Limit laws for Random matrices and free products

Abstract

In earlier articles we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. We prove here that free random variables naturally arise as limits of random matrices and that Wigner's semicircle law is a consequence of the central limit theorem for free random variables. In this way we obtain a non-commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra, Wigner's law being given by the trace of the spectral measure of the selfadjoint component of this operator