Communications in Mathematical Physics

, Volume 164, Issue 3, pp 433–454 | Cite as

Large deviations for ℤ d -Actions

  • A. Eizenberg
  • Y. Kifer
  • B. Weiss


We establish large deviations bounds for translation invariant Gibbs measures of multidimensional subshifts of finite type. This generalizes [FO] and partially [C, O, and B], where only full shifts were considered. Our framework includes, in particular, the hard-core lattice gas models which are outside of the scope of [FO, C, O, and B].


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B] Bryc, W.: On the large deviation principle for stationary weakly dependent random fields. Ann. Prob.20, 1004–1030 (1992)Google Scholar
  2. [C] Comets, F.: Grandes deviations pour des champs de Gibbs zur ℤd. CRAS t.303, No. 11 511–513 (1986)Google Scholar
  3. [DS] Deuschel, J.-D. Stroock, D.W.: Large deviations, Boston: Acad. Press, 1989Google Scholar
  4. [DSZ] Deuschel, J.-D. Stroock, D.W., Zessin, H.: Microcanonical distributions for lattice gases. Commun. Math. Phys.139, 83–101 (1991)Google Scholar
  5. [DV] Donsker, M.D. Varadhan, S.R.S.: Asymptotic evaluation of certain Markov processes expectations for large time, IV. Commun. Pure. Appl. Math.36, 183–212 (1983)Google Scholar
  6. [F] Föllmer, H.: On entropy and information gain in random fields. Z. Wahrsch. verw. Geb.26, 207–217 (1973)Google Scholar
  7. [FO] Föllmer H. Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab.16, 961–977 (1988)Google Scholar
  8. [G] Georgii, H-O.: Gibbs Measures and Phase Transitions. Berlin: W. de Gruyter, 1988Google Scholar
  9. [Ki1] Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc.321, 505–524 (1990)Google Scholar
  10. [Ki2] Kifer, Y.: Large deviations, averaging, and periodic orbits of dynamical systems, Commun. Math. Phys.162, 33–46 (1994)Google Scholar
  11. [Kr] Krengel, U.: Ergodic Theorems. Berlin: W. de Gruyter, 1985Google Scholar
  12. [M] Misiurewicz, M.: A short proof of the variational principle for a ℤ+N action on a compact space. Soc. Math. France Astérisque40, 147–157 (1976)Google Scholar
  13. [O] Olla, S.: Large deviations for Gibbs random fields. Prob. Theory and Rel. Field,77, 343–357 (1988)Google Scholar
  14. [OW] Ornstein, D. Weiss, B.: The Shannon-McMillan-Breinmann theorem for a class of amenable groups. Israel J. Math.44, 53–60 (1983)Google Scholar
  15. [R1] Ruelle, D.: Statistical mechanics on a compact set withZ ν action satisfying expansiveness and specification. Trans. am. Math. Soc.185, 237–251 (1973)Google Scholar
  16. [R2] Ruelle, D.: Thermodynamics Formalism. Reading, Mass: Addison-Wesley, 1978Google Scholar
  17. [S] Schmidt, K.: Algebraic Ideas in Ergodic Theory. CBMS Lecture Notes v.76, AMS, 1990Google Scholar
  18. [T] Tempelman, A.: Ergodic Theorems for Group Actions. Dordrecht: Kluwer, 1992Google Scholar
  19. [W] Weiss, B.: Strictly ergodic models for dynamical systems. Bull. AMS13:2, 143–146 (1985)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • A. Eizenberg
    • 1
  • Y. Kifer
    • 1
  • B. Weiss
    • 1
  1. 1.Institute of MathematicsThe Hebrew UniversityJerusalemIsrael

Personalised recommendations