 Survival and susceptibility thresholds for directed percolation

For the out-cluster $C_x$ in directed percolation, define $p_H=\inf \{p:{\mathbb{P}}_p(|C_x|=\infty )>0\}$ and $p_T=\inf \{p:{\mathbb{E}}_p|C_x|=\infty \}$, with value $1$ when the relevant set is empty below $1$. The first is a survival threshold; the second is a percolation susceptibility threshold. Both depend on the root in a general directed graph. Exploring a site percolation cluster and using each fresh vertex coin for its first tested incoming bond gives a coupling of probability distributions with bond percolation such that ${\theta }_s(p)\le p{\theta }_b(p)$ and ${\chi }_s(p)\le p{\chi }_b(p)$. Consequently $p_H^b\le p_H^s$ and $p_T^b\le p_T^s$.