For an n by n sliding-tile puzzle on a torus, the total variation mixing time is at least order n^4 and at most n^4 log n, with a single tile converging to Brownian motion at an exact rate.
Cut-off for random to random c ard shuffle
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Solution of the 15 puzzle problem
For an n by n sliding-tile puzzle on a torus, the total variation mixing time is at least order n^4 and at most n^4 log n, with a single tile converging to Brownian motion at an exact rate.