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  "body": "\n## Summary\n\nA random walk on a group $G$ starts at the identity and multiplies by independent random elements of $G$.\nAll these elements are drawn from one [[probability measure|059TJN0NZ1]] $\\mu$.\nIt is a [[Markov chain|0G48PYM96Q]] on $G$ whose transition probabilities are invariant under the group.\nWhen $\\mu$ is uniform on a symmetric generating set, it is the simple [[random walk|04VZE3FZXF]] on the [[Cayley graph|0GYQK9NDJF]].\nThe long-run behaviour of the walk reflects the algebra and geometry of the group.\nFor example, let $G$ be finitely generated and let $\\mu$ be symmetric with support generating $G$.\nBy a theorem of Kesten, the return probabilities then decay exponentially exactly when $G$ is not [[amenable|CEAT9BESKK]].\n\n## Explanations\n\n### L1 human\n\nPicture a walker on the grid of integer points in the plane who, at each step, moves one unit north, south, east or west at random.\nThis is a random walk on the group $\\Z^2$, and it comes back to its start with probability $1$.\nShuffling a deck of cards by repeated random swaps is a random walk on the group of permutations of the cards.\nThe question of how many shuffles mix the deck well is a question about this random walk.\n",
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