Open Access
Translator Disclaimer
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

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.

Citation

Download Citation

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. https://doi.org/10.1214/aop/1022677456

Information

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

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

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

Rights: Copyright © 1999 Institute of Mathematical Statistics

JOURNAL ARTICLE
22 PAGES


SHARE
Vol.27 • No. 3 • July 1999
Back to Top