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

TL;DR

This paper studies the averaged characteristic polynomial in Gaussian and chiral Gaussian ensembles with a source, extending results to general beta, and explores duality formulas and edge scaling limits.

Contribution

It introduces a Dyson Brownian motion approach for general beta and derives new duality formulas, providing explicit edge scaling limits in terms of incomplete Airy functions.

Findings

Extended the Dyson Brownian motion model to general beta > 0.

Derived explicit formulas for the averaged characteristic polynomial.

Identified soft edge scaling limits involving incomplete Airy functions.

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 $(\beta = 1)$, 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.