Probability Theory and Related Fields最新文献

{"title":"Phase transition for the smallest eigenvalue of covariance matrices","authors":"Zhigang Bao, Jaehun Lee, Xiaocong Xu","doi":"10.1007/s00440-024-01298-w","DOIUrl":"https://doi.org/10.1007/s00440-024-01298-w","url":null,"abstract":"<p>In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices <span>(mathcal {S}(Y)=YY^*)</span>, where <span>(Y=(y_{ij}))</span> is an <span>(Mtimes N)</span> matrix with iid mean 0 variance <span>(N^{-1})</span> entries. We consider the regime <span>(M=M(N))</span> and <span>(M/Nrightarrow c_infty in mathbb {R}{setminus } {1})</span> as <span>(Nrightarrow infty )</span>. It is known that for the extreme eigenvalues of Wigner matrices and the largest eigenvalue of <span>(mathcal {S}(Y))</span>, a weak 4th moment condition is necessary and sufficient for the Tracy–Widom law (Ding and Yang in Ann Appl Probab 28(3):1679–1738, 2018. https://doi.org/10.1214/17-AAP1341; Lee and Yin in Duke Math J 163(1):117–173, 2014. https://doi.org/10.1215/00127094-2414767). In this paper, we show that the Tracy–Widom law is more robust for the smallest eigenvalue of <span>(mathcal {S}(Y))</span>, by discovering a phase transition induced by the fatness of the tail of <span>(y_{ij})</span>’s. More specifically, we assume that <span>(y_{ij})</span> is symmetrically distributed with tail probability <span>(mathbb {P}(|sqrt{N}y_{ij}|ge x)sim x^{-alpha })</span> when <span>(xrightarrow infty )</span>, for some <span>(alpha in (2,4))</span>. We show the following conclusions: (1) When <span>(alpha &gt;frac{8}{3})</span>, the smallest eigenvalue follows the Tracy–Widom law on scale <span>(N^{-frac{2}{3}})</span>; (2) When <span>(2&lt;alpha &lt;frac{8}{3})</span>, the smallest eigenvalue follows the Gaussian law on scale <span>(N^{-frac{alpha }{4}})</span>; (3) When <span>(alpha =frac{8}{3})</span>, the distribution is given by an interpolation between Tracy–Widom and Gaussian; (4) In case <span>(alpha le frac{10}{3})</span>, in addition to the left edge of the MP law, a deterministic shift of order <span>(N^{1-frac{alpha }{2}})</span> shall be subtracted from the smallest eigenvalue, in both the Tracy–Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by Aggarwal et al. (J Eur Math Soc 23(11):3707–3800, 2021. https://doi.org/10.4171/jems/1089) which is originally done for the bulk regime of the Lévy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"28 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantitative limit theorems and bootstrap approximations for empirical spectral projectors","authors":"Moritz Jirak, Martin Wahl","doi":"10.1007/s00440-024-01290-4","DOIUrl":"https://doi.org/10.1007/s00440-024-01290-4","url":null,"abstract":"<p>Given finite i.i.d. samples in a Hilbert space with zero mean and trace-class covariance operator <span>(Sigma )</span>, the problem of recovering the spectral projectors of <span>(Sigma )</span> naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator <span>({hat{Sigma }})</span>, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of <span>(Sigma )</span>. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"15 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fluctuations of the free energy in p-spin SK models on two scales","authors":"Anton Bovier, Adrien Schertzer","doi":"10.1007/s00440-024-01296-y","DOIUrl":"https://doi.org/10.1007/s00440-024-01296-y","url":null,"abstract":"<p>20 years ago, Bovier, Kurkova, and Löwe (Ann Probab 30(2):605–651, 2002) proved a central limit theorem (CLT) for the fluctuations of the free energy in the <i>p</i>-spin version of the Sherrington–Kirkpatrick model of spin glasses at high temperatures. In this paper we improve their results in two ways. First, we extend the range of temperatures to cover the entire regime where the quenched and annealed free energies are known to coincide. Second, we identify the main source of the fluctuations as a purely coupling dependent term, and we show a further CLT for the deviation of the free energy around this random object.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"44 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effective diffusivities in periodic KPZ","authors":"Yu Gu, Tomasz Komorowski","doi":"10.1007/s00440-024-01297-x","DOIUrl":"https://doi.org/10.1007/s00440-024-01297-x","url":null,"abstract":"<p>For the KPZ equation on a torus with a <span>(1+1)</span> spacetime white noise, it was shown in Dunlap et al. (Commun Pure Appl Math, 2023, https://doi.org/10.1002/cpa.22110) and Gu and Komorowski (Ann Inst H Poincare Prob Stat, 2021, arXiv:2104.13540v2) that the height function satisfies a central limit theorem, and the variance can be written as the expectation of an exponential functional of Brownian bridges. In this paper, we consider another physically relevant quantity, the winding number of the directed polymer on a cylinder, or equivalently, the displacement of the directed polymer endpoint in a spatially periodic random environment. It was shown in Gu and Komorowski (SIAM J Math Anal, arXiv:2207.14091) that the polymer endpoint satisfies a central limit theorem on diffusive scales. The main result of this paper is an explicit expression of the effective diffusivity, in terms of the expectation of another exponential functional of Brownian bridges. Our argument is based on a combination of tools from Malliavin calculus, homogenization, and diffusion in distribution-valued random environments.