Condensation of Non-reversible Zero-Range Processes

  • Insuk SeoEmail author


In this article, we investigate the condensation phenomena for a class of non-reversible zero-range processes on a fixed finite set. By establishing a novel inequality bounding the capacity between two sets, and by developing a robust framework to perform quantitative analysis on the metastability of non-reversible processes, we prove that the condensed site of the corresponding zero-range processes approximately behaves as a Markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk. The results presented in the current paper complete the generalization of the work of Beltran and Landim (Probab Theory Relat Fields 152:781–807,2012) on reversible zero-range processes, and that of Landim (Commun Math Phys 330:1–32,2014) on totally asymmetric zero-range processes on a one-dimensional discrete torus.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



I. Seo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2018R1C1B6006896 and 2017R1A5A1015626). I. Seo wishes to thank Claudio Landim, and Fraydoun Rezakhanlou for providing valuable ideas through numerous discussions. Part of this work was done during the author’s stay at the IMPA for the conference “XXI Escola Brasileira de Probabilidade”. The author thanks IMPA for the hospitality and support for this visit.


  1. 1.
    Armendáriz I., Grosskinsky S., Loulakis M.: Metastability in a condensing zero-range process in the thermodynamic limit. Probab. Theory Relat. Fields169, 105–175 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys.140, 1065–1114 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beltrán J., Landim C.: Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys.149, 598–618 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beltrán J., Landim C.: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Relat. Fields152, 781–807 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berglund N., Gentz B.: Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond. Electron. J. Probab.18, 1–58 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bianchi A., Bovier A., Ioffe D.: Sharp asymptotics for metastability in the random field Curie–Weiss model. Electron. J. Probab.14, 1541–1603 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bianchi A., Dommers S., Giardinà à C.: Metastability in the reversible inclusion process. Electron. J. Probab.22, 1–34 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. Fields119, 99–161 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability in reversible diffusion process I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc.6, 399–424 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bovier, A., den Hollander, F.: Metastability: A Potential-Theoretic Approach. Grundlehren der mathematischen Wissenschaften, vol. 351. Springer, Berlin (2015)Google Scholar
  11. 11.
    Bovier A., Manzo F.: Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Stat. Phys.107, 757–779 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cassandro M., Galves A., Olivieri E., Vares M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys.35, 603–634 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Evans M.R., Hanney T.: Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A38, 195–240 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations. In: Random Perturbations of Dynamical Systems. Grundlehren der mathematischen Wissenschaften, vol. 260. Springer, New York (1998)Google Scholar
  15. 15.
    Gaudillière A., Landim C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields158, 55–89 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Godrèche C., Luck J.M.: Dynamics of the condensate in zero-range processes. J. Phys. A38, 7215–7237 (2005)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Grosskinsky S., Redig F., Vafayi K.: Condensation in the inclusion process and related models. J. Stat. Phys.142, 952–974 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grosskinsky S., Redig F., Vafayi K.: Dynamics of condensation in the symmetric inclusion process. Electron. J. Probab.18, 1–23 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grosskinsky S., Schütz G.M., Spohn H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys.113, 389–410 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jeon I., March P., Pittel B.: Size of the largest cluster under zero-range invariant measures. Ann. Probab.28, 1162–1194 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Landim C.: A topology for limits of Markov chains. Stoch. Proc. Appl.125, 1058–1098 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Landim C., Lemire P.: Metastability of the two-dimensional Blume–Capel model with zero chemical potential and small magnetic field. J. Stat. Phys.164, 346–376 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Landim, C., Loulakis, M., Mourragui, M.: Metastable Markov chains.arXiv:1703.09481 (2017)
  25. 25.
    Landim, C., Mariani, M., Seo, I.: Dirichlet’s and Thomson’s principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes. Arch. Rational Mech. Anal. 231, 887–938 (2019)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Landim C., Misturini R., Tsunoda K.: Metastability of reversible random walks in potential field. J. Stat. Phys.160, 1449–1482 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Landim C., Seo I.: Metastability of non-reversible random walks in a potential field, the Eyring-Kramers transition rate formula. Commun. Pure Appl. Math.71, 203–266 (2018)CrossRefzbMATHGoogle Scholar
  28. 28.
    Landim C., Seo I.: Metastability of non-reversible mean-field Potts model with three spins. J. Stat. Phys.165, 693–726 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nardi, F.R., Zocca, A.: Tunneling behavior of Ising and Potts models in the low-temperature regime.arXiv:1708.09677 (2017)
  30. 30.
    Olivieri, E., Vares, M.E.: Large deviations and metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)Google Scholar
  31. 31.
    Slowik, M.: A note on variational representations of capacities for reversible and nonreversible Markov chains. Unpublished, Technische Universität Berlin (2012)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences and RIMSSeoul National UniversitySeoulRepublic of Korea

Personalised recommendations