Product of Ginibre matrices: Fuss-Catalan and Raney distributions

TL;DR

This paper derives explicit formulas for the Fuss-Catalan and Raney distributions, describing the asymptotic behavior of squared singular values of products of Ginibre matrices, generalizing classical distributions like Marchenko-Pastur and Wigner laws.

Contribution

It provides exact representations of Fuss-Catalan and Raney distributions using hypergeometric functions, extending known distributions to arbitrary parameters.

Findings

Explicit formulas for Fuss-Catalan distributions in terms of hypergeometric functions.

Exact expressions for Raney distributions as a two-parameter generalization.

Connections to classical distributions like Marchenko--Pastur and Wigner semicircle law.

Abstract

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or 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 sF_{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 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.