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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
J.-D. Deuschel and D. W. Stroock,Large Deviations (Academic Press, 1989).
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.
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.
R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: Constructive description,J. Stat. Phys. 46:983–1014 (1987).
R. Holley, Possible rates of convergence in finite range, attractive spin systems,Contemp. Math. 41:215–234 (1985).
R. Holley and D. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models,J. Stat. Phys. 46:1159–1194 (1987).
E. Laroche, Hypercontractivité pour des systèmes de spin de portée infinie,Prob. Theory Related Fields,101:89–132 (1995).
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).
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).
F. Martinelli and E. Olivier, Finite volume mixing conditions for lattice spin systems and exponential approach to equilibrium of Glauber dynamics, Preprint (1994).
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).
S. B. Shlosman and R. H. Schonmann, Complete analyticity for 2D Ising completed, inCommun. Math. Phys., to appear.
D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for continuous spin systems on a lattice,J. Funct. Anal. 104:299–326 (1992).
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).
D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for discrete spin systems on a lattice,Commun. Math. Phys. 149:175–193 (1992).
B. Zegarlinski, Recent progress in hypercontractive semigroups, inProceedings of Ascona Conference (1993).
B. Zegarlinski, Strong decay to equilibrium in one dimensional random spin systems,J. Stat. Phys. 77:717–732 (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stroock, D., Zegarlinski, B. On the ergodic properties of Glauber dynamics. J Stat Phys 81, 1007–1019 (1995). https://doi.org/10.1007/BF02179301
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179301