Asymptotics for products of characteristic polynomials in classical $\beta$-Ensembles

TL;DR

This paper investigates the asymptotic behavior of products of characteristic polynomials in classical $eta$-ensembles, revealing universal scaling limits in the bulk and at the soft edge, with applications to correlation functions.

Contribution

It provides explicit scaling limits of characteristic polynomial products in $eta$-ensembles, including multivariate Airy functions and their asymptotics, extending known results to general $eta$.

Findings

Bulk limit is a hypergeometric function involving Jack polynomials.

Soft edge limit is a multivariate Airy function defined via a Kontsevich integral.

Scaling limits of correlation functions are derived for even $eta$.

Abstract

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as $N\to\infty$. In the bulk of the spectrum of each $\beta$-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_1F_{1}$ 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 $\beta$-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when $\beta$ 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.