Berry–Esseen-rate question for edge universality of Wigner matrices

Determine whether the convergence in distribution of the rescaled largest eigenvalue N^{2/3}(λ_1 − 2) of an N×N Wigner matrix to the Tracy–Widom TW_β law occurs at Kolmogorov–Smirnov distance O(N^{-1}), i.e., at the Berry–Esseen “square-root” rate in the matrix size. Concretely, quantify whether the Kolmogorov–Smirnov distance between the law of N^{2/3}(λ_1 − 2) and TW_β decays as N^{-1} for real symmetric (β=1) and complex Hermitian (β=2) Wigner ensembles.

Background

The Tracy–Widom distribution TW_β describes the fluctuations of the largest eigenvalue of Wigner matrices after edge scaling: N^{2/3}(λ_1 − 2) ⇒ TW_β. In classical probability, the Berry–Esseen theorem provides a universal N^{-1/2} rate for convergence in the central limit theorem, measured in Kolmogorov–Smirnov distance. An analogous quantitative question at the spectral edge asks whether a Berry–Esseen-type rate can hold for the convergence to TW_β.

Prior results show rates derived from exact formulas for the Gaussian ensembles (GOE/GUE) around O(N^{-2/3}), and for general Wigner matrices the sharpest available rate to Tracy–Widom is O(N^{-1/3}). The paper develops edge homogenization for Dyson Brownian motion and obtains optimal control throughout the spectrum, including an N^{-1} rate for the top-gap distribution, but it also presents lower bounds indicating model-dependent limitations for the largest eigenvalue when compared directly to Gaussian ensembles. The overarching question remains whether the TW_β convergence for general Wigner matrices can achieve the Berry–Esseen-type N^{-1} rate.