Closed Form of the Asymptotic Spectral Distribution of Random Matrices Using Free Independence

Abstract

The spectral distribution function of random matrices is an information-carrying object widely studied within Random matrix theory. Random matrix theory is the main eld placing its research interest in the diverse properties of matrices, with a particular emphasis placed on eigenvalue distribution. The aim of this article is to point out some classes of matrices, which have closed form expressions for the asymptotic spectral distribution function. We consider matrices, later denoted by , which can be decomposed into the sum of asymptotically free independent summands. Let  be a probability space. We consider the particular example of a non-commutative space, where  denotes the set of all   random matrices, with entries which are com-plex random variables with finite moments of any order and  is tracial functional. In particular, explicit calculations are performed in order to generalize the theorem given in [15] and illustrate the use of asymptotic free independence to obtain the asymptotic spectral distribution for a particular form of matrix. Finally, the main result is a new theorem pointing out classes of the matrix  which leads to a closed formula for the asymptotic spectral distribution. Formulation of results for matrices with inverse Stieltjes transforms, with respect to the composition, given by a ratio of 1st and 2nd degree polynomials, is provided