Existence of Gibbsian point processes with geometry-dependent interactions

TL;DR

This paper proves the existence of stationary Gibbsian point processes with complex, geometry-dependent interactions, extending classical pair interaction models to include hyperedges like Delaunay edges and Voronoi cliques.

Contribution

It establishes the existence of Gibbsian point processes with geometry-dependent interactions, broadening the scope beyond classical pairwise models.

Findings

Existence of Gibbsian point processes with hyperedge interactions proven.

Applicable to Delaunay edges, Voronoi cliques, and k-nearest neighbor clusters.

Utilizes entropy bounds and stationarity as key tools.

Abstract

We establish the existence of stationary Gibbsian point processes for interactions that act on hyperedges between the points. For example, such interactions can depend on Delaunay edges or triangles, cliques of Voronoi cells or clusters of $k$-nearest neighbors. The classical case of pair interactions is also included. The basic tools are an entropy bound and stationarity.