le interval of asymptotic directions whose length tends to $0$ as its starting point (on $\gamma_{0}$) is far from the origin. The last assertion of Corollary \[corol:skeleton\] can be understood as follows: the expected density of points of $\gamma_{0}$ from which emanates another semi-infinite path is zero. For this purpose, recall that the cardinality of $\gamma_{0}\cap B(O,r)$ is of order $r$ (see [@baccellibordenave Theorem 2.5]).

On the event $\{V_{\infty}^{+}\mbox{ is finite}\}$, let us consider the point $Y$ of $V_{\infty}^{+}$ of highest modulus. Let $Y''$ be the child of $Y$ belonging to $\gamma_0$. From the definition of $V_\infty^+$, the set of points $\{X\in N,\ \mathcal{A}(X)=Y,\ \widehat{Y'' Y X}>0 \mbox{ and }\mathcal{T}_X\mbox{ is unbounded}\}$ is non empty. The points of this set can be ranked by increasing values of $\widehat{Y'' Y X}$s. Let $Y'$ be the point of this set corresponding to the smallest positive angle $\widehat{Y'' Y X}$. In the subtree $\mathcal{T}_{Y'}$, we can define the lowest semi-infinite path $\underline{\gamma}_{Y'}$ as defined in the beginning of Section \[section:directiondeterministe\]. Thanks to Part $(i)$ of Theorem \[HN1\], $\underline{\gamma}_{Y'}$ has an asymptotic direction, say $\theta$. Moreover, using the planarity and the non-crossing property of paths together with the definition of points $Y$, $Y'$, all the paths between $\gamma_0$ and $\underline{\gamma}_{Y'}$ (in the trigonometric sense) are finite. Using Part $(ii)$ of Theorem \[HN1\], we deduce that $\underline{\gamma}_{Y'}$ and $\gamma_0$ have the same asymptotic direction, i.e. $\theta=0$. Now, by Proposition \[prop:&lt;2\], such a situation never occurs. So the set $V_{\infty}^{+}$ is a.s. infinite. The same goes for $V_{\infty}^{-}$.\
Let us prove the second part of Corollary \[corol:skeleton\]. For any $r$, let us consider the point $X$ of smallest modulus among the points of $\gamma_0\cap B(O,r)^{c}$. The semi-infinite paths $\underline{\gamma}_{X}$ and $\overline{\gamma}_{X}$ have a.s. asymptotic