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"1 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds","authors":"Thomas Hotz, Huiling Le, Andrew T. A. Wood","doi":"10.1007/s00440-024-01291-3","DOIUrl":"https://doi.org/10.1007/s00440-024-01291-3","url":null,"abstract":"<p>We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"87 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space","authors":"Arnaud Debussche, Ruoyuan Liu, Nikolay Tzvetkov, Nicola Visciglia","doi":"10.1007/s00440-024-01288-y","DOIUrl":"https://doi.org/10.1007/s00440-024-01288-y","url":null,"abstract":"<p>We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"27 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometry of Gaussian free field sign clusters and random interlacements","authors":"Alexander Drewitz, Alexis Prévost, Pierre-François Rodriguez","doi":"10.1007/s00440-024-01285-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01285-1","url":null,"abstract":"<p>For a large class of amenable transient weighted graphs <i>G</i>, we prove that the sign clusters of the Gaussian free field on <i>G</i> fall into a regime of <i>strong supercriticality</i>, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices such as <span>({mathbb {Z}}^d)</span>, for <span>(dge 3)</span>, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman (Ann Math 171(3):2039–2087, 2010), and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from Sznitman (Invent Math 187(3):645–706, 2012).</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"54 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Characterizing models in regularity structures: a quasilinear case","authors":"Markus Tempelmayr","doi":"10.1007/s00440-024-01292-2","DOIUrl":"https://doi.org/10.1007/s00440-024-01292-2","url":null,"abstract":"<p>We give a novel characterization of the centered model in regularity structures which persists for rough drivers even as a mollification fades away. We present our result for a class of quasilinear equations driven by noise, however we believe that the method is robust and applies to a much broader class of subcritical equations. Furthermore, we prove that a convergent sequence of noise ensembles, satisfying uniformly a spectral gap assumption, implies the corresponding convergence of the associated models. Combined with the characterization, this establishes a universality-type result.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"64 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Benign overfitting and adaptive nonparametric regression","authors":"Julien Chhor, Suzanne Sigalla, Alexandre B. Tsybakov","doi":"10.1007/s00440-024-01278-0","DOIUrl":"https://doi.org/10.1007/s00440-024-01278-0","url":null,"abstract":"<p>We study benign overfitting in the setting of nonparametric regression under mean squared risk, and on the scale of Hölder classes. We construct a local polynomial estimator of the regression function that is minimax optimal on a Hölder class with any given smoothness, and that is a continuous function interpolating the set of observations with high probability. The key element of the construction is the use of singular kernels. Moreover, we prove that adaptation to unknown smoothness is compatible with benign overfitting. Namely, we construct a continuous and interpolating local polynomial estimator attaining the minimax optimal rate in <span>(L_2)</span> adaptively to the unknown Hölder smoothness. Our results highlight the fact that interpolation can be fundamentally decoupled from bias-variance tradeoff in the problem of nonparametric regression.</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"1 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotics of generalized Bessel functions and weight multiplicities via large deviations of radial Dunkl processes","authors":"Jiaoyang Huang, Colin McSwiggen","doi":"10.1007/s00440-024-01282-4","DOIUrl":"https://doi.org/10.1007/s00440-024-01282-4","url":null,"abstract":"<p>This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-<i>N</i> asymptotics of generalized Bessel functions, as well as a large deviations principle for the more general family of radial Heckman–Opdam processes. As an application, we prove a theorem on the asymptotic behavior of weight multiplicities of irreducible representations of compact or complex simple Lie algebras in the limit of large rank. The theorems in this paper generalize several known results describing analogous asymptotics for Dyson Brownian motion, spherical matrix integrals, and Kostka numbers.\u0000</p>","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"28 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}