Gibbsian Representation for Point Processes via Hyperedge Potentials

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...

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Bibliographic Details
Published in:Journal of theoretical probability 2021-03, Vol.34 (1), p.391-417
Main Authors: Jahnel, Benedikt, Külske, Christof
Format: Article
Language:English
Subjects:
Euclidean geometry
Euclidean space
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Representations
Statistical mechanics
Statistics
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Summary: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.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-019-00960-7