Characteristic polynomials

Abstract

This article considers characteristic polynomials and reviews a few useful results obtained in simple Gaussian models of random Hermitian matrices in the presence of an external matrix source. It first considers the products and ratio of characteristic polynomials before discussing the duality theorems for two different characteristic polynomials of Gaussian weights with external sources. It then describes the m-point correlation functions of the eigenvalues in the Gaussian unitary ensemble and how they are deduced from their Fourier transforms U(s1, … , sm). It also analyses the relation of the correlation function of the characteristic polynomials to the standard n-point correlation function using the replica and supersymmetric methods. Finally, it shows how the topological invariants of Riemann surfaces, such as the intersection numbers of the moduli space of curves, may be derived from averaged characteristic polynomials.