Limits for circular Jacobi beta-ensembles

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• 作者：Dang-Zheng Liu ^{dzliu@ustc.edu.cn}
• 关键词：60B20 ; 41A60
• 刊名：Journal of Approximation Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：215
• 期：Complete
• 页码：40-67
• 全文大小：416 K
• 卷排序：215

文摘

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