Steen Thorbjørnsen

Random matrices with complex Gaussian entries

Abstract In this paper we give new and purely analytical proofs of a number of classical results on the asymptotic behavior of large random matrices of complex Wigner type (the GUE-case) or of complex Wishart type: Wigners semi-circle law, the Harer-Zagier recursion formula, the Marchenko-Pastur law, the Geman-Silverstein results on the largest and smallest eigenvalues and other related results. Our approach is based on the derivation of explicit formulae for the moment generating functions for random matrices of the two considered types.

Lévy laws in free probability

This article and its sequel outline recent developments in the theory of infinite divisibility and Lévy processes in free probability, a subject area belonging to noncommutative (or quantum) probability. The present paper discusses the classes of infinitely divisible probability measures in classical and free probability, respectively, via a study of the Bercovici–Pata bijection between these classes.

Classical and Free Infinite Divisibility and Lévy Processes

The present lecture notes have grown out of a wish to understand whether certain important concepts of classical infinite divisibility and Levy processes, such as selfdecomposability and the Levy-Ito decomposition, have natural and interesting analogues in free probability.

ASYMPTOTIC EXPANSIONS FOR THE GAUSSIAN UNITARY ENSEMBLE

Let g:{\mathbb R} --> {\mathbb C} be a C^{\infty}-function with all derivatives bounded and let tr_n denote the normalized trace on the n x n matrices. In the paper [EM] Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr_n(g(X_n))} for a rather general class of random matrices X_n,including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a GUE random matrix X_n that E{tr_n(g(X_n))}= \frac{1}{2\pi}\int_{-2}^2 g(x)\sqrt{4-x^2} dx +\sum_{j=1}^k\frac{\alpha_j(g)}{n^{2j}}+ O(n^{-2k-2}), where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients \alpha_j(g), j\in{\mathbb N}, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr_n[f(X_n)],Tr_n[g(X_n)]}, where f is a function of the same kind as g, and Tr_n=n tr_n. Special focus is drawn to the case where g(x)=1/(z-x) and f(x)=1/(w-x) for non-real complex numbers z and w. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the GUE(n,1/n).

Lévy processes in free probability

This is the continuation of a previous article that studied the relationship between the classes of infinitely divisible probability measures in classical and free probability, respectively, via the Bercovici–Pata bijection. Drawing on the results of the preceding article, the present paper outlines recent developments in the theory of Lévy processes in free probability.