## Electronic Journal of Probability

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

### Mixing Time Bounds for Overlapping Cycles Shuffles

#### 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