Semicircle law and freeness for random matrices with symmetries or correlations

Abstract

For a class of random matrix ensembles with corre- lated matrix elements, it is shown that the density of states is given by the Wigner semi-circle law. This is applied to effective Hamiltonians related to the Anderson model in dimensions greater than or equal to two. 1. The Result It is a classical theorem due to Wigner (Wig) that the density of states of a growing sequence of real symmetric matrices with inde- pendent entries converges in distribution to the semi-circle law. More precisely, this means the following: Consider an n × n random matrix Xn = ( 1 √ n an(p, q))1≤p,q≤n where, apart from the symmetry condition, the entries an(p, q) are independent centered random variables with unit variance (and a growth condition on their moments). Then the expectation value of the moment of Xn of order k ≥ 0 satisfies