Mixed moments of Voiculescu's Gaussian random matrices (Q1585970)

The main result of this important paper is an explicit combinatorial formula for the mixed moments of Gaussian Hermitian matrices, i.e., for \(E(\text{tr}_n(X_{i_1}\cdots X_{i_p}))\) \((i_1,\dots, i_p\in \mathbb{N})\) where \((X_i)_{i\geq 1}\) is a sequence of independent Hermitian random \(n\times n\)-matrices. A result of D. Voiculescu then immediately implies that these moments converge with order \(O(1/n^2)\). This fact is used to prove that \(\text{tr}_n(X_{i_1}\cdots X_{i_p})\) converges almost surely. In fact, the author gives a new short proof of a more general recent almost sure result of F. Hiai and D. Petz which is stated in terms of noncommuting polynomials and noncommutative free probability theory. Moreover, an alternative proof is given of the result of S. Wasserman that free group factors embed into ultra products of matrix algebras. Finally, the almost sure result above is used to obtain asymptotic bounds for the minimum and maximum of the spectrum of \(S^*S\) for Gaussian random matrices \(S\) with operator entries.