Abstract
We prove the Bernoulli property for determinantal point processes on
$ \mathbb{R}^d $
with translation-invariant kernels. For the determinantal point processes on
$ \mathbb{Z}^d $
with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.
Nonequilibrium entropy, Lyapounov variables, and ergodic properties of classical systems