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

Kybernetika (2023)

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

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topPetráková, 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 -

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