Global specifications and nonquasilocality of projections of Gibbs measures

• 成果类型: Article
• 署名作者: Fernandez, R; Pfister, CE
• 署名单位: Universidade de Sao Paulo; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1284-1315
• 关键词: renormalization-group transformations; ising-model; PATHOLOGIES

摘要

We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-sigma-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is shown to be empty whenever a strong uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for nonquasilocality (based on a quantity related to the surface tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our nonquasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).