Quantitative equidistribution of eigenvalues of Random Normal Matrices in the Wasserstein distance

TL;DR

This paper investigates how closely the eigenvalues of random normal matrices and certain point processes approximate their theoretical limits using the 2-Wasserstein distance, introducing a smoothing technique based on the heat equation.

Contribution

It develops a new smoothing method via the heat equation to analyze Wasserstein distances for eigenvalues of random matrices and point processes under broad conditions.

Findings

Established bounds on Wasserstein distances for eigenvalues

Applied smoothing technique to various point processes

Provided quantitative measures of equidistribution

Abstract

The object of study in this paper is the expected $2$-Wasserstein distance between the empirical measures of several point processes and their respective limit. For this, the main tool developed is a smoothing procedure in Euclidean spaces using the heat equation with Neumann boundary conditions. It is applied to the spectrum of Random Normal Matrices with \textit{reasonable} assumptions, as well as to several families of Homogeneous Point Processes such as the infinite Ginibre ensemble, the Bessel ensemble, and the zero set of the planar Gaussian Analytic Function.