ON THE STATISTICAL-MECHANICS APPROACH IN THE RANDOM-MATRIX THEORY - INTEGRATED DENSITY-OF-STATES

We consider the ensemble of random symmetric n x 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 --> infinity and that this limiting distribution is the solution of a certain self-consistent equation.