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

• Published in: Stochastic processes and their applications Vol. 90; no. 1; pp. 67 - 81
• Main Authors: Andjel, E.D., Ferrari, P.A., Guiol, H., Landim, C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2000; Elsevier Science; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Classical ensemble theory; Classical statistical mechanics; Convergence to the maximal invariant measure; Exact sciences and technology; Invariant measures; Mathematics; Physics; Probability and statistics; Probability theory and stochastic processes; Random rates; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical physics, thermodynamics, and nonlinear dynamical systems; Zero-range; Zero-range Random rates Invariant measures Convergence to the maximal invariant measure; Zero-range; secondary 82C22; primary 60K35; Invariant measures; Random rates; Convergence to the maximal invariant measure; Random rate; Invariant measure; Poisson process; Particle system; Marginal distribution; Convergence
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(00)00037-5

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.