Random matrices: Universality of ESDs and the circular law

TLDR: In this article, the authors considered the limiting distribution of the normalized ESD of a random matrix An, where the random variables aij−E(aij) are i.i.d. copies of a fixed random variable x with unit variance.

Abstract: Given an n×n complex matrix A, let $$\mu_{A}(x,y):=\frac{1}{n}|\{1\le i\le n,\operatorname{Re}\lambda_{i}\le x,\operatorname{Im}\lambda_{i}\le y\}|$$ 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 $\mu_{{1}/{\sqrt{n}}A_{n}}$ of a random matrix An=(aij)1≤i, j≤n, where the random variables aij−E(aij) are i.i.d. 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 or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of $\frac{1}{\sqrt{n}}A_{n}-zI$ for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that $\mu_{{1}/{\sqrt{n}}A_{n}}$ converges to the uniform measure on the unit disc when the aij have zero mean.