Journal of Statistical Physics

, 144:1009

Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

Article

Abstract

We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.

Keywords

Path coupling Mixing times Glauber dynamics Large deviations Blume-Capel model Aggregate path coupling 

References

  1. 1.
    Blume, M.: Theory of the first-order magnetic phase change in UO2. Phys. Rev. 141, 517–524 (1966) ADSCrossRefGoogle Scholar
  2. 2.
    Blume, M., Emery, V.J., Griffiths, R.B.: Ising model for the λ transition and phase separation in He3–He4 mixtures. Phys. Rev. A 4, 1071–1077 (1971) ADSCrossRefGoogle Scholar
  3. 3.
    Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pp. 223–231 (1997) CrossRefGoogle Scholar
  4. 4.
    Brémaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics, vol. 31. Springer, New York (1999) Google Scholar
  5. 5.
    Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 32, 966–988 (1966) ADSCrossRefGoogle Scholar
  6. 6.
    Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting II. Physica 33, 295–331 (1967) ADSCrossRefGoogle Scholar
  7. 7.
    Capel, H.W.: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting III. Physica 37, 423–441 (1967) ADSCrossRefGoogle Scholar
  8. 8.
    Costeniuc, M., Ellis, R.S., Touchette, H.: Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46, 063301 (2005) MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Ding, J., Lubetzky, E., Peres, Y.: The mixing time evolution of Glauber dynamics for the mean-field Ising model. Commun. Math. Phys. 289, 725–764 (2009) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Ellis, R.S.: Entropy, Large Deviations and Statistical Mechanics. Springer, New York (1985). Reprinted in 2006 in Classics in Mathematics MATHCrossRefGoogle Scholar
  11. 11.
    Ellis, R.S., Haven, K., Turkington, B.: Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Stat. Phys. 101, 999–1064 (2000) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ellis, R.S., Otto, P.T., Touchette, H.: Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model. Ann. Appl. Probab. 15, 2203–2254 (2005) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ellis, R.S., Machta, J., Otto, P.T.: Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg-Landau polynomials. J. Stat. Phys. 133, 101–129 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Ellis, R.S., Machta, J., Otto, P.T.: Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality. Ann. Appl. Probab. 20, 2118–2161 (2010) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. Am. Math. Soc., Providence (2009) MATHGoogle Scholar
  16. 16.
    Levin, D., Luczak, M., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146 (2010) Google Scholar
  17. 17.
    Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54 235–268 (1982) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Yevgeniy Kovchegov
    • 1
  • Peter T. Otto
    • 2
  • Mathew Titus
    • 1
  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA
  2. 2.Department of MathematicsWillamette UniversitySalemUSA

Personalised recommendations