Mixing times of the biased card shuffling and the asymmetric exclusion process
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1.
Weizmann Institute of Science
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2.
California Institute of Technology
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3.
University of Washington
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4.
University of California, Berkeley
Abstract
Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N. Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p ≠ 1/2, the mixing time of this card shuffling is O(N^2), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.
Additional Information
© 2005 American Mathematical Society. Received by editor(s): June 10, 2003. Received by editor(s) in revised form: June 24, 2003. Posted: March 10, 2005. We thank Persi Diaconis for important comments on a previous version of this paper. We thank Dror Weitz, Prasad Tetali, Thomas Liggett and David Aldous for many interesting discussions. Part of the work was done while Noam Berger was an intern at Microsoft Research. Christopher Hoffman was partially supported by NSF grant #0100445.Attached Files
Published - BENtams05.pdf
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BENtams05.pdf
Additional details
Identifiers
- Eprint ID
- 15016
- Resolver ID
- CaltechAUTHORS:20090813-150839885
Funding
- NSF
- 0100445
Dates
- Created
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2009-08-14Created from EPrint's datestamp field
- Updated
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2021-11-08Created from EPrint's last_modified field