Limits for circular Jacobi beta-ensembles

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.

arXiv comment26 pages

References in corpus (14)

• The Calogero-Sutherland Model and Generalized Classical Polynomials
• A recursion and a combinatorial formula for Jack polynomials
• Universality of random matrices in the microscopic limit and the Dirac operator spectrum
• From Random Matrices to Stochastic Operators
• Products and Ratios of Characteristic Polynomials of Random Hermitian Matrices
• Multicritical Microscopic Spectral Correlators of Hermitian and Complex Matrices
• Duality in random matrix ensembles for all Beta
• Hermite and Laguerre $\beta$-ensembles: asymptotic corrections to the eigenvalue density
• The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I
• Large gaps between random eigenvalues
• The Beta-Wishart Ensemble
• Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials
• Electrostatics of Coulomb gas, lattice paths, and discrete polynuclear growth
• A limiting random analytic function related to the CUE