Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

Abstract

In this paper we study the phase transition of continuum Widom–Rowlinson measures in Rd with q types of particles and random radii. Each particle xi of type i is marked by a random radius ri distributed by a probability measure Qi on R+. The distributions Qi 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 xi and xj with different type i≠ j interact via an exclusion hardcore interaction forcing ri+ rj to be smaller than | xi- xj|. In the symmetric integrable case (i.e. ∫ rdQ1(dr) < + ∞ and Qi= Q1 for every 1 ≤ i≤ 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. ∫ rdQi(dr) = + ∞, 1 ≤ i≤ 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.