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+{"text":"\\section{Introduction}\n\nHigh-mass X-ray binaries (HMXBs) are X-ray sources for which \nhigh-energy emission stems from accretion onto a compact object\n(black hole or neutron star) of material coming from a massive companion\nstar. Until recently, the huge majority of known HMXBs\nwere Be\/X-ray binaries, i.e a neutron star accreting from a disc\naround a Be star. Most of these sources are transient, even if a\nfew are persistent weak X-ray emitters\n($\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{34}$ erg~s$^{-{1}}$). The other\nknown HMXBs were supergiant X-ray binaries (SGXBs), composed of\na compact object orbiting around an early-type supergiant and fed by\naccretion from the strong radiative wind of the companion. These\nobjects are persistent sources\n($\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{36}$ erg~s$^{-{1}}$), and their\nrelative low number compared to the population of Be\/X-ray\nbinaries was explained as the consequence of the short lifetime of\nsupergiant stars.\n\nThe launch of the \\textit{INTErnational Gammay-Ray Astrophysics\nLaboratory}\n\\citep[\\textit{INTEGRAL}, ][]{2003Winkler} in October 2002 completely\nchanged the situation, as many more HMXBs whose companion stars\nare supergiants were discovered during the monitoring\nof the Galactic centre and the Galactic plane using the onboard\nIBIS\/ISGRI instruments \\citep{2003Ubertini, 2003Lebrun}. Most of\nthese sources are reported in\n\\citet{2007Bird} and \\citet{2007Bodaghee}, and their studies\nhave revealed two main features that were not present on previously\nknown SGXBs:\n\n\\begin{itemize} \n\\item first, many of them exhibit a\n\tconsiderable intrinsic absorption, with a column density up to\n\t$N_{\\rm H}\\,\\sim\\,2\\times10^{24}\\,\\textrm{cm}^{-2}$\nin the case of IGR~J16318-4848\n\t\\citep{2003Matt}, which explains why previous\n\thigh-energy missions had not detected them.\n\n\\item second, some of these new sources reveal a transitory\nnature. They are\n\tundetectable most of the time and occasionally present a fast\n\tX-Ray transient activity lasting a few hours. Moreover, they\n\texhibit a quiescent luminosity of\n\t$\\textrm{L}_\\textrm{x}\\,\\sim\\,10^{33}$ erg~s$^{-{1}}$, well below\n\tthe persistent state of other SGXBs.\n\t\\end{itemize}\nIt then appears that the supergiant HMXBs discovered by\n\\textit{INTEGRAL} can be classified in two classes: one\nclass of considerably obscured persistent sources that we will simply\ncall obscured SGXBs in this paper and another of\nsupergiant fast X-ray transients \\citep[SFXTs,][]{2006Negueruelaa}.\n\nHigh-energy observations can give some information about the\ncompact object or about the processes that lead to the emission\nbut do not allow study of the companion star. It is therefore\nvery important to perform multi-wavelength observations of these\nsources - from optical-to-MIR wavelength - as this\nrepresents the only way to characterise the\ncompanion or to detect dust around these highly\nobscured systems. However, positions\ngiven by \\textit{INTEGRAL} are\nnot accurate enough ($\\sim$ 2\\arcmin) to identify their optical\ncounterparts, because of the large number of objects in the\nerror circle. Observations with X-ray telescopes like\n\\textit{XMM-Newton} or \\textit{Chandra} are therefore crucial\nbecause they allow\na localisation with a position accuracy of 4$\\arcsec$ \nor better, which lowers the number of possible optical\ncounterparts.\n\nWe performed optical-to-MIR wavelength observations of\nseveral candidate SGXBs\nrecently discovered with \\textit{INTEGRAL}. Optical and NIR\nobservations were carried out at ESO\/NTT using EMMI\nand SofI instruments and aimed at constraining \nthe spectral type of the companions through accurate astrometry,\nas well as the spectroscopy and photometry of the candidate\ncounterparts. They are reported in the companion\npaper \\citep[][ CHA08 hereafter]{2008Chaty}, and it is shown\nthat most of these sources are actually supergiant stars.\n\nIn this paper, we report MIR photometric observations of the companions\nof twelve \\textit{INTEGRAL} candidate SFXTs and obscured SGXBs\nthat aimed at studying the circumstellar environment of these highly\nabsorbed sources and, more particularly, at detecting any MIR excess in\ntheir emission that could be due to the absorbing material. These\nsources were chosen because they had very accurate positions\nand confirmed 2MASS counterparts.\n\nAll the sources in our sample are listed in Table 1. The total\ngalactic column density of neutral hydrogen\n$N_{\\rm H}(\\ion{H}{i})$ is computed using the web version\nof the $N_{\\rm H}$ FTOOL from HEASARC. This tool\nuses the data from\n\\citet*{1990Dickey}, who performed \\ion{H}{i} observations from\nthe Lyman-$\\alpha$ and 21~cm lines. Moreover,\n$N_{\\rm H}(\\ion{H}{i})$ is\nthe total galactic column density, which means it is\nintegrated along the line of sight over the whole\nGalaxy. Therefore, it is likely to be overestimated compared to the\nreal value at the distance of the sources.\n\nThe total galactic column density of molecular hydrogen\n$N_{\\rm H}(\\textrm{H}_{\\rm 2})$ is computed using the\nvelocity-integrated map (W$_{\\rm CO}$) and the X-ratio given in\n\\citet*{2001Dame}. It is also likely to be overestimated compared to the\nreal value at the distance of the sources, because it is\nintegrated along the line of sight over the whole Galaxy. In\ncontrast, $N_{\\rm Hx}$, the intrinsic X-ray column density of the source, is computed\nfrom the fitting of the high-energy spectral energy distribution\n(SED), so it takes all the \nabsorption into account at the right distance of the source.\n\\newline\n\nUsing these observations, the results reported\nin the companion paper (CHA08), as well as archival\nphotometric data from the\nUSNO, 2MASS, and GLIMPSE\ncatalogues when needed, we performed the broad-band\nSEDs of these sources and fitted them with a two-component\nblack body model to assess the \ncontribution of the star and the enshrouding material in the\nemission. The ESO observations, as well as our model, are described in\nSection 2. In Section 3, results of the fits for each source\nare given and these results are discussed in Section 4. We\nconclude in Section 5. \n\n\n\n\\section{Observations}\n\n\n\\subsection{MIR observations and data reduction}\n\nThe MIR observations were carried out on 2005 June 20-22 and 2006 June\n29-30 using VISIR \\citep{2004Lagage}, \nthe ESO\/VLT mid-infrared imager and spectrograph, composed of an\nimager and a long-slit spectrometer covering several filters in\nN and Q bands and\nmounted on Unit 3 of the VLT (Melipal). The standard ''chopping\nand nodding\" MIR observational technique was used to suppress the\nbackground dominating at these wavelengths. Secondary mirror-chopping was\nperformed in the north-south direction with an amplitude of\n16$\\arcsec$ at a frequency of 0.25 Hz. Nodding technique,\nneeded to compensate for chopping residuals, was chosen as\nparallel to the chopping and applied\nusing telescope offsets of 16$\\arcsec$. Because of the high\nthermal MIR background for ground-based\nobservations, the detector integration\ntime was set to 16~ms.\n\nWe performed broad-band photometry in 3 filters, PAH1\n($\\lambda$=8.59$\\pm$0.42 $\\mu$m), PAH2\n($\\lambda$=11.25$\\pm$0.59 $\\mu$m), and Q2\n($\\lambda$=18.72$\\pm$0.88 $\\mu$m) using the small field in all bands \n(19\\farcs2x19\\farcs2 and 0\\farcs075 plate scale). All\nthe observations were bracketed with standard star\nobservations for flux calibration and\nPSF determination. The weather conditions\nwere good and stable during the observations.\n\nRaw data were reduced using the IDL reduction package written by Eric\nPantin. The elementary images were co-added in real-time to obtain\nchopping-corrected data, then the different nodding positions were\ncombined to form the final image. The VISIR detector is affected by\nstripes randomly triggered by some abnormal high-gain pixels.\nA dedicated destriping method was developed (Pantin 2008,\nin prep.) to suppress them. The MIR fluxes of all observed\nsources including the 1$\\sigma$ errors are listed in Table 2.\n\n\n\\subsection{Archival data}\n\nWhen we did not have optical-to-MIR data for our sources, we\nsearched for the counterparts in 3 catalogues: \n\\begin{itemize}\n\\item in the United States Naval Observatory (USNO)\n\tcatalogues in \\textit{B}, \\textit{R}, and \\textit{I} for USNO-B1.0, \\textit{B} and \\textit{R}\n\tfor USNO-A.2. Positions and fluxes\n\taccuracies are 0\\farcs25 and 0.3 magnitudes in the case of\n\tUSNO-B.1, 0\\farcs2 and 0.5 magnitudes in the case of USNO-A.2.\n\\item in the 2 Micron All Sky Survey (2MASS), in \\textit{J}\n\t(1.25$\\pm$0.16 $\\mu$m), \\textit{H} (1.65$\\pm$0.25 $\\mu$m) and \\textit{Ks}\n\t(2.17$\\pm$0.26 $\\mu$m) bands. Position accuracy is about 0\\farcs2.\n\\item in the \\textit{Spitzer}'s Galactic Legacy Infrared Mid-Plane Survey\n\tExtraordinaire \\citep[GLIMPSE,][]{2003Benjamin}, survey of the\nGalactic plane\n\t($|b|\\,\\leq\\,1^\\circ$ and between \\textit{l}=10$^\\circ$ and\n\\textit{l}=65$^\\circ$ on both sides of the Galactic centre)\n\tperformed with the \\textit{Spitzer Space Telescope}, using the IRAC\n\tcamera in four bands, 3.6$\\pm$0.745 $\\mu$m, 4.5$\\pm$1.023 $\\mu$m,\n\t5.8$\\pm$1.450 $\\mu$m, and 8$\\pm$2.857 $\\mu$m.\n\\end{itemize}\nAll sources had a confirmed 2MASS counterpart and three of them (IGR\nJ16195-4945, IGR~J16207-5129, and IGR~J16318-4848) had a GLIMPSE\ncounterpart given in the literature. We found all the other\nGLIMPSE counterparts using the 2MASS positions and they are listed in\nTable 3. We used all the\nfluxes given in the GLIMPSE catalogue\nexcept in the case of IGR~J17252-3616, IGR~J17391-3021 and\nIGR~J17544-2619 because\ntheir fluxes were not present in the catalogue tables. Nevertheless,\nwe measured their\nfluxes on the archival images directly with aperture\nphotometry. Uncertainties on the\nmeasurements were computed in the same way on the error\nmaps given with the data.\n\n\\subsection{Absorption}\n\nAbsorption at wavelength $\\lambda$, $\\textrm{A}_\\lambda$, is a\ncrucial parameter to fit the SEDs, especially in the\nMIR. Indeed, inappropriate values can lead to a false assessment\nof the MIR excess. Visible\nabsorption A$_\\textrm{v}$ was a free parameter of the fits. An\naccurate interstellar absorption law - i.e. the wavelength\ndependence of\nthe $\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ ratio in\nthe line of sight - was then needed to properly fit the SEDs.\n\nIn the optical bands, we built the function with the analytical\nexpression given in\n\\citet{1989Cardelli} who derived the average\ninterstellar extinction law in the\ndirection of the Galactic centre. From 1.25 $\\mu$m to 8 $\\mu$m,\nwe used the analytical expression given in \\citet[]{2005Indebetouw}. \nThey derived it from the measurements of the mean\nvalues of the colour excess ratios\n$\\frac{(\\textrm{A}_\\lambda-\\textrm{A}_{\\rm K})}\n{(\\textrm{A}_{\\rm J} -\\textrm{A}_{\\rm K})}$\nfrom the colour distributions of observed stars in the direction of the\nGalactic centre. They used archival data from 2MASS\nand GLIMPSE catalogues, which is relevant in our case as we use\nGLIMPSE fluxes. \n\\newline\n\nAbove 8 $\\mu$m, where absorption is dominated by the silicate\nfeatures at 9.7 $\\mu$m and 18 $\\mu$m, we found several\nextinction laws in the\nliterature \\citep{1989Rieke, 1996Lutz, 2001Moneti}, which exhibit some\ndifferences. Considering the high importance of a good assessment\nto correctly fit the MIR excess, we decided to assess the ratio\n$\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ in 2\nVISIR bands - PAH1 and PAH2 - from our data in order to\nbuild the relevant law for our observations. \n\n\\citet*{1985Rieke} gave the interstellar extinction law up to 13\n$\\mu{\\textrm{m}}$, and from their results, we derived \n$0.043\\leq\\frac{\\textrm{A}_{\\rm PAH1}}{\\textrm{A}_\\textrm{v}}\\leq0.074$\nand\n$0.047\\leq\\frac{\\textrm{A}_{\\rm PAH2}}{\\textrm{A}_\\textrm{v}}\\,\\leq0.06$.\nTo get the best values corresponding to our data in PAH1 and\nPAH2, we proceeded\nin 3 steps. \n\\begin{itemize}\n\t\\item We first selected the sources for which we had VISIR\nfluxes in PAH1 and\/or\n\tPAH2 and fitted their SEDs with extinction laws given in\n\t\\citet{1989Cardelli} and \\citet{2005Indebetouw} from 0.36 to 8 $\\mu{\\textrm{m}}$ and\n\thalf-interval values taken in PAH1 and PAH2.\n\t\\item Then, when we did not need any MIR excess to fit the\n\tIRAC fluxes, we adjusted the \n\t$\\frac{\\textrm{A}_{\\rm PAH1}}{\\textrm{A}_\\textrm{v}}$ and\n\t$\\frac{\\textrm{A}_{\\rm PAH2}}{\\textrm{A}_\\textrm{v}}$ ratios to\n\timprove the $\\chi^2$ of our fits.\n\t\\item We finally averaged all the extinction values obtained for all\n\tsources to get what we consider as the right ratios in PAH1 and PAH2\n\tin the direction of the Galactic plane.\n\\end{itemize}\nThe resulting values are in good agreement with those given by\nthe extinction law from \\citet{1996Lutz}, so we chose\ntheir extinction law to fit our SEDs above 8\n$\\mu{\\textrm{m}}$, the Q2 filter\nincluded. The $\\frac{\\textrm{A}_\\lambda}{\\textrm{A}_\\textrm{v}}$ values we\nused in each band are listed in Table 4, and the overall\nextinction law is displayed in Fig 1.\n\n\n\\subsection{SEDs}\n\nWith all the archival and observational data from optical-to-MIR \nwavelength, we built the SEDs for these sources. We fitted them\n(using a $\\chi^2$ minimisation) with a\nmodel combining two absorbed black bodies, one representing the companion\nstar emission and a spherical one representing a possible MIR\nexcess due to the absorbing material enshrouding the companion star:\n\n\\begin{displaymath}\n\t\\lambda{F(\\lambda)}\\,=\\,\\frac{2\\pi{h}{c}^2}{{D_{\\ast}}^2{\\lambda}^4}\n\\,10^{\\textrm{\\normalsize\n\t$-{0.4A_\\lambda}$}}\\left[\\frac{{R_\\ast}^2}{\n\te^{\\textrm{\\large\n\t$\\frac{hc}{{\\lambda}k{T}_\\ast}$}}-1}+\\frac{{R_{\\rm D}}^2}{e^{\\textrm{\\large\n\t$\\frac{hc}{{\\lambda}k{T}_{\\rm D}}$}}-1}\\right]\\,\\,\\,\\,\\,\\,\\,\\textrm{in\n\tW m}^{-2}\n\\end{displaymath}\n\nWe added to the flux uncertainties systematic errors as follows:\n\n\\begin{itemize}\n\t\\item a 2$\\%$ systematic error in each IRAC band as given in\nthe IRAC\nmanual\\footnote{http:\/\/ssc.spitzer.caltech.edu\/documents\/som\/\nsom8.0.irac.pdf}\n\t\\item comparing the variations of the flux calibration\nvalues obtained from standards with VISIR during our\nobservation nights, we figured out that systematic errors\nwith VISIR were about 5$\\%$ at 10 $\\mu{\\textrm{m}}$ and\n10$\\%$ at 20 $\\mu{\\textrm{m}}$.\n\\end{itemize}\nThe free parameters of the fits were the absorption in the\nV-band A$_\\textrm{v}$,\nthe companion star black body temperature T$_\\ast$ and radius\nto distance ratio $\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$, as\nwell as the additional spherical component black body temperature\nand radius T$_{\\rm D}$ and $\\textrm{R}_{\\rm D}$.\\\\\t\n\\newline\nThe best-fitting\nparameters for individual sources, as well as corresponding\n$\\chi^2$ are\nlisted in Table 5 and the fitted SEDs are displayed in Fig 3.\nMoreover, 90\\%-confidence ranges of parameters are listed in\nTable 6. \n\nIn Table 5, along with the best-fitting parameters, we also give\nthe total galactic \nextinctions in magnitudes $\\textrm{A}_{\\ion{H}{i}}$ and\n$\\textrm{A}_{\\rm H_2}$ in the line of sight, as well as the\nX-ray extinction of the source\nin magnitudes $\\textrm{A}_\\textrm{x}$. The values of\n$\\textrm{A}_{\\ion{H}{i}}$, $\\textrm{A}_{\\rm H_2}$, and\n$\\textrm{A}_\\textrm{x}$ are computed from\n$N_{\\rm H}(\\ion{H}{i})$, \n$N_{{\\rm H}}(\\textrm{H}_{\\rm 2})$, and\n$N_{\\rm Hx}$ given in Table 1 using the relation \n$\\textrm{A}_{\\rm H}\\,=\\,\\frac{3.1}{5.8\\times10^{21}\\,\\textrm{cm}^{-{2\n}}}N_{\\rm H}$\n\\citep*{1978Bohlin,1985Rieke}.\n\n\\begin{table*}\n\t\\caption{Sample of sources studied in this paper. \n\tWe give their name, their\n\tcoordinates (J2000 and galactic), the total galactic\n\tcolumn density of neutral hydrogen\n\t($N_{\\rm H}(\\ion{H}{i})$) and the total galactic\n\tcolumn density of molecular hydrogen\n($N_{\\rm H}(\\textrm{H}_{\\rm 2})$) in the line of\n\tsight, the X-ray column\n\tdensity of the source ($N_{\\rm Hx}$), their type\n(SFXT or OBS - obscured sources) and their\n\tspectral type (SpT). Their spectral classifications come from optical\/NIR\n\tspectroscopy, reported in the following references (Ref): c:\n\\citet{2008Chaty}, f: \\citet*{2004Filliatre}, i: \\citet{2006Zand},\n\tn1: \\citet{2005Negueruela}, n2: \\citet{2006Negueruelab}, n3:\n\\citet{2007Nespoli}, p:\n\t\\citet{2006Pellizza}, t: \\citet{2006Tomsick}.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\alpha (\\textrm{J2000})&\\delta\n(\\textrm{J2000})&l&b&N_{\\rm H}(\\ion{H}{i})(10^{22})&\nN_{\\rm H}(\\textrm{H}_{\\rm 2})(10^{22})&N_{\\rm Hx}\n(10^{22})&\\textrm{Type}&\\textrm{SpT}&\\textrm{Ref}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&16\\,\\,19\\,\\,32.20&-49\\,\\,44\\,\\,30.7&3\n33.56&0.339&2.2&5.4&7.0&\\textrm{SFXT ?}&\\textrm{O\/B}&\\textrm{t}\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&16\\,\\,20\\,\\,46.26&-51\\,\\,30\\,\\,06.0&\n332.46&-1.050&1.7&2.2&3.7&\\textrm{OBS}&\\textrm{O\/B}&\\textrm{t}\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&16\\,\\,31\\,\\,48.60&-48\\,\\,49\\,\\,00.0&\n335.62&-0.448&2.1&3.6&200.0&\\textrm{OBS}&\\textrm{sgB[e]}&\\textrm{f}\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&16\\,\\,32\\,\\,01.90&-47\\,\\,52\\,\\,27.0&\n336.30&0.169&2.1&4.4&21.0&\\textrm{OBS}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&16\\,\\,35\\,\\,53.80&-47\\,\\,25\\,\\,41.1&\n337.01&-0.007&2.2&7.3&33.0&\\textrm{OBS}&\\textrm{sgB[e]}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&16\\,\\,41\\,\\,51.00&-45\\,\\,32\\,\\,25.0&\n339.19&0.489&1.9&3.6&10.0&\\textrm{SFXT ?}&\\textrm{O\/B}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&16\\,\\,46\\,\\,35.50&-45\\,\\,07\\,\\,04.0\n&340.05&0.135&2.1&5.9&60.0&\\textrm{SFXT}&\\textrm{B0.5I}&\\textrm{n1}\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&16\\,\\,48\\,\\,06.60&-45\\,\\,12\\,\\,08.0&\n340.16&-0.124&2.1&8.2&7.7&\\textrm{SFXT ?}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&17\\,\\,25\\,\\,11.40&-36\\,\\,16\\,\\,58.6&\n351.50&-0.354&1.6&3.9&15.0&\\textrm{OBS}&\\textrm{O\/BI}&\\textrm{c}\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&17\\,\\,39\\,\\,11.58&-30\\,\\,20\\,\\,37.6&\n358.07&0.445&1.4&4.5&30.0&\\textrm{SFXT}&\\textrm{08Iab(f)}&\\textrm{n2}\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&17\\,\\,54\\,\\,25.28&-26\\,\\,19\\,\\,52.6\n&3.26&-0.336&1.4&7.8&1.4&\\textrm{SFXT}&\\textrm{O9Ib}&\\textrm{p}\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&19\\,\\,14\\,\\,04.23&+09\\,\\,52\\,\\,58.3&\n44.30&-0.469&1.7&3.7&6.0&\\textrm{OBS}&\\textrm{B1I}&\\textrm{n3}\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*}\n\n\\begin{table*}\n\t\\caption{Summary of VISIR observations of newly discovered\n\\textit{INTEGRAL}\n\tsources. We give their MIR fluxes (mJy) in the PAH1 (8.59 $\\mu$m), PAH2\n(11.25 $\\mu$m) and\n\tQ2 (18.72 $\\mu$m) filters. When we did not detect a source,\n\twe give the upper limit. When no flux nor upper limit is\n\tgiven, the source was not observed in the considered filter.}\n\t$$\n\t\\begin{array}{c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{PAH1}&\\textrm{PAH2}&\\textrm{Q2}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&<6.1&<7.8&<50.3\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&21.7\\pm1.4&9.4\\pm1.3&<53.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&426.2\\pm3.0&317.4\\pm3.4&180.7\\pm15.3\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&12.1\\pm1.7&6.3\\pm1.8&\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&<6.9&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&<5.8&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&6.9\\pm1.1&<5.0&\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&10.9\\pm1.2&7.0\\pm1.6&\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&6.1\\pm0.6&<5.0&\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&70.2\\pm1.6&46.5\\pm2.6&\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&46.1\\pm2.8&20.2\\pm2.1&\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&35.2\\pm1.4&19.1\\pm1.4&\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*}\n\n\\begin{table*}\n\t\\caption{List of GLIMPSE counterparts we found for 9\nsources. We give their name, their separation from the 2MASS counterparts\n\tand their fluxes in mJy.}\n\t$$\n\t\\begin{array}{c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{GLIMPSE\ncounterpart}&\\textrm{Separation}&3.6\\,\\mu\\textrm{m}&4.5\\,\\mu\n\\textrm{m}&5.8\\,\\mu\\textrm{m}&8\\,\\mu\\textrm{m}\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&\\textrm{G336.3293+00.1689}&0\\farcs17&\n48.2\\pm1.9&44.3\\pm2.1&36.0\\pm2.0&17.3\\pm1.0\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&\\textrm{G337.0994-00.0062}&0\\farcs46&\n5.9\\pm0.5&5.6\\pm0.6&5.3\\pm1.7&\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&\\textrm{G339.1889+00.4889}&0\\farcs28&\n12.5\\pm0.9&9.5\\pm0.6&5.6\\pm0.6&3.6\\pm0.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&\\textrm{G340.0536+00.1350}&0\\farcs16&\n45.0\\pm2.0&32.6\\pm1.5&22.0\\pm0.9&13.7\\pm0.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&\\textrm{G340.1630-00.1239}&0\\farcs13&\n68.6\\pm3.1&49.6\\pm2.0&41.2\\pm2.3&19.4\\pm0.9\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&&&32.6\\pm3.7&24.7\\pm4.7&21.8\\pm5.0&\n9.6\\pm6.5\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&&&375.9\\pm44.0&297.0\\pm31.0&205.0\\pm33.0&\n111.0\\pm28.0\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&&&213.9\\pm25.4&137.0\\pm18.9&99.6\\pm7.6&\n66.5\\pm12.1\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&\\textrm{G044.2963-00.4688}&0\\farcs48&\n185.0\\pm9.3&152.0\\pm11.1&103.9\\pm5.4&62.0\\pm2.1\\\\\n\t\\hline\n\t\\end{array} \n\t$$\n\\end{table*} \n\n\\begin{table*}\n\t\\caption{Adopted $\\frac{\\textrm{A}_{\\lambda}}{\\textrm{A}_{v}}$ values.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Filters}&\\textrm{\\textit{U}}&\\textrm{\\textit{B}}&\\textrm{\\textit{V}}&\\textrm{\\textit{R}}&\n\\textrm{\\textit{I}}&\\textrm{\\textit{J}}&\\textrm{\\textit{H}}&\\textrm{\\textit{Ks}}&3.6\\,\\mu\\textrm{m}&\n4.5\\,\\mu\\textrm{m}&5.8\\,\\mu\\textrm{m}&8\\,\\mu\\textrm{m}&8.59\\,\\mu\n\\textrm{m}&11.25\\,\\mu\\textrm{m}&18.72\\,\\mu\\textrm{m}\\\\\n\t\\hline\n\t\\frac{\\textrm{A}_{\\lambda}}{\\textrm{A}_{v}}&1.575&1.332&1&0.757&\n0.486&0.289&0.174&0.115&0.0638&0.0539&0.0474&0.0444&0.0595&0.0605&\n0.040\\\\\n\t\\hline\n\t\\end{array} \n\t$$\n\\end{table*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[angle=270,width=10cm]{.\/absorb.ps}\n\t\\caption{Adopted extinction law. We used the law given in\n\t\\citet{1989Cardelli} in the optical, the one given in\n\t\\citet{2005Indebetouw} from 1.25 $\\mu{\\textrm{m}}$ to 8\n$\\mu{\\textrm{m}}$, and the law\n\tfrom \\citet{1996Lutz} above 8 $\\mu{\\textrm{m}}$.}\n\\end{figure*} \n\n\n\\begin{table*}\n\t\\caption{Summary of best-fitting parameters of the SEDs of the\n\tsources. We give the total galactic \n\textinctions in magnitudes $\\textrm{A}_{\\ion{H}{i}}$ and\n\t$\\textrm{A}_{\\rm H_2}$, the X-ray extinction of the source\n\tin magnitudes $\\textrm{A}_\\textrm{x}$ and then the\n\tparameters themselves: the extinction in the optical\n\t$\\textrm{A}_\\textrm{v}$, the temperature\n\t$\\textrm{T}_\\ast$ and the\n\t$\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}$ ratio of the\n\tcompanion and the\n\ttemperature and radius $\\textrm{T}_{\\rm D}$ and\n$\\textrm{R}_{{\\rm D}}$ (in\n\t$\\textrm{R}_{\\ast}$ unit) of the dust component\n\twhen needed. We also add the reduced $\\chi^2$ we reach for\n\teach fit.} \n\t$$\n\t\\begin{array}{c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{A}_{\\ion{H}{i}}&\\textrm{A}_{\\rm H_2}&\n\\textrm{A}_\\textrm{x}&\\textrm{A}_\\textrm{v}&\\textrm{T}_{\\ast}\n(\\textrm{K})&\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}&\\textrm{T}_\n{\\rm D}\n(\\textrm{K})&\\textrm{R}_{\\rm D}\n\\left(\\textrm{R}_{\\ast}\\right)&\\chi^2\/\\textrm\n{dof}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&11.7&28.6&37.4&15.5&23800&5.96\\times\n10^{-{11}}&1160&5.1&3.9\/2\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&9.3&11.8&19.8&10.5&33800&9.42\\times\n10^{-{11}}&&&28.5\/9\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&11.0&19.2&1069.5&17.0&22200&3.74\\times\n10^{-{10}}&1100&10.0&6.6\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&11.4&23.6&112.2&35.4&33000&1.38\\times\n10^{-{10}}&&&7.7\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&11.8&39.4&176.4&17.6&24500&3.16\\times\n10^{-{11}}&810&10.1&3.6\/2\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&10.1&19.3&53.5&14.5&32800&3.77\\times\n10^{-{11}}&&&1.4\/4\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&11.3&31.4&320.7&5.9&25000&6.40\\times\n10^{-{11}}&&&13.9\/7\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&11.4&43.8&41.2&18.5&32800&1.00\\times\n10^{-{10}}&&&7.4\/6\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&8.3&20.1&80.2&20.8&32600&7.57\\times\n10^{-{11}}&&&3.8\/5\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&7.3&23.9&160.4&9.2&31400&1.80\\times\n10^{-{10}}&&&11.7\/10\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&7.7&41.5&7.70&6.1&31000&1.27\\times10^{-{10}}\n&&&6.1\/8\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&9.0&14.0&32.1&16.5&22500&1.92\\times10^\n{-{10}}&&&14.4\/6\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*} \n\n\\begin{table*}\n\t\\caption{Ranges of parameters that give acceptable\n\tfits (90\\%-confidence) for each source.\n\t}\n\t$$\n\t\\begin{array}{c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\Delta\n\t\\textrm{A}_\\textrm{v}&\\Delta\\textrm{T}_{\\ast}&\\Delta\\frac{\\textrm{R\n}_{\\ast}}{\\textrm{D}_{\\ast}}&\\Delta\\textrm{T}_{\\rm D}&\\Delta\\textrm\n{R}_{{\\rm D}}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&14.8-15.8&13100-25900&5.68\\times10^{-{11}}\n-6.68\\times10^{-{11}}&950-1460&3.9-6.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&10.4-10.6&25200-36000&9.36\\times10^{-{11}}\n-1.05\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}&16.6-17.4&19300-24500&3.65\\times10^{-{10}}\n-3.84\\times10^{-{10}}&960-1260&8.8-11.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}&34.8-35.5&22000-35600&1.33\\times10^{-{10}}\n-1.69\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}&17.1-18.1&19500-36000&2.64\\times10^{-{11}}\n-3.52\\times10^{-{11}}&630-1020&8.0-13.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&13.6-14.7&10600-36000&3.58\\times10^{-{11}}\n-5.44\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}&5.0-6.1&15400-33600&5.50\\times10^{-{11}}\n-7.95\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}&18.4-18.8&26200-36000&9.48\\times10^{-{11}}\n-1.13\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}&20.3-21.0&20500-36000&7.17\\times10^{-{11}}\n-9.48\\times10^{-{11}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}&8.8-9.4&16100-32200&1.78\\times10^{-{10}}\n-2.63\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}&6.0-6.2&26700-35500&1.18\\times10^{-{10}}\n-1.38\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}&15.7-16.7&13200-28100&1.71\\times10^{-{10}}\n-2.53\\times10^{-{10}}&&\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table*} \n\n\n\\section{Results}\n\n\\subsection{IGR~J16195-4945}\n\nIGR~J16195-4945 was detected by \\textit{INTEGRAL} during\nobservations carried\nout between 2003 February 27 and October 19 \\citep{2004Walter} and\ncorresponds to the ASCA source AX J161929-4945\n\\citep{2001Sugizaki,2005Sidoli}. \nAnalysing \\textit{INTEGRAL} public data,\n\\citet{2005Sidoli} derive an average flux level of\n$\\sim$ 17~mCrab (20-40~keV). Performing a follow-up with\n\\textit{INTEGRAL}, \\citet{2006Sguera} show it behaves like\nan SFXT and report a peak-flux of $\\sim$ 35~mCrab (20-40~keV).\n\n\\citet{2006Tomsick} observed the source with \\textit{Chandra}\nbetween 2005 April and\nJuly and give its position with 0\\farcs6 accuracy. They fitted its\nhigh-energy emission with an absorbed\npower law and derive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim7\\times10^{22}$~cm$^{-2}$. Moreover,\nusing their accurate localisation, they found its NIR and MIR\ncounterparts in\nthe 2MASS (2MASS~J16193220-4944305) and in the GLIMPSE\n(G333.5571+00.3390) catalogues and performed its NIR photometry\nusing ESO\/NTT observations. They show its spectral type is\ncompatible with an O, B, or A supergiant star. They also found possible\nUSNO-A.2 and USNO-B.1 counterparts. Nevertheless, as already\nsuggested in their paper, the USNO source is\na blended foreground object \\citep{2006Tovmassian}. \n\\newline\n\nWe observed IGR~J16195-4945 on 2006 June 30 in PAH1 during 1200~s, but\ndid not detect it. Typical seeing and airmass were\n0\\farcs88\\ and 1.07. We nevertheless fitted its SED\nusing the NIR and the GLIMPSE flux values given in \\citet{2006Tomsick}\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=15.5,\n$\\textrm{T}_*$=23800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$5.96\\times10^{-{11}}$,\n$\\textrm{T}_{\\rm D}$=1160~K,\n$\\textrm{R}_{\\rm D}$=5.1$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 3.9\/2. \n\\newline\nThe best-fitting parameters without the additional component are\n$\\textrm{A}_\\textrm{v}$=16.1,\n$\\textrm{T}_*$=13800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$8.94\\times10^{-{11}}$,\nand the corresponding reduced $\\chi^2$ is 15.8\/4. We then found\na MIR excess.\n\nThe additional component is then needed to correctly fit the SED, as\nthis source exhibits a MIR excess. Nevertheless, as shown in\nFig 4, this excess is small, and the lack of data above 8\n$\\mu$m does not allow to reach definitive conclusions. Moreover,\nin both cases (with and without dust), the stellar\ncomponent is consistent with an O\/B supergiant, as already\nsuggested in \\citet{2006Tomsick}. \n\n\\subsection{IGR~J16207-5129}\n\nIGR~J16207-5129 is an obscured SGXB that was discovered by\n\\textit{INTEGRAL} during observations carried\nout between 2003 February 27 and October 19 \\citep{2004Walter}.\n\n\\citet{2006Tomsick} observed it with \\textit{Chandra} during the\nsame run as\nIGR~J16195-4945 and give its position with 0\\farcs6\naccuracy. \nThey also fitted its high-energy emission with an absorbed\npower law and derive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim3.7\\times10^{22}$~cm$^{-2}$.\nThanks to their accurate localisation, they found its NIR and MIR\ncounterparts in\nthe 2MASS (2MASS~J16204627-5130060) and in the GLIMPSE\n(G333.4590+01.0501) catalogues and performed its NIR photometry\nusing ESO\/NTT observations. They show its temperature to\nbe $\\geqslant$18000~K, which indicates the system is an\nHMXB. They also found its USNO-B1.0\n(USNO-B1.0~0384-0560875) counterpart.\n\\newline\n\nWe observed IGR~J16207-5129 on 2006 June 29 in PAH1 and PAH2 during\n1200~s in each filter, and in Q2 during 2400~s. Typical seeing and\nairmass were 0\\farcs72 and 1.09. We did not detect it in Q2 but in PAH1\nand PAH2. The fluxes we derived are 21.7$\\pm$1.4~mJy and\n9.4$\\pm$1.3~mJy, respectively.\n\nUsing those values, as well as the fluxes from ESO\/NTT observations and\nthe GLIMPSE archives found in \\citet{2006Tomsick}, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=10.5,\n$\\textrm{T}_*$=33800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$9.42\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is\n28.5\/9. \\citet*{2007Negueruela} find the spectral type is earlier\nthan B1I; our parameters are therefore in good agreement with\ntheir results.\n\nThe best fit with the additional component gives a larger\nreduced $\\chi^2$ of 30\/7\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K, which is not \nsignificant, as the presence of such cold material marginally enhances\nthe MIR flux. We therefore think IGR~J16207-5129 is an O\/B\nmassive star whose enshrouding material marginally contributes\nto its MIR emission.\n\n\n\\subsection{IGR~J16318-4848}\n\nMain high-energy characteristics of this source can be found in\n\\citet*{2003Matt} and \\citet{2003Walter}. IGR~J16318-4848 was\ndiscovered by\n\\textit{INTEGRAL} on 2003 January 29 \\citep{2003Courvoisier} and\nwas then\nobserved with \\textit{XMM-Newton}, which allowed a 4$\\arcsec$ localisation. Those\nobservations showed that the source was exhibiting a strong absorption\nof $N_{\\rm H}\\sim2\\times10^{24}$~cm$^{-2}$, a temperature kT = 9~keV, and a photon index $\\sim\\,2$. \n\nUsing this accurate position, \\citet*{2004Filliatre} discovered its\noptical counterpart and confirmed the NIR counterpart proposed by\n\\citet{2003Walter} (2MASS~J16314831-4849005). They also performed\nphotometry and spectroscopy in optical and NIR on 2003 February 23-25\nat ESO\/NTT and show that the source presents a significant NIR excess\nand that it is strongly absorbed\n($\\textrm{A}_\\textrm{v}\\sim17.4$). The spectroscopy revealed an\nunusual spectrum with a continuum very rich in strong emission\nlines, which, together with the presence of forbidden lines,\npoints towards an sgB[e] companion star \\citep[see e.g. ][ for\ndefinition and characteristics of these stars]{1998Lamers,1999Zickgraf}.\n\nUsing the 2MASS magnitudes, the GLIMPSE (G335.6260-00.4477), and\nthe MSX fluxes,\n\\citet{2006Kaplan} fitted its SED with a combination of a stellar\nand a dust component black bodies, and shows that the presence\nof warm dust around the system was necessary for explaining the\nNIR and MIR excess. From their fit, they derive \n$\\textrm{A}_\\textrm{v}\\sim18.5$, $\\textrm{T}_{\\rm D}$=1030~K,\nand $\\textrm{R}_{\\rm D}$=10$\\textrm{R}_{\\ast}$.\n\\newline\n\nWe observed IGR~J16318-4848 with VISIR twice: \n\\begin{itemize}\n\\item the first time on 2005\n\tJune 21 during 300~s in PAH1 and PAH2, and 600~s in Q2. Typical seeing\n\tand airmass were 0\\farcs81 and 1.14. We detected the source\n\tin all bands, and the derived fluxes are\n\t409.2$\\pm$2.4~mJy, 322.4$\\pm$3.3~mJy, and 172.1$\\pm$14.9~mJy\nin PAH1, PAH2, and Q2, respectively.\n\\item the second on 2006 June 30 during 600~s in all bands. Typical seeing\n\tand airmass were 0\\farcs68 and 1.09. We \n\tdetected the source in all bands, and the derived fluxes are\n\t426.2$\\pm$3.0~mJy, 317.4$\\pm$3.4~mJy, and 180.7$\\pm$15.3~mJy\nin PAH1, PAH2, and Q2, respectively.\n\\end{itemize}\nThose observations show that IGR~J16318-4848 is very bright in\nthe MIR (it is actually the brightest source in our sample) and\nthat its flux was constant within a year, considering VISIR\nsystematic errors.\n\nUsing data from our last run, as well as the magnitudes given in\n\\citet*{2004Filliatre} in the optical and the NIR, and the fluxes from\nthe GLIMPSE archives, we fitted its SED\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=17,\n$\\textrm{T}_*$=22200~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.74\\times10^{-{10}}$,\n$\\textrm{T}_{\\rm D}$=1100~K,\n$\\textrm{R}_{\\rm D}$=10$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 6.6\/6.\nThe best-fitting parameters without the additional component are\n$\\textrm{A}_\\textrm{v}$=17.9,\n$\\textrm{T}_*$=18200~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$5.1\\times10^{-{10}}$,\nand the corresponding reduced $\\chi^2$ is 425\/8. We then confirm\nthat the MIR excess is likely due to the \npresence of warm dust around the system, as already suggested by\n\\citet*{2004Filliatre} and reported in \\citet{2006Kaplan}.\n\n\\subsection{IGR~J16320-4751}\n\nIGR~J16320-4751 was detected by \\textit{INTEGRAL} on 2003\nFebruary \\citep{2003Tomsick} \nand corresponds to the ASCA source AX\nJ1631.9-4752. \\citet{2003Rodrigueza} report observations with\n\\textit{XMM-Newton}. They give an accurate localisation (3$\\arcsec$) and\nfitted its high-energy spectrum with an absorbed power law. They\nderive $\\Gamma\\sim1.6$ and\n$N_{\\rm H}\\sim2.1\\times10^{23}$~cm$^{-2}$. \n\n\\citet{2005Lutovinov} report the discovery of X-Ray pulsations\n(P$\\sim1309$~ s), which proves the compact object is a neutron\nstar. Moreover, \\citet{2004Corbet} obtained the light curve of IGR\nJ16320-4751 between 2004 December 21 and 2005 September 17 with Swift\nand report the discovery of a 8.96 days orbital period. IGR\nJ16320-4751 is then an X-ray binary whose compact object is a\nneutron star.\n\n\\citet*{2007Negueruela} searched for the NIR counterpart of the\nsource \nin the 2MASS catalogue and found its position was consistent with\n2MASS~J16320215-4752289. They also concluded that, if it was an O\/B\nsupergiant, it had to be extremely absorbed. \n\nThe optical and NIR photometry and spectroscopy of this source were\ncarried out at ESO\/NTT, and results are reported in CHA08. It is\nshown that its NIR spectrum is consistent with an O\/B supergiant\nand that its intrinsic absorption is very high, because it was not\ndetected in any of the visible bands. We searched for the MIR\ncounterpart of IGR~J16320-4751 in the GLIMPSE archives and found it\nto be consistent with G336.3293+00.1689.\n\\newline\n\nWe observed IGR~J16320-4751 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the respective exposure\ntimes were 1800~s and 2400~s. Typical seeing and\nairmass were 0\\farcs63 and 1.13. We detected it in both filters, and\nthe respective fluxes are 12.1$\\pm$1.7~mJy and 6.3$\\pm$1.8~mJy.\nUsing the ESO\/NTT NIR magnitudes given in CHA08, as well as the\nGLIMPSE and the VISIR fluxes, we fitted its SED and the best-fitting\nparameters are $\\textrm{A}_\\textrm{v}$=35.4,\n$\\textrm{T}_*$=33000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.38\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is\n7.7\/6. This result is in good agreement with an extremely\nabsorbed O\/B supergiant as reported in CHA08.\n\nThe best fit with the additional component gives a larger\nreduced $\\chi^2$ of 8\/4\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. We therefore\nthink that IGR~J16320-4751 is an O\/B\nsupergiant whose enshrouding material marginally contributes to\nits MIR emission, even if its intrinsic absorption is extremely high.