1203.5838 (Peter J. Forrester)

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

Peter J. Forrester

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 $(\beta = 1)$, complex ($\beta = 2)$ and real quaternion $(\beta = 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $\beta > 0$. In the Gaussian case a further construction valid for $\beta > 0$ 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 a 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.

View original: http://arxiv.org/abs/1203.5838