Nonstandard limit theorems and large deviation for beta -Jacobi ensembles with a different scaling

We consider $\beta$-Jacobi ensembles with parameters $p_1, p_2\geq n.$ We prove that the empirical measure of the rescaled Jacobi ensembles converges weakly to a modified Watcher law via the spectral measure method, which revisits the weak limits obtained in \cite{MaLDPJ} while replacing the condition $\beta n\!>\!> \log n$ by $\beta n\!>\!>1.$ We also provide the central limit theorem and the large deviation for the corresponding rescaled spectral measure.
Comment: 23 pages