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Cluster Expansions for GIBBS Point Processes

Part of: Equilibrium statistical mechanics Special processes Stochastic processes

Published online by Cambridge University Press:  15 November 2019

S. Jansen
S. Jansen*
Affiliation:
Ludwig-Maximilians-Universität München
*
*Postal address: Mathematisches Institut, Ludwig-Maximilians Universität München, Theresienstr. 39, 80333 München, Germany.
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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 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.

Keywords

Gibbs point processcluster expansioncumulantsbranching processBoolean percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B05: Classical equilibrium statistical mechanics (general) 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 1129 - 1178
Copyright
© Applied Probability Trust 2019 

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