RANDOM MATRICES: UNIVERSALITY OF ESDs AND THE CIRCULAR LAW

• 成果类型: Article
• 署名作者: Tao, Terence; Vu, Van; Krishnapur, Manjunath
• 署名单位: University of California System; University of California Los Angeles; Rutgers University System; Rutgers University New Brunswick; Indian Institute of Science (IISC) - Bangalore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP534
• 发表日期: 2010
• 页码: 2023-2065
• 关键词: singular-value; INVERTIBILITY; eigenvalues; inverse

摘要

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.