A note on the limiting mean distribution of singular values for products of two Wishart random matrices

TL;DR

This paper derives the limiting distribution of squared singular values for the product of two complex Gaussian matrices, revealing a connection to multiple orthogonal polynomials and their asymptotic zero distribution.

Contribution

It identifies the biorthogonal polynomials as special cases of multiple orthogonal polynomials related to Macdonald functions, enabling explicit asymptotic analysis.

Findings

Explicit distribution formula for large matrix dimensions

Connection to multiple orthogonal polynomials and Bessel functions

Asymptotic zero distribution results applied

Abstract

The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M=2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.