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

TL;DR

This paper introduces a surface-order entropy for 2D discrete random fields along curves, proves a Shannon-McMillan theorem for it, and applies it to improve large deviation bounds in phase-transition regimes.

Contribution

It defines and proves the existence of a surface-order entropy along curves and uses it to refine large deviation lower bounds for Gibbs measures.

Findings

Existence of surface-order entropy along curves.

Representation of entropy as a mixture along tangent lines.

Refined lower bounds for large deviations in phase transitions.

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.