$e^{-czn}$ (see for instance Grimmet [@G99]).

Now, the number of points in $\R_n$, say $R_n$, is the sum of independent Poisonian random variables. This is less or equal to $M_m$, the maximum of the number of points in $\R$ over all connected regions $\R$ intersecting at most $m$ boxes the origin $\0$. Thus, on the event that $\R_n$ contains less than $zn$ boxes, we have $R_n\leq M_{zn}$. On the other hand, $M_m$ can be seen as a Greedy lattice animal model and for such a model we can also show, for large $\bar{c}>0$, that the probability that $M_{m}\geq \bar{c}m$ decays as $e^{-cm}$ (Lemma 2.3 of [@p-205]).

By cooking together the arguments in these two last paragraphs one obtains that the probability that the number of points in $\R_n$ is greater than $zn$ also decays as $e^{-czn}$, for some constant $c>0$ and sufficiently large $z$.

Let $T_\calD$ denote the graph metric on $\calD$, i.e. for $\v,\bar{\v}\in\calD_v$, $T_\calD(\v,\bar{\v})$ is the minimum number of edges that one path should pass to go from $\v$ to $\bar{\v}$. Notice that $T_\calD(\v,\bar{\v})$ is the first-passage time between $\v$ and $\bar{\v}$ if one associates to each edge $\e$ the passage time value $1$. For each $\A,\B\subseteq\bR^2$ we 