## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 2 (2009), 453-477.

### Improved mixing time bounds for the Thorp shuffle and *L*-reversal chain

#### Abstract

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models.

E. Thorp introduced the following card shuffling model in 1973: Suppose the number of cards *n* is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of *O*(log^{4}*n*). Previously, the best known bound was *O*(log^{29}*n*) and previous proofs were only valid for *n* a power of 2.

We also analyze the following model, called the *L-reversal chain*, introduced by Durrett: There are *n* cards arrayed in a circle. Each step, an interval of cards of length at most *L* is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is *O*(max(*n*, *n*^{3}/*L*^{3})log *n*). We obtain a bound that is within a factor *O*(log^{2}*n*) of this, the first bound within a poly log factor of the conjecture.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 2 (2009), 453-477.

**Dates**

First available in Project Euclid: 30 April 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1241099918

**Digital Object Identifier**

doi:10.1214/08-AOP409

**Mathematical Reviews number (MathSciNet)**

MR2510013

**Zentralblatt MATH identifier**

1170.60028

**Subjects**

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

**Keywords**

Markov chain mixing time

#### Citation

Morris, Ben. Improved mixing time bounds for the Thorp shuffle and L -reversal chain. Ann. Probab. 37 (2009), no. 2, 453--477. doi:10.1214/08-AOP409. https://projecteuclid.org/euclid.aop/1241099918