Armendáriz, Inés, Grosskinsky, Stefan and Loulakis, Michail (2017) Metastability in a condensing zero-range process in the thermodynamic limit. Probability Theory and Related Fields, 169 (1-2). pp. 105-175. doi:10.1007/s00440-016-0728-y

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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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Stochastic processes, Nonequilibrium statistical mechanics, Jump processes, Condensation
Journal or Publication Title:	Probability Theory and Related Fields
Publisher:	Springer
ISSN:	0178-8051
Official Date:	October 2017
Dates:	Date Event October 2017 Published 16 August 2016 Available 12 June 2016 Updated 16 August 2015 Submitted
Volume:	169
Number:	1-2
Number of Pages:	57
Page Range:	pp. 105-175
DOI:	10.1007/s00440-016-0728-y
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), European Social Fund (ESF), National Strategic Reference Framework (Greece) (NSRF), Aristeia 1082
Grant number:	EP/I014799/1 (EPSRC), MIS377291 (NSRF)

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