MIXING TIME FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS IN A RANDOM ENVIRONMENT

Abstract

We consider the simple exclusion process in the integer segment J1, NK with k ≤ N/2 particles and spatially inhomogenous jumping rates. A particle at site x ∈ J1, NK 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 Nβ+o(1) particles where β ∈ [0, 1], we have in the large N asymptotic