The Annals of Applied Probability

Card shuffling and Diophantine approximation

Omer Angel, Yuval Peres, and David B. Wilson

Full-text: Open access

Abstract

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).

Article information

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

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819799

Digital Object Identifier
doi:10.1214/07-AAP484

Mathematical Reviews number (MathSciNet)
MR2418243

Zentralblatt MATH identifier
1142.60046

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

Keywords
Card shuffling Diophantine approximation

Citation

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. https://projecteuclid.org/euclid.aoap/1211819799.


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