Spatial delaunay gibbs point processes

Abstract

This paper studies two types of Delaunay Gibbs point processes originally introduced by Baddeley and M⊘ller (1989) by combining stochastic geometry and computational geometry arguments. The energy function of these processes is for a given point pattern φ given by a sum of potentials associated to a subclass of the cliques, called “Delaunay-cliques”, given by the Delaunay graph correponding to the point pattern (empty set, singletons, Delaunay edges, and Delaunay triangles). This restriction is necessary in order that the Delaunay point process becomes Markov in the sense of Baddeley and M⊘ller (1989). We demonstrate that the Markov property is satisfied when interactions are only permitted for “Delaunay–cliques” and to further establish a Hammersley-Clifford type theorem for the Delaunay Gibbs point processes. Furthermore local stability is studied and simulations are given