On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble

TL;DR

This paper investigates the asymptotic behavior of correlation functions of characteristic polynomials in hermitian Wigner matrices, revealing they match GUE results up to a moment-dependent factor, with higher moments not affecting the limit.

Contribution

It demonstrates that the asymptotics of correlation functions for hermitian Wigner matrices align with GUE results, modulated by a factor depending only on the fourth moment of entry distributions.

Findings

Asymptotics match GUE for even order correlation functions

Higher moments beyond the fourth do not influence the limit

The correlation function asymptotics depend only on the fourth moment

Abstract

We consider the asymptotics of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this for the GUE up to a factor, depending only on the forth moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$, i.e. that the higher moments of $Q$ do not contribute to the above limit.