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

@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 -