# Functional central limit theorems for multivariate Bessel processes in the freezing regime

## TL;DR

This paper establishes functional central limit theorems for multivariate Bessel processes in the freezing regime, revealing Gaussian limits with explicit matrix exponential representations, advancing understanding of their asymptotic behavior.

## Contribution

It introduces functional CLTs for multivariate Bessel processes with specific initial conditions, extending previous fixed-time results to a locally uniform in time setting.

## Key findings

- Gaussian limiting processes with explicit matrix exponential forms
- Functional CLTs valid for initial conditions scaled by k
- Extension of previous fixed-time limit theorems to a dynamic, uniform-in-time context

## Abstract

Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers. Here we provide associated functional central limit theorems which are locally uniform in $t$. The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.