Phase transition for continuum Widom-Rowlinson model with random radii

TL;DR

This paper investigates phase transitions in a continuum Widom-Rowlinson model with random radii, revealing different behaviors in integrable and non-integrable cases and establishing conditions for phase coexistence and uniqueness.

Contribution

It characterizes phase transitions in the Widom-Rowlinson model with random radii, including the existence of multiple phases and their dependence on activity and integrability conditions.

Findings

Standard phase transition with uniqueness at low activity in integrable case.

Existence of multiple extremal phases at low activity in non-integrable case.

Symmetric measure is a mixture of ordered phases at high activity.

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 particles of same type do not interact each other whereas particles $x_i$ and $x_j$ with different type $i \neq j$ interact via an exclusion hardcore interaction forcing $r_i+r_j$ to be smaller than $|x_i-x_j|$. In the integrable case (i.e. $\int r^d Q_i(dr)<+\infty$, $1\le i\le q$), 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. $\int r^d Q_i(dr)=+\infty$, $1\le i \le q$), 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 by showing that the symmetric Widom-Rowlinson measure with free boundary condition is a mixing of the $q$ ordered phases if and only if the activity is large.