Directed percolation

Example

Let $\mathcal L = \set{(x, t) \in \Z \times \Z_{\ge 0} : x + t \text{ even}}$. Draw an oriented edge from each $(x, t) \in \mathcal L$ to $(x - 1, t + 1)$ and to $(x + 1, t + 1)$. Let $p \in [0, 1]$, and declare each oriented edge open with probability $p$, independently, as in bond percolation. The cluster $C$ of the origin is the set of points reached from $(0, 0)$ by a path of open edges that follows the orientation. Along such a path the time coordinate $t$ only increases. Directed bond percolation in $1 + 1$ dimensions asks whether $C$ is infinite with positive probability, that is, whether the set $C_t = \set{x : (x, t) \in C}$ is nonempty for all $t$.

Check of a lower bound for the threshold: each point of $C_t$ has $2$ outgoing edges, so $\mathbb{E} \abs{C_{t+1}} \le 2p \, \mathbb{E} \abs{C_t}$. Hence $\mathbb{E} \abs{C_t} \le (2p)^t \to 0$ for $p < 1/2$, and $C$ is finite almost surely. So the critical value $p_c$ satisfies $p_c \ge 1/2$.