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

TL;DR

This paper studies the metastable behavior of a zero-range process with condensation on a large lattice, showing that the condensate's location converges to a Markov process with independent increments in the thermodynamic limit.

Contribution

It extends previous results to the thermodynamic limit, introduces new estimates for equilibration, and develops a coupling method for bounding exit rates from metastable states.

Findings

The condensate location converges to a Markov process on the unit torus.

Exit rates from metastable wells diverge, requiring regularization.

A coupling construction provides uniform bounds on exit rates.

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.