$G$ at level $p$, and let $\P_p$ be the corresponding probability measure on the subgraphs of $G$. A connected component in the random subgraph obtained in percolation is called a cluster. Let $$p_c(G):=\inf\{p\,:\,\P_p-\mbox{a.s. there is an infinite
  cluster}\}$$ be the critical probability for percolation.

In what follows we will discuss percolation at different levels, and when we do this, we always use the following coupling. To each $e\in
E$ we associate an independent random variable $U_e$ which is uniformly distributed on $[0,1]$. Then say that $e$ is kept at level $p$ if $U_e<p$ and deleted otherwise. Using this construction, we have that if $p_1<p_2$ then any edge kept at level $p_1$ is also kept at level $p_2$. Therefore we call this coupling the monotone coupling.

Now suppose that $p_c<p_1<p_2$ and use the monotone coupling. We say that an infinite cluster at level $p_2$ is $p_1$-stable if it contains an infinite cluster at level $p_1$. Häggström and Peres [@haggperes] showed the following theorem:

\[hpsats\] Suppose $G$ is a transitive unimodular graph and that $p_c(G)<p_1<p_2\le 1$. Then any infinite cluster at level $p_2$ is a.s. $p_1$-stable.

The proof of \[hpsats\] relies on a technique called the mass transport principle, which is not 