Spectral Densities of Singular Values of Products of Gaussian and Truncated Unitary Random Matrices

TL;DR

This paper analyzes the limiting spectral densities of squared singular values of products involving Gaussian and truncated unitary matrices, introducing new integral representations and explicit formulas for special cases.

Contribution

It develops a novel approach for deriving integral representations of spectral densities and provides explicit formulas for cases with at most one Gaussian factor.

Findings

Derived complex integral representations for densities.

Obtained explicit formulas in special cases.

Analyzed boundary behavior of densities.

Abstract

We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.