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  "body": "\n## Summary\n\nThe mixing time of a finite [[Markov chain|0G48PYM96Q]] is the number of steps it needs to come close to its [[stationary distribution|0KRH5157WP]].\nThis must hold whatever the starting state.\nCloseness is measured in total variation distance, usually with threshold $1/4$.\nMixing times measure the cost of Markov chain Monte Carlo and of card shuffling.\nTheir study is a large part of modern Markov chain theory.\n\n## Explanations\n\n### L1 human\n\nShuffle a deck of cards again and again: how many shuffles until the order is close to random?\nThe mixing time answers this kind of question for any chain.\nIt is the first time at which, from the worst starting state, no event has a probability more than $1/4$ away from its probability under the equilibrium law.\nFor a chain that jumps to a uniformly random state at each step, one step is enough, so the mixing time is $1$.\n",
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