Mesoscopic Rates of Convergence for Hermitian Unitary Ensembles
Mengchun Cai, Kyle Taljan
TL;DR
This work establishes mesoscopic convergence rates for eigenvalue DPPs from the GUE, LUE, and JUE to their limiting processes under the $W_1$-Wasserstein distance. It develops a trace-class–norm framework that converts kernel differences into counts of eigenvalues, enabling explicit, scale-aware ROC bounds at bulk, hard-edge, and soft-edge regimes. The authors prove precise rates: $O(N^{-1})$ in the GUE bulk, $O(N^{-2})$ at the LUE hard edge, and $O(N^{-2/3})$ at soft edges across the ensembles, with accompanying distributional implications for extreme eigenvalues. The results integrate kernel decompositions and special-function asymptotics to connect finite-$N$ ensembles with sine, Airy, and Bessel limit processes, contributing a rigorous mesoscopic perspective to random matrix convergence theory.
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.