\n\n\\subsection{IGR~J16358-4726}\n\nIGR~J16358-4726 was detected with \\textit{INTEGRAL} on 2003\nMarch 19 \\citep{2003Revnivtsev} and first observed with\n\\textit{Chandra} on 2003 March 24 \n\\citep{2004Patel}. They give its position with 0\\farcs6 accuracy\nand fitted its high-energy spectrum with an absorbed power law. They\nderive $\\Gamma\\sim0.5$ and\n$N_{\\rm H}\\sim3.3\\times10^{23}$~cm$^{-2}$. They also found a\n5880$\\pm$50~s modulation, which could be either a neutron star\npulsation or an orbital modulation. Nevertheless, \\citet{2006Patel}\nperformed detailed spectral and timing analysis of this source using\nmulti-satellite archival observations and identified a 94~s spin up,\nwhich points to a neutron star origin. Assuming that this spin up was due\nto accretion, they estimate the source magnetic field is between $10^{13}$\nand $10^{15}$ G, which could support a magnetar nature for\nIGR~J16358-4726. \n\n\\citet{2003Kouveliotou} propose 2MASS~J16355369-4725398 as the\npossible NIR counterpart, and NIR spectroscopy and photometry of\nthis counterpart was performed \nat ESO\/NTT and is reported in CHA08. They show that its\nspectrum is consistent with a B supergiant belonging to the\nsame family as IGR~J16318-4848, the so-called B[e]\nsupergiants. We also found its MIR counterpart in the GLIMPSE\narchives (G337.0994-00.0062).\n\\newline\n\nWe observed IGR~J16358-4726 with VISIR on 2006 June 29 but did\nnot detect it in any filter. Using the NIR magnitudes given in CHA08\nand the GLIMPSE fluxes, we fitted its SED\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=17.6,\n$\\textrm{T}_*$=24500~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.16\\times10^{-{11}}$,\n$\\textrm{T}_{\\rm D}$=810~K,\n$\\textrm{R}_{\\rm D}$=10.1$\\textrm{R}_\\ast$, and the reduced\n$\\chi^2$ is 3.6\/2. The best-fitting parameters\nwithout the additional component are $\\textrm{A}_\\textrm{v}$=16.7,\n$\\textrm{T}_*$=9800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$6.05\\times10^{-{11}}$\nand the corresponding reduced $\\chi^2$ is 8.8\/4. \n\nThe additional component is then necessary to correctly fit the\nSED, since\nthis source exhibits a MIR excess (see Fig 4). Even if we lack\nMIR data above 5.8 $\\mu$m, we think this excess is real and\nstems from warm dust, as it\nis consistent with the source being a sgB[e], as reported in CHA08. \n\n\\subsection{IGR~J16418-4532}\n\nIGR~J16418-4532 was discovered with \\textit{INTEGRAL} on 2003\nFebruary 1-5\n\\citep{2004Tomsick}. Using \\textit{INTEGRAL} observations,\n\\citet{2006Sguera} report an SFXT behaviour of this source and a\npeak-flux of $\\sim\\,$80~mCrab (20-30~keV). Moreover, using\n\\textit{XMM-Newton} and\n\\textit{INTEGRAL} observations, \\citet{2006Walter} report a\npulse period of 1246$\\pm$100~s\nand derive $N_{\\rm H}\\sim10^{23}$~cm$^{-2}$. They also\nproposed 2MASS~J16415078-4532253 as its likely NIR counterpart.\nThe NIR photometry of this counterpart was performed at ESO\/NTT and\nis reported in CHA08. We also found the MIR counterpart in the\nGLIMPSE archives (G339.1889+004889).\n\\newline\n\nWe observed IGR~J16418-4532 with VISIR on 2006 June 29 but did not\ndetect it in any filter. Using The NIR magnitudes given in CHA08, as\nwell as the GLIMPSE fluxes, we fitted its SED, and the\nbest-fitting parameters are $\\textrm{A}_\\textrm{v}$=14.5,\n$\\textrm{T}_*$=32800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$3.77\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is 1.4\/4. The best fit with the\nadditional component gives a larger reduced $\\chi^2$ of 3.9\/2\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. \n\nUncertainties on the data are high, which is\nthe reason why the reduced $\\chi^2$ are\nlow. Nevertheless, parameters of the fit, as well as\nthe 90\\%-confidence ranges of parameters listed in Table 6, are\nconsistent with an O\/B massive star nature. The temperature\nof the additional component being insignificant, we\nconclude this source is an O\/B massive star whose enshrouding\nmaterial marginally contributes to its MIR emission.\n\n\\subsection{IGR~J16465-4507}\n\nIGR~J16465-4507 is a transient source discovered with\n\\textit{INTEGRAL} on 2004\nSeptember 6-7 \\citep{2004Lutovinov}. Observations were carried\nout on 2004 September 14 with \\textit{XMM-Newton}, and\n\\citet*{2004Zurita} report a\nposition with 4$\\arcsec$ accuracy, allowing identification of an\nNIR counterpart in the 2MASS catalogue\n(2MASS~J16463526-4507045=USNO-B1.0~0448-00520455). With the ESO\/NTT,\n\\citet{2005Negueruela} performed intermediate-resolution spectroscopy of\nthe source, estimate the spectral type is a B0.5I, and\npropose that it is an SFXT. Using \\textit{XMM-Newton} and\n\\textit{INTEGRAL}, \\citet{2006Walter} find a pulse period of 227$\\pm$5~s\nand derive $N_{\\rm H}\\sim6\\times10^{23}$~cm$^{-2}$.\nWe found its MIR counterpart in the GLIMPSE archives\n(G340.0536+00.1350) using the 2MASS position.\n\\newline\n\nWe observed IGR~J16465-4507 with VISIR twice: \n\\begin{itemize}\n\\item The first one on 2005\n\tJune 20 during 600~s in PAH1. Typical seeing and airmass were\n\t0\\farcs81 and 1.14. We \n\tdetected the source and the derived flux is 8.7$\\pm$1.8~mJy.\n\\item the second on 2006 June 30 during 1200~s in PAH1 and\n\tPAH2. Typical seeing\n\tand airmass were 0\\farcs68 and 1.09. We \n\tdetected the source in PAH1 but not in PAH2. The derived flux is\n6.9$\\pm$1.1\n\tmJy.\n\t\n\\end{itemize}\nThese observations show that the IGR~J16465-4507 MIR flux was constant\nduring the year.\n\nUsing the USNO-B1.0, 2MASS, and GLIMPSE flux values, as well as our\nVISIR data, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=5.9,\n$\\textrm{T}_*$=25000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$6.4\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is\n13.9\/7. The best fit with the additional components gives a larger\nreduced $\\chi^2$ of 20\/5\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. \n\nWe then conclude that no additional component is needed to explain\nthe MIR emission of this source, and the parameters derived\nfrom our fit are in good agreement with IGR~J16465-4507 being a\nB0.5I as reported in \\citet{2005Negueruela}.\n\n\\subsection{IGR~J16479-4514}\n\nIGR~J16479-4514 was discovered with\n\\textit{INTEGRAL} on 2003 August 8-9 \\citep{2003Molkov}. \\citet{\n2005Sguera}\nsuggest it is a fast transient after they detected recurrent\noutbursts, and \\citet{2006Sguera} report a peak-flux of\n$\\sim\\,$120~mCrab (20-60~keV). \\citet{2006Walter} observed it\nwith \\textit{XMM-Newton} and gave\nits position with 4$\\arcsec$ accuracy. Moreover, they derive\n$N_{\\rm H}\\sim7.7\\times10^{22}$~cm$^{-2}$ from\ntheir observations. They also\npropose 2MASS~J16480656-4512068=USNO-B1.0~0447-0531332 as its\nlikely NIR counterpart. The NIR spectroscopy and photometry of\nthis counterpart were performed \nat ESO\/NTT and are reported in CHA08. It is shown that its\nspectrum is consistent with an O\/B supergiant. We also\nfound the MIR counterpart in the GLIMPSE archive\n(G339.1889+004889).\n\\newline\n\nWe observed IGR~J16479-4514 with VISIR on 2006 June 29 in PAH1 and\nPAH2, and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 0\\farcs9 and 1.14. We detected it in both filters, and\nthe respective fluxes are 10.9$\\pm$1.2~mJy and 7.0$\\pm$1.6~mJy.\nUsing the NIR magnitudes given in CHA08, as well as the GLIMPSE\nand the VISIR fluxes, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=18.5,\n$\\textrm{T}_*$=32800~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.00\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 7.4\/6. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 9\/4\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and\nour result is consistent with IGR~J16479-4514 being an\nobscured O\/B supergiant, in good agreement with CHA08. \n\n\\subsection{IGR~J17252-3616}\n\nIGR~J17252-3616 is a heavily-absorbed persistent source\ndiscovered with \\textit{INTEGRAL} on 2004 February 9 and\nreported in \\citet{2004Walter}. It was observed with\n\\textit{XMM-Newton} on 2004 March 21, and \\citet{2006Zurita} give its\nposition with 4$\\arcsec$ accuracy. Using the\n\\textit{XMM-Newton} observations, as well\nas those carried out with \\textit{INTEGRAL}, they show the source was a\nbinary X-ray pulsar with a spin period of $\\sim\\,$413.7~s and an orbital\nperiod of $\\sim\\,9.72$ days, and derive\n$N_{\\rm H}\\sim1.5\\times10^{23}$~cm$^{-2}$. Moreover, they\nfitted its high-energy spectrum with either an absorbed compton\n(kT$\\sim$ 5.5~keV and $\\tau\\sim7.8$) or a flat power law\n($\\Gamma\\sim0.02$). \n\nIn their paper, they\npropose 2MASS~J17251139-3616575 as its \nlikely NIR counterpart, as do \\citet*{2007Negueruela}. The NIR\nspectroscopy and photometry of\nthis counterpart were performed \nat ESO\/NTT and are reported in CHA08, where it is shown that its\nspectrum is consistent with an O\/B supergiant. Using the\n2MASS position, we searched for its MIR counterpart in the GLIMPSE\ncatalogue. Unfortunately, we did not find its IRAC fluxes\nin the database. Nevertheless, we found post-Basic Calibrated\nData (post-BCD) images of the source in all\nfilters. We then reduced those data and derived fluxes directly from\nthe images. They are listed in Table 3.\n\\newline\n\nWe observed IGR~J17252-3616 with VISIR on 2006 June 30 in PAH1 and\nPAH2 and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 0\\farcs97 and 1.09. We detected it in PAH1, and the\nderived\nflux is 6.1$\\pm$0.6~mJy. Using the NIR magnitudes given in CHA08,\nas well as the GLIMPSE and the VISIR fluxes, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=20.8,\n$\\textrm{T}_*$=32600~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$7.57\\times10^{-{11}}$,\nand the reduced $\\chi^2$ is 3.8\/5. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 6.9\/3\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K. We then do not\nneed any additional component to fit the SED, and\nour result is consistent with IGR~J17252-3616 to be an\nobscured O\/B supergiant, in good agreement with CHA08. \n\n\\subsection{IGR~J17391-3021}\n\nIGR~J17391-3021 is a transient source discovered with\n\\textit{INTEGRAL} on 2003\nAugust 26 \\citep{2003Sunyaev} and it corresponds to the \\textit{Rossi\nX-ray Timing Explorer} (\\textit{RXTE}) source\nXTE J1739-302. \\citet{2005Sguera} analysed archival\n\\textit{INTEGRAL} data and\nclassified the source as a fast X-ray transient presenting a typical\nneutron star spectrum. \\citet{2006Smith} observed it with\n\\textit{Chandra} on\n2003 October 15 and give its precise position with\n1$\\arcsec$ accuracy.\nThey also give its optical\/NIR counterpart\n2MASS~J17391155-3020380=USNO-B1.0~0596-0585865 and classify IGR\nJ17391-3021 as an SFXT. \\citet{2006Negueruelab}\nperformed optical\/NIR photometry and spectroscopy of the companion\nusing ESO\/NTT and find it is a O8Iab(f) star whose distance is\n$\\sim$ 2.3 kpc. CHA08 also report optical and NIR spectroscopy\nand photometry of the companion carried out at ESO\/NTT and\nconfirm the nature of the companion. Using the\n2MASS position, we searched for its MIR counterpart in the GLIMPSE\ncatalogue and as for IGR~J17252-3616, we had to reduce post-BCD data\nand derive the fluxes directly from the images. The fluxes are\nlisted in Table 3.\n\\newline\n\nWe observed IGR~J17391-3021 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the exposure\ntime was 600~s in each filter. Typical seeing and\nairmass were 0\\farcs63 and 1.13. We detected it in both filters, and\nthe derived fluxes are 70.2$\\pm$1.6~mJy and 46.5$\\pm$2.6~mJy. Using\nthe optical and the NIR magnitudes given in CHA08, as well as\nthe GLIMPSE and\nthe VISIR fluxes,\nwe fitted its SED and the best-fitting parameters are\n$\\textrm{A}_\\textrm{v}$=9.2,\n$\\textrm{T}_*$=31400~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.8\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 11.7\/10. The best fit with the\nadditional component gives a larger\nreduced $\\chi^2$ of 15.3\/8\nand $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J17391-3021 to be an O8Iab(f) supergiant star, as initially\nreported in \\citet{2006Negueruelab}.\n\n\n\\subsection{IGR~J17544-2619}\n\nIGR~J17544-2619 is a transient source discovered with\n\\textit{INTEGRAL} on 2003\nSeptember 17 \\citep{2003Sunyaev}. \\citet{2004Gonzalez-Riestra}\nobserved it with \\textit{XMM-Newton} and derive\n$N_{\\rm H}\\sim2\\times10^{22}$~cm$^{-2}$. They also confirm the\nassociation of the companion with\n2MASS~J17542527-2619526=USNO-B1.0~0636-0620933, as proposed\nin \\citet{2003Rodriguezb}. \\citet{2005Zand} report on\nobservations performed with \\textit{Chandra}, give its position\nwith 0\\farcs6\naccuracy, $N_{\\rm H}\\sim$ 1.36$\\times10^{22}$~cm$^{-2}$, and show\nthat its high-energy spectrum is typical of an accreting\nneutron star. Moreover, they identify the counterpart as a blue\nsupergiant. \\citet{2006Sguera} report a peak-flux of\n$\\sim\\,$240~mCrab. \nUsing ESO\/NTT, \\citet{2006Pellizza} performed\noptical\/NIR spectroscopy and photometry of the companion, and give\nits spectral type as O9Ib at 2.1-4.2 kpc. Using the 2MASS\nposition, we searched for its MIR counterpart in the GLIMPSE\ncatalogue, and as for IGR~J17252-3616 and IGR~J17391-3021, we had\nto reduce post-BCD data\nand derived the fluxes directly from the images. The fluxes are listed\nin Table 3.\n\\newline\n\nWe observed IGR~J17544-2619 with VISIR on 2005 June 20 in PAH1 and\nPAH2, and the exposure\ntime was 600~s in PAH1 and 1200~s in PAH2. Typical seeing and\nairmass were 0\\farcs64 and 1.13. We detected it in both filters, and\nthe derived fluxes are 46.1$\\pm$2.8~mJy and 20.2$\\pm$2.1~mJy. Using\nthe magnitudes from \\citet{2006Pellizza}, the GLIMPSE and the VISIR fluxes,\nwe fitted its SED, and the best-fitting parameters are\n$\\textrm{A}_\\textrm{v}$=6.1,\n$\\textrm{T}_*$=31000~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.27\\times10^{-{10}}$,\nand the reduced $\\chi^2$ is 6.1\/8. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 9\/6 and $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe then do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J17544-2619 as an O9Ib supergiant star, as initially\nreported in \\citet{2006Pellizza}.\n\n\n\\subsection{IGR~J19140+0951}\n\nIGR~J19140+0951 is a persistent source that was discovered with\n\\textit{INTEGRAL} on 2003 March 6-7\n\\citep{2003Hannikainen}. Observations carried out with\n\\textit{RXTE} allowed $\\Gamma\\sim1.6$\nand $N_{\\rm H}\\sim6\\times10^{22}$~cm$^{-2}$ to be derived \n\\citep*{2003Swank}. Timing analysis of the \\textit{RXTE} data\nshowed a period of 13.55 days\n\\citep{2004Corbet}, which shows the binary nature of the\nsource. After a comprehensive analysis of\n\\textit{INTEGRAL} and \\textit{RXTE} data, \\citet{2005Rodriguez}\nshow the source is spending most of\nits time in a faint state but report high variations in luminosity\nand absorption column density (up to $\\sim\\,10^{23}$~cm$^{-2}$). They\nalso find evidence that the compact object is a neutron star rather\nthan a black hole. Using \\textit{Chandra} observations carried\nout 2004 May 11,\n\\citet{2006Zand} give its position with 0\\farcs6 accuracy.\nThis allowed them to find its NIR counterpart in the 2MASS\ncatalogue (2MASS~J19140422+0952577). Moreover, they searched for\nits MIR counterpart in the Mid-course Space Experiment\n\\citep[MSX][]{1994Mill}\nand found an object at 8.3 $\\mu$m. The NIR photometry and\nspectroscopy of this source were performed at\nESO\/NTT and results are reported in CHA08. It is shown that its\nspectrum is consistent with an O\/B massive star, in good\nagreement with \\citet{2007Nespoli}, who show it is a B1I supergiant.\nUsing the 2MASS position, we also found its MIR\ncounterpart in the GLIMPSE archive (G044.2963-00.4688).\n\\newline\n\nWe observed IGR~J19140+0951 with VISIR on 2006 June 30 in PAH1 and\nPAH2, and the exposure\ntime was 1200~s in each filter. Typical seeing and\nairmass were 1\\farcs12 and 1.17. We detected it in both filters,\nand the derived\nfluxes are 35.2$\\pm$1.4~mJy and 19.1$\\pm$1.4~mJy. We point out that\nthe object given as the MSX counterpart of IGR~J19140+0951 in\n\\citet{2006Zand} is a blended source. Indeed, VISIR\nimages, whose resolution is far better, clearly show there are\ntwo sources in the field,\nIGR~J19140+0951 and a very bright southern source (see Fig 2). \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=5cm]{.\/19140.eps}\n\t\\caption{VISIR image of IGR~J19140+0951 in PAH1 (8.59\n\t$\\mu$m). 19\\farcs2x19\\farcs2 field of view and 0\\farcs075\n\tplate scale. We clearly see the two sources that were blended\n\twith MSX. The MIR counterpart of IGR~J19140+0951 is the\n\tnorthern source.}\n\\end{figure}\n\nUsing the magnitudes\ngiven in CHA08, as well as the fluxes from GLIMPSE and our\nobservations with VISIR, we fitted its SED,\nand the best-fitting parameters are $\\textrm{A}_\\textrm{v}$=16.5,\n$\\textrm{T}_*$=22500~K,\n$\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$=$1.92\\times10^{-{10}}$, and\nthe reduced $\\chi^2$ is 14.4\/6. The best fit with the additional\ncomponent gives a larger\nreduced $\\chi^2$ of 20.2\/4 and $\\textrm{T}_{\\rm D}\\,<\\,200$ K.\n\nWe do not need any additional component to fit the SED, and the\nparameters derived from our fit are in good agreement with\nIGR~J19140+0951 to be an B1I supergiant star, as initially\nreported in \\citet{2007Nespoli}.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16195.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16207.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16318.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16320.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16358.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16418.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16465.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr16479.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17252.eps}\\\\\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17391.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr17544.eps}&\n\t\\includegraphics[angle=-90,width=6.cm]{.\/igr19140.eps}\\\\\n\t\\end{tabular}\n\t\\caption{\\small Optical-to-MIR absorbed (line) and unabsorbed\n\t(dotted-line) SEDs of 12 \\textit{INTEGRAL} sources,\nincluding broad-band photometric data\n\tfrom ESO\/NTT, 2MASS, GLIMPSE, and VISIR.}\n\\end{figure*} \n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16195_star_dust.eps}\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16318_star_dust.eps}\n\t\\includegraphics[angle=-90,width=8.5cm]{.\/igr16358_star_dust.eps}\n\t\\caption{IR SEDs of IGR~J16195-4945,\n\tIGR~J16318-4848, and IGR~J16358-4726 in the NIR and the MIR. We\n\tshow their SEDs including the contribution of the star and the\n\tdust (line), the star only (dashed-line), and the dust only\n(dotted-line).}\n\\end{figure} \n\n\\section{Discussion}\n\n\n\\subsection{B vs B[e] supergiants stellar winds}\n\nAll SEDs were best-fitted without any dust component (even\nthe very absorbed one like IGR~J16320-4751), except three of them\n(IGR~J16195-4945, IGR~J16318-4848, and\nIGR~J16358-4726, see Fig. 3) that exhibit a MIR excess likely\ndue to the presence of dust in their stellar wind. \n\nBlue supergiants are known to\nexhibit a very strong but sparse stellar wind of high\nvelocity ($\\sim$ 1000-2000~km~s$^{-1}$). This has been\nexplained through the so-called\nradiation line-driven CAK model \\citep*{1975Castor} in which the\nwind is driven by\nabsorption in spectral lines. Hot stars emit most of\ntheir radiation in the ultraviolet (UV) where their atmosphere\nhas many absorption lines. Photons coming from the\nphotosphere of the star with the same wavelength are absorbed and\nre-emitted to the expanding medium in a random direction with\nalmost the same\nmomentum, which results in acceleration of the wind. This process\nis very effective because the line\nspectrum of the scattering ions in the wind is Doppler-shifted\ncompared to the stellar rest frame, so the scattering atoms are\nshifted with respect to their neighbours at lower velocities and\ncan interact with an unaffected part of the stellar spectrum.\n\\newline\n\nIGR~J16318-4848 was proven to belong to a particular class of B1\nsupergiants, the B[e] supergiants or sgB[e]\n\\citep*{2004Filliatre}. \nA physical definition of B[e] stars can be found in\n\\citet{1998Lamers}. We just recall two of the\ncharacteristics here: the presence of forbidden emission lines\nof [\\ion{Fe}{ii}] and [\\ion{O}{i}] in the NIR spectrum and \nof a strong MIR excess due to hot circumstellar dust \nthat re-emits the absorbed stellar radiation through\nfree-free emission.\nAn sgB[e] is defined by the B[e] phenomenon, the\nindication of mass-loss in the optical spectrum (P-cygni\nprofiles), and a hybrid spectrum characterised by the\nsimultaneous presence of narrow low-excitation lines and broad\nabsorption features of high-excitation lines. This hybrid nature\nwas empirically explained by the simultaneous presence of a\nnormal supergiant hot polar wind (fast and sparse) and\nresponsible for the broad lines and a cool equatorial\noutflowing disk-like\nwind (slow and dense) responsible for the narrow lines\n\\citep{1983Shore,1985Zickgraf,1987Shore}. This\nempirical model has received some confirmation from polarimetry\n\\citep*{1999Oudjmaier}.\n\\newline\n\nThere are a few models that explain the creation of this outflowing\ndisk, and all of them consider the\nstar rotation to be an important parameter in the process. In\nthis paper, we present only the most consistent of them,\nthe Rotation\nInduced Bi-stability mechanism (RIB), but a review can be found\nin \\citet*{2006Kraus}. \n\\newline\n\nThe lines responsible for the creation of the wind are dependent\non the ionisation structure, and a change in this structure leads to\na change in the radiative flux. This is the bi-stability jump\nfound by \\citet*{1991Lamers}, which appears\nfor B stars with effective temperatures\nof about 23000~K. Above this temperature, the wind tends to be fast\nand sparse. Below, the mass-loss rate if five times higher and\nthe terminal velocity two\ntimes slower, which leads to a wind that is ten times denser. \n\n\\citet*{1997Cassinelli} propose that\nthe same effect is important from polar to equatorial regions\nfor rapidly\nrotating B stars. Indeed, the rapid rotation leads to polar\nbrightening that increases the poles temperature to the hot\nside of the jump. At the same time, the rotation leading to gravity\ndarkening, the equatorial region may be on the cool side of the\njump. Consequently,\nthe wind in the equatorial region is denser than the\nwind in the polar region. Nevertheless, \\citet{2000Pelupessy}\nshow that the rotational velocity of the star should be very\nclose to its critical value to allow the equatorial wind to reach the\ndensity needed to create the disk. However, supergiant stars\ncannot be close to critical rotational velocity because of\nprobable disruption. Additional mechanisms are therefore\nneeded to allow the supergiant star to reach its\ncritical velocity \\citep[see\ne.g. ][]{2006Owocki}. In the particular case of an sgB[e]\nstar in an X-ray binary system, the spin-up should occur during the\nsupergiant phase of the companion, which indicates a different\nevolutionary stage from other HMXBs.\n\nThis disk itself cannot explain the strong MIR excess the sgB[e] stars\nexhibit. Nevertheless, \\citet*{1993Bjorkman} have shown\nthe existence of a zone in the disk (about 50-60 stellar radii\nfrom the star) in which the temperature is below the\ntemperature of sublimation of the dust (about 1500~K) and the density\nhigh enough to allow for its creation.\n\\newline\n\n\nIGR~J16318-4848 is the source in our sample that \nexhibits the strongest MIR excess, and we believe it is due to\nthe sgB[e] nature of its companion star. Indeed, many other\nstrongly absorbed sources in our sample do not present any MIR\nexcess. \n\nMoreover, it is suggested that IGR~J16358-4726, the second\nsource in our sample that exhibits a MIR\nexcess and whose SED needs an additional component to be properly\nfitted is an sgB[e], because its spectrum has all the\ncharacteristic features of supergiant stars plus the [\\ion{Fe}{ii}]\nfeature (CHA08). Our fit is therefore in good\nagreement with their result, and the only other source of our\nsample that definitely exhibits a MIR excess is indeed an sgB[e]\nstar.\n\nIn the case of IGR~J16195-4945, we are more\ncautious concerning the presence of warm dust that could be\nresponsible for a MIR excess, as we lack data above 8\n$\\mu$m. Indeed, Fig 4. shows that these source could exhibit\na MIR excess, but one much lower than the other two. Nevertheless,\nif this excess were to be confirmed, we\nbelieve it would be also due to the sgB[e] nature of the\ncompanion.\n\\newline\n\nWe would like to point out that, because the dust is\nmost located in an equatorial disk in an\nsgB[e] star, the simple model we used to fit the\nSEDs cannot reproduce the complex distribution of\nthe dust around these stars. Nevertheless, it allows the\ndetection of a warm MIR\nexcess because of the presence of dust in the stellar\nwinds. Finally, for all the stars in our sample, we cannot\nexclude the presence of a cold\ncomponent - responsible for their intrinsic absorption - which we\ncannot detect because of the lack of data above 20 $\\mu$m.\n\n\n\\subsection{Spectral type and distance}\n\n\\begin{table}\n\t\\caption{Summary of spectral types (SpT) and distances\n\t($\\textrm{D}_\\ast$) derived from our fits for confirmed\n\tsupergiant stars in our sample. ($^\\ast$)\n\tsources with an accurate spectral type found in the literature\n\t($^\\dagger$) confirmed supergiant stars whose\ntemperature derived from\n\tour fits was used to assess their accurate spectral type. \nReferences to the determination of the spectral type and\/or\nspectral class of these sources are found in Table 1.}\n\t$$\n\t\\begin{array}{c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{SpT}&\\textrm{D}_\\ast(\\textrm{kpc})\\\\\n\t\\hline\n\t\\textrm{IGR~J16318-4848}^\\ast&\\textrm{sgB[e]}&\\sim1.6\\\\\n\t\\hline\n\t\\textrm{IGR~J16320-4751}^\\dagger&\\textrm{O8I}&\\sim3.5\\\\\n\t\\hline\n\t\\textrm{IGR~J16358-4726}^\\ast&\\textrm{sgB[e]}&\\sim18.5\\\\\n\t\\hline\n\t\\textrm{IGR~J16465-4507}^\\ast&\\textrm{B0.5I}&\\sim9.4\\\\\n\t\\hline\n\t\\textrm{IGR~J16479-4514}^\\dagger&\\textrm{O8.5I}&\\sim4.9\\\\\n\t\\hline\n\t\\textrm{IGR~J17252-3616}^\\dagger&\\textrm{O8.5I}&\\sim6.1\\\\\n\t\\hline\n\t\\textrm{IGR~J17391-3021}^\\ast&\\textrm{O8I}&\\sim2.7\\\\\n\t\\hline\n\t\\textrm{IGR~J17544-2619}^\\ast&\\textrm{O9I}&\\sim3.6\\\\\n\t\\hline\n\t\\textrm{IGR~J19140+0951}^\\ast&\\textrm{B1I}&\\sim3.1\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table} \n\n\\begin{table}\n\t\\caption{Summary of the distances\n\t($\\textrm{D}_\\ast$) derived from our fits for sources\n\twith unconfirmed spectral classes.}\n\t$$ \n\t\\begin{array}{c c ccc}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&\\textrm{SpT}&\\multicolumn{3}{c}{\\textrm{D}_\n\\ast(\\textrm{kpc})}\\\\\n\t\\hline\n\t&&\\textrm{V}&\\textrm{III}&\\textrm{I}\\\\\n\t\\hline\n\t\\textrm{IGR~J16195-4945}&\\textrm{B1}&\\sim3.1&\\sim5.7&\\sim9.8\\\\\n\t\\hline\n\t\\textrm{IGR~J16207-5129}&\\textrm{O7.5}&\\sim1.8&\\sim2.8&\\sim4.1\\\\\n\t\\hline\n\t\\textrm{IGR~J16418-4532}&\\textrm{O8.5}&\\sim4.9&\\sim8.3&\\sim13\\\\\n\t\\hline\n\t\\end{array}\n\t$$ \n\\end{table}\n\nIn our sample, six sources are supergiant stars with a known\nspectral type - IGR~J16318-4848 and IGR~J16358-4726 are sgB[e],\nIGR~J16465-4507 is a B0.5I, IGR~J17391-3021 is an 08Iab(f),\nIGR~J17544-2619 is an O9Ib, and IGR J19140+0951 is a B1I - and\nthree are found to be O\/B supergiants whose temperatures derived\nfrom our fits allow an assessment of the spectral types\nusing the classification given in \\citet{2005Martins} and\n\\citet{2006Crowther} for O and B galactic\nsupergiants, respectively, given the uncertainties of\nobservational results\n($\\sim$ 2000~K) and uncertainties on the fits temperatures as given in\nTable 6. We therefore found that IGR~J16320-4751 could be an\n08I and IGR~J16479-4514 and\nIGR~J17252-3616 could be 08.5I stars. \n\\newline\n\nConcerning the last three sources whose spectral class is unknown,\nresults of the fits listed in\nTables 5 and 6 show that they are probably all O\/B massive\nstars, and we also used their derived temperatures to \nassess their spectral type using the\nclassification given in both papers quoted\nabove. IGR~J16195-4945 could be a B1 star, and as already stressed\nabove, it could be an sgB[e] due to its MIR excess, IGR~J16207-5129\nand IGR~J16418-4532 could be O7.5-O8.5 stars. \nNevertheless, even if the high intrinsic X-ray absorption of\ntheir associated compact objects points towards a supergiant\nnature since the\naccretion is likely to be wind-fed, the fits\nthemselves do not allow an assessment of their spectral\nclasses. We then consider they could be either main\nsequence, giant, or supergiant stars.\n\\newline\n\n\\citet*{2006Martins} give a UBVJHK synthetic\nphotometry of galactic OI, OIII, and OV stars, with which one can\nget the expected unabsorbed absolute magnitude in J band\n$\\textrm{M}_{\\rm J}$ for stars having a given spectral\nclassification. Using the absorbed apparent magnitudes\n$\\textrm{m}_{\\rm J}$\nof our sources and the J band absorption we derived\nfrom our fits,\n$\\textrm{A}_{\\rm J}\\,=\\,0.289\\times\\textrm{A}_\\textrm{v}$, it\nis then possible to assess the distance of O stars\nin our sample using the standard relation:\n\n\\begin{displaymath}\n\t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,D_\n\\ast\\,=\\,10^{\\textrm{\\normalsize\n$0.2(m_{\\rm J}-A_{\\rm J}-M_{\\rm J}+5)$}}\\,\\,\\,\\,\\,\\,\\textrm{in\npc}\n\\end{displaymath}\nWe did not find any synthetic\nphotometry for B supergiants. Nevertheless, expected radii of\ngalactic BI, BIII, and BV stars are given in \\citet{1996Vacca},\nand we divided these\nvalues by the $\\frac{\\textrm{R}_\\ast}{\\textrm{D}_{\\ast}}$ ratio\nderived from our fits to get the star distance. The derived\ndistances for the sources whose spectral\nclass is known are listed in Table 7, in Table 8 for the others.\n\n\\subsection{X-ray properties}\n\nExcept in the case of IGR~J17544-2619, X-ray absorptions\nare systematically significantly larger than the visible\nabsorptions. This \nindicates the presence in the system of a two-component\nabsorbing material: one \naround the companion star, responsible for the visible\nabsorption, and a very dense one around the compact object\ncoming from the stellar winds that\naccrete onto the compact object and are responsible for the huge\nX-ray absorption those sources exhibit. \n\nThe obscuration of the compact object by the stellar wind is\ncaused by the photoelectric absorption of the X-ray emission by the\nwind, and this absorption varies along the\norbit of the compact object. This orbital dependence has for\ninstance been observed and modelled on 4U\n1700-37 \\citep*{1989Haberl}. \n\nMoreover, an effect on the X-ray absorption by the photoionisation\nof the stellar wind in the vicinity of the compact object by its\nX-ray emission was predicted by \\citet*{1977Hatchett}. \nIndeed, in SGXBs, the compact object moves through the\nstellar wind of the companion star, and the X-rays are\nresponsible for the enhancement or the depletion of the ionised\natoms responsible\nfor the acceleration of the wind\n(e.g. \\ion{C}{iv} and \\ion{N}{v}). \nThis has a direct consequence on the velocity profile of the\nwind; when the wind enters into an ionised zone, it\nfollows a standard CAK law until it reaches a location in which it\nis enough ionised for no further radiative driving to take\nplace, and the wind velocity is ``frozen'' to a constant value\nfrom this point. This results in a lower wind\nvelocity close to the compact object and consequently a higher\nwind density that leads to a higher obscuration of the compact object.\n\\newline\n\nMost of the sources studied in this\nwork are very absorbed in the high-energy\ndomain. Nevertheless, this absorption may not be always that\nhigh. In\nthe case of very wide eccentric orbits, the column density of\nthe sources could\nnormally vary along their orbit and\nsuddenly increase when very close to the companion star because\nof the wind ionisation. In contrast, if these objects were to\nbe always very absorbed, it could mean that their orbit is \nvery close to the companion star and weakly eccentric. If this\neffect were to be observed, we think\nit could explain the difference in behaviour between\nobscured SGXBs (close quasi-circular orbits) and SFXTs (wide\neccentric orbits).\n\n\\subsection{Optical properties}\n\\begin{table}\n\t\\caption{Sample of parameters we used to fit the SEDs of the\n\tisolated supergiants. We give their\n\tgalactic coordinates, their spectral types, the interstellar\n\textinction in magnitudes $\\textrm{A}_\\textrm{i}$ and then the\n\tparameters themselves: the extinction in the optical\n\t$\\textrm{A}_\\textrm{v}$, the temperature\n\t$\\textrm{T}_\\ast$ and the\n\t$\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}$ ratio of the\n\tstar.}\n\t$$\n\t\\begin{array}{c c c c c c c c c c}\n\t\\hline\n\t\\hline\n\t\\textrm{Sources}&l&b&\\textrm{SpT}&\\textrm{A}_\\textrm{i}&\\textrm{A}\n_\\textrm{v}&\\textrm{T}_{\\ast}\n(\\textrm{K})&\\frac{\\textrm{R}_{\\ast}}{\\textrm{D}_{\\ast}}\\\\\n\t\\hline\n\t\\textrm{HD~144969}&333.18&2.0&\\textrm{B0.5Ia}&3.34&3.9&26000&4.01\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~148422}&329.92&-5.6&\\textrm{B0.5Ib}&0.75&0.9&24700&8.76\n\\times10^{-{11}}\\\\\n\t\\hline\n\t\\textrm{HD~149038}&339.38&2.51&\\textrm{B1Ia}&0.81&1&24000&5.30\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~151804}&343.62&1.94&\\textrm{O8Iaf}&0.83&1.3&32000&4.20\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~152234}&343.46&1.22&\\textrm{B0.5Ia}&1.17&1.5&25100&4.94\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~152235}&343.31&1.1&\\textrm{B1Ia}&3.37&3.9&24500&1.13\n\\times10^{-{9}}\\\\\n\t\\hline\n\t\\textrm{HD~152249}&343.35&1.16&\\textrm{O9Ib}&1.34&1.7&30100&2.89\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\textrm{HD~156201}&351.51&1.49&\\textrm{B0.5Ia}&2.68&2.9&26500&2.90\n\\times10^{-{10}}\\\\\n\t\\hline\n\t\\end{array}\n\t$$\n\\end{table} \n\nWe were able to fit all but three sources\nwith a simple stellar black body model. For these three\nsources, we explained that the MIR excess was probably caused by\nthe warm dust created within the stellar wind due to the\nsgB[e] nature of the companions. Therefore, it seems that the\noptical-to-MIR wavelength emission of these SGXBs corresponds\nto the emission of absorbed blue supergiants or sgB[e]. \n\nMoreover, the results of the fits listed in the Table 5 show\nthat it is \\textit{a priori} impossible to\ndifferentiate an obscured SGXB and an SFXT from their\noptical-to-MIR wavelength SEDs, and it then seems that the difference in\nbehaviour between both kinds of SGXBs only depends on the\ngeometry of the system,\ni.e. its orbital distance or its orbit eccentricity \\citep*{2006Chaty}. \n\\newline\n\nNevertheless, to assess a possible effect of the\ncompact object on the companion star, we took a sample of eight\nisolated O\/B supergiants in the direction of the Galactic\ncentre and fitted their optical-to-NIR wavelength SEDs with an absorbed\nstellar black body. The best-fitting parameters are listed in Table\n9 along to their galactic coordinates, their spectral types and\nthe interstellar \\ion{H}{i} absorption (A$_\\textrm{i}$). The\ndistances of these supergiants are known, which allowed\nus to calculate A$_\\textrm{i}$ out to their\nposition using the tool available on the MAST website\n\\citep{1994Fruscione}.\n\nWe see that their visible absorption is\nof the same order of \nmagnitude as the interstellar \\ion{H}{i} absorption and well\nbelow the level of absorption of our sources. This could mean that \nsome supergiant stars in SGXBs exhibit an excess of absorption due to\na local absorbing component. Unfortunately, the total\ninterstellar absorption out to the distance\nof our sources is unknown, and we cannot compare\ntheir visible absorptions derived from our fits to the total\ninterstellar absorption out to their position.\n\nNevertheless, if this was the case, we think that this excess of \nabsorption could also be caused partly by the photoionisation of the\nwind in the vicinity of the companion star by the high-energy\nemission of the compact object, as this would make their winds\nslower than in isolated supergiant stars. Since the wind velocity\nis lower, the medium is denser and suitable for creating a more\nabsorbant material.\n\nIndeed, in the case of persistent sources with very\nclose and quasi-circular orbits, we think that this\npossible effect could be particularly strong, since the wind around\nthe companion star\nwould be permanently photoionised and would have lower\nvelocities than in isolated supergiants. This could be the\ngeneral scheme of obscured SGXBs.\n\nOn the other hand, in the case of very wide and eccentric\norbits, the compact object would be most of the time far from the\nsecondary and its X-ray emission would not photoionise\nthe wind close to the companion star, which would not exhibit any\nvisible absorption excess until the compact object got\ncloser. This could be the general scheme of SFXTs.\n\nAt last, in both cases, it would be possible to observe a\nvariation in the P-Cygni profiles of the companion star\n(i.e. a variation in the wind velocity) with the phase angle of\nthe compact object along its orbit.\n\\newline\n\nAs a possible confirmation of this general behaviour, we point\nout that the visible absorptions derived from our fits for \nthe companion stars of the only sources in our sample that\nsurely exhibit the SFXT\nbehaviour (IGR~J16465-4945, IGR~J17391-3021, and IGR~J17544-2619)\nare far smaller than the visible absorptions of the\nothers. Moreover, concerning obscured SGXBs, the wind velocity\nof IGR~J16318-4848 was found to be $\\sim$ 410~km~s$^{-{1}}$\n\\citep*{2004Filliatre}, far lower than\nthe expected wind velocity for O\/B supergiants ($\\sim$ 1000-2000~km~s$^{-{1}}$).