Random matrices: Universality of ESDs and the circular law

TL;DR

This paper proves a universality principle for the eigenvalue distributions of large random matrices, showing they converge to the circular law regardless of the distribution of entries, and connects this to singular value laws.

Contribution

It establishes the universality of the empirical spectral distribution for large random matrices with iid entries, confirming the circular law conjecture.

Findings

Eigenvalue distributions are universal across different entry distributions.

The circular law holds for matrices with zero-mean entries.

Distribution convergence is linked to singular value laws.

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 $\lambda_i \in \BBC, i=1, ... n$. 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 $A_n = (a_{ij})_{1 \leq i,j \leq n}$ 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.