Skip to main content
Log in

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baddeley, A.J., Møller, J.: Nearest-neighbour Markov point processes and random sets. Int. Stat. Rev. 57(2), 89–121 (1989)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertin, E., Billiot, J.-M., Drouilhet, R.: Existence of “nearest-neighbour” Gibbs point models. Adv. Appl. Prob. 31, 895–909 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Georgii, H.-O.: Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  8. Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528 (1996)

    Article  MATH  ADS  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Preston, C.J.: Random Fields, vol. 534. Springer, Berlin (1976)

    MATH  Google Scholar 

  14. Ruelle, D.: Statistical Mechanics. Benjamin, New York (1969)

    MATH  Google Scholar 

  15. Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rémy Drouilhet.

Additional information

In memory of Etienne Bertin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bertin, E., Billiot, JM. & Drouilhet, R. R-Local Delaunay Inhibition Model. J Stat Phys 132, 649–667 (2008). https://doi.org/10.1007/s10955-008-9565-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9565-4

Keywords

Navigation