\n\n\\section{Conclusions}\n\nIn this paper, we presented results of observations performed\nat ESO\/VLT with VISIR, which aimed at studying the MIR emission of twelve\n\\textit{INTEGRAL} obscured HMXBs, whose\ncompanions are confirmed or candidate supergiants. Moreover,\nusing the observations performed at ESO\/NTT and reported in the\ncompanion paper (CHA08), previous optical\/NIR observations found\nin the literature and archival data\nfrom USNO, 2MASS, and GLIMPSE, we fitted the broad-band SEDs of\nthese sources using a simple two-component black body model in\norder to obtain their visible absorptions and temperatures, and\nto assess the contribution of their enshrouding material in their\nemission. \n\nWe confirmed that all these sources were likely O\/B supergiant\nstars and that,\nfor most of them, the enshrouding material marginally contributed to the\nemission. Moreover, in the case of IGR~J16318-4848,\nIGR~J16358-4726, and perhaps\nIGR~J16195-4945, the MIR excess\ncould be explained by the sgB[e] nature of the companion stars. \n\nBy comparing the optical and high-energy characteristics of\nthese sources, we showed that the distinction SFXTs\/obscured\nSGXBs does not seem to exist from optical-to-MIR\nwavelength. \nNevertheless, most of the sources in our sample are\nsignificantly absorbed in the optical, and we think that the wind\ncan be denser\naround some supergiants in SGXBs, which could be due to the\nphotoionisation by the high-energy emission \nof the compact object.\n\\newline\n\nSeveral improvements in our study are needed to\nallow definitive conclusions. Indeed, the data used to perform\nthe SEDs were not taken simultaneously, which can for instance\nlead to an incorrect assessment of the MIR excess in the emission.\nMoreover, the lack of optical magnitudes for several sources\ncould have led to an incorrect fitting of their intrinsic visible\nabsorption $\\textrm{A}_\\textrm{v}$.\nFinally, the absence of an accurate measurement of the total\ninterstellar absorption out to the distance of these sources\ndoes not allow us to say whether the\npresence of the compact object can lead to a stellar wind denser\nin some supergiants belonging to SGXBs than in isolated supergiants.\n\\newline\n\nWe then recommend further\noptical investigations of these sources to study any\npossible variation in their P-cygni profile \nwith the phase of the compact object. We also think that the\nmeasurement of the distance of these\nsources is crucial to allow a good assessment of the real\ninterstellar absorption up to their distance, in order to detect\nany local absorbing component around companion stars. Finally,\nwe recommend X-ray monitoring so as to study the\ndependence of their column density on orbital phase angle,\nwhich could help for understanding the difference between obscured\nSGXBs and SFXTs.\n\n\\begin{acknowledgements}\n\t\n\tWe are pleased to thank J\\'er\\^ome Rodriguez for his very\n\tuseful website in which all the \\textit{INTEGRAL} sources are\n\treferenced (\n\thttp:\/\/isdc.unige.ch\/$\\sim$rodrigue\/html\/igrsources.html).\n\t\\newline\n\tBased on observations carried out at the European Southern\n\tObservatory, Chile (through programmes ID. 075.D-0773 and\n\t077.D-0721). This research has made use of NASA's Astrophysics\n\tData System, of the SIMBAD and VizieR databases operated at\n\tthe CDS, Strasbourg, France, of products from the US Naval\n\tObservatory catalogues, of products from the Two\n\tMicron All Sky Survey, which is a\n\tjoint project of the University of Massachusetts and the\n\tInfrared Processing and Analysis Center\/California Institute\n\tof Technology, funded by the National Aeronautics and Space\n\tAdministration and the National Science Foundation as well as\n\tproducts from the Galactic Legacy Infrared Mid-Plane Survey\n\tExtraordinaire, which is a \\textit{Spitzer Space Telescope}\n\tLegacy Science Program.\n\\end{acknowledgements}\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of  Quantum Chromodynamics (QCD) at high temperature\nand\/or baryon density is certainly one of the most active and interesting research topic of\nmodern high energy physics. It has been predicted by means of  first principles calculations \nthat a smooth crossover exists from the low temperature hadron gas phase to a quark-gluon plasma,\nat a (pseudo)-critical temperature $T_c\\approx 150$ \nMeV \\cite{Borsanyi:2010bp,Borsanyi:2010cj,Cheng:2009zi,Bazavov:2011nk,Borsanyi:2013bia}.\nThis crossover is accompanied by the approximate restoration of chiral symmetry.  \nWhile many studies focused on the QCD phase structure in the infinite volume limit,\nit is of a certain interest to investigate the effects of finite size on the critical lines of QCD\nin the temperature-baryon chemical potential plane.\nChiral symmetry restoration at finite temperature and density, for quantization \nin a cubic box of size $L$ with periodic or antiperiodic\nboundary conditions as well as with standing waves conditions, has been treated in a number of studies,\nsee \\cite{Xu:2019gia,Xu:2019kzy,Almasi:2016zqf,Palhares:2009tf,Braun:2011iz,Braun:2005fj,Kiriyama:2006uh,Braun:2005gy,Wang:2018ovx,\nAbreu:2019czp,Magdy:2019frj,Aoki:2002uc,Guagnelli:2004ww,Orth:2005kq,Deb:2020qmx,Zhao:2019ruc,Wang:2018qyq,\nEbert:2010eq,Bhattacharyya:2015zka,Bhattacharyya:2014uxa,Bhattacharyya:2012rp}, \nsee \\cite{Klein:2017shl} for a review. \n  \n\nWe perform a study of chiral symmetry breaking in two-flavor QCD using the renormalized quark-meson (QM) \nmodel \\cite{Skokov:2010sf,Frasca:2011zn,Ruggieri:2013cya},\nwith quantization in a cubic box of size $L$ with periodic boundary conditions. \nOur work differs from previous calculations based on the same model in the way we treat the vacuum quark loop.\nIn fact, we include the vacuum term in the thermodynamic potential:\nsince this is a divergent quantity, care should be put in the regularization and then renormalization of this contribution. \nWe follow the regularization procedure of \\cite{Xu:2019gia} with Pauli-Villars regulators; \nthis regularization is suitable for calculations at finite size\nbecause it cancels the ultraviolet divergences in a very transparent manner and leaves a finite quark loop\nboth in the infinite and the finite $L$ cases. After the regularization has been done,\nwe perform renormalization requiring\nthat in the infinite volume limit, the quark loop does not shift the expectation value of the condensate\nand of the $\\sigma-$meson mass obtained from the classical potential.\nApplying these conditions fixes the counterterms in a way that does not depend of $L$.\nTherefore, the predictions of the model at finite $L$ are unaffected by the ultraviolet divergence.\n\nThe main purpose of this study is the restoration of chiral symmetry at finite quark chemical potential, $\\mu$,\nand low temperature. When the size is small enough, $L\\lesssim 5$ fm, \nan intermediate phase appears between the chiral symmetry broken phase (the hadron gas) \nand chiral symmetry restored phase (normal quark matter). In this intermediate phase,\nthe condensate experiences a decrease but its lowering is not enough to restore chiral symmetry.\nTherefore, the symmetry breaking pattern in this new phase is the same of the hadron gas.\nHowever, this phase differs from the hadron gas phase because it has a nonzero baryon density;\nthe change of density, together with the unchanged symmetry breaking pattern, is reminiscent of a\ngas-to-liquid phase transition.\nMoreover, the correlation domains of the order parameters are larger than those found in the hadron gas\nand normal quark matter phases. \nFor all these reasons, we call this new phase the \n{\\em subcritical liquid phase}. \n\n\nWe also compute $T_c$ versus $L$. Our results  agree with previous studies that implemented effective models\nwith periodic boundary conditions or infrared cutoffs  \\cite{Xu:2019gia,Palhares:2009tf,Wang:2018ovx,Magdy:2019frj}, \nnamely $T_c$ increases with $1\/L$.\nThis is understood in terms of the zero mode that contributes to the  condensate when periodic\nboundary conditions are used,\nas well as of the curvature of the thermodynamic potential which is negative for any $L$\nin the QM model.\n\nThe plan of the article is as follows. In Section II we briefly review the quark-meson model\nand describe the renormalization procedure when periodic boundary conditions are implemented. \nIn Section III we discuss the chiral phase transition at small $T$ and large $\\mu$.\n In Section IV we present a few results for the chiral restoration at finite $T$. \n Finally, in Section V we draw our conclusions.\nWe use the natural units system $\\hbar=c=k_B=1$ throughout this article.\n\n\n\n\n\n\n\\section{The quark-meson model with periodic boundary conditions}\n\n\\subsection{The lagrangian density}\n\nThe QM model is an effective model of QCD with quarks, $\\sigma$ and $\\pi$ mesons\n(in the two-flavor versions that we consider here).\nThe meson lagrangian density is\n\\begin{eqnarray}\n{\\cal L}_\\mathrm{mesons} &=& \\frac{1}{2}\\left(\n\\partial^\\mu\\sigma\\partial_\\mu\\sigma + \\partial^\\mu\\bm\\pi\\cdot\\partial_\\mu\\bm\\pi\n\\right) \\nonumber\\\\\n&&- \\frac{\\lambda}{4}\\left(\\sigma^2 + \\bm\\pi^2 - v^2\\right)^2 +h\\sigma,\\label{eq:ls1_aa}\n\\end{eqnarray}\nwhere $\\bm\\pi = (\\pi_1,\\pi_2,\\pi_3)$ corresponds to the pion isotriplet field. This lagrangian density is\ninvariant under $O(4)$ rotations. On the other hand, as long as $v^2 > 0$ the potential\ndevelops an infinite set of degenerate minima. We choose one ground state, namely\n\\begin{equation}\n\\langle\\bm\\pi\\rangle=0,~~\\langle\\sigma\\rangle\\neq 0.\\label{eq:gs_aa}\n\\end{equation}\nThe ground state~\\ref{eq:gs_aa}\nbreaks the $O(4)$ symmetry down to $O(3)$ since the vacuum is invariant only under the rotations of the pion fields.\nThis is how chiral symmetry is spontaneously broken in this model.\nBesides the spontaneous breaking, chiral symmetry is broken softly but explicitly by the term\n$h\\sigma$ in the lagrangian density; in fact, the pion mass is $m_\\pi^2 = h\/F_\\pi$.\nAt zero temperature and in the chiral limit  $\\langle\\sigma\\rangle=F_\\pi\\approx 93$ MeV \nwhere $F_\\pi$ denotes the pion decay constant in the vacuum.\n\n\nThe quark sector of the QM model is described by the lagrangian density\n\\begin{equation}\n{\\cal L}_\\mathrm{quarks} = \\bar\\psi\\left(\ni\\partial_\\mu\\gamma^\\mu - g(\\sigma +i\\gamma_5 \\bm\\pi\\cdot\\bm\\tau)\n\\right)\\psi,\\label{eq:qlg_aaa}\n\\end{equation}\nwhere $\\bm\\tau$ are Pauli matrices in the flavor space. In the ground state~\\ref{eq:gs_aa} quarks get a dynamical (that is,\na constituent) mass given by\n\\begin{equation}\nM = g\\langle\\sigma\\rangle.\\label{eq:pppAAA}\n\\end{equation}\n We notice that in Eq.~\\ref{eq:qlg_aaa} there is no explicit mass term for the quarks. As a matter of fact,\nin this effective model the explicit breaking of chiral symmetry   is achieved by\n  $h\\neq 0$ in Eq.~\\ref{eq:ls1_aa}.\n Although in Eq.~\\ref{eq:qlg_aaa}  there is no explicit mass term, quarks get a constituent mass because of the spontaneous\nbreaking of the $O(4)$ symmetry in the meson sector: this implies that the quark chiral condensate can be nonzero.\nThe total lagrangian density is given by\n\\begin{equation}\n{\\cal L}_\\mathrm{QM} = {\\cal L}_\\mathrm{quarks} + {\\cal L}_\\mathrm{mesons}.\n\\end{equation}\nIn the following, we will use the notation $\\sigma$ to denote both the field and its expectation value,\nunless from the context it is not clear which of the two we write about.\n\n\n\\subsection{Renormalized thermodynamic potential in the infinite volume limit}\n\n \n\nThe mean field effective potential of the QM model in the infinite volume is given by\n\\begin{equation}\n\\Omega = U + \\Omega_{0,\\infty} + \\Omega_T,\\label{eq:ep1aa}\n\\end{equation}\nwhere\n\\begin{equation}\nU = \\frac{\\lambda}{4}\\left(\\sigma^2 + \\bm\\pi^2 - v^2\\right)^2 - h\\sigma\\label{eq:ls1_aaMMM}\n\\end{equation}\nis the classical potential of the meson fields as it can be read from Eq.~\\ref{eq:ls1_aa}, and\n\\begin{equation}\n\\Omega_{0,\\infty} = -2N_c N_f\\int\\frac{d^3p}{(2\\pi)^3} E_p  \\label{eq:ls1_aaMMMa}\n\\end{equation}\nis the one-loop quark contribution, with\n\\begin{equation}\nE_p = \\sqrt{p^2 +M^2},~~~M=g\\sigma.\n\\end{equation}\nFinally, $\\Omega_T$ corresponds to the finite temperature quarks contribution that we specify later.\nFor regularization and renormalization we can limit ourselves to consider the zero temperature limit of $\\Omega$,\n\\begin{equation}\n\\Omega_0 = U + \\Omega_{0,\\infty}.\\label{eq:ep1}\n\\end{equation}\nEquation~\\ref{eq:ep1} represents the effective potential for the $\\sigma$ field computed at one-loop\nand after renormalization it corresponds to the renormalized condensation energy, namely the difference between\nthe energy of the state with $\\langle\\sigma\\rangle\\neq 0$ and $\\langle\\sigma\\rangle= 0$ at $T=\\mu=0$.\n\nIt is instructive to present the renormalization of $\\Omega_0$ in the infinite volume system firstly.\nIn order to do this, we have to regularize the momentum integral in Eq.~\\ref{eq:ls1_aaMMMa}.\nWe have found that for problems with quantized momenta, in which summations replace integrals, \nrenormalization based on the Pauli-Villars (PV) method is the most convenient one.\nTherefore, also in the infinite volume limit we use PV regularization and renormalization.\nThis is the same method used in \\cite{Xu:2019gia}.\n\nIn the PV scheme we replace Eq.~\\ref{eq:ls1_aaMMMa} with\n\\begin{equation}\n\\Omega_{0,\\infty} = -2N_c N_f\\int\\frac{d^3p}{(2\\pi)^3} \n\\sum_{j=0}^3c_j \\left(E_p^2 + j \\xi^2\\right)^{1\/2},  \\label{eq:ls1_aaMMMaPV}\n\\end{equation}\nwhere $\\{c_j\\}$ is a set of PV coefficients and $\\xi$  is the renormalization scale. \nThe integration is understood cut at the scale $p=\\Lambda$.\nThe coefficient $c_0=1$ by convention;\nthe additional three coefficients are needed to remove the quartic, quadratic and log-type divergences of $\\Omega_{0,\\infty}$\nthat appear in the limit $\\Lambda\\rightarrow\\infty$.\nThe PV coefficients\nwill be chosen so the aforementioned divergences cancel and the final expression does not depend on $\\Lambda$.\nPerforming the integration, it is an easy exercise to see that the choice $c_1=-3$, $c_2=3$ and $c_3=-1$ is\nenough to cancel all the divergences, in agreement with \\cite{Xu:2019gia}. The resulting finite expression\nis\n\\begin{eqnarray}\n\\Omega_{0,\\infty} &=& \\frac{3N_c N_f}{16\\pi^2}\n\\xi^4\\log\\frac{(M^2 + \\xi^2)(M^2 + 3\\xi^2)^3}{(M^2 +2 \\xi^2)^4}\\nonumber\\\\\n&+&\\frac{6N_c N_f}{16\\pi^2}\n\\xi^2 M^2\\log\\frac{(M^2 + \\xi^2)(M^2 + 3\\xi^2)}{(M^2 +2 \\xi^2)^2}\\nonumber\\\\\n&+&\\frac{N_c N_f}{16\\pi^2}\nM^4\\log\\frac{(M^2 + \\xi^2)^3(M^2 + 3\\xi^2)}{M^2(M^2 +2 \\xi^2)^3}.\\nonumber\\\\\n&&\\label{eq:desperate12}\n\\end{eqnarray}\n\nAlthough Eq.~\\ref{eq:desperate12} is finite, it potentially can shift the location of the minimum of the classical potential\nas well as the mass of the $\\sigma-$meson in the vacuum, $m_\\sigma$.\nWhile this would be not a problem since it would require a mere change of the parameters $\\lambda$ and $v$,\nit is easier to work assuming that the quark loop does not shift these quantities.\nTo this end, we add two counterterms,\n\\begin{equation}\n\\Omega_\\mathrm{c.t.} = \\frac{\\delta v}{2}M^2 + \\frac{\\delta\\lambda}{4}M^4,\\label{eq:ctUUU}\n\\end{equation}\nand we impose the renormalization conditions\n\\begin{eqnarray}\n&&\\left.\\frac{\\partial (\\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.})}{\\partial M}\\right|_{M=gF_\\pi}=0,\\label{eq:sh4aF}\\\\\n&&\\left.\\frac{\\partial^2 (\\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.})}{\\partial M^2}\\right|_{M=gF_\\pi}=0.\\label{eq:sh5aF}\n\\end{eqnarray}\nThe first condition imposes  that the quark loop does not change the\nlocation of the minimum of the classical potential, $\\sigma=F_\\pi$,\nwhile the second states that the loop does not shift $m_\\sigma$.\nThe coefficients of the two counterterms can be computed easily,\n\\begin{eqnarray}\n\\delta v &=& -\\frac{3N_c N_f}{4\\pi^2}\\xi^2\\log\\frac{(g^2F_\\pi^2 + \\xi^2)(g^2F_\\pi^2 + 3\\xi^2)}{(g^2F_\\pi^2 + 2\\xi^2)^2},\n\\label{eq:ctdv1}\\\\\n\\delta\\lambda &=&-\\frac{N_c N_f}{4\\pi^2}\n\\log\\frac{(g^2F_\\pi^2 + \\xi^2)^3(g^2F_\\pi^2 + 3\\xi^2)}{g^2F_\\pi^2(F_\\pi^2 + 2\\xi^2)^3}.\\label{eq:ctdl1}\n\\end{eqnarray}\nThe renormalized quark loop is thus given by\n\\begin{equation}\n\\Omega_{0,\\infty}^\\mathrm{ren} = \\Omega_{0,\\infty} + \\Omega_\\mathrm{c.t.}.\n\\label{eq:rty}\n\\end{equation}\n\n\\subsection{Renormalized thermodynamic potential in the finite volume case}\nIn a finite volume $V=L^3$ the $i-$component of momentum is quantized according to\n\\begin{equation}\np_i = \\frac{2\\pi}{L}n_i,~~~n_i=0,\\pm1,\\pm2,\\dots; \\label{eq:mom_quant}\n\\end{equation}\nthis leads to the obvious replacements\n\\begin{eqnarray}\n\\int \\frac{d^3p}{(2\\pi)^3} &\\rightarrow &\\frac{1}{V}\\sum_{n_x,n_y,n_z,}, \\label{eq:mom_repl}\\\\\nE_p &\\rightarrow &E_n = \\left(\nM^2 + \\frac{4\\pi^2}{L^2}n\n\\right)^{1\/2},\\label{eq:Ep_repl}\n\\end{eqnarray}\nwith $n=n_x^2 + n_y^2 + n_z^2$.\nInstead of $\\Omega_{0,\\infty}$ we have  \n\\begin{eqnarray}\n\\Omega_{0,L}(\\Lambda) &=&-\\frac{2N_c N_f}{L^3}|M|\n\\nonumber\\\\\n&& -\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\\sqrt{\\frac{4\\pi^2}{L^2}n+M^2},\n\\label{eq:alpha_2}\n\\end{eqnarray}\nwhere the subscript $L$ reminds that the potential is computed assuming quantization in a box with volume $L^3$;\nthe first addendum on the right hand side of Eq.~\\ref{eq:alpha_2} is the zero mode contribution.\nWe have put $a=\\Lambda^2 L^2\/4\\pi^2$ \nwhere $\\Lambda$ is an UV cutoff that will disappear after \nPV renormalization; \n$r_3(n)$ denotes the sum-of-three-squares function, \nthat counts how many ways it is possible to form $n$ as the sum of the squares of three integers: \nit corresponds to the degeneracy of the level with a given $n$.\nSimilarly, instead of Eq.~\\ref{eq:ep1} we have\n\\begin{equation}\n\\Omega_0(\\Lambda) = U + \\Omega_{0,L}(\\Lambda).\\label{eq:ep1_L}\n\\end{equation}\nThis is the bare potential that is divergent and needs renormalization:\nto make this evident we have made the dependence of $\\Lambda$ explicit. \n\n \n \n \nThe  renormalization  of the thermodynamic potential is performed\nfollowing the PV method delineated in the infinite volume case.\nFirstly, we introduce a set of PV coefficients and \nreplace Eq.~\\ref{eq:alpha_2} with\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\n\\sum_{j=0}^3 c_j\\sqrt{E_n^2 + j \\xi^2}.\n\\label{eq:alpha_2a}\n\\end{eqnarray}\nUsing the coefficients determined in the infinite volume limit is enough to get a finite expression in the $\\Lambda\\rightarrow\\infty$\nlimit. This can be proved by brute force numerically, but can also be understood analytically as follows.\nFor studying the UV divergence, we put  $X_j^2 = M^2 + j\\xi^2$ \nand we extract the $O(X_j^2)$ and $O(X_j^4)$ terms from  $\\Omega_{0,L}$,\nthat will bring a quadratic and log-type divergence respectively. Thus we can write\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3}\\sum_{n=1}^{a}r_3(n)\n\\sum_{j=0}^3 c_j\\left[a_2 X_j^2 + a_4 X_j^4\\right]\\nonumber\\\\\n&&+~\\mathrm{UV~finite~terms},\n\\label{eq:alpha_2a2}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\na_{2} &=&\\frac{L}{4\\pi n^{1\/2}},\\\\\na_{4} &=& -\\frac{L^3}{64\\pi^3 n^{3\/2}}.\n\\end{eqnarray}\nBy virtue of a numerical calculation we prove that in the large $a$ limit\n\\begin{eqnarray}\n\\sum_{n=1}^a \\frac{r_3(n)}{4\\pi n^{1\/2}} &\\approx &\\frac{a}{2},\\\\\n\\sum_{n=1}^a \\frac{r_3(n)}{4\\pi n^{3\/2}} &\\approx & \\frac{\\log a}{2},\n\\end{eqnarray}\nwhich allow to write\n\\begin{eqnarray}\n\\Omega_{0,L} &=& -\\frac{2N_c N_f}{L^3}|M|\\nonumber\\\\\n&&-\\frac{2N_c N_f}{L^3} \n\\sum_{j=0}^3 c_j\\left[a_2 X_j^2\\frac{L a}{2} - a_4 X_j^4\\frac{L^3 \\log a}{32\\pi^2}\\right]\\nonumber\\\\\n&&+~\\mathrm{UV~finite~terms}.\n\\label{eq:alpha_2a3}\n\\end{eqnarray}\nThe above equation shows that $a_2$ and $a_4$ multiply the quadratic and log-type divergence respectively. \nUsing the PV coefficients it is easy to prove that \n\\begin{eqnarray}\n\\sum_{j=0}^3 c_j X_j^2 &=& 0,\\\\\n\\sum_{j=0}^3 c_j X_j^4 &=& 0,\n\\end{eqnarray}\nwhile the $O(X_j^6)$ term is nonzero and UV-finite. \nTherefore, the PV regulator cancels the UV divergence of $\\Omega_{0,L}$\nleaving a UV-finite, $\\xi-$dependent term.\n\nThe counterterms that we have fixed in the infinite volume case can be used here as well:\nthey will implement the conditions that in the large volume limit, we recover\n$\\sigma=F_\\pi$ and the $m_\\sigma$ fixed by the classical potential.\nOn the other hand, for a finite $L$ the $\\Omega_{0,L}$ can shift both the location\nof the minimum of the total potential and $m_\\sigma$. Therefore,\nfor finite $L$ we will use\n\\begin{equation}\n\\Omega_{0,L}^\\mathrm{ren} = \\Omega_{0,L} + \\Omega_\\mathrm{c.t.},\n\\label{eq:rtyAA}\n\\end{equation}\nwith $\\Omega_\\mathrm{c.t.}$ specified bt Eq.~\\ref{eq:ctUUU} with counterterms\ngiven by Eqs.~\\ref{eq:ctdv1} and~\\ref{eq:ctdl1}. \nTaking into account the classical potential, the renormalized thermodynamic potential\nin the vacuum at finite $L$ is\n\\begin{equation}\n\\Omega^\\mathrm{ren} = U + \\Omega_{0,L}+\\Omega_\\mathrm{c.t.}.\n\\label{eq:renpot_1}\n\\end{equation}\n\nWe close this subsection with a short comment on the choice of $\\xi$. \nIn principle, we could change this arbitrarily  at a given $L$\nby requiring that the total derivative of $\\Omega$ with respect to $\\xi$, $d\\Omega\/d\\xi$,  is zero. \nThis would amount to solve a \nRenormalization Group-like equation in which the $\\partial\\Omega\/\\partial\\xi$ is balanced by terms\nproportional to $\\partial\\lambda\/\\partial\\xi$ and $\\partial g\/\\partial\\xi$ so that $d\\Omega\/d\\xi=0$. \nSolving this equation is well beyond the purpose of the study we want to do here.\nTherefore, for a given $L$ we have limited ourselves to inspect the ranges of $\\xi$ that do not change $\\Omega$ \ntoo much. For large $L$ we have found that $\\xi$ can be arbitrarily large. \nOn the other hand, for small $L$ we have found that $\\Omega$ is quite insensitive to the specific value of $\\xi$  \nas long as $\\xi \\lesssim \\gamma F_\\pi $ with $\\gamma=O(1)$. Therefore, we  fix $\\xi=F_\\pi$ in this work.\n\n\n\n\\subsection{The total thermodynamic potential}\nThe finite temperature thermodynamic potential does not need any particular treatment:\nin infinite volume it is given by the standard relativistic fermion gas contribution, namely\n\\begin{equation}\n\\Omega_T = -2N_c N_f T\\sum_{s=\\pm 1} \\int\\frac{d^3p}{(2\\pi)^3}\\log\\left(1 + e^{-\\beta(E_p-s\\mu)}\\right),\n\\end{equation}\nwhere $\\mu$ corresponds to the chemical potential. \nIn finite volume we replace the above equation with\n\\begin{equation}\n\\Omega_T = -2N_c N_f \\frac{T}{L^3}\\sum_{s=\\pm 1}\\sum_{n}r_3(n)\n\\log\\left(1 + e^{-\\beta(E_n-s\\mu)}\\right).\\label{eq:iow}\n\\end{equation}\nPutting all together, we get the renormalized thermodynamic potential of the QM model in a volume $V=L^3$\nwith periodic boundary conditions, that is\n\\begin{equation}\n\\Omega = U + \\Omega_{0,L}+\\Omega_\\mathrm{c.t.}+ \\Omega_T.\n \\label{eq:tot_om_fs}\n\\end{equation}\nFor each value of $\\mu$ and $T$ we determine $\\sigma$ by looking for the global minimum of $\\Omega$. \n\n\\subsection{Parameters of the classical potential}\nThe \nrenormalization procedure outlined above has the advantage that does not require a shift of the \nparameters of the classical potential both in the infinite volume and in the finite size cases. \nTherefore, these parameters can be computed from $U$ and are not affected by $L$.\nThese can be computed easily from the conditions that $\\partial U\/\\partial\\sigma=0$ and\n$\\partial^2 U\/\\partial\\sigma^2 = m_\\sigma^2$, where the derivatives are understood \ncomputed at $\\sigma=F_\\pi$.\nLimiting ourselves to write concrete expressions in the limit $h\\rightarrow 0$ we find\n\\begin{eqnarray}\nv &=& F_\\pi  - \\frac{h}{m_\\sigma^2},\\\\\n\\lambda  &=& \\frac{m_\\sigma^2 }{2 F_\\pi^2 } - \\frac{h}{2F_\\pi^3 }.\n\\end{eqnarray} \n \n\n\n\n \n\n\n\n\n\n\\section{Chiral phase transition at small temperature\\label{Sec:llp}}\n\nIn this section we present the results for the \ncondensate and phase structure at low temperature\nand high density, which is the domain in which the most interesting finite size effects appear. \nOur parameters set is $m_\\sigma=700$ MeV, $F_\\pi=93$ MeV,\n$m_\\pi=138$ MeV, $h=F_\\pi m_\\pi^2$.\nFinally,  we take $\\xi=F_\\pi$ and we assume $M=g F_\\pi=335$ MeV at $\\mu=T=0$ which gives $g=3.6$.\n\n\n\n\n\n\\subsection{Condensate, $m_\\sigma$ and $m_\\pi$ versus size in the vacuum} \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{vac_collect.png}\n\\end{center}\n\\caption{\\label{Fig:vacvac}Quark mass \nand $m_\\sigma$ versus $L$ in the vacuum. \nBoth quantities are measured in units of the infinite volume cases.\nRenormalization scale is $\\xi=F_\\pi$.}\n\\end{figure}  \n\n\nTo begin with, we present the behavior of the condensate, of the $\\sigma-$meson mass and pions mass\nversus $L$ in the vacuum: the results are summarized in Fig.~\\ref{Fig:vacvac}.\nAll quantities are measured in units\nof the infinite volume cases.  We find that both $M$ and $m_\\sigma$ increase with lowering $L$,\nwhile $m_\\pi$ decreases.\nThe effects of a finite $L$ on the physical quantities become noticeable for $L\\lesssim 3$ fm.\nQualitatively the results of the renormalized QM model agree with   \nthe NJL model calculations \\cite{Xu:2019gia}.  \n\n\n\n\n     \n\n\n\\subsection{Transitions at small temperature}\n\n \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{cond_m_L3.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{nb_m_L3.png}\n\\end{center}\n\\caption{\\label{Fig_cc} Condensate (upper panel)\nand $n_B\/\\rho_0$ (lower panel) versus chemical potential, for several values of $T$.\nCalculations correspond to $L=3$ fm.}\n\\end{figure}\n\nIn the upper panel of Fig.~\\ref{Fig_cc} we plot the condensate versus $\\mu$ for several temperatures\nand for $L=3$ fm. The condensate has been computed by a global minimization procedure of $\\Omega$\nfor any $(\\mu,T)$.\nFor $T=0$  $\\sigma$ has a discontinuity for $\\mu\\equiv\\mu_1\\approx 340$ MeV and drops down \nfrom its value in the vacuum \nto a smaller, still substantial value $\\sigma_1\\approx 93$ MeV.\nThis discontinuity agrees with the one found in \\cite{Xu:2019gia} where it has been shown to be driven by the zero mode\n(a direct calculation within the QM model confirms this).\nDespite the discontinuity of $\\sigma$, chiral symmetry is still spontaneously broken \nfor $\\mu>\\mu_1$ since the value of the condensate\nis still large. \nIncreasing $\\mu$ up to a second critical value $\\mu\\equiv\\mu_2\\approx 526$ MeV there is another\njump of $\\sigma$ to a $\\sigma_2\\approx 5$ MeV:\nit is fair to identify this discontinuity with the restoration of chiral symmetry. \nIt is easy to verify that the phase transition happens when the chemical potential is large enough\nto populate the first excited state, $n=1$: in fact,\nusing $\\sigma =\\sigma_2$  the energy of this state is\n$E_1 = \\sqrt{g^2\\sigma^2 + 4\\pi^2\/L^2}\\approx 415$ MeV, therefore the state $n=1$ can be populated\nfor $\\mu \\gtrsim \\mu_2$.\nIncreasing the temperature, the chiral phase transition and the jump of the condensate approach each other;\nmoreover, the discontinuity of $\\sigma$ at $\\mu=\\mu_1$ is smoothed by temperature becoming a crossover.\n\n\nAt low enough $L$ and low $T$  an intermediate phase appears between the chiral symmetry breaking phase\nat low $\\mu$, namely the hadron gas, and the high density phase that is quark matter in which chiral symmetry\nis restored. Even though the passage from the chiral symmetry broken phase to the intermediate one is not \na phase transition because symmetries are broken in the same way in the two phases, for the sake of simplicity we   \nadopt the term transition to discuss also the change of the condensate for $\\mu=\\mu_1$.\nIn fact, we aim to interpret this transition as a gas-to-liquid phase transition.\n\nIt is interesting to examine the behavior of number density around the two transitions.\nTo this end we define the baryon density, $n_B$, as\n\\begin{equation}\nn_B = \\frac{n_u + n_d}{3},\\label{eq:bnm3} \n\\end{equation}\nwhere $n_{u}$ and $n_d$ denote the densities of $u$ and $d$ quarks respectively; it is straightforward to prove that \n\\begin{equation}\nn_B =  -\\frac{1}{3}\\frac{\\partial\\Omega}{\\partial\\mu}.\\label{eq:bnm_inter}\n\\end{equation}\nIn the lower panel of Fig.~\\ref{Fig_cc} we plot $n_B$ versus $\\mu$ for several values of $T$ and $L=3$ fm;\nbaryon density is measured in units of the nuclear saturation density, $\\rho_0= 0.16$ fm$^{-3}$.\nAt small temperature, $n_B$ experiences a first jump from zero to $n_B\\approx 0.95\\rho_0\\equiv n_B^{(1)}$\nfor $\\mu=\\mu_1$, stays constant then experiences another jump to $n_B\\approx 6.46\\rho_0\\equiv n_B^{(2)}$\nfor $\\mu=\\mu_2$.\n \nThe dependence of $n_B$ on $\\mu$ at small temperature can be easily understood.\nAs a matter of fact, at zero (as well as very small but finite) temperature,\nif $\\mu$ is large enough to excite the zero as well as the first mode,\n we have\n\\begin{equation}\nn_B \\approx \\frac{2N_c N_f}{3V}\n\\left[\n\\theta(\\mu-M) + 6\\theta(\\mu-\\sqrt{M^2 + 4\\pi^2\/L^2})\n\\right];\\label{eq:bnm5}\n\\end{equation}\nthe first addendum in the right hand side of the above equation is the contribution of the zero mode,  \nwhile the second addendum\ncorresponds to the first excited state \ncounted with its degeneracy $r_3(1)=6$. \nBaryon density is constant for \n$M <\\mu < \\sqrt{M^2 + 4\\pi^2\/L^2}$ where only the zero mode contributes;\nanalogously, density is constant also for larger values of $\\mu$ until the second excited state can be populated.\nThis is qualitatively different from the behavior $n_B \\propto (\\mu-M)^{3\/2}$ of a \nrelativistic ideal massive gas, because for a finite size system there are only discrete modes in the spectrum,\nand if temperature is low enough only few of them can be occupied giving rise to $\\Omega_T\\propto\\mu$.\nOnly when the degeneracy becomes large  one can approach the continuum limit and eventually recover the aforementioned\ndependence of the density on the chemical potential.\n\n\n\n\n\nWe suggest a similitude between the jump of the chiral condensate and a liquid-gas phase transition.\nAs a matter of fact, chiral symmetry is not restored at $\\mu_1$,\ntherefore the pattern of symmetry breaking is the same at low and intermediate $\\mu$.\nIn addition to this,\nthe quark number density at the first jump has a net increase, \nand the transition is sharp at low temperature then becomes smooth at higher temperatures.\nThese aspects characterize a liquid-gas phase transition,\nwhich corresponds to a change in density and not to a change in the pattern of spontaneous symmetry breaking.\n\n\n\\subsection{Correlation length of the fluctuations of the order parameter}\n\n\n\nWe can further characterize this liquid-gas-like jump of the condensate at $\\mu_1$ by means of the correlation length\nof the static fluctuations of the condensate, that are carried by the $\\sigma-$meson. \nTo do this, firstly we have to compute the in-medium masse of the $\\sigma-$meson.\nComputing this within the quark-meson model is a well established procedure, see for example \\cite{Castorina:2020vbh}\nand references therein, and is straightforward when two-loop contributions are neglected and \nthe Hatree approximation is used to compute the effective 2-particle-irreducible potential. Within these approximations we have\n$M_\\sigma^2 = \\partial^2\\Omega\/\\partial\\sigma^2$,\nwhere the second derivative is understood at the global minimum of $\\Omega$.\nA similar equation holds for the $\\pi-$mesons, $M_\\pi^2 = \\partial^2\\Omega\/\\partial\\pi^2$,\nwhere $\\Omega$ can be augmented with the pion field by the obvious replacement $\\sigma^2\\rightarrow\\sigma^2 + \\pi^2$\nin all but the $h\\sigma$ terms and in the zero mode contribution $M\\rightarrow \\sqrt{M^2 + g^2\\pi^2}$. \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{3fm.png}\n\\end{center}\n\\caption{\\label{Fig:3fm}In-medium $\\pi-$meson and $\\sigma-$meson masses\nversus $\\mu$, for several temperatures and $L=3$ fm. Thin lines correspond to $M_\\pi$ while thick lines\ndenote $M_\\sigma$.\n}\n\\end{figure}\n\nIn   Fig.~\\ref{Fig:3fm} we plot $M_\\sigma$ and $M_\\pi$ versus $\\mu$,\nfor several values of $T$. At $\\mu=\\mu_1$ $M_\\sigma$ drops down. However, \n$M_\\pi$ is almost insensitive to the jump of the condensate. This confirms that\nthe first jump at $\\mu_1$ should not be identified with a real phase transition.\nAt $\\mu=\\mu_2$ where the condensate drops down to almost zero,\n$M_\\sigma$ and $M_\\pi$ join and increase, signaling that the $O(4)$ symmetry is restored and these particles\nbecome heavy enough that decouple from the low energy spectrum dominated by the quarks. \nThis confirms that it is $\\mu_2$ that has to be identified with chiral symmetry restoration.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{correlation2fm.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{correlation3fm.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{correlation6fm.png}\n\\end{center}\n\\caption{\\label{Fig:3fmCOR}Correlation length of the fluctuations of the order parameter versus $\\mu$, for several temperatures.\n}\n\\end{figure}\n\n\nWe can compute the correlation length of the time independent fluctuations of the condensate, \n$\\lambda$, that are transported by the $\\sigma-$meson. In fact, for time independent fluctuations the effective action\nof the $\\sigma-$meson is formally equivalent to that of a Ginzburg-Landau theory\nfor a scalar order parameter with positive squared mass, and it is textbook matter that in this case the correlation length\nof fluctuations is nothing but  $\\lambda=1\/M_\\sigma$. We show $\\lambda\/L$ versus $\\mu$ in  \nFig.~\\ref{Fig:3fmCOR} for three representative values of $L$.\nFor $L=2$ fm, at $T=10$ MeV and $\\mu=\\mu_1$ the correlation length increases and stays constant\nup to $\\mu=\\mu_2$; for larger values of $\\mu$ it decreases and approaches the value it has in the hadron gas phase.\nThe correlation length is frozen in the intermediate phase, due to the fact that only the zero mode is excited. \nQualitatively, this happens also for $T=20$ MeV and $T=40$ MeV.\nAlso notice that in this phase $\\lambda\/L\\approx 0.2$ meaning that one correlation volume occupies\nabout the twenty percent of the volume of the system; around the two transitions $\\lambda$\ndevelops two peaks; in particular, for $T=40$ MeV we find that $\\lambda\/L=O(1)$ that implies\nthat the system is close to criticality. This is what we would expect at a critical endpoint.\nFinally, for $T=60$ MeV there is only one peak of $\\lambda$ in agreement with the fact that\nthe transition to the intermediate phase is smoothed by the temperature;\nnevertheless, $\\lambda$ experiences a net increase in comparison with the value at small $\\mu$,\nthen again $\\lambda\/L=O(1)$ at the chiral phase transition.\nThe qualitative picture is the same at $L=3$ fm, see the middle panel of   Fig.~\\ref{Fig:3fmCOR},\nwhile for a larger value of $L$ the double peak structure as well as the intermediate phase disappear,\nsee the lower panel of Fig.~\\ref{Fig:3fmCOR}.\n\nThe results  summarized in Fig.~\\ref{Fig:3fmCOR} allow to understand better the intermediate phase.\nAs a matter of fact, we learn that beside the characterization of this phase in terms of the baryon density\ndiscussed in the previous subsection,\nwe can distinguish it from the vacuum and the high density normal quark matter also looking at\nthe fluctuations of the order parameter.\nIn particular, the correlations of the order parameters are substantially larger than those in the vacuum \nand in the quark matter phase at high $\\mu$, and do not change by changing $\\mu$\nin this region:  correlation volumes are frozen due to the fact that only the zero mode\nis excited. Moreover, for small $L$ \nthe correlation volumes occupy a substantial portion of the total volume of the system.\nThese facts, together with the increase of density and\nthe symmetry pattern that is unchanged at $\\mu=\\mu_1$, \nsuggest the name of {\\it subcritical liquid} for this intermediate phase.\n\n\n \n\n\n\n\\subsection{Catalysis of chiral symmetry breaking at low temperature}\n\n\n  \n\n \n \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{moresize.png}\n\\end{center}\n\\caption{\\label{Fig:moresize} Condensate versus chemical potential, for several values of $L$\nand $T=10$ MeV.}\n\\end{figure} \n\nIn Fig.~\\ref{Fig:moresize} we plot the condensate versus $\\mu$ at $T=10$ MeV, for several values of $L$.\nFinite size effects are noticeable up to $L\\approx 5$ fm although in this case the transition to the subcritical liquid phase\nis minor. For $L=6$ fm no sign of the subcritical liquid is found, and comparing the results of $L=6$ fm and $L=8$ fm\nwe notice that the effect of the finite size is almost gone and a continuum limit is reached.\nThe results collected in Fig.~\\ref{Fig:moresize}  show that lowering the size catalyzes the spontaneous chiral symmetry \nbreaking by enlarging the subcritical liquid region.  \n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{nbsl.png}\n\\end{center}\n\\caption{\\label{Fig:nbsl}$n_B^\\mathrm{s.l.}\/\\rho_0$ versus $L$ at $T=0$.\n}\n\\end{figure}\n\n\nFor completeness, we report  on\nthe baryon density in the subcritical liquid phase, $n_B^\\mathrm{s.l.}$,\nat $T=0$.\nThis can be estimated quickly because only the zero mode is populated therefore\nwe read its value from Eq.~\\ref{eq:bnm5}, namely\n\\begin{equation}\nn_B^\\mathrm{s.l.} = \\frac{2N_c N_f}{3L^3};\\label{eq:nbsl}\n\\end{equation}\nthis amounts to put $2N_c N_f$ quarks in the volume $L^3$.  \nThe results are collected in Fig.~\\ref{Fig:nbsl} in which we show $n_B^\\mathrm{s.l.}\/\\rho_0$ versus $L$,\nwhere $\\rho_0$ is the nuclear saturation density. In particular, $n_B\\approx \\rho_0$ for $L\\approx 3$ fm.\n\n \n \n \n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{pdl2.png}\n\\end{center}\n\\caption{\\label{Fig:pdl2} Critical lines for $L=2$ fm. \nDotted and dot-dashed lines correspond to smooth crossovers and solid lines to first order phase transitions.\nThe green dot denotes the critical endpoint of the chiral phase transition.\nThe indigo dot corresponds to the critical endpoint for the liquid-gas-like transition to the subcritical liquid phase.\n$\\chi$SR and $\\chi$SB denote the regions in which chiral symmetry is restored and broken respectively.\nWe have shown by blue lines the critical lines for $L=10$ fm for comparison.}\n\\end{figure}  \n\nThe results discussed in this section can be summarized in the form of a phase diagram in the $\\mu-T$ plane.\nIn Fig.~\\ref{Fig:pdl2} we plot the transition lines for the case $L=2$ fm, and for comparison we also show a portion of the\ncritical lines at $L=10$ fm that correspond to the continuum limit.