Isomorphisms between determinantal point processes with translation invariant kernels and Poisson point processes

TL;DR

This paper establishes an isomorphism between translation-invariant determinantal point processes on Euclidean space and Poisson point processes, extending known results from lattice cases and proving Bernoulli properties.

Contribution

It proves the Bernoulli property for continuum determinantal point processes and demonstrates their isomorphism to Poisson processes, extending prior lattice results.

Findings

Proves Bernoulli property for continuum determinantal point processes.

Establishes isomorphism with homogeneous Poisson point processes.

Extends lattice results to Euclidean space.

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 and Shirai and Takahashi. As its continuum version, we prove an isomorphism between the translation-invariant determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels and homogeneous Poisson point processes. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.