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

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