Limit theorems for beta-Jacobi ensembles (Q358145)

The \(\beta\)-Jacobi ensemble with parameters \(\beta>0\) and \(a_1,a_2>\frac{\beta}{2}(n-1)\) is a random vector of eigenvalues \(\lambda=(\lambda_1,\dotsc,\lambda_n)\) taking values in \([0,1]^n\) and having probability density function NEWLINE\[NEWLINE f(\lambda_1,\dotsc,\lambda_n) = \mathrm{const} \cdot \prod_{1\leq i<j\leq n} |\lambda_i - \lambda_j|^{\beta} \cdot \prod_{i=1}^n \lambda_i^{a_1-p} (1-\lambda_i)^{a_2-p}, NEWLINE\]NEWLINE where \(p=1+\frac {\beta} 2 (n-1)\). The author studies the limiting properties of \(\lambda\) in the following limiting regime: NEWLINE\[NEWLINE a_1,a_2, n\to\infty, \; a_1=o(\sqrt{a_2}), \; n=o(\sqrt{a_2}), \; \frac{n\beta}{2a_1}\to \gamma\in (0,1]. NEWLINE\]NEWLINE The author computes the global limiting distribution of the (appropriately normalized) eigenvalues, the limiting distribution of the maximal and minimal eigenvalue (and more generally, the joint limiting distribution of any finite number of extreme order statistics), and proves a central limit theorem for the linear eigenvalue statistics. The proofs are based on the possibility to approximate (in the limiting regime described above) the \(\beta\)-Jacobi ensemble by a \(\beta\)-Laguerre ensemble.