Multivariate Fuss-Narayana polynomials and their application to random matrices

TL;DR

This paper introduces multivariate Fuss-Narayana polynomials derived from noncrossing partition combinatorics and applies them to compute moments in free probability involving random matrices.

Contribution

It explicitly determines multivariate Fuss-Narayana polynomials and links them to limit moments of products of rectangular random matrices, advancing free probability theory.

Findings

Explicit formulas for limit moments of matrix products

Connection between Fuss-Narayana polynomials and noncrossing partitions

Calculation of moments for free multiplicative convolutions

Abstract

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where B(n) is a product of $p$ independent rectangular random matrices, are certain homogenous polynomials in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as multivariate Fuss-Narayana polynomials. Using this result, we compute the moments of the n-fold free multiplicative convolution of Marchenko-Pastur distributions with arbitrary shape parameters.