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

  • C. BahadoranEmail author
  • T. Mountford
  • K. Ravishankar
  • E. Saada


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.


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

Mathematics Subject Classification

60K35 82C22 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Blaise PascalUniversité Clermont AuvergneAubièreFrance
  2. 2.Institut de MathématiquesÉcole Polytechnique FédéraleLausanneSwitzerland
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiShanghaiChina
  4. 4.CNRS, UMR 8145, MAP5Université Paris DescartesParis cedex 06France

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