Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

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