On the statistical mechanics approach in the random matrix theory: Integrated density of states

Abstract

We consider the ensemble of random symmetric n×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit as n→∞ and that this limiting distribution is the solution of a certain self-consistent equation. © 1995 Plenum Publishing Corporation.