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R-Local Delaunay Inhibition Model

  • Etienne Bertin
  • Jean-Michel Billiot
  • Rémy Drouilhet
Article

Abstract

Unlike in the classical framework of Gibbs point processes (usually acting on the complete graph), in the context of nearest-neighbour Gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property. This paper introduces domain-wise (but not pointwise) inhibition stationary Gibbs models based on some tailor-made Delaunay subgraphs. All of them are subgraphs of the R-local Delaunay graph, defined as the Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circles of radii greater than some large positive real value R. The usual relative compactness criterion for point processes needed for the existence result is directly derived from the Ruelle-bound of the correlation functions. Furthermore, assuming only the nonnegativeness of the energy function, we have managed to prove the existence of the existence of R-local Delaunay stationary Gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy.

Keywords

Stationary Gibbs states Inhibition property Delaunay triangulation D.L.R. equations Local specifications Correlation functions 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Etienne Bertin
    • 1
  • Jean-Michel Billiot
    • 1
  • Rémy Drouilhet
    • 1
  1. 1.Laboratory Jean Kuntzman, Department of Statistics, SAGAG teamUniversité Pierre Mendès France, Grenoble IIGrenoble cedex 9France

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