Scaling limits of correlations of characteristic polynomials for the Gaussian beta-ensemble with external source

TL;DR

This paper investigates the asymptotic behavior of characteristic polynomial correlations in large Gaussian beta-ensembles with external sources, revealing phase transitions and involving Airy and Gaussian functions at the spectrum edge.

Contribution

It establishes the limiting correlation structure at the spectrum edge for Gaussian beta-ensembles with external sources, including phase transition phenomena and explicit formulas involving Airy and Gaussian functions.

Findings

Identifies phase transition depending on external source eigenvalues.

Derives limiting correlations involving Airy and Gaussian functions.

Establishes critical external source strength for phase transition.

Abstract

We study the averaged product of characteristic polynomials of large random matrices in the Gaussian beta-ensemble perturbed by an external source of finite rank. We prove that at the edge of the spectrum, the limiting correlations involve two families of multivariate functions of Airy and Gaussian types. The precise form of the limiting correlations depends on the strength of the nonzero eigenvalues of the external source. A critical value for the latter is obtained and a phase transition phenomenon similar to that of arXiv:math/0403022 is established. The derivation of our results relies mainly on previous articles by the authors, which deal with duality formulas arXiv:0801.3438 and asymptotics for Selberg-type integrals arXiv:1112.1119v3.