On the correlation functions of the characteristic polynomials of the hermitian sample covariance ensemble

TL;DR

This paper analyzes the asymptotic behavior of correlation functions of characteristic polynomials of hermitian sample covariance matrices, showing they match Gaussian Unitary Ensemble results up to a moment-dependent factor.

Contribution

It demonstrates that the asymptotics of correlation functions for these matrices are universal and depend only on the fourth moment of entries, extending GUE results.

Findings

Asymptotic behavior matches GUE results up to a factor

Higher moments beyond the fourth do not affect the limit

Results hold for both bulk and edge spectrum regions

Abstract

We consider asymptotic behavior of the correlation functions of the characteristic polynomials of the hermitian sample covariance matrices $H_n=n^{-1}A_{m,n}^*A_{m,n}$, where $A_{m,n}$ is a $m\times n$ complex matrix with independent and identically distributed entries $\Re a_{\alpha j}$ and $\Im a_{\alpha j}$. We show that for the correlation function of any even order the asymptotic behavior in the bulk and at the edge of the spectrum coincides with those for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries $\Re a_{\alpha j}$, $\Im a_{\alpha j}$, i.e. the higher moments do not contribute to the above limit.