On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix

TL;DR

This paper investigates the asymptotic behavior of the second-order correlation function of the characteristic polynomial in Hermitian Wigner matrices, extending known Gaussian Unitary Ensemble results to more general cases.

Contribution

It demonstrates that the known correlation results for GUE also apply to general Hermitian Wigner matrices, using an explicit generating function approach.

Findings

The second-order correlation function asymptotics are similar for GUE and general Hermitian Wigner matrices.

Explicit exponential generating functions are derived for the correlation function.

The results support universality in the spectral properties of Hermitian matrices.

Abstract

We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function of the characteristic polynomial.