Electronic Journal of Probability

Mixing Time Bounds for Overlapping Cycles Shuffles

Johan Jonasson

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Abstract

Consider a deck of n cards. Let $p_1,p_2,\ldots,p_n$ be a probability vector and consider the mixing time of the card shuffle which at each step of time picks a position according to the p<sub>i</sub> 's and move the card in that position to the top. This setup was introduced in [5], where a few special cases were studied. In particular the case $p_{n-k}=p_n=1/2$, $k=\Theta(n)$, turned out to be challenging and only a few lower bounds were produced. These were improved in [1] where it was shown that the relaxation time for the motion of a single card is $\Theta(n^2)$ when $k/n$ approaches a rational number. In this paper we give the first upper bounds. We focus on the case $m:=n-k=\lfloor n/2\rfloor$. It is shown that for the additive symmetrization as well as the lazy version of the shuffle, the mixing time is $O(n^3\log(n))$. We then consider two other modifications of the shuffle. The first one is the case $p_{n-k}=p_{n-k+1}=1/4$ and $p_n=1/2$. Using the entropy technique developed by Morris [7], we show that mixing time is $O(n^2\log^3(n))$ for the shuffle itself as well as for the symmetrization. The second modification is a variant of the first, where the moves are made in pairs so that if the first move involves position $n$ , then the second move must be taken from positions $m$ or $m+1$ and vice versa. Interestingly, this shuffle is much slower; the mixing time is at least of order $n^3\log(n)$ and at most of order $n^3\log^3(n))$. It is also observed that results of [1] can be modified to improve lower bounds for some $k=o(n)$.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 46, 1281-1295.

Dates
Accepted: 11 June 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820215

Digital Object Identifier
doi:10.1214/EJP.v16-912

Mathematical Reviews number (MathSciNet)
MR2827459

Zentralblatt MATH identifier
1231.60066

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

Keywords
comparison technique Wilson's technique relative entropy

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Jonasson, Johan. Mixing Time Bounds for Overlapping Cycles Shuffles. Electron. J. Probab. 16 (2011), paper no. 46, 1281--1295. doi:10.1214/EJP.v16-912. https://projecteuclid.org/euclid.ejp/1464820215


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