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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite connected [[graph|00C0BR7D7H]], let $B \\subseteq V$ be a nonempty set of boundary vertices, and let $U = V \\setminus B$ be nonempty.\nFor $f \\colon V \\to \\R$, let $\\Delta f(x) = \\sum_{y : \\set{x, y} \\in E} (f(y) - f(x))$ be the graph [[Laplacian|0E0B6RA8K0]], and let $\\Delta_U$ be its restriction to functions that vanish on $B$, viewed as a $U \\times U$ matrix.\nThe *discrete Gaussian free field* on $G$ with zero boundary values on $B$ is a random function $h \\colon V \\to \\R$ with $h = 0$ on $B$.\nIts values $(h(x))_{x \\in U}$ have the following density on $\\R^U$:\n$$\\frac{1}{Z} \\exp\\Big( -\\frac12 \\sum_{\\set{x, y} \\in E} \\big(h(x) - h(y)\\big)^2 \\Big),$$\nwhere $Z$ is the normalising constant.\n\n## Remarks\n\nThe exponent equals $-\\frac12 \\langle h, -\\Delta_U h \\rangle$, and $-\\Delta_U$ is positive definite because $B$ is nonempty and $G$ is connected.\nSo $h$ restricted to $U$ is a centred [[Gaussian process|22WEM6RTW0]] with [[covariance|0DJ8W3HP2D]] matrix $(-\\Delta_U)^{-1}$.\nThis matrix is the Green function of the continuous-time random walk that crosses each edge at rate $1$ and is killed on $B$.\nExample: let $V = \\set{a, x, b}$ with edges $\\set{a, x}$ and $\\set{x, b}$, and $B = \\set{a, b}$.\nThe exponent is $-\\frac12 (h(x)^2 + h(x)^2) = -h(x)^2$, so $h(x)$ is normal with mean $0$ and variance $1/2$, and indeed $-\\Delta_U = (2)$ has inverse $1/2$.\nNon-example: with $B$ empty, the exponent is unchanged when a constant is added to $h$, so no normalising constant $Z$ exists.\nThe continuum Gaussian free field on a domain $D \\subseteq \\R^d$ is the centred Gaussian random generalised function whose covariance is the Dirichlet Green function of $-\\Delta$ on $D$.\nIt is not a function when $d \\ge 2$.\n",
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