Limit Theorems and Large Deviations for β-Jacobi Ensembles at Scaling Temperatures

Let λ = (λ₁,…,λ_n) be β-Jacobi ensembles with parameters p₁, p₂, n and β while β varying with n. Set \(\gamma = {\lim _{n \to \infty }}{n \over {{p_1}}}\) and \(\sigma = {\lim _{n \to \infty }}{{{p_1}} \over {{p_2}}}\). In this paper, supposing \({\lim _{n \to \infty }}{{\log n} \over {\beta n}} = 0\), we prove that the empirical measures of different scaled λ converge weakly to a Wachter distribution, a Marchenko–Pastur law and a semicircle law corresponding to σγ > 0, σ = 0 or γ = 0, respectively. We also offer a full large deviation principle with speed βn² and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of β-Jacobi ensembles is obtained.