\nThe regions denoted with $\\chi$SR and $\\chi$SB denote the portions of the phase diagram \nin which chiral symmetry is restored and broken respectively. \nThe dots denote critical endpoints. In the figure we focus on the subcritical region phase\nthat appears as an intermediate phase between $\\chi$SB and $\\chi$SR phases.\nComparing the critical lines for $L=2$ fm and $L=10$ fm the catalysis of symmetry breaking is evident. \nWe also notice that the critical endpoint for chiral symmetry restoration moves towards higher values of $\\mu$\nand lower $T$ with the lowering of $L$.\n\nWe have verified the stability of our results by changing the number of colors:\nin particular, for $N_c=2$ the picture is unchanged. Since QCD with $N_c=2$\nand finite $\\mu$ can be simulated on the lattice, the predictions of this article can be tested by means\nof first principle calculations.\n\n \n\\section{Chiral phase transition at high temperature}\nThe chiral  phase transition at finite temperature and low $\\mu$ has been more studied in the literature,\ntherefore we limit ourselves to present a few results and compare them with those of\nother effective models. In particular, our results agree with those of \\cite{Xu:2019gia,Xu:2019kzy}\nwhere the NJL model with PV regulators has been used.\n\n\\subsection{Numerical computation of $T_c$ versus $L$}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{masses_comparison.png}\\\\\n\\includegraphics[width=0.45\\textwidth]{masses_comparison_m300.png}\n\\end{center}\n\\caption{\\label{Fig_mm}Condensate $\\sigma$ versus temperature, for several values of $L$\nand two representative values of $\\mu$.\n}\n\\end{figure}\n\nIn Fig.~\\ref{Fig_mm} we plot the condensate, $\\sigma$, versus temperature, for $\\mu=0$ (upper panel) and $\\mu=300$ MeV\n(lower panel) and several values of $L$. \nWe can define a pseudo-critical temperature, $T_c$, by looking at the location of the maximum variation of\n$d\\sigma\/d\\beta$. At $\\mu=0$ the condensate increases with $1\/L$ and this pattern remains stable in the whole\ntemperature range examined. We conclude that our picture is consistent with the catalysis of chiral symmetry breaking\ninduced by lowering $L$. The catalysis remains also for higher values of $\\mu$, see the lower panel \nof Fig.~\\ref{Fig_mm}, in agreement with the results presented in section~\\ref{Sec:llp}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{tctc.png}\n\\end{center}\n\\caption{\\label{Fig:tctc}Critical temperature for chiral symmetry restoration versus size at $\\mu=0$.\nThe line represents a crossover for any value of $L$.\n}\n\\end{figure} \n \nWe show $T_c$  versus $L$ at $\\mu=0$ in Fig.~\\ref{Fig:tctc}.\nThe behavior for other values of $\\mu$ can be easily guessed from the results that we have shown before.\nIn the infinite volume limit $T_c\\approx 180$ MeV. The catalysis of chiral symmetry breaking is\nclear in Fig.~\\ref{Fig:tctc}. For example, for $L=3$ fm the increase of critical temperature is $\\approx 23\\%$,\nwhile it becomes $\\approx 55\\%$ for $L=2$ fm.  The qualitative behavior of $T_c$ is in agreement \nwith previous studies within the QM model~\\cite{Palhares:2009tf,Magdy:2019frj}\nas well as the NJL model with periodic boundary conditions \\cite{Xu:2019gia,Xu:2019kzy,Wang:2018ovx}.\n\n\n\n\n\n \n\n\\subsection{Critical temperature from the Ginzburg-Landau potential}\nIn this subsection we clarify the role of the zero mode \non the behavior of $T_c$ versus $L$. We do this in the chiral limit, $h=0$, and at $\\mu=0$, \nto make the discussion more transparent. \nIn these conditions the chiral transition is of the second order and the critical temperature is given by the zero\nof the coefficient $\\alpha_2$ of the Ginzburg-Landau potential,\n\\begin{equation}\n\\alpha_2 = \\left.\\frac{\\partial^2\\Omega}{\\partial\\sigma^2}\\right|_{\\sigma=0},\n\\end{equation}\nwhere  $\\Omega$ is given by Eq.~\\ref{eq:tot_om_fs}. \nThe coefficient $\\alpha_2$ corresponds to the curvature of $\\Omega$ at $\\sigma=0$.\nWe feel that this discussion is necessary because in the literature some confusion arises when different models \nare compared to each other.\nWe keep the parameters of the classical potential,\n$v$ and $\\lambda$, unchanged by $L$. This is natural within the renormalization scheme \nthat we have adopted in this article, because the finite size corrections to $\\Omega$\nare finite and do not require any additional renormalization condition. \n\n\n \nStarting from Eq.~\\ref{eq:tot_om_fs}  we get\n\\begin{eqnarray}\n\\alpha_2^\\mathrm{QM} &=& C_2^\\mathrm{QM} \\nonumber\\\\\n&&+\\frac{4 N_c N_f g^2}{2\\pi L^2}\\sum_{n=1}^\\infty r_3(n)\\frac{e^{-\\beta\\varepsilon_n}}{\\sqrt{n}(1+e^{-\\beta\\varepsilon_n})}\n\\nonumber\\\\\n&&\n-\\frac{4 N_c N_f }{ L^3 }\\frac{g^2}{4T},\\label{eq:a2QM}\n\\end{eqnarray} \nwhere we have put $\\varepsilon_n = 2\\pi\\sqrt{n}\/L$,\nand $C_2^\\mathrm{QM}$ denotes the curvature of $\\Omega$ at $T=0$ and $\\sigma=0$,\n\\begin{eqnarray}\nC_2^\\mathrm{QM} &=& -v^2\\lambda + g^2\\delta v \\nonumber\\\\\n&&-\\frac{2 N_c N_f g^2}{L^3}\n\\sum_{n=1}^\\infty r_3(n)\\sum_{j=0}^3\\frac{c_j}{\\sqrt{j\\xi^2 + 4\\pi^2n\/L^2}}.\\nonumber\\\\\n&&\\label{eq:C2qm}\n\\end{eqnarray} \nWe remind that $m_\\sigma^2=2v^2\\lambda$ corresponds to the $\\sigma-$meson mass\nin the vacuum and in the infinite volume limit.\nThe last addendum on the right hand side of \nEq.~\\ref{eq:a2QM} is the zero mode contribution to $\\alpha_2$,\nwhile the summation represents the contribution of the higher modes.\nThe zero mode contribution is negative while the sum over the higher modes is positive:\nwhile thermal fluctuations increase $\\alpha_2$ making the broken phase less stable,\nthe zero mode lowers $\\alpha_2$ causing the broken phase to be more stable.\nA numerical inspection shows that $C_2^\\mathrm{QM}$ is quite insensitive of $L$\nbecause it is dominated by the classical contribution. Moreover $C_2^\\mathrm{QM}<0$.\n\n\n  \n\nLowering $L$, the zero mode contribution grows up in magnitude, therefore it is necessary to increase\n$T_c$ to get a positive contribution from the higher modes that overcomes both the zero mode and $C_2^\\mathrm{QM}$\nto satisfy $\\alpha_2^\\mathrm{QM}(T_c)=0$. Thus $T_c$ increases by lowering $L$.\n \n\nIf we removed the zero mode from Eq.~\\ref{eq:a2QM}, for example by imposing antiperiodic boundary condtions\nor an infrared cutoff,\nwe would be left with\n\\begin{eqnarray}\n\\alpha_{2,\\mathrm{nzm}}^\\mathrm{QM} &=& C_2^\\mathrm{QM}   \\nonumber\\\\\n&&+\\frac{4 N_c N_f g^2}{2\\pi L^2}\\sum_{n=1}^\\infty r_3(n)\\frac{e^{-\\beta\\varepsilon_n}}{\\sqrt{n}(1+e^{-\\beta\\varepsilon_n})}.\n\\label{eq:a2QMapbc}\n\\end{eqnarray}\nEven in this case, the requirement $\\alpha_{2,\\mathrm{nzm}}^\\mathrm{QM}(T_c)=0$\nimplies that\nlowering $L$ has to be balanced by the increase of $T_c$,\nbecause $ C_2^\\mathrm{QM}<0$ and almost insensitive to $L$.\nThus in the QM model even without the zero mode,  $T_c$ has to increase with  $1\/L$,\nin agreement with~\\cite{Palhares:2009tf,Magdy:2019frj}.\nThe arguments above would apply also if we had not implemented\nrenormalization and had regularized the divergent quark loop via an effective cutoff: in this case,\nas long as $-v^2\\lambda <0$, $C_2^\\mathrm{QM}<0$ for any $L$. \n\n \n\nSummarizing, we have shown that within the QM model in the chiral limit and at $\\mu=0$,\n$T_c$ increases with $1\/L$ regardless of the presence of the zero mode in the spectrum or not.\nThis happens because $C_2^\\mathrm{QM}<0$ for any $L$.\nThis behavior of $T_c$ and its explanation has not been stressed enough in the literature.\n\nThe increase of $T_c$ with $1\/L$ when periodic boundary conditions are implemented is in agreement with\nthe results of the NJL model, see for example \\cite{Xu:2019gia,Wang:2018ovx}. On the other hand,\nwhen antiperiodic boundary conditions are implemented within the NJL model, $T_c$ is found to decrease with $1\/L$.\nThis is in qualitative disagreement with the QM model and needs to be clarified. \nThe very reason of the different behavior cannot be traced back to the lack of the zero mode only:\nafter all, in the QM model $T_c$ increases with $1\/L$ even when the zero mode is removed from the spectrum.\nIn fact, a difference between the QM and NJL models is the curvature of the potential\nat $\\sigma=0$ and $T=0$, due to the different classical potentials in the two models.\nIn the QM model $C_2^\\mathrm{QM}<0$.\nOn the other hand, in the NJL model there are no mesons at the tree level and the classical potential\nis merely the mean field term $\\sigma^2\/4G$.  \nThe divergent quark loop is necessary to make $\\alpha_2$ negative and break chiral symmetry:\n while this is enough to guarantee a negative curvature in the infinite volume limit,\nit is not guaranteed that it remains negative for any $L$. \n\nTo keep the treatment simple, we use the common hard cutoff scheme for the NJL model;\nin fact, the results we find here agree with those obtained within\nPV regularization \\cite{Xu:2019gia}.\nUsing a  cutoff $\\Lambda$ the curvature at $T=0$ in NJL is\n\\begin{equation}\nC_2^\\mathrm{NJL} = \\frac{1}{2G}  -\\frac{N_c N_f}{\\pi L^2}\\sum_{n=1}^a \\frac{r_3(n)}{\\sqrt{n}},\n\\end{equation}\nwhere $a=\\Lambda^2 L^2\/4\\pi^2$ and $G$ is the NJL coupling. The contribution of the thermal fluctuations to $\\alpha_2$ in NJL\nis formally equivalent to that of QM model and is not repeated here.\nWe notice that  $C_2^\\mathrm{NJL}$\nbecomes positive for small enough $L$ because the quark loop shrinks. \nWhen the zero mode is removed from the spectrum, \nthe quark loop in $C_2^\\mathrm{NJL}$ is the only source for a negative $\\alpha_2$:\nsince lowering $L$ shrinks the loop, the contribution of the excited states at $T_c$ has to be lowered to get a vanishing $\\alpha_2$\nwhich implies that $T_c$ decreases when $L$ decreases. \n\n \nThe message of this section is that the zero mode alone is not enough to explain the behavior of $T_c$ versus $L$\nin chiral models: the difference between QM and NJL appears even when the zero mode is absent in the\nspectrum of both models. The curvature of the potential is another necessary ingredient for $T_c$ and it is precisely the\ndifferent curvature that leads to different predictions of $T_c$ versus $L$ in the two models.\n\nWe remark that if we had assumed \na dependence of the classical potential of $L$, the behavior of $T_c$ versus $L$ might have been more difficult to predict\nwithin the Ginzburg-Landau coefficient because $T_c(L)$ would have depended also on \nthe additional functions $v=v(L)$ and $\\lambda=\\lambda(L)$.\n \n \n   \n\n\n\n\n\n\n\\section{Summary and Conclusions}\nWe have studied the effect of periodic boundary conditions on chiral symmetry breaking and its restoration in QCD.\nAs an effective model of the effective potential for the quark condensate, we have used the quark-meson model \nwhich couples quarks to background meson fields.\nWe have implemented periodic boundary conditions on the effective potential for a cubic box of size $L^3$,\nthen we have performed the renormalization of the divergent vacuum term in the box;\nwe have computed the behavior of the condensate at finite temperature, $T$, and quark chemical potential, $\\mu$.\nFor the implementation of the renormalization conditions at finite $L$ we have \nadopted the Pauli-Villars regulators as in \\cite{Xu:2019gia}, that are enough to cancel the divergent contributions\nin the infinite volume as well as at finite $L$.\n\nThe most interesting effects happen for the chiral phase transition at small temperature and finite chemical potential.\nWe have found that for $L\\lesssim 5$ fm, increasing $\\mu$ up to a critical value, $\\mu_1$, \nresults in a jump of the condensate\nto lower but finite values. This jump is due to the population of the zero mode.\nThe contribution of the zero mode at such moderate size is not very strong, therefore its excitation is not enough\nto restore chiral symmetry. Increasing $\\mu$ to higher values, the first mode is excited eventually\nand chiral symmetry is restored at $\\mu=\\mu_2$.\n\nWe have suggested a similitude between the jump of the condensate at $\\mu=\\mu_1$ and a liquid-gas phase transition.\nIn fact, chiral symmetry is not restored at $\\mu_1$,\ntherefore symmetries are broken in the same way at low and intermediate $\\mu$.\nMoreover, the quark number density at the first jump has a net increase.\nBoth these aspects are common to the liquid-gas phase transition.\nWe have further characterized the jump of the condensate at $\\mu_1$ by means of the correlation length\nof the fluctuations of the condensate, that are carried by the $\\sigma-$meson; in particular,\nwe have identified $\\lambda=1\/M_\\sigma$ with $\\lambda$ the correlation length and $M_\\sigma$ the in-medium\nmass of the $\\sigma-$meson. We have found that at low temperature and $\\mu=\\mu_1$ the correlation length\nincreases then stays constant up to $\\mu=\\mu_2$ where chiral symmetry is restored. Increasing temperature\nbrings the system close to criticality and this is confirmed by the increase of $\\lambda$.\n\nWe name the intermediate phase as {\\em subcritical liquid phase} because even though\nthe system is not critical in the whole $(\\mu-T)$ window, the correlation domains in this phase\nare larger than those in the hadron gas and quark matter phases, respectively at small and large $\\mu$,\nas if the system was approaching a critical point; in addition to this, baryon density is finite due to the occupation\nof the zero mode, and symmetries are broken in the same way of the hadron phase, as it would happen in the gas-to-liquid\ntransition. \n\n\nOverall, we have found that lowering $L$  and imposing periodic boundary conditions\ncatalyzes the spontaneous breaking of chiral symmetry. This has been understood as  the result of the excitation \nof the zero mode at intermediate values of $\\mu$, that lowers a bit the value of the condensate without restoring chiral symmetry, \nand the need of a large $\\mu$ to excite the first mode\nthat leads to the definitive lowering of the condensate; the smaller $L$ the larger is the value of $\\mu$ needed to excite the first mode,\nthus leading to the catalysis of chiral symmetry breaking.\n\n\nWe have completed the study by computing \nthe critical temperature,\n$T_c(L)$, versus $L$. We have found that $T_c$ decreases with $L$ \nthus supporting the catalysis of chiral symmetry breaking \nfound in previous studies where periodic boundary conditions, or effective infrared\ncutoffs in the QM model, have been implemented \\cite{Xu:2019gia,Palhares:2009tf,Wang:2018ovx,Magdy:2019frj}. \n\nThere are several ways to continue the work presented here. A straightforward extension of the work is the \ninclusion of meson fluctuations on the same line of \\cite{Castorina:2020vbh}.\nIt would be interesting to include the possibility of inhomogeneous condensates\n\\cite{Abuki:2018iqp,Takeda:2018ldi,Abuki:2013pla,\nAbuki:2013vwa,Buballa:2020xaa,Carignano:2019ivp,Buballa:2018hux,Carignano:2017meb,Nickel:2009ke,Nickel:2009wj}.\nThe inclusion of the Polyakov loop  to take trace of confinement-deconfinement phase transition via a collective field\nwould also be possible \\cite{Fukushima:2003fw,Ratti:2005jh,Abuki:2008nm}. \nFinally, it is of a certain interest to analyze the thermodynamic geometry\n\\cite{Weinhold:1975get,Weinhold:1975gtii,\nRuppeiner:1979trg,Ruppeiner:1983ntp,Ruppeiner:1983tcf,Ruppeiner:1985tcv,Ruppeiner:1995rgf,Ruppeiner:1998rgc,\n Castorina:2019jzw,Castorina:2020vbh,Zhang:2019neb} \nof the effective models of chiral symmetry breaking with finite size.\nWe plan to report on these topics in the near future.\nWe have also investigated the stability of our results by changing the number of colors:\nin particular, we have verified that for $N_c=2$ the picture is unchanged. Since QCD with $N_c=2$\nand finite $\\mu$ can be simulated on the lattice, the predictions of this article can be tested by means\nof first principle calculations.\n\n \n\n\n \n\n\n\n\n\n\n\\section*{ACKNOWLEDGEMENTS}\nM. R. acknowledges John Petrucci for inspiration.\nThe work of the authors is supported by the National Science Foundation of China (Grants No.11805087 and No. 11875153).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe goal of information theory is to find the fundamental limits imposed on information processing and transmission by the laws of physics. One of the early breakthroughs in quantum information theory was the characterisation of the capacity of a classical-quantum (c-q) channel to transmit classical information by Holevo~\\cite{holevo98,holevo73b} and Schumacher--Westmoreland~\\cite{schumacher97}. The \\emph{classical capacity} of a quantum channel is defined as the maximal rate (in bits per channel use) at which we can transmit information such that the decoding error vanishes asymptotically as the length of the code increases. However, for many practical applications there are natural restrictions on the code length imposed, for example, by limitations on how much quantum information can be processed coherently. Therefore it is crucial to go beyond the asymptotic treatment and understand the intricate tradeoff between decoding error {probability}, code rate and code length. \n\nFor this purpose, we will study families of codes that have {\\em both} a rate approaching the capacity and an {error probability} that vanishes asymptotically as the code length $n$ increases. The following tradeoff relation gives a rough illustration of our main result: if the code rate approaches capacity as $\\Theta(n^{-t})$ for some $t \\in (0,1\/2)$, then the decoding error cannot be smaller than $\\exp(- \\Theta(n^{1-2t}))$. In fact, we will show that the constants implicit in the $\\Theta$ notation are determined by a second channel parameter {beyond} the capacity, called the \\emph{channel dispersion}. We will also show that this relation is tight, i.e., there exist families of codes achieving equality {asymptotically}. \n\nOur work thus complements previous work on the boundary cases corresponding to {$t  \\in \\{0, 1\/2\\}$}. The error exponent (or reliability function) of c-q channels (see, e.g., Refs.~\\cite{holevo00,hayashi07,dalai13}) corresponds to the case $t = 0$ where the rate is bounded away from capacity and the error probability vanishes exponentially in $n$. This is also called the \\emph{large deviations} regime.  Moreover, the second-order asymptotics of c-q channels were evaluated by Tomamichel and Tan~\\cite{tomamicheltan14}. They correspond to the case $t = 1\/2$ where the rate approaches capacity as $\\Theta(n^{-1\/2})$ and the  error probability  is non-vanishing. This is also called the \\emph{small deviations} regime. \n\nIn the present work, we consider the entire regime in between, which is dubbed the \\emph{moderate deviation} regime.\\footnote{In the technical analysis, we are considering moderate deviations from the mean of a sum of independent  {log-likelihood} ratios, thus justifying the name emanating from {statistics~\\cite[Theorem~3.7.1]{dembo98}}.} The different parameter regimes are illustrated in Fig.~\\ref{fig:regimes}.\n\n\\begin{figure}[t!]\n\t\\centering\n\n\t\\includegraphics{figure-compressed.pdf}\n\t\\begin{tabular}{|l|c|c|c|c|c|}\n\t\t\\hline\n\t\t& (I) & (II) & (III) & (IV) & (V) \\\\\n\t\t\\hline\n\t\t\\multirow{2}{*}{regime} & error  & \\!moderate deviation\\! & constant error & moderate deviation & strong converse \\\\\n\t\t& exponent & (below capacity) & (second-order) & (above capacity) & exponent \\\\\n\t\t\\hline\n\t\terror prob.\\ & \\footnotesize $\\!\\exp(-\\Theta(n))\\!$ & \\footnotesize $\\exp(-o(n))$ \\& $\\omega(1)$ & \\footnotesize $\\Theta(1)$ & \\footnotesize \\!$1 - \\exp(-o(n))$ \\& \\footnotesize $1 - \\omega(1)$\\! & \\footnotesize $1 - \\exp(-\\Theta(n))$ \\\\\n\t\t\\hline\n\t\tcode rate & \\footnotesize $C - \\Theta(1)$ & \\footnotesize $C - o(1)$ \\& $C - \\omega\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C - \\Theta\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C + o(1)$ \\& $C + \\omega\\big(n^{-\\frac12}\\big)$ & \\footnotesize $C + \\Theta(1)$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{The figure shows the optimal error probability as a function of the rate, for different block lengths. Darker lines correspond to longer block lengths, and the capacity is denoted by $C$. The table shows the asymptotics in each region, as the blocklength $n$ goes to infinity. The functions of $n$ implicit in the $\\Theta$, $o$, and $\\omega$ notation are assumed to be positive-valued.}\n\t\\label{fig:regimes}\n\\end{figure}\n\n\\paragraph*{Main results.} \nBefore we present our main results, let us introduce the notion of a \n\\emph{moderate sequence} of real numbers, $\\{ x_n\\}_n$ for $n \\in \\mathbb{N}$, \nwhose defining properties are that $x_n \\searrow 0$ and $\\sqrt{n}\\, x_n \\to \n+\\infty$ as $n \\to \\infty$.\\footnote{As mentioned above an archetypical \n\tmoderate sequence is $x_n = \\Theta(n^{-t})$ for some $t \\in (0, \\frac12)$. The \n\tboundary cases are not included\\,---\\,in fact $t = 0$ requires a large \n\tdeviation analysis whereas $t = \\frac12$ requires a small deviation analysis.} \nOur two main results concern binary asymmetric quantum hypothesis testing and \nc-q channel coding.\n\n\\begin{enumerate}\n\t\\item The first result, presented in detail in Sect.~\\ref{sec:hypo}, concerns binary quantum hypothesis testing between a pair of quantum states $\\rho$ and~$\\sigma$. We show that for any moderate sequence $x_n$, there exists a sequence of tests $\\{ Q_n \\}_n$ such that the two kinds of errors satisfy\n\t\\begin{align}\n\t\t\\Tr \\rho^{\\otimes n} (1 - Q_n) = e^{-nx_n^2}  &~\\textrm{and}~ \n\t\t\\Tr \\sigma^{\\otimes n} Q_n = \\exp\\Big(-n \\Big( D(\\rho\\|\\sigma) - \\sqrt{2 V(\\rho\\|\\sigma) }\\, x_n  + o(x_n) \\Big)\\Big) \\,,\n\t\t\\intertext{and another sequence of tests $\\{ Q_n' \\}_n$ such that the errors satisfy\n\t\t}\n\t\t\\Tr \\rho^{\\otimes n} (1-Q_n')  = 1-e^{- n x_n^2} &~\\textrm{and}~  \n\t\t\\Tr \\sigma^{\\otimes n} Q_n' = \\exp\\Big(-n \\Big( D(\\rho\\|\\sigma) + \\sqrt{2 V(\\rho\\|\\sigma) }\\, x_n  + o(x_n) \\Big)\\Big) \\,,\n\t\\end{align}\n\twhere $D(\\cdot\\|\\cdot)$ and $V(\\cdot\\|\\cdot)$ denote the relative entropy~\\cite{umegaki62} and relative entropy variance~\\cite{tomamichel12,li12}, respectively. (The reader is referred to the next section for formal definitions of all concepts discussed here.)\n\tMost importantly, we show that both of these tradeoffs are in fact optimal.\n\t\n\t\\item The main result, covered in Sect.~\\ref{sec:channels}, concerns coding over a memoryless classical-quantum channel $\\mathcal{W}$. Let us denote by $M^*(\\mathcal{W};n,\\varepsilon)$ the maximum $M \\in \\mathbb{N}$ such that there exists a code transmitting one out of $M$ messages over $n$ uses of the channel $\\mathcal{W}$ such that the average probability of error does not exceed $\\varepsilon$. For any sequence of tolerated error probabilities $\\{\\varepsilon_n\\}_n$ vanishing sub-exponentially with $\\varepsilon_n = e^{-nx_n^2}$, we find that\n\t\\begin{align}\n\t\t\\label{eq:moderatesecondorder1}\n\t\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, \\varepsilon_n) &= C(\\mathcal{W}) - \\sqrt{2 V_{\\min}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\\\\\n\t\t\\label{eq:moderatesecondorder2}\n\t\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, 1-\\varepsilon_n) &= C(\\mathcal{W}) + \\sqrt{2 V_{\\max}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\n\t\\end{align}\n\twhere $C(\\cdot)$ denotes the channel capacity and~$V_{\\min}(\\cdot)$ and $V_{\\max}(\\cdot)$ denote the minimal and maximal channel dispersion as defined in~Ref.~\\cite{tomamicheltan14}, respectively.\n\tThis result holds very generally for channels with arbitrary input alphabet and without restriction on the channel dispersion, strengthening also the best known results for classical channels. Moreover, as in~Ref.~\\cite{tomamicheltan14}, this generality allows us to lift the above result to a statement about coding classical information over image-additive quantum channels and general channels as long as the encoders are restricted to prepare separable states.\n\\end{enumerate}\n\nSince quantum hypothesis testing underlies many other quantum information processing tasks such as entanglement-assisted classical communication as well as private and quantum communication, we expect that our techniques will have further applications in quantum information theory.\n\n\\paragraph*{Related work.}\n\nFor classical channels, Alt\\u{u}g and Wagner~\\cite{altug14} first established the best decay rate of the average error probability for a class of discrete memoryless channels (DMCs) when the code rate approaches capacity at a rate slower than $\\Theta(n^{-1\/2})$. Shortly after the conference version of Ref.~\\cite{altug14}, Polyanskiy and Verd\\'u~\\cite{polyanskiy10c} relaxed some of the  conditions on the class of DMCs and also established the moderate deviations asymptotics for other important classical channels such as the additive white Gaussian noise channel. The other main contributions to the analysis of  hypothesis testing, channel coding, quantum hypothesis testing, and c-q channel coding in the different parameter regimes are summarised in Table~\\ref{tb:relatedwork}. \n\nFrom a technical perspective the moderate deviations regime can be approached via a refined large deviations analysis (as was done in Ref.~\\cite{altug14}) or via a variation of second-order analysis via the information spectrum method (as was proposed in Ref.~\\cite{polyanskiy10c}). In our work, we mostly follow the latter approach, interspersed with ideas from large deviation theory. In particular, we build on bounds from one-shot information theory by Wang and Renner~\\cite{wang10} and use techniques developed for the second-order asymptotics in Ref.~\\cite{tomamicheltan14}. In concurrent work, Cheng and Hsieh~\\cite{cheng17} provide a moderate deviation analysis for c-q channels via a refined error exponent analysis. Their result holds for c-q channels with finite input alphabets and their techniques are complementary to ours.\n\n\\begin{table}\n\t\\begin{tabular}{|l|c|c|c|c|}\n\t\t\\hline\n\t\t&  asymmetric binary  &  channel coding &  quantum hypothesis &  classical-quantum  \\\\\n\t\t& hypothesis testing &   &  testing & channel coding \\\\\n\t\t\\hline\n\t\tlarge deviation ($<$) & \\cite{hoeffding65} & \\cite{gallager68, csiszar11} & \\cite{hayashi07,nagaoka06} & unknown\\footnotemark \\\\\n\t\t\\hline\n\t\tmoderate deviation ($<$) & \\cite{Sas11} & \\cite{altug14, polyanskiy10c} &  \\em this work & \\em this work  \\\\\n\t\t\\hline\n\t\tsmall deviation & \\cite{strassen62} & \\cite{strassen62,hayashi09,polyanskiy10}  & \\cite{li12,tomamichel12} & \\cite{tomamicheltan14}  \\\\\n\t\t\\hline\n\t\tmoderate deviation ($>$) & \\em this work & \\em this work  & \\em this work & \\em this work \\\\\n\t\t\\hline\n\t\tlarge deviation ($>$) & \\cite{csiszar71,han89} & \\cite{arimoto73, dueck79} & \\cite{mosonyiogawa13,mosonyi14} & \\cite{mosonyi14-2} \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Exposition of related work on finite resource analysis of hypothesis testing and channel coding problems. The rows correspond to different parameter regimes, labelled by the deviation from the critical rate (i.e., the relative entropy for hypothesis testing and the capacity for channel coding problems).}\n\t\\label{tb:relatedwork}\n\\end{table}\n\n\\footnotetext{In contrast to classical channels a tight characterisation of the error exponent of c-q channels remains elusive to date even for high rates. See, e.g., Refs.~\\cite{holevo00,hayashi07,dalai13} for partial progress.}\n\n\n\\section{Preliminaries}\n\n\\subsection{Notation and classical coding over quantum channels}\n\nLet $\\mathcal{H}$ be a finite-dimensional Hilbert space and denote by $\\mathcal{S}:=\\lbrace \\rho\\in\\mathcal{H}\\,|\\,\\Tr\\rho=1,\\rho\\geq 0\\rbrace$ the quantum states on $\\mathcal{H}$. We take $\\exp(\\cdot)$ and $\\log(\\cdot)$ to be in an arbitrary but compatible base (such that they are inverses), and denote the natural logarithm by $\\ln(\\cdot)$. For convenience, we will consider the dimension of this Hilbert space to be a fixed constant, and omit any dependence constants may have on this dimension. For $\\rho, \\sigma \\in \\mathcal{S}$ we write $\\rho \\ll \\sigma$ if the support of $\\rho$ is contained in the support of $\\sigma$. For any closed subset $\\mathcal{S}_{\\circ}\\subseteq \\mathcal{S}$, we will denote by $\\mathcal{P}(\\mathcal{S}_{\\circ})$ the space of probability distributions supported on $\\mathcal{S}_{\\circ}$. We equip $\\mathcal{S}$ with the trace metric $\\delta_{\\Tr}(\\rho,\\rho'):=\\frac{1}{2}\\norm{\\rho-\\rho'}_1$ and $\\mathcal{P}(\\mathcal{S})$ with a weak-convergence metric\\footnote{An example of which is the induced L\\'evy--Prokhorov metric (see, e.g., Section 6 and Theorem 6.4 in Ref.~\\cite{partha67}).} $\\delta_\\text{wc}$, such that both are compact metric spaces with\n\\begin{align}\nf:\\mathcal{S}\\to \\mathbb{R}\\text{ continuous} \\qquad\\implies\\qquad \\mathbb{P}\\mapsto \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,f(\\rho)\\text{ continuous}.\n\\end{align}\n\nWe will use the \\emph{cumulative standard normal distribution} function $\\Phi$ is defined as\n\\begin{align}\n\t\\Phi(a)&:=\\int_{-\\infty}^{a}\\frac{1}{\\sqrt{2\\pi}}\\, e^{-\\frac{x^2}2}\\, \\mathrm{d}x.\n\\end{align}\n\nFollowing Ref.~\\cite{tomamicheltan14}, we consider a general \\emph{classical-quantum channel} $\\mathcal{W}: \\mathcal{X} \\to \\mathcal{S}$ where $\\mathcal{X}$ is any set (without further structure). We define the \\emph{image of the channel} as the set $\\mathop{\\mathrm{im}} \\mathcal{W} \\subset \\mathcal{S}$ of all quantum states $\\rho$ such that $\\rho = \\mathcal{W}(x)$ for some $x \\in \\mathcal{X}$. For convenience we assume that our Hilbert space satisfies\n\\begin{align}\n\t\\mathcal{H} = \\mathop{\\mathrm{Span}}_{\\rho \\in \\mathop{\\mathrm{im}} \\mathcal{W}} \\mathrm{supp}(\\rho)\n\\end{align}\nsuch that $\\sigma>0$ is equivalent to $\\rho\\ll\\sigma$ for all $\\rho\\in \\mathop{\\mathrm{im}} \\mathcal{W}$.\n\nFor $M, n \\in \\mathbb{N}$, an \\emph{$(n,M)$-code} for a classical-quantum \nchannel $\\mathcal{W}$ is comprised of an encoder and a decoder. The \\emph{encoder} is a \nmap $E: \\{1, 2, \\ldots, M \\} \\to \\mathcal{X}^n$ and the \\emph{decoder} is a positive \noperator-valued measure $\\{ D_m \\}_{m=1}^M$ on $\\mathcal{H}^{\\otimes n}$. Moreover, an \n\\emph{$(n,M,\\varepsilon)$-code} is an \\emph{$(n,M)$-code} that satisfies\n\\begin{align}\n\t\\frac{1}{M} \\sum_{m=1}^M \\Tr \\bigg( \\bigotimes_{i=1}^n \\mathcal{W}\\bigl(E_i(m)\\bigr) \n\tD_m \\bigg) \\geq 1 - \\varepsilon \\,,\n\\end{align}\ni.e.\\ the average probability of error does not exceed $\\varepsilon$.\nThe finite blocklength achievable region for a channel $\\mathcal{W}$ is the set of triples $(n,M,\\varepsilon)$ for which there exists an \\emph{$(n,M,\\varepsilon)$-code} on $\\mathcal{W}$. We are particularly interested in the boundary\n\\begin{align}\n\tM^*(\\mathcal{W};n,\\varepsilon) := \\max \\big\\{ M \\in \\mathbb{N} : \\exists \\textrm{ a $(n,M,\\varepsilon)$-code for } \\mathcal{W} \\big\\} \\,.\n\\end{align}\nSpecifically we are going to be concerned with the behaviour of the \\emph{maximum rate}, which is defined as $R^*(\\mathcal{W};n,\\varepsilon) := \\frac{1}{n}\\log M^*(\\mathcal{W};n,\\varepsilon)$.\n\n\\subsection{Channel parameters}\n\nAn important parameter of a channel is the largest rate such that there exists a code of vanishing error probability in the large blocklength limit. This critical rate is known as the \\emph{capacity} of a channel $C(\\mathcal{W})$, which is defined as\n\\begin{align}\n\tC(\\mathcal{W}):=\\inf_{\\epsilon>0}\\liminf\\limits_{n\\to\\infty}R^*(\\mathcal{W};n,\\epsilon).\n\\end{align}\nFor classical-quantum channels there exists a \\emph{strong converse} bound, which states that the capacity described the asymptotic rate not just for vanishing error probability, but those for non-zero fixed error probabilities as well~\\cite{winter99,ogawa99}. Together with the original channel coding theorem~\\cite{schumacher01,holevo98}, this yields\n\\begin{align}\n\t\\lim\\limits_{n\\to\\infty}R^*(\\mathcal{W};n,\\epsilon)=C(\\mathcal{W})\\quad \\text{for all }\\epsilon\\in (0,1).\n\\end{align}\n\nIn essence the strong converse tells us that the capacity entirely dictates the asymptotic behaviour of the maximum rate at a fixed error probability. How quickly the rate approaches this asymptotic value for arbitrarily low and high error probabilities are described by the channel \\emph{min-dispersion} $V_{\\min}(\\mathcal{W})$ and \\emph{max-dispersion} $V_{\\max}(\\mathcal{W})$, which are defined respectively as\n\\begin{align}\n\tV_{\\min}(\\mathcal{W})&:=\\inf_{\\epsilon> 0}\\limsup_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2,\\\\\n\tV_{\\max}(\\mathcal{W})&:=\\sup_{\\epsilon<1}\\limsup_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2.\n\\end{align}\nAs with the strong converse, the min and max-dispersions also describe the dispersion at other fixed error probabilities~\\cite{tomamicheltan14}:\n\\begin{align}\n\t\\lim_{n\\to\\infty}\\left(\\frac{C(\\mathcal{W})-R^*(\\mathcal{W};n,\\epsilon)}{\\Phi^{-1}(\\epsilon)\/\\sqrt{n}}\\right)^2=\\begin{dcases}\n\t\tV_{\\min}(\\mathcal{W}) &\\epsilon\\in(0,1\/2)\\\\\n\t\tV_{\\max}(\\mathcal{W}) &\\epsilon\\in(1\/2,1)\n\t\\end{dcases}.\n\\end{align}\n\n\\subsection{Information quantities}\n\nClassically, for two distributions $P$ and $Q$, the \\emph{relative entropy} $D(P\\|Q)$ and \\emph{relative entropy variance} $V(P\\|Q)$ are both defined as the mean and variance of the log-likelihood ratio $\\log \\bigl(P\/Q\\bigr)$ with respect to the distribution $P$. In the non-commutative case, for $\\rho,\\sigma \\in \\mathcal{S}$ with $\\rho \\ll \\sigma$, these definitions are generalised as~\\cite{umegaki62,li12,tomamichel12}\n\\begin{align}\n\tD(\\rho\\|\\sigma)&:=\\Tr\\rho\\left(\\log\\rho-\\log\\sigma\\right), \\\\\n\tV(\\rho\\|\\sigma)&:=\\Tr\\rho\\bigl(\\log\\rho-\\log\\sigma-D\\left(\\rho\\|\\sigma\\right)\\cdot\\mathrm{id}\\bigr)^2\\,.\n\\end{align}\nIf $\\rho \\not\\ll \\sigma$ both quantities are set to $+\\infty$. \n\nFollowing Ref.~\\cite{tomamicheltan14}, for a closed set $\\mathcal{S}_{\\circ}\\in \\mathcal{S}$, the \\emph{divergence radius}\\footnote{Whilst \\cref{eqn:radius} characterises the divergence radius, we will mostly rely on a more useful form presented in \\cref{defn:radius}.} $\\chi(\\mathcal{S}_{\\circ})$ is given by\n\\begin{align}\n\t\\label{eqn:radius}\n\t\\chi(\\mathcal{S}_{\\circ})\n\t=\\sup_{\\mathbb{P}\\in \\mathcal{P}(\\mathcal{S}_{\\circ})}\\int\\mathrm{d}\\mathbb{P}(\\rho)\\,D\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right).\n\\end{align}\nwhere $\\mathcal{P}(\\mathcal{S}_{\\circ})$ denotes the space of distributions on $\\mathcal{S}_{\\circ}$. If we let $\\Pi(\\mathcal{S}_{\\circ})$ denote the distributions which achieve the above supremum, we also define the \\emph{minimal and maximal peripheral variance}, $v_{\\min}(\\mathcal{S}_{\\circ})$ and $v_{\\max}(\\mathcal{S}_{\\circ})$, as\n\\begin{align}\n\tv_{\\min}(\\mathcal{S}_{\\circ}):=\\inf_{\\mathbb{P}\\in \\Pi(\\mathcal{S}_{\\circ})} \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,V\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right),\\\\\n\tv_{\\max}(\\mathcal{S}_{\\circ}):=\\sup_{\\mathbb{P}\\in \\Pi(\\mathcal{S}_{\\circ})} \\int\\mathrm{d}\\mathbb{P}(\\rho)\\,V\\left( \\rho\\, \\middle\\| \\, \\int\\mathrm{d}\\mathbb{P}(\\rho')\\,\\rho' \\right).\n\\end{align}\n\nFor the image of a quantum channel, the above three information quantities correspond exactly to the three previously defined channel parameters~\\cite{tomamicheltan14}. Specifically, for $\\mathcal{S}_{\\circ}=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}}$, we have\n\\begin{align}\n\tC(\\mathcal{W})=\\chi(\\mathcal{S}_{\\circ}),\\qquad\\quad\n\tV_{\\min}(\\mathcal{W})=v_{\\min}(\\mathcal{S}_{\\circ}),\\qquad\\quad\n\tV_{\\max}(\\mathcal{W})=v_{\\max}(\\mathcal{S}_{\\circ}).\n\\end{align}\n\n\\subsection{Moderate deviation tail bounds}\n\\label{subsec:moddev}\n\nWe now discuss the relevant tail bounds we will require in the moderate deviation regime. Let $\\lbrace X_{i,n}\\rbrace_{i\\leq n}$ be independent zero-mean random variables, and define the average variance as\n\\begin{align}\n\tV_n:=\\frac{1}{n}\\sum_{i=1}^{n}\\Var[X_{i,n}].\n\\end{align}\n\nRecall that a sequence $\\lbrace t_n\\rbrace_n$ is moderate if $x_n\\searrow 0$ \nand $\\sqrt nx_n\\to+\\infty$ as $n\\to\\infty$. Given certain bounds on the moments \nand cumulants \nof these variables, which we will make explicit below, we will see that the \nprobability that the average variable $\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}$ \ndeviates from the mean by a moderate sequence $\\lbrace t_n\\rbrace_n$ decays \nasymptotically as\n\\begin{align}\n\t\\label{eqn:asymp}\n\t\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\right]=-\\bigl(1+o(1)\\bigr)\\frac{nt_n^2}{2V_n}.\n\\end{align}\n\n\\begin{lem}[Moderate deviation lower bound]\n\t\\label{lem:moddev lower}\n\tIf there exist constants $\\nu>0$ and $\\tau$ such that $\\nu\\leq V_n$ and \n\t\\begin{align}\n\t\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}\\left[\\abs{X_{i,n}}^3\\right]\\leq \\tau\n\t\\end{align} \n\tfor all $n$, then for any $\\eta>0$ there exists a constant $N(\\lbrace t_i\\rbrace,\\nu,\\tau,\\eta)\n\t$ such that, for all $n\\geq N$, the probability of a moderate deviation is lower bounded as\n\t\\begin{align}\n\t\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^nX_{i,n}\\geq t_n\\right]\\geq \n\t-(1+\\eta)\\frac{nt_n^2}{2V_n}.\n\t\\end{align}\n\\end{lem}\n\n\\begin{lem}[Moderate deviation upper bound]\n\t\\label{lem:moddev upper}\n\tIf there exists a constant $\\gamma$ such that \n\t\\begin{align}\n\t\\frac{1}{n}\\sum_{i=1}^{n}\\sup_{s\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{sX_{i,n}}\\right]}\\leq \\gamma,\n\t\\end{align}\n\tfor all $n$, then for any $\\eta>0$ there exists a constant $N(\\lbrace t_i\\rbrace, \\gamma,\\eta)\n\t$ such that, for all $n\\geq N$, the probability of a moderate deviation is upper bounded as\n\t\\begin{align}\n\t\\ln \\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\right]\\leq -\\frac{nt_n^2}{2V_n+\\eta}.\n\t\\end{align}\n\\end{lem}\n\nIf $V_n$ has a uniform lower bound, then as $\\eta\\searrow 0$ the above two bounds sandwich together, giving the two-sided asymptotic scaling of Eq.