Universality at the edge of the spectrum in Wigner random matrices

TL;DR

This paper proves that the eigenvalue distributions at the spectrum edge of large Wigner matrices follow universal patterns, matching Tracy-Widom distributions, regardless of matrix specifics.

Contribution

It establishes universality at the spectrum edge for Wigner matrices, extending known results to broader classes of random matrices.

Findings

Eigenvalues at the spectrum edge follow Tracy-Widom distributions.

Universality holds for both Hermitian and real symmetric Wigner matrices.

Results apply as matrix size tends to infinity.

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.