E.D. Andjel, P.A. Ferrari, H. Guiol, C. Landim
Convergence to the maximal invariant measure
for a zero-range process with random rates.
(43K, LaTeX2e)
ABSTRACT. We consider a one-dimensional totally asymmetric
nearest-neighbor zero-range process with site-dependent jump-rates
---an \emph{environment}. For each environment $p$ we prove that the
set of all invariant measures is the convex hull of a set of product
measures with geometric marginals. As a consequence we show that
for environments $p$ satisfying certain asymptotic
property, there are no invariant measures concentrating on
configurations with critical density bigger than $\rho^*(p)$, a
critical value. If $\rho^*(p)$ is finite we say that there is
phase-transition on the density. In this case we prove that if the
initial configuration has asymptotic density strictly above
$\rho^*(p)$, then the process converges to the
maximal invariant measure.