Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian -Ensemble with External Source

Abstract

We study the averaged product of characteristic polynomials of large random matrices in the Gaussian β-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 [2] is established. The derivation of our results relies mainly on previous articles by the authors, which deal with duality formulas [18] and asymptotics for Selberg-type integrals [22].