Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

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

2008-11-06

arxiv.org/abs/0811.0979v1

Mathematics

Probability

15 pages

Scientific paper

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\Delta_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\mathcal N}_n(\Delta_{i,n}), 1\leq i\leq p)$, 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 condition number of a matrix from the GUE.

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