Central limit theorem for linear eigenvalue statistics of orthogonally invariant matrix models

Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models

We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n×n Hermitian matrices as n → ∞. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of ...

Fluctuations for Analytic Test Functions in the Single Ring Theorem

We consider a non-Hermitian random matrix A whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni [31], states that the empirical eigenvalue distribution of A converges to a limit measure supported by an annulus S. In this text, we establish the convergence in distribution of random v...

Central and Local Limit Theories in Markov Dependent Random Variables

Extreme eigenvalue fluctutations for GUE

Random matrices were introduced in multivariate statistics, in the thirties by Wishart [Wis] and in theoretical physics by Wigner [Wig] in the fifties. Since then, the theory developed in a wide range of mathematics fields and physical mathematics. These lectures give a brief introduction to a well studied model : the Gaussian Unitary ensemble (GUE). The GUE is both a Wigner matrix (independent...

Central Limit Theorem for Traces of Large Random Symmetric Matrices with Independent Matrix Elements

We study Wigner ensembles of symmetric random matrices A = (a ij) i; j = 1; : : : ; n with matrix elements a ij ; i j being independent symmetrically distributed random variables a ji = ij n 1 2 : We assume that Var ij = 1 4 , for i < j, Var ii 6const and that all higher moments of ij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit t...