Disagreement percolation for Gibbs ball models

TL;DR

This paper extends disagreement percolation techniques to Gibbs point processes with variable radii, enabling proofs of uniqueness and decay of correlations by comparing with sub-critical models, with applications to continuum models.

Contribution

It introduces a generalized disagreement percolation method for Gibbs ball models with varying radii, providing new tools for analyzing continuum Gibbs processes.

Findings

Established uniqueness of Gibbs measures in low activity regimes.

Proved exponential decay of pair correlations.

Applied methods to continuum random cluster and Quermass-interaction models.

Abstract

We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process.