On the existence and nonexistence of finitary codings for a class of random fields

• 成果类型: Article
• 署名作者: van den Berg, J; Steif, JE
• 署名单位: Centrum Wiskunde & Informatica (CWI); Chalmers University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1999
• 页码: 1501-1522
• 关键词: maximal entropy; models; shifts

摘要

We study the existence of finitary codings (also called finitary homomorphisms or finitary factor maps) from a finite-valued i.i.d. process to certain random fields. For Markov random fields we show, using ideas of Marton and Shields, that the presence of a phase transition is an obstruction for the existence of the above coding; this yields a large class of Bernoulli shifts for which no such coding exists. Conversely, we show that, for the stationary distribution of a monotone exponentially ergodic probabilistic cellular automaton, such a coding does exist. The construction of the coding is partially inspired by the Propp-Wilson algorithm for exact simulation. In particular, combining our results with a theorem of Martinelli and Olivieri, we obtain the fact that for the plus state for the ferromagnetic Ising model on Z(d), d greater than or equal to 2, there is such a coding when the interaction parameter is below its critical value and there is no such coding when the interaction parameter is above its critical value.