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2010 Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble
Pascal Bianchi, Mérouane Debbah, Jamal Najim
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Electron. Commun. Probab. 15: 376-395 (2010). DOI: 10.1214/ECP.v15-1568


Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$ 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 ${\mathcal N}_n(\Delta_{i,n}), (1\leq i\leq p)$, 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 ratio of the extreme eigenvalues of a matrix from the GUE.


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Pascal Bianchi. Mérouane Debbah. Jamal Najim. "Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble." Electron. Commun. Probab. 15 376 - 395, 2010.


Accepted: 26 September 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1225.15031
MathSciNet: MR2726085
Digital Object Identifier: 10.1214/ECP.v15-1568

Primary: 15B52
Secondary: 15A18‎ , 60F05

Keywords: Asymptotic independence , Eigenvalues , Gaussian unitary ensemble , Random matrix

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