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 the simple exclusion process in ballistic random environment, results on the mixing time were so far only available in the non-nestling case. For any (deterministic) environment with a uniform bound on the drift, Miracle and Streib showed that the mixing time is linear in the size of the segment using path coupling, see @cite_11 .
@cite_11
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In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i, j) with probability (p_ i,j ). Our goal is to identify the class of parameter sets ( P = p_ i,j ) for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing.