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 $λ_xg(k)p(y-x)$, 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 $λ=\{λ_x\}$ 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 $\{ν_{λ, v}:0\le v\le c\}$ 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 ${ν_{λ, c}}$, 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 $S_λ(t)$ the semigroup of the process, we prove that all weak limits of $\{μS_λ(t),t\ge 0\}$ as $t\to \infty$ are dominated, in the natural partial order, by ${ν_{λ, c}}$. In particular, if $μ$ dominates ${ν_{λ, c}}$, then $μS_λ(t)$ converges to ${ν_{λ, c}}$. The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.