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.
Highlights
It is well-known that stationary renewal processes with a reasonable spacing distribution can be characterised as Gibbs processes for an interaction between nearest-neighbour pairs of points [16, Section 6]
We consider an analogue in two dimensions, viz. Gibbsian point processes on 2 with an interaction depending on nearest-neighbour triples of points, where the nearest-neighbour triples are defined in terms of the Delaunay triangulation
Recall that the Delaunay triangulation is dual to the Voronoi tessellation, in the sense that two points are connected by a Delaunay edge if and only if their Voronoi cells have a common edge
Summary
It is well-known that stationary renewal processes with a reasonable spacing distribution can be characterised as Gibbs processes for an interaction between nearest-neighbour pairs of points [16, Section 6]. In the Delaunay case, the particle at x gives rise to some new tiles of the Delaunay triangulation, and destroys some other tiles that were present in the triangulation of ω This so-called non-hereditary nature of the Delaunay triangulation blurs the usual distinction between attractive and repulsive interactions and makes it difficult to use a local characterisation of Gibbs measures in terms of their Campbell measures and Papangelou intensities. Such a local approach to the existence of Gibbs measures for Delaunay interactions was used in the previous work [2; 3; 5; 7] and made it necessary to impose geometric constraints on the interaction by removing triangles with small angles or large circumcircles. Let us emphasise that we make repeated use of Euler’s polyhedral formula and the resulting linear complexity of the Delaunay triangulations, and are limited to two dimensions, as was already the case in the previous papers mentioned above
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