Abstract
It has been observed (Evans in Braz J Phys 30:42–57, 2000; Jeon et al. in Ann Probab 28:1162–1194, 2000) that some zero-range processes exhibit condensation, a macroscopic fraction of particles concentrates on one single site. We examined in (Beltrán and Landim in Probab Theory Relat Fields 152:781–807, 2012) the asymptotic evolution of the condensate in the case where the dynamics is reversible, the number of sites is fixed, and the total number of particles diverges. We proved in that paper that in an appropriate time-scale the condensate evolves according to a symmetric random walk whose transition rates are proportional to the capacities of the underlying random walk. In this article, we extend this result to the condensing totally asymmetric zero-range process, a non-reversible dynamics.
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References
Armendáriz I., Loulakis M.: Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Relat. Fields 145, 175–188 (2009)
Armendáriz I., Loulakis M.: Conditional distribution of heavy tailed random variables on large deviations of their sum. Stoch. Proc. Appl. 121, 1138–1147 (2011)
Armendáriz I., Großkinsky S., Loulakis M.: Zero range condensation at criticality. Stoch. Process. Appl. 123, 346–3496 (2013)
Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)
Beltrán J., Landim C.: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Relat. Fields 152, 781–807 (2012)
Beltrán J., Landim C.: Metastability of reversible finite state Markov processes. Stoch. Proc. Appl. 121, 1633–1677 (2011)
Beltrán, J., Landim, C.: Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. To appear in Ann. Inst. H. Poincaré, Probab. Statist. (2014)
Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains II, the nonreversible case. J. Stat. Phys. 149, 598–618 (2012)
Beltrán, J., Landim, C.: A martingale approach to metastability. To appear in Probab. Theory Related Fields (2014)
Bianchi, A., Gaudillière, A.: Metastable states, quasi-stationary and soft measures, mixing time asymptotics via variational principles. arXiv:1103.1143 (2011)
Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability in stochastic dynamics of disordered mean field models. Probab. Theory Relat. Fields 119, 99–161 (2001)
Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys. 228, 219–255 (2002)
Cassandro M., Galves A., Olivieri E., Vares M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984)
Doyle, P.: Energy for Markov Chains. Preprint http://math.dartmouth.edu/doyle/:16 (1994)
Evans M.R.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30, 42–57 (2000)
Evans M.R., Hanney T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A 38(19), R195–R240 (2005)
Ferrari P.A., Landim C., Sisko V.V.: Condensation for a fixed number of independent random variables. J. Stat. Phys. 128, 1153–1158 (2007)
Gaudillière, A.: Condenser physics applied to Markov chains: A brief introduction to potential theory. Online http://arxiv.org/abs/0901.3053
Gaudillière A., Landim C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields 158, 55–89 (2014)
Gois, B., Landim, C.: Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus. To appear in Ann. Probab. (2014)
Godrèche C., Luck J.M.: Dynamics of the condensate in zero-range processes. J. Phys. A 38, 7215–7237 (2005)
Großkinsky S., Schütz G.M., Spohn H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003)
Jara M., Landim C., Teixeira A.: Quenched scaling limits of trap models. Ann. Probab. 39, 176–223 (2011)
Jara, M., Landim, C., Teixeira, A.: Universality of trap models in the ergodic time scale. To appear in Annals of Probability (2014)
Jeon I., March P., Pittel B.: Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28, 1162–1194 (2000)
Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov processes. Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345. Springer, Heidelberg, (2012)
Lacoin, H., Teixeira, A.: A Mathematical Perspective on Metastable Wetting. arXiv:1312.7732 (2013)
Landim, C.: A Topology for Limits of Markov Chains. arXiv:1310.3646 (2013)
Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)
Slowik, M.: A Note on Variational Representations of Capacities for Reversible and Non-reversible Markov Chains. Preprint (2013)
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Landim, C. Metastability for a Non-reversible Dynamics: The Evolution of the Condensate in Totally Asymmetric Zero Range Processes. Commun. Math. Phys. 330, 1–32 (2014). https://doi.org/10.1007/s00220-014-2072-3
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DOI: https://doi.org/10.1007/s00220-014-2072-3