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June 2008 Card shuffling and Diophantine approximation
Omer Angel, Yuval Peres, David B. Wilson
Ann. Appl. Probab. 18(3): 1215-1231 (June 2008). DOI: 10.1214/07-AAP484


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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Omer Angel. Yuval Peres. David B. Wilson. "Card shuffling and Diophantine approximation." Ann. Appl. Probab. 18 (3) 1215 - 1231, June 2008.


Published: June 2008
First available in Project Euclid: 26 May 2008

zbMATH: 1142.60046
MathSciNet: MR2418243
Digital Object Identifier: 10.1214/07-AAP484

Primary: 60J10
Secondary: 60C05

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.18 • No. 3 • June 2008
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