Escape of Mass in Zero-Range Processes with Random Rates

• Published in: Lecture notes-monograph series 2007-01, Vol.55, p.108-120
• Main Authors: Ferrari, Pablo A., Sisko, Valentin V.
• Format: Article
• Language: English
• Subjects: Average linear density; Density; Infinity; Particle density; Particle interactions; Poisson equation; Poisson process; Random variables; Random walk; Semigroups
• ISSN: 0749-2170
• DOI: 10.1214/074921707000000300

We consider zero-range processes in${\Bbb Z}^{d}$with site dependent jump rates. The rate for a particle jump from site x to y in${\Bbb Z}^{d}$is given by$\lambda _{x}g(k)p(y-x)$, where p(·) is a probability in${\Bbb Z}^{d}$, g(k) is a bounded nondecreasing function of the number k of particles in x and$\lambda =\{\lambda _{x}\}$is a collection of i.i.d. random variables with values in (c, 1], for some c > 0. For almost every realization of the environment λ the zero-range process has product invariant measures$\{\nu _{\lambda ,v}$: 0 ≤ ν ≤ c} parametrized by ν, the average total jump rate from any given site. The density of a measure, defined by the asymptotic average number of particles per site, is an increasing function of ν. There exists a product invariant measure$\nu _{\lambda ,c}$, with maximal density. Let μ be a probability measure concentrating mass on configurations whose number of particles at site x grows less than exponentially with ∥x∥. Denoting by$S_{\lambda (t)}$the semigroup of the process, we prove that all weak limits of {$\mu S_{\lambda (t)}$, t ≥ 0} as t → ∞ are dominated, in the natural partial order, by$\nu _{\lambda ,c}$. In particular, if μ dominates$\nu _{\lambda ,c}$, then$\mu S_{\lambda (t)}$converges to$\nu _{\lambda ,c}$. The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.