward of 100[$\mu$m]{} (Fig.\[SED\]). This model would physically be more attractive, but by its ad hoc nature would be of course not unique, and better constraints on the physical parameters would require a better sampling of the SED. Relation to the planet [q$^{1}$Eri]{}b -------------------------------------- An index of the order of $-3.5$, used by @miyake1993, could indicate that the size distribution resulted from a collisional cascade [for a discussion, see @thebault2007]. The observed absence of small debris ($a$ of the order of 1[$\mu$m]{} or smaller) in the [q$^{1}$Eri]{} disk suggests that its production has ceased and that it had diminished on a short time scale compared to the age of the system. Remarkably, the density parameter of @wyatt2005 has a value that, given the age of [q$^{1}$Eri]{}, is atypically large ($\eta_0 \sim 1000$ for the correct form of Eq.7), which may mean that the observed absence of warm material is caused by radiation pressure blow-out, rather than by the action of a planet, hindering the migration inward toward the star. Anyway, at 2AU distance, the Jupiter-size planet [q$^{1}$Eri]{}b could hardly have had any influence on particles out to the innermost edge at 25AU and on orbits far beyond that [@wyatt2005]. The existence of this belt of large grains may point to the presence of another major planet. It would therefore seem important to verify or to disprove the existence of [q$^{1}$Eri]{}c. Conclusions =========== Below, our main conclusions from this work are summarized: - Observations of the solar-type star [q$^{1}$Eri]{} and its planet [q$^{1}$Eri]{}b at 870[$\mu$m]{} revealed a source with peak emission at the position of the star. The source appears extended to the LABOCA beam, i.e., elongated in roughly the East-West direction (640AU) but essentially unresolved in the perpendicular direction. - At an age exceeding 1Gyr, the fractional luminosity of the infrared excess is very high ($\ge 10^{-4}$). The entire SED of this excess emission, extending from [$\stackrel {>}{_{\sim}}$]{}20[$\mu$m]{} to 1mm, is fit by a single-temperature blackbody ($T_{\rm bb}=60$K). - This would imply that the emitting regions are located at about 25AU from the star and in addition, very limited in spatial extent (ring-like). Exhibiting a grey opacity over the entire wavelength range, the emitting particles must be large ($>>100$[$\mu$m]{}). Using the theoretically derived maximum grain opacity for 1mm-size icy particles, we estimate a minimum mass of the dust belt of 0.04[$M_{\oplus}$]{}. - It seems highly unlikely that the planet [q$^{1}$Eri]{}b at 2AU would be responsible for the clearing of the region from small dust particles interior to 25AU, but may hint at the existence of another planet. - Taking the observed extent of the 870[$\mu$m]{} source into account leads to an emission model in which an outer cold (17K) dust belt needs to be included. This belt would be centered on the radial distance of 300AU and have a width of 60AU. This belt is inclined at $i \ge 52$[$^{\circ}$]{}, possibly viewed at an angle close to edge-on. - The extreme character of the debris disk around the relatively old star-planet system [q$^{1}$Eri]{} provides yet another example of the large diversity of such disks. 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--- abstract: 'We present a parameterization of the non-collinear (virtual) Compton scattering tensor in terms of form factors, in which the Lorentz tensor associated with each form factor possesses manifest electromagnetic gauge invariance. The main finding is that in a well-defined form factor expansion of the scattering tensor, the form factors are either symmetric or antisymmetric under the exchange of two Mandelstam variables, $s$ and $u$. Our decomposition can be used to organize complicated higher-order and higher-twist contributions in the study of the virtual Compton scattering off the proton. Such procedures are illustrated by use of the virtual Compton scattering off the lepton. In passing, we note the general symmetry constraints on Ji’s off-forward parton distributions and Radyushkin’s double distributions.' address: | Centre de Physique Théorique[^1], Ecole Polytechnique\ 91128 Palaiseau Cedex, France\ [ ]{} author: - Wei Lu date: July 1997 title: 'FORM FACTOR DESCRIPTION OF THE NON-COLLINEAR COMPTON SCATTERING TENSOR' --- introduction ============ Recently, there is [@ji1; @ra1; @hyde; @chen; @pire] much revived interest in the virtual Compton scattering (VCS). By VCS, people usually mean the the scattering of a virtual photon into a real photon off a proton target $$\gamma^\ast (q) + N(P,S) \to \gamma^\ast(q^\prime) +N(P^\prime, S^\prime) \ .$$ As usual, three Mandelstam variables are defined for this process: $s \equiv (q+P)^2, $ $ t \equiv (q-q^\prime)^2 ,$ $ u \equiv(P-q^\prime )^2$. Due to the momentum conservation $ P + q = P^\prime + q^\prime, $ there is the following constraint: $$s+t +u =q^2 + q^{\prime 2} + 2m^2,$$ where $m$ is the proton mass. The object of study is the following scattering tensor $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime,S^\prime) = i\int d^4 \xi e^{iq^\prime\cdot \xi} \langle P^\prime,S^\prime| {\rm T}[J^\mu (\xi) J^\nu (0)]|P,S\rangle \ ,$$ where $J$ is the quark electromagnetic current in the proton and T stands for the time-ordering of the operators. At present, much of interest is focused on the deeply VCS (DVCS), which is a very special kinematic region of the generic VCS. It has been claimed that the dominant mechanism in the DVCS is the VCS off a massless quark [@ji1]. Correspondingly, two different approaches to the DVCS tensor have been developed: the Feynman diagram expansion [@ji1; @ra1] and operator product expansion (OPE) [@wana; @chen]. A careful reader might be aware of such a fact: At the leading twist expansion of the DVCS tensor, both in the Feynman diagram expansion and in OPE approach, the resultant expressions do not possess manifest electromagnetic gauge invariance. The purpose of this paper is to remedy the case by presenting a full form factor parameterization of the non-collinear Compton scattering tensor. With the help of our decomposition of the scattering tensor, one can safely ignore the higher-twist terms at leading-twist expansion and recover the electromagnetic gauge invariance by brute force. Hopefully, our form factor description can be used to organize complicated calculations as one goes beyond leading twist and/or leading order. We confess that we are not the very first to attempt to develop a form factor parameterization of the VCS tensor. As early as in 1960s, Berg and Lindner [@lindner] ever reported a parameterization of the VCS tensor in terms of form factors. The virtue, also an implicit assumption, in their decomposition is that the scattering tensor can be put into a form of direct products of the Lorentz tensors and Dirac bilinears, i.e., the Lorentz index of the VCS tensor is $not$ carried by the gamma matrices. In fact, all the leading twist expansions of the DVCS tensor so far assume such a factorized form. A drawback of the Berg-Lindner decomposition is that they employed a lot of momentum combinations which have no specific crossing and time reversal transformation properties. As a consequence, the form factors they defined possess no specific symmetry properties under crossing and time reversal transformations. Moreover, their decomposition lacks a term associated with the Lorentz structure $\epsilon^{\mu\nu\alpha\beta}q_\alpha q^\prime_\beta$, which has been shown by recent researches to be a carrier of leading-twist contributions. It should be stressed that there is no unique decomposition of the Compton tensor. A few years ago, Guichon, Liu and Thomas [@pierre] worked out a general decomposition of the VCS tensor, which contains no explicit proton spinors. Their decomposition is nice for the discussion of the generalized proton polarizabilities, as has done in Ref. [@pierre]. Recently, the decomposition of the VCS tensor of such type has been refined by Drechsel et al. [@Drechsel] within more extensive contexts. However, a decomposition of the VCS tensor without explicit Dirac bilinear structures is of very limited use for the present Feynman diagram expansion and OPE analysis of the DVCS tensor. Hence, it is desirable to reconstruct a parameterization of the Compton scattering tensor in terms of form factors with explicit Dirac structures, which constructs the subject of this paper. To make our arguments more transparent, we will first consider the Lorentz decomposition for the double VCS off a lepton, then transplant our results onto the proton case. \[By double VCS we mean that both the initial- and final-state photons are virtual. Correspondingly, we will refer the usual VCS to as the single VCS in distinction.\] The reason for adopting such a strategy is that in quantum electrodynamics, it is more convenient to discuss the chiral properties of the Dirac bilinears. At the later stage, we will reduce our results for the double VCS to the real Compton scattering (RCS) as well as the single VCS. Such a procedure will greatly facilitate the discussion of the symmetry properties of the single VCS form factors. The decomposition of the Compton tensor is essentially subject to the symmetries that it observes, so we begin with a brief discussion of the symmetry properties of the Compton scattering. First, the current conservation requires that $$q_\mu T^{\mu\nu}(q, P,S; q^\prime, P^\prime,S^\prime) = q^\prime_\nu T^{\mu\nu}(q, P,S; q^\prime, P^\prime,S^\prime) =0\ . \label{gauge}$$ Second, parity conservation tells us $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime,S^\prime) = T_{\mu\nu}( \tilde q,\tilde P, - \tilde S; \tilde q^\prime, \tilde P^\prime, - \tilde S^\prime) \ , \label{p}$$ where $\tilde q^\mu \equiv q_\mu$, and so on. Third, time reversal invariance demands $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime, S^\prime) = T_{\nu\mu}(\tilde q^\prime, \tilde P^\prime,\tilde S^\prime; \tilde q, \tilde P,\tilde S )\ . \label{t}$$ Fourthly, there is a crossing symmetry for the Compton scattering, namely, $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime, S^\prime) = T^{\nu\mu}(-q^\prime, P,S; -q, P^\prime, S^\prime)\ . \label{cr}$$ By combining (\[p\]) with (\[t\]), we have $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime, S^\prime) = T^{\nu\mu}(q^\prime, P^\prime, -S^\prime ; q, P,-S)\ .\label{pt}$$ That is to say, the adjoint parity-time-reversal transformation amounts to $\mu \leftrightarrow \nu$, $q \leftrightarrow q^\prime $, $P \leftrightarrow P^\prime $, $S\to -S^\prime$ and $S^\prime \to -S$. Furthermore, combining (\[cr\]) with (\[pt\]) yields $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime, S^\prime) = T^{\mu\nu}(-q,P^\prime,- S^\prime ; -q^\prime, P,-S )\ . \label{crpt}$$ In fact, the Compton scattering respects more symmetries than summarized above. For example, it is subject to the momentum and angular momentum conservations. In the case of collinear scattering, the angular momentum conservation exerts further constraints on the Compton scattering. To show this, we digress to the helicity amplitude description of the Compton scattering. In the expansion of the Compton scattering amplitude, a fundamental question that must be answered in advance is that how many independent state vectors there are in a complete basis. This can be most naturally done by counting independent helicity amplitudes. Here we stress that each independent helicity amplitude corresponds to one $observable$ independent form factor in the Compton scattering tensor, while there is no simple one-to-one correspondence between helicity amplitudes and form factors. Let us consider the most general non-collinear double VCS off a massive lepton ($l$). Because the massive lepton and the virtual photon have 2 and 3 helicity states respectively, there are $2\times 3 \times 2 \times 3=36$ helicity amplitudes. By parity conservation, only half of the these helicity amplitudes are independent. In the non-collinear case, the other symmetries cannot further reduce the number of independent helicity amplitudes. Similarly, there are 12 and 8 independent helicity amplitudes for the non-collinear single VCS and RCS off the massive lepton. In the collinear scattering limits, however, time reversal invariance and angular momentum conservation impose further constrains on the Compton scattering. We denote a generic Compton scattering helicity amplitude $A(\lambda_q, \lambda_l; \lambda_{q^\prime}, \lambda_{l^\prime})$, where $\lambda$’s are the helicities of the corresponding particles. By time reversal invariance, there is $$A(\lambda_q, \lambda_l; \lambda_{q^\prime}, \lambda_{l^\prime}) = A( \lambda_{q^\prime}, \lambda_{l^\prime}; \lambda_q, \lambda_l)\ .