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

TL;DR

This paper investigates the asymptotic behavior of the second-order correlation function of the characteristic polynomial in real symmetric Wigner matrices, extending known results from Gaussian ensembles to more general cases.

Contribution

It demonstrates that the asymptotic properties of the correlation function for Gaussian Orthogonal Ensemble matrices also apply to general real symmetric Wigner matrices.

Findings

Asymptotic behavior matches that of Gaussian Orthogonal Ensemble

Results extend to a broader class of real symmetric matrices

Supports universality in spectral statistics

Abstract

We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble essentially continues to hold for a general real symmetric Wigner matrix.