Summary
A large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on {0, 1{Z d is proven. In this way a pseudo-free energy and pressure is defined. For ergodic systems the large deviations property holds with the usual scaling. The case of non ergodic systems is also discussed. A similar result holds for occupation times. The perturbation by an external field is treated.
Article PDF
Similar content being viewed by others
References
Accardi, L., Olla, S.: Donsker and Varadhan's theory for stationary processes. Preprint
Bramson, M., Cox, T., Griffeath, D.: Consolidation rates for two interacting systems in the plane. Preprint
Bramson, M., Cox, T., Griffeath, D.: Occupation time large deviations of the voter model. Preprint
Bricmont, J., Kuruda, K., Lebowitz, J.L.: First order phase transitions in lattice and continuous systems: extension of Pirogov-Sinai theory. Commun. Math. Phys.101, 501–538 (1985)
Cox, T., Griffeath, D.: Large deviations for Poisson systems of independent random walks. Z. Wahrscheinlichkeitstheor. Verw. Geb.66, 543–558 (1984)
Cox, T., Griffeath, D.: Occupation times for critical branching Brownian motions. Ann. Probab.13, 1108–1132 (1985)
Durrett, R., Griffeath, D.: Supercritical contact processes on Z. Ann. Probab.11, 1–15 (1983)
Ellis, R.: Entropy, large deviations and statistical mechanics. Berlin Heidelberg New York: Springer 1985
Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
Goldstein, S., Kipnis, C., Ianiro, N.: Stationary states for mechanical systems with stochastic boundary conditions. J. Stat. Phys.41, 915–939 (1985)
Goldstein, S., Lebowitz, J.L., Presutti, E.: Mechanical systems with stochastic boundaries. In: Fritz, J., Lebowitz, J.L., Szasz, D. (eds.) Random Fields. Amsterdam: North-Holland 1981
Gray, L.: The critical behavior of a class of simple interacting systems — a few answers and a lot of questions. In: Durrett, R. (ed.) Contemporary mathematics, vol. 41, pp. 149–160. Providence: Am. Math. Soc. 1985
Griffeath, D.: Additive and cancelative interacting particle systems. Lect. Notes Math., vol. 724. Berlin Heidelberg New York: Springer 1979
Katz, S., Lebowitz, J.L., Spohn, H.: Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors. J. Stat. Phys.34, 497–537 (1984)
Künsch, H.: Non reversible stationary measures for infinite interacting particle systems. Z. Wahrscheinlichkeitstheor. Verw. Geb.66, 407–424 (1984)
Lanford, O.E.: Entropy and equilibrium states in classical statistical mechanics. Lect. Notes, Phys., vol.20, pp.1–113. Berlin Heidelberg New York: Springer 1971
Lebowitz, J.L.: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Commun. Math. Phys.28, 313–321 (1972)
Lebowitz, J.L.: Exact results in nonequilibrium statistical mechanics: Where do we stand? Prog. Theor. Phys.64, 35–49 (1978)
Liggett, T.M.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985
Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.74, 119–228 (1980)
Olla, S.: Large deviations for almost Markovian processes. Probab. Th. Rel. Fields16, 395–409
Orey, S.: Large deviations and Shanon-McMillan theorems. Seminar on Stochastic Processes, 1984. Boston: Birkäuser 1986
Plachky, D.: On a theorem by G.L. Sievers. Ann. Math. Statist.42, 1442–1443 (1971)
Plachky, D., Steinbach, J.: A theorem about probabilities of large deviations with an application to queuing theory. Per. Math. Hungarica6, 343–345 (1975)
Ruelle, D.: Statistical mechanics. Rigorous results. Menlo Park: Benjamin 1969
Ruelle, D.: Thermodynamic formalism. Encyclopedia of Mathematics and Its Applications, vol. 5. Reading, Mass.: Addison-Wesley 1978
Sievers, G.L.: On the probability of large deviations and exact slopes. Ann. Math. Statist.40, 1908–1921 (1969)
Sinai, Ya.G.: Theory of phase transitions: Rigorous results. Oxford New York: Pergamon Press 1982
Slawny, J.: Low temperature properties of classical lattice systems: phase transitions and phase diagrams. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, Vol. 11. Amsterdam: North Holland 1987
Sullivan, W.G.: Potentials for almost Markovian random fields. Commun. Math. Phys.33, 61–74 (1973)
Author information
Authors and Affiliations
Additional information
Work partially supported by NSF-DMR81-14726 (USA) and CNPq (Brazil)
Also Department of Physics
Rights and permissions
About this article
Cite this article
Lebowitz, J.L., Schonmann, R.H. Pseudo-free energies and large deviations for non gibbsian FKG measures. Probab. Th. Rel. Fields 77, 49–64 (1988). https://doi.org/10.1007/BF01848130
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01848130