Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T15:21:28.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Percolation results for the continuum random cluster model

Part of: Equilibrium statistical mechanics Special processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  20 March 2018

Pierre Houdebert
Pierre Houdebert*
Affiliation:
Université Lille 1
*
* Current address: Centre de Mathématiques et Informatique (CMI), Aix-Marseille Université, Technopôle Château-Gombert, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

Keywords

Gibbs point processWidom–Rowlinson modelstochastic comparisonFortuin–Kasteleyn representation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G10: Stationary processes 60G55: Point processes 60G57: Random measures 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.) 82B26: Phase transitions (general) 82B43: Percolation
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 231 - 244
Copyright
Copyright © Applied Probability Trust 2018 

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.)

References

[1] Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Commun. Math. Phys. 121, 501–505. Google Scholar
[2] Chayes, J. T., Chayes, L. and Kotecký, R. (1995). The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551–569. Google Scholar
[3] Coupier, D. and Dereudre, D. (2014). Continuum percolation for quermass interaction model. Electron. J. Prob. 19, 35. Google Scholar
[4] Dereudre, D. and Houdebert, P. (2015). Infinite volume continuum random cluster model. Electron. J. Prob. 20, 125. Google Scholar
[5] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd edn. De Gruyter, Berlin. CrossRefGoogle Scholar
[6] Georgii, H.-O. and Häggström, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528. CrossRefGoogle Scholar
[7] Georgii, H.-O. and Küneth, T. (1997). Stochastic comparison of point random fields. J. Appl. Prob. 34, 868–881. Google Scholar
[8] Georgii, H.-O., Häggström, O. and Maes, C. (2001). The random geometry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18, Academic Press, San Diego, CA, pp. 1–142. Google Scholar
[9] Gouéré, J.-B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Prob. 36, 1209–1220. CrossRefGoogle Scholar
[10] Grimmett, G. (2006). The Random-Cluster Model. Springer, Berlin. Google Scholar
[11] Jansen, S. (2016). Continuum percolation for Gibbsian point processes with attractive interactions. Electron. J. Prob. 21, 47. Google Scholar
[12] Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95. Google Scholar
[13] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press. Google Scholar
[14] Möller, J. and Helisová, K. (2008). Power diagrams and interaction processes for unions of discs. Adv. Appl. Prob. 40, 321–347. Google Scholar
[15] Möller, J. and Helisová, K. (2010). Likelihood inference for unions of interacting discs. Scand. J. Statist. 37, 365–381. CrossRefGoogle Scholar
[16] Nguyen, X.-X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115. Google Scholar
[17] Stucki, K. (2013). Continuum percolation for Gibbs point processes. Electron. Commun. Prob. 18, 67. Google Scholar
[18] Van den Berg, J. and Maes, C. (1994). Disagreement percolation in the study of Markov fields. Ann. Prob. 22, 749–763. Google Scholar
[19] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684. Google Scholar