R-local Delaunay inhibition model

TL;DR

This paper introduces the R-local Delaunay inhibition model, a modified Gibbs point process with a new local stability property, enabling the proof of existence of Gibbs states despite challenges in traditional methods.

Contribution

It defines the R-local Delaunay graph and demonstrates its use in establishing Gibbs state existence under a new R-local stability condition.

Findings

Introduces R-local Delaunay graph as a modification of Delaunay subgraph.

Establishes R-local stability as a key property for Gibbs state existence.

Proves existence of Gibbs states for the R-local Delaunay inhibition model.

Abstract

Let us consider the local specification system of Gibbs point process with inhib ition pairwise interaction acting on some Delaunay subgraph specifically not con taining the edges of Delaunay triangles with circumscribed circle of radius grea ter than some fixed positive real value $R$. Even if we think that there exists at least a stationary Gibbs state associated to such system, we do not know yet how to prove it mainly due to some uncontrolled "negative" contribution in the expression of the local energy needed to insert any number of points in some large enough empty region of the space. This is solved by introducing some subgraph, called the $R$-local Delaunay graph, which is a slight but tailored modification of the previous one. This kind of model does not inherit the local stability property but satisfies s ome new extension called $R$-local stability. This weakened property combined with the local property provides the existence o f Gibbs state.