On the extremal eigenvalues of Jacobi ensembles at zero temperature

TL;DR

This paper investigates the behavior of the largest and smallest eigenvalues in frozen Jacobi ensembles as the dimension grows, revealing new covariance limits expressed through Bessel functions, extending prior results at zero temperature.

Contribution

It derives the asymptotic covariance limits for extremal eigenvalues of frozen Jacobi ensembles as the dimension tends to infinity, connecting to Bessel functions and extending previous finite beta results.

Findings

Covariance limits for extremal eigenvalues expressed via Bessel functions

Extension of hard edge analysis to the zero temperature limit

Connections to prior results in Laguerre ensembles and finite beta cases

Abstract

For the $\beta$-Hermite, Laguerre, and Jacobi ensembles of dimension $N$ there exist central limit theorems for the freezing case $\beta\to\infty$ 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 $N\to\infty$ 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 $\beta$.