A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

Abstract

We present simple proofs of several basic facts of the global regime (the existence and the form of the non-random limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices, whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare - Nash inequality, allowing us to obtain the O(n - 2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter.