Gibbsian Representation for Point Processes via Hyperedge Potentials

Benedikt Jahnel () and Christof Külske ()
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Benedikt Jahnel: Weierstrass Institute Berlin
Christof Külske: Ruhr-Universität Bochum

Journal of Theoretical Probability, 2021, vol. 34, issue 1, 391-417

Abstract: Abstract We consider marked point processes on the d-dimensional Euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We construct absolutely summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. (Probab Theory Relat Fields 153(3–4):643–670, 2012), which are useful in models of stochastic geometry. These potentials allow for weak non-localities and are a natural generalization of the usual physical multi-body potentials, which are strictly local. Our proof relies on regrouping arguments, which use the possibility of controlled non-localities in the class of hyperedge potentials. As an illustration, we also provide such representations for the Widom–Rowlinson model under independent spin-flip time evolution. With this work, we aim to draw a link between the abstract theory of point processes in infinite volume, the study of measures under transformations and statistical mechanics of systems of point particles.

Keywords: Gibbsian point processes; Kozlov theorem; Sullivan theorem; Hyperedge potentials; Widom–Rowlinson model; 82B21; 60K35 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-019-00960-7

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