$$ To discuss the constraints from angular momentum conservation, we need to distinguish two collinear limits: $$\lambda_q -\lambda_l=\pm (\lambda_{q^\prime} -\lambda_{l^\prime}) \ .$$ where $\pm$ corresponds to the forward and backward collinear scattering, respectively. As a result, only a small fraction of the Compton helicity amplitudes survive in the collinear limits. We summarize those surviving (independent) helicity amplitudes in Table 1. The above helicity anplitude analysis implies a thorny fact: As one approaches the collinear limits, there is significant degeneracy in the form factor parameterization of the Compton scattering tensor. Here we emphasize that one must avoid the over-degeneracy of the form factor description in the collinear cases as much as possible. As will be clarified, some form factor parameterizations of the Compton scattering tensor, albeit applicable in the non-collinear cases, might become ill-defined in the collinear scattering limits. Now we investigate the general structure of the non-collinear Compton tensor. As stated before, the Berg-Lindner decomposition assumes the following form $$T^{\mu\nu}(q, P,S; q^\prime, P^\prime, S^\prime) = \sum_{i} \bar U(P^\prime, S^\prime) \Gamma_i U(P,S)t^{\mu \nu}_i F_i \ , \label{6}$$ where $\Gamma_i$s are gamma matrices (saturated with particle momenta if carrying Lorentz indices), $t^{\mu \nu}_i$s Lorentz (pseudo)-tensors constructed from the relevant particle momenta (the metric tensor and the Levi-Civita tensor may be involved), and $F_i$s Lorentz invariant form-factor like objects. As a matter of fact, all of the recent research results about the DVCS tensor can be tailored into the form of Eq. (\[6\]). Now we justify Eq. (\[6\]) for the non-collinear Compton scattering. In principle, one can assume a decomposition for the Compton tensor in which there is no explicit Dirac bilinears. Then, the Lorentz indices of the scattering tensor can be carried by the metric tensor, the Levi-Civita tensor, the particle momenta and lepton spin four-vectors. We will not write down any such decompositions. Rather, we note that the lepton spin four-vector can carry the free Lorentz index now. The spin four-vector $S$ of a lepton of momentum $P$ is subject to $S\cdot P=0$, so it can be expressed in terms of three $non$-$collinear$ particle momenta. For the non-collinear scattering, one can write down, say $$S^{\mu}=(S\cdot K_1) P^{\prime \mu} +(S\cdot K_2) q^\mu + (S\cdot K_3)q^{\prime \mu}, \label{7}$$ where $K_1,$ $K_2$ and $K_3$ are three momentum combinations whose expressions we do not need. As a consequence, one can eliminate the lepton spin four-vectors from the building blocks that carry the free Lorentz indices and lump $S\cdot K_j$ into the form factors. At this stage, if one displays the Dirac bilinears, the decomposition of the Compton scattering tensor assumes the structure of Eq. (\[6\]). From the above justification, we see that some subtleties will arise as one goes to the collinear limits of the Compton scattering. That is, Eq. (\[6\]) is inapplicable to the discussion of the transverse proton spin dependence of the Compton scattering amplitude in the collinear limits. Fortunately, the collinear scattering are only very special kinematic limits of the VCS. So it is still desirable to develop a form factor parameterization of the non-collinear Compton scattering with the general structure of Eq. (\[6\]). The symmetries impose further constraints on the decomposition of Eq. (\[6\]). For a generic VCS, its form factors depend on 4 independent kinematical variables. Though there is much interest in the small-$|t|$ limit behavior [@ji1; @pire] of the single VCS off the proton, we insist in choosing $s$, $u$, $q^2$ and $q^{\prime 2}$ ($t$ being an auxiliary quantity) as four independent kinematical variables for the form factors. The reason for doing so is that the crossing transformation of the Compton scattering essentially relates its $s$- to $u$-channel contributions or vice versa. Under the crossing transformation, $ s \leftrightarrow u$ and $q^2 \leftrightarrow q^{\prime 2}$. Further, under the time reversal (or the adjoint parity-time-reversal) transformation, there is $q^2 \leftrightarrow q^{\prime 2}$. So, it is a natural choice for us to define the form factors in such a way that all of them possess specific symmetry properties under the crossing and time-reversal transformations. To this end, we demand that the Lorentz tensor and Dirac bilinear associated with each form factor are either symmetric or antisymmetric under the crossing and time-reversal transformations. At this stage, we recognize that it is more useful to talk about the adjoint crossing-(parity)-time-reversal transformation properties of the single VCS form factors, for which people usually take into account the on-shell condition $q^{\prime 2}=0$ of the final photon in practical calculations. Now we set about the construction of the form factor description of the non-collinear Compton scattering tensor. As claimed earlier, we begin with the double VCS off a lepton. We first consider the case of a massless lepton. Then, its helicity and chirality coincide with each other. For the massless lepton, the chiral symmetry holds exactly, so the lepton helicity is conserved in the Compton scattering. As a consequence, there are only 9 independent helicity amplitudes and accordingly 9 complex form factors for the non-collinear double VCS off the massless lepton. All the spinor bilinears must be chiral-even, so only the vector and axial-vector Dirac structures, $\gamma_\alpha $ and $ \gamma_\alpha\gamma_5,$ get into work. To saturate the Lorentz indices carried by the Dirac matrices, we choose $q+q^\prime$, $P$ and $P^\prime$ as 3 independent momenta. Obviously, there are only two nontrivial, independent Dirac structures: $ \bar U(P^\prime,S^\prime)(\rlap/q+\rlap/q^\prime) U(P,S)$ and $ \bar U(P^\prime,S^\prime)(\rlap/q+\rlap/q^\prime) \gamma_5 U(P,S)$. Now we construct proper gauge-invariant Lorentz tensors to match $ \bar U(P^\prime,S^\prime)(\rlap/q+\rlap/q^\prime) U(P,S)$. In doing so, we keep it in mind to render our choices of independent Lorentz tensors possess specific crossing and parity-time-reversal transformation properties. Using the metric tensor, we can write down $ -(q\cdot q^\prime)g^{\mu\nu}+q^{\prime\mu} q^\nu$. As index $\mu$ is carried by particle momenta, we can write down only two independent momentum combinations because of the momentum conservation and gauge invariance. Similarly for index $\nu$. So, we have 4 more independent tensors without invoking the metric tensor. Our choices are $ A^\mu B^\nu$, $ A^\mu_1 B^\nu + A^\mu B^\nu_1$, $ A^\mu_1 B^\nu - A^\mu B^\nu_1$, and $ A^\mu_1 B^\nu_1,$ where $$\begin{aligned} A^\mu&=& (q^{\prime \mu} - \frac{q\cdot q^\prime}{P\cdot q } P^{ \mu}) +(q^{\prime \mu} - \frac{q\cdot q^\prime}{P^\prime\cdot q } P^{ \prime \mu}) \ ,\\ A^\mu_1&=&q^\mu -\frac{q^2}{q\cdot q^\prime}q^{\prime\mu}\ ,\\ B^\nu&=&(q^{ \nu} -\frac {q\cdot q^\prime }{P^\prime \cdot q^\prime }P^{ \prime \nu}) +(q^{ \nu} -\frac {q\cdot q^\prime }{P \cdot q^\prime }P^{ \nu}) \ , \\ B^\nu_1&=&q^{\prime \nu} -\frac{q^{\prime 2} }{q\cdot q^\prime}q^{\nu}\ . \end{aligned}$$ By construction, $ A^\mu B^\nu$, $ A^\mu_1 B^\nu + A^\mu B^\nu_1$, $ A^\mu_1 B^\nu - A^\mu B^\nu_1$, and $ A^\mu_1 B^\nu_1 $ have specific symmetry properties under the crossing and time-reversal transformations. To match $ \bar U(P^\prime,S^\prime)(\rlap/q +\rlap/q^\prime) \gamma_5 U(P,S)$, we need to invoke one Levi-Civita tensor. If the Levi-Civita tensor is demanded to carry two free Lorentz indices, we have $\epsilon^{\mu\nu\alpha \beta }q_\alpha q^\prime_\beta$ by gauge invariance. As one of the Lorentz indices is carried by the particle momentum, at our disposal are $A^\mu D^\nu +B^\nu C^\mu,$ $A^\mu D^\nu -B^\nu C^\mu,$ $A^\mu_1 D^\nu+B^\nu_1 C^\mu$, $A^\mu_1 D^\nu-B^\nu_1 C^\mu$, where $$\begin{aligned} C^\mu&=&\epsilon^{\mu \alpha \beta \gamma }q_\alpha P_\beta P ^\prime_\gamma\ ,\\ D^\nu&=&\epsilon^{\nu \alpha \beta \gamma }q^\prime_\alpha P_\beta P ^\prime_\gamma\ . \end{aligned}$$ Again, $A^\mu D^\nu +B^\nu C^\mu,$ $A^\mu D^\nu -B^\nu C^\mu,$ $A^\mu_1 D^\nu+B^\nu_1 C^\mu$, $A^\mu_1 D^\nu-B^\nu_1 C^\mu$ have specific symmetry properties under the crossing and time-reversal transformations. By definition, $A^\mu D^\nu,$ $A^\mu_1 D^\nu,$ $B^\nu C^\mu$, and $B^\nu_1 C^\mu$ are independent of each other. On the other hand, the antisymmetric property of $\epsilon^{\mu\nu\alpha \beta } q_\alpha q^\prime_\beta$ tells us it that may have 6 non-vanishing components. Due to the current conservation conditions, Eq. (\[gauge\]), only 4 of them are independent. Therefore, $\epsilon^{\mu\nu\alpha \beta } q_\alpha q^\prime_\beta$ can be expanded in terms of $A^\mu D^\nu,$ $A^\mu_1 D^\nu, $ $ C^\mu B^\nu$, and $ C^\mu B^\nu_1$. In fact, one can directly construct the following identity: $$\begin{aligned} \epsilon^{\mu\nu\alpha \beta } q_\alpha q^\prime_\beta &= & \frac{( P\cdot A_1 -P^\prime\cdot A_1 ) A^\mu D^\nu -(P\cdot A -P^\prime\cdot A ) A^\mu_1 D^\nu} {(P\cdot A) (P^\prime \cdot A_1)- (P^\prime \cdot A) (P \cdot A_1) } \nonumber\\ & & + \frac{ - ( P\cdot B_1 -P^\prime\cdot B_1 ) C^\mu B^\nu +(P\cdot B -P^\prime\cdot B ) C^\mu B^\nu_1} {(P\cdot B) (P^\prime \cdot B_1)- (P^\prime \cdot B) (P \cdot B_1)}\ . \end{aligned}$$ Notice that this identity holds only for the non-forward Compton scattering. This is where the subtleties arise in choosing four Lorentz pseudo-tensors to match $ \bar U(P^\prime,S^\prime)(\rlap/q +\rlap/q^\prime) \gamma_5 U(P,S)$. If one selects $A^\mu D^\nu$, $ B^\nu C^\mu,$ $A^\mu_1 D^\nu,$ and $ C^\mu B^\nu_1$, all of them will drop out in the collinear limits. As we stressed, we need to avoid the over-degeneracy in the collinear limits as much as possible. On the other hand, all of recent studies indicate that the $\epsilon^{\mu\nu\alpha \beta }q_\alpha q^\prime_\beta$ term incorporates the leading twist contributions. Hence, we choose $\epsilon^{\mu\nu\alpha \beta } q_\alpha q^\prime_\beta$, $A^\mu D^\nu+ B^\nu C^\mu,$ $A^\mu_1 D^\nu+ C^\mu B^\nu_1 ,$ and $A^\mu_1 D^\nu- C^\mu B^\nu_1 $ as 4 independent Lorentz pseudo-tensors to match with $ \bar U(P^\prime,S^\prime) (\rlap/q +\rlap/q^\prime)\gamma_5 U(P,S).$ \[In the Berg-Lindner decomposition, there is no $\epsilon^{\mu\nu\alpha \beta }q_\alpha q^\prime_\beta$ term.