Relative entropy and variational properties of generalized Gibbsian measures

Abstract

We study the relative entropy density for generalized Gibbs measures. We first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of ``almost Gibbsian measures'' (almost sure continuity of conditional probabilities). For quasilocal measures, we obtain a full variational principle. For the joint measures of the random field Ising model, we show that the weak Gibbs property holds, with an almost surely rapidly decaying translation-invariant potential. For these measures we show that the variational principle fails as soon as the measures lose the almost Gibbs property. These examples suggest that the class of weakly Gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism.

Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000342

References in corpus (5)

• Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
• Critical Behaviour of the Randomly Spin--Diluted 2-d Ising Model --- A Grand Ensemble Approach
• Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures
• The restriction of the Ising model to a layer
• Comment on: Critical behavior on the spin diluted 2D Ising model: A grand ensemble approach, by R. Kuehn

Cited by in corpus (7)

• Low-temperature dynamics of the Curie-Weiss Model: Periodic orbits, multiple histories, and loss of Gibbsianness
• Discrete approximations to vector spin models
• Two connections between random systems and non-Gibbsian measures
• A class of nonergodic interacting particle systems with unique invariant measure
• Variational principle for weakly dependent random fields
• Dynamical Gibbs-non-Gibbs transitions in lattice Widom-Rowlinson models with hard-core and soft-core interactions
• Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3