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 β-Jacobi ensembles on the alcoves

A := {x N 1 x 1 xN 1}

with parameters k1,k2,k3 0. In the freezing case (k1,k2,k3) = κ (a,b,1) with a,b > 0 fixed and κ , 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 β-Hermite and Laguerre ensembles for β .

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
Milestones
Received: 23 November 2020
Revised: 18 December 2020
Accepted: 5 January 2021
Published: 20 October 2021
Authors
Kilian Hermann
Fakultät Mathematik
Technische Universität Dortmund
Germany
Michael Voit
Fakultät Mathematik
Technische Universität Dortmund
Germany