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Universality at the edge of the spectrum in Wigner random matrices. (English) Zbl 1062.82502

Summary: 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 C. A. Tracy and H. Widom in 1994 [Commun. Math. Phys. 159, No. 1, 151–174 (1994; Zbl 0789.35152)] (for Hermitian matrices)in the G.U.E. case and in 1996 [Commun. Math. Phys. 177, No. 3, 727–754 (1996; Zbl 0851.60101)] (for symmetric matrices) in G.O.E. case.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60B99 Probability theory on algebraic and topological structures
60F99 Limit theorems in probability theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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