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

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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 (β = 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 derivations of the general case of Desrosiers' dualities are given. A soft edge scaling limit of the averaged characteristic polynomial is identified, and an explicit evaluation in terms of so-called incomplete Airy functions is obtained.