Beta Jacobi Ensembles and Associated Jacobi Polynomials

• Published in: Journal of statistical physics Vol. 185; no. 1
• Main Authors: Trinh, Hoang Dung, Trinh, Khanh Duy
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer; Springer Nature B.V
• Subjects: Constraining; Eigenvalues; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Polynomials; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Primary 60F05; Beta Jacobi processes; Associated Jacobi polynomials; 60K35; Secondary 60B20; Beta Jacobi ensembles
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-021-02832-z

Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as the eigenvalues of certain tridiagonal random matrices. The paper deals with beta Jacobi ensembles, the type with the Jacobi weight. Making use of the random matrix model, we show that in the regime where β N → c o n s t ∈ [ 0 , ∞ ) , with N the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials. This is analogous to the existing results for the other two classical weights. We also study the limiting behavior of the empirical measure process of beta Jacobi processes in the same regime and obtain a dynamical version of the above.