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 N cards numbered 1 through N N . Fix a parameter p 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 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 p\ne 1/2 , the mixing time of this card shuffling is O ( N 2 ) 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.