Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

2008/11/06 by P. Bianchi, Bianchi, P., M. Debbah +3
Mathematics · Physics and Astronomy · #15A52 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:15A52

paper · pdf · doi:10.48550/arxiv.0811.0979

15 pages

arxiv created 2008/11/06 · arxiv updated 2009/12/01

Abstract

Consider a n × n matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets (Δ_i,n, 1≤ i≤ 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(Δ_i,n), 1≤ i≤ p), where \mathcal N_n(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 condition number of a matrix from the GUE.

Related

• Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble
• Generalized Gaussian Random Unitary Matrices Ensemble
• Gaussian fluctuations and moderate deviations of eigenvalues in unitary invariant ensembles
• Bulk universality for generalized Wigner matrices
• On Universality of Bulk Local Regime of the Deformed Gaussian Unitary Ensemble