Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-08T20:51:37.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuum percolation in the Gabriel graph

Part of: Sufficiency and information Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Etienne Bertin ,
Jean-Michel Billiot  and
Rémy Drouilhet
Etienne Bertin*
Affiliation:
Université Pierre Mendès France
Jean-Michel Billiot*
Affiliation:
Université Pierre Mendès France
Rémy Drouilhet*
Affiliation:
Université Pierre Mendès France
*
∗ Postal address: Labsad, BSHM, Université Pierre Mendès France, 1251 avenue centrale, BP 47, 38040 Grenoble Cedex 9, France.
∗ Postal address: Labsad, BSHM, Université Pierre Mendès France, 1251 avenue centrale, BP 47, 38040 Grenoble Cedex 9, France.
∗ Postal address: Labsad, BSHM, Université Pierre Mendès France, 1251 avenue centrale, BP 47, 38040 Grenoble Cedex 9, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present study, we establish the existence of site percolation in the Gabriel graph for Poisson and hard-core stationary point processes.

Keywords

PercolationGabriel graphhard-core point processPoisson point process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62B05: Sufficient statistics and fields

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 689 - 701
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Baddeley, A. and Moller, J. (1989). Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 57, 89–121.CrossRefGoogle Scholar
[2] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). Existence of Delaunay pairwise Gibbs point processes with superstable component. J. Statist. Phys. 95, 719–744.Google Scholar
[3] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). Existence of ‘nearest-neighbour’ spatial Gibbs models. Adv. Appl. Prob. 31, 895–909.Google Scholar
[4] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). k-nearest-neighbour Gibbs point processes. Markov Process. Relat. Fields 5, 219–234.Google Scholar
[5] Bertin, E., Billiot, J.-M. and Drouilhet, R. (1999). Spatial Delaunay Gibbs point processes. Stoch. Models 15, 181–199.Google Scholar
[6] Bertin, E., Billiot, J.-M. and Drouilhet, R. (2002). Phase transition in the Delaunay continuum Potts models. Submitted.Google Scholar
[7] Chayes, J., Chayes, L. and Kotecky, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569.CrossRefGoogle Scholar
[8] Döge, G., et al. (2000). Grand canonical simulations of hard-disk systems by simulated tempering. Tech. Rep. R-00-2003, Aalborg University.Google Scholar
[9] Georgii, H.-O. (2000). Phase transition and percolation in Gibbsian particle models. In Statistical Physics and Spatial Statistics (Lecture Notes Phys. 554), eds Mecke, K. R. and Stoyan, D., Springer, Berlin, pp. 267–294.CrossRefGoogle Scholar
[10] Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528.Google Scholar
[11] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18, eds Domb, C. and Lebowitz, J., Academic Press, London, pp. 1–142.CrossRefGoogle Scholar
[12] Geyer, C. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry, Likelihood and Computation, eds Kendall, W. S., Barndoff-Nielsen, O. E. and van Lieshout, M. N. M., Chapman and Hall, London, pp. 141–172.Google Scholar
[13] Geyer, C. and Moller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
[14] Grimmet, G. (1995). The stochastic random-cluster process, and uniqueness of random cluster measures. Ann. Prob. 23, 1461–1510.CrossRefGoogle Scholar
[15] Grimmet, G. (1999). Percolation, 2nd edn. Springer, New York.Google Scholar
[16] Häggström, O., (2000). Markov random fields and percolation on general graphs. Adv. Appl. Prob. 32, 39–66.Google Scholar
[17] Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9, 295–315.3.0.CO;2-S>CrossRefGoogle Scholar
[18] Lyons, R. and Peres, Y. (2002). Probability on Trees and Networks. Cambridge University Press.Google Scholar
[19] Mase, S. et al., (2001). Packing densities and simulated tempering for hard core Gibbs point processes. Ann. Inst. Statist. Math. 53, 661–680.CrossRefGoogle Scholar
[20] Meester, R. W. J. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.CrossRefGoogle Scholar
[21] Möller, J., (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
[22] Ruelle, D. (1969). Statistical Mechanics. Benjamin, New York.Google Scholar
[23] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159.CrossRefGoogle Scholar
[24] Sakamoto, S., Yonezawa, F. and Hori, M. (1989). A proposal for the estimation of percolation thresholds in two-dimensional lattices. J. Phys. A 22, L699–L704.CrossRefGoogle Scholar