The Annals of Applied Probability

Card shuffling and Diophantine approximation

Omer Angel, Yuval Peres, and David B. Wilson

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The “overlapping-cycles shuffle” mixes a deck of n cards by moving either the nth card or the (nk)th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of k and n, has surprising behavior. For example, suppose k is the closest integer to αn for a fixed real α∈(0, 1). Then for rational α the spectral gap is Θ(n−2), while for poorly approximable irrational numbers α, such as the reciprocal of the golden ratio, the spectral gap is Θ(n−3/2).

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Ann. Appl. Probab. Volume 18, Number 3 (2008), 1215-1231.

First available in Project Euclid: 26 May 2008

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60C05: Combinatorial probability

Card shuffling Diophantine approximation


Angel, Omer; Peres, Yuval; Wilson, David B. Card shuffling and Diophantine approximation. Ann. Appl. Probab. 18 (2008), no. 3, 1215--1231. doi:10.1214/07-AAP484.

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