Jacobi polynomial moments and products of random matrices

TL;DR

This paper explores the distributions from products of complex Gaussian and Haar unitary matrices, characterizing them through Jacobi polynomial moments and connecting to free probability theory.

Contribution

It introduces a general class of measures characterized by Jacobi polynomial moments and links algebraic equations to random matrix distributions.

Findings

Characterization of matrix product distributions via Jacobi polynomial moments

Solution of a moment problem using Riemann surface analysis

Establishment of connections to free probability theory

Abstract

Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class of measures and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters. Solving this moment problem requires a study of the Riemann surfaces associated to a class of algebraic equations. The connection to random matrix theory is then established using methods from free probability.