Random matrices: Universality of ESDs and the circular law

2008/07/30 by Terence Tao, Van Vu, Tao, Terence +3 · 20 citations
Mathematics · Physics and Astronomy · #15A52 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:15A52

paper · pdf · doi:10.48550/arxiv.0807.4898

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

openalex publication_date 2008/07/30 · arxiv created 2009/04/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an n × n complex matrix A, let μ_A(x,y):= (1)/(n) |\1≤ i ≤ n, \Re λ_i ≤ x, \Im λ_i ≤ y\| be the empirical spectral distribution (ESD) of its eigenvalues λ_i ∈ \BBC, i=1, ... n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD μ_{(1)/(√(n)) An} of a random matrix A_n = (a_ij)_{1 ≤ i,j ≤ 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 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 singular value distribution of (1)/(√(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_{(1)/(√(n)) An} converges to the uniform measure on the unit disk when the a_ij have zero mean.

Citations

• On the Distribution of the Roots of Certain Symmetric Matrices

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Related

• Random Matrices: The circular Law
• Circular law for random discrete matrices of given row sum
• From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices
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