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

TL;DR

This paper extends existence results for infinite-volume marked Gibbs point processes, demonstrating their application in stochastic geometry models like Gibbs facet processes and Gibbs-Laguerre tessellations.

Contribution

It generalizes a recent existence theorem and applies it to new models, showing existence under specific conditions and highlighting limitations for certain interactions.

Findings

Existence of Gibbs facet processes with repulsive interactions in .

Finite-volume Gibbs facet processes with attractive interactions may not exist.

Existence of an infinite-volume Gibbs-Laguerre process under specific assumptions.

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.