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  "body": "\n## Summary\n\nA discrete harmonic function is a function on the vertices of a graph equal at each vertex to a weighted average of its neighbouring values.\nIt is the discrete analogue of a [[harmonic function|0DEB0JMN0W]], and it is exactly a function killed by the [[graph Laplacian|TN6M6DVZRP]].\nDiscrete harmonic functions describe voltages in an [[electrical network|T7KRE7J3B4]] and hitting probabilities of a [[random walk|04VZE3FZXF]].\nThey satisfy discrete versions of the mean value property and the maximum principle.\n\n## Explanations\n\n### L1 human\n\nPut a number on every vertex of a graph.\nThe function is harmonic at a vertex if the number there is the average of the numbers at the neighbouring vertices.\nOn the integers, with neighbours $n - 1$ and $n + 1$, the harmonic functions are the straight lines $f(n) = an + b$.\nA harmonic function cannot have a strict maximum at an interior vertex, since an average cannot exceed all its terms.\nIf a random walker moves to a uniformly chosen neighbour at each step, then a harmonic function keeps its expected value along the walk.\n",
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