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

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 [J. Beltran, C. Landim, Probab. Theory Related 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.

55 pages, 1 figure

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• Condensation and extremes for a fluctuating number of independent random variables
• Derivation of mean-field equations for stochastic particle systems
• The boundary driven zero-range process
• Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes
• Size-biased diffusion limits and the inclusion process
• Condensation of the invariant measures of the supercritical zero range processes
• Second Time Scale of the Metastability of Reversible Inclusion Processes
• Condensation in zero-range processes with a fast rate