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

Benjamini, Itai; Berger, Noam; Hoffman, Christopher; Mossel, Elchanan

doi:10.1090/S0002-9947-05-03610-X

Mixing times of the biased card shuffling and the asymmetric exclusion process
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by Itai Benjamini, Noam Berger, Christopher Hoffman and Elchanan Mossel PDF

Trans. Amer. Math. Soc. 357 (2005), 3013-3029 Request permission

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 $p\ne 1/2$, 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.

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