at the same time as $\TT$: begin with the root, and generate children according to $Z$, giving each new edge a $(0,1)$ weight uniformly and independently. If there exists a weight below $p$, take the edge of minimal weight among all boundary edges and generate children according to $Z$, giving each new edge a weight as before. Continue this process; if it terminates, then choose the minimal weight overall (which will necessarily be greater than $p$), generate children and assign weights as before. Repeat this process.

Each event measurable with respect to the law described above has the same probability as with respect to $\P$. At the step after each termination—when all available weights are greater than $p$—the event that the newly generated vertex has an infinite subtree is independent from all that came before and has positive probability $g(p)$. By the second Borel-Cantelli lemma, there will necessarily be an invaded edge with an infinite subtree below it with weights less than $p$. $\Cox$

\[cor-finite\] For any $p > p_c$, the number of edges in $I$ with weight greater than $p$ is almost surely finite.

This was proven for a large class of graphs by Häggström, Peres and Schonmann [@HPS], 