Heat Kernel Empirical Laws on $\mathbb{U}_N$ and $\mathbb{GL}_N$

TL;DR

This paper investigates the behavior of eigenvalues and singular values of random matrices from heat kernel measures on unitary and general linear groups, providing new convergence results and identifying limiting distributions.

Contribution

It establishes the strongest known convergence results for eigenvalues on $_N$ and the first almost sure convergence results for eigenvalues and singular values on $ ext{GL}_N$, along with identifying their limits.

Findings

Strongest known convergence results for eigenvalues on $_N$

First almost sure convergence results for eigenvalues and singular values on $ ext{GL}_N$

Identification of the limit noncommutative distribution as a flow on an infinite-dimensional polynomial space

Abstract

This paper studies the empirical measures of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups $\mathbb{U}_N$ and the general linear groups $\mathbb{GL}_N$, for $N\in\mathbb{N}$. It establishes the strongest known convergence results for the empirical eigenvalues in the $\mathbb{U}_N$ case, and the first known almost sure convergence results for the eigenvalues and singular values in the $\mathbb{GL}_N$ case. The limit noncommutative distribution associated to the heat kernel measure on $\mathbb{GL}_N$ is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to $L^p$ estimates for even integers $p$.