\] Notice that $\epsilon^{\mu\nu\alpha \beta }q_\alpha q^\prime_\beta$ survives the collinear limits. Thus, we have identified 9 independent structures for the non-collinear double VCS off the massless lepton. Now we take into the lepton mass effects. Then, the helicity of a massive lepton is no longer in coincidence with its chirality. The chiral-even lepton state is roughly in the helicity-$+\frac{1}{2}$ state, with a helicity-$-\frac{1}{2}$ contamination of ${\cal O}(m/Q)$, where $m$ and $Q$ are the lepton mass and a high energy interaction scale, respectively. The inclusion of the lepton mass effect will generate 9 more structures, which flip the lepton helicity. Writing down the Dirac bilinears is essentially an expansion according to the chiral structure. Although the expansion according to the lepton helicity is not coincident with that according to the chiral structure, the number of independent terms in any complementary expansion should be equal. Hence, the lepton mass will generate 9 more independent chiral-odd structures in the general decomposition. In constructing independent chiral-odd Dirac bilinears, we have $1$, $\gamma_5$ and $\sigma^{\alpha\beta}$ at our disposal. We first consider the pseudo-scalar Dirac structure $ \bar U(P^\prime,S^\prime)\gamma_5 U(P,S)$. Remind that there should be at least two spin-dependent form factors in the collinear scattering limit, because 4 independent helicity amplitudes survive in the collinear limits of double VCS. To avoid possible over-degeneracy in the collinear limits, we select again $\epsilon^{\mu\nu\alpha \beta } q_\alpha q^\prime_\beta$, $A^\mu D^\nu+ B^\nu C^\mu,$ $A^\mu_1 D^\nu+ C^\mu B^\nu_1 ,$ and $A^\mu_1 D^\nu- C^\mu B^\nu_1 $ as 4 independent Lorentz pseudo-tensors to match with $ \bar U(P^\prime,S^\prime)\gamma_5 U(P,S)$. Now we consider the tensor Dirac structures. By choosing $q+q^\prime$, $P$ and $P^\prime$ as 3 independent particle momenta, we have $ \bar U(P^\prime,S^\prime)\sigma^{\alpha\beta}P_\alpha P^\prime_\beta U(P,S),$ $\bar U(P^\prime,S^\prime)\sigma^{\alpha\beta}(q +q^\prime) _\alpha P^\prime_\beta U(P,S),$ $\bar U(P^\prime,S^\prime)\sigma^{\alpha\beta} P_\alpha (q+q^\prime) _\beta U(P,S)$. By use of the Dirac equation, one can show that 1) $\bar U(P^\prime,S^\prime)\sigma^{\alpha\beta}P_\alpha P^\prime_\beta U(P,S)$ is equivalent to the scalar Dirac structure; and 2) Both $\bar U(P^\prime,S^\prime)\sigma^{\alpha\beta}(q+q^\prime) _\alpha P^\prime_\beta U(P,S)$ and $ \bar U(P^\prime,S^\prime)\sigma^{\alpha\beta}P_\alpha (q+q^\prime)_\beta U(P,S)$ reduce to a combination of the the vector and scalar Dirac structures. We choose $\bar U(P^\prime,S^\prime) \sigma^{\alpha\beta} P _\alpha P^\prime_\beta U(P,S)$ as an independent Dirac structure, which can be matched with $ -(q\cdot q^\prime)g^{\mu\nu}+q^{\prime\mu} q^\nu,$ $A^\mu B^\nu,$ $ A^\mu_1 B^\nu+ A^\mu B^\nu_1,$ $ A^\mu_1 B^\nu- A^\mu B^\nu_1,$ and $ A^\mu_1 B^\nu_1.$ Thus, we have specified 9 independent chiral-odd structures for the non-collinear double VCS off the massive lepton. Now we are in a position to make our suggestion about the Lorentz decomposition of the non-collinear double VCS off the massive lepton: [ $$\begin{aligned} T^{\mu\nu}& = & \frac{ -(q\cdot q^\prime) g^{\mu\nu} +q^{\prime\mu} q^\nu }{su} \bar U(P^\prime,S^\prime) \Big(f_1 ( \rlap/q + \rlap/q^\prime) +f_2 \frac{ i \sigma^{\alpha\beta}P _\alpha P^\prime_\beta }{ 2m } \Big)U(P,S) \nonumber \\ && + \frac{A^\mu B^\nu}{s u } \bar U(P^\prime,S^\prime)\Big(f_3 ( \rlap/q + \rlap/q^\prime) +f_4 \frac{ i \sigma^{\alpha\beta}P _\alpha P^\prime_\beta }{ 2 m } \Big)U(P,S) \nonumber \\ && + \frac{A^\mu_1 B^\nu + A^\mu B^\nu_1}{s u } \bar U(P^\prime,S^\prime)\Big(f_5 ( \rlap/q + \rlap/q^\prime) +f_6 \frac{ i \sigma^{\alpha\beta}P _\alpha P^\prime_\beta }{ 2 m } \Big)U(P,S) \nonumber \\ && +\frac{A^\mu_1 B^\nu - A^\mu B^\nu_1}{ su } \bar U(P^\prime,S^\prime)\Big(f_7 ( \rlap/q + \rlap/q^\prime) +f_8 \frac{ i \sigma^{\alpha\beta}P _\alpha P^\prime_\beta }{ 2 m } \Big)U(P,S) \nonumber \\ && +\frac{A^\mu_1 B^\nu_1}{ s u } \bar U(P^\prime,S^\prime)\Big(f_9 ( \rlap/q + \rlap/q^\prime) +f_{10} \frac{ i \sigma^{\alpha\beta}P _\alpha P^\prime_\beta }{ 2 m } \Big)U(P,S) \nonumber \\ &&+\frac{i \epsilon^{\mu\nu \alpha \beta} q_\alpha q^\prime_\beta}{ s u } \bar U(P^\prime,S^\prime) \Big(g_1 ( \rlap/q + \rlap/q^\prime) \gamma_5 + g_2 \frac{ (P\cdot P^\prime)\gamma_5}{2m} \Big)U(P,S) \nonumber \\ &&+ \frac{ i( A^\mu D^\nu + C^\mu B^\nu) }{(P\cdot P^\prime) s u } \bar U(P^\prime,S^\prime) \Big(g_3 ( \rlap/q + \rlap/q^\prime) \gamma_5 +g_4 \frac{ (P\cdot P^\prime)\gamma_5}{2m} \Big) U(P,S) \nonumber \\ &&+\frac{i(A^\mu_1 D^\nu + C^\mu B^\nu_1) } {(P\cdot P^\prime) s u } \bar U(P^\prime,S^\prime) \Big(g_5 ( \rlap/q + \rlap/q^\prime) \gamma_5 + g_6\frac{(P\cdot P^\prime)\gamma_5}{2m} \Big)U(P,S) \nonumber \\ &&+\frac{i( A^\mu_1 D^\nu -C^\mu B^\nu_1) } {(P\cdot P^\prime) s u } \bar U(P^\prime,S^\prime) \Big(g_7 ( \rlap/q + \rlap/q^\prime) \gamma_5 + g_8\frac{ (P\cdot P^\prime)\gamma_5}{2m} \Big) U(P,S) , \label{decom} \end{aligned}$$ ]{} where $f_i$ and $g_i$ are dimensionless complex form factors, dependent on $s$, $u$, $q^2$ and $q^{\prime 2}$. From our procedure to establish the above decomposition, the reader can convince himself that our decomposition is complementary in the non-collinear case. The $s$ and $u$ factors in Eq. (\[decom\]) can be understood as the remnants of the $s$- and $u$-channel propagators. The Compton scattering off the proton and that off the lepton observe the same symmetries, while the proton is a composite object. Therefore, Eq. (\[decom\]) applies as well to the non-collinear double VCS off the proton. The soft physics in the proton, in relation to the chiral symmetry breaking, does not bring about extra problems in decomposing the VCS amplitude. From now on, we understand Eq. (\[decom\]) to be the general decomposition of the double VCS tensor for the proton. By construction, the form factors in Eq. (\[decom\]) are either symmetric or antisymmetric under the crossing and time reversal transformations. The crossing symmetry properties of the form factors can be read off straightforwardly. To obtain the time-reversal transformation properties of the form factors, just note that the form factors are functions of $s$, $u$, $q^2$ and $q^{\prime 2}$, irrelevant of the spin state of the proton. Then, one can put each of the protons in a specific helicity state to show how the form factors transform under the adjoint parity-time-reversal transformations. We summarize the crossing and parity-time-reversal transformation properties of the form factors in Table 2. The various symmetry transformation properties of the form factors can be employed to perform consistency check of theoretical calculations. Of more interest are the symmetry properties of the form factors under the adjoint crossing, parity and time-reversal transformations, where $s$ and $u$ exchange their roles. They are especially useful for the study of the single VCS, where the on-shell condition $q^{\prime 2}=0$ of the final photon is usually implicit in the calculations. Now we address the reduction of Eq. (\[decom\]) to the cases of the single VCS. Regarding the single VCS, the on-shell condition of the final-state photon implies as well the Lorentz condition of the final-state photon. Therefore, all of its $B^\nu_1$-related terms become unobservable. As a consequence, we are left with 12 independent, observable form factors $f_{1, 2, 3, 4,5+7, 6+8},$ and $g_{1, 2, 3, 4,5+7, 6+8}$. For convenience, we introduce the shorthand $f_{i\pm j}=f_i \pm f_j$. Similarly for the $g$-type form factors. Notice that $f_{5+7,6+8}$ and $g_{5+7, 6+8}$ have no specific transformation properties under individual crossing and parity-time-reversal transformation. However, they are symmetric under the adjoint crossing-parity-time-reversal transformation. More concretely, they are symmetric under $s \leftrightarrow u$. Notice that the single VCS tensor contains more information than the corresponding transition amplitude. Generally speaking, $f_{5-7, 6-8, 9,10}$ and $g_{5-7, 6-8, 9, 10}$ do not vanish at $q^{\prime 2}=0$ on their own. They do not make contributions to the single VCS amplitude simply because the contraction of their associated Lorentz tensors with the polarization vector of the final-state photon vanish. To obtain these form factors, one could extrapolate the data from $q^{\prime 2}\neq 0$, which is beyond the scope of this work. Another interesting reduction of Eq. (\[decom\]) is to go to the non-collinear RCS. Imposing the on-shell condition both on the initial and final-state photons, we are left with 8 $observable$, independent form factors: $f_{1, 2, 3, 4}$ and $g_{1, 2, 3, 4}$. In other words, we can reach the non-collinear RCS by simply dropping those terms constructed with $A^\mu_1$ and/or $B^\nu_1$. Our conclusion is consistent with the independent helicity counting by Kroll, Schürmann and Guichon [@kroll], but disagrees with the claim made by Berg and Lindner that there are only 6 non-vanishing form factors for the non-collinear RCS. Here we remark that the Berg-Lindner claim was incorrect, because it was based on an abuse of the crossing symmetry of the Compton scattering. It is literally true that the single VCS form factors depend on only 3 independent kinematical variables if the on-shell condition of the final photon is taken into account. As far as the crossing symmetry properties are concerned, however, $q^{\prime 2}$ must be taken as an independent kinematical variable as the single VCS is discussed. In Ref. [@lindner] it is assumed that the single VCS form factors are three-argument functions, so its discussion about the crossing symmetry properties are incorrect. In addition, the crossing symmetry for the proton was employed in Ref.[@lindner] to eliminate two form factors. We note that the fermion crossing symmetry, which is essentially a charge conjugation symmetry, can be used to relate the Compton scattering off the proton to that off the anti-proton, so it does not generate any constraints on the VCS form factors. A straightforward application of Eq. (\[decom\]) is to recover the manifest electromagnetic gauge invariance in the leading twist expansion of the DVCS tensor [@ji1; @ra1; @chen]. Notice that in these leading twist expansions, all of the involved nonperturbative matrix elements, such as Ji’s off-forward parton distributions (OFPD) and Radyushkin’s double distributions, are color gauge invariant by definition. In addition, all the leading twist contributions in these expansions are associated with the Lorentz structure of types $g^{\mu\nu} +\cdots $ and $\epsilon^{\mu\nu \cdots}$. To recover the electromagnetic gauge invariance, one can simply replace in Refs. [@ji1; @ra1; @chen] $g^{\mu\nu} +\cdots $ and $\epsilon^{\mu\nu \cdots}$ by $ g^{\mu\nu} -q^{\prime\mu} q^\nu/(q\cdot q^\prime)$ and $\epsilon^{\mu\nu \alpha \beta} q_\alpha q^\prime_\beta$ respectively, without need to look into non-leading terms. The usefulness of our decomposition of the non-collinear Compton scattering tensor is more than above. It can be expected that the study of the VCS will inevitably go beyond leading twist and leading order. In fact, some progress along this line has been witnessed [@os]. The basic use of Eq. (\[decom\]) lies in helping theoreticians organize complicated calculations in the study of higher-twist and higher-order contributions. In the following, we illustrate such procedures by expanding the Born-level amplitude for the double VCS off the massless lepton in terms of form factors. Since we are working with the massless lepton, there are only 9 double VCS form factors. Namely, all the form factors with an even subscript drop out in Eq. (\[decom\]) now. From this heuristic example, we will verify our analysis of the symmetry properties of the Compton form factors. In addition, we will learn that there do exist some physical quantities in Nature that cannot be accessed directly by experiments but can, in principle, be extracted by extrapolation. To project out the form factors, we first multiply both sides of Eq. (\[decom\]) with the complex conjugate of $\bar u(p^\prime,s^\prime) ( \rlap/q + \rlap/q^\prime) u(p,s) $, and perform the spin sum over the initial and final-state leptons so as to eliminate the Dirac spinors. Here the $g$-type form factors drop out because the lepton has been assumed to be massless. Then, we saturate the Lorentz indices of the resulting tensor equations in turn with the Lorentz tensors associated with each form factor. As a result, we obtain for the $f$-type form factors the algebraic equations of the following form $$\left( \begin{array}{c} c_1\\ c_3\\ c_{5+7} \\ c_{5-7} \\ c_9 \end{array} \right)= \left( \begin{array}{ccccc} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\ \end{array} \right) \left( \begin{array}{c} f_1\\ f_3\\ f_{5+7} \\ f_{5-7} \\ f_9 \end{array} \right) \ .