Abstract
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 \(\beta N \rightarrow const \in [0, \infty )\), 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.
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This work is supported by JSPS KAKENHI Grant Number JP19K14547 (K.D.T.). The authors would like to thank a referee for helpful comments.
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Communicated by Eric A. Carlen.
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Trinh, H.D., Trinh, K.D. Beta Jacobi Ensembles and Associated Jacobi Polynomials. J Stat Phys 185, 4 (2021). https://doi.org/10.1007/s10955-021-02832-z
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DOI: https://doi.org/10.1007/s10955-021-02832-z