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

Martina Petráková

Kybernetika (2023)

  • Volume: 59, Issue: 1, page 130-159
  • ISSN: 0023-5954

Abstract

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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 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 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.

How to cite

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Petráková, Martina. "A note on the existence of Gibbs marked point processes with applications in stochastic geometry." Kybernetika 59.1 (2023): 130-159. <http://eudml.org/doc/299084>.

@article{Petráková2023,
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.},
author = {Petráková, Martina},
journal = {Kybernetika},
keywords = {infinite-volume Gibbs measure; existence; Gibbs facet process; Gibbs–Laguerre tessellation},
language = {eng},
number = {1},
pages = {130-159},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on the existence of Gibbs marked point processes with applications in stochastic geometry},
url = {http://eudml.org/doc/299084},
volume = {59},
year = {2023},
}

TY - JOUR
AU - Petráková, Martina
TI - A note on the existence of Gibbs marked point processes with applications in stochastic geometry
JO - Kybernetika
PY - 2023
PB - Institute of Information Theory and Automation AS CR
VL - 59
IS - 1
SP - 130
EP - 159
AB - 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.
LA - eng
KW - infinite-volume Gibbs measure; existence; Gibbs facet process; Gibbs–Laguerre tessellation
UR - http://eudml.org/doc/299084
ER -

References

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  10. Ruelle, D., Statistical Mechanics: Rigorous Results., W. A. Benjamin, Inc., New York - Amsterdam 1969. MR0289084
  11. Schneider, R., Convex Bodies: the Brunn-Minkowski Theory., Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge 1993. MR1216521
  12. Schneider, R., Weil, W., Stochastic and Integral Geometry., Probability and its Applications (New York). Springer-Verlag, Berlin 2008. Zbl1175.60003MR2455326
  13. Večeřa, J., Beneš, V., , Methodology Computing Appl. Probab. 18 (2016), 4, 1217-1239. MR3564860DOI
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