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 Electronic Communications in Probability > Vol. 15(2010) > Paper 35 open journal systems 


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.


Full text: PDF

Pages: 376-395

Published on: September 26, 2010
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Electronic Communications in Probability. ISSN: 1083-589X