on the fluctuations of $T$, which is exactly Corollary 1.1 in [@p-205]:

\[l2\] Under , for all $\kappa\in(1/2,1)$ there exist constants $\delta,c_j>0$ such that for all $r\geq 1$ and $s\in [c_1 (\log r)^{1/\delta},c_2 r^{\kappa}]$ $$\bP\big(|T(\0,r\vec{e}_1)-\mu r|\geq sr^{\kappa}\big)\leq e^{-c_3 s^{\delta}}\,.$$

The second step is to parallel Newman and Piza [@np95] to prove that the control of the fluctuations of $T$ can give the control of the fluctuations of a minimizing path connecting $\0$ to $r\vec{e}_1$ about the line segment $[\0,r\vec{e}_1]$. Precisely, for $\xi\in (0,1)$ let $$\C_r^\xi:=\{\x\in\bR^2\,:\,d(\x,[\0,r\vec{e}_1])\leq r^\xi\}\,,$$ where $[\x,\y]$ denotes the line segment connecting $\x$ to $\y$ and $d(\x,\A)$ denotes the euclidean distance between $\x$ and $\A\subseteq\bR^2$.

\[l1\] For all $\xi\in(3/4,1)$ there exist constants $c,\delta>0$ such that for all $r\geq 1$ $$\bP\big(\rho(\0,r\vec{e}_1)\not\subseteq\C_r^\xi\big)\leq e^{-cr^\delta}\,.$$

Let $\kappa\in(1/2,1)$, $\tilde{\kappa}\in(\kappa,1)$ and set $\xi\:=(\tilde{\kappa}+1)/2$. Let $$\C^{1,\xi}_{r}:=\{\x\in\bR^{2}\backslash \C^{\xi}_{r}\,:\, d(\x,\C^{\xi}_{r})<r^{\xi}\}\,.$$ Denote by $F_{r}$ the event defined by the following properties:

-   $\v_{\0},\v_{r\vec{e}_1}\in \C_r^{\xi}$

-   for all edges $\e=(\v,\tilde{\v})$ with $|\v|\leq 2r$ or $|\tilde{\v}|\leq 2r$ we have that $|\v-\tilde{\v}|\leq r^{\xi}$.

Notice that $F_r^c\subseteq A_{\xi,r/3}$ (as in Lemma \[5.2\]), and thus $$\label{HN1}
\bP\big(F_r^c\big)\leq c_1e^{-(r/3)^{2\xi}}\,.$$

For each $\z\in\bZ^2$ consider the random variable $$T_{\z}:=\max_{|\v-\z|\leq 1}\{T(\z,\v)\}\,.$$ We claim that, under , for some constants $c_2,c_3>0$ $$\label{e7t3}
\bP(T_{\z}\geq r^{\kappa})=\bP\big(T_{\0}\geq r^{\kappa}\big)\leq c_2e^{-c_3r^{\kappa}}\,.$$ To see this, 