  We consider the hard-core model in $\mathbb{R}^2$, in which a random set of
non-intersecting unit disks is sampled with an intensity parameter $\lambda$.
Given $\varepsilon>0$ we consider the graph in which two disks are adjacent if
they are at distance $\leq \varepsilon$ from each other. We prove that this
graph, $G$, is highly connected when $\lambda$ is greater than a certain
threshold depending on $\varepsilon$. Namely, given a square annulus with inner
radius $L_1$ and outer radius $L_2$, the probability that the annulus is
crossed by $G$ is at least $1 - C \exp(-cL_1)$. As a corollary we prove that a
Gibbs state admits an infinite component of $G$ if the intensity $\lambda$ is
large enough, depending on $\varepsilon$.
