Limits for circular Jacobi beta-ensembles

TL;DR

This paper investigates the spectral limits of circular Jacobi beta-ensembles, revealing new multivariate functions at spectrum singularities and connecting these limits to classical beta-ensembles, with implications for correlation functions and hypergeometric functions.

Contribution

It computes new scaling limits for characteristic polynomials of circular Jacobi beta-ensembles, including a novel multivariate function at spectrum singularities and links to classical ensembles.

Findings

New multivariate function at spectrum singularity

Scaling limits match classical ensembles in certain regimes

Transition observed from spectrum singularity to soft edge as parameter grows

Abstract

Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi $\beta$-ensemble, which is a generalization of the Dyson circular $\beta$-ensemble but equipped with an additional parameter $b$, and further studied its limiting spectral measure. We calculate the scaling limits for expected products of characteristic polynomials of circular Jacobi $\beta$-ensembles. For the fixed constant $b$, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When $b=\beta Nd/2$, the scaling limits in the bulk and at the soft edge agree with those of the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\beta$-ensembles proved in the joint work with P Desrosiers "Asymptotics for products of characteristic polynomials in classical beta-ensembles", Constr. Approx. 39 (2014), arXiv:1112.1119v3. As corollaries, for even $\beta$ the scaling limits of point correlation functions for the ensemble are given. Besides, a transition from the spectrum singularity to the soft edge limit is observed as $b$ goes to infinity. The positivity of two special multivariate hypergeometric functions, which appear as one factor of the joint eigenvalue densities for spiked Jacobi/Wishart $\beta$-ensembles and Gaussian $\beta$-ensembles with source, will also be shown.