Electronic Communications in Probability

Rapid mixing of dealer shuffles and clumpy shuffles

Johan Jonasson and Benjamin Morris

Full-text: Open access

Abstract

A famous result of Bayer and Diaconis is that the Gilbert-Shannon-Reeds (GSR) model for the riffle shuffle of n cards mixes in $\frac{3}{2}\log_2(n)$ steps and that for $52$ cards about $7$ shuffles suffices to mix the deck. In this paper,we study variants of the GSR shuffle that have been proposed to model more realistically how people actually shuffle a deck of cards. The clumpy riffle shuffle and dealer riffle shuffle differ from the GSR model in that when a card is dropped from one hand, the conditional probability that the next card is dropped from the same hand is higher/lower than for the GSR model. Until now, no nontrivial rigorous results have been known for the clumpy shuffle or dealer shuffle. In this paper we show that the mixing time is $O(\log^4(n))$.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 120, 11 pp.

Dates
Accepted: 27 February 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320947

Digital Object Identifier
doi:10.1214/ECP.v20-3682

Mathematical Reviews number (MathSciNet)
MR3320408

Zentralblatt MATH identifier
1328.60172

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

Keywords
riffle shuffle entropy technique collisions

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

Citation

Jonasson, Johan; Morris, Benjamin. Rapid mixing of dealer shuffles and clumpy shuffles. Electron. Commun. Probab. 20 (2015), paper no. 120, 11 pp. doi:10.1214/ECP.v20-3682. https://projecteuclid.org/euclid.ecp/1465320947


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