Characteristic Polynomials over Random Matrix Ensembles through a Grassman Integration Approach

Abstract

We introduce Grassman integration to derive the asymptotic behavior of the correlation function of the product of two characteristic polynomials for various matrix ensembles. We first derive the result for the Gaussian Unitary and Gaussian Orthogonal ensembles (GUE and GOE), before moving onto general real symmetric Wigner matrices. The main result is to show that a Grassman integration approach can be used to derive a GOE universality result for second order correlation functions. While the resulting asymptotics are already known, it is unclear how to generalize any previous approach to study universality of higher order correlations. This work yields the same result for the second order correlations, and previous work on GUE universality suggests that the method can be generalized to higher order correlations over GOE as well.