Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process

Abstract

We consider finite-range, nonzero mean, one-dimensional exclusion processes on Z. We show that, if the initial configuration has "density" α, then the process converges in distribution to the product Bernoulli measure with mean density α. From this we deduce the strong form of local equilibrium in the hydrodynamic limit for non-product initial measures.

Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

“…Similar local equilibrium and convergence results were obtained in [6] for the one-dimensional ASEP without disorder. However, in the approach used there, a key ingredient was the strict concavity of the flux function f .…”

“…This process has the property that if η 0 ∈ X, then almost surely, one has η t ∈ X for every t > 0. In this case, it may be considered as a Markov process on X with generator (6) restricted to functions f : X → R.…”

Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model

“…In this section we describe and provide properties of several couplings that will be used in the proof of Theorem 6.1. We begin by stating the following proposition, whose analog for the ASEP was established as Proposition 12 of [8]. It essentially states that a stochastic six-vertex model with an approximately constant initial density profile will "almost" couple with a stationary stochastic six-vertex model, after being run for a sufficiently long time.…”

“…We will only show the first of these two equalities, as the proof of the latter is entirely analogous. Given the content of the previous sections, this will follow Section 4 of [8].…”

“…This statement can be viewed as a generalization of the two-block estimate (see, for instance, Lemma 3.2 in Chapter 5 of the book [46] by Kipnis) for interacting particle systems. Indeed, in the case of the ASEP, [8] essentially reduced it to a two-block estimate that had earlier been proven by Kosygina [48]. In our setting of the stochastic six-vertex model, we will establish the approximate coupling statement directly, through a suitable adaptation of the method implemented in [48] of equating two different approximations for the total current of the model.…”