Limit Theorems for Random Matrices

A. Boutet de Monvel, A. Khorunzhy

1998, v.4, Issue 2, 175-197

ABSTRACT

We consider ensembles of random real symmetric $N\times N$ matrices $H_N$ whose entries are weakly dependent Gaussian random variables with the covariance matrix $V$. We study convergence of the normalized eigenvalue counting function $\sigma(\lambda;H_N )$ to a non-random limit as $N\to\infty $. We prove that if $g_N(z)$ is the Stieltjes transform of $\sigma(\lambda;H_N)$, and $|\Im z|\geq w_ 0$ for a positive and $N$-independent $w_0$, then the centered random variable $N[g_N(z) - E g_N(z)]$ converges in distribution to a Gaussian random variable as $N\to\infty $. We find explicitly the leading term of the correlator $$E g_N(z_1) g_N(z_2) - E g_N(z_1) E g_N(z_2) $$ and show that in the ``scaling'' limit this term converges to the expression that does not depend on the particular form of $V$. This can be regarded as an evidence that the ensembles of random matrices with weakly dependent entries belong to the universality class of the Gaussian Orthogonal Ensemble.

Keywords: random matrices,eigenvalue distribution,strong self-averaging,universality conjecture

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