Central limit theorems for multivariate Bessel processes in the freezing regime

TL;DR

This paper establishes central limit theorems for multivariate Bessel processes associated with root systems A, B, and D, revealing normal and freezing limits, and connecting to beta-ensemble results.

Contribution

It provides elementary proofs of CLTs for these processes, including novel freezing limits in the B-case, extending prior weak law results.

Findings

Normal distribution limits for types A and D.

Freezing limits involving half-space distributions in type B.

Connections established with Dumitriu and Edelman's CLTs for beta-ensembles.

Abstract

Multivariate Bessel processes are classified via associated root systems and positive multiplicity constants. They describe the dynamics of interacting particle systems of Calogero-Moser-Sutherland type. Recently, Andraus, Katori, and Miyashita derived some weak laws of large numbers for these processes for fixed positive times and multiplicities tending to infinity. In this paper we derive associated central limit theorems for the root systems of types A, B and D in an elementary way. In most cases, the limits will be normal distributions, but in the B-case there are freezing limits where distributions associated with the root system A or one-sided normal distributions on half-spaces appear. Our results are connected to central limit theorems of Dumitriu and Edelman for beta-Hermite and beta-Laguerre ensembles.