the same vertex are not allowed to cross each other and, if a semi-infinite geodesics is caught between two semi-infinite geodesics with the same asymptotic orientation $e^{i\alpha}$ then it must have the asymptotic orientation $e^{i\alpha}$ (by planarity). Therefore, if we denote by $d_{\v}$ the degree of the site $\v=U_{k,l}$ then $$|\{\alpha\in[0,2\pi)\,:\,\1_{A_\alpha(k,l)}(\omega)=1\}|\leq d_{\v}(\omega)\,.$$ ($|A|$ is the cardinality of the set $A$). In particular, almost surely, $$\int_{[0,2\pi)}\1_{A_\alpha(k,l)}d\alpha =0\,,$$ and so, by Fubini’s theorem, $$0\leq \int_{[0,2\pi)}\bP\Big(\big(\Omega_3(\alpha)\big)^c\Big)d\alpha=\int_{\Omega}\big(\int_{[0,2\pi)}\1_{\big(\Omega_3(\alpha)\big)^c}d\alpha\big)d\bP\leq$$ $$\int_{\Omega}\big(\int_{[0,2\pi)}\sum_{(k,l)}\1_{A_\alpha(k,l)}d\alpha\big)d\bP=\int_{\Omega}\big(\sum_{(k,l)}\int_{[0,2\pi)}\1_{A_\alpha(k,l)}d\alpha\big)d\bP=0\,.$$ Consequently, there exists $I\subseteq[0,2\pi)$ with total Lebesgue measure so that for all $\alpha\in I$, $\bP\big(\Omega_3(\alpha)\big)=1$. Since $\bP\big(\Omega_3(\alpha)\big)$ does not depend on $\alpha$, this yields Proposition \[puni\].

The last result we require to construct the Busemann function is the coalescence behavior of semi-infinite geodesics with the same asymptotic direction:

\[pcoal\] For $\alpha\in[0,2\pi)$ let $\Omega_4(\alpha)\subseteq\Omega_3(\alpha)$ be the event that for all $\v,\bar{\v}\in\calD_v$, if $\rho_{\v}(\alpha)$ and $\rho_{\bar{\v}}(\alpha)$ do exist (and are unique) then they must coalesce, i.e. there exists $\c=\c(\v,\bar{\v},\alpha)\in\calD_v$ such that $$\rho_{\v}(\alpha)=\rho(\v,\c)\cup\rho_{\c}(\alpha)\,\mbox{ and }\,\rho_{\bar{\v}}(\alpha)=\rho(\bar{\v},\c)\cup\rho_{\c}(\alpha)\,.$$ Assume only that $\bF$ is continuous. Then $\bP\big(\Omega_4(\alpha)\big)=1$.

We note that the almost sure statement in Proposition \[pcoal\] is for fixed $\alpha\in[0,2\pi)$. As we will see later, almost surely, there exists a random direction $\theta$ so that neither uniqueness nor coalescence hold. Indeed, we 