# Central limit theorems for multivariate Bessel processes in the freezing regime II: the covariance matrices

## TL;DR

This paper extends central limit theorems for multivariate Bessel processes in the freezing regime, analyzing covariance matrices and eigenstructures, with applications to particle systems and random matrix ensembles.

## Contribution

It generalizes CLTs for Bessel processes to arbitrary starting distributions and characterizes the covariance matrices' eigenvalues and eigenvectors.

## Key findings

- Eigenvalues and eigenvectors of covariance matrices identified
- Extended CLT to arbitrary starting distributions
- Applications to intermediate particle CLTs for large parameters

## Abstract

Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to\infty$ and then $N\to\infty$.