Phase transition for continuum Widom-Rowlinson model with random radii

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.