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  "body": "\n## Summary\n\nPercolation theory studies the connected clusters of a random subgraph of a large graph, where each edge or vertex is kept independently with probability $p$.\nIts main question is whether an infinite cluster appears.\nOn many graphs there is a sharp [[percolation threshold|MP8VSZK19A]]: below it all clusters are finite, above it an infinite cluster exists.\nIt is a basic model of a phase transition and of flow through a porous medium.\n\n## Explanations\n\n### L1 human\n\nTake a huge square grid of pipes and block each pipe at random, keeping it open with probability $p$.\nPour water at one point and ask whether it can flow arbitrarily far.\nIf $p$ is small, the water stays in a small region; if $p$ is close to $1$, it can reach infinity with positive probability.\nThe change happens suddenly at a critical value of $p$, which for this grid is $1/2$, by [[Kesten's theorem|B8Y0ZRQP17]].\n",
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