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Shannon–McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations
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  • Published: 28 November 2007

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

  • Julia Brettschneider1,2 

Probability Theory and Related Fields volume 142, pages 443–473 (2008)Cite this article

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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Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    Julia Brettschneider

  2. Department of Community Health and Epidemiology and Cancer Research Institute Division of Cancer Care and Epidemiology, Queen’s University, Kingston, ON, K7L 3N6, Canada

    Julia Brettschneider

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  1. Julia Brettschneider
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Correspondence to Julia Brettschneider.

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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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  • Received: 26 March 2006

  • Revised: 10 October 2007

  • Published: 28 November 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0112-z

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Mathematical Subject Classification (2000)

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