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Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

   
@article{ECP1568,
	author = {Pascal Bianchi and Mérouane Debbah and Jamal Najim},
	title = {Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {15},
	year = {2010},
	keywords = {Random matrix; eigenvalues; asymptotic independence; 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.},
	pages = {no. 35, 376-395},
	issn = {1083-589X},
	doi = {10.1214/ECP.v15-1568},    
        url = {http://ecp.ejpecp.org/article/view/1568}}