The Annals of Probability

Random matrices: Universality of ESDs and the circular law

Terence Tao, Van Vu, and Manjunath Krishnapur

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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.

Article information

Ann. Probab. Volume 38, Number 5 (2010), 2023-2065.

First available in Project Euclid: 17 August 2010

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Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 15A52 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems

Circular law eigenvalues random matrices universality


Tao, Terence; Vu, Van; Krishnapur, Manjunath. Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 (2010), no. 5, 2023--2065. doi:10.1214/10-AOP534.

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