~\\ref{eqn:asymp}. In this case we can see that\n\\begin{align}\n\t\\sigma\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\right]=\\sqrt{V_n\/n}=\\Theta(1\/\\sqrt{n})\n\t\\qquad\\text{and}\\qquad\n\t\\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}\\sigma^2\\left[X_i\\right]}=\\sqrt{V_n}=\\Theta(1),\n\\end{align}\nwhere $\\sigma\\left[\\cdot\\right]$ denotes the standard deviation. If we interpret the standard deviation as setting the `length-scale' on which a distribution decays, then the above two quantities---the deviation of the average, and average\\footnote{More specifically the root-mean-square} of the deviation---set the length-scales of small and large deviation bounds respectively. Using this intuition, we can generalise moderate deviation bounds to give tight two-sided bounds for distributions with arbitrary normalisation, in which $V_n$ is no longer bounded. To do this we will tail bound for deviations which are moderate, \\emph{in units of }$\\sqrt{V_n}$.\n\n\\begin{cor}[Dimensionless moderate deviation bound]\n\t\\label{cor:moddev nondim}\n\tIf there exists a $\\gamma$ such that\n\t\\begin{align}\n\t\t\\frac{1}{nV_n^{3\/2}}\\sum_{i=1}^{n}\\sup_{s\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{sX_{i,n}}\\right]}\\leq \\gamma,\n\t\\end{align}\n\tfor all $n$,\n\tthen there exists a constant $N(\\lbrace t_i\\rbrace,\\gamma)$ such that, for all $n\\geq N$, we have the two-sided bound \n\t\\begin{align}\n\t\t-(1+\\eta)\\frac{nt_n^2}{2}\\leq \\ln \\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_{i,n}\\geq t_n\\sqrt{V_n}\\right]\\leq -(1-\\eta)\\frac{nt_n^2}{2}.\n\t\\end{align}\n\\end{cor}\n\nWe present proofs of these lemmas in \\cref{app:tailbounds}.\n\n\\subsection{Reversing lemma}\n\nIntuitively one might expect that moderate deviation bounds can be `reversed' e.g.\\ that the bound on the probability given the deviation (see Lemmas \\ref{lem:moddev lower} and \\ref{lem:moddev upper}) of the form\n\\begin{align}\n\t\\lim\\limits_{n\\to \\infty}\\frac{V_n}{nt_n^2}\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\geq t_n\\right]=-\\frac{1}{2},\n\\end{align}\nis equivalent to a bound on the deviation given the probability\n\\begin{align}\n\t\\lim\\limits_{n\\to\\infty}\\frac{1}{t_n}\\inf\\left\\lbrace t\\in\\mathbb{R} \\,\\middle|\\, \\frac{V_n}{nt_n^2}\\ln\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^{n}X_i\\geq t\\right]\\leq -\\frac{1}{2} \\right\\rbrace=\n\\end{align}\n\nWe will now see that such an ability to `reverse' moderate deviation bounds is generic. We do this by considering two quantities $A$ and $B$ defined on the same domain, and considering the infimum value of each quantity for a fixed value of the other. \n\n\\begin{lem}[Reversing Lemma]\n\t\\label{lem:reverse}\n\tLet $\\lbrace A_i\\rbrace_i$ and $\\lbrace B_i\\rbrace_i$ be sequences of real functions with $\\inf_{t} A_i(t)\\leq 0$ and $\\inf_{t}B_i(t)\\leq 0$ for all $i$. If we define $\\hat{A}_n(b):=\\inf_{t} \\left\\lbrace A_n(t) \\middle| B_n(t)\\leq b \\right\\rbrace$ and $\\hat{B}_n(a):=\\inf_{t} \\left\\lbrace B_n(t) \\middle| A_n(t)\\leq a \\right\\rbrace$, then\n\t\\begin{align}\n\t\t\\lim\\limits_{n\\to\\infty}\\frac{\\hat{A}_n(b_n)}{b_n}=1, \\quad\\forall \\lbrace b_n\\rbrace\\text{ moderate}\n\t\t~~\\qquad&\\Longleftrightarrow\\qquad~~ \n\t\t\\lim\\limits_{n\\to\\infty}\\frac{\\hat{B}_n(a_n)}{a_n}=1, \\quad\\forall \\lbrace a_n\\rbrace\\text{ moderate}.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tSee \\cref{app:reverse}.\n\\end{proof}\n\n\n\\section{Hypothesis testing}\n\\label{sec:hypo}\n\nWhilst the divergence radius characterises the channel capacity, one-shot channel bounds are characterised by a quantity known as the \\emph{$\\epsilon$-hypothesis testing divergence}~\\cite{wang10}. As the name suggests, as well as being relevant to one-shot channel coding bounds, the hypothesis testing divergence also has an operational interpretation in the context of hypothesis testing of quantum states. We will start by considering a moderate deviation analysis of this quantity.\n\n\\subsection{Hypothesis testing divergence}\n\nConsider a hypothesis testing problem, in which $\\rho$ and $\\sigma$ correspond to the null and alternative hypotheses respectively. A test between these hypotheses will take the form of a POVM $\\lbrace Q,I-Q\\rbrace$, where $0\\leq Q\\leq I$. For a given $Q$, the type-I and type-II error probabilities are given by \n\\begin{align}\n\t\\alpha(Q;\\rho,\\sigma):=\\Tr (I-Q)\\rho,\\qquad\\qquad\n\t\\beta(Q;\\rho,\\sigma):=\\Tr Q\\sigma.\n\\end{align}\nIf we define the smallest possible type-II error given a type-I error at most $\\epsilon$ as\n\\begin{align}\n\t\\beta_\\epsilon(\\rho\\|\\sigma):=\\min_{0\\leq Q\\leq \\mathbb{I}}\\left\\lbrace \\beta(Q;\\rho,\\sigma) \\,\\middle|\\, \\alpha(Q;\\rho,\\sigma)\\leq\\epsilon \\right\\rbrace,\n\\end{align}\nthen the $\\epsilon$-hypothesis testing divergence is defined as\n\\begin{align}\n\tD^\\epsilon_{\\mathrm{h}}(\\rho\\|\\sigma):=-\\log\\frac{\\beta_{\\epsilon}(\\rho\\|\\sigma)}{1-\\epsilon}.\n\\end{align}\nWe note that the denominator of $1-\\epsilon$ follows the normalisation in~\\cite{dupuis12} such that $D_{\\mathrm{h}}^\\epsilon(\\rho\\|\\rho)=0$ for all $\\rho$.\n\nAn obvious extension of this hypothesis problem is to the case of $n$ copies of each state, i.e.\\ a hypothesis test between $\\rho^{\\otimes n}$ and $\\sigma^{\\otimes n}$, or more generally between two product states $\\otimes_{i=1}^n\\rho_i$ and $\\otimes_{i=1}^n\\sigma_i$. A second-order analysis of the $\\epsilon$-hypothesis testing divergence for a non-vanishing $\\varepsilon$ was given in~\\cite{li12,tomamichel12}.\n\n\\begin{thm}[Moderate deviation of the hypothesis testing divergence]\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and states $\\lbrace \\rho_n\\rbrace_n$ and $\\lbrace \\sigma_n\\rbrace_n$ such that both $\\lambda_{\\min}(\\sigma_i)$ and $V(\\rho_i\\|\\sigma_i)$ are both uniformly bounded away from zero, the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences of non-uniform product states for $\\epsilon_n=e^{-na_n^2}$ scale as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&=D_n-\\sqrt{2V_n}\\,a_n+o(a_n), \\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&=D_n+\\sqrt{2V_n}\\,a_n+o(a_n),\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$ and $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$. More specifically for any $\\rho$ and $\\sigma$ such that $\\rho\\ll \\sigma$, the hypothesis testing divergences of uniform product states scale as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right)&=D(\\rho\\|\\sigma)-\\sqrt{2V(\\rho\\|\\sigma)}\\,a_n+o(a_n), \\label{eq:hypo-mod}\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right)&=D(\\rho\\|\\sigma)+\\sqrt{2V(\\rho\\|\\sigma)}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{thm}\n\nIn Sect.~\\ref{subsec:inward} we will bound the regularised hypothesis testing divergences towards the relative entropy (the \\emph{inward bound}), and in Sect.~\\ref{subsec:outward} we will bound them away (the \\emph{outward bound}).\n\n\n\\begin{remark}\n\tFor sequences $\\varepsilon_n$ bounded away from zero and one the second-order expansion in Refs.~\\cite{li12,tomamichel12} yields \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\rho^{\\otimes n} \\middle\\|\\sigma^{\\otimes n}\\right) = D(\\rho\\|\\sigma) + \\sqrt{\\frac{V(\\rho\\|\\sigma)}{n}}\\, \\Phi^{-1}(\\varepsilon_n)  + O\\left(\\frac{\\log n}{n}\\right) , \\label{eq:hypo-so}\n\t\\end{align}\n\twhere $\\Phi$ denotes the cumulative distribution function of the standard normal.\n\tAs already pointed out in Ref.~\\cite{polyanskiy10c}, for small $\\varepsilon_n$ we have $\\Phi^{-1}(\\varepsilon_n) \\approx \\sqrt{- 2 \\ln \\varepsilon_n}$. Ignoring all higher order terms, the substitution $\\varepsilon_n = e^{-n a_n^2}$ into~\\eqref{eq:hypo-so} then recovers the expression in~\\eqref{eq:hypo-mod}. In this sense the two results thus agree at the boundary between small and moderate deviations.\n\\end{remark}\n\n\\begin{remark}\n\tA similar argument can be sketched at the boundary between moderate and large \n\tdeviations.\n\tThe quantum Hoeffding bound~\\cite{nagaoka06,hayashi07} states that if $\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}(\\rho^{\\otimes n}\\|\\sigma^{\\otimes n})\\leq D(\\rho\\|\\sigma) - r$ for some small $r > 0$ then $\\varepsilon_n$ drops exponentially in $n$ with the exponent given by\n\t\\begin{align}\n\t\t\\sup_{0\\leq\\alpha<1}\\frac{\\alpha-1}{\\alpha}\\Bigl[D(\\rho\\|\\sigma)-r-D_{\\alpha}(\\rho\\|\\sigma)\\Bigr] , \\label{eqn:expr1}\n\t\\end{align}\n\twhere $D_{\\alpha}(\\rho\\|\\sigma)$ is the Petz' quantum R\\'enyi relative entropy~\\cite{petz86}. For sufficiently small $r$, the expression in~\\eqref{eqn:expr1} attains its supremum close to $\\alpha = 1$ and we can thus approximate $D_{\\alpha}(\\rho\\|\\sigma) \\approx D(\\rho\\|\\sigma) + \\frac{\\alpha-1}{2} V(\\rho\\|\\sigma)$ by its Taylor expansion~\\cite{lintomamichel14}. Evaluating this approximate expression yields \n\t\\begin{align}\n\t\t\\varepsilon_n = e^{- n\\frac{r^2}{2V(\\rho\\|\\sigma)}} \\,. \\label{eqn:expr2}\n\t\\end{align} \n\tup to leading order in $r$. Substituting $r = \\sqrt{2V(\\rho\\|\\sigma)}a_n$ into~\\eqref{eqn:expr2} then recovers~\\eqref{eq:hypo-mod}.\n\tAn essentially equivalent argument is also applicable to the strong converse exponent derived in Ref.~\\cite{mosonyiogawa13}.\n\\end{remark}\n\n\\subsection{Nussbaum--Szko\\l a distributions}\n\nTo allow us to apply a moderate deviation analysis to the quantum hypothesis testing divergence, we leverage the results of Ref.~\\cite{tomamichel12} which allow us to reduce the hypothesis testing divergence of quantum states to a quantity known as the information spectrum divergence of certain classical distributions, known as the Nussbaum--Szko\\l a distributions. \n\n\\begin{defn}[Nussbaum--Szko\\l a distributions~\\cite{NussbaumSzkola2009}]\n\tThe \\emph{Nussbaum--Szko\\l a distributions} for a pair of states $\\rho$ and $\\sigma$ are given by\n\t\\begin{align}\n\t\tP^{\\rho,\\sigma}(a,b)=r_a\\abs{\\braket{\\phi_a}{\\psi_b}}^2\\qquad\\text{and}\\qquad Q^{\\rho,\\sigma}(a,b)=s_b\\abs{\\braket{\\phi_a}{\\psi_b}}^2\n\t\\end{align}\n\twhere the states are eigendecomposed as $\\rho=\\sum_ar_a\\ketbra{\\phi_a}{\\phi_a}$ and $\\sigma=\\sum_bs_b\\ketbra{\\psi_b}{\\psi_b}$. \n\\end{defn}\n\nThe power of the Nussbaum--Szko\\l a distributions lies in their ability to reproduce both the divergence and variance of the underlying quantum states\n\\begin{align}\n\tD(\\rho\\|\\sigma)=D(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})\n\t,\n\t\\qquad \\text{and}\\qquad \n\tV(\\rho\\|\\sigma)=V(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})\n\t.\n\\end{align}\nAs well as capturing these asymptotic quantities, the hypothesis testing relative entropy, which arises one-shot channel coding  bounds, can also be captured by the Nussbaum--Szko\\l a distributions. Specifically this is done via the \\emph{information spectrum divergence}, which is defined for two classical distributions $P$ and $Q$ by a tail bound on the log-likelihood ratio as\n\\begin{align}\n\tD_{\\mathrm{s}}^{\\epsilon}(P\\|Q):=\\sup\\left\\lbrace R ~\\middle|~ \\Pr_{X\\leftarrow P}\\left[\\log \\frac{P(X)}{Q(X)}\\leq R\\right]\\leq \\epsilon \\right\\rbrace.\n\\end{align}\nInserting the Nussbaum--Skzo\\l a distributions, we find that the (classical) information spectrum divergence approximates the (quantum) hypothesis testing divergence.\n\\begin{lem}[Thm.~14, Ref.~\\cite{tomamichel12}]\n\t\\label{lem:infospecdiv}\n\tThere exists a universal constant $K$ such that for any states $\\rho$ and $\\sigma$ with $\\lambda_{\\min}(\\sigma)\\geq \\lambda$ and $\\epsilon< 1\/2$, we find that $D_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)$ is bounded as\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)&\\leq D_{\\mathrm{s}}^{2\\epsilon}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})+\\log\\frac{1-\\epsilon}{\\epsilon^3(1-2\\epsilon)}+\\log K \\lceil \\ln(1\/\\lambda)\\rceil\\\\\n\t\tD_{\\mathrm{h}}^\\epsilon(\\rho\\|\\sigma)&\\geq D_{\\mathrm{s}}^{\\epsilon\/2}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})-\\log\\frac{1}{\\epsilon(1-\\epsilon)}-\\log K \\lceil \\ln(1\/\\lambda)\\rceil,\n\t\\end{align}\n\tand $D_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)$ is bounded as\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)&\\leq D_{\\mathrm{s}}^{1-\\epsilon\/2}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})+\\log\\frac{1-\\epsilon\/2}{\\epsilon^4}+\\log K \\lceil \\ln(1\/\\lambda)\\rceil\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon}(\\rho\\|\\sigma)&\\geq D_{\\mathrm{s}}^{1-2\\epsilon}(P^{\\rho,\\sigma}\\|Q^{\\rho,\\sigma})-\\log\\frac{1}{\\epsilon^2}-\\log K \\lceil \\ln(1\/\\lambda)\\rceil.\n\t\\end{align}\n\\end{lem}\n\nAs the information spectrum divergence is defined in terms of a tail bound, we will bound these quantities using the moderate deviation tail bounds of Sect.~\\ref{subsec:moddev}. To do this, we will start by showing that the log-likelihood ratio of Nussbaum--Skzo\\l a distributions is sufficiently well behaved, specifically that its cumulant generating function has bounded derivatives.\n\n\\begin{lem}[Bounded cumulants]\n\t\\label{lem:boundedcumulants}\n\tFor $\\lambda>0$, there exists constants $C_k(\\lambda)$ such that the cumulant generating function $h(t):=\\ln \\mathbb{E}\\left[e^{tZ}\\right]$ of the log-likelihood ratio $Z:=\\log P^{\\rho,\\sigma}\/Q^{\\rho,\\sigma}$ for $\\lambda_{\\min}(\\sigma)\\geq \\lambda$ is smooth and has uniformly bounded derivatives in a neighbourhood of the origin\n\t\\begin{align}\n\t\\sup_{\\abs{t}\\leq 1\/2}\\abs{\\frac{\\partial^k}{\\partial t^k}h(t)} \\leq C_k.\n\t\\end{align}\n\\end{lem}\nWe present a proof of this lemma in \\cref{app:cumulant}.\n\n\\subsection{Inward bound}\n\\label{subsec:inward}\n\\begin{prop}[Inward bound]\n\t\\label{prop:HTD-inward}\n\tFor any constants $\\lambda,\\eta>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N$, the hypothesis testing divergence can be bounded for any states $\\lbrace\\rho_i\\rbrace_i$ and $\\lbrace \\sigma_i\\rbrace_i$ with\n\t$\\lambda_{\\min}(\\sigma_i)\\geq\\lambda$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\sqrt{2V_n}a_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\sqrt{2V_n}\\,a_n+\\eta a_n.\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$ and $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$.\n\\end{prop}\n\\begin{proof}\n\tFirstly, let $Z_i$ be the log-likelihood ratios \n\t\\begin{align}\n\tZ_i:=\\log\\frac{P^{\\rho_i,\\sigma_i}(A_i,B_i)}{Q^{\\rho_i,\\sigma_i}(A_i,B_i)}, \\qquad (A_i,B_i)\\leftarrow P^{\\rho_i,\\sigma_i}.\n\t\\end{align}\n\tIn terms of these log-likelihood ratios, the lower and upper bound on the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences respectively from \\cref{lem:infospecdiv} become\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq \\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\log \\frac{1}{\\epsilon_n(1-\\epsilon_n)}-\\log Kn\\lceil\\ln(1\/\\lambda) \\rceil,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq \\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq 1-\\epsilon_n\/2 \\right\\rbrace+\\log \\frac{1-\\epsilon_n\/2}{\\epsilon_n^4}+\\log Kn\\lceil\\ln(1\/\\lambda) \\rceil.\n\t\\end{align}\n\t\n\tRecalling that $\\epsilon_n:=e^{-na_n^2}$, we can see that in both cases the error terms scale like $\\Theta(na_n^2)$ and $\\Theta(\\log n)$ respectively, which are both $o(na_n)$. As such, there must exist an $N_1(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_1$, these error terms are bounded by $\\eta na_n\/2$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq \\frac{1}{n}\\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq \\frac{1}{n}\\sup\\left\\lbrace R~\\middle|~ \\Pr\\Biggl[\\sum_{i=1}^n Z_i\\leq R\\Biggr]\\leq 1-\\epsilon_n\/2 \\right\\rbrace+\\eta a_n \/2.\n\t\\end{align}\n\t\n\tNext we want to apply the tail bounds of Sect.~\\ref{subsec:moddev}. To this end, we will start by defining zero-mean variables $X_i:=Z_i-D(\\rho_i\\|\\sigma_i)$. In terms of these variables, the above bounds take the form\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (-X_i)\\geq t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace-\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(+X_i\\right)\\geq  t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace+\\eta a_n\/2.\n\t\\end{align}\n\t\n\tBy \\cref{lem:boundedcumulants} there exists constants $\\bar V(\\lambda)$ and $\\gamma(\\lambda)$, such that $V_i\\leq \\bar V$ and\n\t\\begin{align}\n\t\\sup_{t\\in[0,1\/2]}\\abs{\\frac{\\mathrm{d}^3}{\\mathrm{d}s^3}\\ln\\mathbb{E}\\left[e^{s(\\pm X_i)}\\right]}\\leq \\gamma\n\t\\end{align}\n\tfor all $i$. If we let $t_n:=\\left(\\sqrt{2V_n}+\\eta\/2\\right)a_n$, then \\cref{lem:moddev upper} gives an $N_2(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_2$, the tail probability is bounded as\n\t\\begin{align}\n\t\t\\ln\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t_n\\Biggr]\n\t\t&\\leq \\frac{-nt_n^2}{2V_n+\\eta^2 \/3}\\\\\n\t\t&\\leq -\\frac{\\left(\\sqrt{2V_n}+\\eta\/2\\right)^2}{2V_n+\\eta^2\/5}na_n^2\\\\\n\t\t&\\leq -\\frac{2V_n+\\eta^2\/4}{2V_n+\\eta^2\/5}na_n^2\\\\\n\t\t&= -\\left(1+\\frac{\\eta^2}{40V_n+4\\eta^2}\\right)na_n^2\\\\\n\t\t&\\leq -\\left(1+\\frac{\\eta^2}{40\\bar{V}+4\\eta^2}\\right)na_n^2.\n\t\\end{align}\n\tAs $\\eta^2\/(40\\bar{V}+4\\eta)$ is a constant and $na_n^2\\to \\infty$, there must exist a constant $N_3(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such $n\\geq N_3$ implies\n\t\\begin{align}\n\t\t\\ln\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t_n\\Biggr]\\leq -na_n^2-1=\\ln (\\epsilon_n\/2),\n\t\\end{align}\n\tand therefore that \n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq t\\Biggr]\\leq \\epsilon_n\/2 \\right\\rbrace\\leq t_n.\n\t\\end{align}\n\tPutting everything together, we get that for any $n\\geq N(\\lbrace a_i\\rbrace,\\lambda,\\eta):=\\max\\lbrace N_1,N_2,N_3\\rbrace$ we have\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\sqrt{2V_n}\\,a_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\sqrt{2V_n}\\,a_n+\\eta a_n.\n\t\\end{align}\n\tas required.\n\\end{proof}\n\n\\subsection{Outward bound}\n\\label{subsec:outward}\n\n\\begin{prop}[Outward bound]\n\t\\label{prop:HTD-outward}\n\tFor any constants $\\lambda,\\eta>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N$, the hypothesis testing divergence can be bounded for any states $\\lbrace\\rho_i\\rbrace_i$ and $\\lbrace\\sigma_i\\rbrace_i$ with\n\t$\\lambda_{\\min}(\\sigma_i)\\geq\\lambda$ as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n-\\eta\\,a_n.\n\t\\end{align}\n\twhere $D_n:=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma_i)$. Moreover, if we \n\tlet $V_n:=\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma_i)$, and there also \n\texists a constant $\\nu>0$ such that $V_i\\geq \\nu$ for all $i$, then there \n\texists an $N'(\\lbrace a_i\\rbrace,\\lambda,\\nu,\\eta)$ such that, for $n\\geq \n\tN'$, the hypothesis testing divergence is more tightly bounded as\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\sqrt{2V_n}a_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\sqrt{2V_n}\\,a_n-\\eta a_n.\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tSimilar to \\cref{prop:HTD-inward}, we will start by taking the upper and lower bounds on the $\\epsilon_n$- and $(1-\\epsilon_n)$-hypothesis testing divergences respectively from \\cref{lem:infospecdiv}. \n\tThis gives that there exists an $N_1(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that, for $n\\geq N_1$, we have\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (-X_i)\\geq t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace+\\eta a_n\/2,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(+X_i\\right)\\geq  t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace-\\eta a_n\/2.\n\t\\end{align}\n\twhere $X_i:=Z_i-D(\\rho_i\\|\\sigma_i)$.\n\t\n\tFirstly, applying Chebyshev's inequality two standard deviations below the mean gives us that\n\t\\begin{align}\n\t\t\\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\\geq  -2\\sqrt{V_n\/n}\\Biggr]\\geq 3\/4\\geq2\\epsilon_n,\n\t\\end{align}\n\tand so we conclude that\n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n \\left(\\pm X_i\\right)\\geq  t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace\\geq -2\\sqrt{V_n\/n}.\n\t\\end{align}\n\tBy \\cref{lem:boundedcumulants}, $V_n$ must be bounded $V_n\\leq \\bar{V}(\\lambda)$, and thus $\\sqrt{V_n\/n}=\\mathcal{O}(1\/\\sqrt{n})=o(a_n)$. As such, there must exist an $N_2(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that $n\\geq N_2$ implies $2\\sqrt{V_n\/n}\\leq \\eta a_n\/2$. Inserting this tail bound, we get that for any $n\\geq N(\\lbrace a_i\\rbrace,\\lambda,\\eta):=\\max\\lbrace N_1,N_2\\rbrace$ that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\eta a_n,\n\t\\end{align}\n\tas required.\n\t\n\tIf there also exists an $\\nu > 0$ such that $V_i\\geq \\nu$, then we can use a more refined moderate deviation bound. Specifically, \\cref{lem:boundedcumulants} gives us a bound on the absolute third moment of $X_i$, which allows us to apply \\cref{lem:moddev lower}. If we let $t_n:=(\\sqrt{2V_n}-\\eta\/2)a_n$ and assume $\\eta<\\sqrt{8\\nu}$ such that $\\lbrace t_n\\rbrace_n$ is moderate, then this gives us that there exists an $N_3(\\lbrace a_i\\rbrace, \\lambda,\\nu,\\eta)$ such that, for any $n\\geq N_3$, the tail probabilities are bounded\n\t\\begin{align}\n\t\t\\ln \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\t\\geq t_n\\Biggr]\n\t\t&\\geq -\\Bigl(1+\\eta\/\\sqrt{2\\bar{V}}\\,\\Bigr)\\frac{nt_n^2}{2V_n} \\\\\n\t\t&\\geq -\\frac{\\left(1+\\eta\/\\sqrt{2\\bar{V}}\\right)\\left(\\sqrt{2V_n}-\\eta\/2\\right)^2}{2V_n}na_n^2 \\\\\n\t\t&\\geq -\\frac{\\left(1+\\eta\/\\sqrt{2V_n}\\right)\\left(\\sqrt{2V_n}-\\eta\/2\\right)^2}{2V_n}na_n^2 \\\\\n\t\t&\\geq -\\left(1-\\frac{5\\eta^2}{8\\bar V}\\right)na_n^2.\n\t\\end{align}\n\tOnce again, the second term in the parenthesis is a non-zero constant, and thus there must exist an $N_4(\\lbrace a_i\\rbrace,\\lambda,\\eta)$ such that\n\t\\begin{align}\n\t\t\\log \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\t\\geq t_n\\Biggr]\\geq -na_n^2+1=\\ln 2\\epsilon_n,\n\t\\end{align}\n\tallowing us to conclude $\\Pr\\left[\\frac{1}{n}\\sum_{i=1}^n (\\pm X_i)\n\t\\geq t_n\\right]\\geq 2\\epsilon_n$, and therefore \n\t\\begin{align}\n\t\t\\inf\\left\\lbrace t~\\middle|~ \\Pr\\Biggl[\\frac{1}{n}\\sum_{i=1}^n X_i\\geq t\\Biggr]\\leq 2\\epsilon_n \\right\\rbrace\\geq t_n.\n\t\\end{align}\n\tInserting this into the above bounds, we find that for any $n\\geq N'(\\lbrace a_i\\rbrace,\\lambda,\\nu,\\eta):=\\max\\lbrace N_1,N_3,N_4\\rbrace$, we have the desired final bound\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\leq D_n-\\sqrt{2V_n}\\,a_n+\\eta a_n,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\|\\,\\bigotimes_{i=1}^n\\sigma_i\\right)&\\geq D_n+\\sqrt{2V_n}\\,a_n-\\eta a_n.\n\t\\end{align}\n\\end{proof}\n\n\n\\section{Channel Coding}\n\\label{sec:channels}\n\nWe are now going to show how the above moderate deviation bounds can be applied to the capacity of a classical-quantum channel. \n\n\\begin{thm}[Moderate deviation of c-q channels]\n\t\\label{thm:moddevcode}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and memoryless c-q channel \n\t$\\mathcal{W}$ with capacity $C(\\mathcal{W})$ and min-dispersion $V_{\\min}(\\mathcal{W})$, being \n\toperated at error probability no larger than $\\epsilon_n:=e^{-na_n^2}$, the \n\toptimal rate deviates below the capacity as\n\t\\begin{align}\n\t\tR^*\\left(\\mathcal{W} ;n,\\epsilon_n\\right)=C(\\mathcal{W})-\\sqrt{2V_{\\min}(\\mathcal{W})}\\,a_n+o(a_n).\n\t\\end{align}\n\tConversely, if the channel has max-dispersion $V_{\\max}$ and is operated at \n\terror probability no larger than $1-\\epsilon_n$, then the optimal rate \n\tdeviates above the capacity as\n\t\\begin{align}\n\t\tR^*\\left(\\mathcal{W};n,1-\\epsilon_n\\right)=C(\\mathcal{W})+\\sqrt{2V_{\\max}(\\mathcal{W})}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{thm}\nIf either the min- or max-dispersion is non-zero, an application of \\cref{lem:reverse} gives an equivalent formulation in terms of the minimal error probability at a given rate.\n\n\\begin{cor}\n\tFor any moderate sequence $\\lbrace s_n\\rbrace_n$, the error probability for a code with min-dispersion $V_{\\min}>0$ deviating below capacity by $s_n$ scales as\n\t\\begin{align}\n\t\t\\lim\\limits_{n\\to \\infty}\\frac{1}{ns_n^2}\\ln\\epsilon^*(\\mathcal{W};n,C-s_n)=-\\frac{1}{2V_{\\min}}.\n\t\\end{align}\n\tSimilarly, for a code with max-dispersion $V_{\\max}>0$ deviating above capacity by $s_n$, the error probability scales\n\t\\begin{align}\n\t\\lim\\limits_{n\\to \\infty}\\frac{1}{ns_n^2}\\ln\\bigl( 1-\\epsilon^*(\\mathcal{W};n,C + s_n)\\bigr)=-\\frac{1}{2V_{\\max}}.\n\t\\end{align}\n\\end{cor}\n\n\\begin{remark}\n\tRecall that our definition of c-q channels does not put any restriction on the input set. In particular, this set may be comprised of quantum states itself such that the c-q channel is just a representation of a quantum channel. Hence, as pointed out in Ref.~\\cite{tomamicheltan14}, our results immediately also apply to classical communication over general image-additive channels~\\cite{wolf14} as well as classical communication over quantum channels with encoders restricted to prepare separable states. We refer the reader to Corollaries 6 and 7 of Ref.~\\cite{tomamicheltan14} for details. \n\\end{remark}\n\nWe will split the proof of \\cref{thm:moddevcode} in two, in Sect.~\\ref{subsec:coding1} we will prove a lower bound on the maximum rate (`achievability'), followed in Sect.~\\ref{subsec:coding2} by a corresponding the upper bound (`optimality'). For the rest of this section, we will fix the channel $\\mathcal{W}$, and omit any dependencies on $\\mathcal{W}$ from here on for notational convenience.\n\n\\subsection{Achievability}\n\\label{subsec:coding1}\n\nFor achievability, we will use a lower bound on the $\\epsilon$-one-shot rate that is essentially due to Hayashi and Nagaoka~\\cite{hayashi03} who analysed the coding problem using the information spectrum method.\n\n\\begin{lem}[Theorem 1 of Ref.~\\cite{wang10}]\n\t\\label{lem:wangrenner}\n\tIf we have a c-q channel which maps from a finite message space $Y$ as $y\\mapsto \\rho^{(y)}$ , then the maximum rate with error probability at most $\\epsilon$ and $1-\\epsilon$, $R^*(\\epsilon)$ and $R^*(1-\\epsilon)$ respectively, are lower bounded\n\t\\begin{align}\n\t\tR^*(\\epsilon) &\\geq \\sup_{P_Y}\n\t\tD^{\\epsilon\/2}_{\\mathrm{h}} \\left( \\pi_{YZ} \\middle\\| \\pi_Y\\otimes \\pi_Z \\right) -\\log\\frac{8(2-\\epsilon)}{\\epsilon}\\\\\n\t\tR^*(1-\\epsilon) &\\geq \\sup_{P_{Y}}\n\t\tD^{1-2\\epsilon}_{\\mathrm{h}} \\left( \\pi_{YZ} \\middle\\| \\pi_Y\\otimes\\pi_Z \\right) -\\log\\frac{8(1-\\epsilon)}{\\epsilon}\n\t\\end{align}\n\twhere $\\pi_{YZ}$ is the joint state of the input and output, with inputs chosen according to the distribution $P_{Y}$\n\t\\begin{align}\n\t\t\\pi_{YZ}:=\\sum_{y\\in Y}P_{Y}(y)\\ketbra{y}{y}_{Y}\\otimes \\rho^{(y)}_{Z}.\n\t\\end{align}\n\\end{lem}\n\n\\begin{prop}[Channel coding: Achievability]\n\t\\label{prop:codingachievability}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and error probability $\\epsilon_n:=e^{-na_n^2}$, the rate is at least\n\t\\begin{align}\n\t\tR^*\\left(n,\\epsilon_n\\right)\\geq C-\\sqrt{2V_{\\min}}a_n+o(a_n).\n\t\\end{align}\n\tSimilarly, at error probability $1-\\epsilon_n$, the rate is at least\n\t\\begin{align}\n\t\tR^*\\left(n,1-\\epsilon_n\\right)\\geq C+\\sqrt{2V_{\\max}}a_n+o(a_n).\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tLet $X$ be our, possibly infinite, message space. By Lemma 3 of Ref.~\\cite{tomamicheltan14}, there exists a finite subset $Y\\subseteq X$, and a distribution $Q_{Y}$ thereon, such that $D(\\rho\\|\\sigma)=C$ and $V(\\rho\\|\\sigma)=V_{\\min}$ for states\n\t\\begin{align}\n\t\t\\rho:=\\sum_{y\\in Y}Q_{Y}(y)\\ketbra{y}{y}\\otimes \\rho^{(y)} \n\t\t\\qquad \\text{and} \\qquad \\sigma:=\\sum_{y\\in Y}Q_{Y}(y)\\ketbra{y}{y}\\otimes \\sum_{y'\\in Y}Q_{Y}(y')\\rho^{(y')}.\n\t\\end{align}\n\t\n\tClearly by restricting the message space we can only ever decrease the rate. By applying \\cref{lem:wangrenner} to the restriction of the message space to $Y$, we can lower bound the maximum rate of the full code. Applying this reasoning to $n$ memoryless applications of our channel we find\n\t\\begin{align}\n\t\tnR^*(n,\\epsilon_n) \n\t\t&\\geq \\sup_{P_{{Y}^n}} D^{\\epsilon_n\/2}_{\\mathrm{h}} \\left( \\pi_{Y^nZ^n} \\middle\\| \\pi_{Y^n}\\otimes \\pi_{Z^n} \\right) -\\log\\frac{8(2-\\epsilon_n)}{\\epsilon_n}.\n\t\\end{align}\n\tSubstituting in both the error probability, which is no larger than $\\epsilon_n=e^{-na_n^2}$, and a product distribution $Q_{{Y}^n}(\\vec{y}):= \\prod_{i=1}^n Q_{Y}(y_i)$ then we get\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n) \\geq \n\t\t\\frac{1}{n}D^{\\epsilon_n\/2}_{\\mathrm{h}} \\left( \\rho^{\\otimes n} \\middle\\| \\sigma^{\\otimes n} \\right)+\\mathcal{O}(a_n^2).\\label{ineq:rate}\n\t\\end{align}\n\tApplying \\cref{prop:HTD-inward}, we get an overall bound on the rate of\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n) \\geq \n\t\tC-\\sqrt{2V_{\\min}}\\,a_n+o(a_n).\n\t\\end{align}\n\tIf instead we were to take a distribution $Q_{Y}$ such that $V(\\rho\\|\\sigma)=V_{\\max}$, then the same arguments would allow us to use \\cref{prop:HTD-outward} to analogously give\n\t\\begin{align}\n\t\tR^*(n,1-\\epsilon_n) \\geq \n\t\tC+\\sqrt{2V_{\\max}}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{proof}\n\n\\subsection{Optimality}\n\\label{subsec:coding2}\nSimilar to the second-order analysis of Ref.~\\cite{tomamicheltan14}, we are going to do this by relating the capacity and one-shot maximum rates to geometric quantities known as the divergence radius and divergence centre. \n\n\\begin{defn}[Divergence radius and centre]\n\t\\label{defn:radius}\n\tFor some set of states $\\mathcal{S}_0\\subseteq\\mathcal{S}$, the \\emph{divergence radius} $\\chi(\\mathcal{S}_0)$ and \\emph{divergence centre} $\\sigma^*(\\mathcal{S}_0)$ are defined as \n\t\\begin{align}\n\t\t\\chi(\\mathcal{S}_0):=\\mathop{\\mathrm{inf}\\vphantom{p}}_{\\sigma\\in\\mathcal{S}} \\sup_{\\rho\\in\\mathcal{S}_0}~D(\\rho\\|\\sigma), \\qquad\\qquad \n\t\t\\sigma^*(\\mathcal{S}_0):=\\mathop\\mathrm{arg~min}\\limits_{\\sigma\\in \\mathcal{S}}\\sup_{\\rho\\in\\mathcal{S}_0}D(\\rho\\|\\sigma).\n\t\\end{align}\n\tSimilarly the\n\t$\\epsilon$-\\emph{hypothesis testing divergence radius} $\\chi_{\\mathrm{h}}^\\epsilon(\\mathcal{S}_0)$ is defined as \n\t\\begin{align}\n\t\t\\chi_{\\mathrm{h}}^{\\epsilon}(\\mathcal{S}_0):=\\mathop{\\mathrm{inf}\\phantom{p}}_{\\sigma\\in\\mathcal{S}} \\sup_{\\rho\\in\\mathcal{S}_0}~D_{h}^{\\epsilon}(\\rho\\|\\sigma).\n\t\\end{align}\n\\end{defn}\n\nWhilst we have seen that the divergence radius captures the capacity of a channel, the $\\epsilon$-hypothesis testing divergence radius approximates the one-shot capacity.\n\n\\begin{lem}[Proposition 5 of \\cite{tomamicheltan14}]\n\t\\label{lem:rate}\n\tFor $\\mathcal{I}:=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}}$, the maximum rate with error probability at most $\\epsilon$, $R^*(\\epsilon)$, is upper bounded as\n\t\\begin{align}\n\t\tR^*(\\epsilon)\\leq \\chi_{\\mathrm{h}}^{2\\epsilon}(I)+\\log\\frac{2}{1-2\\epsilon}.\n\t\\end{align}\n\tSimilarly for an error probability $1-\\epsilon$, the maximum rate is upper bounded as\n\t\\begin{align}\n\t\tR^*(1-\\epsilon)\\leq \\chi_{\\mathrm{h}}^{1-\\epsilon\/2}(I)+\\log\\frac{2(2-\\epsilon)}{\\epsilon^2}.\n\t\\end{align}\n\\end{lem}\n\nIf we take $\\mathcal{I}_n:=\\overline{\\mathop{\\mathrm{im}} \\mathcal{W}^{\\otimes n}}$ to be the closure of the image of $n$ uses of this channel, then we can extend this bound on the one-shot rate to the $n$-shot rate as\n\\begin{align}\n\tnR^*(n,\\epsilon_n)&\\leq \\chi_{\\mathrm{h}}^{2\\epsilon_n}(\\mathcal{I}_n)+\\log\\frac{2}{1-\\epsilon_n},\\\\\n\tnR^*(n,1-\\epsilon_n)&\\leq \\chi_{\\mathrm{h}}^{2\\epsilon_n}(\\mathcal{I}_n)+\\log\\frac{2(2-\\epsilon_n)}{\\epsilon_n^2}.\n\\end{align}\nAs we are considering memoryless c-q channels, $\\mathcal{I}_n$ simply consists of elementwise tensor products of $\\mathcal{I}$\n\\begin{align}\n\t\\mathcal{I}_n=\\left\\lbrace \\bigotimes_{i=1}^n\\rho_i \\,\\middle|\\, \\rho_i\\in \\mathcal{I} \\right\\rbrace.\n\\end{align}\n\nOnce again we are going to take $a_n$ to be an arbitrary moderate sequence, and $\\epsilon_n:=e^{-na_n^2}$. Expanding this out, this gives bounds on the rate of\n\\begin{align}\n\tR^*(n,\\epsilon_n)&\\leq \\inf_{\\sigma^n}\\sup_{\\lbrace \\rho_i\\rbrace\\subseteq \\mathcal{I}}~\\frac{1}{n}D^{2\\epsilon_n}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \\middle\\|\\sigma^n \\right) +\\frac{1}{n}\\log \\frac{2}{1-\\epsilon_n},\\label{ineq:converse}\\\\\n\tR^*(n,1-\\epsilon_n)&\\leq \\inf_{\\sigma^n}\\sup_{\\lbrace \\rho_i\\rbrace\\subseteq \\mathcal{I}}~\\frac{1}{n}D^{1-\\epsilon_n\/2}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \\middle\\|\\sigma^n \\right) +\\frac{1}{n}\\log \\frac{2(2-\\epsilon)}{\\epsilon_n^2}.\n\\end{align}\n\nA standard approach now is to pick a state $\\sigma^n$, such that we can bound \nthe above quantities for arbitrary sequences $\\lbrace \\rho_i\\rbrace$ using the \nmoderate deviation analysis of the hypothesis testing divergence presented in \nSect.~\\ref{sec:hypo}. To do this we need to consider two cases. The \\emph{high cases} are those in which the empirical relative entropy corresponding to \n$\\lbrace \\rho_i\\rbrace _{i=1}^n$ is close to capacity, and the \\emph{low cases} \nare those in which the empirical relative entropy corresponding to $\\lbrace \n\\rho_i\\rbrace _{i=1}^n$ is far from capacity. Specifically, for some constant \n$\\gamma$ that will be chosen later, the $n$ which correspond to high and low \ncases are denoted by $H(\\lbrace\\rho_i\\rbrace,\\gamma)$ and \n$L(\\lbrace\\rho_i\\rbrace,\\gamma)$, respectively. They are defined as\n\\begin{align}\n\tH(\\lbrace\\rho_i\\rbrace,\\gamma):=\\left\\lbrace n~ \\middle| \n\t\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)\\geq C-\\gamma \\right\\rbrace\n\t\\quad\\text{and}\\quad\n\tL(\\lbrace\\rho_i\\rbrace,\\gamma):=\\left\\lbrace n~ \\middle| \n\t\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)< C-\\gamma \\right\\rbrace\n\\end{align}\nsuch that $H(\\lbrace\\rho_i\\rbrace,\\gamma)$ and $L(\\lbrace\\rho_i\\rbrace,\\gamma)$ \nbipartition $\\mathbb{N}$ for all $\\gamma$.\n\nBefore employing a moderate deviation bound, we are going to construct a separable state $\\sigma^n$ that will allow us two different moderate deviation analyses for low and high sequences, such that we can obtain the required bounds in both cases. A convenient choice of $\\sigma^n$ would be  $\\sigma^n={\\bar\\rho_n}^{\\otimes n}$ where $\\bar\\rho_n:=\\frac{1}{n}\\sum_{i=1}^n\\rho_i$, but the order of the infimum and supremum require $\\sigma^n$ to be chosen to be independent of the sequence $\\lbrace \\rho_i\\rbrace$. Instead we are going to construct $\\sigma^n$ from a mixture of states that lie in a covering of $\\mathcal{S}$, and the divergence centre $\\sigma^*(\\mathcal{I})$. \n\nThe following lemma is based on a construction in Lemma II.4 of Ref.~\\cite{hayden04b}.\n\n\\begin{lem}[Lemma 18 of Ref.~\\cite{tomamicheltan14}]\n\tFor every $\\delta\\in(0,1)$, there exists a set $\\mathcal{C}^\\delta\\subset\\mathcal{S}$ of size\n\t\\begin{align}\n\t\\abs{\\mathcal{C}^\\delta}\\leq \\left(\\frac{20(2d+1)}{\\delta}\\right)^{2d^2}\\left(\\frac{8d(2d+1)}{\\delta}+2\\right)^{d-1}\\leq\\left(\\frac{90d}{\\delta^2}\\right)^{2d^2}\n\t\\end{align}\n\tsuch that, for every $\\rho\\in \\mathcal{S}$ there exists a state $\\tau\\in\\mathcal{C}^\\delta$ such that\n\t\\begin{align}\n\tD(\\rho\\|\\tau)\\leq \\delta\\qquad\n\t\\textrm{and}\\qquad\\lambda_{\\mathrm{min}}(\\tau)\\geq \\frac{\\delta}{8d(2d+1)+\\delta}\\geq \\frac{\\delta}{25d^2}.\n\t\\end{align} \n\\end{lem}\n\nGiven this covering upon states, we now want to take $\\sigma^n$ to be the separable state given by a mixture over such a covering, and the divergence centre\n\\begin{align}\n\t\\sigma^n(\\gamma):=\n\t\\frac{1}{2}\\sigma^*(I)^{\\otimes n}+\n\t\\frac{1}{2\\abs{\\mathcal{C}^{\\gamma\/4}}}\\sum_{\\tau\\in\\mathcal{C}^{\\gamma\/4}} \\tau^{\\otimes n}.\\label{eqn:sigmadef}\n\\end{align}\nUsing the inequality \n\\begin{align}\n\tD_{\\mathrm{h}}^\\epsilon\\bigl(\\rho\\,\\big\\|\\,\\mu\\sigma+(1-\\mu)\\sigma'\\bigr)\\leq D^\\epsilon_{\\mathrm{h}}(\\rho\\|\\sigma)-\\log \\mu\n\\end{align}\nwe will be able to bound divergences with respect to $\\sigma^n$ by those divergences with respect to either elements of $\\mathcal{C}^{\\gamma\/4}$, or $\\sigma^*$.\n\nWe will start by considering the low case. We will see that this case only accounts for hypothesis testing relative entropies which are below the capacity by a constant amount.\n\n\\begin{lem}[Low case]\n\t\\label{lem:low}\n\tFor any $\\gamma>0$, there exists a constant $N(\\lbrace a_i\\rbrace,\\gamma)$ such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq C-\\gamma\/4,\\\\\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq C-\\gamma\/4,\n\t\\end{align}\n\tfor any $\\lbrace\\rho_i\\rbrace_i\\subset I$, $n\\in \n\tL(\\lbrace\\rho_i\\rbrace,\\gamma)$ and $n\\geq N$.\n\\end{lem}\n\\begin{proof}\n\tWe are going to start by considering the $\\epsilon_n$-hypothesis testing divergence. Take $\\tau_n$ to be the closest element in $\\mathcal{C}^{\\gamma\/4}$ to $\\bar{\\rho}_n$, such that $D(\\bar{\\rho}_n\\|\\tau_n)\\leq \\gamma\/4$. Splitting out the $\\tau_n$ term from $\\sigma_n(\\gamma)$, we have\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\gamma)\\right)\n\t\t&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\tau_n^{\\otimes n}\\right)+\\log 2\\abs{\\mathcal{C}^\\gamma}\\\\\n\t\t&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\tau_n^{\\otimes n}\\right)+2d^2\\log\\left(\\frac{120d}{\\gamma^2}\\right).\n\t\\end{align}\n\tAs the final term depending on $\\abs{\\mathcal{C}^{\\gamma\/4}}$ is independent of $n$, there must exist a constant $N_1(\\gamma)$ such that $2d^2\\log(120d\/\\gamma^2)\\leq n\\gamma\/4$ for any $n\\geq N_1$, and thus that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma)\\right)\n\t\t\\leq \\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes n}\\right)+\\gamma\/4.