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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $e_1, \\dots, e_d$ be the standard unit vectors of $\\Z^d$.\nThe *discrete Laplace operator* sends a [[function|0EPAT51DXQ]] $u \\colon \\Z^d \\to \\R$ to the function $\\Delta u \\colon \\Z^d \\to \\R$ given by\n$$(\\Delta u)(m) = \\sum_{i=1}^{d} \\big( u(m + e_i) - 2u(m) + u(m - e_i) \\big) = \\sum_{\\abs{m' - m} = 1} \\big( u(m') - u(m) \\big).$$\nMore generally, let a graph have vertex set $V$, and let every vertex have finitely many neighbours.\nThen the operator sends $u \\colon V \\to \\R$ to\n$$(\\Delta u)(v) = \\sum_{w \\sim v} \\big( u(w) - u(v) \\big), \\qquad v \\in V,$$\nwhere $w \\sim v$ means that $w$ and $v$ are joined by an edge.\n\n## Remarks\n\nExample: on $\\Z^2$, the function $u(m, n) = m^2 + n^2$ has $(\\Delta u)(m, n) = 2 + 2 = 4$ everywhere, matching $\\Delta(x^2 + y^2) = 4$ in the plane.\nThe function $u(m, n) = mn$ has $\\Delta u = 0$, so it is *discrete harmonic*.\n\nNon-example of a discrete harmonic function: $u(m) = 2^m$ on $\\Z$ has $(\\Delta u)(m) = 2^{m+1} - 2^{m+1} + 2^{m-1} = 2^{m-1} \\ne 0$.\n\nOn a grid of mesh $h \u003e 0$ one uses $h^{-2} \\Delta$, which tends to the Laplacian of $u$ as $h \\to 0$ when $u$ is a smooth function sampled on the grid.\nOn a finite graph with [[Laplacian matrix|TN6M6DVZRP]] $L = D - A$, one has $\\Delta u = -L u$ when $u$ is written as a column vector.\nSome authors use the opposite sign or divide by the degree.\n",
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