FINITARY CODINGS FOR SPATIAL MIXING MARKOV RANDOM FIELDS

• 成果类型: Article
• 署名作者: Spinka, Yinon
• 署名单位: University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1405
• 发表日期: 2020
• 页码: 1557-1591
• 关键词: lattice spin systems; one-phase region; hard-core; perfect simulation; bernoulli-shifts; glauber dynamics; gibbs measures; potts models; entropy; equilibrium

摘要

It has been shown by van den Berg and Steif (Ann. Probab. 27 (1999) 1501-1522) that the subcritical and critical Ising model on Z(d) is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom-Rowlinson model and the beach model. For instance, for the ferromagnetic q-state Potts model on Z(d) at inverse temperature beta, we show that it is ffiid with exponential tails if beta is sufficiently small, it is ffiid if beta < beta(c) ( q , d), it is not ffiid if beta >beta(c)( q , d) and, when d = 2 and beta =beta(c) (q, d), it is ffiid if and only if q <= 4.