  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.
