Asymptotic freeness of Gaussian random matrices

Abstract

In this chapter we shall introduce a principal object of study: Gaussian random matrices. This is one of the few ensembles of random matrices for which one can do explicit calculations of the eigenvalue distribution. For this reason the Gaussian ensemble is one of the best understood. Information about the distribution of the eigenvalues is carried by it moments: {E(tr(Xk))}k where E is the expectation, tr denotes the normalized trace (i.e. tr(IN) = 1), and X is an N × N random matrix.