Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions

TL;DR

This paper establishes large-dimensional limit laws for Bessel and Dunkl processes, connecting them with free convolutions and classical random matrix results, and introduces new limit distributions, especially in the frozen case.

Contribution

It derives new limit laws for Bessel and Dunkl processes of types A and B, linking them to free convolutions and providing simplified proofs of classical results.

Findings

Analogues of Wigner's semicircle law for large N

Marchenko-Pastur law for large N

New non-symmetric semicircle-type distributions for Dunkl processes of type B

Abstract

We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_N$ these Bessel processes are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the particles with arbitrary initial empirical distributions by using free convolutions. In particular, for Dunkl processes of type B new non-symmetric semicircle-type limit distributions on $\mathbb R$ appear. Our results imply that the form of the limiting measures is already completely determined by the frozen processes. Moreover, in the frozen cases, our approach leads to a new simple proof of the semicircle and Marchenko-Pastur limit laws for the empirical measures of the zeroes of Hermite and Laguerre polynomials respectively.