Asymptotic independence in the spectrum of the gaussian unitary ensemble

Consider a nn matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets (Δ_{i, n}, 1 ≥ i ≥ 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 we prove that the related counting measures N_n(Δ_{i, n}), (1 ≤ i ≤ P), where 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 ratio of the extreme eigenvalues of a matrix from the GUE.