Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

Abstract

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral), and Jacobi β-ensembles of N×N random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N→∞. In the bulk of the spectrum of each β-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_{1}F_{1}$ , whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre β-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when β is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson’s lemma and the steepest descent method for integrals of Selberg type.