Mixing time for the asymmetric simple exclusion process in a random environment

Abstract

We consider the simple exclusion process in the integer segment 1, N with k ≤ N/2 particles and spatially inhomogenous jumping rates. A particle at site x ∈ 1, N jumps to site x−1 (if x ≥ 2) at rate 1−ωx and to site x+1 (if x ≤ N −1) at rate ωx if the target site is not occupied. The sequence ω = (ωx) x∈Z is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume E[log ρ1] < 0 where ρ1 := (1 − ω1)/ω1, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of N . More precisely, for the exclusion process with N β+o(1) particles where β ∈ [0, 1), we have in the large N asymptoticthe equation has no positive root) and C is a constant which depends on the distribution of ω. We conjecture that our lower bound is sharp up to sub-polynomial correction.

Mixing time and cutoff for one dimensional particle systems

“…That is to say p varies with i. In [42,29] the case of IID random biases has been considered. This is a multi-particle version of the the classical Random Walk in a Random Environment (RWRE) (see e.g.…”

“…[44,20] for seminal references and [14] for a study of mixing time of RWRE). The works [29,42] show that the presence of inhomogeneities in the environment can slow down the convergence to equilibrium.…”

Mixing times for the TASEP in the maximal current phase

“…Moreover, for several instances of exclusion processes, the limit profiles of the mixing time are studied; see [19,39,42]. Very recently, mixing times are also investigated for simple exclusion processes with a size-dependent bias, and in random environments [40,43,51].…”