say that $\Lambda:=(\B_{\z_1}^{L},\dots,\B_{\z_k}^{L})$ is a circuit of boxes if $(\z_1,\dots,\z_k)$ is a circuit in $\bZ^2$ (in the usual sense). Let $\lambda$ be the closed polygonal path composed by the line segments connecting $L\z_j$ to $L\z_{j+1}$, where $j=1,\dots,k-1$, together with $[\z_k,\z_1]$. To each circuit $\Lambda$ we associate two subsets of the plane: $\Lambda^{in}$ denotes the interior of the bounded component of $\bR^2\backslash\cup_{j=1}^k\B_{\z_j}^{L}$ while $\lambda^{in}$ will denote the interior of the bounded component of $\bR^2\backslash\lambda$. Now, assume that $\Lambda:=(\B_{\z_j}^{L})_{j=1}^k$ is a circuit composed by full boxes. By Lemma 2.1 in [@p-205], we have the following geometrical property: if $\C_{\v}\cap \Lambda^{in}\neq\emptyset$ then $\C_{\v}\subseteq \lambda^{in}$. One important consequence of this is that the set of vertices used by $\gamma(\0,n\vec{e}_1)$ or by $\calE_n$ are both contained in the region $\R_n$ limited by the smallest circuit of full boxes surrounding the line segment $[\0,n\vec{e}_1]$. Therefore to show Lemma \[lgraph2\] is enough to prove the analog decay for the number of Poissonian points in $\R_n$ [^2].

Notice that, since each box is full independently of each other and the probability that it occurs goes to $1$ when $L$ goes to infinity, for a fixed large $L_0>0$, the probability that $\R_n$ contains more than $zn$ boxes decays as 