Limits for circular Jacobi beta-ensembles

Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi β-ensemble. This is a generalization of the Dyson circular β-ensemble but equipped with an additional parameter b controlling the order of a spectrum singularity. We calculate the scaling limits for expected products of characteristic polynomials of circular Jacobi β-ensembles. For a fixed constant b, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When b=βNd/2, the scaling limits in the bulk and at the soft edge agree with those of the Hermite (Gaussian), Laguerre (chiral) and Jacobi β-ensembles proved in Desrosiers and Liu (2014). As corollaries, for even β 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 Jacobi/Wishart β-ensembles with general covariance and Gaussian β-ensembles with source, will also be shown.