Stationary States of the One-dimensional Facilitated Asymmetric Exclusion Process

Abstract

We describe the translation invariant stationary states (TIS) of the one-dimensional facilitated asymmetric exclusion process in continuous time, in which a particle at site $i\in\mathbb{Z}$ jumps to site $i+1$ (respectively $i-1$) with rate $p$ (resp. $1-p$), provided that site $i-1$ (resp. $i+1$) is occupied and site $i+1$ (resp. $i-1$) is empty. All TIS states with density $ρ\le1/2$ are supported on trapped configurations in which no two adjacent sites are occupied; we prove that if in this case the initial state is i.i.d.~Bernoulli then the final state is independent of $p$. This independence also holds for the system on a finite ring. For $ρ>1/2$ there is only one TIS. It is the infinite volume limit of the probability distribution that gives uniform weight to all configurations in which no two holes are adjacent, and is isomorphic to the Gibbs measure for hard core particles with nearest neighbor exclusion.