Random matrices: Universality of ESDs and the circular law

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dc.creatorTao, Terence
dc.creatorVu, Van
dc.creatorKrishnapur, Manjunath
dc.date2008-07-30
dc.date2009-04-23
dc.date.accessioned2026-07-07T13:07:03Z
dc.date.available2026-07-07T13:07:03Z
dc.descriptionGiven an $n \times n$ complex matrix $A$, let $$μ_{A}(x,y):= \frac{1}{n} |\{1\le i \le n, \Re λ_i \le x, \Im λ_i \le y\}|$$ 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 $μ_{\frac{1}{\sqrt{n}} A_n}$ 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 $μ_{\frac{1}{\sqrt{n}} A_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean.
dc.description45 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
dc.identifierhttps://arxiv.org/abs/0807.4898
dc.identifierhttp://arxiv.org/abs/0807.4898
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/227972
dc.subjectProbability
dc.subject15A52
dc.titleRandom matrices: Universality of ESDs and the circular law
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