Convergence to the maximal invariant measure for a zero-range process with random rates

We consider a one-dimensional totally asymmetric nearest-neighbor zero-range process with site-dependent jump-rates - an 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 $ρ^*(p)$, a critical value. If $ρ^*(p)$ 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 $ρ^*(p)$, then the process converges to the maximal invariant measure.

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