Random matrices: Universality of ESDs and the circular law

Abstract

Given an $n\times n$ complex matrix $A$, let $${\mu }_A(x,y):=\frac{1}{n}|\{1\le i\le n,\Re {\lambda }_i\le x,\Im {\lambda }_i\le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $λ_i \in \BBC, i=1, ... n$. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD ${\mu }_{\frac{1}{\sqrt{n}}{A}_n}$ of a random matrix $A_n=(a_{ij}{)}_{1\le i,j\le n}$ where the random variables $a_{ij} - \E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove a \emph{universality principle} for such ensembles, namely that the limit distribution in question is {\it 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 \emph{singular} value distribution of $\frac{1}{\sqrt{n}}{A}_n-zI$ for complex $z$. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that ${\mu }_{\frac{1}{\sqrt{n}}{A}_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean.

45 pages, 8 figures, submitted, Acta Math. The main article is by Tao and Vu, the appendix is by Krishnapur, and the figures are by Phillip Wood. A simplified proof of the replacement principle added; some other corrections

References in corpus (2)

• A simple invariance theorem
• Circular law, Extreme Singular values and Potential theory

Cited by in corpus (1)

• Random matrices: The distribution of the smallest singular values