Lemma \[l1\] and the concept of *$\delta$-straightness* for trees discussed before.

Combining Lemma \[l1\] and Lemma \[5.2\] with the Borel-Cantelli’s lemma, one has that for all $\delta=1-\xi\in(0,1/4)$, almost surely, the assumptions of Lemma \[hn-101\] hold for all semi-infinite path (geodesic) $(\v_n)_{n\geq 1}$ in $\calT_\v$. So, $\calT_\v$ is $\delta$-straight at $\v$. Since, with probability one, a realization of the Poisson point process is omnidirectional, together with Lemma \[hn-201\] this yields Proposition \[pexis\].

\[r-straight\] Let $\xi\in(3/4,1)$. The almost sure $(1-\xi)$-straightness of the tree of infection also implies that for all $\alpha\in[0,2\pi)$, if $(\v_1,\v_2,\dots)$ is a semi-infinite geodesic with asymptotic orientation $e^{i\alpha}$ then $$ang(\v_n,e^{i\alpha})\leq c|\v_n|^{\xi-1}$$ for sufficiently large $n$.

Semi-infinite geodesics: uniqueness and coalescence
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Concerning uniqueness of semi-infinite geodesics we have:

\[puni\] For $\alpha\in[0,2\pi)$ let $\Omega_3(\alpha)$ be the event that for all $\v\in \calD_v$ there exists at most one geodesic starting from $\v$ and with asymptotic orientation $e^{i\alpha}$. Assume only that $\bF$ is continuous. Then $\bP\big(\Omega_3(\alpha)\big)=1$

For $(k,l)\in\bN^2$, let $A_\alpha(k,l)$ be the event that $U_{k,l}\in \calD_v$ (or equivalently, $N_k\geq l$) and there exists two semi-infinite geodesics starting from $\v=U_{k,l}$, with asymptotic orientation $e^{i\alpha}$, and such that after $\v$ they do not intersect each other. Thus, $$\big(\Omega_3(\alpha)\big)^c\subseteq \cup_{(k,l)\in\bN^2}A_\alpha(k,l)\,.$$ Now, semi-infinite geodesics starting from 