\n\t\\end{align}\n\t\n\tApplying \\cref{prop:HTD-outward} to the $\\epsilon_n$-hypothesis testing relative entropy with respect to $\\tau_n$ we get that there exists an $N_2(\\lbrace a_i\\rbrace,\\gamma)$ such that \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes} \\right)\\leq \\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\tau_n)+\\gamma\/4,\n\t\\end{align}\n\tfor any $n\\geq N_2$. As for the divergence terms given with respect to $\\tau_n$, we can rearrange them in terms of divergences relative to the sequence mean $\\bar\\rho_n$ using the information geometric Pythagorean theorem, yielding\n\t\\begin{align}\n\t\t\\sum_{i=1}^{n}D(\\rho_i\\|\\tau_n)\n\t\t&=\\sum_{i=1}^{n}\\Tr \\rho_i(\\log\\rho_i-\\log\\bar\\rho_n)+\\sum_{i=1}^{n}\\Tr\\rho_i(\\log\\bar\\rho_n-\\log \\tau_n)\\label{eqn:line2}\\\\\n\t\t&=\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)+nD(\\bar\\rho_n\\|\\tau_n)\\\\\n\t\t&\\leq\\sum_{i=1}^{n}D(\\rho_i\\|\\bar\\rho_n)+n\\gamma\/4.\n\t\\end{align}\n\t\n\tIf we let $N(\\lbrace a_i\\rbrace, \\gamma):=\\max\\lbrace N_1,N_2\\rbrace$, then pulling the above results together we see that for any $n\\geq N$ \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\n\t\t&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\tau_n^{\\otimes n}\\right)+\\gamma\/4\\\\\n\t\t&\\leq\\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\tau_n)+2\\gamma\/4\\\\\n\t\t&\\leq\\frac{1}{n}\\sum_{i=1}^nD(\\rho_i\\|\\bar{\\rho}_n)+3\\gamma\/4.\n\t\\end{align}\n\tFinally, since $n\\in L(\\lbrace\\rho_i\\rbrace,\\gamma)$ the average relative \n\tentropy is bounded away from capacity, and we arrive at the bound:\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\\leq C-\\gamma\/4.\n\t\\end{align}\n\t\n\tAs we only relied on \\cref{prop:HTD-outward} to bound the regularised \n\thypothesis testing divergence to within a constant of the average relative \n\tentropy, we could perform a similar analysis for the \n\t$(1-\\epsilon_n)$-hypothesis testing divergence using \\cref{prop:HTD-inward} \n\tinstead, which gives\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\middle\\| \\sigma_n(\\gamma) \\right)\\leq C-\\gamma\/4.\n\t\\end{align}\n\\end{proof}\n\nNow that we have dealt with cases far from capacity, we turn our attention to the high cases.\n\n\\begin{lem}[High case]\n\t\\label{lem:high}\n\tFor any $\\eta>0$, there exist constants $\\Gamma(\\eta)$ and $N(\\lbrace a_i\\rbrace,\\eta)$, such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\Gamma)\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n\n\t\\end{align}\n\tfor any $\\lbrace\\rho_i\\rbrace_i\\subset I$, $n\\in \n\tH(\\lbrace\\rho_i\\rbrace,\\Gamma)$ and $n\\geq N$. Similarly, the \n\t$(1-\\epsilon_n)$-hypothesis testing relative entropy is bounded\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\Gamma)\\right)\\leq C+\\sqrt{2V_{\\max}}a_n+\\eta a_n.\n\t\\end{align}\n\\end{lem}\n\\begin{proof}\n\tSplitting out the $\\sigma^*$ factor within $\\sigma_n(\\gamma)$ gives\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\log 2,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\log 2.\n\t\\end{align}\n\tAs $\\frac{1}{n}\\log 2=o(a_n)$, there exists an $N_1(\\lbrace a_i\\rbrace)$ such that $n\\geq N_1$ implies \n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3,\\\\\n\t\tD_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3.\n\t\\end{align}\n\tWe now wish to employ a moderate deviation result. We will start by addressing the $\\epsilon_n$-hypothesis testing divergence. For the weaker bound of \\cref{prop:HTD-outward} we will have no required bounds on $\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)$, but for the stronger bound we will need a uniform lower bound.\n\t\n\tIf $V_{\\min}\\leq \\eta^2\/18$, then the weakened bound of \\cref{prop:HTD-outward} is sufficient, giving an $N_2(\\lbrace a_n\\rbrace,\\eta)$ such that $n\\geq N_2$ implies\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)+2\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\t\n\tNext we need to consider the case where $V_{\\min}>\\eta^2\/18$. To do this, we will need to establish a lower bound on $\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)$, which places it near $V_{\\min}$. The min-dispersion is defined for distributions which exactly achieve capacity; we will now consider an analogous quantity for distributions which are \\emph{near} capacity. Specifically\n\t\\begin{align}\n\t\tV_{\\min}(\\gamma):=\\inf_{P\\in\\mathcal{P}(\\mathcal{I})}\\left\\lbrace \n\t\t\\int\\mathrm{d}P(\\rho)~V\\left(\\rho\\middle\\|\\sigma^*\\right) \n\t\t~\\middle|~\n\t\t\\int\\mathrm{d}P(\\rho)~D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P(\\rho')~\\rho'\\right)\\geq C-\\gamma\n\t\t\\right\\rbrace.\n\t\\end{align}\n\tBy definition of the channel dispersion we have that $V_{\\min}(0)=V_{\\min}$. By Lemma 22 of Ref.~\\cite{tomamicheltan14} we can strengthen this to $\\lim_{\\gamma\\to 0^+}V_{\\min}(\\gamma)=V_{\\min}$, and so for any $\\eta>0$ there must exist a constant $\\Gamma(\\eta)$ such that \n\t\\begin{align}\n\t\t\\sqrt{2V_{\\min}(\\Gamma)}\\geq \\sqrt{2V_{\\min}}-\\eta\/3.\\label{eqn:V}\n\t\\end{align} \n\tAs $V_{\\min}\\geq \\eta^2\/18$, this implies that $V_{\\min}(\\Gamma)>0$.\n\t\n\tNext, let $P_n$ be the empirical distribution corresponding to the set \n\t$\\lbrace\\rho_i\\rbrace_{i=1}^n$, i.e.\\ \n\t$P_n(\\rho):=\\frac{1}{n}\\sum_{i=1}^{n}\\delta(\\rho-\\rho_i)$. For all $n\\in \n\tH(\\lbrace\\rho_i\\rbrace,\\Gamma)$, these distributions are near capacity\n\t\\begin{align}\n\t\t\\int\\mathrm{d}P_n(\\rho)~ D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P_n(\\rho')~\\rho'\\right)=\\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\bar{\\rho}_n)\\geq C-\\Gamma,\n\t\\end{align}\n\tand so we can lower bound the average variance with respect to the divergence centre\n\t\\begin{align}\n\t\t\\frac{1}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)=\\int\\mathrm{d}P(\\rho)~ V(\\rho\\|\\sigma^*)\\geq V_{\\min}(\\Gamma)>0.\n\t\\end{align}\n\tUsing this lower bound, we can apply the stronger bound from \n\t\\cref{prop:HTD-outward} to give a constant $N_3(\\lbrace a_i\\rbrace,\\eta)$, \n\tsuch that, for every $n\\in H(\\lbrace\\rho_i\\rbrace,\\Gamma)$ and $n\\geq N_3$, \n\tthe hypothesis testing divergence is upper bounded\n\t\\begin{align}\n\t\tD_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)&\\leq\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,{\\sigma^*}^{\\otimes n}\\right)+\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{\\frac{2}{n}\\sum_{i=1}^{n}V(\\rho_i\\|\\sigma^*)}a_n+2\\eta a_n\/3\\\\\n\t\t&\\leq \\frac{1}{n}\\sum_{i=1}^{n}D(\\rho_i\\|\\sigma^*)-\\sqrt{2V_{\\min}(\\Gamma)}\\,a_n+2\\eta a_n\/3\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\t\n\tPerforming a similar argument for $V_{\\max}$, we construct a function\n\t\\begin{align}\n\t\tV_{\\max}(\\gamma):=\\sup_{P\\in\\mathcal{P}(\\mathcal{I})}\\left\\lbrace \n\t\t\\int\\mathrm{d}P(\\rho)~V\\left(\\rho\\middle\\|\\sigma^*\\right) \n\t\t~\\middle|~\n\t\t\\int\\mathrm{d}P(\\rho)~D\\left(\\rho~\\middle\\|\\int\\mathrm{d}P(\\rho')~\\rho'\\right)\\geq C-\\gamma\n\t\t\\right\\rbrace,\n\t\\end{align}\n\tand define a $\\Gamma$ such that\n\t\\begin{align}\n\t\\sqrt{2V_{\\max}(\\Gamma)}\\leq \\sqrt{2V_{\\max}}+\\eta\/3.\n\t\\end{align} \n\tFollowing through the rest of the argument, and employing \\cref{prop:HTD-inward}, we also get a bound on the $(1-\\epsilon_n)$-hypothesis testing divergence\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{1-\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\,\\sigma_n(\\gamma)\\right)\n\t\t&\\leq C+\\sqrt{2V_{\\max}}a_n+\\eta a_n.\n\t\\end{align} \n\\end{proof}\n\n\\begin{prop}[Channel coding: Optimality]\n\t\\label{prop:codingconverse}\n\tFor any moderate sequence $\\lbrace a_n\\rbrace_n$ and error probability $\\epsilon_n:=e^{-na_n^2}$, the rate is upper bounded as\n\t\\begin{align}\n\t\tR^*\\left(n,\\epsilon_n\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+o(a_n).\n\t\\end{align}\n\tFor error probability $(1-\\epsilon_n)$ the rate is similarly upper bound as\n\t\\begin{align}\n\t\tR^*\\left(n,1-\\epsilon_n\\right)\\leq C+\\sqrt{2V_{\\max}}\\,a_n+o(a_n).\n\t\\end{align}\n\\end{prop}\n\\begin{proof}\n\tApplying Lemmas \\ref{lem:low} and \\ref{lem:high}, we get that there exist constants $\\Gamma(\\eta)$ and $N_1(\\lbrace a_i\\rbrace,\\eta)$ such that\n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\Gamma)\\right)\\leq \n\t\t\\begin{dcases}\n\t\t\tC-\\Gamma\/4 & n\\in L(\\lbrace\\rho_i\\rbrace,\\Gamma)\\\\\n\t\t\tC-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n & n\\in H(\\lbrace\\rho_i\\rbrace,\\Gamma)\n\t\t\\end{dcases}\n\t\\end{align}\n\tfor any $n\\geq N_1$. As $\\Gamma$ is a constant, there must exist some $N_2(\\lbrace a_i\\rbrace,\\eta)$ such that $\\Gamma\/4\\geq \\sqrt{2V_{\\min}}a_n$. As such, for any $n\\geq \\max\\lbrace N_1,N_2\\rbrace$, high or low, we have \n\t\\begin{align}\n\t\t\\frac{1}{n}D_{\\mathrm{h}}^{\\epsilon_n}\\left( \\bigotimes_{i=1}^n\\rho_i \\,\\middle\\|\\, \\sigma_n(\\Gamma)\\right)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+\\eta a_n.\n\t\\end{align}\n\tPulling this bound back to Eq.~\\ref{ineq:converse}, we have\n\t\\begin{align}\n\t\tR^*(n,\\epsilon_n)\n\t\t&\\leq \\sup_{\\lbrace \\rho_i\\rbrace\\subseteq I}\\frac{1}{n}D^{2\\epsilon_n}_{\\mathrm{h}}\\left(\\bigotimes_{i=1}^n\\rho_i \n\t\t\\middle\\|\\sigma^n(\\Gamma) \\right) +\\frac{1}{n}\\log \n\t\t\\frac{2}{1-\\epsilon_n}\\\\\n\t\t&\\leq C-\\sqrt{2V_{\\min}}a_n+\\eta a_n+\\frac{1}{n}\\log \\frac{2}{1-\\epsilon_n}.\n\t\\end{align}\n\tFinally, noting that $1\/n=o(a_n)$, there must exist a constant $N_3 (\\lbrace a_i\\rbrace,\\eta)$ such that $n\\geq N_3$ implies\n\t\\begin{align}\n\t\\frac{1}{n}\\log\\frac{2}{1-\\epsilon_n}\\leq \\eta a_n.\n\t\\end{align}\n\tWe can therefore conclude that, for $n\\geq \\max\\lbrace N_1,N_2,N_3\\rbrace $, we get the overall upper bound\n\t\\begin{align}\n\tR^*(n,\\epsilon_n)\\leq C-\\sqrt{2V_{\\min}}\\,a_n+2\\eta a_n.\n\t\\end{align}\n\tAs this is true for arbitrary $\\eta>0$, we can take $\\eta \\searrow0$ and conclude \n\t\\begin{align}\n\tR^*(n,\\epsilon_n)\\leq C-\\sqrt{2V_{\\min}}a_n+o(a_n)\n\t\\end{align}\n\tas required. A similar analysis for the $(1-\\epsilon_n)$-error regime shows\n\t\\begin{align}\n\tR^*\\left(n,1-\\epsilon_n\\right)\\leq C+\\sqrt{2V_{\\max}}a_n+o(a_n).\n\t\\end{align}\n\\end{proof}\n\n\n\\section{Conclusion}\n\nThe main result of this paper is to give a second order approximation of the non-asymptotic fundamental limit for classical information transmission over a quantum channel in the moderate deviations regime, as in Eqs.~\\ref{eq:moderatesecondorder1} and \\ref{eq:moderatesecondorder2}:\n\\begin{align}\n\t\\frac{1}{n} \\log M^*(\\mathcal{W};n, \\varepsilon_n) &= C(\\mathcal{W}) - \\sqrt{2 V_{\\min}(\\mathcal{W})}\\, x_n + o(x_n) \\,,\\\\\n\t\\frac{1}{n} \\log M^*(\\mathcal{W};n,1- \\varepsilon_n) &= C(\\mathcal{W}) + \\sqrt{2 V_{\\max}(\\mathcal{W})}\\, x_n + o(x_n) \\,.\n\\end{align}\nAlong the lines of third and fourth order approximations for classical channel coding in the fixed error regime (see, e.g., Refs.~\\cite{polyanskiythesis10,tomamicheltan12,moulin12}), a natural question to ask is whether we can expand this further and resolve the term $o(x_n)$. A preliminary investigation suggests the conjecture that $o(x_n) = O(x_n^2) + O(\\log n)$ and that at least some of the implicit constants can be determined precisely. We leave this for future work.\n\nDue to the central importance of binary asymmetric quantum hypothesis testing \nwe expect our techniques to have applications also to other quantum channel \ncoding tasks. In particular, source coding~\\cite{datta15,leditzky16}, \nentanglement-assisted classical coding~\\cite{datta14} as well as \nquantum~\\cite{tomamichel16} and private coding~\\cite{wilde16} over quantum \nchannels have recently been analysed in the small deviations regime by relating \nthe problem to quantum hypothesis testing. An extension of these results to \nmoderate deviations using our techniques thus appears feasible.\n\n\\paragraph*{Acknowledgements.} MT is funded by an Australian Research Council Discovery Early Career Researcher Award (DECRA) fellowship (Grant Nos. CE110001013, DE160100821). Both MT and CTC acknowledge support from the ARC Centre of Excellence for Engineered Quantum Systems (EQuS). We also thank Hao-Chung Cheng and Min-Hsiu Hsieh for useful discussions and insightful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhite dwarfs (WDs) are degenerate stellar nuclei with a mass roughly that of the\nSun and radii one hundredth that of the Sun; consequently,\ntheir surface gravity is $\\sim$$10^4$ greater than the\nSun's. \\cite{lie84} identified some odd WDs with metal-rich\natmospheres. With such a powerful gravity pulling the chemical\nelements toward the stellar nucleus, it is somewhat unusual to have a metal-rich\natmosphere. In fact, the timescales for an element heavier than\nhydrogen or helium to sink are small: $\\sim$$10^2$ yr in WDs with\nhydrogen atmospheres (DAs) and $\\sim$$10^5$ yr with helium atmospheres\n(DBs) \\citep{jur08,von07,paq86}.\n\nInterstellar material accretion onto the WD surface was one of the first\nexplanations for the metal-rich atmospheres. Knowing the diffusion\ntimescales of metals in the stellar atmospheres and the metallicity of a given\nstar, it is possible to calculate the necessary accretion rate to keep this\nmetallicity constant in time \\citep{koe06}. Typical values are\n$10^{-18}$ to $10^{-15}$ M$_\\odot$yr$^{-1}$. These values are too high to be\nexplained exclusively by interstellar accretion. Furthermore, if there were\naccretion from the interstellar medium onto the DBs, there should be a large\namount of hydrogen pollution in their spectra, but this pollution has not been\ndetected \\citep{dup1993,far2008}.\n\n\\cite{zuc87} observed an IR~excess in the spectrum of G29-38. The shape of this\nIR~excess is a bump which peaks at $\\sim$$10~\\mu m$. Its width is roughly\n$20~\\mu m$ and can be fitted with a blackbody of $T_{\\mathrm{eff}} \\sim 10^3$ K.\n\\cite{gra90} argued that an asteroid closely approaching G29-38 could explain\nthis infrared excess. When the asteroid orbit reaches the Roche radius it is\ndisrupted and forms a disk around the star. This disk is heated up by the\nstellar radiation and emits in the infrared. The disk material falls down\ncontinuously on the WD giving rise to the observed metal-rich atmospheres.\n\nUsing a disk model, Jura \\& collaborators (2003,2007,2008) were able to fit\nthe IR excess of many WDs. Through these fits they determined disk physical\nparameters. With a different approach, \\cite{rea05} fit the spectrum to a thin\ndust shell model.\n\nWhile searching for interacting binary WDs, \n\\cite{gan2006,gan2007,gan2008} observed double peaks in calcium lines\nin a few DAZs and DBZs.  Although these observations strongly suggest\nthe disk hypothesis,it is still possible that in some cases the emitting\nregion could be a torus or a shell \\citep{rea09} rather than a disk.\n\nPrevious works focused on the IR emission properties of\ndebris disks around the WD. In this work, we propose a new and complementary\nobservational test looking at the absorption and scattering properties instead.\nWe develop a simple theoretical framework to predict what will be observable\nand measurable according to the properties of the system. We also suggest an\nobservational program to reach our goal.\n\nWe investigate the possibility of detection of debris disks effects\nin the near-UV and optical. We start the analysis in the limit of an optically\nthick disk in Section~\\ref{sec:od} and in Section~\\ref{sec:ot} we extend this to\nthe optically thin limit case. In Section~\\ref{ObsTest}, we discuss the\nobservational predictions of our models. Our conclusions are presented in\nSection~\\ref{sec:conc}.\n\n\n\n\\section{Opaque disk}\n\\label{sec:od}\n\n  The optically thick limit is the natural first approach to investigate the\npossible effects of a debris disk in the spectrum of a WD. This limit can be\nachieved not only in massive disks but also in certain regions of all disks\nspecially if the disk has some gas like some recently discovered gaseous disks\n\\citep{gan2006,gan2007,gan2008}. Also, the mathematical treatment developed in\nthis section will be used in the next section when dealing with the\noptically thin limit.\n\nA completely opaque disk will not have any\nspectral features because it is totally opaque and absorbs any photon whatever\nits energy. The only effect of an obscuring disk will be a decrease in the\nreceived flux from the star. The most the disk can obscure the star is half of\nthe projected stellar surface, $\\pi R_{wd}^2\/2$. The increase in the apparent\nmagnitude of the star will be $0.75$ mag.\n\nThis value is much higher than current photometric accuracy, and if present,\nwould have been detected for those WDs which have an observed parallax, a good\nspectrum and an IR~excess. The ``good spectrum' permits a determination of\n$T_{\\mathrm{eff}}$ and $\\log g$ and hence a luminosity. Clearly the luminosity\ninferred  from the parallax should agree with the luminosity inferred from the\nspectral fit. At the very least, this procedure will allow us to affirm that\nthere is not a big opaque disk in the known WDs with IR~excess or, at least,\nthis disk is not in a favorable inclination.\n\n  For the general case of a disk with arbitrary inclination and any combination\nof inner and outer radii, the flux received at the Earth from the system\n(target) is,\n\n\\begin{equation}\n  \\begin{aligned}\n  F^{target} &=\\lefteqn { \\int_{\\Omega_{wd}} I \\cos \\theta d\\Omega }\\\\\n     &= I \\left(\\frac{R_*}{D}\\right)^2\n      \\int_{0}^{\\pi\/2} \\int_{\\phi_{min}}^{\\phi_{max}}\n      \\sin \\theta \\cos \\theta d \\phi d \\theta \n  \\end{aligned}\n\\end{equation}\n\n\\noindent\nwhere we assumed that the intensity (I) is uniform over the stellar surface.\nThe stellar radius is $R_*$ and $D$ is the distance from the Earth to the system.\nThere are three possible projections for the disk, as seen in Figure~\\ref{3disks}.\nThis gives different values of $\\phi_{min}$ and $\\phi_{max}$:\n\n\n\\begin{figure}\n\\epsscale{0.6}\n\\plotone{3disks2.ps}\n\\caption{Debris disk and WD. For a given size and inclination of the disk,\n         different amounts of it obscure the star. The $x-$ and $y-$axes are in\n         the sky-plane and the observer is over the $z-$axis which makes an\n         angle $i$ with the normal of the disk. The\n         angles $\\phi_i$ and $\\phi_e$ show where the disk starts and stops\n         obscuring the star.\n}\n\\label{3disks}\n\\end{figure}\n\n\n\n  \\begin{itemize}\n  \\item \n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\left[ \\int_{-\\pi\/2}^{\\phi_i} d \\phi + \\int_{\\phi_e}^{\\pi\/2} d \\phi\n      \\right ]\n      \\\\= 2\\pi - 2( \\phi_e-\\phi_i )\n  $,\n\n  \\item\n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\int_{-\\pi\/2}^{\\phi_i} d \\phi    =\n      \\pi + 2\\phi_i\n  $,\n\n  \\item\n  $\\int_{\\phi_{min}}^{\\phi_{max}} d \\phi =\n      2 \\int_{-\\pi\/2}^{\\pi\/2} d \\phi     =\n      2 \\pi\n  $.\n\n  \\end{itemize}\n\n  Using the dimensionless radii $r_{\\{i\/e\\}} \\equiv R_{\\{i\/e\\}}\/R_{star}$ we\ndefine the function $g \\equiv g(\\theta,r_i,r_e)$:\n\n  \\begin{equation}\n  g = \\left\\{\n   \\begin{array}{l l}\n      \\pi - (\\phi_e-\\phi_i) & \\mbox{, $r_e \\cos i < \\sin \\theta $}\\\\\n      \\pi\/2 + \\phi_i        & \\mbox{, $r_i \\cos i < \\sin \\theta \\leq r_e\n                                                             \\cos i $}\\\\\n      \\pi                   & \\mbox{, $r_i \\cos i \\geq \\sin \\theta$}\\\\\n    \\end{array}\n    \\right.\n  \\label{eqg}\n  \\end{equation}\n\n\n\\noindent\nand write the flux:\n\n  \\begin{equation}\n  F^{target} = I \\left(\\frac{R_*}{D}\\right)^2\n            \\int_{0}^{\\pi\/2} \\sin(2\\theta) g(\\theta,r_i,r_e) d \\theta\n  \\label{eqF}\n  \\end{equation}\n\n\\noindent\nwhere\n\n  \\begin{equation}\n  \\phi_{i,e} = \\arctan \\left[ \\cos i \\arccos\\left(\n               \\frac{\\sqrt{ \\sin^2\\theta\/r_{i,e}^2 - \\cos^2 i}}{\\sin i}\n               \\right) \\right ]\n  \\label{eqphi}\n  \\end{equation}\n\n\n  The total flux received from an unobscured system is $\\pi I\n  (R_*\/D)^2$.  We call the hypothetical unobscured star ``template''\n  and the obscured star ``target''. Defining $p$ as the ratio of the\n  obscured to the total projected area:\n\n\n\n\\begin{equation}\n p \\equiv \\frac{A_{obscured}}{A_{total}}  = \\frac{A_{target}}{A_{template}}\n\\label{def:p}\n\\end{equation}\n\n\\noindent\nwe write the increase in magnitude as:\n\n\\begin{equation}\n \\Delta m = -2.5 \\log(1-p)\n\\label{eq:mp}\n\\end{equation}\n\n\nIn the completely opaque hypothesis we may obviously write that,\n\n\\begin{equation}\n  \\begin{aligned}\n p &=\\lefteqn { \\frac{F^{target}}{F^{template}}\n   = \\frac{F^{target}}{\\pi I} \\left(\\frac{D}{R_*}\\right)^2 }\\\\\n   &= \\frac{1}{\\pi} \\int_{0}^{\\pi\/2} \\sin(2\\theta) g(\\theta,r_i,r_e) d \\theta\n  \\end{aligned}\n\\label{eq:p}\n\\end{equation}\n\n  The solution of Equation \\ref{eq:p} with Equations \\ref{eqg} and \\ref{eqphi}\ngives the flux received from the system, as can been seen in\nFigure~\\ref{fig:hidden}. The probability of finding a system more inclined than\na given angle is the ratio of the solid angle occupied by these systems\nto the total solid angle: $P(i>i_0) = \\cos i_0$.\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{hiddenFlux.ps}\n\\caption{Increase in magnitude vs. inclination for opaque disks for\n   different inner ($r_i$) and outer ($r_e$) disk radius combinations.\n   Systems seen face on have $i=0^\\circ$. The fast decrease of $\\Delta m$ for\n   $i \\rightarrow 90^\\circ$ is an artifact of the mathematical model that\n   assumes an infinitely thin disk. For real disks, there would be a plateau\n   lower than curve peak, but the disk should be really thin indeed, so the\n   plateau is very close to $\\Delta m = 0$ and it actually is not a plateau,\n   but rather a single point.\n   The top axis label shows the percentage of systems more inclined than $i$.\n   The right axis label shows $p$, the ratio of the obscured to the total\n   projected area (Equation~\\ref{def:p}).}\n\\label{fig:hidden}\n\\end{figure}\n\n  Figure~\\ref{fig:hidden} shows that it is possible to detect a completely\nopaque debris disk. For an inclination angle causing any blocking, the bigger\nthe disk is, the easier it is to detect the effect. The inner and outer radii\nare based on physical constrains. If the inner radius is small, the dust\nparticles sublimate because the\ntemperature exceeds $\\sim 1200$K~\\citep{jur07a}. On the other hand, if the\nouter edge of the disk is big ($\\gtrsim 100 R_{wd}$), the dust grains will be\ncold and any emission will be undetectable in practice. The exact\nvalues depend on the dust type and grain size.\n\n  The presence of an obscuring disk might be inferred by comparing the expected\nincrease in stellar magnitude with the luminosity derived from\n$T_{\\mathrm{eff}}$ and $\\log g$ and parallax. If the observed and the expected\nmagnitudes are correct and not equal, the flux deficiency can probably be\nexplained by obscuration from a debris disk.\n\n  We use the measured parallax of GD~362 to illustrate the previous analysis\nwith one real case. \\cite{kil08} obtained d~$= 50.6_{-3.1}^{+3.5}$~pc for\nGD~362. Using simple error propagation, we have a rough estimate for the highest\nacceptable difference between expected and measured magnitude: $\\sigma_m\n\\approx 0.15$ mag.  \\cite{kil08} did not find any discrepancies between parallax\nand flux, implying no obscuration of the star.\n\nFrom a geometrical perspective this is expected since from\nFigure~\\ref{fig:hidden} $\\Delta m\\approx0.15$ mag implies an inclination higher\nthan $\\sim$$80^\\circ$ and less than $\\sim$$20\\%$ of the systems will be more\ninclined than this. Indeed, \\cite{jur07b} showed that GD~362 must be seen\nnearly face on to be able to reproduce its IR~excess flux with physically\nreasonable inner and outer radii. Assuming an almost edge on system would\nrequire a big disk and an unusual mechanism to heat it to reproduce the\nmeasured IR~excess flux. Therefore, our work is in agreement with the previous\nresults and this analysis illustrates what kind of study must be done with\nother systems which may be found to be nearly edge on.\n\n\\section{Optically thin dust disk}\n\\label{sec:ot}\n\n  After having derived the basic concepts of the problem with the optically\n  thick limit we generalize the equations to the optically thin limit.\n  The ratio ($\\xi_\\nu$) of the flux from obscured (target) to the equivalent\nstar with no obscuration (template) is composed of three main components:\n\n\\begin{equation}\n\\begin{aligned}\n   \\xi_\\nu &\\equiv \\lefteqn {\\frac{F^{target}_\\nu}{F^{template}_\\nu}}\\\\ &=\n   \\left. \\xi_\\nu \\right|_{unobscured} +\n   \\left. \\xi_\\nu \\right|_{obscured} +\n   \\left. \\xi_\\nu \\right|_{scattered}\n\\end{aligned}\n\\label{eq:3comp}\n\\end{equation}\n\n  The unobscured ratio component is simply $1-p$. The obscured ratio\n  is given by $p e^{-\\tau_\\nu^{ext} \/\\cos{i}}$. The extinction optical depth\n  $\\tau_\\nu^{ext}$ of the disk regions obscuring the star accounts for the\n  absorbed and scattered light along the line of sight to the star.\n  The scattered component comes from the disk regions which do not obscure the\n  star but scatter photons to the line of sight. There is no emission component\n  because the dust temperature is lower than the dust sublimation temperature\n  ($\\sim$1200~K) and thus the dust emission only contributes in the infrared.\n\n  Using the dimensionless extinction efficiency ($Q^{ext}_\\nu$) instead of\nextinction cross section ($[C^{ext}_\\nu] = cm^2$) we write the differential\nextinction optical depth in the disk as:\n\n\\begin{equation}\n d\\tau_\\nu = n C^{ext}_\\nu dz = n Q^{ext}_\\nu \\pi a^2 dz,\n\\end{equation}\n\n\\noindent\nwhere $n$ (cm$^{-3}$) is the number of dust grains per unit volume, $a$ (cm)\nis the grain radius and $z$ (cm) is the vertical dimension of the disk\n\n The disk volume density ($\\rho$) is related to the density of a typical dust\ngrain ($\\rho_d$) through,\n\n\\begin{equation}\n \\rho = \\frac{4}{3} \\pi a^3 \\rho_d n .\n\\label{rho}\n\\end{equation}\n\n  Assuming the disk to be vertically uniform we integrate to write\n\n\\begin{equation}\n \\tau_\\nu^{ext} = \\int^{H\/2}_{-H\/2} \\frac{3 Q^{ext}_\\nu \\rho}{4 a \\rho_d} dz\n          = \\frac{3 Q^{ext}_\\nu \\rho}{4 a \\rho_d} H\n          = \\frac{3 Q^{ext}_\\nu \\Sigma}{4 a \\rho_d},\n\\label{eq:tau}\n\\end{equation}\n\n\\noindent\nwhere $\\Sigma$ (g\/cm$^2$) is the disk surface density and $H$ is the disk\nheight.\n\n  We define:\n\n\\begin{equation}\n \\tau_0 = \\frac{3 \\Sigma}{4 a \\rho_d}\n\\label{eqt0}\n\\end{equation}\n\n\\noindent\nand write Equation~\\ref{eq:3comp} as\n\n\\begin{equation}\n \\xi_\\nu = (1-p) +\n               p e^{ -\\tau_0 Q^{ext}_\\nu \/ \\cos{i} } +\n               \\left. \\xi_\\nu \\right|_{scattered}.\n\\label{eqFd}\n\\end{equation}\n\n  To simplify the scattering term, we assume isotropic and\ncoherent scattering and also that the light is not attenuated before and after\nbeing scattered by the disk. The last hypothesis is valid in the optically thin\ncase and causes an overestimation of the scattering because we ignore the\nabsorbed photons. The scattered intensity is given by\n\n\\begin{equation}\n I^{sca}_\\nu = \\epsilon_\\nu \\frac{H}{\\cos{i}}\n             = \\pi a^2 Q^{sca}_\\nu n J^{wd}_\\nu \\frac{H}{\\cos{i}},\n\\end{equation}\n\n\\noindent\nwhere $\\epsilon_\\nu$ is the emissivity, $Q^{sca}_\\nu$ is the scattering\nefficiency and $J^{wd}_\\nu$ is the mean stellar intensity\n\n\\begin{equation}\n  J^{wd}_\\nu = \\frac{I^{wd}_\\nu \\pi R_{wd}^2}{4 \\pi r^2}\n\\end{equation}\n\n  Ignoring the disk regions hidden by the star we integrate over the disk\nsurface to get the flux:\n\n\n\\begin{equation}\n  F_\\nu = \\frac{1}{2} \\frac{3 \\Sigma}{4 a \\rho_d} Q^{sca}_\\nu\n          \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i}\n          \\, \\pi I^{wd}_\\nu \\left( \\frac{R_{wd}}{D} \\right)^2\n\\end{equation}\n\n\\noindent\nusing Equation~\\ref{eqt0} and dividing by the template flux,\n\n\n\\begin{equation}\n  \\left. \\xi_\\nu \\right|_{scattered} = \\frac{1}{2} \\tau_0 Q^{sca}_\\nu\n          \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i},\n\\end{equation}\n\n\\noindent\nwhich allows us to write Equation~\\ref{eqFd} as\n\n\\begin{equation}\n \\xi_\\nu = (1-p) +\n               p e^{ -\\tau_0 Q^{ext}_\\nu \/ \\cos{i} } +\n               \\frac{1}{2} \\tau_0 Q^{sca}_\\nu\n               \\ln \\left( \\frac{re}{ri} \\right ) \\cos{i}\n\\label{eqFdcp}\n\\end{equation}\n\n\nBesides the parameters $p$ and $\\tau_0$, we have the absorption efficiencies\nwhich are characteristic of the dust type. We used the tables of optical\nconstants of silicate glasses from \\cite{dor95}. The\nauthors prepared two different glasses in laboratory: pyroxene,\nMg$_x$Fe$_{1-x}$SiO$_3$, with $x$=0.4, 0.5, 0.6, 0.7, 0.8, 0.95, 1.0 and olivine,\nMg$_{2x}$Fe$_{2-2x}$SiO$_4$, with $x$=0.4 and 0.5.\n\nFigures \\ref{pyrmg70} and \\ref{pyrmg70sca} display the results from\nEquation~\\ref{eqFdcp} for different optical depths and system\ngeometries. Figure \\ref{pyrmg70} represents inclinations where the disk\nobscures the star, and light is absorbed and Figure\n\\ref{pyrmg70sca} when there is no obscuration and we see only\nscattering plus the WD light.  For observational tests, the region\nfrom $\\textrm{3000\\AA\\ to 5000\\AA}$ is the most interesting, because it\nshows a sharp change in the ratio between the target and the template\nwhich cannot be easily discarded as bad flux calibrations.\n\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{pyrmg70_sca.ps}\n\\caption{Expected ratio between the target and the template star\n          (Equation~\\ref{eqFdcp}) when the inclination is such that the disk\n          obscures\n          a part of the WD as in 1 and 2 in Figure \\ref{3disks}. The disk is\n          composed of olivine Mg$_{0.8}$Fe$_{1.2}$SiO$_4$ dust grains. To\n          calculate the scattering we used\n          $\\ln(r_e\/r_i) \\lesssim \\ln(100\/5) = 3$\n          as a superior limit in Equation~\\ref{eqFdcp} and adopted $i=85^\\circ$.}\n\\label{pyrmg70}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{pyrmg70_pureSca.ps}\n\\caption{Expected ratio between the target and the template star\n          (Equation~\\ref{eqFdcp}) when the disk does not obscure the WD\n          and we see the scattering plus the stellar light. The disk is\n          composed of olivine Mg$_{0.8}$Fe$_{1.2}$SiO$_4$ dust grains. All the\n          curves were calculated with $\\tau_0 = 0.1$ because the pure scattering\n          term in Equation~\\ref{eqFdcp} is linear in $\\tau_0$.}\n\\label{pyrmg70sca}\n\\end{figure}\n\n\n\n\\section{Observational test}\n\\label{ObsTest}\n\n  Our modeling gives rise to direct observational tests. We can test the\n  effects of the disk in the near-UV and the optical, dividing the\n  spectrum of the target by the template. In the optically thin case, the\n  result will be color dependent and can provide physical parameters for\n  the disk structure.\n\n  The template star should be as similar to the target star as\n  possible. Ideally, it would be the same WD without the obscuring\n  disk. As that is not possible to have, we need a similar WD without\n  any peculiarity in the spectrum.  Any other WD will differ from the\n  target star in $T_{\\mathrm{eff}}$ and $\\log g$ and this difference\n  can make the division of the spectra resemble the expected disk\n  effects.\n\n  In Figure~\\ref{wdRatio}, we present these effects using theoretical WD\n  spectra from \\cite{koe08}.  We assume a target star of\n  $T_{\\mathrm{eff}} = 12,000$~K and $\\log g = 8.0$, similar to G29-38\n  \\citep{rea09}. In the upper panel, we keep $T_{\\mathrm{eff}}$ fixed\n  and vary $\\log g$ by $0.05$~dex and in the lower we keep $\\log g$\n  fixed and vary $T_{\\mathrm{eff}}$ by $150$~K. According to \\cite{lie05},\n  the uncertainties in temperature are of the order of $1.2 \\% \\approx 150$~K\n  and 0.038 in $\\log g$. So, larger temperature or $\\log g$ differences would be\n  readily noticed. One can see from Figure~\\ref{wdRatio} that modification of UV\n  flux densities by disks that absorb or scatter light, as in\n  Figures \\ref{pyrmg70} and \\ref{pyrmg70sca}, can be distinguished from\n  observational uncertainties in template stars.\n\n  The White Dwarf Catalog \\citep{mcc09} currently lists 12,456 stars.\n  Therefore it is not too hard to find a template star with a temperature\n  similar to the target.  As an example, G29-38 and Ross~548 have\n  exactly the same temperature and $\\log g$.  One needs to be careful\n  about this comparison as these values of $T_{\\mathrm{eff}}$ and\n  $\\log g$ were obtained from different determinations.  When\n  comparing the target with template it will be necessary to use\n  $T_{\\mathrm{eff}}$ and $\\log g$ obtained from similar data and the\n  same models.\n\n\\begin{figure}\n\\epsscale{1}\n\\plotone{wdRatio.ps}\n\\caption{Comparison of the effects from debris disk obscuration and the effects\n   of small differences in $T_{\\mathrm{eff}}$ and $\\log g$ between the target and\n   the template stars. The gray lines show the expected effects shown in Figures\n   \\ref{pyrmg70} and \\ref{pyrmg70sca}. In the upper panel the black solid\n   lines show the ratio of WD spectra with fixed $T_{\\mathrm{eff}}$ and\n   varying $\\log g$. In the lower panel, we fixed $\\log g$ and\n   varied $T_{\\mathrm{eff}}$.}\n\\label{wdRatio}\n\\end{figure}\n\n\\subsection{Parameter determination}\n\n  It is difficult to compare directly the definition of $p$ and $\\tau_0$ with\nthe expected values for real disks. We use disk parameters obtained in earlier\nworks to give observational expectations and also to help design future\nobservations.\n  \n  For disks, the fraction of the obscured to the total projected area, $p$\n(Equation~\\ref{def:p}), varies between $0$ when there is no obscuration and 0.5 for\na disk which obscures half of the stellar surface.\nHowever, if the infrared emission region is not a disk but a shell around the\nstar \\citep{rea05} $p$ will be always $1$. Hence, this work provides an\nindependent method to test the disk hypothesis.\n\n  In addition to $p$, we can also determine $\\tau_0$. Estimates for the expected\nvalues are more uncertain, but we can get a rough idea by using some mean values\nfor the dust and disk properties. \\cite{kru03} gives $2.5$~g\/cm$^3$ as a typical\nvalue for the interstellar dust and we assume it as a good order of magnitude\nvalue for the dust in the disk. \\cite{jur07b} constrain the disk mass of GD~362\nbetween $10^{18}$ and $10^{24}$~g. Using typical disk sizes of $10~R_{wd}$ and\n$100~R_{wd}$ for the inner and outer disk radii, respectively, we get a range\nof $\\tau_0$ from $10^{-4}$ to somewhat greater than $1$, depending on the disk\nmass and the type of dust \\citep{dor95}. Therefore the parameters used in Figure\n\\ref{pyrmg70} are realistic.\n\n  The disk inclination angle can be inferred from the presence of an flux excess\ndue to scattering into the line of sight. Figure \\ref{pyrmg70sca} shows this for\ninclinations of $0^\\circ$ and $60^\\circ$. Flux excess in near-UV has already\nbeen detected by \\cite{gan2006} in SDSS~1228+1040 and could be caused by light\nscattering. For larger inclinations there is a flux deficiency due to\nabsorption and scattering out the line of sight, as shown in\nFigure \\ref{pyrmg70} computed for $i=85^\\circ$. The dividing line between\nthe first or the second case is $\\sim$$80^\\circ$.\n\n\\section{Conclusions}\n\\label{sec:conc}\n\nIn this work, we introduce a new way of looking at the cause of IR excess in\nwhite dwarf stars.\nBy looking in the near-UV and optical instead of IR we add a new constraint to\ntest the disk hypothesis.\n\nOne important distinction of our method, is the fact that the presence of disks\nwould cause flux deficiencies in some systems and flux excess in others. We also\npoint out that shells would only introduce flux deficiencies effects, and these\neffects would be detectable in all shells. If we find flux deficiencies in every\nstar we observe this would strongly indicate the presence of shells rather than\ndisks. Flux deficiencies in only some objects and flux excess in others\ncorroborate the idea of a disk.\n\nIf we are convinced disk models are more adequate, detailed\ncomparisons between disk models and data will provide disk \nmass \\citep{jur07b}, composition, optical depth and inclination\nrelative to the line of sight.\n\n\n\n\\begin{acknowledgements}\n  The authors acknowledge financial support from CNPq-MCT\/Brazil. We thank\nPaola D'Alessio, Detlev Koester, and Don Winget for helpful discussions,\nWilliam Reach for providing data, and Shashi Kanbur for reading the manuscript.\nWe are also grateful to Nikolai Voshchinnikov for his Mie-Theory program.\nFinally, we thank the anonymous referee for helpful comments on paper.\n\\end{acknowledgements}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of random walks on groups has a long history and multiple connections to almost all areas of mathematics. It is therefore natural that from the early days of the theory of topological quantum groups, random walks on them were considered. The point of view was often that of discrete groups (the problem is interesting even for duals of classical compact Lie groups). In particular, the study of probabilistic boundaries has been the subject of several works and is still an active area of research. However, there is an aspect which has attracted no attention up to very recently, even though it is an important part of the subject for classical groups : the search for explicit estimates of convergence of random walks.