Product of Ginibre matrices: Fuss-Catalan and Raney distributions

Abstract

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.