Free products of large random matrices - a short review of recent developments

TL;DR

This paper reviews recent methods for calculating eigenvalue distributions of large random matrix products, including non-Hermitian cases, with practical examples involving Gaussian and Ginibre matrices.

Contribution

It introduces a generalization of free multiplication law to non-Hermitian matrices and demonstrates its application through explicit eigenvalue density calculations.

Findings

Eigenvalue densities for products of GUE and Ginibre matrices computed

Extension of free multiplication law to non-Hermitian matrices demonstrated

Practical methods for eigenvalue distribution analysis provided

Abstract

We review methods to calculate eigenvalue distributions of products of large random matrices. We discuss a generalization of the law of free multiplication to non-Hermitian matrices and give a couple of examples illustrating how to use these methods in practice. In particular we calculate eigenvalue densities of products of Gaussian Hermitian and non-Hermitian matrices including combinations of GUE and Ginibre matrices.