Disagreement coupling of Gibbs processes with an application to Poisson approximation

TL;DR

This paper introduces a coupling method for finite Gibbs processes, enabling Poisson approximation of certain infinite-volume Gibbs processes and analyzing their empty space probabilities.

Contribution

It extends existing coupling techniques to finite Gibbs processes with specified intensities, facilitating Poisson approximation and analysis of empty space probabilities.

Findings

Established Poisson approximation for point processes from Gibbs processes.

Developed a coupling method for Gibbs processes with different boundary conditions.

Analyzed empty space probabilities of specific Gibbs processes.

Abstract

We discuss a thinning and an embedding procedure to construct finite Gibbs processes with a given Papangelou intensity. Extending the approach in Hofer-Temmel (2019) and Hofer-Temmel and Houdebert (2019) we will use this to couple two finite Gibbs processes with different boundary conditions. As one application we will establish Poisson approximation of point processes derived from certain infinite volume Gibbs processes via dependent thinning. As another application we shall discuss empty space probabilities of certain Gibbs processes.