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Cluster expansions for Gibbs point processes. (English) Zbl 1427.60202
Summary: 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 R. Fernández and A. Procacci [Commun. Math. Phys. 274, No. 1, 123–140 (2007; Zbl 1206.82148)], the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by R. Bissacot et al. [J. Stat. Phys. 139, No. 4, 598–617 (2010; Zbl 1196.82135)]. 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 [G. Peccati and M. S. Taqqu, Wiener chaos: Moments, cumulants and diagrams. A survey with computer implementation. Milano: Bocconi University Press; Milano: Springer (2011; Zbl 1231.60003)]. 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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