Monday, May 4th, 2015
2:30pm - 3:30pm, in McCormack 1-208

Anna Lytova

University of Alberta, Canada

The Central Limit Theorems for Linear Eigenvalue Statistics for Some Ensembles of Random Matrices

Abstract: We consider ensembles of Wigner and Sample Covariance random matrices with independent or weakly dependent entries. The central object of interest is the distribution of eigenvalues $\{\lambda_j\}_{j=1}^n$ of a random nxn matrix $M_n$ as n goes to infinity. The first step in asymptotic study of the eigenvalue distribution consists in finding the limit of the normalized counting measures of eigenvalues as n goes to infinity. More generally, one is interested in finding the limit of the normalized linear eigenvalue statistic $n^{-1}\sum_{j=1}^n \varphi(\lambda_j)$ corresponding to a test-function $\varphi$. The natural second step is to find the limiting probability law for the fluctuations of linear eigenvalue statistic. It appears that for many ensembles of random matrices the limiting probability law for fluctuations of linear eigenvalue statistics is Gaussian. These two steps correspond in classical probability theory to the Law of Large Numbers and the Central Limit Theorem for sums of independent (or weakly dependent) random variables. The main difference with the classical probability theory is that here we deal with sums of strongly dependent random variables, since the eigenvalues of a random matrix are strongly dependent even if its entries are independent.