Matrices aléatoires: Statistique asymptotique des valeurs propres

Abstract

Ce cours † est une brève introductionà la théorie de la distribution asymptotique des valeurs propres des matrices aléatoires symétriques réelles ou hermitiennes de grande taille. Cette théorie a connu récemment beaucoup d'évolutions motivées par diverses branches des mathématiques et de la physique. L'étude de quelques régimes asymptotiques mèneà des résultats età des techniques intéressants.Keywords: Matrice aléatoire, valeur propre, ensemble orthogonal gaussien, loi du demi-cercle Abstract: Ce cours † est une brève introductionà la théorie de la distribution asymptotique des valeurs propres des matrices aléatoires symétriques réelles ou hermitiennes de grande taille. Cette théorie a connu récemment beaucoup d'évolutions motivées par diverses branches des mathématiques et de la physique. L'étude de quelques régimes asymptotiques mèneà des résultats età des techniques intéressants.

Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices

“…✷ Remark 4.1. By combining the estimation proved above for the difference between g N and the Stieltjes transform of µ σ ⊞ µ AN with some classical arguments developed in [29], one can recover the almost sure convergence of the spectral distribution of M N to the free convolution µ σ ⊞ ν. The proof still uses the ideas of [23] and [31] but, since µ σ ⊞ µ AN depends on N , we need here to apply the inverse Stieltjes tranform to functions depending on N .…”

“…✷ Remark 4.1. By combining the estimation proved above for the difference between g N and the Stieltjes transform of µ σ ⊞ µ AN with some classical arguments developed in [29], one can recover the almost sure convergence of the spectral distribution of M N to the free convolution µ σ ⊞ ν.…”

Rectangular random matrices, related convolution

“…the uniform distribution on the F. Benaych-Georges (B) LPMA, UPMC Univ Paris 6, Case Courier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France e-mail: florent.benaych@gmail.com set of eigenvalues) of these random matrices converges to the so-called semicircle law. This result was improved, and other results giving the asymptotic spectral law of random matrices were proved (see, among many other sources [23]). In the same time, people studied the local structure of the spectrum of random matrices (see, e.g.…”

“…We can define a distance on the set of probability measures on the real line with the Cauchy transform by (σ 1 , σ 2 ) → sup{ G σ 1 (z) − G σ 2 (z) ; z ≥ 1}. This distance defines the topology of weak convergence [1,23]. The Cauchy transform of the spectral distribution of an hermitian matrix M is the normalized trace of its resolvant R z (M) = (z − M) −1 .…”

A CLT for a band matrix model

“…The interest in the limiting properties of the empirical distribution of eigenvalues of large symmetric random matrices can be traced back to [Wis28] and to the pathbreaking article of Wigner [Wig55]. We refer to [Ba99], [De00], [HP00], [Me91] and [PL03] for partial overview and some of the recent spectacular progress in this field.…”