On the extremal eigenvalues of Jacobi ensembles at zero temperature

For the $\beta$-Hermite, Laguerre, and Jacobi ensembles of dimension $N$ there exist central limit theorems for the freezing case $\beta \to \mathrm{\infty }$ such that the associated means and covariances can be expressed in terms of the associated Hermite, Laguerre, and Jacobi polynomials of order $N$ respectively as well as via the associated dual polynomials in the sense of de Boor and Saff. In this paper we derive limits for $N\to \mathrm{\infty }$ for the covariances of the $r\in \mathbb{N}$ largest (and smallest) eigenvalues for these frozen Jacobi ensembles in terms of Bessel functions. These results correspond to the hard edge analysis in the frozen Laguerre cases by Andraus and Lerner-Brecher and to known results for finite $\beta$.