$$ Similarly, by employing $\bar u(p^\prime,s^\prime) ( \rlap/q + \rlap/q^\prime) \gamma_5 u(p,s) $, we have for the $g$-type form factors, $$\left( \begin{array}{c} d_1\\ d_3\\ d_{5+7} \\ d_{5-7} \end{array} \right)=\left( \begin{array}{ccccc} b_{11} & b_{12} & b_{13} & b_{14} \\ b_{21} & b_{22} & b_{23} & b_{24} \\ b_{31} & b_{32} & b_{33} & b_{34} \\ b_{41} & b_{42} & b_{43} & b_{44} \end{array} \right) \left( \begin{array}{c} g_1\\ g_3\\ g_{5+7} \\ g_{5-7} \end{array} \right) \ ,$$ To save space, we omit the concrete expressions for $c_i$, $d_i$, $a_{ij}$ and $b_{ij}$. Straightforward algebra gives us $$\begin{aligned} f^{(0)}_1(s,u,q^2,q^{\prime 2})&=& \frac{s - u}{s+u}\ ,\label{lept10} \\ f^{(0)}_3(s,u,q^2,q^{\prime 2}) &=& \frac{(s-q^2) (u-q^2) (s-q^{\prime 2} ) (u-q^{\prime 2} )} {(s^2 - u^2) (su- q^2q^{\prime 2})}\ , \\ f^{(0)}_{5+7}(s,u,q^2,q^{\prime 2}) &=& -\frac{ (s-q^{\prime 2} ) (u-q^{\prime 2} ) (s + u -q^2) } { (s-u)(su- q^2q^{\prime 2})}\ , \\ f^{(0)}_{5-7}(s,u,q^2,q^{\prime 2}) &=& -\frac{ (s-q^{\prime 2} ) (u-q^{\prime 2} ) (s + u -q^{\prime 2}) } { (s-u)(su- q^2q^{\prime 2})}\ , \\ f^{(0)}_9(s,u,q^2,q^{\prime 2}) &=&\frac{ (s+u) [s^2 +u^2 + su -(q^2 +q^{\prime 2} ) ( s+u) + q^2q^{\prime 2}]} { (s-u)(su- q^2q^{\prime 2})}\ , \\ g^{(0)}_1(s,u,q^2,q^{\prime 2})&=& \frac{ (q^2 -q^{\prime 2} ) (s+u)[s^2 +u^2 - (q^2 +q^{\prime 2} ) ( s+u) +2 q^2q^{\prime 2}]} { [2 s u- (q^2 +q^{\prime 2} ) ( s+u)+ q^4 + q^{\prime 4}] [s^2 +u^2 + 2 su - 4 q^2q^{\prime 2}]} \ , \\ g^{(0)}_3(s,u,q^2,q^{\prime 2}) &=&\frac{ (s-q^{\prime 2} ) (u-q^{\prime 2} )(s-q^2) (u-q^2) (s + u -q^2 -q^{\prime 2}) } {(s-u) (su - q^2 q^{\prime 2}) [2 s u- (q^2 +q^{\prime 2} ) ( s+u)+ q^4 + q^{\prime 4}]}\ , \\ g^{(0)}_{5+7}(s,u,q^2,q^{\prime 2}) &=&\Big\{ (s + u)(s + u -q^2 -q^{\prime 2}) [s^2 +u^2 - (q^2 +q^{\prime 2} ) ( s+u) + q^2q^{\prime 2}]\nonumber \\ & & ~~~~~ [-su (s +u) +q^{\prime 2} (s^2 +u^2) +q^2q^{\prime 2}(s +u) + 2 q^2q^{\prime 4} ] \Big\}\nonumber \\ & & \times \Big\{ (s-u)(s u - q^2q^{\prime 2})[s^2 +u^2 + 2 su - 4 q^2q^{\prime 2}] \nonumber \\ && ~~~~~ [2 su - (q^2 +q^{\prime 2} ) ( s+u)+ q^4 + q^{\prime 4}] \Big\} ^{-1} \ , \\ g^{(0)}_{5-7}(s,u,q^2,q^{\prime 2}) &=&\Big\{ (s + u)(s + u -q^2-q^{\prime 2}) [s^2 +u^2 - (q^2 +q^{\prime 2} ) ( s+u) + q^2q^{\prime 2}]\nonumber \\ & &~~~~~ [-su(s +u) +q^{ 2} (s^2 +u^2) +q^2q^{\prime 2}(s +u) + 2 q^4q^{\prime 2} ] \Big\}\nonumber \\ & & \times \Big\{(s-u) (s u - q^2q^{\prime 2}) [s^2 +u^2 + 2 su - 4 q^2q^{\prime 2}] \nonumber \\ && ~~~~~ [2 su - (q^2 +q^{\prime 2} ) ( s+u)+ q^4 + q^{\prime 4}] \Big\} ^{-1} \ , \label{lept20} \end{aligned}$$ where the superscript $(0)$ labels the lowest-order results. Obviously, Eqs. (\[lept10\]-\[lept20\]) are consistent with our general symmetry analysis summarized in Table. 2. Letting $q^{\prime 2}=0$ in Eqs. (\[lept10\]-\[lept20\]), we obtain the form factors for the single VCS off the massless lepton. Evidently, in the single VCS case, $f_{5-7}$, $f_9$ and $g_{5-7}$ do not vanish themselves. Moreover, all the form factors have specific symmetry properties under $s\leftrightarrow u$, while $f_{5-7}$ and $g_{5-7}$ have no specific symmetry properties under $q^2 \leftrightarrow q^{\prime 2}$. As we have pointed out, $f_{5-7}$, $f_9$ and $g_{5-7}$ do not come into action in the single VCS amplitude, due to the Lorentz conditions of the final photon. Such a fact informs us that they can be replaced by arbitrary numbers, in any complete, gauge-invariant expansions of the single VCS tensor for the massless lepton (quark). In another word, even $f_{5-7}$, $f_9$ and $g_{5-7}$ are included explicitly in the bases for expanding the single VCS tensor, they cannot be solved out uniquely. By explicit calculations, one can easily show that all $a_{ij}$ and $b_{ij}$ with index 4 and/or 5 vanish in the case of the single VCS. Hence, $c_{5-7} $, $c_9$ and $d_{5-7} $ must vanish, which is a manifestation of the electromagnetic gauge invariance. The above fact sounds trivial to the expansion of the Born amplitudes, but serves as a very useful consistency check in the practical calculations of loop corrections. Note that one can also construct projectors (with Lorentz tensor and Dirac bilinear structures) for all of the form factors that function in the single VCS. Our form factor decomposition of the non-collinear Compton scattering tensor has further implications to the study of Ji’s OFPDs and Radyushkin’s double distributions. In the leading-twist Feynman diagram expansion of the DVCS off the proton, the underlying dynamics is believed to be the single VCS off the massless quark. A virtue of our decomposition, Eq. (\[decom\]), is that the Lorenz tensors for single VCS tensor of the massless quark are the same as those in the DVCS tensor of the proton. Since the VCS off the quark and that off the proton are subject to the same symmetry constraints, one can naturally conclude that both the OFPDs and double distributions possess some symmetry properties. These symmetry properties will impose some constraints as one attempts to model the OFPDs and double distributions. In Ji’s expansion [@ji1] of the DVCS tensor, a light-like momentum $p^\mu$ in connection with the average of the initial- and final-state proton momenta is introduced. Then, the momenta of the initial- and final-state protons are approximated as $(1+\xi )p$ and $(1-\xi )p$, respectively, where $\xi$ ($0<\xi <1$) is the analog of the Bjorken variable. Correspondingly, the momenta of the initial- and final-state partons participating in the hard single VCS are effectively taken as $(x+\xi )p$ and $(x-\xi )p$. One can easily show that two partonic Mandelstam variables are related to their hadronic counterparts via $$\begin{aligned} \hat s & \equiv & [q^\prime +(x-\xi )p]^2 \simeq \frac{x-\xi}{1-\xi} s \ , \\ \hat u & \equiv & [q^\prime -(x+\xi )p]^2\simeq \frac{x+\xi}{1+\xi} u \ . \end{aligned}$$ A DVCS form factor of the proton can be roughly thought of as the convolution of the corresponding parton form factor with an appropriate OFPD. The quark form factor is either symmetric or antisymmetric under $\hat s \leftrightarrow \hat u$, which amounts to $s\leftrightarrow u$ and $\xi \to -\xi$. Hence, the symmetry properties of the proton form factor demands that the OFPDs satisfy the following relations: $$\begin{aligned} H(x, \xi, \Delta^2)&= &H(x, -\xi, \Delta^2)\ , \label{01} \\ E(x, \xi, \Delta^2)&= & E(x, -\xi, \Delta^2)\ , \\ \tilde H(x, \xi, \Delta^2)&= &\tilde H(x, -\xi, \Delta^2)\ , \\ \tilde E(x, \xi, \Delta^2)&= &\tilde E(x, -\xi, \Delta^2)\ , \label{04} \end{aligned}$$ where $\Delta^2\equiv (P^\prime -P)^2$ is the Mandelstam variable $t$. In fact, the above relations can be derived from the definitions of these OFPDs directly by time reversal invariance. Further, one can show [@ji] that all of the OFPDs are real, with the help of Eqs. (\[01\]-\[04\]). Now let us consider Radyushkin’s expansion [@ra1] of the DVCS tensor, in which the momenta of the initial- and final-state partons are approximated as $xp +yr$ and $xp -\bar y r$ respectively, with $r\equiv P-P^\prime$ and $\bar y= 1-y$. Here two partonic Mandelstam variables read $$\begin{aligned} \hat s & \equiv & [q^\prime +xp-(1-y)r]^2 \simeq \bar y (s+u) - x u \ , \\ \hat u & \equiv & [q^\prime -xp-yr]^2\simeq -y (s +u) - x u \ . \end{aligned}$$ Obviously, $\hat s \leftrightarrow \hat u$ implies $y\to - \bar y$ and $\bar y \to -y$. Now, the nonperturbative physics is incorporated by two double distributions $F(x,y)$ and $G(x,y)$. Consequently, $F(x,y)$ and $G(x,y)$ must be invariant under the transformation $y\to - \bar y$ and $\bar y \to -y$. Here we recall that $F(x,y)$ is actually defined by the following leading-twist expansion of the proton matrix: $$\int \frac{d \lambda d \eta} {(2\pi)^2} e^{i \lambda (x+\zeta y) -i \eta (x-\bar y \zeta)} \langle P^\prime,S^\prime| \bar \psi (\lambda n) \gamma^\alpha \psi(\eta n)|P, S\rangle = \bar U(P^\prime,S^\prime) \gamma^\alpha U(P,S) F(x, y) +\cdots , \label{Fxy}$$ where $\zeta\equiv r\cdot n$ and $ n$ is a light-like vector with an inverse momentum dimension. Hence, we can effectively write down $$\begin{aligned} F(x,y)\equiv F(x;y,\bar y), \end{aligned}$$ That is to say, $y$ and $\bar y$ function in the double distributions as if they were two independent variables. To examine the symmetry properties of $F(x,y)$, one can put each of the protons in a helicity eigenstate. Then, by use of time reversal invariance, one can quickly show $$\begin{aligned} F(x;y,\bar y)&=&F(x; -\bar y \ , - y ) \ , \label{sy1}\end{aligned}$$ Similarly, there is $$\begin{aligned} G(x;y,\bar y)&=&G(x; -\bar y \ , - y ) \ . \label{sy2}\end{aligned}$$ Equations (\[sy1\]-\[sy2\]) are a useful guide as one parameterizes $F(x,y)$ and $G(x,y)$. In Ref. [@ra1], there was an observation that the double distributions are purely real in some toy models. In fact, this is generally true in QCD. To show this, just take the complex conjugate of Eq. (\[Fxy\]). There will be $F^\ast(x;y,\bar y)=F(x; -\bar y \ , - y )$. Combining this with Eq. (\[sy1\]), we know that $F(x,y)$ is real. The proof that $G(x,y)$ is real goes the same way. In closing, we remark the limitations of our form factor description of the Compton scattering tensor. It is applicable to the non-collinear Compton scattering, both real and virtual. As going to the collinear limits, however, it becomes pathological. The case is the worst as one attempts to discuss the transverse proton spin dependence of the Compton amplitude in the collinear limits. There is no remedy in our present scheme to parameterize the Compton scattering tensor in terms of form factors. In fact, the drawbacks of our decomposition are shared by all of the present Feynman diagram expansions and OPE analyses of the proton DVCS tensor. To develop a form factor description of the Compton scattering tensor suitable for taking the collinear limits, one can demand that the gamma matrices carry free Lorentz indices. At present, we have not seen any advantages in adopting such a scenario. The author thanks M. Anselmino, M. Diehl, M. Glück, T. Gousset, P.A.M. Guichon, Xiangdong Ji, Boqiang Ma, B. Pire, A. Radyushkin, J.P. Ralston and E. Reya for useful discussions and/or correspondence. In particular, he is grateful to M. Diehl and B. Pire for helping clarify the constraints of the crossing symmetry on the Compton scattering tensor. Table 1. Surviving independent helicity amplitudes of the various Compton scattering in the collinear scattering limits. Table 2. Crossing and parity-time-reversal transformation properties of the double VCS form factors. The plus and minus signs represent that the form factor is symmetric and antisymmetric, respectively. form factor $f_1$ $f_2$ $f_3$ $f_4$ $f_5$ $f_6$ $f_7$ $f_8$ $f_9$ $f_{10}$ $g_1$ $g_2$ $g_3$ $g_4$ $g_5$ $g_6$ $g_7$ $g_8$ ------------------------------------------------------------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ---------- ------- ------- ------- ------- ------- ------- ------- ------- $s \leftrightarrow u $ & $ q^2\leftrightarrow q^{\prime 2}$ $-$ $+$ $-$ $+$ $-$ $+$ + $ -$ $ -$ $+$ $-$ $+$ $ -$ $+$ $ -$ + + $ -$ $q^2 \leftrightarrow q^{\prime 2} $ $+$ $ +$ $+$ $ +$ $+$ $+$ $-$ $-$ $+$ $ +$ $-$ $+$ $+$ $ -$ $+$ $-$ $-$ $+$ $s \leftrightarrow u $ $-$ $+$ $-$ $ +$ $ -$ $+$ $-$ $+$ $-$ $ +$ $+$ $+$ $-$ $ -$ $ -$ $-$ $-$ $-$ X. Ji, Phys. Rev. Lett. [**78**]{}, 610 (1997); Phys. Rev. [**D**]{}53, 7114 (1997). A.V. Radyushkin, Phys. Lett. [**380B**]{}, 417 (1996); Phys. Lett. [**385B**]{}, 333 (1996); hep-ph/9704207 to appear in Phys. Rev. D. C. Hyde-Wright, Proceedings of the second ELFE workshop, Saint Malo, France, 1996, eds. N. d’Hose at al., to be published in Nucl. Phys. [**A**]{} (1997). Z. Chen, Columbia preprint CU-TP-835, hep-ph/9705279. M. Diehl, T. Gousset, B. Pire, and J. P. Ralston, hep-ph/9706344. K. Watanabe, Prog. Th. Phys. [**67**]{}, 1834 (1982). R.A. Berg and C.N. Lindner, Nucl. Phys. [**26**]{}, 259 (1961). P.A.M. Guichon, G.Q. Liu and A. W. Thomas, Nucl. Phys. [**A591**]{}, 606 (1995). D. Drechsel, G. Knöchlein, A. Yu. Korchin, A. Metz and S. Scherer, MKPH-T-97-11, nucl-th/9704064. P. Kroll, M. Schürmann and P.A.M. Guichon, Nucl. Phys. [**A598**]{}, 435 (1996). X. Ji and J. Osborne, hep-ph/9707254. X. Ji, private communication. [^1]: Unité propre 14 du Centre National de la Recherche Scientifique.

