Kybernetika 59 no. 1, 130-159, 2023

A note on the existence of Gibbs marked point processes with applications in stochastic geometry

Martina PetrákováDOI: 10.14736/kyb-2023-1-0130

Abstract:

This paper generalizes a recent existence result for infinite-volume marked Gibbs point processes. We try to use the existence theorem for two models from stochastic geometry. First, we show the existence of Gibbs facet processes in~$\mathbb{R}^d$ with repulsive interactions. We also prove that the~finite-volume Gibbs facet processes with attractive interactions need not exist. Afterwards, we study Gibbs--Laguerre tessellations of $\mathbb{R}^2$. The mentioned existence result cannot be used, since one of its assumptions is not satisfied for tessellations, but we are able to show the existence of an infinite-volume Gibbs--Laguerre process with a particular energy function, under the assumption that we almost surely see a~point.

Keywords:

existence, Gibbs-Laguerre tessellation, infinite-volume Gibbs measure, Gibbs facet process

Classification:

60D05, 60G55

paper.pdf

References:

  1. D. Dereudre: The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. Appl.Probab. 41 (2009), 3, 664-681.   DOI:10.1017/S0001867800003517
  2. D. Dereudre: Introduction to the theory of Gibbs point processes. In: Stochastic Geometry: Modern Research Frontiers, (D. Coupier, ed.), Springer International Publishing, Cham 2019, pp 181-229.   DOI:10.1007/978-3-030-13547-8\_5
  3. D. Dereudre, R. Drouilhet and H. O. Georgii: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Related Fields 153 (2012), 3, 643-670.   DOI:10.1007/s00440-011-0356-5
  4. H. O. Georgii and H. Zessin: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Related Fields 96 (1993), 2, 177-204.   DOI:10.1007/BF01192132
  5. D. Jahn and F. Seitl: Existence and simulation of Gibbs-Delaunay-Laguerre tessellations. Kybernetika 56 (2020), 4, 617-645.   DOI:10.14736/kyb-2020-4-0617
  6. C. Lautensack: Random Laguerre Tessellations. PhD Thesis, University of Karlsruhe, 2007.   CrossRef
  7. J. Moller: Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics, Springer-Verlag, New York 1994.   CrossRef
  8. J. Moller and R. P. Waagepetersen: Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability. Chapman and Hall/CRC, Boca Raton 2004.   CrossRef
  9. S. Roelly and A. Zass: Marked Gibbs point processes with unbounded interaction: an existence result. J. Statist. Physics 179 (2020), 4, 972-996.   DOI:10.1007/s10955-020-02559-3
  10. D. Ruelle: Statistical Mechanics: Rigorous Results. W. A. Benjamin, Inc., New York - Amsterdam 1969.   CrossRef
  11. R. Schneider: Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge 1993.   CrossRef
  12. R. Schneider and W. Weil: Stochastic and Integral Geometry. Probability and its Applications (New York). Springer-Verlag, Berlin 2008.   CrossRef
  13. J. Večeřa and V. Beneš: Interaction processes for unions of facets, the asymptotic behaviour with increasing intensity. Methodology Computing Appl. Probab. 18 (2016), 4, 1217-1239.   DOI:10.1007/s11009-016-9485-8
  14. H. Zessin: Point processes in general position. J. Contempor. Math. Anal. 43 (2008), 1, 59-65.   DOI:10.3103/s11957-008-1005-x