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

Shota Osada

doi:10.1017/etds.2020.123

Abstract

AbstractWe 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.

Highlights

Introduction and the main resultWe consider an isomorphism problem of measure-preserving dynamical systems among translation-invariant point processes on Rd such as the homogeneous Poisson point processes and the determinantal point processes with translation-invariant kernel functions.The homogeneous Poisson point process is a point process in which numbers of particles on disjoint subsets obey independently Poisson distributions
For determinantal point processes on Zd with translation-invariant kernel and the counting measure, Lyons and Steif [5] and Shirai and Takahashi [12] independently proved the Bernoulli property, the latter giving a sufficient condition for the weak Bernoulli property under the assumption K : Zd × Zd → C satisfying (1), (2), Spec(K) ⊂ (0, 1), and (4)
We prove the Bernoulli property of the discrete point processes

Summary

Introduction

Introduction and the main resultWe consider an isomorphism problem of measure-preserving dynamical systems among translation-invariant point processes on Rd such as the homogeneous Poisson point processes and the determinantal point processes with translation-invariant kernel functions.The homogeneous Poisson point process is a point process in which numbers of particles on disjoint subsets obey independently Poisson distributions. We call a Borel probability measure μ on Conf(X) a point process on X. Let μK be a determinantal point process on Rd with translation-invariant kernel K such that Under assumptions (1)–(3), there exists a unique (K, λ)-determinantal point process μ with the kernel function K [11, 13].

Results

Conclusion