Mixed Moments of Voiculescuʹs Gaussian Random Matrices

We prove an explicit combinatorial formula for the mixed moments of a family of independent selfadjoint Gaussian random matrices. As a corollary we show that the convergence of such mixed moments, proved by Voiculescu, is as fast as O(1/n2), and we use this fact to give a short proof of the fact, recently proved by Hiai and Petz, that the convergence in Voiculescuʹs result actually holds almost surely. This latter result is then used to give an alternative proof of the result by S. Wassermann, that free group factors embed into ultra products of matrix algebras. Moreover, the almost sure convergence result is used to obtain asymptotic lower (respectively upper) bounds on the maximum (respectively minimum) of the spectrum of S*S, for Gaussian random matrices S with operator entries; in particular those studied by U. Haagerup and the author (1999, Documenta Math.4, 341–450).