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Asymmetric conservative processes with random rates

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  • Benjamini, I.
  • Ferrari, P. A.
  • Landim, C.

Abstract

We study a one-dimensional nearest neighbor simple exclusion process for which the rates of jump are chosen randomly at time zero and fixed for the rest of the evolution. The ith particle's right and left jump rates are denoted pi and qi respectively; pi+ qi = 1. We fix c [epsilon] (1/2, 1) and assume that pi [epsilon] [c, 1] is a stationary ergodic process. We show that there exists a critical density [varrho]* depending only on the distribution of {{pi}} such that for almost all choices of the rates: (a) if [varrho] [epsilon] [[varrho]*, 1], then there exists a product invariant distribution for the process as seen from a tagged particle with asymptotic density [varrho]; (b) if [varrho] [epsilon] [0, [varrho]*), then there are no product measures invariant for the process. We give a necessary and sufficient condition for [varrho]* > 0 in the iid case. We also show that under a product invariant distribution, the position Xt of the tagged particle at time t can be sharply approximated by a Poisson process. Finally, we prove the hydrodynamical limit for zero range processes with random rate jumps.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:61:y:1996:i:2:p:181-204
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    References listed on IDEAS

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    1. Benois, O. & Kipnis, C. & Landim, C., 1995. "Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes," Stochastic Processes and their Applications, Elsevier, vol. 55(1), pages 65-89, January.
    2. James R. Jackson, 1963. "Jobshop-Like Queueing Systems," Management Science, INFORMS, vol. 10(1), pages 131-142, October.
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    1. 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.
    2. Beltrán, Johel, 2005. "Regularity of diffusion coefficient for nearest neighbor asymmetric simple exclusion on," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1451-1474, September.
    3. Großkinsky, Stefan, 2008. "Equivalence of ensembles for two-species zero-range invariant measures," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1322-1350, August.
    4. 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.
    5. Belitsky, V. & Schütz, G.M., 2018. "Self-duality and shock dynamics in the n-species priority ASEP," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1165-1207.
    6. 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.

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