Limit theorems for Jacobi ensembles with large parameters

Abstract

Consider Jacobi random matrix ensembles with the distributions $$c_{k_1,k_2,k_3}\prod _{1\le i<j\le N}{\left(x_j-x_i\right)}^{k_3}\prod _{i=1}^N{\left(1-x_i\right)}^{\frac{k_1+k_2}{2}-\frac{1}{2}}{\left(1+x_i\right)}^{\frac{k_2}{2}-\frac{1}{2}}dx$$ of the eigenvalues on the alcoves $$A:=\{x\in {\mathbb{R}}^N|-1\le x_1\le ...\le x_N\le 1\}.$$ 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.

The presentation of the results is improved, and additional references are added

References in corpus (3)

• Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices
• Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Functional central limit theorems for multivariate Bessel processes in the freezing regime

Cited by in corpus (2)

• Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Distances of roots of classical orthogonal polynomials