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  "body": "\n## Statement\n\nLet $\\Z^2$ be the square lattice.\nIts vertices are the points of $\\R^2$ with integer coordinates, and its edges join points at distance $1$.\nThen the [[bond percolation threshold|MP8VSZK19A]] of $\\Z^2$ is\n$$p_c(\\Z^2) = \\frac12.$$\n\n## Remarks\n\nThe key tool is self-duality.\nThe dual lattice, with a vertex at the centre of each square face, is a translate of $\\Z^2$.\nA dual edge is declared open when the primal edge it crosses is closed.\nAt $p = 1/2$ the primal and dual configurations have the same law.\nA box crossing argument then shows that a suitable rectangle is crossed by an open path with probability exactly $1/2$.\nHarris proved in 1960 that $\\theta(1/2) = 0$, which gives $p_c \\ge 1/2$; Kesten proved in 1980 that $p_c \\le 1/2$.\nSo there is no infinite open cluster at $p = 1/2$ itself, while for every $p \u003e 1/2$ there is one almost surely.\n\n## Explanations\n\n### L1 human\n\nKeep each edge of the infinite square grid with probability $p$, independently.\nThe theorem says that an infinite connected piece appears exactly when $p$ is larger than $1/2$.\nThe number $1/2$ comes from a symmetry: the open edges and the closed edges of the grid play the same role in the plane.\n\n### L2 human\n\nIt is the model case of an exact percolation threshold computed by planar duality.\nThe same duality idea gives $p_c = 1/2$ for site percolation on the triangular lattice.\nEstimates on the probability that open paths cross long rectangles are the main tool of the proof.\nIn dimensions $d \\ge 3$ no exact value of $p_c(\\Z^d)$ is known.\n",
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