期刊
ANNALS OF PROBABILITY
卷 38, 期 5, 页码 2023-2065出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/10-AOP534
关键词
Circular law; eigenvalues; random matrices; universality
资金
- MacArthur Foundation
- NSF [DMS-06-49473, DMS-09-01216]
- DOD [AFOSAR-FA-9550-09-1-0167]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1212424] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0901216] Funding Source: National Science Foundation
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.
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