Electronic Journal of Probability

Continuum percolation for quermass interaction model

David Coupier and David Dereudre

Full-text: Open access

Abstract

The continuum percolation for Markov (or Gibbs) germ-grain models in dimension 2 is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincaré characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type quermass model is given.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 35, 19 pp.

Dates
Accepted: 19 March 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065677

Digital Object Identifier
doi:10.1214/EJP.v19-2298

Mathematical Reviews number (MathSciNet)
MR3183579

Zentralblatt MATH identifier
1291.60201

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B05: Classical equilibrium statistical mechanics (general) 82B21: Continuum models (systems of particles, etc.) 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Keywords
Stochastic geometry Gibbs point process germ-grain model Quermass interaction percolation phase transition

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Coupier, David; Dereudre, David. Continuum percolation for quermass interaction model. Electron. J. Probab. 19 (2014), paper no. 35, 19 pp. doi:10.1214/EJP.v19-2298. https://projecteuclid.org/euclid.ejp/1465065677


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References

  • Baddeley, A. J.; van Lieshout, M. N. M. Area-interaction point processes. Ann. Inst. Statist. Math. 47 (1995), no. 4, 601–619.
  • Chayes, J. T.; Chayes, L.; Kotecky, R. The analysis of the Widom-Rowlinson model by stochastic geometric methods. Comm. Math. Phys. 172 (1995), no. 3, 551–569.
  • Dereudre, David. The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. in Appl. Probab. 41 (2009), no. 3, 664–681.
  • Georgii, H.-O.; Haggstrom, O. Phase transition in continuum Potts models. Comm. Math. Phys. 181 (1996), no. 2, 507–528.
  • Georgii, Hans-Otto; Kuneth, Torsten. Stochastic comparison of point random fields. J. Appl. Probab. 34 (1997), no. 4, 868–881.
  • Giacomin, G.; Lebowitz, J. L.; Maes, C. Agreement percolation and phase coexistence in some Gibbs systems. J. Statist. Phys. 80 (1995), no. 5-6, 1379–1403.
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
  • Hall, Peter. On continuum percolation. Ann. Probab. 13 (1985), no. 4, 1250–1266.
  • Kendall, W. S.; van Lieshout, M. N. M.; Baddeley, A. J. Quermass-interaction processes: conditions for stability. Adv. in Appl. Probab. 31 (1999), no. 2, 315–342.
  • Liggett, T. M.; Schonmann, R. H.; Stacey, A. M. Domination by product measures. Ann. Probab. 25 (1997), no. 1, 71–95.
  • C. N. Likos, K. R. Mecke, and H. Wagner. Statistical morphology of random interface microemulsions. J. Chem. Phys., pages 9350–9361, 1995.
  • K. R. Mecke. A morphological model for complex fluids. J. Phys. Condens. Matter 8, pages 9663–9667, 1996.
  • Meester, Ronald; Roy, Rahul. Uniqueness of unbounded occupied and vacant components in Boolean models. Ann. Appl. Probab. 4 (1994), no. 3, 933–951.
  • Meester, Ronald; Roy, Rahul. Continuum percolation. Cambridge Tracts in Mathematics, 119. Cambridge University Press, Cambridge, 1996. x+238 pp. ISBN: 0-521-47504-X
  • Preston, Chris. Random fields. Lecture Notes in Mathematics, Vol. 534. Springer-Verlag, Berlin-New York, 1976. ii+200 pp.
  • B. Widom and J. S. Rowlinson. New model for the study of liquid-vapor phase transitions. J. Chem. Phys., pages 1670–1684, 1970.