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

Let $C_x^b$ and $C_x^s$ be the out-clusters in directed percolation, with the site percolation cluster empty when $x$ is closed. Write ${\theta }_b(p)={\mathbb{P}}_p(|C_x^b|=\infty )$ and ${\chi }_b(p)={\mathbb{E}}_p|C_x^b|$, and similarly for site percolation. The survival and susceptibility thresholds for directed percolation are the infima of the parameters at which these quantities become positive and infinite, respectively.

We construct a coupling of probability distributions that compares the whole clusters. First condition the site percolation root to be open. Explore its out-cluster, using a fixed order of the finitely many outgoing bonds at each vertex. When a previously untested vertex $v$ is first encountered along a bond $u\to v$ from an already reached vertex, use a fresh Bernoulli random variable of parameter $p$ both for the state of $v$ and for the state of this particular bond. If $v$ is open, add it to the exploration; if it is closed, mark it as tested and never add it. Assign fresh independent random variables of the same parameter to every other bond, including bonds leading to vertices already tested. The choice of the next unassigned bond depends only on previously revealed information. Its assigned coin is therefore independent of that information, so the resulting bond configuration has exactly the product bond percolation law. Parallel bonds cause no difficulty; loops may be discarded.

Every reached site has an open parent bond leading from a previously reached site. Consequently the conditional site percolation cluster is contained in the bond percolation cluster. The connected graph is countable because it is a locally finite graph; taking increasing finite stages of the exploration proves the same containment for infinite clusters. Removing the conditioning on the root givesThus positive site percolation survival implies positive bond percolation survival at the same parameter, and infinite percolation susceptibility for site percolation implies infinite percolation susceptibility for bond percolation. Taking the infima in the definitions proves both comparisons:

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.

No. The parallel-bond ray with different site and bond thresholds supplies a locally finite graph counterexample. Take vertices $v_0,v_1,\dots$ and $n+1$ parallel directed edges from $v_n$ to $v_{n+1}$. Its underlying multigraph is connected and every individual degree is finite, but there is no uniform incoming-degree bound.

For bond percolation rooted at $v_0$, survival means that every successive group of parallel bonds contains at least one open bond. Independence givesFor $p>0$, the sum of the failure probabilities is ${\sum }_{n\ge 0}(1-p{)}^{n+1}=(1-p)/p<\infty$. Eventually these terms are at most $1/2$, and $\log (1-t)\ge -2t$ for $0\le t\le 1/2$. The tail of the infinite product is therefore bounded below by a positive exponential; its finitely many initial factors are positive as well. Thus ${\theta }_b(p)>0$ for every $p>0$.

For site percolation, however, every vertex of the unique underlying ray must be open. For $p<1$, the probability that its first $N$ vertices are all open is $p^N\to 0$. At $p=1$ the ray is open. Consequently