\n\nIn the case of classical finite groups, the first important results for us are due to P. Diaconis and his coauthors in the eighties and reveal a surprising behaviour called the \\emph{cut-off phenomenon} : for a number of steps, the total variation distance (see Subsection \\ref{subsec:randomwalks} for the definition) between the random walk and the uniform distribution stays close to one and then it suddenly drops and converges exponentially to $0$. This triggered numerous works yielding more and more examples of cut-off in various settings, but also counter-examples so that the question of why and when this happens stays largely unanswered. In the quantum setting, the only results up to now are contained in the recent thesis of J.P. McCarthy \\cite{maccarthy2017random} which studies convergence of random walks on finite quantum groups. There, the author gives explicit bounds for families of random walks on the Kac-Paljutkin and Sekine quantum groups, as well as on duals of symmetric groups. Unfortunately, the estimates are not tight enough to yield a complete cut-off statement for these examples.\n\nIn the present work, we turn to the case of infinite compact quantum groups. In particular, we will show that a specific random walk on the free orthogonal quantum groups $O_{N}^{+}$, coming from random rotations on $SO(N)$, has a cut-off with the same threshold as in the classical case, namely $N\\ln(N)\/2(1-\\cos(\\theta))$. This is the first complete cut-off result for a compact quantum group and the statement is all the more surprising that the computations involve mainly representation theory, which is very different for $SO(N)$ and $O_{N}^{+}$. Moreover, the representation theory of $O_{N}^{+}$ being in a sense simpler than that of $SO(N)$ we are able to give very precise statements for the bounds (not only up to some order) and the conditions under which they hold.\n\nUsing techniques from \\cite{hough2017cut}, we can extend our result to random mixtures of rotations provided that the support of the measure governing the random choice of angle is bounded away from $0$. We also consider other examples involving the free symmetric quantum groups $S_{N}^{+}$. In that case, the previous techniques often prove useless. One way round the problem is to compare the corresponding transition operators, which are always well-defined. There is then several options for the choice of a norm and we give results for one of the simplest choices, namely the norm as operators on the $L^{2}$ space.\n\nLet us conclude this introduction with an overview of the organization of this work. In Section \\ref{sec:preliminaries} we give some preliminaries concerning compact quantum groups and random walks on them. We have tried to remain as elementary as possible so that the paper could be readable for people outside the field of quantum groups. In Section \\ref{sec:orthogonal} we study central random walks associated to pure states on free orthogonal quantum groups and prove a kind of cut-off phenomenon in Theorem \\ref{thm:estimategreaterthan2} : for a number of steps, the walk is not comparable in total variation distance with the Haar measure and as soon as it is, it converges exponentially. Using this, we show in Theorem \\ref{thm:randomrotation} that the uniform plane Kac walk on $O_{N}^{+}$ has a cut-off with the same threshold as in the classical case. Eventually, we give in Section \\ref{sec:further} other examples connected to free symmetric quantum groups and illustrate the analytical issue mentioned above.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nIn this section we recall the basic notions concerning compact quantum groups and random walks on them. Since the abstract setting is not really needed to perform concrete computations, we will mainly set notations and give some fundamental results.\n\n\\subsection{Compact quantum groups}\n\nCompact quantum groups are objects of noncommutative topological nature and therefore belong to the world of operator algebras. However, in the present work most things can be treated at an algebraic level which is slightly simpler to describe. We will therefore first give the main definitions in the setting of Hopf algebras and then briefly introduce in the end of this subsection the related analytical objects. We refer the reader to Parts I and II of \\cite{timmermann2008invitation} for a detailed treatment of the algebraic theory of compact quantum groups and its link to the operator algebraic theory.\n\nThe basic example to keep in mind is of course that of a classical compact group $G$. In that case, the corresponding algebraic object is the complex algebra $\\O(G)$ of \\emph{regular functions}, i.e. coefficients of unitary representations. This is a Hopf algebra with an involution given by $f^{*}(g) = \\overline{f(g)}$. Moreover, the Haar measure on $G$ yields by integration a linear form $h$ on $\\O(G)$ which is positive ($h(a^{*}a) \\geqslant 0$) and invariant under translation. Abstracting these properties leads to the following notion (with $\\otimes$ denoting the algebraic tensor product over $\\mathbb{C}$) :\n\n\\begin{de}\nA compact quantum group $\\mathbb{G}$ is given by a Hopf algebra $\\O(\\mathbb{G})$ with an involution and a unital positive linear map $h : \\O(\\mathbb{G})\\to \\mathbb{C}$ which is invariant in the sense that for all $a\\in \\O(\\mathbb{G})$,\n\\begin{equation*}\n(h\\otimes\\id)\\circ\\Delta(a) = h(a).1 = (\\id\\otimes h)\\circ\\Delta(a),\n\\end{equation*}\nwhere $\\Delta : \\O(\\mathbb{G})\\to \\O(\\mathbb{G})\\otimes \\O(\\mathbb{G})$ is the coproduct.\n\\end{de}\n\nIn the present work we will always assume that $\\mathbb{G}$ is \\emph{of Kac type}, meaning that for all $a, b\\in \\O(\\mathbb{G})$ $h(ab) = h(ba)$ (the Haar state is then said to be \\emph{tracial}). Since the fundamental work of P. Diaconis and M. Shahshahani \\cite{diaconis1981generating}, it is known that convergence of random walks can be controlled using representation theory and we will see that the same is true in the quantum setting. As for classical compact groups, the results of \\cite{woronowicz1987compact} imply that any representation of a compact quantum group is equivalent to a direct sum of finite-dimensional unitary ones, so that we will only define the latter.\n\n\\begin{de}\nA \\emph{unitary representation of dimension $n$} of $\\mathbb{G}$ is a unitary element $u\\in M_{n}(\\O(\\mathbb{G}))$ such that for all $1\\leqslant i, j\\leqslant N$,\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{n}u_{ik}\\otimes u_{kj}.\n\\end{equation*}\nA \\emph{morphism} between representations $u$ and $v$ of dimension $n$ and $m$ respectively is a linear map $T : \\mathbb{C}^{n}\\rightarrow \\mathbb{C}^{m}$ such that $(T\\otimes \\id)u = v(T\\otimes \\id)$. Two representations are said to be \\emph{equivalent} if there is a bijective morphism between them. A representation $u$ is said to be \\emph{irreducible} if the only morphisms between $u$ and itself are the scalar multiples of the identity.\n\\end{de}\n\nWe will denote by $\\Irr(\\mathbb{G})$ the set of equivalence classes of irreducible representations of $\\mathbb{G}$ and for each $\\alpha\\in \\Irr(\\mathbb{G})$ we fix a representative $u^{\\alpha}$ and denote by $d_{\\alpha}$ its dimension (which does not depend on the chosen representative). It then follows that $\\O(\\mathbb{G})$ is spanned by the coefficients $u^{\\alpha}_{ij}$ of all the $u^{\\alpha}$'s. Moreover, the Haar state induces an inner product on $\\O(\\mathbb{G})$ for which the basis of coefficients is orthogonal. More precisely, it was proven in \\cite{woronowicz1987compact} that for any $\\alpha, \\beta\\in \\Irr(\\mathbb{G})$ and $1\\leqslant i, j\\leqslant d_{\\alpha}$, $1\\leqslant k, l\\leqslant d_{\\beta}$,\n\\begin{equation*}\nh(u^{\\alpha}_{ij}u^{\\beta\\ast}_{kl}) = \\delta_{\\alpha, \\beta}\\frac{\\delta_{i, k}\\delta_{j, l}}{d_{\\alpha}}.\n\\end{equation*}\nThe key object for computations with random walks is characters of irreducible representations. Let us therefore define these :\n\n\\begin{de}\nThe \\emph{character} of a representation $u^{\\alpha}$ of a compact quantum group $\\mathbb{G}$ is defined as\n\\begin{equation*}\n\\chi_{\\alpha} = \\sum_{i=1}^{d_{\\alpha}}u_{ii}^{\\alpha}\\in \\O(\\mathbb{G}).\n\\end{equation*}\nMoreover, it only depends on $\\alpha$ and not on the chosen representative.\n\\end{de}\n\nWe conclude this subsection with some analysis. As already mentioned, the bilinear map $(a, b)\\mapsto h(b^{*}a)$ defines an inner product on $\\O(\\mathbb{G})$ and the corresponding completion is a Hilbert space denoted by $L^{2}(\\mathbb{G})$. For any element of $\\O(\\mathbb{G})$, left multiplication extends to a bounded operator on $L^{2}(\\mathbb{G})$, yielding an injective $*$-homomorphism $\\O(\\mathbb{G})\\to B(L^{2}(\\mathbb{G}))$. The closure of the image of this map with respect to the weak operator topology is a von Neumann algebra denoted by $L^{\\infty}(\\mathbb{G})$. We will also need the analogue of $L^{1}$ functions. For $a\\in L^{\\infty}(\\mathbb{G})$, set $\\|a\\|_{1} = h(\\vert a\\vert)$ where $\\vert a\\vert = \\sqrt{a^{*}a}$ is defined through functional calculus. Then, $L^{1}(\\mathbb{G})$ is defined to be the completion of $L^{\\infty}(\\mathbb{G})$ with respect to this norm.\n\n\\subsection{Random walks and central states}\\label{subsec:randomwalks}\n\nWe will now introduce some material concerning random walks on compact quantum groups and the total variation distance. For finite quantum groups the subject has been treated in great detail by J.P. MacCarthy in \\cite{maccarthy2017random}. The generalization to the compact case is not difficult so that this subsection will be rather expository. If $G$ is a compact group and $\\mu$ is a measure on $G$, the associated random walk consists in picking elements of $G$ at random according to $\\mu$ and then multiplying them. The probability of being in some measurable set after $k$ steps is then given by the $k$-th convolution power $\\mu^{\\ast k}$ of $\\mu$, which can be expressed at the level of functions as\n\\begin{equation*}\n\\int_{G}f(g) \\mathrm{d}\\mu^{\\ast k}(g) = \\int_{G^{k}}f(g_{k}\\cdots g_{1})\\mathrm{d}\\mu(g_{1})\\cdots \\mathrm{d}\\mu(g_{k}).\n\\end{equation*}\nStudying the random walk associated to $\\mu$ is therefore the same as studying the sequence of measures $(\\mu^{\\ast k})_{k\\in \\mathbb{N}}$.\n\nTurning to quantum groups, first note that measures yield through integration linear forms on $\\O(G)$. If the measure is moreover positive, then so is the linear form and if its total mass is $1$ then the linear form sends the unit of $\\O(G)$ to $1$. Thus, probability measures yield \\emph{states} in the following sense :\n\n\\begin{de}\nA state on an involutive unital algebra $A$ is a linear form $\\varphi : A\\to \\mathbb{C}$ such that $\\varphi(1) = 1$ and $\\varphi(a^{*}a) \\geqslant 0$ for all $a\\in A$.\n\\end{de}\n\nA random walk on a compact quantum group $\\mathbb{G}$ is therefore given by a state $\\varphi$ on $\\O(\\mathbb{G})$. The definition of convolution translates straightforwardly to this setting and one can for instance define $\\varphi^{\\ast k}$ by induction through the formula\n\\begin{equation*}\n\\varphi^{\\ast (k+1)} = (\\varphi\\otimes \\varphi^{\\ast k})\\circ\\Delta = (\\varphi^{\\ast k}\\otimes\\varphi)\\circ\\Delta.\n\\end{equation*}\n\nThe key tool to estimate the rate of convergence of a random walk is a fundamental result of P. Diaconis and M. Sashahani \\cite{diaconis1981generating} bounding the \\emph{total variation distance} of the difference between a measure and the uniform one. Classically, if $\\mu$ and $\\nu$ are any two Borel probability measures on $G$, then\n\\begin{equation*}\n\\|\\mu - \\nu\\|_{TV} =\\sup_{E}\\vert \\mu(E) - \\nu(E)\\vert\n\\end{equation*}\nwhere the supremum is over all Borel subsets $E\\subset G$. This definition can be extended to quantum groups thanks to the fact that Borel subsets correspond to projections in the associated von Neumann algebra. This however requires that the states extend to $L^{\\infty}(\\mathbb{G})$, which may not be the case (see for instance Lemma \\ref{lem:boundednesscriterion}).\n\n\\begin{de}\\label{de:totalvariation}\nThe total variation distance between two states $\\varphi$ and $\\psi$ on $L^{\\infty}(\\mathbb{G})$ is defined by\n\\begin{equation*}\n\\|\\varphi - \\psi\\|_{TV} = \\sup_{p\\in \\mathcal{P}(L^{\\infty}(\\mathbb{G}))}\\vert \\varphi(p) - \\psi(p)\\vert,\n\\end{equation*}\nwhere $\\mathcal{P}(L^{\\infty}(\\mathbb{G})) = \\{p\\in L^{\\infty}(\\mathbb{G}) \\mid p^{2} = p = p^{*}\\}$.\n\\end{de}\n\nAssume now that we consider a state $\\varphi$ which is \\emph{absolutely continuous} with respect to the Haar state $h$, in the sense that there exists an element $a_{\\varphi}\\in L^{1}(\\mathbb{G})$ such that $\\varphi(x) = h(a_{\\varphi}x)$ for all $x\\in \\O(\\mathbb{G})$. Then, the total variation distance can be expressed in terms of $a_{\\varphi}$. Note that the proof below uses the traciality of the Haar state.\n\n\\begin{lem}\\label{lem:totalvariation}\nIf $\\varphi$ is a state on $L^{\\infty}(\\mathbb{G})$ with an $L^{1}$-density $a_{\\varphi}\\in L^{1}(\\mathbb{G})$, then\n\\begin{equation*}\n\\|\\varphi - h\\|_{TV} = \\frac{1}{2}\\|a_{\\varphi} - 1\\|_{1}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $\\mathbf{1}_{\\mathbb{R}_{+}}$ be the indicator function of the positive real numbers and define a projection $p_{+} = \\mathbf{1}_{\\mathbb{R}_{+}}(a_{\\varphi} - 1)$ through functional calculus. We claim that the supremum in Definition \\ref{de:totalvariation} is attained at $p_{+}$. Indeed, for any projection $q\\in \\mathcal{P}(L^{\\infty}(\\mathbb{G}))$, setting $b_{\\varphi} = a_{\\varphi} - 1$ we have\n\\begin{equation*}\n\\vert\\varphi-h\\vert(q) = \\vert h(b_{\\varphi}p_{+}q) + h(b_{\\varphi}(1-p_{+})q)\\vert \\leqslant \\max(h(b_{\\varphi}p_{+}q), h(b_{\\varphi}(p_{+}-1)q))\n\\end{equation*}\nand observing that $p_{+}$ commutes with $b_{\\varphi}$ we get\n\\begin{equation*}\n\\vert\\varphi-h\\vert(q) \\leqslant \\max(h(b_{\\varphi}p_{+}qp_{+}), h(b_{\\varphi}(p_{+}-1)q(p_{+}-1))) \\leqslant \\max(h(b_{\\varphi}p_{+}), h(b_{\\varphi}(p_{+}-1))).\n\\end{equation*}\nWe conclude using the fact that $h(b_{\\varphi}) = (\\varphi - h)(1) = 1$. Now since\n\\begin{equation*}\n\\vert b_{\\varphi}\\vert = \\vert b_{\\varphi}\\vert p_{+} + \\vert b_{\\varphi}\\vert(1-p_{+}) = 2b_{\\varphi}p_{+} - b_{\\varphi}\n\\end{equation*}\nwe get\n\\begin{equation*}\n\\|a_{\\varphi} - 1\\|_{1} = h(\\vert b_{\\varphi}\\vert) = 2h(b_{\\varphi}p_{+}) - h(b_{\\varphi}) = 2(\\varphi - h)(p_{+}) = 2\\|\\varphi - h\\|_{TV}.\n\\end{equation*}\n\\end{proof}\n\nThis equality is the trick leading to the Diaconis-Shahshahani upper bound lemma which, in the end, does not involve $a_{\\varphi}$ any more. To state this result, let us first introduce a notation : if $\\varphi$ is a state and $\\alpha\\in \\Irr(\\mathbb{G})$, we denote by $\\widehat{\\varphi}(\\alpha)$ the matrix with coefficients $\\varphi(u^{\\alpha}_{ij})$ (this does not depend on the choice of a representative of $\\alpha$). We can then consider $\\widehat{\\varphi}$ as an element of the $\\ell^{\\infty}$-sum of the matrix algebras $B(H_{\\alpha})$, denoted by $\\ell^{\\infty}(\\widehat{\\mathbb{G}})$.\n\n\\begin{lem}[Upper bound lemma]\\label{lem:upperbound}\nLet $\\mathbb{G}$ be a compact quantum group and let $\\varphi$ be a state on $\\mathbb{G}$ which is absolutely continuous with respect to the Haar state. Then,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV}^{2} \\leqslant \\frac{1}{4}\\sum_{\\alpha\\in \\Irr(\\mathbb{G})\\setminus\\{\\varepsilon\\}}d_{\\alpha}\\Tr\\left(\\widehat{\\varphi}(\\alpha)^{* k}\\widehat{\\varphi}(\\alpha)^{k}\\right),\n\\end{equation*}\nwhere $\\varepsilon = 1\\in M_{1}(\\O(\\mathbb{G}))$ denotes the trivial representation.\n\\end{lem}\n\n\\begin{proof}\nThe proof for compact groups (assuming that the measure is central) was given in \\cite[Lem 4.3]{rosenthal1994random} and the proof for finite quantum groups was given in \\cite[Lem 5.3.8]{maccarthy2017random}. The argument here is the same so that we simply sketch it. The Cauchy-Schwartz inequality yields\n\\begin{equation*}\n\\|a_{\\varphi} - 1\\|_{1}^{2} = h(\\vert a_{\\varphi} - 1\\vert)^{2} \\leqslant h(1^{*}1)h\\left((a_{\\varphi}-1)^{*}(a_{\\varphi}-1)\\right) = \\|a_{\\varphi}-1\\|_{2}^{2}\n\\end{equation*}\nMoreover, the formula\n\\begin{equation*}\n\\widehat{h}(x) = \\sum_{\\alpha\\in \\Irr(\\mathbb{G})}d_{\\alpha}\\Tr(x)\n\\end{equation*}\ndefines a positive weight on $\\ell^{\\infty}(\\widehat{\\mathbb{G}})$. This is the analogue of the counting measure on a discrete group and one can define a Fourier transform $\\mathcal{F} : L^{2}(\\mathbb{G})\\rightarrow \\ell^{2}(\\mathbb{G})$ (see for instance \\cite[Sec 2]{podles1990quantum}) which is isometric. The conclusion now follows from the fact that the Fourier transform of $a_{\\varphi}$ is $\\widehat{\\varphi}$ and the relationship between convolution and Fourier transform.\n\\end{proof}\n\n\\begin{rem}\nBecause of the Cauchy-Schwartz inequality, $L^{2}(\\mathbb{G})\\subset L^{1}(\\mathbb{G})$ so that if $\\varphi$ is not absolutely continuous with respect to $h$, then the right-hand side of the inequality is infinite and the inequality trivially holds.\n\\end{rem}\n\nOur goal is therefore to bound $\\sum d_{\\alpha}\\Tr(\\varphi(\\alpha)^{*k}\\varphi(\\alpha)^{k})$ by an explicit function of $k$. This requires the computation of the trace of arbitrary powers of matrices which can be very complicated. As already observed in \\cite{rosenthal1994random}, things get more tractable when the measure is assumed to be central, i.e. invariant under the adjoint action, since then the Fourier transform of its density consists in scalar multiples of identity matrices. The same is true in the quantum setting, thanks to \\cite[Prop 6.9]{cipriani2012symmetries} which we recall here for convenience.\n\n\\begin{prop}\nLet $\\mathbb{G}$ be a compact quantum group and let $\\varphi : \\O(\\mathbb{G})\\to \\mathbb{C}$ be a state. Then, $\\varphi$ is invariant under the adjoint action if and only if for any irreducible representation $\\alpha\\in \\Irr(\\mathbb{G})$, there exists $\\varphi(\\alpha) \\in \\mathbb{C}$ such that $\\varphi(u_{ij}^{\\alpha}) = \\varphi(\\alpha)\\delta_{ij}$.\n\\end{prop}\n\nSuch \\emph{central states} are completely determined by their restriction to the so-called \\emph{central algebra} of $\\mathbb{G}$, which is simply the algebra $\\O(\\mathbb{G})_{0}$ generated by the characters, thanks to the equality\n\\begin{equation*}\n\\varphi(\\chi_{\\alpha}) = \\sum_{i=1}^{d_{\\alpha}}\\varphi(u_{ii}^{\\alpha}) = d_{\\alpha}\\varphi(\\alpha).\n\\end{equation*}\nIn several key examples, the central algebra is commutative, hence states exactly correspond to measures on its spectrum. A particular case is that of Dirac measures, i.e. evaluation at one point. This setting covers natural analogues of the random walk associated to the uniform measure on a conjugacy class. To see this, assume that $\\O(\\mathbb{G})$ is generated by the coefficients of a representation $u$ of dimension $N$ and let $G$ be the abelianization of $\\mathbb{G}$, that is to say the compact group such that $\\O(G)$ is the maximal abelian quotient of $\\O(\\mathbb{G})$. By construction, $G$ is realized as a group of $N\\times N$ matrices. Let $g\\in G$ and let $\\ev_{g} : \\O(\\mathbb{G})\\to \\mathbb{C}$ be the algebra map sending $u_{ij}$ to $g_{ij}$. Then,\n\\begin{equation*}\n\\varphi_{g} = h\\circ m^{(2)}\\circ(\\id\\otimes \\ev_{g}\\otimes S)\\circ\\Delta^{(2)}\n\\end{equation*}\nis a state on $\\O(\\mathbb{G})$, where $\\Delta^{(2)} = (\\id\\otimes\\Delta)\\circ\\Delta$, $m^{(2)} = m\\circ(\\id\\otimes m)$ and $S$ is the antipode. If $\\mathbb{G}$ is classical, then for any function $f$ one has\n\\begin{equation*}\n\\varphi_{g}(f) = \\int_{G}f(kgk^{-1})\\mathrm{d} k\n\\end{equation*}\nso that $\\varphi_{g}$ is the uniform measure on the conjugacy class of $g$. In the general case, the centrality of $\\varphi_{g}$ is easily checked :\n\\begin{align*}\n\\varphi_{g}(u^{\\alpha}_{ij}) & = \\sum_{k,l=1}^{d_{\\alpha}}h(u^{\\alpha}_{ik}\\ev_{g}(u^{\\alpha}_{kl})u^{\\alpha\\ast}_{jl}) = \\sum_{k,l=1}^{d_{\\alpha}}\\ev_{g}(u^{\\alpha}_{kl})h(u^{\\alpha}_{ik}u^{\\alpha\\ast}_{jl}) \\\\\n& = \\sum_{k,l=1}^{d_{\\alpha}}\\ev_{g}(u^{\\alpha}_{kl})\\frac{\\delta_{ij}\\delta_{kl}}{d_{\\alpha}} = \\delta_{ij}\\frac{\\ev_{g}(\\chi_{\\alpha})}{d_{\\alpha}}.\n\\end{align*}\nSeveral interesting random walks on compact Lie groups are of this type and we will study them in the quantum setting in the next sections.\n\n\\section{Free orthogonal quantum groups}\\label{sec:orthogonal}\n\nThe main example which we will study in this work is free orthogonal quantum groups. These objects, denoted by $O_{N}^{+}$, were first introduced by S. Wang in \\cite{wang1995free}. Here is how the associated involutive Hopf algebra is defined :\n\n\\begin{de}\nLet $\\O(O_{N}^{+})$ be the universal $*$-algebra generated by $N^{2}$ \\emph{self-adjoint} elements $u_{ij}$ such that for all $1\\leqslant i, j \\leqslant N$,\n\\begin{equation*}\n\\sum_{k=1}^{N}u_{ik}u_{jk} = \\delta_{ij} = \\sum_{k=1}^{N}u_{ki}u_{kj}.\n\\end{equation*}\nThe formula\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{N}u_{ik}\\otimes u_{kj}\n\\end{equation*}\nextends to a $*$-algebra homomorphism $\\Delta : \\O(O_{N}^{+})\\to \\O(O_{N}^{+})\\otimes \\O(O_{N}^{+})$ and this can be completed into a compact quantum group structure.\n\\end{de}\n\nThe relations defining $\\O(O_{N}^{+})$ are equivalent to requiring that the matrix $[u_{ij}]_{1\\leqslant i, j\\leqslant N}$ is orthogonal. Using this, it is easy to see that the abelianization of $O_{N}^{+}$ is the orthogonal group $O_{N}$. To compute upper bounds for random walks, we need a description of the representation theory of these objects. In fact, since we will only consider central states, all we need is a description of the central algebra which comes from the work of T. Banica \\cite{banica1996theorie}.\n\n\\begin{thm}[Banica]\nThe irreducible representations of $O_{N}^{+}$ can be labelled by positive integers, with $u^{0}$ being the trivial representation and $u^{1} = [u_{ij}]_{1\\leqslant i, j\\leqslant N}$. Moreover, the characters satisfy the following recursion relation :\n\\begin{equation}\\label{eq:charactersorthogonal}\n\\chi_{1}\\chi_{n} = \\chi_{n+1} + \\chi_{n-1}.\n\\end{equation}\n\\end{thm}\n\nIn particular, the central algebra $\\O(\\mathbb{G})_{0}$ is abelian and in fact isomorphic to $\\mathbb{C}[X]$. Moreover, the recursion relation \\eqref{eq:charactersorthogonal} is reminiscent of that of Chebyshev polynomials of the second kind. Indeed, let $U_{n}$ be these polynomials, i.e. $U_{n}(\\sin(\\theta)) = \\sin(n\\theta)$ for all $\\theta\\in \\mathbb{R}$. Then, $u_{n}(x) = U_{n}(x\/2)$ satisfies Equation \\eqref{eq:charactersorthogonal}. With this in hand, it can be proven that the map sending $\\chi_{n}$ to $u_{n}$ is an isomorphism. Moreover, we have the equality $d_{n} = u_{n}(N)$.\n\n\\subsection{Pure state random walks}\\label{subsec:purestates}\n\nA state is said to be \\emph{pure} if it cannot be written as a non-trivial convex combination of other states. Moreover, pure states on $\\O(O_{N}^{+})$ are still pure when restricted to the central algebra and it is well-known that pure states on an abelian algebra are given by evaluation at points of the spectrum. For $O_{N}^{+}$, the spectrum of $\\chi_{1}$ in the enveloping C*-algebra of $\\O(O_{N}^{+})$ is $[-N, N]$ by \\cite[Lem 4.2]{brannan2011approximation} so that for any $t\\in [-N, N]$ there is a central state $\\varphi_{t}$ on $\\O(O_{N}^{+})$ defined by $\\varphi_{t}(n) = u_{n}(t)\/d_{n}\\in \\mathbb{R}$. It follows from the definition that for a central state $\\varphi$, we have $\\varphi^{\\ast k}(n) = \\varphi(n)^{k}$, so that by Lemma \\ref{lem:upperbound}\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}^{2} \\leqslant \\frac{1}{4}\\sum_{n=1}^{+\\infty}d_{n}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-1}} = \\frac{1}{4}\\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-2}}\n\\end{equation*}\nand we will have to bound specific values of the polynomials $u_{n}$. It turns out that the behaviour of Chebyshev polynomials is very different if the argument is less than or greater than one and this will be reflected in the existence or absence of a kind of cut-off phenomenon for the associated random walks. Before turning to this, let us give general tools for the computations. Assume that $t > 2$ and let $0 < q(t) < 1$ be such that $t = q(t) + q(t)^{-1}$, i.e.\n\\begin{equation*}\nq(t) = \\frac{t - \\sqrt{t^{2} - 4}}{2}.\n\\end{equation*}\nThen, it can be shown by induction that\n\\begin{equation*}\nu_{n}(t) = \\frac{q(t)^{-n-1} - q(t)^{n+1}}{q(t)^{-1} - q(t)}.\n\\end{equation*}\nThis writing enables to efficiently bound $u_{n}(t)$ :\n\n\\begin{lem}\\label{lem:encadrement}\nFor all $n\\geqslant 1$ and $t\\geqslant 2$,\n\\begin{equation*}\ntq(t)^{-(n-1)} \\leqslant u_{n}(t)\\leqslant\\frac{q(t)^{-n}}{1-q(t)^{2}}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nConsider the sequence $a_{n} = u_{n}(t)q(t)^{n}$. Then,\n\\begin{equation*}\n\\frac{a_{n+1}}{a_{n}} = q(t)\\frac{u_{n+1}(t)}{u_{n}(t)} = q(t)\\frac{q(t)^{-n-1} - q(t)^{n+1}}{q(t)^{-n} - q(t)^{n}} = \\frac{q(t)^{-n} - q(t)^{n+2}}{q(t)^{-n} - q(t)^{n}} > 1\n\\end{equation*}\nso that $(a_{n})_{n\\in \\mathbb{N}}$ is increasing. It is therefore always greater than its first term, which is $q(t)t = 1+q(t)^{2}$ and always less than its limit, which is\n\\begin{equation*}\n\\frac{q(t)^{-1}}{q(t)^{-1} - q(t)} = \\frac{1}{1-q(t)^{2}}.\n\\end{equation*}\n\\end{proof}\n\nTo lighten notations, let us set\n\\begin{equation*}\nA_{k}(t) = \\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{d_{n}^{2k-2}} = \\sum_{n=1}^{+\\infty}\\frac{u_{n}(t)^{2k}}{u_{n}(N)^{2k-2}}\n\\end{equation*}\n\n\\subsubsection{Random walks associated to small pure states}\n\nWe start with the case where $t$ is less than $2$. As we will see, things are then rather simple.\n\n\\begin{prop}\\label{prop:upperboundlessthantwo}\nLet $\\vert t\\vert < 2$ be fixed. Then, for any $k\\geqslant 2$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{N}{2\\sqrt{1-q(N)^{2}}}\\left(\\frac{1}{N\\sqrt{1 - t^{2}\/4}}\\right)^{k}\n\\end{equation*}\nIn particular, if $t<2\\sqrt{1-N^{-2}}$ then the random walk associated to $\\varphi_{t}$ converges exponentially.\n\\end{prop}\n\n\\begin{proof}\nBecause $\\vert t\\vert\\leqslant 2$, there exists $\\theta$ such that $t = 2\\cos(\\theta)$. Thus,\n\\begin{equation*}\n\\vert u_{n}(t)\\vert = \\vert U_{n}(\\cos(\\theta))\\vert = \\left\\vert \\frac{\\sin\\left((n+1)\\theta\\right)}{\\sin(\\theta)}\\right\\vert \\leqslant \\frac{1}{\\vert\\sin(\\theta)\\vert}\n\\end{equation*}\nand\n\\begin{align*}\nA_{k}(t) & \\leqslant \\sum_{n=1}^{+\\infty}\\frac{1}{\\vert\\sin(\\theta)^{2k}\\vert u_{n}(N)^{2k-2}} \\\\\n& \\leqslant \\frac{1}{\\vert\\sin(\\theta)\\vert^{2k}}\\sum_{n=1}^{+\\infty}\\left(\\frac{q(N)^{n-1}}{N}\\right)^{2k-2} \\\\\n& = \\frac{1}{\\vert\\sin(\\theta)\\vert^{2k}N^{2k-2}}\\frac{1}{1-q(N)^{2k-2}} \\\\\n& \\leqslant \\frac{N^{2}}{1-q(N)^{2}}\\left(\\frac{1}{N\\vert\\sin(\\theta)\\vert}\\right)^{2k} \\\\\n\\end{align*}\nThe result now follows from Lemma \\ref{lem:upperbound} and the fact that $\\vert\\sin(\\theta)\\vert = \\sqrt{1-t^{2}\/4}$. Note that for $k = 1$, we get the sum of $\\vert \\sin((n+1)\\theta)\\vert\/\\vert\\sin(\\theta)\\vert$ which need not converge even though $\\varphi_{t}$ is bounded on $L^{\\infty}(O_{N}^{+})$.\n\\end{proof}\n\nProposition \\ref{prop:upperboundlessthantwo} shows that for a fixed $t$, the distance to the Haar state decreases exponentially provided $N$ is large enough and it is natural to wonder how optimal the rate $1\/N\\sqrt{1-t^{2}\/4}$ is. We give a partial answer through a lower obtained by the duality between the noncommutative $L^{1}$ and $L^{\\infty}$ spaces of a tracial von Neumann algebra. Concretely, this means that for any $a\\in L^{1}(\\mathbb{G})$,\n\\begin{equation*}\n\\|a\\|_{1} = \\sup\\{h(ax)\\mid \\|x\\|_{\\infty}\\leqslant 1\\} = \\|\\varphi\\|.\n\\end{equation*}\n\n\\begin{prop}\\label{prop:lowerbound}\nFor any $t\\in [-N, N]$ and any $k\\geqslant 1$,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV} \\geqslant \\frac{N}{4}\\left(\\frac{t}{N}\\right)^{k}.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nRecall that $\\varphi^{\\ast k}(n) = \\varphi(n)^{k}$ and that $h(\\chi_{n}) = 0$ for all $n\\geqslant 1$. Thus,\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\| \\geqslant \\frac{1}{2}\\sup_{n \\geqslant 1}\\frac{\\varphi^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} = \\frac{1}{2}\\sup_{n \\geqslant 1}\\frac{d_{n}}{\\|\\chi_{n}\\|_{\\infty}}\\left(\\frac{u_{n}(t)}{d_{n}}\\right)^{k}.\n\\end{equation*}\nTaking $n=1$ and using $\\|\\chi_{1}\\|_{\\infty} = 2$ then yields the result.\n\\end{proof}\n\nEven though this bound is very general since it works for all $t$, it yields the same exponential rate as Proposition \\ref{prop:upperboundlessthantwo} for $t = \\pm \\sqrt{2}$, meaning that the bound of Proposition \\ref{prop:upperboundlessthantwo} is rather tight.\n\n\\subsubsection{The cut-off phenomenon}\n\nWe now turn to the case when $\\vert t\\vert$ is larger than two. The corresponding states will exhibit a kind of cut-off phenomenon : for a number of steps (depending on $t$ and $N$), the state is not absolutely continuous and as soon as it is, it converges exponentially. Let us first consider the boundedness problem.\n\n\\begin{lem}\\label{lem:boundednesscriterion}\nLet $\\vert t\\vert > 2$ be fixed. Then, $\\varphi_{t}^{\\ast k}$ extends to $L^{\\infty}(O_{N}^{+})$ if and only if $q(t) > q(N)^{1-1\/k}$. Moreover, it then has an $L^{1}$-density with respect to $h$.\n\\end{lem}\n\n\\begin{proof}\nBecause $\\|\\chi_{n}\\|_{\\infty} = n+1$,\n\\begin{equation*}\n\\frac{\\varphi_{t}^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} = \\frac{1}{n+1}\\frac{u_{n}(t)^{k}}{u_{n}(N)^{k-1}}\n\\end{equation*}\nso that Lemma \\ref{lem:encadrement} yields\n\\begin{align*}\n\\frac{\\varphi_{t}^{\\ast k}(\\chi_{n})}{\\|\\chi_{n}\\|_{\\infty}} & \\geqslant \\frac{1}{n+1}\\left(q(N)^{n}(1-q(N)^{2})\\right)^{k-1}\\left(q(t)^{-n+1}t\\right)^{k} \\\\\n& = \\left(\\frac{q(N)^{k-1}}{q(t)^{k}}\\right)^{n}\\frac{(tq(t))^{k}(1-q(N)^{2})^{k-1}}{n+1}\n\\end{align*}\nand this is not bounded in $n$ if $q(t) < q(N)^{1-1\/k}$. If now $k$ satisfies the inequality in the statement, then a similar estimate shows that the sequence\n\\begin{equation*}\na_{t, p} = \\sum_{n=0}^{p}\\frac{u_{n}(t)^{k}}{u_{n}(N)^{k-1}}\\chi_{n}\n\\end{equation*}\nconverges in $L^{\\infty}(\\mathbb{G})\\subset L^{1}(\\mathbb{G})$ and its limit is the density of $\\varphi_{t}^{\\ast k}$.\n\\end{proof}\n\nThe previous statement may be disappointing in that for a fixed $t$, the number $k$ goes to $1$ as $N$ goes to infinity. However, in the cases coming from classical random walks, $t$ depends on $N$ and we then get a cut-off parameter which also depends on $N$, see Subsection \\ref{subsec:randomrotations}. Let us now prove that as soon as $\\varphi_{t}^{\\ast k}$ is absolutely continuous, it converges exponentially to the Haar state. This is the main result of this section.\n\n\\begin{thm}\\label{thm:estimategreaterthan2}\nLet $\\vert t\\vert > 2$ be fixed and let $k_{0}$ be the smallest integer such that $q(t) > q(N)^{1-1\/k_{0}}$. Then, for any $k\\geqslant k_{0}$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\frac{Nq(t)^{k_{0}}}{\\sqrt{q(t)^{2k_{0}} - q(N)^{2k_{0}-2}}}\\left(\\frac{1}{Nq(t)(1-q(t)^{2})}\\right)^{k}\n\\end{equation*}\nMoreover, there exists $t_{0}$ and $t_{1}$ depending on $N$ satisfying $2 < t_{0} < 4\/\\sqrt{3} < t_{1} < N$ and such that if $t_{0} < \\vert t\\vert < t_{1}$, then the random walk associated to $\\varphi_{t}$ converges exponentially after $k_{0}$ steps.\n\\end{thm}\n\n\\begin{proof}\nWe start the computation by using Lemma \\ref{lem:encadrement} :\n\\begin{align*}\n\\frac{u_{n}(t)^{2k}}{u_{n}(N)^{2k-2}} & \\leqslant \\left(\\frac{1}{q(t)^{n}(1-q(t)^{2})}\\right)^{2k}\\left(\\frac{q(N)^{n-1}}{N}\\right)^{2k-2} \\\\\n& = \\left(\\frac{q(N)^{2k-2}}{q(t)^{2k}}\\right)^{n-1}\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}.\n\\end{align*}\nBy assumption, $q(t) > q(N)^{1-1\/k}$ so that\n\\begin{align*}\nA_{k}(t) & \\leqslant\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}\\frac{1}{1-\\frac{q(N)^{2k-2}}{q(t)^{2k}}} \\\\\n& \\leqslant\\frac{1}{N^{2k-2}(q(t)(1-q(t)^{2}))^{2k}}\\frac{1}{1-\\frac{q(N)^{2k_{0}-2}}{q(t)^{2k_{0}}}} \\\\\n& = \\frac{N^{2}q(t)^{2k_{0}}}{q(t)^{2k_{0}} - q(N)^{2k_{0}-2}}\\left(\\frac{1}{Nq(t)(1-q(t)^{2})}\\right)^{2k} \\\\\n\\end{align*}\nand the result follows by Lemma \\ref{lem:upperbound}.\n\nThe condition for exponential convergence is $q(t)(1-q(t)^{2}) > N^{-1}$. Consider the function $f : x\\mapsto x(1-x^{2})$. Elementary calculus shows that its maximum is\n\\begin{equation*}\nf\\left(\\frac{1}{\\sqrt{3}}\\right) = \\frac{2}{3\\sqrt{3}} > \\frac{1}{3} \\geqslant \\frac{1}{N}.\n\\end{equation*}\nThus, there exists an open interval $I$ containing $1\/\\sqrt{3}$ such that $f(q(t)) > 1\/N$ as soon as $q(t)$ is in $I$. Since $q(t) = 1\/\\sqrt{3}$ corresponds to $t = q(t) + q(t)^{-1} = 4\/\\sqrt{3}$, the proof is complete.\n\\end{proof}\n\nSo far our use of the term cut-off has been a little improper since we did not provide an upper bound for the total variation distance depending only on $(k-k_{0})\/N$. We will see however that when considering particular values of $t$ related to classical random walks, one can sharpen the previous result into a genuine cut-off statement. To conclude this section, let us give an explicit formula for the threshold $k_{0}$. Taking the logarithm of both sides of the equality $q(t)^{k} > q(N)^{k-1}$ and noting that $q(t) > q(N)$ yields\n\\begin{equation*}\nk_{0} = \\left\\lceil-\\frac{\\ln(q(N))}{\\ln(q(t)\/q(N))}\\right\\rceil.\n\\end{equation*}\n\n\\subsection{The quantum uniform plan Kac walk}\\label{subsec:randomrotations}\n\nIn this section we will give an explicit example of cut-off phenomenon by considering the quantum analogue of the \\emph{uniform plane Kac walk} on $SO(N)$. In the classical case, this was studied by J. Rosenthal in \\cite{rosenthal1994random} and by B. Hough and Y. Jiang in \\cite{hough2017cut} (who coined the name). In this model, a random rotation is obtained by randomly choosing a plane in $\\mathbb{R}^{N}$ and then performing a rotation of some fixed angle $\\theta$ in that plane. The corresponding measure is the uniform measure on the conjugacy class of a matrix $R_{\\theta}$ corresponding to a rotation in a plane (they are all conjugate once the angle $\\theta$ is fixed, so that the choice of the plane does not matter). As explained in Subsection \\ref{subsec:randomwalks}, this uniform measure has a natural analogue on $O_{N}^{+}$. In a sense, we are now \"quantum rotating\" the plane of $R_{\\theta}$ and the corresponding state is $\\varphi_{R_{\\theta}}$. Since\n\\begin{equation*}\n\\ev_{R_{\\theta}}(\\chi_{1}) = \\Tr(R_{\\theta}) = N-2+2\\cos(\\theta) = u_{1}(N-2+2\\cos(\\theta)),\n\\end{equation*}\nit follows by induction that $\\ev_{R_{\\theta}}(\\chi_{n}) = u_{n}(N-2+2\\cos(\\theta))$ so that $\\varphi_{R_{\\theta}} = \\varphi_{N-2+2\\cos(\\theta)}$. Note that this in a sense means that two classical orthogonal matrices are quantum conjugate if and only if they have the same trace since the uniform measure on their conjugacy classes then coincide. This illustrates the fact that there is no \"quantum $SO(N)$\" subgroup in $O_{N}^{+}$.\n\nAssuming that $\\theta$ is fixed once and for all, we will show that the corresponding random walk has a cut-off. This means that we have to prove that there exists $k_{1}$ such that for $k_{1} + cN$ steps the total variation distance decreases exponentially in $c$ while for $k_{1} - cN$ steps it is bounded below by a function which decreases slowly in $c$. We will therefore split the arguments into two parts. To simplify notations let us set $\\tau = 2(1-\\cos(\\theta))$.\n\n\\subsubsection{Upper bound}\n\nWe start with the upper bound. Since the parameter $t$ now depends on $N$, it is not even clear that the convolution powers of the state will ever extend to $L^{\\infty}(\\mathbb{G})$. To get some insight into this problem, let us first consider the threshold for $t = N-\\tau$ obtained in the previous section. For large $N$,\n\\begin{equation*}\n-\\frac{\\ln(q(N))}{\\ln(q(N-\\tau)\/q(N))} \\sim \\frac{N\\ln(N)}{\\tau}\n\\end{equation*}\nwhich is exactly the cut-off parameter conjecture by J. Rosenthal the classical case (and proven there to be valid for $\\theta = \\pi$) and later confirmed by B. Hough and Y. Jiang in \\cite{hough2017cut}. This suggests that the same phenomenon should occur for $O_{N}^{+}$. However, proving it requires some suitable estimates on the function $q(t)$. We start by giving some elementary inequalities.\n\n\\begin{lem}\\label{lem:variousbounds}\nThe following inequalities hold for all $t, N\\geqslant 4$ :\n\\begin{enumerate}\n\\item $\\displaystyle\\frac{q(N)}{q(N-\\tau)} \\leqslant \\frac{N-\\tau}{N}$,\n\\item $q(N) > 1\/N$,\n\\item $N\\ln\\displaystyle\\left(1-\\frac{\\tau}{N}\\right) \\leqslant -\\tau$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nConsider the function $f : t\\mapsto tq(t)$. Noticing that\n\\begin{equation*}\nq'(t) = \\frac{1}{2} - \\frac{1}{2}\\frac{t}{\\sqrt{t^{2} - 4}} = \\frac{\\sqrt{t^{2} - 4} - t}{2\\sqrt{t^{2}-4}} = \\frac{-q(t)}{\\sqrt{t^{2}-4}},\n\\end{equation*}\nwe see that\n\\begin{equation*}\nf'(t) = q(t) + tq'(t) = \\left(1-\\frac{t}{\\sqrt{t^{2}-4}}\\right)q(t) < 0\n\\end{equation*}\nso that $f$ is decreasing. Applying this to $N > N-\\tau$ yields the first inequality while $f(t) > \\lim_{+\\infty} f(t) = 1$ yields the second one. The third equality follows from the well-known bound $\\ln(1-x) \\leqslant -x$ valid for any $0 \\leqslant x < 1$.\n\\end{proof}\n\nIn the proof of Theorem \\ref{thm:estimategreaterthan2} we saw that the total variation distance can be bounded by\n\\begin{equation*}\n\\frac{Nq(N-\\tau)^{k}}{2\\sqrt{q(N-\\tau)^{2k} - q(N)^{2k-1}}}\\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k}\n\\end{equation*}\nand the inequalities above will enable us to bound the first part of this expression. For the second part, we need to study $Nq(N-\\tau)(1-q(N-\\tau)^{2})$. Note that it is not even clear that this is greater than one and in fact, for $\\tau = 0$ it equals $1-q(N)^{4} < 1$. However, as soon as $\\tau > 0$, assuming that $N$ is large enough everything will work. To show this we will first prove two computational lemmata.\n\n\\begin{lem}\\label{lem:threefunctions}\nConsider the following functions defined for $t > 2$ :\n\\begin{equation*}\nf(t) = \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} \\text{ and } g(t) = \\frac{16}{5}\\frac{1}{t^{3}(t^{2}-4)}\n\\end{equation*}\nand set\n\\begin{equation*}\nC(\\tau) = \\frac{2}{\\tau\\sqrt{5}}(2+\\sqrt{2+9\\tau^{2}})\n\\end{equation*}\nThen, $f(t) \\geqslant g(t)$ as soon as $t\\geqslant C(\\tau)$.