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The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains.

We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.

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