Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

TL;DR

This paper proves that eigenvalue counts in disjoint regions of GUE matrices become independent as matrix size grows, and explores related asymptotic independence of extreme eigenvalues and fluctuations of the condition number.

Contribution

It establishes asymptotic independence of eigenvalue counts in disjoint sets and of extreme eigenvalues in GUE matrices, extending understanding of spectral behavior.

Findings

Eigenvalue counts in disjoint regions are asymptotically independent.

Largest and smallest eigenvalues are asymptotically independent.

Fluctuations of the GUE matrix condition number are characterized.

Abstract

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$, where ${\mathcal N}_n(\Delta)$ represents the number of eigenvalues within $\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.