On the spectral distribution of Gaussian random matrices

Abstract

We consider the empirical spectral distribution (ESD) of a random matrix from the Gaussian Unitary Ensemble. Based on the Plancherel-Rotach approximation formula for Hermite polynomials, we prove that the expected empirical spectral distribution converges at the rate of O(n -1) to the Wigner distribution function uniformly on every compact intervals [u, v] within the limiting support (-1, 1). Furthermore, the variance of the ESD for such an interval is proved to be (πn)-2 log n asymptotically which surprisingly enough, does not depend on the details (e. g. length or location) of the interval. This property allows us to determine completely the covariance function between the values of the ESD on two intervals. © Springer-Verlag 2006.