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

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.

20 pages. v3: slightly longer version with improved notation

References in corpus (7)

• Limits of spiked random matrices I
• Intersection theory from duality and replica
• The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I
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Cited by in corpus (5)

• Probability densities and distributions for spiked and general variance Wishart $\beta$-ensembles
• The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
• Loop Equation Analysis of the Circular $\beta$ Ensembles
• Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model
• Phase transitions for products of characteristic polynomials under Dyson Brownian motion