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

We consider a one-dimensional totally asymmetric nearest-neighbor zero-rang e process with site-dependent jump-rates - an environment. For each environ ment p we prove that the set of all invariant measures is the convex hull o f a set of product measures with geometric marginals. As a consequence we s how that for environments p satisfying certain asymptotic property, there a re no invariant measures concentrating on configurations with density bigge r 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 initia l configuration has asymptotic density strictly above rho*(p), then the pro cess converges to the maximal invariant measure. (C) 2000 Elsevier Science B.V. All rights reserved.