Singular values of products of random matrices and polynomial ensembles

TL;DR

This paper explores the singular values of products of random matrices, introducing polynomial ensembles, deriving new integral representations, and demonstrating the universality of certain scaling limits across different matrix ensembles.

Contribution

It introduces polynomial ensembles as a framework for analyzing singular values of matrix products and derives new integral formulas and universality results for their scaling limits.

Findings

Squared singular values form polynomial ensembles.

Derived double integral representation for correlation kernels.

Identified universal scaling limits at the hard edge.

Abstract

Akemann, Ipsen, and Kieburg showed recently that the squared singular values of a product of M complex Ginibre matrices are distributed according to a determinantal point process. We introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. We also show that the squared singular values of the product of M-1 complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and we derive a double integral representation for the correlation kernel associated with this ensemble. We use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found by Kuijlaars and Zhang for the squared singular values of a product of M complex Ginibre matrices. Our final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles of Borodin with parameter theta > 0, in case theta or 1/theta is an integer. This further supports the conjecture that these kernels have a universal character.