## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 55, 2007, 108-120

### Escape of mass in zero-range processes with random rates

Pablo A. Ferrari and Valentin V. Sisko

#### Abstract

We consider zero-range processes in $\Z^d$ with site dependent jump rates. The rate for a particle jump from site $x$ to $y$ in $\Z^d$ is given by $\lambda_x g(k) p(y-x)$, where $p(\cdot)$ is a probability in $\Z^d$, $g(k)$ 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 $\{\nlv: 0\le v \le c\}$ 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 $\nlc$, 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 $\|x\|$. 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 $\nlc$. In particular, if $\mu$ dominates $\nlc$, then $\mu S_{\lambda}(t)$ converges to $\nlc$. The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.

#### Chapter information

**Source***Asymptotics: Particles, Processes and Inverse Problems: Festschrift for Piet Groeneboom* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007)

**Dates**

First available in Project Euclid: 4 December 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196797071

**Digital Object Identifier**

doi:10.1214/074921707000000300

**Mathematical Reviews number (MathSciNet)**

MR2459934

**Zentralblatt MATH identifier**

1205.60169

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

**Keywords**

random environment zero-range process

**Rights**

Copyright © 2007, Institute of Mathematical Statistics

#### Citation

Ferrari, Pablo A.; Sisko, Valentin V. Escape of mass in zero-range processes with random rates. Asymptotics: Particles, Processes and Inverse Problems, 108--120, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000300. https://projecteuclid.org/euclid.lnms/1196797071