Relative entropy and variational properties of generalized Gibbsian measures

• 成果类型: Article
• 署名作者: Külske, C; Le Ny, A; Redig, F
• 署名单位: Technical University of Berlin; Universite Paris Saclay; Eindhoven University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000342
• 发表日期: 2004
• 页码: 1691-1726
• 关键词: 2d ising-model; critical-behavior; PATHOLOGIES; PRINCIPLE

摘要

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.