set $T_\calD(\A,\B)$ to be the minimum of $T_\calD(\v,\tilde{\v})$ over all pairs $\v$ and $\tilde{\v}$ such that $\C_\v\cap\A\neq\emptyset$ and $\C_{\tilde{\v}}\cap\B\neq\emptyset$. By the shape theorem, we have:

\[lgraph1\] There exists $\nu\in(0,\infty)$ such that almost surely $$\lim_{n\to\infty}\frac{T_\calD(\A,ne^{i\alpha}+\B)}{n}=\nu\,.$$

We notice that $\nu$ does not depend either on $\A$ and $\B$ or on $\alpha\in[0,2\pi)$. One can also see that, if we denote $\lambda=\lambda(\bF)$ the supremum of the support of $\bF$ then $$\mu(\bF)\leq \bE(\tau_e)\nu <\lambda\nu\,$$ (one must assume that $\bF$ is not concentrate in one point, which is the case since $\bF$ is continuous).

We shall also use the following lemma, which is (5.2) of Lemma 5.2 in [@hn01]:

\[5.2\] For $\xi\in (0,1)$ and $r>0$ let $A_{\xi,r}$ be the event that there exists $\x\in\bR^2$ with $|\x|\leq 2r$ and $|\x-\v(\x)|\geq r^\xi$. Then, for some constant $c_1>0$, $$\bP\big(A_{\xi,r}\big)\leq c_1 e^{-r^{2\xi}}$$

$\delta$-straightness of semi-infinite paths {#pre-del}
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Recall that for $\alpha\in[0,2\pi)$ we have defined that a self-avoiding path $(\x_{n})_{n\geq 1}$, with vertices in $\bR^2$ and such that $|\x_n|\to\infty$ when $n\to\infty$, is a $\alpha$-path if $$\lim_{n\to\infty}\frac{\x_{n}}{|\x_{n}|}=e^{i\alpha}:=(\cos\alpha,\sin\alpha)\,,$$ and that a sufficient condition for a path $(\x_{n})_{n\geq 1}$ to be a $\alpha$-path for some $\alpha\in[0,2\pi)$ is that, for some fixed $\delta\in(0,1)$ and $c>0$, and for large enough $n$ $$ang(\x_{n},\x_m)\leq |\x_n|^{-\delta}\,\mbox{ whenever }m>n$$ 