The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

Abstract

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real (= 1), complex (β = 2) and real quaternion (β = 4) elements. We use the Dyson Brownian motion model to give a meaning for general β > 0. In the Gaussian case a further construction valid for β > 0 is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. The resulting functional forms are shown to be special cases of duality formulas due to Desrosiers. New deriv..