Mixing times via super-fast coupling

Details Mixing times via super-fast coupling Mixing times via super-fast coupling

2006-09-20

arxiv.org/abs/math/0609568v3

Mathematics

Probability

Scientific paper

We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate Cn(log{n}) with a moderate constant, C. This rate was determined by Diaconis and Shahshahani, but the question of a natural probabilistic coupling proof has been missing, and questions of its existence have been raised. The proof, and indeed any proof, requires that we enlarge the methodology of coupling to include intuitive but non-adapted coupling rules, because a typical Markovian coupling is incapable of resolving finer questions of rates.

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