Gibbsian Representation for Point Processes via Hyperedge Potentials
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Gibbsian Representation for Point Processes via Hyperedge Potentials Benedikt Jahnel1
· Christof Külske2
Received: 11 June 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
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 nonlocalities 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 Mathematics Subject Classification (2010) 82B21 · 60K35
This work is dedicated to the memory of Professor Hans-Otto Georgii. Benedikt Jahnel thanks the Leibniz program ’Probabilistic methods for mobile ad hoc networks’ for their support. Christof Külske thanks the Weierstrass Institute for its hospitality.
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Benedikt Jahnel [email protected] Christof Külske [email protected]
1
Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany
2
Fakultät für Mathematik, Ruhr-Universität Bochum, 44801 Bochum, Germany
123
Journal of Theoretical Probability
1 Introduction In this note, we study models for not necessarily translation-invariant Poisson point processes (PPP) in Euclidean space Rd with general marks. Such models are the subject in the infinite-volume statistical mechanics of classical point particles which interact via potentials. They are already very interesting when there are no marks (or internal states of particles), and only the positions of the colorless point particles are relevant. Potentials coming from physics are often pair potentials. Take as an example the famous Lennard–Jones potential. For results on existence of such models in the infinite volume (see [15,17]). Also more general potentials than pair potentials appear, describing interactions between finite collections of particles. These are quite relevant in physics as well; see, for instance, the proof of a phase transition for a long-range (but finite) potential involving 4-body interactions in [14]. For models from statistical physics with marks; see, e.g., the Potts gas in [5]. The famous Widom–Rowlinson model (WRM) is a specific example for this which is proved to have a phase transition in the
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