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

2007/09/17 by Julia Brettschneider, Brettschneider, Julia
Mathematics · Physics and Astronomy · #60F10 (Primary) #60G60 #82B20 (Secondary) #82B26 #94A17 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #math.ST #msc:60F10 #msc:60G60 #msc:82B20 #msc:82B26 #msc:94A17 #stat.TH

paper · pdf · doi:10.48550/arxiv.0709.2662

25 pages, correction of typos

openalex publication_date 2007/09/17 · arxiv created 2007/10/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Foellmer and Ort's lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behavior in the phase-transition regime.

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