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 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.