Probability Theory and Related Fields

, Volume 142, Issue 3–4, pp 443–473 | Cite as

Shannon–McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations

  • Julia BrettschneiderEmail author


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.

Mathematical Subject Classification (2000)

Primary: 60F10 Secondary: 60G60 94A17 82B26 82B20 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of WarwickCoventryUK
  2. 2.Department of Community Health and Epidemiology and Cancer Research Institute Division of Cancer Care and EpidemiologyQueen’s UniversityKingstonCanada

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