Advertisement

Gibbsian Representation for Point Processes via Hyperedge Potentials

  • Benedikt JahnelEmail author
  • Christof Külske
Article
  • 8 Downloads

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.

Keywords

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

Mathematics Subject Classification (2010)

82B21 60K35 

Notes

References

  1. 1.
    Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172(3), 551–569 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dereudre, D., Drouilhet, R., Georgii, H.-O.: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153(3–4), 643–670 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dereudre, D., Houdebert, P.: Sharp phase transition for the continuum Widom–Rowlinson model (2018). arXiv preprint arXiv:1807.04988
  4. 4.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions, vol. 9, 2nd edn. Walter de Gruyter & Co., Berlin (2011)CrossRefGoogle Scholar
  5. 5.
    Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181(2), 507–528 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Giacomin, G., Lebowitz, J.L., Maes, C.: Agreement percolation and phase coexistence in some Gibbs systems. J. Stat. Phys. 80(5–6), 1379–1403 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Grimmett, G.R.: A theorem about random fields. Bull. Lond. Math. Soc. 5, 81–84 (1973)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hirsch, C., Jahnel, B.: Large deviations for the capacity in spatial relay networks. Markov Process. Relat. Fields 25(1), 33–73 (2019)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hirsch, C., Jahnel, B., Patterson, R.I.A., Keeler, P.: Traffic flow densities in large transport networks. Adv. Appl. Probab. 49(4), 1091–1115 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jahnel, B., Külske, C.: The Widom–Rowlinson model under spin flip: immediate loss and sharp recovery of quasilocality. Ann. Appl. Probab. 27(6), 3845–3892 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kozlov, O.K.: A Gibbs description of a system of random variables. Problemy Peredači Informacii 10(3), 94–103 (1974)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kozlov, O.K.: A Gibbsian description of point random fields. Teor. Verojatnost. i Primenen. 21(2), 348–365 (1976)MathSciNetGoogle Scholar
  13. 13.
    Külske, C.: Weakly Gibbsian representations for joint measures of quenched lattice spin models. Probab. Theory Relat. Fields 119(1), 1–30 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lebowitz, J.L., Mazel, A., Presutti, E.: Liquid-vapor phase transitions for systems with finite-range interactions. J. Stat. Phys. 94(5–6), 955–1025 (1999)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27(16), 1040–1041 (1971)CrossRefGoogle Scholar
  17. 17.
    Ruelle, D.: Statistical Mechanics. World Scientific Publishing Co., Inc., River Edge; Imperial College Press, London (1999). Rigorous results, Reprint of the 1989 editionGoogle Scholar
  18. 18.
    Sullivan, W.G.: Potentials for almost Markovian random fields. Commun. Math. Phys. 33, 61–74 (1973)MathSciNetCrossRefGoogle Scholar
  19. 19.
    van Enter, A.C.D., Fernández, R., den Hollander, F., Redig, F.: Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures. Commun. Math. Phys. 226(1), 101–130 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52(4), 1670–1684 (1970)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations