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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 anyneighbourhood of the support of Wigner's semi-circle law, we prove that therelated 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 finallydescribe the fluctuations of the condition number of a matrix from the GUE.

Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble