Random matrices: Universality of ESDs and the circular law

Abstract

Given an n × n complex matrix A, letbe the empirical spectral distribution (ESD) of its eigenvalues λ i ∈ C, i = 1, . . . n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD µ 1 √ n An of a random matrix A n = (a ij ) 1≤i,j≤n where the random variables a ij − E(a ij ) are iid copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1

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