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Percolation results for the continuum random cluster model

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

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: pierre.houdebert@gmail.com
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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 

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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