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Journal of Statistical Physics

, Volume 174, Issue 1, pp 56–76 | Cite as

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

  • David DereudreEmail author
  • Pierre Houdebert
Article

Abstract

In this paper we study the phase transition of continuum Widom–Rowlinson measures in \(\mathbb {R}^d\) with \(q\) types of particles and random radii. Each particle \(x_i\) of type i is marked by a random radius \(r_i\) distributed by a probability measure \(Q_i\) on \(\mathbb {R}^+\). The distributions \(Q_i\) may be different for different i, this setting is called the non-symmetric case. The particles of same type do not interact with each other whereas a particle \(x_i\) and \(x_j\) with different type \(i\ne j\) interact via an exclusion hardcore interaction forcing \(r_i+r_j\) to be smaller than \(|x_i-x_j|\). In the symmetric integrable case (i.e. \(\int r^dQ_1(dr)<+\infty \) and \(Q_i=Q_1\) for every \(1\le i\le q\)), we show that the Widom–Rowlinson measures exhibit a standard phase transition providing uniqueness, when the activity is small, and co-existence of q ordered phases, when the activity is large. In the non-integrable case (i.e. \(\int r^dQ_i(dr)=+\infty \), \(1\le i \le q\)), we show another type of phase transition. We prove, when the activity is small, the existence of at least \(q+1\) extremal phases and we conjecture that, when the activity is large, only the q ordered phases subsist. We prove a weak version of this conjecture in the symmetric case by showing that the Widom–Rowlinson measure with free boundary condition is a mixing of the \(q\) ordered phases if and only if the activity is large.

Keywords

Gibbs point process DLR equation Boolean model Continuum percolation Random cluster model Fortuin–Kasteleyn representation 

Notes

Acknowledgements

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the GDR 3477 Geosto and the ANR project PPPP (ANR-16-CE40-0016).

References

  1. 1.
    Bricmont, J., Kuroda, K., Lebowitz, J.L.: The structure of Gibbs states and phase coexistence for nonsymmetric continuum Widom-Rowlinson models. Z. Wahrsch. Verw. Gebiete 67(2), 121–138 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chayes, J.T., Chayes, L., Kotecký, R.: The analysis of the Widom–Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172(3), 551–569 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley Series in Probability and Statistics, 3rd edn. Wiley, Chichester (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer, New York (1988)zbMATHGoogle Scholar
  5. 5.
    Dereudre, D.: The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. Appl. Probab. 41(3), 664–681 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dereudre, D., Houdebert, P.: Infinite volume continuum random cluster model. Electron. J. Probab. 20(125), 24 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dereudre, D., Drouilhet, R., Georgii, H.-O.: Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153(3–4), 643–670 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fitzsimmons, P.J., Fristedt, B., Shepp, L.A.: The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete 70(2), 175–189 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, vol. 2, 9th edn. Walter de Gruyter & Co., Berlin (2011)CrossRefGoogle Scholar
  10. 10.
    Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181(2), 507–528 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Georgii, H.-O., Küneth, J.M.: Stochastic comparison of point random fields. J. Appl. Probab. 34(4), 868–881 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Georgii, H.-O., Zessin, H.: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96(2), 177–204 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gouéré, J.-B.: Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36(4), 1209–1220 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hofer-Temmel, C., Houdebert, P.: Disagreement percolation for marked Gibbs point processes. ArXiv e-prints (2017)Google Scholar
  15. 15.
    Houdebert, P.: Continuum random cluster model. PhD thesis (2017)Google Scholar
  16. 16.
    Houdebert, P.: Percolation results for the continuum random cluster model. Adv. Appl. Probab. 50(1), 231–244 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mazel, A., Suhov, Y., Stuhl, I.: A classical WR model with $q$ particle types. J. Stat. Phys. 159(5), 1040–1086 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ruelle, D.: Statistical Mechanics: Rigorous Results. W. A. Benjamin Inc., New York (1969)zbMATHGoogle Scholar
  19. 19.
    Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041 (1971)ADSCrossRefGoogle Scholar
  20. 20.
    Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684 (1970)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Paul PainlevéUniversity of LilleVilleneuve d’AscqFrance
  2. 2.Aix Marseille University, CNRS, Centrale Marseille, I2MMarseilleFrance

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