R-Local Delaunay Inhibition Model

  • Etienne Bertin
  • Jean-Michel Billiot
  • Rémy Drouilhet


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.


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


  1. 1.
    Baddeley, A.J., Møller, J.: Nearest-neighbour Markov point processes and random sets. Int. Stat. Rev. 57(2), 89–121 (1989) zbMATHCrossRefGoogle Scholar
  2. 2.
    Bertin, E., Billiot, J.-M., Drouilhet, R.: Existence of Delaunay pairwise Gibbs point processes with superstable component. J. Stat. Phys. 95, 719–744 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertin, E., Billiot, J.-M., Drouilhet, R.: Existence of “nearest-neighbour” Gibbs point models. Adv. Appl. Prob. 31, 895–909 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertin, E., Billiot, J.-M., Drouilhet, R.: Phase transition in nearest-neighbour continuum Potts models. J. Stat. Phys. 114(1/2), 79–100 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Connelly, R., Rybnikov, K., Volkov, S.: Percolation of the loss of tension in an infinite triangular lattice. J. Stat. Phys. 105(1/2), 143–171 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer, New York (1988) Google Scholar
  7. 7.
    Georgii, H.-O.: Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51 (1976) CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528 (1996) zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Georgii, H.-O., Zagrebnov, V.A.: On the interplay of magnetic and molecular forces in Curie-Weiss ferrofluid models. J. Stat. Phys. 93, 79–107 (1998) zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Georgii, H.-O., Zessin, H.: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theor. Relat. Fields 96, 177–204 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kendall, W.S., van Lieshout, M.N.M., Baddeley, A.J.: Quermass-interaction processes: conditions for stability. Adv. Appl. Probab. 31, 315–342 (1999) zbMATHCrossRefGoogle Scholar
  12. 12.
    Menshikov, M., Rybnikov, K., Volkov, S.: The loss of tension in an infinite membrane with holes distributed according to a Poisson law. Adv. Appl. Probab. 34(2), 292–312 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Preston, C.J.: Random Fields, vol. 534. Springer, Berlin (1976) zbMATHGoogle Scholar
  14. 14.
    Ruelle, D.: Statistical Mechanics. Benjamin, New York (1969) zbMATHGoogle Scholar
  15. 15.
    Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970) zbMATHCrossRefADSMathSciNetGoogle Scholar

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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