Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 15 / 1 / ii / Solution

Use the incoming-edge coupling of site and bond percolation. Start with independent random variables indicating open bonds, each with probability $q$. For a vertex $v$ of incoming degree $d_v$, let $J_v$ indicate that at least one incoming bond is open. ThenThe indicators $J_v$ are independent: the sets of bonds entering different vertices are disjoint. Using an additional independent coin at each vertex, turn some zeros into ones so that the resulting indicators $S_v$ are independent with common probability $r$. Explicitly, when $J_v=0$, set $S_v=1$ with conditional probability $(r-\mathbb{P}(J_v=1))/(1-\mathbb{P}(J_v=1))$.

Every vertex other than $x$ in an open bond percolation out-cluster has an open incoming bond, and is consequently an open site. On $\{S_x=1\}$, the whole bond out-cluster is therefore contained in the site percolation out-cluster. The event of infinite bond reachability does not depend on bonds entering $x$: delete cycles from any finite open directed path starting at $x$. It is thus independent of $S_x$, including its additional coin. HenceFor every $q>p_H^b$ below $1$, monotonicity and the definition of the survival threshold give ${\theta }_b(q)>0$. Letting $q\downarrow p_H^b$ yields the quantitative boundIn particular, if $p_H^b<1$, the right-hand side is strictly less than $1$. If $p_H^b=1$, no separation from $1$ is implied or generally possible.