Gibbsian Representation for Point Processes via Hyperedge Potentials

  • Benedikt JahnelEmail author
  • Christof Külske


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.


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

Mathematics Subject Classification (2010)

82B21 60K35 



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Authors and Affiliations

  1. 1.Weierstrass Institute BerlinBerlinGermany
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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