The Annals of Probability
- Ann. Probab.
- Volume 38, Number 5 (2010), 2023-2065.
Random matrices: Universality of ESDs and the circular law
Given an n×n complex matrix A, let
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 of a random matrix An=(aij)1≤i, j≤n, where the random variables aij−E(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 for complex z.
As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that converges to the uniform measure on the unit disc when the aij have zero mean.
Ann. Probab. Volume 38, Number 5 (2010), 2023-2065.
First available in Project Euclid: 17 August 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 15A52 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems
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. https://projecteuclid.org/euclid.aop/1282053780.