Moments of Characteristic Polynomials of Non-Symmetric Random Matrices

Abstract: We study the moments of the absolute characteristic polynomial of the real elliptic ensemble, including the case of the real Ginibre ensemble. We obtain asymptotics for all integral moments inside the real bulk to order 1 + o(1). In particular, for the real Ginibre ensemble, this extends known computations for even moments, and confirms a recent conjecture of Serebryakov and Simm [48] in the integral case. For the elliptic case, this generalizes computations of first two moments by Fyodorov [25] and Fyodorov and Tarnowski [31]. We additionally find uniform asymptotics for the multi-point correlations of the absolute characteristic polynomial. Our proof relies on a relation between expectations for the absolute characteristic polynomial and the real correlation functions, as well as an algebraic method of obtaining asymptotics for the behavior of these correlation functions near the diagonal.