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  "body": "\n## Example\n\nLet $G = (V, E)$ be a finite [[graph|00C0BR7D7H]], and let $J \u003e 0$, $h \\in \\R$ and $\\beta \\ge 0$ be real numbers.\nA spin configuration is a map $\\sigma \\colon V \\to \\set{-1, +1}$.\nIts energy is\n$$H(\\sigma) = -J \\sum_{\\set{u, v} \\in E} \\sigma(u) \\sigma(v) - h \\sum_{v \\in V} \\sigma(v).$$\nThe *Ising model* on $G$ at inverse temperature $\\beta$ is the [[probability measure|059TJN0NZ1]] on $\\set{-1, +1}^V$ given by\n$$\\mu_\\beta(\\sigma) = \\frac{e^{-\\beta H(\\sigma)}}{Z_\\beta}, \\qquad Z_\\beta = \\sum_{\\tau \\in \\set{-1, +1}^V} e^{-\\beta H(\\tau)}.$$\nCheck: each weight $e^{-\\beta H(\\sigma)}$ is positive and there are finitely many configurations.\nSo $Z_\\beta \u003e 0$, and the values of $\\mu_\\beta$ are positive and sum to $1$.\n\n## Remarks\n\nWith $J \u003e 0$, neighbouring spins prefer to agree, and the measure favours aligned configurations more strongly as $\\beta$ grows.\nAt $\\beta = 0$ the spins are independent and uniform.\nOn the infinite lattice $\\Z^d$, limits of these measures as $G$ grows are Gibbs measures.\nFor $h = 0$ and $d \\ge 2$ there is a phase transition: there is more than one Gibbs measure for large $\\beta$ and only one for small $\\beta$.\nFor $d = 1$ there is no phase transition.\nFor $d = 2$ and $h = 0$, Onsager's solution gives the critical value $\\beta_c$ by $\\sinh(2 \\beta_c J) = 1$, that is $\\beta_c = \\frac{\\ln(1 + \\sqrt 2)}{2J}$.\nIts Glauber dynamics, in which single spins flip at random, is an [[interacting particle system|GR49AM3PSZ]] with $\\mu_\\beta$ as invariant measure.\n\n## Explanations\n\n### L1 human\n\nPicture a magnet as a grid of tiny arrows, each pointing up or down.\nNeighbouring arrows like to point the same way, and heat shakes them at random.\nAt high temperature the arrows are disordered; at low temperature, in two or more dimensions, most of them line up and the material is magnetised.\nFor two spins joined by one edge with $h = 0$, the aligned configurations each get weight $e^{\\beta J}$ and the opposite ones $e^{-\\beta J}$.\n",
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