The Annals of Applied Probability

Analysis of top to bottom-k shuffles

Sharad Goel

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Abstract

A deck of n cards is shuffled by repeatedly moving the top card to one of the bottom kn positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as kn ranges from a constant to n. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom kn positions or a random card from the bottom kn positions is moved to the top. For this reversible shuffle we derive bounds on the L2 mixing time. Finally, we transfer mixing time estimates for the above shuffles to the lazy top to bottom-k walks that move with probability 1/2 at each step.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 1 (2006), 30-55.

Dates
First available in Project Euclid: 6 March 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1141654280

Digital Object Identifier
doi:10.1214/10505160500000062

Mathematical Reviews number (MathSciNet)
MR2209335

Zentralblatt MATH identifier
1096.60004

Subjects
Primary: 60
Secondary: 68

Keywords
Finite Markov chains mixing time card shuffling Rudvalis shuffle

Citation

Goel, Sharad. Analysis of top to bottom- k shuffles. Ann. Appl. Probab. 16 (2006), no. 1, 30--55. doi:10.1214/10505160500000062. https://projecteuclid.org/euclid.aoap/1141654280


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The Annals of Applied Probability

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