Limits for circular Jacobi beta-ensembles
Abstract
Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi β-ensemble, which is a generalization of the Dyson circular β-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 β-ensembles. For the fixed constant b, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When b=β Nd/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 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 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 spiked Jacobi/Wishart β-ensembles and Gaussian β-ensembles with source, will also be shown.