Freezing Limits for Beta-Cauchy Ensembles

TL;DR

This paper investigates the behavior of beta-Cauchy ensembles derived from Bessel processes associated with root systems, providing explicit distributions and central limit theorems in the freezing regime where parameters grow large.

Contribution

It introduces explicit distributions for beta-Cauchy ensembles and establishes central limit theorems in the freezing regime, extending known results for related ensembles.

Findings

Explicit distributions for beta-Cauchy ensembles derived.

Central limit theorems established in the freezing regime.

Connections made to known freezing results for other ensembles.

Abstract

Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $\beta$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $\beta$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing results for $\beta$-Hermite and Laguerre ensembles and for Bessel processes.