An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions

TL;DR

This paper establishes an explicit region where Gibbs point processes with repulsive interactions are unique, using a Dobrushin criterion adapted for continuous settings, and compares it with other methods.

Contribution

It introduces a new explicit uniqueness region for Gibbs point processes with non-negative pair potentials, employing a classical Dobrushin criterion in the continuous setting.

Findings

Derived an explicit uniqueness condition in terms of activity and inverse temperature.

Compared the Dobrushin criterion with cluster expansion and disagreement percolation methods.

Demonstrated the applicability of the criterion to processes with repulsive interactions.

Abstract

We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity $z$ and inverse temperature $\beta$. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interactions.