A Note on the Existence of Gibbs Marked Point Processes with Applications in Stochastic Geometry

Martina Petráková

Published 2022-08-04Version 1

This paper considers a recent existence result for infinite-volume marked Gibbs point processes. We reformulate its problematic assumption and 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.