The Annals of Probability

On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields

J. E. Steif and J. van den Berg

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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 $\mathbf{Z}^d, d \geq 2$, 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.

Article information

Ann. Probab., Volume 27, Number 3 (1999), 1501-1522.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28D99: None of the above, but in this section
Secondary: 60K35 82B20 82B26: Phase transitions (general)

Ising model random fields phase transitions finitary coding


van den Berg, J.; Steif, J. E. On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields. Ann. Probab. 27 (1999), no. 3, 1501--1522. doi:10.1214/aop/1022677456.

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