Journal of Statistical Physics

, Volume 165, Issue 4, pp 693–726 | Cite as

Metastability of Non-reversible, Mean-Field Potts Model with Three Spins

  • C. Landim
  • I. Seo


We examine a non-reversible, mean-field Potts model with three spins on a set with \(N\uparrow \infty \) points. Without an external field, there are three critical temperatures and five different metastable regimes. The analysis can be extended by a perturbative argument to the case of small external fields, and it can be carried out in the case where the external field is in the direction or in the opposite direction to one of the values of the spins. Numerical computations permit to identify other phenomena which are not present in the previous situations.


Metastability Tunneling behavior Mean-field Potts model Non-reversible Markov chains 


  1. 1.
    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beltrán, J., Landim, C.: A Martingale approach to metastability. Probab. Theory Relat. Fields 161, 267–307 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berglund, N.: Kramers’ law: validity, derivations and generalisations. Markov Process. Relat. Fields 19, 459–490 (2013)MathSciNetMATHGoogle Scholar
  5. 5.
    Bouchet, F., Reygner, J.: Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes. Preprint (2015). arXiv:1507.02104
  6. 6.
    Bovier, A., den Hollander, F.: Metastability: A Potential-Theoretic Approach. Grundlehren der mathematischen Wissenschaften, vol. 351. Springer, Berlin (2015)Google Scholar
  7. 7.
    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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cuff, P., Ding, J., Louidor, O., Lubetzky, E., Peres, Y., Sly, A.: Glauber dynamics for the mean-field Potts model. J. Stat. Phys. 149, 432–477 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gaudillière, A., Landim, C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields 158, 55–89 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Landim, C.: A topology for limits of Markov chains. Stoch. Proc. Appl. 125, 1058–1098 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Landim, C., Seo, I.: Metastability of non-reversible random walks in a potential field, the Eyring–Kramers transition rate formula (2016, submitted). arXiv:1605.01009
  14. 14.
    Landim, C., Misturini, R., Tsunoda, K.: Metastability of reversible random walks in potential field. J. Stat. Phys. 160, 1449–1482 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Misturini, R.: Evolution of the ABC model among the segregated configurations in the zero temperature limit. Ann. Inst. H. Poincaré Probab. Stat. 52, 669–702 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Potts, R.B.: Mathematical investigation of some cooperative phenomena. Ph.D. Thesis, University of Oxford (1950)Google Scholar
  17. 17.
    Slowik, M.: A note on variational representations of capacities for reversible and nonreversible Markov chains. Unpublished, Technische Universität Berlin (2012)Google Scholar
  18. 18.
    Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.CNRS UMR 6085, Université de RouenSaint-Étienne-du-RouvrayFrance
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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