variable with the same distribution as the conditional distribution of $\delta_k(x)$ given the event $D_k(x)$. By the above observations we get that $$\P[D_{a_{k+1}}(x)^c|D_{a_k}(x)]\ge\P[S(0,R+\max(0,1-\Delta_k))\subset
  C^{\lambda}[S(0,R)]]\ge c_2$$ for some constant $c_2>0$ for all $k$. This implies $\lim_{k\rightarrow \infty}\P[D_{a_k}(x)]=0$ and consequently $\P[D(x)]=0$.

[*Proof of Theorem \[uniqmon1\]:*]{}

We consider the monotone coupling of the model at intensities $\lambda_1<\lambda_2$, and we write $C=(C^{\lambda_1},C^{\lambda_2})$. Let $$E(x):=\{x\mbox{ is in an unbounded $C^{\lambda_2}$ component which
  is $\lambda_1$-unstable}.\}$$ Let $$E_1(x):=E(x)\cap \{\tilde{D}(x)\le 3\}\mbox{ and
}E_2(x):=E(x)\cap\{\tilde{D}(x)>2\},$$ where $\tilde{D}$ is defined as in the proof of Theorem \[uniqmon1\].

Finally let $E$, $E_1$ and $E_2$ be the events that $E(x)$, $E_1(x)$ and $E_2(x)$ respectively happen for some $x$.

We will first show that $\P[E_2(x)]=0$. Pick $a$ and $R=R(a)$ so that $$\P[S(x,R)\mbox{ intersects an unbounded component of
  $C^{\lambda_1}$}]\ge 1-a$$

Let $Z^{'}=(Z^{'\lambda_1},Z^{'\lambda_2})$ and $Z^{''}=(Z^{''\lambda_1},Z^{''\lambda_2})$ be two independent copies of $C$, and let $X^{'}=(X^{'\lambda_1},X^{'\lambda_2})$ and $X^{''}=(X^{''\lambda_1},X^{''\lambda_2})$ be their underlying Poisson processes. A prime will be used to denote objects relating to $Z^{'}$ and a double prime will be used to denote objects relating to $Z^{''}$.

Grow the component of $Z^{'\lambda_2}$ containing $x$ as described above, but if at time $t$ we find that a ball of radius $R$ is contained in $Z^{'\lambda_2}[L_t^{'\lambda_2}(x)]$ we stop the process. 