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


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.


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



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).


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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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