Universality at the edge of the spectrum in Wigner random matrices

Abstract

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit $n\to +\infty$. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.

We corrected several misprints and little mistakes (the most important one is formula (1.15)). We also reformulated two auxiliary theorems (Theorem 2 and Theorem 3) to emphasize the asymptotic nature of the results