Random matrices: Universality of ESDs and the circular law

• Van Vu (Rutgers University)
• Thursday 05 February 2009, 14:00-15:00
• MR12.

Given an n by n complex matrix $A$, let \mu(A) be the empirical spectral distribution (ESD) of its eigenvalues $\lambda_i, i=1, ..., n$.

We consider the limiting distribution of the normalized ESD $\mu_{\frac{1}{\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})$ where the random variables $a{ij} – \E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove the universality principle”, 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 corollary we establish the Circular Law conjecture in full generality. The proof combines ideas from several areas of mathematics: additive combinatorics, theoretical computer science, probability and high dimensional geometry.

Joint work with Terence Tao.

This talk is part of the Combinatorics Seminar series.