{ for any } 1\leq k\leq 5 , \mbox{ there exists a semi-infinite path } \gamma_{k} \\
\mbox{ included in the cone } C_{2k\pi/5,\varepsilon,r_{k}} \mbox{ and starting from } \\
\mbox{ a vertex } X_{k} \mbox{ satisfying } r_{k}<|X_{k}|\leq r_{k}+1
\end{array} \right\} ~,$$ we get that for all $\varepsilon>0$, there exist some (deterministic) radii $r_{1},\ldots,r_{5}\in{{\mathbb N}}^*$ such that $A_{\varepsilon}(r_{1},\ldots,r_{5})$ occurs with positive probability.\
Let $R=\max\{r_{k}+1 ; 1\leq k\leq 5\}$ and $V_{\varepsilon}(r_{1},\ldots,r_{5})$ be the complementary set of the five cones in the ball $B(O,R)$: $$V_{\varepsilon}(r_{1},\ldots,r_{5}) = B(O,R) \setminus \Big[\Big( \cup_{k=1}^{5} C_{2k\pi/5,\varepsilon,r_{k}} \Big) \cup \{O\} \Big] ~.$$ Now, we are going to change the configuration of the PPP $N$ in $V_{\varepsilon}(r_{1},\ldots,r_{5})$ in such a way that the $X_k$’s are all of different colors. Let $\widetilde{N}=N\cap V_{\varepsilon}^{c}(r_{1},\ldots,r_{5})$ be the thinned PPP obtained by deleting all the points of $N$ belonging to $V_{\varepsilon}(r_{1},\ldots,r_{5})$ (Jacod and Shiryaev [@jacod], II.4.b). It is crucial to remark that deleting the points of $V_{\varepsilon}(r_{1},\ldots,r_{5})$ does not affect the occurrence of $A_{\varepsilon}(r_{1},\ldots,r_{5})$. In other words, if $N$ satisfies the event $A_{\varepsilon}(r_{1},\ldots,r_{5})$, so does $\widetilde{N}$; $${{\mathbb P}}\left( \widetilde{N} \in A_{\varepsilon}(r_{1},\ldots,r_{5}) \right) \geq {{\mathbb P}}\left( N \in A_{\varepsilon}(r_{1},\ldots,r_{5}) \right) > 0 ~.$$ Now, let us consider a PPP $\hat{N}$ on $V_{\varepsilon}(r_{1},\ldots,r_{5})$ with intensity $1$. Let us denote by $r$ the minimum of the $r_k$’s.\
The event $\hat{N}\in B_{\varepsilon}(r_{1},\ldots,r_{5})$ is defined by the three following conditions.

-   For any $k\in\{1,\dots,5\}$, if $r_{k}>r$ then for all integers $r \leq n \leq r_{k}-1$, $$\hat{N} \left( B(n e^{\i 2k\pi/5} , \varepsilon) \right) = 1 ~,$$ else $$\hat{N} \left( B(r e^{\i 2k\pi/5} , \varepsilon