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  "body": "\n## Definition\n\nLet $G$ be a countable [[group|G5S1M9]] with identity $e$, and let $\\mu$ be a [[probability measure|059TJN0NZ1]] on $G$.\nLet $X_1, X_2, \\dots$ be independent [[random variables|0PNRKMRC55]] with values in $G$, each with distribution $\\mu$.\nThe *random walk* on $G$ with step distribution $\\mu$ is the sequence $(Z_n)_{n \\ge 0}$ with\n$$Z_0 = e, \\qquad Z_n = X_1 X_2 \\cdots X_n \\quad (n \\ge 1).$$\n\n## Remarks\n\nThe sequence $(Z_n)$ is a [[Markov chain|0G48PYM96Q]] on $G$ with transition probabilities $p(g, h) = \\mu(g^{-1} h)$.\nThese satisfy $p(kg, kh) = p(g, h)$ for all $k \\in G$.\nThe distribution of $Z_n$ is the $n$-fold convolution $\\mu^{*n}$.\n\nExample: on $\\Z^d$, written additively, let $\\mu$ give mass $\\frac{1}{2d}$ to each of the $2d$ vectors $\\pm e_1, \\dots, \\pm e_d$.\nThen $Z_n = X_1 + \\cdots + X_n$ is the simple random walk on the lattice $\\Z^d$, which is the walk on the [[Cayley graph|0GYQK9NDJF]] for the standard generators.\n\nNon-example: on $\\Z$, let $X_1$ be $\\pm 1$ with probability $\\frac{1}{2}$ each, and let $X_n = X_1$ for all $n$.\nEach $X_n$ has the same distribution, but they are not independent, so $Z_n = n X_1$ is not a random walk on $\\Z$ in this sense.\n",
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