#### Vol. 3, No. 4, 2021

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Limit theorems for Jacobi ensembles with large parameters

### Kilian Hermann and Michael Voit

Vol. 3 (2021), No. 4, 843–860
##### Abstract

Consider $\beta$-Jacobi ensembles on the alcoves

$A:=\left\{x\in {ℝ}^{N}\mid -1\le {x}_{1}\le \cdots \le {x}_{N}\le 1\right\}$

with parameters ${k}_{1},{k}_{2},{k}_{3}\ge 0$. In the freezing case $\left({k}_{1},{k}_{2},{k}_{3}\right)=\kappa \cdot \left(a,b,1\right)$ with $a,b>0$ fixed and $\kappa \to \infty$, 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 $\beta$-Hermite and Laguerre ensembles for $\beta \to \infty$.

##### Keywords
$\beta$-Jacobi ensembles, freezing, central limit theorems, zeros of Jacobi polynomials, eigenvalues of covariance matrices
##### Mathematical Subject Classification
Primary: 33C45, 60B20, 60F05, 70F10
Secondary: 33C67, 82C22