Asymptotic Independence of the Extreme Eigenvalues of GUE

TL;DR

This paper provides an operator-theoretic proof of the asymptotic independence of the minimal and maximal eigenvalues in GUE, revealing specific constants that distinguish GUE from other Wigner ensembles.

Contribution

It introduces a novel operator-based proof for eigenvalue independence in GUE and identifies a specific constant in their correlation decay.

Findings

Asymptotic independence of extreme eigenvalues in GUE proven

Correlation between eigenvalues decays as n^{-2/3}/4σ^2

Constant in decay rate distinguishes GUE from other ensembles

Abstract

We give a short, operator-theoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n \times n Gaussian Unitary Ensemble in the large matrix limit n \to \infty. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n^{-2/3}/4\sigma^2, where \sigma^2 denotes the variance of the Tracy--Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random hermitian matrix ensembles in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian Unitary Ensemble statistically from other Wigner ensembles.