Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Abstract

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$ with positive distance from one another, eventually included in any neighbourhood of the support of Wigner's semi-circle law and properly rescaled (with respective lengths $n^{-1}$ in the bulk and $n^{-2/3}$ around the edges), 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 ratio of the extreme eigenvalues of a matrix from the GUE.