Mixed Moments for the Product of Ginibre Matrices

TL;DR

This paper analyzes the spectral properties of products of Ginibre matrices, revealing that their mixed moments at large size are governed by non-crossing pairings and Fuss-Catalan numbers, with the ensemble exhibiting Gaussian behavior.

Contribution

It introduces a new approach to compute mixed moments of Ginibre matrix products, connecting them to non-crossing pairings and Fuss-Catalan combinatorics.

Findings

Mixed moments at large N are given by non-crossing pairings.

The ensemble of matrix products is Gaussian with a variance averaged over a multi-Wishart ensemble.

Fuss-Catalan numbers weight the enumeration of pairings in the moments calculation.

Abstract

We study the ensemble of a product of n complex Gaussian i.i.d. matrices. We find this ensemble is Gaussian with a variance matrix which is averaged over a multi-Wishart ensemble. We compute the mixed moments and find that at large $N$, they are given by an enumeration of non-crossing pairings weighted by Fuss-Catalan numbers.