Improved concentration of Laguerre and Jacobi ensembles

TL;DR

This paper enhances the understanding of how Laguerre and Jacobi ensembles concentrate around polynomial zeros in asymptotic limits, providing sharper bounds and simpler proofs for their concentration properties.

Contribution

It improves the concentration bounds and error analysis for Laguerre and Jacobi ensembles, refining previous results and simplifying the proofs.

Findings

Sharper concentration bounds for ensembles

Simplified proof techniques

Improved bounds for moments of Jacobi ensemble

Abstract

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.