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

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  • Andjel, E. D.
  • Ferrari, P. A.
  • Guiol, H.
  • Landim *, C.

Abstract

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 [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.

Suggested Citation

  • Andjel, E. D. & Ferrari, P. A. & Guiol, H. & Landim *, C., 2000. "Convergence to the maximal invariant measure for a zero-range process with random rates," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 67-81, November.
    • Handle: RePEc:eee:spapps:v:90:y:2000:i:1:p:67-81
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    References listed on IDEAS

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    1. James R. Jackson, 1957. "Networks of Waiting Lines," Operations Research, INFORMS, vol. 5(4), pages 518-521, August.
    2. Koukkous, A., 1999. "Hydrodynamic behavior of symmetric zero-range processes with random rates," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 297-312, December.
    3. Gielis, G. & Koukkous, A. & Landim, C., 1998. "Equilibrium fluctuations for zero range processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 77(2), pages 187-205, September.
    4. Benjamini, I. & Ferrari, P. A. & Landim, C., 1996. "Asymmetric conservative processes with random rates," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 181-204, February.
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    Cited by:

    1. Mailler, Cécile & Mörters, Peter & Ueltschi, Daniel, 2016. "Condensation and symmetry-breaking in the zero-range process with weak site disorder," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3283-3309.
    2. Lin, Hao & Seppäläinen, Timo, 2012. "Properties of the limit shape for some last-passage growth models in random environments," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 498-521.
    3. Armendáriz, Inés & Grosskinsky, Stefan & Loulakis, Michail, 2013. "Zero-range condensation at criticality," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3466-3496.

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