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  "body": "\n## Definition\n\nLet $G = (V, E)$ be an infinite connected [[graph|00C0BR7D7H]] with countably many vertices, each of finite degree, and let $o \\in V$.\nFor $p \\in [0, 1]$, let $\\theta(p)$ be the probability that the open cluster of $o$ is infinite in Bernoulli bond [[percolation|BH7YPXCYDK]] on $G$ with parameter $p$.\nThe *bond percolation threshold* of $G$ is\n$$p_c(G) = \\sup \\set{p \\in [0, 1] : \\theta(p) = 0}.$$\nThe site percolation threshold is defined in the same way with site percolation.\n\n## Remarks\n\nSince $G$ is connected, $p_c$ does not depend on the choice of $o$.\nBy Kolmogorov's zero–one law, an infinite open cluster exists with probability $0$ when $p \u003c p_c$ and with probability $1$ when $p \u003e p_c$.\nExample: for the line graph $\\Z$, $\\theta(p) = 0$ for every $p \u003c 1$, so $p_c(\\Z) = 1$.\nExample: take the infinite tree in which every vertex has $d \\ge 3$ neighbours.\nAfter the first step, the open cluster of $o$ grows as a branching process with mean offspring $(d-1)p$, so $p_c = 1/(d-1)$.\nNon-example: a finite graph does not satisfy the hypothesis; there every cluster is finite, so $\\theta \\equiv 0$ and the supremum carries no information.\nFor the lattice $\\Z^d$ with $d \\ge 2$, one has $0 \u003c p_c \u003c 1$.\n",
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