## The Annals of Probability

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

#### Abstract

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

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

Dates
First available in Project Euclid: 29 May 2002

https://projecteuclid.org/euclid.aop/1022677456

Digital Object Identifier
doi:10.1214/aop/1022677456

Mathematical Reviews number (MathSciNet)
MR1733157

Zentralblatt MATH identifier
0968.60091

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1022677456

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