unique nearest neighbor of $\v_l$, different from $\v_{l-1}$, such that the edge of $\C_{\v_l}$ that is perpendicular to $[\v_l,\v_{l+1}]$ cross $[\x,\y]$; otherwise we set $k:=l$ and the construction is finished. We denote $|\gamma(\0,n\vec{e}_1)|$ the number of edges in $\gamma(\0,n\vec{e}_1)$.

For $\z\in\bR^2$ and $L>0$ let $$\B_\z^{L}:=L\z+[-L/2,L/2]\,$$ For $n>0$ consider the set $\calE_n$ composed of edges $(\v,\bar{\v})\in\calD_e$ with $\C_\v\cap\B_\z^{1}\neq\emptyset$ or $\C_{\bar\v}\cap\B_\z^{1}\neq\emptyset$ for some $\z\in[\0,n\vec{e}_1]$. We denote $|\calE_n|$ the number of edges in $\calE_n$.

\[lgraph2\] There exists constants $z_j,c_j>0$ such that for all $n\geq 1$, $$\label{egraph2*}
\bP\big(|\gamma(\0,n\vec{e}_1)|\geq z n\big)\leq e^{-c_1 zn}\,\mbox{ whenever }z\geq z_0\,,$$ and $$\label{egraph3*}
\bP\big(|\calE_n|\geq z n\big)\leq e^{-c_2 zn}\,\mbox{ whenever }z\geq z_1\,.$$

The proof of this lemma is performed through renormalization ideas developed in [@p-205]. To avoid some repetitions we give a sketch of the proof and leave the details to the reader, which can be filled by following the arguments in the proof of Proposition 2.2 in [@p-205] (which is exactly the proof in ).

For $\z\in\bZ^2$ and $L>0$ divide a square box $\B_\z^{L}$ into thirty-six sub boxes of the same length, say $\B_1,\dots,\B_{36}$. We stipulate $\B$ is a full box if all those thirty-six sub boxes have at least one Poissonian point (Figure \[ffull\]).

![Renormalization: a full box[]{data-label="ffull"}](full){width="20.00000%"}

We 