Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Bianchi, P.; Debbah, M.; Najim, J.

Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Authors

Bianchi, P.

Debbah, M.

Najim, J.

Description

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