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

Moderate deviations for the eigenvalue counting function of Wigner matrices

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUEmatrices due to Gustavsson. The extension to certain families of Wigner matrices is based on the Tao and Vu Four Moment Theorem and applies localization results by Erdös, Yau and Yin. Mor...

Central and Local Limit Theories in Markov Dependent Random Variables

Limit Theorems for Spectra of Random Matrices with Martingale Structure

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semi-circular law.

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...

Properties of eigenvalue function

For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.