A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin

Eigenvalue variance bounds for covariance matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum. In a preceding paper, the author established analogous results for Wigner matrices and stated the results for covariance matrices. They are proved in the present paper. Relying on the LUE example, which needs to be investigated first, the main bounds are extended to complex covariance matrices by means of the Tao, Vu and Wang Four Moment Theorem and recent localization results by Pillai and Yin. The case of real covariance matrices is obtained from interlacing formulas

On the Central Limit Theorem for the Eigenvalue Counting Function of Wigner and Covariance matrices

This note presents some central limit theorems for the eigenvalue counting function of Wigner matrices in the form of suitable translations of results by Gustavsson and O'Rourke on the limiting behavior of eigenvalues inside the bulk of the semicircle law for Gaussian matrices. The theorems are then extended to large families of Wigner matrices by the Tao and Vu Four Moment Theorem. Similar results are developed for covariance matrices

FLUCTUATIONS OF LINEAR SPECTRAL STATISTICS OF DEFORMED WIGNER MATRICES

We investigate the fluctuations of linear spectral statistics of a Wigner matrix $W_N$ deformed by a deterministic diagonal perturbation $D_N$, around a deterministic equivalent which can be expressed in terms of the free convolution between a semicircular distribution and the empirical spectral measure of $D_N$. We obtain Gaussian fluctuations for test functions in $\mathcal{C}_c^7(\R)$ ($\mathcal{C}_c^2(\R)$ for fluctuations around the mean). Furthermore, we provide as a tool a general method inspired from Shcherbina and Johansson to extend the convergence of the bias if there is a bound on the bias of the trace of the resolvent of a random matrix. Finally, we state and prove an asymptotic infinitesimal freeness result for independent GUE matrices together with a family of deterministic matrices, generalizing the main result from [Shl18]

Quelques aspects de l'étude quantitative de la fonction de comptage et des valeurs propres de matrices aléatoires

Cette thèse est consacrée à l'étude quantitative de la fonction de comptage et des valeurs propres de matrices aléatoires. Initialement introduites dans le cadre de la physique statistique, ces matrices servent de modèles pour des opérateurs de dimension infinie. Leurs propriétés asymptotiques ont donc été particulièrement étudiées. Devenues populaires grâce au phénomène d'universalité, i.e. le fait que ces propriétés asymptotiques ne dépendent pas de la loi des coefficients de la matrice, leur étude a intéressé de nombreux domaines pour lesquels les propriétés à taille de matrice fixée sont plus exploitables que les propriétés asymptotiques. Nous nous sommes intéressés à ce pan de l'étude des matrices aléatoires par le biais de la fonction de comptage des valeurs propres, c'est-à-dire du nombre de valeurs propres d'une matrice présentes dans un intervalle I. Après avoir introduit les modèles de matrices aléatoires que nous étudions dans cette thèse, les matrices de Wigner et de covariance, nous présentons les principaux résultats asymptotiques en lien avec la fonction de comptage et plus globalement avec les valeurs propres de ces matrices. Les outils permettant d'établir ces résultats d'universalité sont ensuite détaillés. L'accent est mis sur ceux qui peuvent être utilisés dans le cadre de l'étude quantitative, notamment les résultats récents de Erdös et al. d'une part et de Tao et Vu d'autre part, qui ont permis une avancée spectaculaire des études asymptotiques et non asymptotiques. Nous discutons ensuite les enjeux de l'étude non asymptotique et présentons les travaux effectués durant cette thèse. Dans une première étude, nous établissons un théorème central limite pour la fonction de comptage des valeurs propres de matrices de Wigner et de covariance et nous obtenons des estimées quantitatives sur l'espérance et la variance de cette fonction de comptage. Ces résultats se basent sur les résultats précédemment établis par Gustavsson et Su dans le cas de matrices gaussiennes et sont étendus à des familles plus générales de matrices de Wigner et de covariance par le biais de travaux récents de Erdös, Yau et Yin, Pillai et Yin et de Tao et Vu. Dans une seconde étude, nous établissons des bornes quantitatives sur la variance des valeurs propres de matrices de Wigner. En s'appuyant sur les propriétés de la fonction de comptage, nous montrons d'abord une inégalité de déviation pour les valeurs propres individuelles à l'intérieur du spectre dans le cas de matrices gaussiennes et nous en déduisons les bornes dans ce cas. Afin d'étendre ces bornes au cas des matrices de Wigner plus générales, nous combinons de nouveau les outils de Erdös, Yau et Yin et de Tao et Vu. Nous établissons également des résultats analogues pour les matrices de covariance, en utilisant la même démarche.This thesis is concerned by the quantitative study of the counting function and the eigenvalues of random matrices. These matrices were first introduced for use in statistical physics and are models for infinite dimensional operators. Therefore a thorough study of their asymptotic properties was initiated. They became popular due to the universality phenomenon, i.e. the fact that these asymptotic properties do not depend on the distribution of the entries. Lots of fields became interested in random matrices. Several of these fields need quantitative results rather than asymptotic ones. In this thesis, we are interested in the non asymptotic study of random matrices, through the study of the eigenvalue counting function, which is the number of eigenvalues which are in an interval I. We first introduce models of random matrices which are considered in this thesis, Wigner matrices and covariance matrices. The main asymptotic results regarding the counting function and more generally regarding the eigenvalues of these matrices are presented. Tools to reach universality results are then described. We insist on those which are useful to obtain non asymptotic results as well, such as recent results by Erdös et al. on the one hand and by Tao and Vu on the other hand. We then discuss non asymptotic issues and present the work we did during this thesis. In a first part, a Central Limit Theorem for the counting function is established for Wigner and covariance matrices. Quantitative asymptotics for the mean and the variance of the counting function are obtained. These results are based on previous results by Gustavsson and Su for Gaussian matrices and are extended to large families of Wigner and covariance matrices by recent results of Erdös, Yau and Yin, Pillai and Yin and of Tao and Vu. In a second part, quantitative bounds on the variance of individual eigenvalues of Wigner matrices are established. Relying on properties of the counting function, a deviation inequality for eigenvalues in the bulk is proved for Gaussian matrices. The bounds on the variance in this case are then easily derived. These bounds are again extended to families of Wigner matrices by a combination of tools of Erdös, Yau and Yin and of Tao and Vu. Following the same scheme, analogous results are then established for covariance matrices

Quelques aspects de l'étude quantitative de la fonction de comptage et des valeurs propres de matrices aléatoires

TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF

Sparse Recovery from Extreme Eigenvalues Deviation Inequalities

33 pages, 1 figureInternational audienceThis article provides a new toolbox to derive sparse recovery guarantees - that is referred to as ‘‘stable and robust sparse regression'' (SRSR) ' from deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices models. In this paper, we show upper bounds on RICs for Gaussian and Rademacher matrices using state-of-the-art deviation estimates on their extreme eigenvalues. This allows us to derive a lower bound on the probability of getting SRSR. One benefit of this paper is a direct and explicit derivation of upper bounds on RICs and lower bounds on SRSR from deviations on the extreme eigenvalues given by Random Matrix theory

FLUCTUATIONS OF THE STIELTJES TRANSFORM OF THE EMPIRICAL SPECTRAL DISTRIBUTION OF SELFADJOINT POLYNOMIALS IN WIGNER AND DETERMINISTIC DIAGONAL MATRICES

We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions

FLUCTUATIONS OF THE STIELTJES TRANSFORM OF THE EMPIRICAL SPECTRAL DISTRIBUTION OF SELFADJOINT POLYNOMIALS IN WIGNER AND DETERMINISTIC DIAGONAL MATRICES

We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions