discrete case, see for example Lemma 2.6 in [@jonasson2]. As in the discrete case, we introduce two critical intensities. Let $$\lambda_c(M):=\inf\{\lambda\,:\,N_C>0\mbox{ a.s.}\} \mbox{ and
}\lambda_u(M):=\inf\{\lambda\,:\,N_C=1\mbox{ a.s.}\}$$

be the critical intensity for percolation and the uniqueness treshold for the Poisson Boolean model.

[**Remark.**]{} Obviously it is only interesting to study what happens at and above $\lambda_u$ when $\lambda_u<\infty$. For example this is case for $\hyp\times {\mathbb R}$ and may be proved by adjusting the arguments for the $\hyp$ case, see [@tykesson]. Simple modifications (just embed a different graph in the space) of the arguments in Lemma 4.8 in [@tykesson] shows that for $\lambda$ large enough there are a.s. unbounded components in $C^{\lambda}$ but a.s. no unbounded components in $V^{\lambda}$. Since any two unbounded components in $C^{\lambda}$ must be separated by some unbounded component in $V^{\lambda}$ it follows that for $\lambda$ large enough there is a.s. a unique unbounded component in $C^{\lambda}$.

We will often work with the model at several different intensities at the same time. Suppose we do this at the intensities $\lambda_1<\lambda_2<...<\lambda_n$. Then we will always assume that $C^{\lambda_{i+1}}$ is the union of $C^{\lambda_i}$ and balls centered at the points of a Poisson process with intensity $\lambda_{i+1}-\lambda_{i}$. We call 