--- abstract: 'Primordial black holes can represent all or most of the dark matter in the window $10^{17}-10^{22}\,$g. Here we present an extension of the constraints on PBHs of masses $10^{13}-10^{18}\,$g arising from the isotropic diffuse gamma ray background. Primordial black holes evaporate by emitting Hawking radiation that should not exceed the observed background. Generalizing from monochromatic distributions of Schwarzschild black holes to extended mass functions of Kerr rotating black holes, we show that the lower part of this mass window can be closed for near-extremal black holes.' author: - Alexandre Arbey - Jérémy Auffinger - Joseph Silk bibliography: - 'biblio.bib' title: Constraining primordial black hole masses with the isotropic gamma ray background --- CERN-TH-2019-084 \[sec:intro\]Introduction ========================= Primordial Black Holes (PBHs) are the only candidate able to solve the Dark Matter (DM) issue without invoking new physics. Two mass windows are still open for the PBHs to contribute to all or most of the DM: the $10^{17} - 10^{19}\,$g range, recently re-opened by [@Katz2018] after revisiting the $\gamma$-ray femtolensing constraint, and the $10^{20}-10^{22}\,$g range [@Niikura2017], from HST microlensing probes of M31. PBHs are believed to have formed during the post-inflationary era, and subsequently evolved through accretion, mergers and Hawking Radiation (HR). If the PBHs were sufficiently numerous, that is to say if they contribute to a large fraction of DM, HR from PBHs may be the source of observable background radiation. In this Letter, we update the constraints on the number density of PBHs by observations of the diffuse Isotropic Gamma Ray Background (IGRB) [@Carr2010], taking into account the latest FERMI-LAT data and, as new constraints, the spin of PBHs and extension of the PBH mass function. Our assumption is that part of the IGRB comes from the time-stacked, redshifted HR produced by evaporating PBHs distributed isotropically in the extragalactic Universe. Those PBHs must have survived at least until the epoch of CMB transparency for the HR to be able to propagate in the intergalactic medium. This sets the lower boundary on the PBH mass $M{_{\rm min}} \approx 5\times10^{13}\,$g. Furthermore, the HR peaks at an energy which decreases when the PBH mass increases. This sets the upper boundary for the PBH mass $M{_{\rm max}} \approx 10^{18}\,$g as the IGRB emission does not constrain the photon flux below $100\,$keV. This Letter is organized as follows: Section \[sec:Hawking\] gives a brief reminder of HR physics, Section \[sec:IGRB\] describes the IGRB flux computation and Section \[sec:results\] presents the new constraints obtained with Kerr and extended mass function PBHs. \[sec:Hawking\]Kerr PBH Hawking radiation ========================================= Black Holes (BHs) emit radiation and particles similar to blackbody radiation [@Hawking1975] with a temperature linked to their mass $M$ and spin parameter $a \equiv J/M \in [0,M]$ ($J$ is the BH angular momentum) through $$T \equiv \dfrac{1}{2\pi}\left( \dfrac{r_+ - M}{r_+^2 + a^2} \right)\,, \label{eq:temperature}$$ where $r_+ \equiv M + \sqrt{M^2-a^2}$ and we have chosen a natural system of units with $G = \hbar = k{_{\rm B}} = c = 1$. The number of particles $N_i$ emitted per units of energy and time is given by $$\dfrac{{{\rm d}}^2N_i}{{{\rm d}}t{{\rm d}}E} = \dfrac{1}{2\pi}\sum_{\rm dof} \dfrac{\Gamma_i(E,M,a^*)}{e^{E^\prime/T}\pm 1}\,, \label{eq:hawking}$$ where $E^\prime \equiv E - m\Omega$ is the total energy of the particle taking into account the BH horizon rotation velocity $\Omega \equiv a^*/(2r_+)$, $a^* \equiv a/M \in [0,1]$ is the reduced spin parameter, $m$ is the projection of the particle angular momentum $l$ and the sum is over the degrees of freedom (dof) of the particle (color and helicity multiplicities). The $\pm$ signs are for fermions and bosons, respectively. The greybody factor $\Gamma_i(E,M,a^*)$ encodes the probability that a Hawking particle evades the gravitational well of the BH. This emission can be integrated over all energies to obtain equations for the evolution of both PBH mass and spin [@PageII1976] $$\dfrac{{{\rm d}}M}{{{\rm d}}t} = -\dfrac{f(M,a^*)}{M^2}\,, \label{eq:diffM}$$ and $$\dfrac{{{\rm d}}a^*}{{{\rm d}}t} = \dfrac{a^*(2f(M,a^*) - g(M,a^*))}{M^3}\,, \label{eq:diffa}$$ where $$\begin{aligned} f(M,a^*) &\equiv -M^2 \dfrac{{{\rm d}}M}{{{\rm d}}t}\label{eq:fM} \\ &= M^2\int_{0}^{+\infty} \sum_{\rm dof} \dfrac{E}{2\pi}\dfrac{\Gamma(E,M,a^*)}{e^{E^\prime/T}\pm 1} {{\rm d}}E \,, \nonumber \end{aligned}$$ $$\begin{aligned} g(M,a^*) &\equiv -\dfrac{M}{a^*} \dfrac{{{\rm d}}J}{{{\rm d}}t}\label{eq:gM} \\ &= \dfrac{M}{a^*}\int_{0}^{+\infty} \sum_{\rm dof}\dfrac{m}{2\pi} \dfrac{\Gamma(E,M,a^*)}{e^{E^\prime/T}\pm 1}{{\rm d}}E \,. \nonumber \end{aligned}$$ There are two main effects coming from the PBH spin that play a role in the IGRB. Firstly, a Kerr PBH with a near-extremal spin $a^* \lesssim 1$ radiates more photons than a Schwarzschild one ($a^* = 0$). This is due to the coupling between the PBH rotation and the particle angular momentum for high-spin particles [@Chandra4]. We thus expect the constraints to be more stringent. Secondly, a near-extremal Kerr PBH will evaporate faster than a Schwarzschild PBH with the same initial mass due to this enhanced HR [@Taylor1998]. Hence, we expect that the constraints will be shifted toward higher PBH masses when the reduced spin parameter $a^*$ increases. \[sec:IGRB\]Isotropic Gamma Ray Background ========================================== Many objects in the Universe produce gamma rays, such as Active Galactic Nuclei (AGN) and gamma ray bursts [@Ackermann2015]. The IGRB is the diffuse radiation that fills the intergalactic medium once all point-sources have been identified and removed from the measured photon flux. This background might come from unresolved sources, or more speculatively from DM decays or annihilations. [Fig. \[fig:data\_IGRB\]]{} shows the IGRB measured by four experiments (HEAO1+balloon, COMPTEL, EGRET and FERMI-LAT) over a wide range of energies between 100 keV and 820 GeV. If we consider the simplifying hypothesis that DM is distributed isotropically at sufficiently large scales, then its annihilations/decays should produce, at each epoch of the Universe since transparency, an isotropic flux of photons. Thus, the flux measured along some line of sight should be the redshifted sum over all epoch emissions. Following Carr [*et al.*]{} [@Carr2010], we estimate the flux at energy $E$ to be $$\begin{aligned} I &\equiv E \dfrac{{{\rm d}}F}{{{\rm d}}E} \label{eq:flux_IGRB} \\ &\approx \dfrac{1}{4\pi} n{_{\rm BH}}(t_0) E \int_{t{_{\rm min}}}^{t{_{\rm max}}} (1+z(t)) \dfrac{{{\rm d}}^2 N}{{{\rm d}}t{{\rm d}}E}((1+z(t))E){{\rm d}}t\,, \nonumber\end{aligned}$$ where $n{_{\rm PBH}}(t_0)$ is the number density of PBHs of a given mass $M$ today, $z(t)$ is the redshift and the time integral runs from $t{_{\rm min}} = 380\,000\,$years at last scattering of the CMB to $t{_{\rm max}} = {\rm Max}(\tau(M),t_0)$ where $\tau(M)\sim M^3$ is the PBH lifetime and $t_0$ is the age of the Universe. As the Universe is expanding, the number density of PBHs evolves as $(1+z(t))^{-3}$, and the energy of the emitted photons evolves as $(1+z(t))^{-1}$. A last factor $(1+z(t))$ comes from the change of integrand variable from the line of sight to the present time. \[sec:results\]Results ====================== We have used the new public code `BlackHawk` [@BlackHawk] to compute the HR of [Eq. (\[eq:hawking\])]{} and the PBH evolution given by Eqs.  and . We consider monochromatic PBH distributions of masses comprised between $M{_{\rm min}} = 10^{13}\,$g and $M{_{\rm max}} = 10^{18}\,$g and initial spin parameters between $a_{i{\rm,min}}^* = 0$ and $a_{i{\rm,max}}^* = 0.9999$, and compute the integral of [Eq. (\[eq:flux\_IGRB\])]{} over the redshift (matter-dominated era) $$z(t) = \left( \dfrac{1}{H_0 t} \right)^{2/3} -1\,, \label{eq:redshift}$$ where $H_0$ is the present Hubble parameter. We then compare the result of the integral to the measured IGRB and find the maximum allowed value of the present PBH number density $n{_{\rm PBH}}(t_0)$ at a given PBH mass $M$, with a conservative approach taking into account the most stringent constraints (e.g. FERMI-LAT$_{\rm C}$ at $E = 1\,$GeV). The corresponding limit on the DM fraction $f$ constituted of PBHs of mass $M$ is obtained through $ n{_{\rm PBH}}(t_0) = f{\rho{_{\rm DM}}}/{M}\,, $ where $\rho{_{\rm DM}} \approx 0.264\times\rho{_{\rm tot}} \approx 2.65\times 10^{-30}\,$g$\cdot$cm$^{-3}$ is the current average DM density in the Universe [@Planck2018]. If the maximum allowed fraction $f$ is greater than 1, we set it to $1$ in order not to exceed the observed DM density, meaning that the IGRB does not constrain $f$ for the given PBH mass. Monochromatic PBH distribution ------------------------------ [Fig. \[fig:results\_mono\]]{} shows the resulting constraints for the DM fraction $f$ in PBHs of mass $M_*$ for initial spins $a^*_i \in \{0,0.9,0.9999\}$. First, we see that the $a^*_i = 0$ constraints are comparable with those of [@Carr2010]. Our results do not present the feature just after the peak linked to primary/secondary photons domination explained in this article because we compute the secondary spectrum for all PBH masses. As a consequence, the peak is smoothed out. We see the second effect anticipated in Section \[sec:Hawking\], that is to say the shifting of the constraint toward higher masses as the initial PBH spin parameter $a_i^*$ increases. This is due to the fact that Kerr PBHs with high initial spin evaporate faster. Thus, in order to have the same kind of HR time-distribution as a Schwarzschild PBH, the PBH must have a higher initial mass. However, this is not accompanied with a more stringent constraint linked to the enhanced emission for Kerr PBHs. We understand this as follows: PBHs with a higher mass emit photons at lower energies (cf. the temperature-mass relation [Eq. (\[eq:temperature\])]{}) where the IGRB constraints are less severe. The two effects approximately cancel. The main result that we find is that if PBHs have a high initial spin parameter $a^*\lesssim 1$, the “small-mass" window $10^{17}-10^{19}\,$g can be closed up to almost one order of magnitude on its lower boundary. For the possible existence of such high spins, see for example [@ExtremalSpins]. Extended PBH distribution {#sect:extended} ------------------------- We also obtained constraints for extended mass functions to study the effects related to the width of the peak. Some pioneering work has been done in [@Carr2017; @Kuhnel2017; @Bellomo2018; @Lehmann2018] concerning extended mass functions, predicting that the constraints on an extended distribution should be more stringent than the expected constraint resulting from the addition of monochromatic distributions. The conclusion of these papers is that a simple conversion from monochromatic to extended mass functions is not analytically trivial. We thus derive the extended mass function constraints by computing the full Hawking spectra associated to them before applying the constraints. We considered extended mass functions of log-normal form $$\dfrac{{{\rm d}}n}{{{\rm d}}M} = \frac{A}{\sqrt{2\pi}\sigma M}\exp\left( -\dfrac{(\ln(M/M_*))^2}{2\sigma^2} \right)\,, \label{eq:ext_dis}$$ [*i.e.