Cluster expansions for Gibbs point processes

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{á}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{á}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.

arXiv comment41 pages

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Cited by in corpus (10)

• Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics
• On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential
• An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
• Revisiting Groeneveld's approach to the virial expansion
• A variational framework for the inverse Henderson problem of statistical mechanics
• Strong spatial mixing for repulsive point processes
• Cluster expansions: Necessary and sufficient convergence conditions
• Convergence of cluster and virial expansions for repulsive classical gases
• Structural properties of Gibbsian point processes in abstract spaces
• Potential-weighted connective constants and uniqueness of Gibbs measures