Fast approach to the Tracy-Widom law at the edge of GOE and GUE

TL;DR

This paper presents a method to achieve an optimal convergence rate for the largest eigenvalue distributions in GOE and GUE to Tracy-Widom limits, with implications for other ensembles.

Contribution

It introduces specific centering and scaling constants that improve convergence rates to the Tracy-Widom law for Gaussian ensembles.

Findings

Achieves an $O(N^{-2/3})$ convergence rate.

Constants provide good approximations for small N.

Method extends to Laguerre and Jacobi ensembles.

Abstract

We study the rate of convergence for the largest eigenvalue distributions in the Gaussian unitary and orthogonal ensembles to their Tracy-Widom limits. We show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the centering and scaling constants. The arguments here also shed light on more complicated cases of Laguerre and Jacobi ensembles, in both unitary and orthogonal versions. Numerical work shows that the suggested constants yield reasonable approximations, even for surprisingly small values of N.