Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Publication Type:

Journal Article

Source:

Electronic Communications in Probability, Volume (15) 2010, p.376-395 (2010)

Abstract:

Consider a n × n matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets (∆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 (Nn(∆i,n), 1 ≤ i ≤ p), where Nn(∆) represents the number of eigenvalues within ∆, 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.

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