, Volume 12, Issue 4, pp 903-932

Rifle Shuffles and Their Associated Dynamical Systems

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Abstract

It is shown that for every stationary sequence of random riffle permutations there is a natural associated dynamical system consisting of random orbits in the space of sequences from a finite alphabet. For many interesting models of card-shuffling, the associated dynamical systems have simple descriptions in terms of random or deterministic measure-preserving maps of the unit interval. It is shown that the rate of mixing for a card-shuffling process is constrained by the fiber entropy h of this map: at least (log N)/h repetitions of the shuffle are needed to randomize a deck of size N, when N is large.