Singular value statistics of matrix products with truncated unitary matrices

TL;DR

This paper analyzes the singular value distribution of matrix products involving truncated Haar unitary matrices, revealing polynomial ensemble structures and deriving explicit joint density formulas with applications to scaling limits.

Contribution

It proves that the squared singular values form a polynomial ensemble and derives the joint density and correlation kernel for products of truncated unitary matrices.

Findings

Squared singular values follow a polynomial ensemble.

Derived explicit joint density and correlation kernel.

Identified new finite rank perturbations of Meijer G-kernels.

Abstract

We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix with a random matrix. We show that the structure of polynomial ensembles and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint singular value density of a product of truncated unitary matrices and its corresponding correlation kernel which can be written as a double contour integral. This leads to hard edge scaling limits that also include new finite rank perturbations of the Meijer G-kernels found for products of complex Ginibre random matrices.