*]{} a Gaussian distribution in logarithmic scale for the density. $A$ is some amplitude, linked to the fraction of DM into PBHs. This distribution is normalized for $A = 1$. To compute the spectra, `BlackHawk_tot` [@BlackHawk] was used with $\texttt{spectrum\_choice} = 5$, and 10 different PBH masses scanning the whole peak width. We do not assume any model of PBH formation to justify this distribution, which is based on the fact that a Gaussian peak can mimic any peak in the PBH distribution resulting from a particular mechanism of formation, but we note that this mass distribution – with some variations – has been used in various works linked to different PBH formation mechanisms and mass ranges [@Green2016; @Kannike2017; @Calcino2018; @Boudaud2019; @DeRocco2019; @Laha2019]. We have done a similar scan to the one described in the previous section, with $M_*$ the mean of the Gaussian distribution ranging from $10^{13}\,$g to $10^{18}\,$g (cf. the Introduction for the PBH mass bounds), and its width $\sigma \in \{0.1, 0.5, 1\}$. [Fig. \[fig:ext\_dis\]]{} shows examples of these distributions for $M_* = 3\times10^{15}\,$g. [Eq. (\[eq:flux\_IGRB\])]{} must be modified to obtain the fraction for an extended mass function. The flux is now given by $$\begin{aligned} I &\approx \dfrac{1}{4\pi} E \int_{t{_{\rm min}}}^{t{_{\rm max}}} (1+z(t)) \dfrac{{{\rm d}}^2 n}{{{\rm d}}t{{\rm d}}E}((1+z(t))E){{\rm d}}t \label{eq:flux_IGRB_ext} \\ &\approx \dfrac{1}{4\pi} E \int_{t{_{\rm min}}}^{t{_{\rm max}}} (1+z(t)) \nonumber \\ &\times \int_{M{_{\rm min}}}^{M{_{\rm max}}}\left[\dfrac{{{\rm d}}n}{{{\rm d}}M}\dfrac{{{\rm d}}^2 N}{{{\rm d}}t{{\rm d}}E}(M,(1+z(t))E)\,{{\rm d}}M\right]{{\rm d}}t\,. \nonumber\end{aligned}$$ with ${{\rm d}}n/{{\rm d}}M$ given by [Eq. (\[eq:ext\_dis\])]{}. The fraction of DM in form of PBHs is obtained by maximizing this flux (increasing the normalization constant $A$) while respecting all the IGRB constraints, and given by $$\begin{aligned} f &\equiv \dfrac{\rho{_{\rm PBH}}}{\rho{_{\rm DM}}} \\ &= \dfrac{A}{\rho{_{\rm DM}}\sqrt{2\pi}\sigma} \int_{M{_{\rm min}}}^{M{_{\rm max}}} \exp\left( -\dfrac{\log(M/M_*)^2}{2\sigma^2} \right){{\rm d}}M\,.\end{aligned}$$ It is again limited to 1 in order not to exceed the DM content of the universe. Even if the IGRB constraints valid at $M_*$ prevent $A$ from exceeding its maximum value when $\sigma \rightarrow 0$ (monochromatic distribution), we expect that when the distribution width $\sigma$ increases, monochromatic IGRB constraints from $M \lesssim M_*$ and $M\gtrsim M_*$ will become more and more important, thus limiting $A$. On the other hand, if $\sigma$ increases, the full distribution integral that contributes to the DM fraction $f$ increases as well because of the $M\lesssim M_*$ and $M\gtrsim M_*$ contributions. The competition between the two effects is difficult to forecast. [Fig. \[fig:results\]]{} (panels $b$, $c$ and $d$) shows the constraints for distribution widths $\sigma \in \{0.1,0.5,1\}$ (respectively) and $a^*_i \in \{0,0.9,0.9999\}$. There are 3 kinds of observations to be considered. 1\) For a fixed PBH initial spin $a^*$, when the width of the distribution $\sigma$ increases, the excluded region widens. This effect is sensible when $\sigma \gtrsim 0.5$. Indeed, for a sharp distribution, the IGRB constraints that play a role in limiting $f$ are those close to the central mass $M_*$. When the distribution gets wider, constraints from masses far from the central mass are important. As the constraints are the most severe for $M{_{\rm peak}}\lesssim 10^{15}\,$g, wide distributions centered on $M_* \ll M{_{\rm peak}}$ and $M_* \gg M{_{\rm peak}}$, which have a tail reaching the peak mass, are severely constrained. This extends the excluded region to $M \ll M{_{\rm peak}}$ and $M \gg M{_{\rm peak}}$ and closes the $10^{17} - 10^{18}\,$g window for all DM made of PBHs. 2\) For a fixed PBH initial spin $a^*$, when the width of the distribution $\sigma$ increases, the constraint on $f$ close to the peak $M{_{\rm peak}}$ decreases. This is due to the fact that the amplitude $A$ of the mass distribution is most severely constrained by the $M\approx M{_{\rm peak}}$ contribution. If we extend the mass distribution around $M{_{\rm peak}}$, we do not add new strong IGRB constraints, but we increase the mass fraction $f$ of DM into PBHs. 3\) For a fixed width of the distribution $\sigma$, when the initial spin $a^*$ of the PBHs increases, the constraints are shifted toward higher central masses while being slightly more stringent. This is coherent with the results of [Fig. \[fig:results\]]{} (panel $a$) for the monochromatic distributions. We can sum up these observations in the following way. For an extended PBH mass function, the overall constraint comes from the PBHs evaporating today in this distribution with initial mass $M = M{_{\rm peak}}$. Distributions centered away from $M{_{\rm peak}}$ are more and more constrained as the tail of the distribution is important at $M{_{\rm peak}}$: $f$ decreases as $\sigma$ increases because the maximum value of $A$ decreases. Distributions centered close to $M{_{\rm peak}}$ are not much more constrained when the distribution expands, the maximum value of $A$ remains quite the same: $f$ increases as $\sigma$ increases because the distribution integral increases. The very same effects can be observed on the right panel of Fig. 2 of [@Boudaud2019]. Comparison to other constraints ------------------------------- Recent works have tried to close the very same mass range for PBHs constituting all of the dark matter, showing that this scenario is attracting much attention. Future femtolensing [@Katz2018] or X-ray [@Ballestros2019] surveys as well as galactic positrons data [@Boudaud2019; @DeRocco2019; @Laha2019] all constrain the same $M = 10^{16} - 10^{18}\,$g mass range. [Fig. \[fig:comparison\]]{} compares the limits derived here and those based on galactic positrons [@Boudaud2019; @DeRocco2019] (we do not include the extended mass function limits of [@Laha2019] because they result from a convolution of monochromatic limits with the mass function, a method which was decried – see Section \[sect:extended\]). The limits obtained with a local measurement of the positron flux by Voyager 1, which has recently leaved the heliosphere and is capable of detecting low-energy positrons, are somewhat of the same order of magnitude as ours for widths $\sigma \lesssim 0.5$, while becoming significantly more stringent for $\sigma \gtrsim 1$. The limits derived from electron-positron annihilation in the galactic bulge – thus contributing to the $511\,$keV photon line – are more severe than the 2 others in the central mass region $M \sim 10^{17}\,$g, but the authors claim that the Voyager limits are more restrictive when $\sigma \gtrsim 1$. As those limits come from totally different galactic and extra-galactic measurements, we consider them as most interestingly independent and complementary, increasing the robustness of the conclusion concerning PBHs not constituting all of the DM in the $10^{16} - 10^{17}\,$g mass range. Conclusion ========== In this Letter, we have updated the IGRB constraint on PBH evaporation for monochromatic Schwarzschild PBH distributions, using the latest FERMI-LAT data and the new code `BlackHawk`. This has resulted in enhancing the constraint on the masses of presently evaporating PBHs, and reducing the constraint on $M{_{\rm peak}} \lesssim 10^{15}\,$g. Our main result is the extension of the IGRB constraint from Schwarzschild to Kerr PBHs, and from monochromatic to extended mass functions. We have shown that increasing the initial spin parameter $a_i^*$ of PBHs to near extremal values can close the mass window $10^{17} - 10^{19}\,$g (where PBHs could still represent all of the DM). We have also demonstrated that extended mass functions can allow a greater fraction of DM in the form of PBHs when they are centered close to the strongest monochromatic constraint, while they are more severely constrained when centered away from this peak. In this case, the allowed mass window can be reduced even with Schwarzschild PBHs, complementing previous work in the same mass range with positron emission by evaporating PBHs. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank P. Graham and W. DeRocco for useful private discussion and for pointing out an error (corrected).

--- abstract: 'The first generation quantum computer will be implemented in the cloud style, since only few groups will be able to access such an expensive and high-maintenance machine. How the privacy of the client can be protected in such a cloud quantum computing? It was theoretically shown \[A. Broadbent, J. F. Fitzsimons, and E. Kashefi, Proceedings of the 50th Annual IEEE Symposium on Foundation of Computer Science, 517 (2009)\], and experimentally demonstrated \[S. Barz, E. Kashefi, A. Broadbent, J. F. Fitzsimons, A. Zeilinger, and P. Walther, Science [**335**]{}, 303 (2012)\] that a client who can generate randomly-rotated single qubit states can delegate her quantum computing to a remote quantum server without leaking any privacy. The generation of a single qubit state is not too much burden for the client, and therefore we can say that “almost classical client" can enjoy the secure cloud quantum computing. However, isn’t is possible to realize a secure cloud quantum computing for a client who is completely free from any quantum technology? Here we show that perfectly-secure cloud quantum computing is impossible for a completely classical client unless classical computing can simulate quantum computing, or a breakthrough is brought in classical cryptography.' author: - Tomoyuki Morimae - Takeshi Koshiba title: Impossibility of secure cloud quantum computing for classical client --- Introduction ============ Imagine that Alice who does not have any sophisticated quantum technology wants to factor a large integer. She has a rich friend, Bob, who has a full-fledged quantum computer. Alice asks Bob to perform her quantum computing on his quantum computer. However, the problem is that Bob is not a reliable person, and therefore she does not want to reveal her input (the large integer), output (a prime factor), and the program (Shor’s algorithm), to Bob. Can she delegate her quantum computing to Bob while keeping her privacy? Recently, it was theoretically shown that such a secure cloud quantum computing is indeed possible [@BFK]. (A proof-of-principle experiment was also demonstrated with photonic qubits [@Barz].) In the protocol of Ref. [@BFK] (Fig. \[fig1\]), Alice, a client, has a device that emits randomly rotated single qubit states. She sends these states to Bob, the server, who has the full quantum technology. Alice and Bob are also connected with a classical channel. Bob performs quantum computing by using qubits sent from Alice, and classical messages exchanging with Alice via the classical channel. After finishing his quantum computation, Bob sends the output of his computation, which is a classical message, to Alice. This message encrypts the result of Alice’s quantum computing, which is not accessible to Bob. Alice decrypts the message, and obtains the desired result of her quantum computing. It was shown that whatever Bob does, he cannot learn anything about the input, the program, and the output of Alice’s computation [@BFK; @Vedrancomposability] (except for some unavoidable leakage, such as the upperbound of the input size, etc.). ![ The secure cloud quantum computing protocol proposed in Ref. [@BFK]. Alice possesses a device that emits randomly-rotated single-qubit states. Bob has a universal quantum computer. Alice and Bob share a two-way classical channel. []{data-label="fig1"}](BFK.eps){width="40.00000%"} In this protocol, the client has to possess a device that generates single qubit states. Generation of single qubit states is ubiquitous in today’s laboratories, and therefore not too much burden for the client. In other words, “almost classical" client can enjoy secure cloud quantum computing. However, isn’t it possible to realize secure cloud quantum computing for a completely classical client (Fig. \[classical\])? Many variant protocols of secure cloud quantum computing have been proposed recently [@MABQC; @BarzNP; @FK; @Vedran; @AKLTblind; @topoblind; @CVblind; @Lorenzo; @Joe_intern; @Sueki; @tri]. For example, it was shown that, in stead of single-qubit states, the client has only to generate weak coherent pulse states if we add more burden to the server [@Vedran]. Coherent states are considered as “more