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 $bbbz^d$, $d geq 2$, 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.

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CWI. Probability, Networks and Algorithms [PNA]

van den Berg, R., & Steif, J. E. (1998). On the existence and non-existence of finitary codings for a class of random fields. CWI. Probability, Networks and Algorithms [PNA]. CWI.