Metastability in a Condensing Zero-Range Process in the Thermodynamic Limit

STEFAN GROSSKINSKY; MICHAIL LOULAKIS; INÉS ARMENDÁRIZ

PROBABILITY THEORY AND RELATED FIELDS

SPRINGER

Lugar: Berlin; Año: 2017 vol. 169 p. 105 - 175

0178-8051

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finitefraction of all particles concentrates on a single lattice site when the total density exceeds a criticalvalue. We study such a process on a one-dimensional lattice with periodic boundary conditionsin the thermodynamic limit with fixed, super-critical particle density. We show that the processexhibits metastability with respect to the condensate location, i.e. the suitably accelerated processof the rescaled location converges to a limiting Markov process on the unit torus. This process hasstationary, independent increments and the rates are characterized by the scaling limit of capacitiesof a single random walker on the lattice. Our result extends previous work for fixed lattices anddiverging 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. Besidesadditional technical difficulties in estimating error bounds for transition rates, the thermodynamiclimit requires new estimates for equilibration towards a suitably defined distribution in metastablewells, corresponding to a typical set of configurations with a particular condensate location. Thetotal exit rates from individual wells turn out to diverge in the limit, which requires an intermediateregularization 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 theexit rates from metastable wells, which is of a general nature and can be adapted to other models.