Random matrices: the four-moment theorem for Wigner ensembles

Summary

We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focussing in particular on the four-moment theorem and its applications.

1. Introduction

This paper surveys the four-moment theorem and its applications in understanding the asymptotic spectral properties of random matrix ensembles of Wigner type. Due to limitations of space, this survey will be far from exhaustive; an extended version will appear elsewhere. (See also [Erdʺos 2011; Guionnet 2011; Schlein 2011] for some recent surveys in this area.)

To simplify the exposition (at the expense of stating the results in maximum generality), we shall restrict attention to a model class of random matrix ensembles, in which we assume somewhat more decay and identical distribution hypotheses than are strictly necessary for the main results.

Definition 1 (Wigner matrices). Let n ≥ 1 be an integer (which we view as a parameter going off to infinity). An n × n Wigner Hermitian matrix M_n is defined to be a random Hermitian n × n matrix M_n = (ξ_ij1≤i, j≤n), in which the ξ_ij for 1 ≤ i ≤ j ≤ n are jointly independent with ξ_ij ≡ ξ_ij (in particular, the _ii are real-valued). For 1 ≤ i < j ≤ n, we require that the ij have mean zero and variance one, while for 1 ≤ i = j ≤ n we require that the _ij have mean zero and variance σ² for some σ² > 0 independent of i, j, n. To simplify some of the statements of the results here, we will also assume that the ξ_ij ≡ ξ_ij are identically distributed for i < j, and the ξ_ij ≡ ξ' are also identically distributed for i ≡ j, and furthermore that the real and imaginary parts of ξ are independent. We refer to the distributions Reξ , Im ξ, and ξ' as the atom distributions of M_n.