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

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.