Percolation theory

Definition

Let $G = (V, E)$ be a graph with countably many vertices, each of finite degree, and let $p \in [0, 1]$. Bernoulli bond percolation on $G$ with parameter $p$ is the product probability measure $P_p$ on $\Omega = \set{0, 1}^E$. Under $P_p$ the coordinates $\omega(e)$, $e \in E$, are independent and $P_p(\omega(e) = 1) = p$. An edge $e$ is open if $\omega(e) = 1$ and closed otherwise. The open cluster $C(x)$ of a vertex $x$ is the vertex set of the connected component of $x$ in the subgraph $(V, \set{e : \omega(e) = 1})$. For a fixed vertex $o$, the percolation probability is $\theta(p) = P_p(\abs{C(o)} = \infty)$. Site percolation is defined in the same way, with open and closed vertices in place of edges. Percolation theory is the study of these random clusters.