Products of Random Matrices from Polynomial Ensembles

TL;DR

This paper explores the spectral properties of products of bi-unitarily invariant random matrices, deriving transformation formulas for their joint densities and kernels, and connecting to Lyapunov exponents in large matrix products.

Contribution

It introduces a transformation formula for joint densities of products of polynomial ensemble matrices and generalizes existing results in random matrix theory.

Findings

Derived a formula for joint densities of matrix products.

Constructed a family of ensembles interpolating between different matrix products.

Connected spectral distributions to Lyapunov exponents in large matrix limits.

Abstract

Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ensemble and the second from a polynomial ensemble of derivative type. This allows us to re-derive and generalize a number of recent results in random matrix theory, including a transformation formula for the kernels of the corresponding determinantal point processes. Starting from these results, we construct a continuous family of random matrix ensembles interpolating between the products of different numbers of Ginibre matrices and inverse Ginibre matrices. Furthermore, we make contact to the asymptotic distribution of the Lyapunov exponents of the products of a large number of bi-unitarily invariant random matrices of fixed dimension.