Recent exact and asymptotic results for products of independent random matrices

TL;DR

This review summarizes recent advances in understanding the eigenvalues and singular values of products of independent random matrices, highlighting exact formulas, asymptotic behaviors, and universality classes.

Contribution

It provides a comprehensive overview of new exact and asymptotic results for various ensembles of random matrices, including universality classes and limiting distributions.

Findings

Exact determinantal and Pfaffian formulas derived

New microscopic universality classes identified

Lyapunov and stability exponents follow normal distribution in the limit

Abstract

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are given in terms of the respective kernels of orthogonal polynomials or functions. Here we list all known cases and some straightforward generalisations. The asymptotic results for large matrix size include new microscopic universality classes at the origin and a generalisation of weak non-unitarity close to the unit circle. So far in all other parts of the spectrum the known standard universality classes have been identified. In the limit of infinite products the Lyapunov and stability exponents share the same normal distribution. To leading order they both follow a permanental point processes. Our focus is on presenting recent developments in this rapidly evolving area of research.