Normal and Poisson approximation for Gibbs point processes with pair potentials

TL;DR

This paper develops new approximation and limit theorems for Gibbs point processes with pairwise interactions, extending previous results to unbounded interaction ranges and marked spaces, using advanced coupling techniques.

Contribution

It introduces generalized disagreement coupling methods for Gibbs processes with unbounded interactions and extends approximation results to infinite-volume and marked spaces.

Findings

Poisson approximation for dependent thinnings of Gibbs processes

Quantitative central limit theorems for geometric functionals

Extension to processes with unbounded interaction range

Abstract

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.