On the extremal eigenvalues of Jacobi ensembles at zero temperature

TL;DR

The paper analyzes extremal eigenvalue fluctuations in frozen Jacobi ensembles and hard-edge Laguerre ensembles as $N\to\infty$. It leverages dual orthogonal polynomial frameworks (de Boor–Saff) and a central limit regime to express inverse-covariance structures in terms of Jacobi and Laguerre polynomials, yielding explicit limiting covariances for the $r$-th largest eigenvalues. The main results express these limits via integrals involving the Bessel function $J_\alpha$ (and $J_{\nu-1}$ for Laguerre), with zeros $j_{\alpha,r}$ and $j_{\nu-1,r}$, and show how algebraic and trigonometric coordinate representations converge to the same Bessel-based kernels. This work solidifies a Bessel-universality pattern at the hard edge and provides concrete asymptotic formulas that connect Jacobi and Laguerre freezing limits through dual polynomial structures.

Abstract

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

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Lemma 2.6
• proof : Proof of Theorem \ref{['asymptotics-jacobi']}