  The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point
process in $\mathbb{R}^d$ with the formal Hamiltonian
$H(\omega)=\text{Volume}(\cup_{x\in\omega} B_1(x))$, where $\omega$ is a
locally finite configuration of points and $B_1(x)$ denotes the unit closed
ball centred at $x$. The model is tuned by two parameters: the activity $z>0$
and the inverse temperature $\beta\ge 0$. We investigate the phase transition
of the model in the point of view of percolation theory and the liquid-gas
transition. First, considering the graph connecting points with distance
smaller than $2r>0$, we show that for any $\beta>0$, there exists
$0<\tilde{z}^a(\beta, r)<+\infty$ such that an exponential decay of
connectivity at distance $n$ occurs in the subcritical phase and a linear lower
bound of the connection at infinity holds in the supercritical case. Secondly
we study a standard liquid-gas phase transition related to the
uniqueness/non-uniqueness of Gibbs states depending on the parameters
$z,\beta$. Old results claim that a non-uniqueness regime occurs for $z=\beta$
large enough and it is conjectured that the uniqueness should hold outside such
an half line ($z=\beta\ge \beta_c>0$). We solve partially this conjecture by
showing that for $\beta$ large enough the non-uniqueness holds if and only if
$z=\beta$. We show also that this critical value $z=\beta$ corresponds to the
percolation threshold $ \tilde{z}^a(\beta, r)=\beta$ for $\beta$ large enough,
providing a straight connection between these two notions of phase transition.
