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 $({\Delta }_{i,n},1\le i\le p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\mathcal{N}}_n({\Delta }_{i,n}),1\le i\le p)$, 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.

15 pages

Cited by in corpus (1)

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