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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a [[graph|00C0BR7D7H]] with countably many vertices, each of finite degree, and let $p \\in [0, 1]$.\n*Bernoulli bond percolation* on $G$ with parameter $p$ is the product [[probability measure|059TJN0NZ1]] $P_p$ on $\\Omega = \\set{0, 1}^E$.\nUnder $P_p$ the coordinates $\\omega(e)$, $e \\in E$, are independent and $P_p(\\omega(e) = 1) = p$.\nAn edge $e$ is *open* if $\\omega(e) = 1$ and *closed* otherwise.\nThe *open cluster* $C(x)$ of a vertex $x$ is the vertex set of the [[connected component|0JG9W3KA6B]] of $x$ in the subgraph $(V, \\set{e : \\omega(e) = 1})$.\nFor a fixed vertex $o$, the *percolation probability* is $\\theta(p) = P_p(\\abs{C(o)} = \\infty)$.\nSite percolation is defined in the same way, with open and closed vertices in place of edges.\nPercolation theory is the study of these random clusters.\n\n## Remarks\n\nExample: on the line graph $\\Z$, a cluster is infinite only if all edges on one side of $o$ up to infinity are open.\nThis has probability $\\lim_n p^n = 0$ for $p \u003c 1$, so $\\theta(p) = 0$ for $p \u003c 1$ and $\\theta(1) = 1$.\nNon-example: a random subgraph in which two edges at the same vertex are never both open is not Bernoulli percolation, since its edge states are not independent.\nThe function $\\theta$ is nondecreasing in $p$, by coupling all $P_p$ with one family of uniform random variables on $[0, 1]$.\n",
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