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Metastability in a condensing zero-range process in the thermodynamic limit

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Abstract

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density [In: Beltran and Landim, Probab Theory Relat Fields 152(3–4):781–807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.

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Acknowledgments

The authors are grateful to Martin Slowik and Claudio Landim for useful discussions. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) – Grant No. EP/I014799/1, grant PICT 2012-2744 Stochastic Processes and Statistical Mechanics, the European Social Fund and Greek national funds through the Operational Programme “Education and Lifelong learning” -NSRF Research Funding Programmes Thales MIS377291 and Aristeia 1082.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad de Buenos Aires, C1428EGA, Buenos Aires, Argentina

    Inés Armendáriz

  2. Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK

    Stefan Grosskinsky

  3. School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780, Athens, Greece

    Michail Loulakis

Authors
  1. Inés Armendáriz
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  2. Stefan Grosskinsky
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  3. Michail Loulakis
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Correspondence to Michail Loulakis.

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Armendáriz, I., Grosskinsky, S. & Loulakis, M. Metastability in a condensing zero-range process in the thermodynamic limit. Probab. Theory Relat. Fields 169, 105–175 (2017). https://doi.org/10.1007/s00440-016-0728-y

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  • DOI: https://doi.org/10.1007/s00440-016-0728-y

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