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September 2010 Random matrices: Universality of ESDs and the circular law
Terence Tao, Van Vu, Manjunath Krishnapur
Ann. Probab. 38(5): 2023-2065 (September 2010). DOI: 10.1214/10-AOP534


Given an n×n complex matrix A, let $$\mu_{A}(x,y):=\frac{1}{n}|\{1\le i\le n,\operatorname{Re}\lambda_{i}\le x,\operatorname{Im}\lambda_{i}\le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues λi∈ℂ, i=1, …, n.

We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $\mu_{{1}/{\sqrt{n}}A_{n}}$ of a random matrix An=(aij)1≤i, jn, where the random variables aijE(aij) are i.i.d. 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 independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of $\frac{1}{\sqrt{n}}A_{n}-zI$ for complex z.

As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that $\mu_{{1}/{\sqrt{n}}A_{n}}$ converges to the uniform measure on the unit disc when the aij have zero mean.


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Terence Tao. Van Vu. Manjunath Krishnapur. "Random matrices: Universality of ESDs and the circular law." Ann. Probab. 38 (5) 2023 - 2065, September 2010.


Published: September 2010
First available in Project Euclid: 17 August 2010

zbMATH: 1203.15025
MathSciNet: MR2722794
Digital Object Identifier: 10.1214/10-AOP534

Primary: 15A52 , 60F17
Secondary: 60F15

Keywords: circular law , Eigenvalues , random matrices , Universality

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 5 • September 2010
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