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  "body": "\n## Example\n\nLet $d \\ge 1$.\nThe *voter model* on $\\Z^d$ is the spin system on $\\set{0, 1}^{\\Z^d}$ with flip rates\n$$c(x, \\eta) = \\frac{1}{2d} \\, \\#\\set{y \\in \\Z^d : \\abs{y - x} = 1, \\ \\eta(y) \\ne \\eta(x)}.$$\nHere $\\eta(x)$ is the opinion of the voter at site $x$.\nAt rate $1$, each voter picks one of its $2d$ neighbours uniformly at random and adopts that neighbour's opinion.\nThe rates lie in $[0, 1]$ and depend on $2d + 1$ sites, so it is an [[interacting particle system|GR49AM3PSZ]].\n\n## Remarks\n\nThe two constant configurations, all $0$ and all $1$, are absorbing.\nThe model is dual to a system of coalescing [[random walks|04VZE3FZXF]].\nTracing back whose opinion a voter holds gives a random walk path, and two such paths merge when they meet.\nBecause of this duality, the long-time behaviour depends on whether simple random walk on $\\Z^d$ is recurrent, which by [[Pólya's recurrence theorem|BEQH6AZPYM]] holds exactly for $d \\le 2$.\nStarted from independent opinions with density $\\rho \\in (0, 1)$, for $d \\le 2$ any two sites tend to agree, $P(\\eta_t(x) \\ne \\eta_t(y)) \\to 0$ as $t \\to \\infty$.\nFor $d \\ge 3$ there is a one-parameter family of invariant measures in which both opinions keep a positive density.\n\n## Explanations\n\n### L1 human\n\nEach person on a grid holds one of two opinions.\nFrom time to time a person asks a random neighbour and copies their opinion.\nNobody is persuaded by arguments; opinions simply spread by imitation.\nOn a line or in the plane, large regions of agreement form and grow, while in three or more dimensions disagreement can persist forever.\n",
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