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Gibbsian Representation for Point Processes via Hyperedge Potentials

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.

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Correspondence to Benedikt Jahnel.

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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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Jahnel, B., Külske, C. Gibbsian Representation for Point Processes via Hyperedge Potentials. J Theor Probab 34, 391–417 (2021). https://doi.org/10.1007/s10959-019-00960-7

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Keywords

  • Gibbsian point processes
  • Kozlov theorem
  • Sullivan theorem
  • Hyperedge potentials
  • Widom–Rowlinson model

Mathematics Subject Classification (2010)

  • 82B21
  • 60K35