The Annals of Probability

Improved mixing time bounds for the Thorp shuffle and L-reversal chain

Ben Morris

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We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models.

E. Thorp introduced the following card shuffling model in 1973: Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log4n). Previously, the best known bound was O(log29n) and previous proofs were only valid for n a power of 2.

We also analyze the following model, called the L-reversal chain, introduced by Durrett: There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n3/L3)log n). We obtain a bound that is within a factor O(log2n) of this, the first bound within a poly log factor of the conjecture.

Article information

Ann. Probab., Volume 37, Number 2 (2009), 453-477.

First available in Project Euclid: 30 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Markov chain mixing time


Morris, Ben. Improved mixing time bounds for the Thorp shuffle and L -reversal chain. Ann. Probab. 37 (2009), no. 2, 453--477. doi:10.1214/08-AOP409.

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