classical" than single-photon states, and therefore it enables secure cloud quantum computing for “more classical" client. It was also shown that secure cloud quantum computing is possible for a client who can only measure states [@MABQC] (Fig. \[measuringAlice\]). A measurement of a bulk state with a threshold detector is sometimes much easier than the single-photon generation, and therefore the protocol also enables “more classical" client. However, these two protocols still require the client to have some minimum quantum technologies, namely the generation of weak coherent pulses or measurements of quantum states. In fact, all protocols proposed so far require the client to have some quantum ability, such as generation, measurement, or routing of quantum states [@MABQC; @BarzNP; @FK; @Vedran; @AKLTblind; @topoblind; @CVblind; @Lorenzo; @Joe_intern; @Sueki; @tri]. (It is known that [@BFK] if we have two quantum servers, a completely classical client can delegate her quantum computing. However, in this case, we have to assume that two servers cannot communicate with each other.) In other words, the possibility of the perfectly secure cloud quantum computing for a completely classical client has been an open problem. ![ The secure cloud quantum computing for a classical client. Alice has only a classical computer, whereas Bob has a universal quantum computer. Alice and Bob share a two-way classical channel. []{data-label="classical"}](classical.eps){width="40.00000%"} ![ The secure cloud quantum computing protocol proposed in Ref. [@MABQC]. Alice possesses a device that measure qubits. Bob has the ability of entangling operations and quantum memory. []{data-label="measuringAlice"}](measuringAlice.eps){width="40.00000%"} In this paper, we show that perfectly-secure cloud quantum computing for a completely classical client is unlikely possible. Here, the perfect security means that an encrypted text gives no information about the plain text [@nonlinearcrypto]. It is a typical security notion in the information theoretical security. The idea of the proof is as follows. Since no non-affine cryptography is known to be perfectly secure [@nonlinearcrypto], we assume that the client uses an affine cryptography. We then show that if the cloud quantum computing can be done in the perfectly secure way for a completely classical client, classical computing can efficiently simulate quantum computing. Although the conjecture of $\mbox{BPP}\subsetneq\mbox{BQP}$ is not so solid as $\mbox{P}\ne\mbox{NP}$ or that the polynomial hierarchy does not collapse, researchers in quantum computing believe $\mbox{BPP}\subsetneq\mbox{BQP}$. Therefore, we conclude that perfectly-secure cloud quantum computing is impossible for a completely classical client (unless a non-affine cryptography is shown to be perfectly secure or classical computing can efficiently simulate quantum computing). Result ====== Our setup is given in Fig. \[classical\]. Alice has only a classical computer (more precisely, the probabilistic polynomial time Turing machine), whereas Bob has a universal quantum computer. Furthermore, Alice and Bob share a two-way classical channel. Let $U$ be the $n$-qubit unitary operator that Alice wants to implement in her quantum computing, where $n$ is a polynomial of the size of the input of her problem. (More precisely, she choses a unitary from the finite set $\{U_j\}_{j=1}^r$ of unitaries, since the capacity of the classical channel between Alice and Bob is finite, and a set of finite unitaries is sufficient for universal quantum computing.) Without loss of generality, we can assume that the initial state of her quantum computing is the standard state $|0\rangle^{\otimes n}$. In other words, if she wants to start with a certain input state $|\psi\rangle$, the preparation of it is included in $U$. (In the secure cloud quantum computing protocol of Ref. [@BFK], Alice can use unknown quantum state as the input, such as a given state from another person. However, in the present setup, by definition, Alice’s input is restricted to classical information. Therefore we assume that the input state is a standard one or she knows the classical description of the input quantum state.) Therefore, what Alice wants to hide from Bob are the classical description $[U]$ of the unitary $U$, and the output of the computation, which is the computational basis measurement result on $U|0\rangle^{\otimes n}$. (The protocol of Ref. [@BFK] allows Alice to finally obtain an output quantum state. However, again, we assume that Alice’s output is a classical information, since she is completely classical.) In the setup of Fig. \[classical\], what Alice and Bob can do is the following protocol. - Alice sends Bob the classical message $$\begin{aligned} a=E([U],k)\end{aligned}$$ that encrypts the classical description $[U]$ of the unitary $U$ with the private key $k\in K$, where $K$ is the set of keys and $E$ is an encrypting operation. Since the encryption is done by Alice, we assume that the key generation and the encrypting operation $E$ can be done with a classical computer in $poly(n)$ time. We also assume that the encryption operation $E$ is an affine one. - Bob performs quantum computing by using $a$, and obtains the output $$\begin{aligned} b=Q(a)\end{aligned}$$ of his quantum computation, which is a classical message. Here, $Q$ is an operation that can be done with a quantum computer in $poly(n)$ time. Bob sends $b$ to Alice. The message $b$ is an encrypted text of her quantum computing result, i.e., the measurement result on $U|0\rangle^{\otimes n}$. - Alice decrypts $b$ by calculating $$\begin{aligned} c=D([U],k,b), \end{aligned}$$ where $D$ is the decrypting operation. Again, we assume that the decrypting operation $D$ can be done with a classical computer in $poly(n)$ time, since the decryption is done by Alice. The value $c$ is the desired result of her quantum computing, i.e., the result of the measurement on $U|0\rangle^{\otimes n}$. Now we show that if the above protocol can be done in the secure way, classical computing can efficiently simulate quantum computing by using a similar argument of Ref. [@inspire]. Here, secure means the perfect security, i.e., Bob cannot gain any information from $a$ about the classical description $[U]$ of the unitary $U$ (except for some necessarily leaking information such as the maximum size of $[U]$, etc.). Let us assume that in the above protocol Alice wants to delegate a unitary $U$. Let us define $$\begin{aligned} \xi\equiv E([I^{\otimes n}],k_0) \end{aligned}$$ for a certain $k_0\in K$, where $I\equiv|0\rangle\langle0|+|1\rangle\langle1|$ is the two-dimensional identity operator. For any unitary $U$, there must exist a key $k_U\in K$ such that $$\begin{aligned} E([U],k_U)=\xi, \label{key}\end{aligned}$$ since otherwise Bob can learn that Alice’s computation is not $U$ when he receives $\xi$ from Alice. This means that Bob can gain some information about Alice’s unitary, which contradicts to the assumption of the perfect security [@nonlinearcrypto]. Since we assume that $E$ is an affine encryption, Alice can calculate $k_U$ which satisfies Eq. (\[key\]) by herself for any $U$ [@inversion]. Furthermore, for the fixed value $$\begin{aligned} Q(\xi)=Q(E([I^{\otimes n}],k_0)),\end{aligned}$$ Alice can efficiently calculate $$\begin{aligned} D([U],k_U,Q(\xi))\end{aligned}$$ by herself by assumption. However, since $$\begin{aligned} D([U],k_U,Q(\xi)) =D([U],k_U,Q(E([U],k_U))),\end{aligned}$$ this means that Alice can efficiently calculate $$\begin{aligned} D([U],k_U,Q(E([U],k_U)))\end{aligned}$$ by herself, which is the result of the quantum computing $U$, i.e., the result of the measurement on $U|0\rangle^{\otimes n}$. This means that classical computing can efficiently simulate quantum computing. Therefore, if the above protocol can be done in the perfect secure way, classical computing can efficiently simulate quantum computing, which shows our claim. Discussion ========== In this paper, we have shown that perfectly-secure cloud quantum computing is impossible for a completely classical client unless a non-affine cryptography is shown to be perfectly secure or classical computing can efficiently simulate quantum computing. We think a majority of researchers in quantum information believe quantum computing is more powerful than classical computing. Furthermore, if classical computing could efficiently simulate quantum computing, a classical client can achieve perfectly secure cloud quantum computing in the following trivial way, and therefore our question, namely the possibility of the perfect-secure cloud quantum computing for a classical client, is trivialized. - The client encrypts a message which contains information about client’s desired quantum computation, and sends it to the server via the classical channel. - The server returns the message to the client without doing anything. - The client decrypts the message, and does the quantum computing by herself. At this moment, we do not know whether a perfectly secure non-affine cryptography is possible or not. Our result therefore does not exclude the possibility that in a future a perfectly-secure non-affine cryptography is found, and the non-affine cryptography enables perfectly-secure cloud quantum computing for a completely classical client in some clever way. We also note that if we relax the requirement of the perfect security, there might be several possibilities of secure cloud quantum computing for a classical client. For example, we require not the perfect security but only the computation-theoretical security, fully-homomorphic encryption scheme [@Gentry] would be able to achieve secure cloud quantum computing. TM is supported by the Tenure Track System by MEXT Japan and KAKENHI 26730003 by JSPS. TK is supported by KAKENHI 26540002, 24106008, 24240001, 23246071 by JSPS. [00]{} A. Broadbent, J. F. Fitzsimons, and E. Kashefi, Universal blind quantum computation. Proc. of the 50th Annual IEEE Sympo. on Found. of Comput. Sci. 517 (2009). S. Barz, E. Kashefi, A. Broadbent, J. F. Fitzsimons, A. Zeilinger, and P. Walther, Demonstration of blind quantum computing. Science [**335**]{}, 303 (2012). V. Dunjko, J. F. Fitzsimons, C. Portmann, and R. Renner, Composable security of delegated quantum computation. arXiv:1301.3662 T. Morimae and K. Fujii, Blind quantum computation for Alice who does only measurements. Phys. Rev. A [**87**]{}, 050301(R) (2013). S. Barz, J. F. Fitzsimons, E. Kashefi, and P. Walther, Experimental verification of quantum computation. Nature Phys. [**9**]{}, 727 (2013). J. F. Fitzsimons and E. Kashefi, Unconditionally verifiable blind computation. arXiv:1203.5217. V. Dunjko, E. Kashefi, and A. Leverrier, Blind quantum computing with weak coherent pulses. Phys. Rev. Lett. [**108**]{}, 200502 (2012). T. Morimae, V. Dunjko, and E. Kashefi, Ground state blind quantum computation on AKLT state. arXiv:1009.3486. T. Morimae and K. Fujii, Blind topological measurement-based quantum computation. Nature Communications [**3**]{}, 1036 (2012). T. Morimae, Continuous-variable blind quantum computation. Phys. Rev. Lett. [**109**]{}, 230502 (2012). V. Giovannetti, L. Maccone, T. Morimae, and T. G. Rudolph, Efficient universal blind computation. Phys. Rev. Lett. [**111**]{}, 230501 (2013). A. Mantri, C. Pérez-Delgado, and J. F. Fitzsimons, Optimal blind quantum computation. Phys. Rev. Lett. [**111**]{}, 230502 (2013). Q. Li, W. H. Chan, C. Wu, and Z. Wen, Triple-server blind quantum computation using entanglement swapping. Phys. Rev. A [**89**]{}, 040302(R) (2014). T. Sueki, T. Koshiba, and T. Morimae, Ancilla-driven universal blind quantum computation. Phys. Rev. A [**87**]{}, 060301(R) (2013). T. Morimae and K. Fujii, Secure entanglement distillation for double-server blind quantum computation. Phys. Rev. Lett. [**111**]{}, 020502 (2013). D. R. Stinson, [*Cryptography: Theory and Practice*]{}, (Chapman `&` Hall / CRC, 2006). V. Dunjko, T. Kapourniotis, and E. Kashefi, arXiv:1405.4558 An affine transformation is a linear transformation plus a parallel translation. A candidate of the vector $v(k_U)$ corresponding to a key $k_U$ that satisfies $E([U],k_U)=\xi$ is $v(k_U)=v(\xi)-v([U])$, where $v([U])$ and $v(\xi)$ are vectors corresponding to $[U]$ and $\xi$, respectively. The vector (key) $v(k_U)$ is a valid key, since a parallel translation is an invertible affine transformation. (An affine transformation is invertible if and only if the linear transformation is invertible. In this case, the linear transformation is the identity operation, and therefore invertible.) 0 [ Any affine transformation can be written as a linear transformation by using augmented matrix. A pesude-inverse matrix of the linear transformation can be calculated efficiently by calculating the singular value decomposition of it. (Calculating a singular value decomposition can be done in a polynomial time of the matrix size). As we have mentioned, there always exists $k_U$ that satisfies $E([U],k_U)=\xi$ for any $[U]$ and $\xi$. Therefore, the “solution" obtained by the pseudo-inverse matrix, which gives the minimum distance $|E([U],k_U)-\xi|$, gives $|E([U],k_U)-\xi|=0$, i.e., the solution is exact. ]{} C. Gentry, Fully homomorphic encryption using ideal lattices, Symposium on the Theory of Computing (STOC) pp.169 (2009).

