Equidistribution of points in the Harmonic ensemble for the Wasserstein distance

TL;DR

This paper analyzes the convergence rates of the Wasserstein distance between empirical measures and background volume forms for various point processes, highlighting optimal rates for the harmonic ensemble on certain manifolds.

Contribution

It establishes the optimal convergence rates of the Wasserstein distance for the harmonic ensemble and other point processes on specific manifolds, advancing understanding of point distribution uniformity.

Findings

Optimal convergence rate for harmonic ensemble on homogeneous manifolds of dimension d≥3

Optimal rate for the spherical ensemble and zeros of Gaussian Analytic Functions

Discussion of variations of the process on the torus

Abstract

We study the asymptotics of the expected Wasserstein distance between the empirical measure of a Point Process and the background volume form. The main DPP studied is the harmonic ensemble, where we get the optimal rate of convergence for homogeneous manifolds of dimension $d\geq 3$, and for two-point homogeneous manifolds. We also discuss some variations of this process on the torus. Regarding other point processes, we find the optimal rate for the spherical ensemble and the zeros of Gaussian Analytic Functions.