we will denote again by $\check{\chi}_r$ the last random variable. Notice also that the length of $I_r$ is bounded by 2 as soon as $r$ is sufficiently large. Let $c>2+3\sqrt{2}$ (a technical condition needed in the sequel), and let us use $\oplus$ for the Minkowski addition. Then: $$\check{\chi}_r=\check{\chi}_r^{\leq c}+\check{\chi}_r^{>c} ~\mbox{ a.s. } ,\label{decomp}$$ where $\check{\chi}_r^{\leq c}$ (resp. $\check{\chi}_r^{>c}$) denotes the number of these edges with descendants belonging to $I_r\oplus B(O,c)$ (resp. being at a distance at least $c$ from $I_r$). $\check{\chi}_r^{\leq c}$ is upper bounded by $\card(N\cap (I_r\oplus B(O,c)))$ and admits a moment of order 2 that is bounded independently of $r$. It remains to study $\check{\chi}_r^{>c}$. Our idea is that each long edge is accompanied by a large empty space, so that it is rare that many long edges intersect $I_r$.\
Let us consider an edge $[\mathcal{A}(X),X]$ that crosses $I_r$ at $J(X)$ and such that the distance from $X$ to $I_r$ is larger than $c$. If $X=(x,y)$, then let us consider $z(X)$ the point with coordinates $([x]+1,\sign(y)[|y|])$, where $[x]$ denotes the integer part of $x$. Among the points with integer coordinates that have an abscissa larger than $x$ and which are closer to the abscissa axis than $X$, $z(X)$ is the point that is the closest to $X$. See the right part of Figure \[fig:arc\].\
By construction, $|X-z(X)|\leq \sqrt{2}$. Hence $B(O,|z(X)|-\sqrt{2})\subset B(O,|X|)$, where the radius of the first ball is positive as soon as $r\geq \sqrt{2}$. Let us consider the ball $B(z(X),|z(X)-(r,0)|-2-2\sqrt{2})$. For our choice of $c$, $|z(X)-(r,0)|-2-2\sqrt{2}\geq c-\sqrt{2}-2-2\sqrt{2}\geq 0$ and the radius is positive. If $U\in B(z(X),|z(X)-(r,0)|-2-2\sqrt{2})$, then $$|U-X|\leq |U-z(X)|+\sqrt{2}\leq |z(X)-(r,0)|-2-\sqrt{2}\leq |X-J(X)|\leq |X-\mathcal{A}(X)|,$$and thus $B(z(X),|z(X)-(r,0)|-2-2\sqrt{2})\subset B(X,|X-\mathcal{A}(X)|)$. As a consequence, if we introduce $\Lambda(z,r)=B(O,|z|)\cap B(z,|z-(r,0)|-2-2\sqrt{2})$ for $z