Freezing Limits for Beta-Cauchy Ensembles

Bessel processes associated with the root systems AN-1 and BN describe inter-acting particle systems with N particles on R; they form dynamic versions of the classical 0-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to 0-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 0-Hermite and Laguerre ensembles and for Bessel processes.