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2022 Hydrodynamic limit of the zero range process on a randomly oriented graph
Márton Balázs, Felix Maxey-Hawkins
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Electron. J. Probab. 27: 1-29 (2022). DOI: 10.1214/22-EJP753


We prove the hydrodynamic limit of a totally asymmetric zero range process on a torus with two lanes and randomly oriented edges. The asymmetry implies that the model is non-reversible. The random orientation of the edges is constructed in a bistochastic fashion which keeps the usual product distribution stationary for the quenched zero range model. It is also arranged to have no overall drift along the Z direction, which suggests diffusive scaling despite the asymmetry present in the dynamics. Indeed, using the relative entropy method, we prove the quenched hydrodynamic limit to be the heat equation with a diffusion coefficient depending on ergodic properties of the orientation of the edges.

The zero range process on this graph turns out to be non-gradient. Our main novelty is the introduction of a local equilibrium measure which decomposes the vertices of the graph into components of constant density. A clever choice of these components eliminates the non-gradient problems that normally arise during the hydrodynamic limit procedure.

Funding Statement

Felix Maxey-Hawkins was supported by an EPSRC studentship and Márton Balázs was partially supported by the EPSRC EP/R021449/1 Standard Grant of the UK. This study did not involve any underlying data.


The authors would like to thank Bálint Tóth for his suggestion to study the hydrodynamic limit of a totally asymmetric interacting particle system in a random environment using the relative entropy method, Stefano Olla for his suggestion to consider a zero range process, and anonymous Referees for valuable suggestions to improve the manuscript.


Download Citation

Márton Balázs. Felix Maxey-Hawkins. "Hydrodynamic limit of the zero range process on a randomly oriented graph." Electron. J. Probab. 27 1 - 29, 2022.


Received: 30 May 2020; Accepted: 5 February 2022; Published: 2022
First available in Project Euclid: 11 February 2022

arXiv: 2002.09214
Digital Object Identifier: 10.1214/22-EJP753

Primary: 60K35
Secondary: 60K37

Keywords: Hydrodynamic limit , random environment , Relative entropy , Zero range process


Vol.27 • 2022
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