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Gibbsian Representation for Point Processes via Hyperedge Potentials

Author

Listed:
  • Benedikt Jahnel

    (Weierstrass Institute Berlin)

  • Christof Külske

    (Ruhr-Universität Bochum)

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.

Suggested Citation

  • Benedikt Jahnel & Christof Külske, 2021. "Gibbsian Representation for Point Processes via Hyperedge Potentials," Journal of Theoretical Probability, Springer, vol. 34(1), pages 391-417, March.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:1:d:10.1007_s10959-019-00960-7
    DOI: 10.1007/s10959-019-00960-7
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