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 $(\beta =1)$, 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.

21 pages

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• Universal shocks in the Wishart random-matrix ensemble - a sequel
• Exact spectral densities of complex noise-plus-structure random matrices
• Note on a duality of topological branes
• Three Problems Related to the Eigenvalues of Complex Non-central Wishart Matrices with a Rank-1 Mean
• Symmetric Function Theory and Unitary Invariant Ensembles
• Duality and integrability of supermatrix model with external source
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion
• Optimization landscape in the simplest constrained random least-square problem
• Three-Parametric Marcenko-Pastur Density