Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published March 10, 2005 | Published
Journal Article Open

Mixing times of the biased card shuffling and the asymmetric exclusion process

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

Files

BENtams05.pdf
Files (253.4 kB)
Name Size Download all
md5:c07e6c1184e0a8eede8623e015ac41e0
253.4 kB Preview Download

Additional details

Created:
August 19, 2023
Modified:
October 18, 2023