### Mixing Time Bounds for Overlapping Cycles Shuffles

**Johan Jonasson**

*(Chalmers University of Technology and GĂ¶teborg University)*

#### 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

_{i}'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 $k=o(n)$.Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1281-1295

Publication Date: June 11, 2011

DOI: 10.1214/EJP.v16-912

#### References

- Angel, Omer; Peres, Yuval; Wilson, David B. Card shuffling and Diophantine approximation.
*Ann. Appl. Probab.*18 (2008), no. 3, 1215--1231. MR2418243 (2009c:60013) - Diaconis, Persi; Saloff-Coste, Laurent. Comparison techniques for random walk on finite groups.
*Ann. Probab.*21 (1993), no. 4, 2131--2156. MR1245303 (95a:60009) - Diaconis, Persi; Shahshahani, Mehrdad. Generating a random permutation with random transpositions.
*Z. Wahrsch. Verw. Gebiete*57 (1981), no. 2, 159--179. MR0626813 (82h:60024) - Durrett, Rick. Shuffling chromosomes.
*J. Theoret. Probab.*16 (2003), no. 3, 725--750. MR2009200 (2004j:92045) - Jonasson, Johan. Biased random-to-top shuffling.
*Ann. Appl. Probab.*16 (2006), no. 2, 1034--1058. MR2244440 (2008g:60218) - Jonasson, Johan. The overhand shuffle mixes in $Theta(nsp 2log n)$ steps.
*Ann. Appl. Probab.*16 (2006), no. 1, 231--243. MR2209341 (2006m:60016) - Morris, Ben. Improved mixing time bounds for the Thorp shuffle and $L$-reversal chain.
*Ann. Probab.*37 (2009), no. 2, 453--477. MR2510013 (2010e:60156) - Saloff-Coste, Laurent. Random walks on finite groups.
*Probability on discrete structures,*263--346, Encyclopaedia Math. Sci., 110,*Springer, Berlin,*2004. MR2023654 (2004k:60133) - Wilson, David Bruce. Mixing times of Lozenge tiling and card shuffling Markov chains.
*Ann. Appl. Probab.*14 (2004), no. 1, 274--325. MR2023023 (2004m:60155) - Wilson, David Bruce. Mixing time of the Rudvalis shuffle.
*Electron. Comm. Probab.*8 (2003), 77--85. MR1987096 (2004d:60191)

This work is licensed under a Creative Commons Attribution 3.0 License.