Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices

摘要

It is a classical result of Wigner that for an hermitian matrix withindependent entries on and above the diagonal, the mean empirical eigenvaluedistribution converges weakly to the semicircle law as matrix size tends toinfinity. In this paper, we prove analogs of Wigner's theorem for randommatrices taken from all infinitesimal versions of classical symmetric spaces.This is a class of models which contains those studied by Wigner and Dyson,along with seven others arising in condensed matter physics. Like Wigner's, ourresults are universal in that they only depend on certain assumptions about themoments of the matrix entries, but not on the specifics of their distributions.What is more, we allow for a certain amount of dependence among the matrixentries, in the spirit of a recent generalization of Wigner's theorem, due toSchenker and Schulz-Baldes. As a byproduct, we obtain a universality result forsample covariance matrices with dependent entries.