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Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder

Abstract

We study asymmetric zero-range processes on \(\mathbb {Z}\) with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In particular, we relax the weak convexity assumption of Bahadoran et al. (Braz J Probab Stat 29(2):313–335, 2015; Ann Inst Henri Poincaré Probab Stat 53(2):766–801, 2017) for the escape of mass property.

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Acknowledgements

We thank Gunter Schütz for many interesting discussions. This work was partially supported by laboratoire MAP5, Grants ANR-15-CE40-0020-02 and ANR-14-CE25-0011, LabEx CARMIN (ANR-10-LABX-59-01), Simons Foundation Collaboration Grant 281207 awarded to K. Ravishankar. We thank Universités Clermont Auvergne and Paris Descartes for hospitality. This work was partially carried out during C.B.’s 2017–2018 délégation CNRS, whose support is acknowledged. Part of it was done during the authors’ stay at the Institut Henri Poincaré (UMS 5208 CNRS-Sorbonne Université) - Centre Emile Borel for the trimester “Stochastic Dynamics Out of Equilibrium”, and during the authors’ stay at NYU Shanghai. The authors thank these institutions for hospitality and support.

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Bahadoran, C., Mountford, T., Ravishankar, K. et al. Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder. Probab. Theory Relat. Fields 176, 149–202 (2020). https://doi.org/10.1007/s00440-019-00916-2

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  • DOI: https://doi.org/10.1007/s00440-019-00916-2

Keywords

  • Asymmetric zero-range process
  • Site disorder
  • Phase transition
  • Condensation
  • Hydrodynamic limit
  • Strong local equilibrium
  • Large-time convergence

Mathematics Subject Classification

  • 60K35
  • 82C22