Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices

Abstract

We present a new approach of asymptotic freeness in mean and almost everywhere for independent Gaussian or Wishart complex matrices together with some independent set {A i | i ∈ N*} of random matrices. This approach is based on the convergence of the generalized moments together with an iterative argument on the number of involved Gaussian or Wishart matrices. In the Gaussian case, a first proof is realized without any counting argument. Nevertheless in both cases, a precise computation of these moments leads asymptotically to the convolution relation set up by Speicher and Nica between the moments of free variables. We draw a sufficient condition on Hermitian matricial models for asymptotic freeness. The asymptotic freeness is then proved between independent Beta and Wishart matrices, and the limiting eigenvalues distribution of the last ones is determined.