result to show Theorem \[t2\] is the following theorem, which is a small step towards the above conjecture.

\[tBuse-2\] For $\alpha\in[0,\pi/2)$ let $\Omega_1(\alpha)\subseteq\Omega_0(\alpha)$ be the event that $$\label{eBuse-2}
-\mu(\bF)\leq \lim\inf_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_{1},\0)}{n}\leq \limsup_{n\to\infty}\frac{H^{\alpha}(n\vec{e}_{1},\0)}{n}\leq -\mu(\bF)\frac{\cos\alpha}{1+\sin\alpha}\,.$$ Under , $\bP\big(\Omega_1(\alpha)\big)=1$. In particular, with probability one, $$\lim_{n\to\infty}\frac{H^{0}(n\vec{e}_{1},\0)}{n}=-\mu(\bF)\,.$$

Overview {#overview .unnumbered}
--------

In Section \[multi\] we will deduce Theorem \[t1\] and Theorem \[t2\] from Theorem \[tBuse-1\] and Theorem \[tBuse-2\]. In Section \[pre\] we will start by defining the probability space where our model takes place and we will show a modification lemma that will play an important rule in the study of coalescence of semi-infinite geodesics. After that we will study some geometrical aspects of Voronoi tilings. We note that in the Delaunay triangulation context some technical difficulties are imposed by its long range dependence. Some of them will be avoided by making references to results of a previous work of the author [@p-105; @p-205]. In the third part we will recall some geometrical lemmas concerning the $\delta$-straightness of semi-infinite paths. Finally, in Section \[geo\] will study existence and coalescence of semi-infinite geodesics to show Theorem \[tBuse-1\] and Theorem \[tBuse-2\]. It will largely parallel the analog study develop by Newman et al [@hn01; @ln96; @n95; @np95] in 