## Electronic Communications in Probability

### Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 35, 376-395.

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

https://projecteuclid.org/euclid.ecp/1465243978

Digital Object Identifier
doi:10.1214/ECP.v15-1568

Mathematical Reviews number (MathSciNet)
MR2726085

Zentralblatt MATH identifier
1225.15031

Rights

#### Citation

Bianchi, Pascal; Debbah, Mérouane; Najim, Jamal. Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble. Electron. Commun. Probab. 15 (2010), paper no. 35, 376--395. doi:10.1214/ECP.v15-1568. https://projecteuclid.org/euclid.ecp/1465243978

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