Open Access
July 1999 On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields
J. E. Steif, J. van den Berg
Ann. Probab. 27(3): 1501-1522 (July 1999). DOI: 10.1214/aop/1022677456


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.


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J. E. Steif. J. van den Berg. "On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields." Ann. Probab. 27 (3) 1501 - 1522, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0968.60091
MathSciNet: MR1733157
Digital Object Identifier: 10.1214/aop/1022677456

Primary: 28D99
Secondary: 60K35 , 82B20 , 82B26

Keywords: finitary coding , Ising model , Phase transitions , Random fields

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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