Variational characterisation of Gibbs measures with Delaunay triangle interaction

TL;DR

This paper characterizes Gibbs point processes on the plane with Delaunay triangle interactions, showing they include all free energy minimizers and establishing conditions for their existence and optimality.

Contribution

It provides a variational characterization of Gibbs measures with Delaunay triangle interactions, linking free energy minimizers to the class of such Gibbs processes.

Findings

Gibbs processes with Delaunay triangle interactions include all free energy minimizers.

Existence of these Gibbs processes is guaranteed under certain conditions.

Minimizers of free energy density correspond to specific Gibbs measures.

Abstract

This paper deals with stationary Gibbsian point processes on the plane with an interaction that depends on the tiles of the Delaunay triangulation of points via a bounded triangle potential. It is shown that the class of these Gibbs processes includes all minimisers of the associated free energy density and is therefore nonempty. Conversely, each such Gibbs process minimises the free energy density, provided the potential satisfies a weak long-range assumption.