Cluster expansions for Gibbs point processes

TL;DR

This paper establishes conditions for the uniqueness of Gibbs point processes with non-negative interactions and provides convergent cluster expansions for various functionals, connecting them to diagrammatic and combinatorial structures.

Contribution

It introduces a continuum convergence criterion for Gibbs processes, extending previous discrete results, and offers explicit formulas linking cluster expansions to multigraph and partition-based diagrammatic methods.

Findings

Provided a sufficient condition for uniqueness of Gibbs point processes.

Derived convergent expansions for the log-Laplace functional and correlation functions.

Connected cluster expansions to diagrammatic and combinatorial representations.

Abstract

We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fern{\'a}ndez and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fern{\'a}ndez and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.