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 Fernández and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández and Procacci (2010). In addition, we provide formulas for 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.

Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

“…In this subsection we present some fundamental facts on Gibbs processes in a general setting. We base our definition of a Gibbs process on the following GNZ-equation, referring to [14], for example, for a discussion of the literature. Definition 3.…”

“…(By assumption (2.4) we have that μ( ) < ∞ for any ball ⊂ C (d) .) Under these assumptions it has been shown in [14] that a Gibbs process exists.…”

“…In the recent paper [14] uniqueness of the distribution of a Gibbs process on R d has been proved with very different methods, based on a fixed point argument for correlation functions. It is an interesting problem to explore the relationship between this method and disagreement percolation.…”

“…It is an interesting problem to explore the relationship between this method and disagreement percolation. Some initial answers are given in [14].…”

Cluster expansions: Necessary and sufficient convergence conditions

“…Overall, in the last two decades, a notable effort was made to generalize classical sufficient conditions in the abstract polymer setup (including the condition by Fernández and Procacci) to hold in continuous spaces and for systems with soft-core (or even more general) interactions, see [32,24,5,14,22].…”