notice that $T_\0\leq \sum_{\e\in\calE_1}\tau_\e$, where $\calE_1$ is the set of edges $\e=(\v,\bar{\v})$ in $\calD_e$ with $\C_{\v}\cap \B_{\0}^1\neq\emptyset$ or $\C_{\bar{\v}}\cap \B_{\0}^1\neq\emptyset$. By Lemma \[lgraph2\], $\bE\big(\exp(a|\calE_1|)\big)< \infty$ for some $a>0$. Combining this with assumption and the independence between the Poisson point process and the passage time distribution, one obtains .

Now, $$\big[\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big]\cap F_{r}\subseteq$$ $$\label{e1t3} 
 \big[\exists v\in\calD_v\cap \C_{r}^{1,\xi}\,:\, T(0,\v)+T(\v,r\vec{e}_1)=T(0,r\vec{e}_1)\big]\subseteq A(r)\,,$$ where $$A(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\, T(\0,\z)+T(\z,r\vec{e}_1)\leq T(\0,r\vec{e}_1)+2T_{\z}\big]\,.$$ Let $$\Delta(\z,r\vec{e}_1):= \mu|\z-r\vec{e}_1|+\mu|\z|-\mu|r\vec{e}_1|\,.$$ Thus $$T(\0,\z)+T(\z,r\vec{e}_1)\leq T(\0,r\vec{e}_1)+2T_{\z}$$ if and only if, $$\Delta(\z,r\vec{e}_1)\leq \big(T(\0,r\vec{e}_1)-\mu r\big)+\big(\mu|\z|-T(\0,\z)\big)+$$ $$\big(\mu|\z-r\vec{e}_1|-T(\z,r\vec{e}_1)\big)+2T_{\z}\,.$$ This implies that $A(r)\subseteq\cup_{j=0}^{3}A_{j}(r)$, where $$A_{0}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,T_{\z}\geq \frac{\Delta(\z,r\vec{e}_1)}{8}\big]\,,$$ $$A_{1}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\z,r\vec{e}_1)-\mu|\z-r\vec{e}_1||\geq \frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,,$$ $$A_{2}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\0,\z)-\mu|\z||\geq\frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,,$$ $$A_{3}(r):=\big[\exists \z\in\bZ^{2}\cap \C_{r}^{1,\xi}\,:\,|T(\0,r\vec{e}_1)-\mu|r\vec{e}_1||\geq\frac{\Delta(\z,r\vec{e}_1)}{4}\big]\,.$$ Combining this with one gets that $$\label{e3t3}
 \bP\big(\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big)\leq \bP\big(F_{r}^{c}\big)+ \sum_{j=0}^{3}\bP\big(A_{j}(r)\big)\,.$$

Notice there exist constants $b_{1},b_{2}>0$ such that for sufficiently large $r>0$ and $\z\in\bZ^{2}\cap \C_{r}^{1,\xi}$ we have that $$\label{e4*t3}
 b_{1}r^{\tilde{\kappa}}=b_{1}r^{2\xi-1}\leq \Delta(\z,r\vec{e}_1)\leq b_{2}r^{\xi}=b_{2}r^{\frac{\tilde{\kappa}+1}{2}}\,,$$ and $$\label{e4t3}
 r^{\xi}\leq |\z|,|\z-r\vec{e}_1|\leq 2r\,.$$

Together with Lemma , and yield that for some constant $c_1 >0$ $$\label{e5t3}
 \bP\big(A_{j}(r)\big)\leq e^{-c_{1}r^{\delta}}\,.$$ Combining with , , and one can finish the proof of this lemma.

For $\v\in\calD_v$ let $\calT_{\v}$ be the union over all $\tilde{\v}\in \calD_v$ of the unique geodesic between $\v$ and $\tilde{\v}$ (the tree of infection at $\v$). Therefore, $\calT_{\v}$ is a tree spanning all $\calD_v$. Thus, the third step is to use 