Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 069, 25 pages      arXiv:2205.08153      https://doi.org/10.3842/SIGMA.2022.069

Freezing Limits for Beta-Cauchy Ensembles

Michael Voit
Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221 Dortmund, Germany

Received May 19, 2022, in final form September 15, 2022; Published online September 28, 2022

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.

Key words: Cauchy processes; Bessel processes; $\beta$-Hermite ensembles; $\beta$-Laguerre ensembles; freezing; zeros of classical orthogonal polynomials; Calogero-Moser-Sutherland particle models.

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