Asymptotics of Characteristic Polynomials of Wigner Matrices at the Edge of the Spectrum

TL;DR

This paper studies the asymptotic behavior of the characteristic polynomial correlations of Wigner matrices at the spectrum edge, showing they converge to the Airy kernel, extending known results from GUE to more general ensembles.

Contribution

It generalizes the asymptotic correlation results at the spectrum edge from GUE to real-symmetric Wigner matrices, revealing universal behavior.

Findings

Rescaled correlation functions converge to the Airy kernel

Results apply to both Hermitian and real-symmetric Wigner matrices

Extends universality of edge behavior beyond GUE

Abstract

We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.