Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble
Pascal Bianchi, Télécom Paristech
Mérouane Debbah, Alcatel-Lucent chair on flexible radio, SUPELEC
Jamal Najim, CNRS and Télécom Paristech
Abstract
Consider a n×n matrix from the Gaussian Unitary Ensemble
(GUE). Given a finite collection of bounded disjoint real Borel sets
(Δi,n, 1≤ i≤ 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 Νn(Δi,n),
(1≤ i≤ p), where Νn(Δ) represents the
number of eigenvalues within Δ, 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.
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