Escape of mass in zero-range processes with random rates

Abstract

We consider zero-range processes in ${\mathbb{Z}}^d$ with site dependent jump rates. The rate for a particle jump from site $x$ to $y$ in ${\mathbb{Z}}^d$ is given by ${\lambda }_{x}g(k)p(y-x)$, where $p(\cdot )$ is a probability in ${\mathbb{Z}}^d$, $g(k)$ 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 $\lambda$ the zero-range process has product invariant measures $\{{\nu }_{\lambda ,v}:0\le v\le c\}$ parametrized by $v$, 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 $v$. There exists a product invariant measure ${\nu }_{\lambda ,c}$, with maximal density. Let $\mu$ be a probability measure concentrating mass on configurations whose number of particles at site $x$ grows less than exponentially with $\Vert x\Vert$. Denoting by $S_{\lambda }(t)$ the semigroup of the process, we prove that all weak limits of $\{\mu S_{\lambda }(t),t\ge 0\}$ as $t\to \infty$ are dominated, in the natural partial order, by ${\nu }_{\lambda ,c}$. In particular, if $\mu$ 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.

Published at http://dx.doi.org/10.1214/074921707000000300 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)