Abstract
The notion of a surface-order specific entropy h c (P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon–McMillan theorem. We obtain a representation of h c (P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Föllmer and Ort’s lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behaviour in the phase-transition regime.
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Brettschneider, J.: Shannon–McMillan theorems for random fields along curves and lower bounds for surface-order large deviations. Eurandom. Technical report 2001-018
Comets, F.: Grandes deviations pour des champs de gibbs sur F d. C. R. Acad. Sci. Paris Ser. I 303, 511–513 (1986)
Deuschel, J.-D., Stroock, D.W.: Large Deviations. Academic, Boston (1989)
Dobrushin, R., Kotecky, R., Shlosman, S.: Wulff construction. A global shape from local interaction. In: AMS Translations of Mathematical Monographs, vol. 104. Providence (1992)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time. II. Commun. Pure Appl. Math. 28, 279–301 (1975)
Föllmer, H.: On entropy and information gain in random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb. 26, 207–217 (1973)
Föllmer, H.: On the global Markov property. In: Quantum Fields—Algebras, Processes, Proc. Symp., Bielefeld 1978, pp. 293–302. Springer, Heidelberg (1980)
Föllmer, H., Orey, S.: Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16(3), 961–977 (1988)
Föllmer, H., Ort, M.: Large deviations and surface entropy for Markov fields. Astérisque 288, 173–190 (1988)
Georgii, H.-O.: Gibbs Measures and Phase Transitions. W. de Gruyter, Berlin (1988)
Ioffe, D.: Large deviations for the 2D Ising model: a lower bound without cluster expansion. J. Stat. Phys. 74, 411–432 (1994)
Ioffe, D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Relat. Fields 102(3), 313–330 (1995)
Israel, R.B.: Some examples concerning the global Markov property. Commun. Math. Phys. 105, 669–673 (1986)
Krengel, U.: Ergodic Theorems. W. de Gruyter, Berlin (1985)
Lanford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)
Milnor, J.: Directional entropies of cellular automaton-maps. In: Disorderd Systems and Biological Organization (Les Houches, 1985), vol. 20, pp. 113–115. NATO Adv. Sci. Inst. Ser. F: Comut. Systems Sci., New York (1986)
Milnor, J.: On the entropy geometry of cellular automata. Complex Syst. 2(3), 357–385 (1988)
Olla, S.: Large deviations for Gibbs random fields. Probab. Theory Relat. Fields 77(3), 343–357 (1988)
Park, K.K.: Continuity of directional entropy. Osaka J. Math. 31(3), 613–628 (1994)
Park, K.K.: Continuity of directional entropy for a class of \({\mathbb{Z}^2}\) -actions. J. Korean Math. Soc. 32(3), 573–582 (1995)
Park, K.K.: Entropy of a skew product with a \({\mathbb{Z}^2}\) -action. Pac. J. Math. 172(1), 227–241 (1996)
Park, K.K.: On directional entropy functions. Isr. J. Math. 113, 243–267 (1999)
Preston, C.: Random Fields. In: LNM, vol. 534. Springer, New York (1976)
Schonmann, R.: Second order large deviation estimates for ferromagnetic systems in the phase coexistence region. Commun. Math. Phys. 112(3), 409–422 (1987)
Sinai, Ya.G.: An answer to a question by J. Milnor. Comment. Math. Helv. 60, 173–178 (1985)
Sinai, Ya.G.: Topics in Ergodic Theory. Princeton University Press, Princeton (1994)
Thouvenot, J.-P.: Convergence en moyenne de l’information pour l’action de \({\mathbb{Z}^2}\) . Z. Wahrscheinlichkeitstheor. Verw. Geb. 24, 135–137 (1972)
Weizsäcker, H.V.: Exchanging the order of taking suprema and countable intersections of σ-algebras. Ann. Inst. Henri Poincaré B 19(1), 91–100 (1983)
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Brettschneider, J. Shannon–McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations. Probab. Theory Relat. Fields 142, 443–473 (2008). https://doi.org/10.1007/s00440-007-0112-z
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DOI: https://doi.org/10.1007/s00440-007-0112-z