Correlation kernels for sums and products of random matrices

TL;DR

This paper derives explicit formulas for the correlation kernels of squared singular values and eigenvalues in various random matrix models involving products and sums, extending polynomial ensemble characterizations.

Contribution

It provides new double contour integral formulas for correlation kernels of singular values and eigenvalues in complex matrix products and sums, and characterizes conditions for polynomial ensemble structures.

Findings

Derived formulas for correlation kernels of products involving Ginibre and truncated unitary matrices.

Identified conditions under which the squared singular values form a polynomial ensemble.

Connected eigenvalue distributions of matrix sums to the two-matrix model.

Abstract

Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum $H + M$ where $H$ is a GUE matrix and $M$ is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of $H + M$ follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.