Geodesic
A note on the existence of Gibbs marked point processes with applications in stochastic geometry
Kybernetika, Tome 59 (2023) no. 1, pp. 130-159
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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 $\mathbb{R}^d$ 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.
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.
DOI : 10.14736/kyb-2023-1-0130
Classification : 60D05, 60G55
Keywords: infinite-volume Gibbs measure; existence; Gibbs facet process; Gibbs–Laguerre tessellation 
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Petráková, Martina. A note on the existence of Gibbs marked point processes with applications in stochastic geometry. Kybernetika, Tome 59 (2023) no. 1, pp. 130-159. doi: 10.14736/kyb-2023-1-0130

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