Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case

TL;DR

This paper investigates the asymptotic behavior of the characteristic polynomial of large non-square sample covariance matrices with i.i.d. entries, revealing universal kernel limits in the bulk and at the edge of the spectrum.

Contribution

It extends the understanding of characteristic polynomials to non-square matrices, showing universal sine and Airy kernel limits in the asymptotic regime.

Findings

Second-order correlation function converges to the sine kernel in the bulk.

At the spectrum edge, the correlation function converges to the Airy kernel.

Results apply to both complex and real sample covariance matrices.

Abstract

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices.