= Percolation probability
{title2=$\theta(p)$}
For independent <bond percolation> on an infinite <graph> with a specified root $o$, let $C_o$ be its <percolation cluster>. The percolation probability is
$$
\theta(p)=\mathbb P_p(|C_o|=\infty).
$$
It concerns the root, rather than the existence of an <infinite percolation cluster> somewhere else. Even above the <percolation critical probability> of the <square lattice>, an isolated root has <probability> $(1-p)^4$, because its four incident <edges> can all be closed. Thus $\theta(p)\leq1-(1-p)^4<1$ when $p<1$. Positive percolation probability does not mean every <vertex> is connected to infinity.
On the <square lattice>, the <events> that $o$ is joined by an <open path> to the boundary of $[-n,n]^2$ decrease as $n$ increases. Their <probabilities> converge to $\theta(p)$ by <continuity from above of a measure>. This relates an infinite question to finite observations, without equating any one finite boundary connection with survival forever. The <monotone coupling of Bernoulli percolation> shows that $\theta$ is a <monotone function> of $p$.
Back to article page
