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

Let λ = (λ1,…,λn) be β-Jacobi ensembles with parameters p1, p2, n and β while β varying with n. Set γ=limn→∞np1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = {\lim _{n \to \infty }}{n \over {{p_1}}}$$\end{document} and σ=limn→∞p1p2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = {\lim _{n \to \infty }}{{{p_1}} \over {{p_2}}}$$\end{document}. In this paper, supposing limn→∞lognβn=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lim _{n \to \infty }}{{\log n} \over {\beta n}} = 0$$\end{document}, 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.