--- author: - | Marwa Hadj SalahDidier Schwab Hervé Blanchon Mounir Zrigui\ [ (1) LIG-GETALP, Univ. Grenoble Alpes, France\ `Prénom.Nom@univ-grenoble-alpes.fr ` (2) LaTICE, Tunis, 1008, Tunisie\ `Prénom.Nom@fsm.rnu.tn` ]{} bibliography: - 'biblio.bib' title: 'Système de traduction automatique statistique Anglais-Arabe' --- Introduction ============ La traduction automatique (TA) est le processus qui consiste à traduire un texte rédigé dans une langue source vers un texte dans une langue cible. Dans cet article, nous présentons notre système de traduction automatique statistique anglais-arabe. Dans un premier temps, nous présentons le processus général pour mettre en place un système de traduction automatique statistique, ensuite nous décrivons les outils ainsi que les différents corpus que nous avons utilisés pour construire notre système de TA. Traduction automatique ====================== Traduction automatique statistique ----------------------------------- La traduction automatique statistique (TAS) est une approche très utilisée dans la TA et qui se base sur l’apprentissage de modèles statistiques à partir de corpus parallèles. En effet, comme il est montré dans la figure \[FigureTA\], la traduction automatique statistique se base essentiellement sur: Un modèle de langage (ML), un modèle de traduction (MT) et un décodeur. ![Processus de la traduction automatique statistique[]{data-label="FigureTA"}](smt.png) ### Modèle de langage Parmi les modèles de langages utilisés dans les systèmes de TAS les principaux sont le modèle n-gramme, le modèle Cache [@kuhn1990cache] et le modèle Trigger [@lau1993trigger]. Le modèle Cache repose sur les dépendances des mots non contigus. Quant à lui, le modèle Trigger consiste à déterminer le couple de mots (X, Y) où la présence de X dans l’historique déclenche l’apparition de Y. Toutefois, le modèle n-gramme (1$\leq$n$\leq$5) reste le plus utilisé dans les systèmes de traduction actuels et plus précisément le modèle trigramme ( -gramme pour le traitement des langues européennes. En effet, le modèle n-gramme permet d’estimer la vraisemblance d’une suite de mots en lui attribuant une probabilité. Soit $\textit{t} = w_{1}w_{2} . . . w_{k}$ une séquence de k mots dans une langue donnée et n la taille maximale des n-gramme (1$\leq$n$\leq$5, la formule de p(t est exprimée en : $$P(t)=\prod_{i=1}^{k} (w_{i}|w_{i-1}w_{i-2} ... w_{i-n+1})$$ ### Modèle de traduction à base de segments Pour construire un modèle de traduction à base de segments [@och2003systematic] , il est nécessaire de passer par trois étapes indispensables: - Segmentation de la phrase en séquences de mots - Traduction des séquences de mots en se fondant sur la table de traduction - Ré-ordonnancement des séquences de mots à l’aide d’un modèle de distorsion ### Décodeur Moses [@koehn2007moses] est une boite à outils disponible sous licence libre GPL, basée sur des approches statistiques de la traduction automatique. En effet, Moses nous permet de développer et manipuler un système de traduction selon nos besoins grâce à ses nombreuses caractéristiques, telle que la production du modèle de traduction et le modèle de réordonnance à partir des corpus volumineux.\ Parmi les principaux modules du Moses, on trouve : - **Train** : permet de construire des modèles de traduction ainsi que des modèles de réordonnance. - **Mert** : permet d’ajuster les poids des différents modèles afin d’optimiser et maximiser la qualité de traduction en utilisant les données de développement (DEV) . - **Décodage** : ce module contient des scripts et des excusables permettant de trouver la traduction la plus probable d’une phrase source en consultant les modèles du module Train. Outils ------ ### Le décodeur Moses Moses [@koehn2007moses] est une boite à outils disponible sous licence libre GPL, basée sur des approches statistiques de la traduction automatique. En effet, Moses nous permet de développer et manipuler un système de traduction selon nos besoins grâce à ses nombreuses caractéristiques, telle que la production du modèle de traduction et le modèle de réordonnance à partir des corpus volumineux.\ Parmi les principaux modules du Moses, on trouve : - **Train** : permet de construire des modèles de traduction ainsi que des modèles de réordonnance. - **Mert** : permet d’ajuster les poids des différents modèles afin d’optimiser et maximiser la qualité de traduction en utilisant les données de développement (DEV) . - **Décodage** : ce module contient des scripts et des excusables permettant de trouver la traduction la plus probable d’une phrase source en consultant les modèles du module Train. ### IRSTLM IRSTLM [@federico2007efficient] est une boite à outils utilisée pour la construction des modèles de langage statistiques. L’avantage de cette boite à outils est de réduire les besoins de stockage ainsi que la mémoire lors de décodage. Par conséquent, cet outil nous permet de gagner du temps pour le chargement du modèle de langage. ### BLEU:Métrique d’évaluation automatique Le score BLEU (en anglais : Bilingual Evaluation Understudy) a initialement été proposé par [@papineni2002bleu].C’est un algorithme utilisé en vue d’évaluer la qualité des hypothèses de sortie produites par un système de traduction automatique. En effet, le concept est fondé sur l’idée de comparer l’hypothèse de traduction avec une ou plusieurs références au niveau des mots, des bigrammes, trigrammes etc. Le score BLEU est normalisé entre 0 et 1, et il est exprimé généralement en pourcentage. Notons qu’une traduction humaine peut parfois obtenir un mauvais score BLEU , si elle s’écarte de la référence. ### MADAMIRA L’analyseur morphologique MADAMIRA [@pasha2014madamira] : est un système d’analyse morphologique et de désambiguïsation de l’arabe qui exploite certains des meilleurs aspects des deux systèmes existants et les plus utilisés pour le traitement automatique de la langue arabe que sont : MADA ([@habash2005arabic]; [@habash2009mada+];. [@habash2013morphological]) et AMIRA [@diab2009second]. En effet, MADAMIRA permet la tokenisation, la lemmatisation, le racinisation, l’étiquetage morpho-syntaxique, la désambiguïsation morphologique, la diacritisation, la reconnaissance des entités nommées, etc. MADAMIRA propose les deux shémas de tokenisation suivants: - **ATB:** consiste à segmenter touts les clitiques excepté les articles définis, de même elle consiste à normaliser les caractères ALIF et YA en utilisant le caractère ’+’ comme étant un marqueur de clitiques. - **MyD3:** consiste à tokeniser les proclitiques QUES, CONJ, les clitiques PART, ainsi que touts les articles et enclitiques. En outre, elle normalise les caractères ALIF et YA après la dévoyelisation des caractères arabes. Corpus parallèles ------------------ ### LDC-Ummah Ummah (LDC2004T18) est un corpus de news historique arabe aligné avec des traductions Anglais collectées via le service de presse *Ummah* de Janvier 2001 à Septembre 2004. Il totalise 8.439 paires histoire, 68,685 paires de phrases, de mots arabes et 2M mots 2,5M anglais. Le corpus est aligné au niveau de la phrase. Tous les fichiers de données sont des documents SGML. Nombre de mots Nombre de lignes --------- ---------------- ------------------ -- arabe 2M 68,6 K Anglais 2,4M 68,6 K : Description des corpus Ummah ### LDC-News le corpus LDC-News (Arabic News Translation Text Part 1) a été produit par *LDC* (Linguistic Data Consortium) sous le numéro de catalogue LDC2004T17. Trois sources de texte journalistique arabe ont été sélectionnés pour produire ce corpus arabe - Service des nouvelles *AFP*: 250 nouvelles, 44 193 mots arabes, octobre 1998 - décembre 1998 - - Service des ouvelles *Xinhua*: 670 nouvelles histoires, 99 514 mots arabes, Novembre 2001 - Mars 2002 - An Nahar : 606 nouvelles, 297 533 mots arabes, de Octobre 2001 - Décembre 2002 Nombre de mots Nombre de lignes --------- ---------------- ------------------ -- arabe 441 K 18,6 K Anglais 581 K 18,6 K : Description des corpus LDC-News ### News Commentary Le corpus News commentary est un corpus parallèle aligné au niveau des phrases. Ce corpus contient des extraits de diverses publications de presse et de commentaires du projet *Syndicate* et il est disponible dans plusieurs langues (arabe, anglais, français, espagnol, allemand, et tchèque, etc). Nombre de mots Nombre de lignes --------- ---------------- ------------------ -- arabe 3,9 M 174,5 K Anglais 4,1 M 174,5 K : Description du corpus News Commentary ### TED Talks TED Talks est un ensemble de transcriptions des conférences en anglais présentés sous format vidéo sur le site officiel de TED. Ces transcriptions ont été traduites par les bénévoles pour plus de 70 autres langues (arabe, français, italien, coréen, portugais, etc.). Nombre de mots Nombre de lignes --------- ---------------- ------------------ -- arabe 416 K 29,7 K Anglais 501 K 29,7 K : Description du corpus TED Mise en place du système de TA anglais-arabe ============================================ En arabe nous trouvons plusieurs clitiques qui se collent au mot, conduisant à des ambiguïtés morphologiques et orthographiques. Ainsi, pour construire un système de traduction Anglais-arabe, il est nécessaire de passer par une étape de segmentation du corpus au niveau des mots en pré-traitement (avant de construire le système de traduction) ainsi qu’une étape de détokenisation en post-traitement (après la traduction d’un corpus tokenisé). De ce fait, il est important de trouver le bon schéma de tokenisation à suivre qui ne se trompe pas en détectant le token et les clitiques, et de réussir à retourner après le format initial au texte arabe traduit. Diverses approches ont été proposées pour faire face aux problèmes (d’ambiguïté morphologique en arabe) de tokenisation et détokenisation en arabe. Dans l’un des premiers ouvrages, et d’ailleurs l’un des plus connus dans ce domaine [@habash2006arabic] ont présenté différents schémas de tokenisation pour le pré-traitement de l’arabe en vue de voir quelle est la méthode de segmentation la plus utile pour la TAS. Ces schémas sont disponibles dans l’outil MADAMIRA que nous avons utilisé. Nous avons construit un système de traduction automatique statistique à l’aide de la boite à outils Moses ainsi que IRSTLM pour créer notre modèle de langage 5-grammes, et en utilisant les corpus parallèles décrits précédemment (LDC-Ummah, LDC-News, News Commentary, TED Talks). Nous avons évalué notre système en termes du score BLEU (score de 24,51). Conclusion et Perspectives ========================== Dans cet article, nous avons présenté notre système de traduction anglais-arabe basé sur la boite à outils Moses, construit à l’aide d’un modèle de langage 5-grammes et en utilisant différents corpus parallèles que nous avons décrits. Nous envisageons d’exploiter notre système pour traduire de grands corpus de l’anglais vers l’arabe.