Random Matrices: Universality of Esds and the Circular Law

Given an n× n complex matrix A, let μA(x, y) := 1 n |{1 ≤ i ≤ n,Reλi ≤ x, Imλi ≤ y}| be the empirical spectral distribution (ESD) of its eigenvalues λi ∈ 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 An = (aij)1≤i,j≤n where the random variables aij − E(aij) 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 √ n An − zI for complex z. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that μ 1 √ n An converges to the uniform measure on the unit disk when the aij have zero mean.