Beta Jacobi Ensembles and Associated Jacobi Polynomials

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...

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Bibliographic Details
Published in:Journal of statistical physics 2021-10, Vol.185 (1), Article 4
Main Authors: Trinh, Hoang Dung, Trinh, Khanh Duy
Format: Article
Language:English
Subjects:
Constraining
Eigenvalues
Mathematical and Computational Physics
Physical Chemistry
Physics
Physics and Astronomy
Polynomials
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
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Summary: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.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-021-02832-z