Journal of Statistical Physics

, Volume 81, Issue 5–6, pp 1007–1019 | Cite as

On the ergodic properties of Glauber dynamics

  • D. Stroock
  • B. Zegarlinski
Articles

Abstract

We show that if there is an infinite volume Gibbs measure which satisfies a logarithmic Sobolev inequality with local coefficients of moderate growth, then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm.

Key Words

Glauber dynamics logarithmic Sobolev inequality ergodic properties 

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References

  1. 1.
    M. Aizenman and R. A. Holley, Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin-Shlosman regime, inPercolation Theory and Ergodic Theory of Infinite Particle Systems, H. Kesten, ed. (Springer-Verlag, 1987), pp. 1–11.Google Scholar
  2. 2.
    J.-D. Deuschel and D. W. Stroock,Large Deviations (Academic Press, 1989).Google Scholar
  3. 3.
    R. L. Dobrushin and S. B. Shlosman, Constructive criterion for the uniqueness of Gibbs field, inStatistical Physics and Dynamical Systems, Rigorous Results, Fritz, Jaffe, and Szasz, eds. (Birkhäuser, 1985), pp. 347–370.Google Scholar
  4. 4.
    R. L. Dobrushin and S. B. Shlosman, Completely analytical Gibbs fields, inStatistical Physics and Dynamical Systems, Rigorous Results, Fritz, Jaffe, and Szasz, eds. (Birkhäuser, 1985), pp. 371–403.Google Scholar
  5. 5.
    R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: Constructive description,J. Stat. Phys. 46:983–1014 (1987).Google Scholar
  6. 6.
    R. Holley, Possible rates of convergence in finite range, attractive spin systems,Contemp. Math. 41:215–234 (1985).Google Scholar
  7. 7.
    R. Holley and D. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models,J. Stat. Phys. 46:1159–1194 (1987).Google Scholar
  8. 8.
    E. Laroche, Hypercontractivité pour des systèmes de spin de portée infinie,Prob. Theory Related Fields,101:89–132 (1995).Google Scholar
  9. 9.
    Sheng Lin Lu and Horng-Tzer Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics,Commun. Math. Phys. 156:399–433 (1993).Google Scholar
  10. 10.
    F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region: I. The attractive case/II. The general case,Commun. Math. Phys. 161:447–486/487–514 (1994).Google Scholar
  11. 11.
    F. Martinelli and E. Olivier, Finite volume mixing conditions for lattice spin systems and exponential approach to equilibrium of Glauber dynamics, Preprint (1994).Google Scholar
  12. 12.
    F. Martinelli, E. Olivieri, and R. H. Schonmann, For 2-D lattice spin systems weak mixing implies strong mixing,Commun. Math. Phys. 165:33–47 (1994).Google Scholar
  13. 13.
    S. B. Shlosman and R. H. Schonmann, Complete analyticity for 2D Ising completed, inCommun. Math. Phys., to appear.Google Scholar
  14. 14.
    D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for continuous spin systems on a lattice,J. Funct. Anal. 104:299–326 (1992).Google Scholar
  15. 15.
    D. W. Stroock and B. Zegarlinski, The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition,Commun. Math. Phys. 144:303–323 (1992).Google Scholar
  16. 16.
    D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for discrete spin systems on a lattice,Commun. Math. Phys. 149:175–193 (1992).Google Scholar
  17. 17.
    B. Zegarlinski, Recent progress in hypercontractive semigroups, inProceedings of Ascona Conference (1993).Google Scholar
  18. 18.
    B. Zegarlinski, Strong decay to equilibrium in one dimensional random spin systems,J. Stat. Phys. 77:717–732 (1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • D. Stroock
    • 1
  • B. Zegarlinski
    • 2
  1. 1.MITCambridge
  2. 2.Imperial CollegeLondonUK

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