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  "body": "\n## Statement\n\nLet $(G, c)$ be an infinite [[electrical network|T7KRE7J3B4]] with vertex set $V$, and let $c(x) = \\sum_y c(x, y)$.\nLet $(X_n)$ be the [[random walk|04VZE3FZXF]] on $V$ with transition probabilities $p(x, y) = c(x, y)/c(x)$.\nFix $a \\in V$ and finite sets $V_1 \\subseteq V_2 \\subseteq \\cdots$ with union $V$ and $a \\in V_1$.\nLet $G_n$ be the finite network obtained from $G$ by gluing all vertices outside $V_n$ into one vertex $z_n$.\nEdges with both ends outside $V_n$ are deleted, and the edge from $x \\in V_n$ to $z_n$ gets the sum of the conductances $c(x, y)$ over $y \\notin V_n$.\nThen the [[effective resistances|XAE5YJ3H68]] $R_{\\mathrm{eff}}(a, z_n)$ in $G_n$ increase to a limit $R(a \\leftrightarrow \\infty) \\in (0, \\infty]$ that does not depend on the choice of the sets $V_n$.\nMoreover\n$$\\mathbb{P}_a\\big[X_n \\ne a \\text{ for all } n \\ge 1\\big] = \\frac{1}{c(a)\\, R(a \\leftrightarrow \\infty)},$$\nwith $1/\\infty = 0$.\nIn particular the walk is [[recurrent|PBZTBWZ4GS]] if and only if $R(a \\leftrightarrow \\infty) = \\infty$, that is, if and only if the capacity $1/R(a \\leftrightarrow \\infty)$ of $a$ is $0$.\n\n## Remarks\n\nFor a finite network, the probability that the walk from $a$ reaches $b$ before returning to $a$ is $1/\\big(c(a) R_{\\mathrm{eff}}(a, b)\\big)$.\nThis follows from the fact that $x \\mapsto \\mathbb{P}_x[\\text{hit } b \\text{ before } a]$ is harmonic off $\\set{a, b}$, so it is $1 - v$ for the voltage $v$ with $v(a) = 1$ and $v(b) = 0$.\nThe statement follows by applying this to $G_n$ and letting $n \\to \\infty$.\nThe limit increases because $G_{n+1}$ arises from $G_n$ by un-gluing vertices, which raises resistance by the [[Rayleigh monotonicity law|KX4152NVW8]].\nExample: on $\\Z^2$ with unit conductances, the edges between the boundary of the box $[-k, k]^2$ and the next box form $4(2k + 1)$ disjoint cutsets.\nThe Nash-Williams inequality gives $R(0 \\leftrightarrow \\infty) \\ge \\sum_{k \\ge 0} \\frac{1}{4(2k + 1)} = \\infty$, so the simple random walk on $\\Z^2$ is recurrent.\nOn $\\Z^3$ the resistance to infinity is finite, which gives the transience part of [[Pólya's recurrence theorem|BEQH6AZPYM]].\n\n## Explanations\n\n### L1 human\n\nImagine the graph as an infinite network of wires and connect one pole of a battery to a vertex $a$ and the other pole to infinity.\nIf the resistance between $a$ and infinity is infinite, no current escapes, and a random walker started at $a$ comes back to $a$ with probability $1$.\nIf the resistance is finite, current escapes, and the walker has a positive chance never to return.\nThis turns a question about random paths into a question about circuits.\n\n### L2 human\n\nThe theorem lets one decide recurrence with the tools of electrical networks.\nLower bounds on resistance come from gluing vertices, as in the Nash-Williams inequality; upper bounds come from flows of finite energy, by [[Thomson's principle|8BPMHA5MT4]].\nRecurrence is a property of the graph and the conductances, not of the starting vertex.\nIndeed $R(a \\leftrightarrow \\infty)$ is finite for one vertex exactly when it is finite for all vertices of the connected graph.\nComparisons between graphs, for example under bounded changes of conductances, preserve recurrence and transience.\n",
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