Limit theorems for Jacobi ensembles with large parameters

Consider β-Jacobi ensembles on the alcoves A:={x∈ℝN∣−1≤x1≤⋯≤xN≤1} with parameters k1,k2,k3≥0. In the freezing case (k1,k2,k3)=κ⋅(a,b,1) with a,b>0 fixed and κ→∞, we derive a central limit theorem. The drift and covariance matrix of the limit are expressed via the zeros of classical Jacobi polynomials. We also determine the eigenvalues and eigenvectors of the covariance matrices. Our results are related to corresponding limits for β-Hermite and Laguerre ensembles for β→∞.