and determined by the underlying Poisson processes in the model, $D$ contains unbounded components with probability $1$.

So we can find an infinite sequence of points $u_1,u_2,...\in \hyp$ such that they are all in the same component of $D$, $d(u_i,u_{i+1})<1/2$ for all $i$ and $d(u_1,u_i)\rightarrow \infty$ as $i\rightarrow \infty$. Since $(u_i,y_1)\in A_1$ there is some ball $s_i$ in $C^{\lambda_0}_{\infty}$ centered within distance $r_1+1$ from $(u_i,y_1)$. Since $d((u_i,y_1),(u_{i+1},y_1))<1/2$ and $(u_i,y_1)\in A_2$ for all $i$ there is a sequence of at most $n$ balls in $C^{\lambda_0}_{\infty}$ connecting $s_i$ to $s_{i+1}$. Since the distance between the center of any ball in this sequence and $(u_i,y_1)$ is at most $r_1+2n$ and $(u_i,y_1)\in A_3$, all balls in the sequence is present also at level $\lambda_1$. Thus there is an unbounded component in $C^{\lambda_1}$ that comes within distance $r_1$ from $(u_i,y_1)$ for all $i$. In the same way there is an unbounded component in $C^{\lambda_1}$ that comes within distance $r_1$ from $(u_i,y_2)$ for all $i$.

Now choose $\lambda_2$ and $\lambda_3$ so that $\lambda_1<\lambda_2<\lambda_3<\lambda_*$. For $x\in\hyp$ let $D(x)$ be the component of $D$ containing $x$. Then we have from the above that $$\label{ball1}\P[S((x,y_1),r_1)\leftrightarrow
  S((x,y_2),r_1)\mbox{ in }C^{\lambda_2}|\mu(D(x))=\infty]=1.$$ This follows from the fact that the two unbounded components at 