We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.
For @math , we refer to the resulting process as asymmetric simple exclusion process (ASEP). studied the mixing time of the ASEP in the context of biased card shuffling and showed that it is asymptotically of order @math for any number of particles @math , see @cite_2 . We will follow their approach for an upper bound on the mixing time in the marginal nestling case. Labbé and Lacoin @cite_0 refined this bound for the ASEP and proved cutoff. The proof relies on an explicit calculation of the spectral gap which was independently obtained by Levin and Peres @cite_15 . They studied the mixing time of the ASEP when the bias vanishes for @math going to infinity. Recently, cutoff results were established in this regime by Labbé and Lacoin @cite_12 .
@cite_0 @cite_15 @cite_12 @cite_2
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We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times, thus showing that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunction, stochastic comparisons and concentration inequalities. We analyze the mixing behavior of the biased exclusion process on a path of length n as the bias ( _n ) tends to 0 as (n ). We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when ( _n ) is of order 1 n, and the other when ( _n ) is order ( n n ). We consider the simple exclusion process with @math particles on a segment of length @math performing random walks with transition @math to the right and @math to the left. We focus on the case where the asymmetry in the jump rates @math vanishes in the limit when @math and @math tend to infinity, and obtain sharp asymptotics for the mixing times of this sequence of Markov chains in the two cases where the asymmetry is either much larger or much smaller than @math . We show that in the former case ( @math ), the mixing time corresponds to the time needed to reach macroscopic equilibrium, like for the strongly asymmetric (i.e. constant @math ) case studied in [LL18], while the latter case ( @math ) macroscopic equilibrium is not sufficient for mixing and one must wait till local fluctuations equilibrate, similarly to what happens in the symmetric case worked out in [Lac16b]. In both cases, convergence to equilibrium is abrupt: we have a cutoff phenomenon for the total-variation distance. We present a conjecture for the remaining regime when the asymmetry is of order @math . 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.