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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

Abstract

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.

Acknowledgments

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.

Citation

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

Information

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

Subjects:
Primary: 60K35
Secondary: 60K37

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

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