\n\\end{lem}\n\n\\begin{proof}\nThe inequality $f(t)\\geqslant g(t)$ can be written as\n\\begin{equation}\\label{eq:functionalinequality}\n5\\tau^{2}t^{2}(t^{2}-4)\\geqslant 32(t+\\tau)^{2}.\n\\end{equation}\nBecause $t^{2}\\geqslant t^{2}-4$, the left-hand side is greater than $[\\sqrt{5}\\tau(t^{2}-4)]^{2}$ and \\eqref{eq:functionalinequality} will be satisfied as soon as $\\sqrt{5}\\tau(t^{2}-4)\\geqslant 4\\sqrt{2}(t+\\tau)$, which amounts to the quadratic inequality\n\\begin{equation*}\n\\sqrt{5}\\tau t^{2} - 4\\sqrt{2}t - 4\\tau(\\sqrt{5} + \\sqrt{2})\\geqslant 0.\n\\end{equation*}\nThe discriminant is $32 + 16\\tau^{2}(5+\\sqrt{10}) \\leqslant 32 + 16\\times 9\\times \\tau^{2}$ so that \\eqref{eq:functionalinequality} is satisfied as soon as\n\\begin{equation*}\nt\\geqslant \\frac{2\\sqrt{2}}{\\sqrt{5}\\tau} + \\frac{2}{\\sqrt{5}\\tau}\\sqrt{2+9\\tau^{2}}.\n\\end{equation*}\nThe result now follows from the observation that $C(\\tau)$ is greater than the right-hand side because $2\/3\\geqslant \\sqrt{2\/5}$.\n\\end{proof}\n\nWith this in hand we can prove the main inequality that we need.\n\n\\begin{lem}\\label{lem:hardlowerbound}\nLet $0 < \\tau \\leqslant 4$. Then, for any $N\\geqslant \\tau + C(\\tau)$,\n\\begin{equation*}\nq(N-\\tau)(1-q(N-\\tau)^{2}) \\geqslant \\frac{e^{\\tau\/N}}{N}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet us set $a_{0} = 1$, $a_{1} = 1\/2$ and for $n\\geqslant 2$,\n\\begin{equation*}\na_{n} = \\frac{1\\times 3\\times\\cdots\\times(2n-3)}{2\\times 4\\times\\cdots\\times 2n} = \\frac{(2n-3)!}{2^{n-2}(n-2)!\\times 2^{n}(n!)} = \\frac{1}{n4^{n-1}}{\\binom{2n-3}{n-1}}\n\\end{equation*}\nso that $\\sqrt{1+x} = \\sum_{n}(-1)^{n+1}a_{n}x^{n}$. It follows that\n\\begin{equation*}\nq(t) = \\frac{t}{2}\\left(1 - \\sqrt{1-\\frac{4}{t^{2}}}\\right) = \\frac{1}{t} + \\sum_{n=2}^{+\\infty}a_{n}\\left(\\frac{2}{t}\\right)^{2n-1}.\n\\end{equation*}\nMoreover, using twice the identity $1+q(t)^{2} = tq(t)$, we see that\n\\begin{equation*}\nq(t)(1-q(t)^{2}) = q(t)(2-tq(t)) = 2q(t) - t(tq(t) - 1) = 2q(t) - t^{2}q(t) + t\n\\end{equation*}\nand we deduce from this a series expansion, namely (setting $t = N-\\tau$)\n\\begin{align*}\nq(N-\\tau)(1-q(N-\\tau)^{2}) & = \\frac{2}{t} + \\sum_{n=2}^{+\\infty}2a_{n}\\left(\\frac{2}{t}\\right)^{2n-1} - t - \\sum_{n=2}^{+\\infty}a_{n}\\frac{2^{2n-1}}{t^{2n-3}} + t \\\\\n& = \\frac{1}{t} + \\sum_{n=2}^{+\\infty}(2a_{n} - 4a_{n+1})\\left(\\frac{2}{t}\\right)^{2n-1} \\\\\n& = \\frac{1}{t} - \\sum_{n=2}^{+\\infty}b_{n}\\left(\\frac{2}{t}\\right)^{2n-1}\n\\end{align*}\nwith $b_{n} = -2(a_{n} - 2a_{n+1}) = 2a_{n}(n-2)\/(n+1) > 0$. We have to find an upper bound for the sum in this expression. Using the fact that $\\binom{a}{b}\\leqslant 2^{a}$, we see that for $n\\geqslant 4$\n\\begin{equation*}\nb_{n}\\leqslant 2\\frac{n-2}{n+1}\\frac{1}{n4^{n-1}}2^{2n-3} = \\frac{n-2}{n(n+1)} \\leqslant \\frac{1}{10}\n\\end{equation*}\nwhere the last inequality comes from the fact that the sequence $(n-2)\/n(n+1)$ is decreasing for $n\\geqslant 4$. Since $b_{2}=0$ and $b_{3} = 1\/32 < 1\/10$, we have for all $t > 2$\n\\begin{align*}\n\\sum_{n=2}^{+\\infty}b_{n}\\left(\\frac{2}{t}\\right)^{2n-1} \\leqslant \\frac{1}{10}\\sum_{n=3}^{+\\infty}\\left(\\frac{2}{t}\\right)^{2n-1} = \\frac{16}{5}\\frac{1}{t^{3}(t^{2}-4)} = g(t).\n\\end{align*}\nMoreover,\n\\begin{align*}\n\\frac{1}{t} - \\frac{e^{\\tau\/(t+\\tau)}}{t+\\tau} & = \\frac{1}{t} - \\frac{1}{t+\\tau} - \\frac{\\tau}{(t+\\tau)^{2}} - \\frac{\\tau^{2}}{2(t+\\tau)^{3}} - \\sum_{k=3}^{+\\infty}\\frac{\\tau^{k}}{k!(t+\\tau)^{k+1}} \\\\\n& \\geqslant \\frac{\\tau^{2}}{(t+\\tau)^{2}}\\left(\\frac{1}{t} - \\frac{1}{2(t+\\tau)}\\right) - \\frac{\\tau^{3}}{(t+\\tau)^{4}}\\sum_{k=3}^{+\\infty}\\frac{1}{k!} \\\\\n& = \\frac{\\tau^{2}}{(t+\\tau)^{2}}\\left(\\frac{1}{2t} + \\frac{\\tau}{2t(t+\\tau)}\\right) - \\frac{\\tau^{3}}{(t+\\tau)^{4}}\\sum_{k=3}^{+\\infty}\\frac{1}{k!} \\\\\n&  \\geqslant \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} + \\frac{\\tau^{3}}{2t(t+\\tau)^{3}} - \\left(e-\\frac{5}{2}\\right)\\frac{\\tau^{3}}{(t+\\tau)^{4}} \\\\\n& \\geqslant \\frac{\\tau^{2}}{2t(t+\\tau)^{2}} = f(t)\n\\end{align*}\nSumming up, by Lemma \\ref{lem:threefunctions},\n\\begin{equation}\\label{eq:hardlowerbound}\nq(t)(1-q(t)^{2}) - \\frac{e^{\\tau\/(t+\\tau)}}{t+\\tau} \\geqslant f(t) - g(t)\\geqslant 0\n\\end{equation}\nas soon as $t\\geqslant C(\\tau)$, i.e. $N\\geqslant \\tau + C(\\tau)$\n\\end{proof}\n\n\\begin{rem}\nThe condition $N\\geqslant \\tau + C(\\tau)$ could probably be sharpened by considering better bounds for the binomial coefficients and improving Lemma \\ref{lem:threefunctions}. However, it is already quite good since for instance for $\\tau = 4$ it yields $N\\geqslant 8$ and for $\\tau=2$ it yields $N\\geqslant 6$.\n\\end{rem}\n\nWe are now ready to establish the upper bound for the cut-off phenomenon announced in the beginning of this section, which is the main result of this work.\n\n\\begin{thm}\\label{thm:randomrotation}\nThe random walk associated to $0 < \\theta\\leqslant \\pi$ has an upper cut-off at\n\\begin{equation*}\n\\frac{N\\ln(N)}{2(1-\\cos(\\theta))}\n\\end{equation*}\nsteps in the following sense : if $N\\geqslant \\tau + C(\\tau)$, then for any $c_{0} > 0$ and all $c\\geqslant c_{0}$, after\n\\begin{equation*}\nk = \\frac{N\\ln(N)}{2(1-\\cos(\\theta))} + cN\n\\end{equation*}\nsteps we have\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\leqslant \\frac{1}{2\\sqrt{1-e^{-4c_{0}(1-\\cos(\\theta))}}}e^{-2c(1-\\cos(\\theta))}.\n\\end{equation*}\n\\end{thm}\n\n\\begin{proof}\nLet us set $k_{1} = N\\ln(N)\/\\tau$,\n\\begin{equation*}\nB_{k}(N) = \\frac{Nq(N-\\tau)^{k}}{2\\sqrt{q(N-\\tau)^{2k} - q(N)^{2k-2}}} \\text{ and } B_{k}'(N) = \\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k}.\n\\end{equation*}\nWe will bound each part separately and then combine them to get the desired estimate. First, using Lemma \\ref{lem:variousbounds} we see that\n\\begin{equation*}\nN\\ln\\left(\\frac{q(N)}{q(N-\\tau)}\\right) \\leqslant N\\ln\\left(\\frac{N-\\tau}{N}\\right) \\leqslant -\\tau,\n\\end{equation*}\nso that\n\\begin{equation*}\n(2k_{1} + 2cN)\\ln\\left(\\frac{q(N)}{q(N-\\tau)}\\right) - 2\\ln(q(N)) \\leqslant - 2\\tau c\n\\end{equation*}\nand it then follows that\n\\begin{align*}\nB_{k}(N)\\leqslant \\frac{N}{2\\sqrt{1-e^{-2\\tau c}}} \\leqslant \\frac{N}{2\\sqrt{1-e^{-2\\tau c_{0}}}}.\n\\end{align*}\nTurning now to $B_{k}'(N)$, we have by Lemma \\ref{lem:hardlowerbound}\n\\begin{align*}\n(k_{1} + cN)\\left(-\\ln(Nq(N-\\tau)(1-q(N-\\tau)^{2}))\\right) & \\leqslant \\ln(N) - \\tau c\n\\end{align*}\nso that\n\\begin{equation*}\nB_{k}'(N) \\leqslant \\frac{1}{N}e^{-\\tau c}\n\\end{equation*}\nGathering both inequalities eventually yields the announced estimate.\n\\end{proof}\n\n\\begin{rem}\nThe fact that the statement is not uniform in $c$ may be disappointing, but we cannot do better with the upper bound lemma in the sense that for $k = k_{1} + cN$, all the inequalities we used become equivalences as $N$ goes to infinity, so that $e^{c\\tau}A_{k}(N-\\tau)$ is not bounded above uniformly in $c$. Note that the result could also be stated in the following way : for any $\\epsilon > 0$, there is a uniform (in $c$) upper cut-off at $k_{1}(1+\\epsilon)$ steps.\n\\end{rem}\n\n\\subsubsection{Lower bound}\n\nTo have a genuine cut-off phenomenon, we must now show that if $k = N\\ln(N)\/\\tau - cN$, then the total variation distance is bounded below by something which is almost constant. Usually, such bounds are proven using the Chebyshev inequality. Note that if $x$ is a self-adjoint element in a von Neumann algebra, then it generates an abelian subalgebra which is therefore isomorphic to $L^{\\infty}(X)$ for some space $X$. Then, any state on the original algebra restricts to a measure on $X$ so that it makes sense to apply the Chebyshev inequality to $x$. In our case, we will apply it to $x = \\chi_{1}$, so that we have to estimate the expectation and variance of this element under the state $\\varphi_{R_{\\theta}}^{\\ast k}$. To keep things clear, we first give these estimates in a lemma.\n\n\\begin{lem}\\label{lem:lowerboundquantumrotation}\nFor $k = N\\ln(N)\/\\tau - cN$ and $N\\geqslant 5$, we have\n\\begin{equation*}\n\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) \\geqslant \\frac{e^{c\\tau}}{5} \\text{ and } \\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) \\leqslant 1.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nAs explained in the proof of \\cite[Thm 2.1]{rosenthal1994random}, for any $N\\geqslant 5$ and any $-4 \\leqslant \\tau \\leqslant 4$,\n\\begin{equation*}\nN\\left(1 - \\frac{\\tau}{N}\\right)^{N\\ln(N)\/\\tau}\\geqslant \\frac{1}{5}\n\\end{equation*}\nso that\n\\begin{equation*}\n\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) = \\frac{(N-\\tau)^{k}}{N^{k-1}} = N\\left(1-\\frac{\\tau}{N}\\right)^{N\\ln(N)\/\\tau}\\left(1-\\frac{\\tau}{N}\\right)^{-cN}\\geqslant \\frac{e^{c\\tau}}{5}.\n\\end{equation*}\nAs for the second inequality, first note that $\\chi_{1}^{2} = \\chi_{2} + 1$ so that\n\\begin{align*}\n\\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) & = 1 + \\frac{((N-\\tau)^{2} - 1)^{k}}{(N^{2}-1)^{k-1}} - \\left(\\frac{(N-\\tau)^{k}}{N^{k-1}}\\right)^{2} \\\\\n& \\leqslant 1 + \\frac{(N-\\tau)^{2k}}{N^{2k-2}}\\left(\\frac{\\left(1-(N-\\tau)^{-2}\\right)^{k}}{(1-N^{-2})^{k-1}} - 1\\right) \\\\\n& \\leqslant 1.\n\\end{align*}\n\\end{proof}\n\nWe are now ready for the proof of the lower bound.\n\n\\begin{prop}\\label{prop:lowercutoff}\nThe random walk associated to $0 < \\theta\\leqslant \\pi$ has a lower cut-off at\n\\begin{equation*}\n\\frac{N\\ln(N)}{2(1-\\cos(\\theta))}\n\\end{equation*}\nsteps in the following sense : for any $c > 0$, at\n\\begin{equation*}\nk = \\frac{N\\ln(N)}{2(1-\\cos(\\theta))} - cN\n\\end{equation*}\nsteps we have\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\geqslant 1-200e^{-2c\\tau}\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nWe will evaluate the states at projections obtained by functional calculus and use the original definition of the total variation distance. Let us denote by $\\mathbf{1}_{S}$ the indicator function of a subset $S$ of $\\mathbb{R}$. The proof relies on the same trick as in the classical case (see for instance \\cite{rosenthal1994random}) using Chebyshev's inequality : noticing that because of the first inequality of Lemma \\ref{lem:lowerboundquantumrotation}, $\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert \\chi_{1}\\vert)\\leqslant \\mathbf{1}_{[e^{c\\tau}\/10, +\\infty]}(\\vert\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) - \\chi_{1}\\vert)$, we have\n\\begin{align*}\n\\varphi_{R_{\\theta}}^{\\ast k}\\left(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)\\right) & \\leqslant \\varphi_{R_{\\theta}}^{\\ast k}(\\mathbf{1}_{[e^{c\\tau}\/10, +\\infty]}\\left(\\vert\\varphi_{R_{\\theta}}^{\\ast k}(\\chi_{1}) - \\chi_{1}\\vert)\\right) \\\\\n& \\leqslant 100e^{-2c\\tau}\\var_{\\varphi_{R_{\\theta}}^{\\ast k}}(\\chi_{1}) \\\\\n& \\leqslant 100e^{-2c\\tau}\n\\end{align*}\nOn the other hand, since $h(\\chi_{1}) = 0$ and $h(\\chi_{1}^{2}) = 1$,\n\\begin{equation*}\nh\\left(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)\\right) = 1 - h\\left(\\mathbf{1}_{]e^{c\\tau}\/10, +\\infty[}(\\vert\\chi_{1}\\vert)\\right) \\geqslant 1 - 100e^{-2c\\tau}.\n\\end{equation*}\nGathering these facts, we get\n\\begin{equation*}\n\\|\\varphi_{R_{\\theta}}^{\\ast k} - h\\|_{TV} \\geqslant \\frac{1}{2}\\left\\vert h(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert)) - \\varphi_{R_{\\theta}}^{\\ast k}(\\mathbf{1}_{[0, e^{c\\tau}\/10]}(\\vert\\chi_{1}\\vert))\\right\\vert \\geqslant 1-200e^{-2c\\tau}\n\\end{equation*}\n\\end{proof}\n\nThe combination of Theorem \\ref{thm:randomrotation} and Proposition \\ref{prop:lowerbound} establishes the announced cut-off phenomenon. For $\\theta = \\pi$, J. Rosenthal proved \\cite{rosenthal1994random} that $N\\ln(N)\/4$ steps suffice to get exponential convergence, in accordance with our result (for $N\\geqslant 8$). For $\\theta\\neq \\pi$, he could only show that at least $N\\ln(N)\/2(1-\\cos(\\theta))$ steps are required and the sufficiency was proved by B. Hough and Y. Jiang in \\cite{hough2017cut}. One can also consider the random walk given by a random reflection since they form a conjugacy class. Noting that any reflection has trace $N-2$ the previous argument shows that for $N\\geqslant 6$ there is a cut-off with parameter $N\\ln(N)\/2$, in accordance with the results of U. Porod in the classical case \\cite{porod1996cut}. Note that we could also use the same computations to obtain a cut-off for $\\varphi_{t}$ as soon as $t > 2$, but we decided to stick to random walks which are connected to important classical examples.\n\nThe reader may be surprised that our random walk is defined on an analogue of $O_{N}$ instead of $SO(N)$ since we are considering the conjugacy class of a matrix with determinant one. This comes from the fact that $O_{N}^{+}$ is in a sense connected as a compact quantum group so that it has no \"$SO^{+}_{N}$\" quantum subgroup (recall that matrices with opposite determinant are quantum conjugate if they have the same trace) and this allows the random walk to spread on the whole of $O_{N}^{+}$. There is another quantum group linked to $SO(N)$, which is the quantum group of trace-preserving automorphisms of the algebra $M_{N}(\\mathbb{C})$ of $N$ by $N$ matrices. We will see in Subsection \\ref{subsec:quantumautomorphisms} that the uniform plane Kac walk on it has a cut-off with the same parameter as for $O_{N}^{+}$.\n\n\\subsection{Mixed rotations}\\label{subsec:mixedrotations}\n\nOne may also consider random walks associated to states which are \"mixed\" instead of being pure. For instance, let $\\nu$ be a probability measure on the circle $\\mathbb{T}$, and set\n\\begin{equation*}\n\\varphi_{\\nu}(x) = \\int_{\\mathbb{T}}\\varphi_{R_{\\theta}}(x)\\mathrm{d}\\nu(\\theta).\n\\end{equation*}\nThis defines a central state $\\varphi_{\\nu}$ on $O_{N}^{+}$ corresponding to a random walk where $\\theta$ is chosen randomly according to $\\nu$ and then $R_{\\theta}$ is randomly conjugated. B. Hough and Y. Jiang obtained in \\cite{hough2017cut} cut-off results for these random walks with the sole restriction that $\\nu\\neq\\delta_{0}$. In our context, a stronger assumption will be needed, due to an analytic issue. Assume for instance $\\nu(\\{0\\}) = p > 0$, then $\\varphi_{\\nu}$ can be written as\n\\begin{equation*}\n\\varphi_{\\nu} = p\\varphi_{N} + (1-p)\\varphi_{\\nu'}\n\\end{equation*}\nwhere $\\nu' = (1-p)^{-1}(\\nu - p.\\delta_{0})$. But $\\varphi_{N}$ is a very particular map called the \\emph{co-unit} and for a compact quantum group, the co-unit is bounded on $L^{\\infty}(\\mathbb{G})$ if and only if $\\mathbb{G}$ is \\emph{co-amenable}. Since co-amenability is known to fail for $O_{N}^{+}$ by \\cite[Cor 1]{banica1997groupe} and $\\varphi_{N}^{\\ast k} = \\varphi_{N}$ for any $k$, we see that $\\varphi_{\\nu}^{\\ast k}\\geqslant p^{k}\\varphi_{N}$ never extends to $L^{\\infty}(O_{N}^{+})$ so that the total variation distance between $\\varphi_{\\nu}$ and $h$ is not defined.\n\nThis suggests to assume that $\\nu(\\{0\\}) = 0$, but we were not able to prove a cut-off in this generality. However, if we assume that the support of $\\nu$ is bounded away for $0$, then everything works. The proof closely follows that of B. Hough and Y. Jiang in \\cite{hough2017cut} except for some computations, which we first treat separately.\n\n\\begin{lem}\\label{lem:boundsformixedrotations}\nFor any $N\\geqslant \\tau + C(\\tau)$ and any $0 < \\tau\\leqslant 4$,\n\\begin{equation*}\n\\varphi_{N-\\tau}(n)^{N\\ln(N)\/\\tau}\\leqslant d_{n}^{-1}.\n\\end{equation*}\nMoreover, for any $N\\geqslant 3$ and any $\\lambda > 0$,\n\\begin{equation*}\n\\sum_{n=1}^{+\\infty}d_{n}^{-\\lambda\/\\ln(N)} \\leqslant \\frac{e^{-\\lambda\/2}}{1-e^{-\\lambda\/2}}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nSetting $k = N\\ln(N)\/\\tau$, we have by Lemma \\ref{lem:hardlowerbound} and the bounds of Lemma \\ref{lem:encadrement} and Lemma \\ref{lem:variousbounds},\n\\begin{align*}\n\\varphi_{N-\\tau}(n)^{k} & = \\left(\\frac{u_{n}(N-\\tau)}{u_{n}(N)}\\right)^{k} \\leqslant \\left(\\frac{q(N)^{n-1}}{q(N-\\tau)^{n}}\\frac{1}{N(1-q(N-\\tau)^{2})}\\right)^{k} \\\\\n& \\leqslant \\left(\\frac{q(N)}{q(N-\\tau)}\\right)^{k(n-1)}\\left(\\frac{1}{Nq(N-\\tau)(1-q(N-\\tau)^{2})}\\right)^{k} \\\\\n& \\leqslant \\frac{1}{N}\\left(1-\\frac{\\tau}{N}\\right)^{k(n-1)} \\leqslant \\frac{1}{N^{n}}\\leqslant \\frac{1}{d_{n}}.\n\\end{align*}\nFor the second inequality, we use again Lemma \\ref{lem:encadrement} to get\n\\begin{equation*}\nd_{n}^{-\\lambda\/\\ln(N)} \\leqslant \\left(\\frac{q(N)^{n-1}}{N}\\right)^{\\lambda\/\\ln(N)} = e^{-\\lambda}q(N)^{(n-1)\\lambda\/\\ln(N)}.\n\\end{equation*}\nUsing $\\sqrt{x}\\leqslant 1\/2  + x\/2$, we see that $q(N) \\leqslant 2\/N$ for all $N\\geqslant 4$, so that the right-hand side of the above inequality is bounded by\n\\begin{equation*}\ne^{-\\lambda}\\exp\\left((n-1)\\lambda\\left(\\frac{\\ln(2)}{\\ln(N)} - 1\\right)\\right) \\leqslant e^{-\\lambda(n+1)\/2},\n\\end{equation*}\nfrom which the result follows.\n\\end{proof}\n\nWe are now ready for the proof of the cut-off phenomenon. For convenience, we will rather consider a measure $\\mu$ on the interval $[0, 4]$ and set\n\\begin{equation*}\n\\varphi_{\\mu} = \\int_{0}^{4}\\varphi_{N-\\tau}\\mathrm{d}\\mu(\\tau).\n\\end{equation*}\n\n\\begin{thm}\\label{thm:mixedrotations}\nLet $\\mu$ be a measure on $[0, 4]$ such that there exists $\\delta > 0$ satisfying $\\mu([\\delta, 4]) = 1$ and set $\\eta = \\int\\tau\\mathrm{d}\\mu$. Then, for any $N\\geqslant \\max_{\\tau\\in [\\delta, 4]}(\\delta + C(\\delta))$, the random walk associated to $\\varphi_{\\mu}$ has a cut-off at $N\\ln(N)\/\\eta$ steps.\n\\end{thm}\n\n\\begin{proof}\nThe proof closely follows the argument of \\cite[Sec 4]{hough2017cut} and we first treat the upper bound. Let us set $k = N\\ln(N)\/\\eta + cN$. We start by the straightforward inequality\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast k} - h\\|_{TV}\\leqslant \\int_{[\\delta, 4]^{k}}\\|\\varphi_{N-\\tau_{k}}\\ast\\cdots\\ast\\varphi_{N-\\tau_{1}} - h\\|_{TV}\\;\\mathrm{d}\\mu(\\tau_{1})\\cdots\\mathrm{d}\\mu(\\tau_{n})\n\\end{equation*}\nand set $E = \\{(\\tau_{1}, \\cdots, \\tau_{k})\\in [\\delta, 4]^{k}\\mid \\sum_{i=1}^{k}\\tau_{i}\\leqslant N\\ln(N) + c\\eta N\/2\\}$. Consider the random variable $X = \\sum_{i=1}^{k}\\tau_{i}$, which has expectation $k\\eta$ under $\\mu$. The measurable set $E$ corresponds to the event\n\\begin{equation*}\nX\\leqslant \\mathbb{E}(X)\\left(1-\\frac{c\\eta}{2(\\ln(N) + c\\eta)}\\right)\n\\end{equation*}\nso that by Hoeffding's inequality (using the fact that $0\\leqslant \\tau\\leqslant 4$),\n\\begin{equation*}\n\\mu^{\\otimes k}(E)\\leqslant \\exp\\left(-\\frac{2k}{16}\\left(\\frac{c\\eta}{2(\\ln(N) + c\\eta)}\\right)^{2}\\right) = \\exp\\left(-\\frac{c^{2}\\eta N}{32(\\ln(N)+c)}\\right).\n\\end{equation*}\nThe function $x\\mapsto x\/(\\ln(x)+c)$ is increasing as soon as $x\\geqslant e\\geqslant e^{1-c}$. In particular, for $N\\geqslant 3$ it can be bounded below by $3\/(\\ln(3)+c)$. Moreover, $c^{2}\/(\\ln(3)+c) > c-\\ln(3)$ so that\n\\begin{equation*}\n\\mu^{\\otimes k}(E)\\leqslant 3^{\\eta\/32}e^{-c\\eta\/32}\\leqslant 3^{1\/8}e^{-\\eta c\/32}.\n\\end{equation*}\nWe still have to bound the integral on the complement of $E$. To do this, we apply Lemma \\ref{lem:upperbound} and Lemma \\ref{lem:boundsformixedrotations} to the integrand (recall that $\\tau_{i} \\geqslant \\delta$ for all $i$), which is therefore less than\n\\begin{equation*}\n\\frac{1}{2}\\sqrt{\\sum_{n=1}^{+\\infty}d_{n}^{2}\\prod_{i=1}^{k}\\varphi_{N-\\tau_{i}}(n)^{2}}\\leqslant \\frac{1}{2}\\sqrt{\\sum_{n=1}^{+\\infty}\\exp\\left(2\\ln(d_{n})\\left(1-\\frac{1}{N\\ln(N)}\\sum_{i=1}^{k}\\tau_{i}\\right)\\right)}.\n\\end{equation*}\nBy definition of the complement of $E$, each term is bounded by $\\exp(-\\ln(d_{n})c\\eta\/\\ln(N))$ and by Lemma \\ref{lem:boundsformixedrotations} we conclude that\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast k} - h\\|_{TV}\\leqslant 3^{1\/8}e^{-\\eta c\/32} + \\frac{e^{-c\\eta\/4}}{2\\sqrt{1-e^{-c\\eta\/2}}}.\n\\end{equation*}\n\nFor the lower bound, we proceed as in Proposition \\ref{prop:lowercutoff} and all that is needed is estimates of the mean and variance of $\\chi_{1}$. Noticing that\n\\begin{equation*}\n\\varphi_{\\mu}(1) = \\frac{1}{N}\\int_{0}^{4}\\chi_{1}(N-\\tau)\\mathrm{d}\\mu = 1 - \\frac{\\eta}{N},\n\\end{equation*}\nwe get $\\varphi_{\\mu}^{\\ast k}(\\chi_{1}) = d_{1}\\varphi_{\\mu}(1)^{k} = N(1-\\eta\/N)^{k}$ and as before we conclude that this is greater than or equal to $e^{\\eta c}\/5$ for any $N\\geqslant 5$. As for the variance, it follows from Popoviciu's inequality (see \\cite[Thm 2]{bhatia2000better} for an operator algebraic statement and proof), that since $-2 < \\chi_{1} < 2$,\n\\begin{equation*}\n\\var_{\\psi}(\\chi_{1})\\leqslant (2 - (-2))^{2}\/4 = 4\n\\end{equation*}\nfor any state $\\psi$. Applying this to $\\varphi_{\\mu}^{\\ast k}$ and using the same argument as in Proposition \\ref{prop:lowercutoff} then yields\n\\begin{equation*}\n\\|\\varphi_{\\mu}^{\\ast (N\\ln(N)\/4 - cN)} - h\\|_{TV} \\geqslant 1 - 500e^{-2\\eta c}.\n\\end{equation*}\n\\end{proof}\n\nExtending the previous result seems impossible with the techniques of the present work since it is clear that our estimates for fixed $\\tau$ can only be valid for $N$ larger than a function of $\\tau$ going to infinity as $\\tau$ goes to $0$.\n\n\\section{Further examples}\\label{sec:further}\n\nIn this section we will consider random walks on other compact quantum groups which were also introduced by S. Wang in \\cite{wang1998quantum} and called \\emph{free symmetric quantum groups}. As before, we define them through a universal algebra :\n\n\\begin{de}\nLet $\\O(S_{N}^{+})$ be the universal $*$-algebra generated by $N^{2}$ \\emph{self-adjoint} elements $u_{ij}$ such that for all $1\\leqslant i, j \\leqslant N$,\n\\begin{equation*}\nu_{ij}^{2} = u_{ij} \\text{ and } \\displaystyle\\sum_{k=1}^{N}u_{ik} = 1 = \\displaystyle\\sum_{k=1}^{N}u_{kj}.\n\\end{equation*}\nThe formula\n\\begin{equation*}\n\\Delta(u_{ij}) = \\sum_{k=1}^{N}u_{ik}\\otimes u_{kj}\n\\end{equation*}\nextends to a $*$-algebra homomorphism $\\Delta : \\O(S_{N}^{+})\\to \\O(S_{N}^{+})\\otimes \\O(S_{N}^{+})$ and this can be completed into a compact quantum group structure.\n\\end{de}\n\nAs the name and notation suggest, the abelianization of $\\O(S_{N}^{+})$ is exactly $\\O(S_{N})$ and the two even coincide for $N\\leqslant 3$. However, as soon as $N\\geqslant 4$ the compact quantum group $S_{N}^{+}$ is infinite (in the sense that the algebra $\\O(S_{N}^{+})$ is infinite-dimensional) and therefore behaves very differently from the classical symmetric group. This will raise an analytic issue in the sequel. Let us now describe the representation theory of $S_{N}^{+}$, which is quite close to that of $O_{N}^{+}$. The irreducible representations are still labelled by nonnegative integers but this time the recursion relation for characters is\n\\begin{equation}\\label{eq:recursionsymetric}\n\\chi_{1}\\chi_{n} = \\chi_{n+1} + \\chi_{n} + \\chi_{n-1}.\n\\end{equation}\nTo translate this into an explicit isomorphism with $\\mathbb{C}[X]$, first note that keeping the notations of Section \\ref{sec:orthogonal}, $u_{2n}(X)$ has only even powers of $X$ for any $n$. Thus, $v_{n}(X) = u_{2n}(\\sqrt{X})$ is a polynomial in $X$ and it is easily checked that this new sequence satisfies the above recursion relation. Once again, one has $d_{n} = v_{n}(N) = u_{2n}(\\sqrt{N})$.\n\n\\subsection{Pure state random walks on free symmetric quantum groups}\n\nAs for free orthogonal quantum groups, we can study pure state random walks. In view of the link between the polynomials $u_{n}$ and $v_{n}$, estimates of the total variation distance for the random walk associated to a pure state on $S_{N}^{+}$ can be easily deduced from Proposition \\ref{prop:upperboundlessthantwo} and Theorem \\ref{thm:estimategreaterthan2}. We will therefore simply give the statements, starting with the case of small $t$.\n\n\\begin{prop}\nLet $\\vert t\\vert < 4$ be fixed. Then, for any $k\\geqslant 2$,\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\sqrt{\\frac{N}{q(\\sqrt{N})^{2}(1-q(\\sqrt{N})^{4})}}\\left(\\frac{q(\\sqrt{N})}{N\\sqrt{1 - t^{2}\/4}}\\right)^{k}\n\\end{equation*}\nIn particular, if $t<2\\sqrt{1-\\left(\\frac{q(\\sqrt{N})}{N}\\right)^{2}}$ then the random walks converges exponentially.\n\\end{prop}\n\nOne can also get a lower bound like in Proposition \\ref{prop:lowerbound} : noticing that $u_{2}(X) = X^{2} - 1$ yields\n\\begin{equation*}\n\\|\\varphi^{\\ast k} - h\\|_{TV} \\geqslant \\frac{N-1}{6}\\left(\\frac{t-1}{N-1}\\right)^{k}.\n\\end{equation*}\n\nFor larger $t$, the proof is also the same as in Theorem \\ref{thm:estimategreaterthan2}.\n\n\\begin{prop}\nLet $\\vert t\\vert > 4$ and let $k_{0}$ be the smallest integer such that $q(t) > q(N)^{1-1\/k_{0}}$. If $k\\leqslant k_{0}$ then the state $\\varphi_{t}^{\\ast k}$ is not bounded on $L^{\\infty}(S_{N}^{+})$ and otherwise\n\\begin{equation*}\n\\|\\varphi_{t}^{\\ast k} - h\\|_{TV}\\leqslant \\frac{1}{2}\\sqrt{\\frac{N^{2}q(\\sqrt{t})^{4k_{0}}}{q(\\sqrt{t})^{4k_{0}} - q(\\sqrt{N})^{4k_{0}-4}}}\\left(\\frac{q(\\sqrt{N})}{\\sqrt{N}q(\\sqrt{t})^{2}(1-q(\\sqrt{t})^{2})}\\right)^{k}\n\\end{equation*}\n\\end{prop}\n\nThe main point in the above statement is that the threshold $k_{0}$ is the same as for $O_{N}^{+}$, so that the cut-off parameter of a uniform random walk on a conjugacy class should be given by the same formula as before. One of the simplest examples of such a random walk is the one associated to the uniform measure on the set of transpositions, or equivalently on the conjugacy class of a transposition. Since the trace of a transposition matrix is $N-2$, this is given by the state $\\varphi_{N-2}$ and the expected cut-off parameter is $N\\ln(N)\/2$. This can be proven by the same strategy as for Theorem \\ref{thm:randomrotation} but the computations are more involved.\n\n\\begin{thm}\\label{thm:randomtranspositions}\nFor any $N\\geqslant 16$, the random walk associated to $\\varphi_{N-2}$ on $S_{N}^{+}$ has a cut-off at $N\\ln(N)\/2$ steps.\n\\end{thm}\n\n\\begin{proof}\nFor the upper bound, the part concerning $B_{k}$ is the same as in the proof of Theorem \\ref{thm:randomrotation}, so let us focus on $B_{k}'$. It is enough to prove that\n\\begin{equation*}\n\\frac{\\sqrt{N}}{q(\\sqrt{N})}q(\\sqrt{N-2})^{2}\\left(1-q(\\sqrt{N-2})^{2}\\right) \\geqslant e^{2\/N}.\n\\end{equation*}\nWriting $q(t)^{2}(1-q(t)^{2}) = (3t-t^{3})q(t) + t^{2} - 2$ and expanding we get\n\\begin{align*}\nq(t)^{2}(1-q(t)^{2}) = t\\sum_{n=2}^{+\\infty}(3a_{n} - 4a_{n+1})\\left(\\frac{2}{t}\\right)^{2n-1} \\\\\n\\end{align*}\nsince $c_{n} = 4a_{n+1} - 3a_{n} = (n-5)a_{n}\/(n+1)$, the sum splits as\n\\begin{equation*}\n\\frac{1}{t^{2}} + \\frac{1}{t^{4}} + \\frac{1}{t^{6}} - t\\sum_{n=6}^{+\\infty}c_{n}\\left(\\frac{2}{t}\\right)^{2n-1}.\n\\end{equation*}\nMoreover, the same estimate as for $b_{n}$ yields $c_{n}\\leqslant (n-5)\/2n(n+1)$ and the sequence on the right-hand side is increasing up to $n=10$ and then decreasing. Its maximum is therefore $1\/44$. Using $\\sqrt{t}$ instead of $t$ and the fact that $c_{n} \\leqslant 1\/44$, we eventually get\n\\begin{equation*}\nq(\\sqrt{t})^{2}(1-q(\\sqrt{t})^{2}) = \\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{t^{3}} - \\sum_{n=6}^{+\\infty}2c_{n}\\left(\\frac{4}{t}\\right)^{n-1}\\geqslant \\frac{t^{2} + t + 1}{t^{3}} - \\frac{4^{5}}{22\\times t^{4}(t-4)} .\n\\end{equation*}\nOn the other hand,\n\\begin{equation*}\ne^{2\/(t+2)} \\leqslant \\sum_{k=0}^{+\\infty}\\left(\\frac{2}{t+2}\\right)^{k} = \\frac{t+2}{t} = 1+\\frac{2}{t}\n\\end{equation*}\nso that it is enough to have (noticing that $q(x)^{-1} = (x+\\sqrt{x^{2}-4})\/2$)\n\\begin{align*}\n& \\sqrt{t+2}\\frac{\\sqrt{t+2} + \\sqrt{t-2}}{2}\\left(\\frac{t^{2} + t + 1}{t^{3}} - \\frac{4^{5}}{22\\times t^{4}(t-4)}\\right) - 1 - \\frac{2}{t} \\geqslant 0.\n\\end{align*}\nTo see when this inequality holds, let us first prove that for $t\\geqslant 12$, $1\/2t^{3} \\geqslant 4^{5}\/22(t^{4}(t-4))$. Proceeding as in the proof of Lemma \\ref{lem:threefunctions}, we reduce the problem to $11t(t-4)\\geqslant 4^{5}$, i.e.\n\\begin{equation*}\nt^{2} - 4t - \\frac{4^{5}}{11} \\geqslant 0.\n\\end{equation*}\nwhich is satisfied as soon as $t$ is greater than $2 + 2\\sqrt{1+16^{2}\/11}\\leqslant 12$. Using this, it is now enough to check that\n\\begin{align*}\n1 + \\frac{2}{t} & \\leqslant \\left(1 + \\frac{t}{2} + \\frac{\\sqrt{t^{2}-4}}{2}\\right)\\left(\\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{2t^{3}}\\right) \\\\\n& = \\frac{1}{t} + \\frac{1}{t^{2}} + \\frac{1}{2t^{3}} + \\frac{1}{2} + \\frac{1}{2t} + \\frac{1}{4t^{2}} + \\frac{\\sqrt{t^{2}-4}}{2t} + \\frac{\\sqrt{t^{2}-4}}{2t^{2}} + \\frac{\\sqrt{t^{2}-4}}{4t^{3}}.\n\\end{align*}\nWe will prove the stronger inequality obtained by removing the terms with $t^{3}$ at the denominator in the right-hand side. After simplifying and multiplying by $2t^{2}$ we get the inequality\n\\begin{equation*}\nt^{2} + t \\leqslant \\frac{5}{2} + t\\sqrt{t^{2}-4} + \\sqrt{t^{2}-4}.\n\\end{equation*}\nNow, the function $f: t\\mapsto (t+1)(t-\\sqrt{t^{2}-4})$ satisfies\n\\begin{equation*}\nf'(t) = t-\\sqrt{t^{2}-4} + (t+1)\\left(1-\\frac{t}{\\sqrt{t^{2}-4}}\\right) = \\frac{t-\\sqrt{t^{2}-4}}{\\sqrt{t^{2} - 4}}\\left(\\sqrt{t^{2}-4} - (t+1)\\right).\n\\end{equation*}\nThis is negative, thus $f$ is decreasing and for $t\\geqslant 14$ it is smaller than $f(14) \\approx 2.15 < 5\/2$.\n\nConcerning the lower bound, first note that the expectation and variance of $\\chi_{1}$ with respect to $h$ are respectively equal to $0$ and $1$. Moreover, by the same argument as for $O_{N}^{+}$,\n\\begin{equation*}\n\\varphi_{N-2}(\\chi_{1}) = \\frac{(N-3)^{k}}{(N-1)^{k}} \\geqslant \\frac{e^{2c}}{5}\n\\end{equation*}\nfor $k = N\\ln(N)\/2 - c$ and the variance can be bounded independently from $N$ by Popoviciu's inequality.\n\\end{proof}\n\n\\begin{rem}\nPlotting the function appearing in the study of the upper bound suggests that it is positive as soon as $t\\geqslant 12$, which would give a cut-off for all $N\\geqslant 14$. This indicates that even though they look loose, our estimates are close to optimal.\n\\end{rem}\n\nThe cut-off parameter is the same as in the classical case (see \\cite{diaconis1981generating}). We can even consider the conjugacy class of $m$-cycles for any integer $m$ and, for $N$ large enough, the cut-off will appear at $N\\ln(N)\/m$ steps, again as in the classical case \\cite{hough2016random}.\n\n\\subsection{Mixed states and transition operators}\n\nThere are many examples of mixed states on $S_{N}^{+}$ coming from classical random walks. However, their study in the quantum case is prevented by the fact that $S_{N}^{+}$ is not amenable for $N\\geqslant 5$, a phenomenon which was alluded to for $O_{N}^{+}$ in Subsection \\ref{subsec:mixedrotations}. We will now illustrate this in more details on a simple example with random transpositions as follows : assume you have a deck of $N$ cards and spread them on a table. Randomly select one card uniformly (i.e. with probability $1\/N$ for each card) and then select another one in the same way. If the same card has been selected twice, nothing is done. Otherwise, the two cards are swapped. This corresponds to the measure on $S_{N}$ giving probability $1\/N^{2}$ to all transpositions and $1\/N$ to the identity. Since transpositions form a conjugacy class, the measure can be restated as being\n\\begin{equation*}\n\\mu_{\\text{rt}} = \\frac{N-1}{N}\\mu_{\\text{tran}} + \\frac{1}{N}\\delta_{\\id}.\n\\end{equation*}\nwhere $\\mu_{\\text{tran}}$ is the uniform measure on the set of transpositions. The equation above directly gives the state on $\\O(S_{N}^{+})$ corresponding to \"random quantum transposition\" :\n\\begin{equation*}\n\\varphi_{\\text{rt}} = \\frac{N-1}{N}\\varphi_{N-2} + \\frac{1}{N}\\varphi_{N}.\n\\end{equation*}\nThe state $\\varphi_{N-2}^{\\ast k}$ is bounded on $L^{\\infty}(S_{N}^{+})$ for $k$ large enough but not $\\varphi_{N}^{\\ast k}$ since it is the co-unit and $S_{N}^{+}$ is not co-amenable for $N\\geqslant 5$. This implies that no convolution power of $\\varphi_{\\text{rt}}$ is bounded on $L^{\\infty}(S_{N}^{+})$ so that the total variation distance is never defined (it is clear that the sum in the upper bound lemma diverges since each term is greater than $N^{-k}$). This is in sharp contrast with the classical case (a finite quantum group is always amenable).\n\nHowever, it is known (see for instance \\cite[Lem 3.4]{brannan2011approximation}) that the associated transition operator $P_{\\varphi_{\\text{rt}}} = (\\id\\otimes \\varphi_{\\text{rt}})\\circ\\Delta$ always extends to a bounded linear map on $L^{\\infty}(\\mathbb{G})$. We can therefore compare it with $P_{h}$ using operator norms. In particular, we can see them as operators on $L^{2}(S_{N}^{+})$ and the corresponding norm is then easy to compute :\n\n\\begin{lem}\nLet $\\psi$ be any central linear form on a compact quantum group $\\mathbb{G}$. Then,\n\\begin{equation*}\n\\|P_{\\psi}\\|_{B(L^{2}(\\mathbb{G}))} = \\sup_{\\alpha\\in \\Irr(\\mathbb{G})}\\vert \\psi(\\alpha)\\vert.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nBy Woronowicz' Peter-Weyl theorem, the elements $u^{\\alpha}_{ij}$ form an orthogonal basis of $L^{2}(\\mathbb{G})$. Moreover, a straightforward calculation yields\n\\begin{equation*}\nP_{\\psi}(u^{\\alpha}_{ij}) = \\psi(\\alpha)u^{\\alpha}_{ij}\n\\end{equation*}\nso that $P_{\\psi}$ is diagonal in this basis and the result follows.\n\\end{proof}\n\nSince $P_{h}$ is the projection onto the linear span of $1$, the above Lemma means that the distance in operator norm is exactly given by the spectral gap of the operator $P_{\\varphi}$. In this setting it is not very difficult to prove that there is a cut-off phenomenon.\n\n\\begin{prop}\nThe random walk associated to $\\varphi_{\\text{rt}}$ has a cut-off in the $L^{2}$-operator norm at $k = N\/2$ steps.\n\\end{prop}\n\n\\begin{proof}\nWe first show that the supremum of\n\\begin{equation*}\n\\left(\\frac{N-1}{N}\\frac{v_{n}(N-2)}{v_{n}(N)} + \\frac{1}{N}\\right)^{k}\n\\end{equation*}\nis attained at $n = 1$. Let us set, for $n\\geqslant 1$, $a_{n}(t) = u_{n+1}(t)\/u_{n}(t)$. The recursion relation \\eqref{eq:recursionsymetric} implies\n\\begin{equation*}\na_{n+1}(t) = t - \\frac{1}{a_{n}(t)}\n\\end{equation*}\nfrom which it follows by induction that for all $n$, $t-1\/t \\leqslant a_{n}(t)\\leqslant t$. Using this, we see that\n\\begin{equation*}\n\\frac{u_{n+1}(\\sqrt{N-2})}{u_{n+1}(\\sqrt{N})}\\frac{u_{n}(\\sqrt{N})}{u_{n}(\\sqrt{N-2})} = \\frac{a_{n}(\\sqrt{N-2})}{a_{n}(\\sqrt{N})} \\leqslant \\frac{\\sqrt{N(N-2)}}{N-1} < 1.\n\\end{equation*}\nThus, the sequence $u_{n}(\\sqrt{N-2})\/u_{n}(\\sqrt{N})$ is decreasing and the claim is proved. As a consequence,\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}^{\\ast k}} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))} = \\left(\\frac{N-1}{N}\\frac{N-3}{N-1} + \\frac{1}{N}1\\right)^{k} = \\left(1 - \\frac{2}{N}\\right)^{k}.\n\\end{equation*}\nBecause $1-x\\leqslant e^{-x}$, for any $c > 0$\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}}^{\\ast (N\/2+cN)} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))}\\leqslant e^{-1}e^{-2c},\n\\end{equation*}\nyielding the upper bound.\n\nAs for the lower bound, using an estimate already mentioned in Lemma \\ref{lem:lowerboundquantumrotation} we have for $N\\geqslant 5$\n\\begin{align*}\n\\left(1-\\frac{2}{N}\\right)^{N\/2}\\geqslant \\left(\\frac{1}{5N}\\right)^{1\/\\ln(N)} = e^{-1-\\ln(5)\/\\ln(N)} \\geqslant e^{-2}\n\\end{align*}\nso that for $c<1$,\n\\begin{equation*}\n\\|P_{\\varphi_{\\text{rt}}}^{\\ast (N\/2-cN)} - P_{h}\\|_{B(L^{2}(S_{N}^{+}))}\\geqslant e^{2c-2}\\geqslant e^{-2}(1-e^{-2c}).\n\\end{equation*}\n\\end{proof}\n\nThe cut-off in total variation distance for the classical random walk associated to $\\mu_{\\text{rt}}$ occurs at $N\\ln(N)\/2$ steps (see \\cite{diaconis1981generating}) and was one of the first important results of the theory. Since we considered a weaker norm, we get a better cut-off parameter. However, there are other norms available for operators on a von Neumann algebra which may be closer to the total variation distance and therefore yield a different cut-off parameter. In particular, since transition operators are completely positive, it would be interesting to have estimates for the \\emph{completely bounded norm} of $P_{\\varphi_{\\text{rt}}^{\\ast k}} - P_{h}$.\n\n\\subsection{Quantum automorphisms of matrices}\\label{subsec:quantumautomorphisms}\n\nAs mentioned in the end of Subsection \\ref{subsec:randomrotations}, apart from $O_{N}^{+}$ there is another quantum generalization of $SO(N)$, called the \\emph{quantum automorphism group of $(M_{N}(\\mathbb{C}), \\mathrm{tr})$}. This means that it is a universal object in the category of compact quantum groups acting on $M_{N}(\\mathbb{C})$ in a trace-preserving way. For $N=2$, this is known to be isomorphic to $SO(3)$.\n\nIt was shown in \\cite{banica1999symmetries} that the representation theory of this quantum group is the same as $S_{N}^{+}$. The only difference is that the dimensions are given by $u_{2n}(N) = v_{n}(N^{2})$. We can therefore consider the pure states $\\varphi_{t}$ as before for $0 \\leqslant t < N^{2}$ and the same arguments as in Theorem \\ref{thm:randomtranspositions} would show that the random walk associated to random rotations with a fixed angle $\\theta$ has a cut-off at $N\\ln(N)\/2(1-\\cos(\\theta))$ steps. There is however a quicker way to this. Consider the subalgebra of $\\O(O_{N}^{+})$ generated by all products $u_{ij}u_{kl}$ of two generators. Then, this is isomorphic to the Hopf algebra of the quantum automorphism group of $(M_{N}(\\mathbb{C}), \\mathrm{tr})$. The random walk can therefore be obtained by simply restricting the state to this subalgebra and as far as Lemma \\ref{lem:upperbound} is concerned this is just restricting to the sum of even terms. The upper bound for the cut-off then trivially follows from Theorem \\ref{thm:randomrotation}. As for the lower bound, it is a computation similar to that of Proposition \\ref{prop:lowerbound} using $\\chi_{2}$ instead of $\\chi_{1}$.\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}