Supercritical behavior of asymmetric zero-range process with sitewise disorder

Abstract

We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for one-dimensional asymmetric nearest-neighbour zero-range processes with non-homogeneous jump rates. The class of "environments" considered is close to that considered by [1], while our class of processes is broader. We also give in arbitrary dimension a simpler proof of the result of [19] with weaker assumptions.

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

“…For the totally asymmetric M/M/1 model, it was shown in [4] that the system converges to the maximal invariant measure (thereby implying a loss of mass). This was established in [7,8] for the general nearest-neighbour model under a weak convexity assumption, and we showed that this could fail for non nearestneighbour jump kernels.…”

“…The purpose of this paper is to introduce a new approach for the derivation of quenched strong local equilibrium in order to address this question, which was left open by the previous works [10,21]. In the case of supercritical hydrodynamic density ρ(t, x) > ρ c , we prove that the local equilibrium property fails, and that, locally around "typical points" of the environment, the distribution of the microscopic state is close to the critical measure with density ρ c : this can be viewed as a dynamic version of the loss of mass property studied in [4,7,8]. The dynamic loss of mass that we establish here allows us to remove the convexity assumption used in [7,8], but for a slightly less general class of initial configurations.…”

“…For the existence and uniqueness of (η α t ) t≥0 see [8,Appendix B]. Recall from [1] that, since g is nondecreasing, (η α t ) t≥0 is attractive, i.e.…”

“…As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular, we relax the weak convexity assumption of [7,8] for the escape of mass property.MSC 2010 subject classification: 60K35, 82C22.…”

Hydrodynamics in a condensation regime: The disordered asymmetric zero-range process

“…The measure (6) is always supported on X if β ∈ (0, c) ∪ {0} (11) When β = c > 0, conventions (9)-(10) yield a measure supported on configurations with infinitely many particles at all sites x ∈ Z that achieve the infimum in (7), and finitely many particles at other sites. In particular, this measure is supported on X when the infimum in (7) is not achieved.…”