# Limit theorems for Jacobi ensembles with large parameters

## TL;DR

This paper establishes a central limit theorem for Jacobi ensembles with large parameters, describing the asymptotic distribution of eigenvalues and connecting it to classical orthogonal polynomials and other random matrix ensembles.

## Contribution

It derives a CLT for Jacobi ensembles with large parameters, expressing the limit covariance in terms of Jacobi polynomial zeros and relating results to other beta-ensembles.

## Key findings

- CLT for Jacobi ensembles with large parameters
- Limit covariance expressed via Jacobi polynomial zeros
- Eigenvalues and eigenvectors of the covariance matrix identified

## Abstract

Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod_{1\leq i< j \leq N}\left(x_j-x_i\right)^{k_3}\prod_{i=1}^N \left(1-x_i\right)^{\frac{k_1+k_2}{2}-\frac{1}{2}}\left(1+x_i\right)^{\frac{k_2}{2}-\frac{1}{2}} dx$$ of the eigenvalues on the alcoves $$A:=\{x\in\mathbb R^N| \> -1\leq x_1\le ...\le x_N\leq 1\}.$$ For $(k_1,k_2,k_3)=\kappa\cdot (a,b,1)$ with $a,b>0$ fixed, we derive a central limit theorem for the distributions above for $\kappa\to\infty$. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomials. We also rewrite the CLT in trigonometric form and determine the eigenvalues and eigenvectors of the limit covariance matrices. These results are related to corresponding limits for $\beta$-Hermite and $\beta$-Laguerre ensembles for $\beta\to\infty$ by Dumitriu and Edelman and by Voit.