Communications in Mathematical Physics

, Volume 144, Issue 2, pp 303–323 | Cite as

The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition

  • Daniel W. Stroock
  • Boguslaw Zegarlinski


Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Time Distribution 
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  1. [A&H] Aizenman, M., Holley, R.: Rapid convergence to equilibrium of stochastic Ising Models in the Dobrushin Shlosman régime, Percolation Theory and Ergodic Theory of Infinite Particle Systems, H. Kesten, (ed.), IMS Volumes in Math. and Appl. vol.8, pp. 1–11. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  2. [D] Dobrushin, R.: Markov processes with a large number of locally interacting components-Existence of the limiting process and its ergodicity. Probl. Peredaci Inform.7, 70–87 (1971)Google Scholar
  3. [D&S] Deuschel, J.-D., Stroock, D.: Large Deviations, Pure and Appl. Math. Series, vol.137. Boston: Academic Press 1989Google Scholar
  4. [Dob&S,1] Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of Gibbs field, Statistical Physics and Dynamical Systems, Rigorous Results, pp. 347–370. Fritz, Jaffe, and Szasz (eds.). Basel-Boston: Birkhauser 1985Google Scholar
  5. [Dob&S,2] —: Completely analytical Gibbs fields. Statistical Physics and Dynamical Systems, Rigorous Results, pp. 371–403. Fritz, Jaffe, and Szasz, (eds.), Basel-Boston: Birkhäuser 1985Google Scholar
  6. [Dob&S,3] —: Completely analytical interactions: Constructive description. J. Stat. Phys.46, 983–1014 (1987)Google Scholar
  7. [F] Federbush, P.: Partially alternative derivation of a result of Nelson. J. Math. Phys.10 (1), 50–52 (1989)Google Scholar
  8. [G, 1] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 553–586 (1979)Google Scholar
  9. [G, 2] —: Absence of second-order phase transitions in the Dobrushin uniqueness region. J. Stat. Phys.25 (1), 57–72 (1981)Google Scholar
  10. [H] Holley, R.: Possible rates of convergence in finite range, attractive spin systems. Contemp. Math.41, 215–234 (1985)Google Scholar
  11. [H&S, 1] Holley, R., Stroock, D.: Applications of the stochastic Ising Model to the Gibbs states. Commun. Math. Phys.48, 249–265 (1967)Google Scholar
  12. [H&S, 2] —: Uniform andL 2 convergence in one dimensional stochastic ising models. Commun. Math. Phys.123, 85–93 (1989)Google Scholar
  13. [L] Liggett, T.: Infinite Particle Systems. Grundlehren Series, vol.276, Berlin, Heidelberg, New York: Springer 1985Google Scholar
  14. [S&Z] Stroock, D., Zegarlinski, B.: The logarithmic Sobolev inequality for lattice gases with continuous spins. J. Funct. Anal. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Daniel W. Stroock
    • 1
  • Boguslaw Zegarlinski
    • 1
    • 2
  1. 1.2-272, Mathematics DepartmentM.I.T.CambridgeUSA
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochum 1Germany

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