are centered at a Poisson process of intensity $\lambda_2-\lambda_1>0$ in this region, and this Poisson process is independent of everything else we have previously conditioned on. Thus if there are infinitely many balls in $C^{\lambda_2}(x)$ centered at distance between $r+1$ and $r+3/2$ from unbounded components in $C^{\lambda_1}$, then balls centered at the points of the previously mentioned Poisson process will almost surely connect $C^{\lambda_2}(x)$ to some unbounded component in $C^{\lambda_1}$. Thus $\P[B^{\infty}(x)|\tilde{D}(x)=r]=0$ for any $r$ and consequently $\P[B^{\infty}(x)]=0$. [$\Box$ ]{}

For the second proof of Theorem \[uniqmon1\], we need some preliminary results. First we describe a method to find the component of $C^{\lambda}$ containing $x$. This may be considered to be the continuum version of the algorithm described in for example [@schonmann] for finding the cluster of a given vertex in discrete percolation.

At $x$, we grow a ball with unit speed until it has radius $1$, when the growth of the ball stops. Whenever the boundary of this ball hits a Poisson point, a new ball starts to grow with unit speed at this point until it has radius $2$. In the same way, every time a new Poisson point (which has not already been found) is hit by 