Beta Jacobi ensembles and associated Jacobi polynomials, II

Abstract

In a high temperature regime, it was shown in Trinh--Trinh (\emph{J.\ Stat.\ Phys.}\ \textbf{185}(1), Paper No.\ 4, 15 (2021)) that the empirical distribution of beta Jacobi ensembles converges to a limiting probability measure which is related to Model III of associated Jacobi polynomials. In this paper, we establish Gaussian fluctuations around the limit whose statement involves orthogonal polynomials. For the proofs, we refine a moment method at the process level which has been used to deal with Gaussian beta ensembles and beta Laguerre ensembles.