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# Random matrices: Universality of ESDs and the circular law

Tao, Terence and Vu, Van and Krishnapur, Manjunath (2010) Random matrices: Universality of ESDs and the circular law. In: Annals of Probability, 38 (5). pp. 2023-2065.

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## Abstract

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1/root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) 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 1/root 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/root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.

Item Type: Journal Article Copyright of this article belong to Institute of Mathematical Statistics. Circular law;eigenvalues;random matrices;universality. Division of Physical & Mathematical Sciences > Mathematics Id for Latest eprints 20 Sep 2010 07:45 20 Sep 2010 07:45 http://eprints.iisc.ac.in/id/eprint/32260 View Item