Mesoscopic Rates of Convergence for Hermitian Unitary Ensembles

TL;DR

This paper establishes mesoscopic convergence rates in Wasserstein distance for eigenvalue processes of major Hermitian unitary ensembles, analyzing bulk and edge behaviors to compare finite and limiting spectra.

Contribution

It provides the first mesoscopic convergence rates for eigenvalue distributions of GUE, LUE, and JUE ensembles in various spectral regions.

Findings

Derived Wasserstein convergence rates for GUE, LUE, JUE

Controlled trace class norms of DPP kernels

Compared point counts between finite and limiting processes

Abstract

This paper provides mesoscopic rates of convergence (ROC) with respect to the $L^1$-Wasserstein distance for the eigenvalue determinantal point processes (DPPs) from the three major Hermitian unitary ensembles, the Gaussian Unitary Ensemble (GUE), the Laguerre Unitary Ensemble (LUE), and the Jacobi Unitary Ensemble (JUE) to their limiting point processes. We prove ROCs for the bulk of the GUE spectrum, the hard edge of the LUE spectrum, and the soft edges of the GUE, LUE, and JUE spectrums. These results are called mesoscopic because we are able to directly compare the point counts between the converging and limit DPPs in a range of scales. We are able to achieve these results by controlling the trace class norm of the integral operators determined by the DPP kernels.