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**On the existence and nonexistence of finitary codings for a class of random fields.**
*(English)*
Zbl 0968.60091

For a finite set \(A\), consider the class of stationary random fields on \({\mathbb Z}^d\) (\(d\geq 2\)) that are defined as probability measures \(\mu\) on \(A^{{\mathbb Z}^d}\) invariant w.r.t. the natural \({\mathbb Z}^d\)-action. Two such fields \(\mu\) and \(\nu\) (corresponding to finite sets \(A\) and \(B\)) are called isomorphic if there exists an invertible measure preserving map from \((A^{{\mathbb Z}^d},\mu)\) to \((B^{{\mathbb Z}^d},\nu)\) which is defined a.e. and which commutes with all \({\mathbb Z}^d\)-shifts. Any such field that is isomorphic to a stationary i.i.d. field is called a Bernoulli shift. An interesting result due to Ornstein and Weiss claims that the plus state in the ferromagnetic Ising model in \({\mathbb Z}^d\), \(d\geq 1\), (at zero external field) is a Bernoulli shift for any positive temperature, and thus the considered isomorphisms are insensitive to the phase transition taking place in dimensions \(d\geq 2\).

A measurable map from \((A^{{\mathbb Z}^d},\mu)\) to \((B^{{\mathbb Z}^d},\nu)\) is called a finitary coding if it commutes with \({\mathbb Z}^d\)-action and is continuous a.e. (w.r.t. the natural product topology). The main question addressed in the paper is: Which random fields can be obtained as finitary codings by finite-valued i.i.d. random fields? The authors show that the stationary distributions of a monotone exponentially ergodic probabilistic cellular automaton does admit such a coding. On the other hand, for the Markov random field in the phase transition regime such coding does not exist. Applying the obtained results to the ferromagnetic Ising model in \({\mathbb Z}^d\) (at zero external field), the authors deduce that the corresponding plus state admits a finitary coding by a finite-valued i.i.d. random field only in the high-temperature (i.e. uniqueness) region.

A measurable map from \((A^{{\mathbb Z}^d},\mu)\) to \((B^{{\mathbb Z}^d},\nu)\) is called a finitary coding if it commutes with \({\mathbb Z}^d\)-action and is continuous a.e. (w.r.t. the natural product topology). The main question addressed in the paper is: Which random fields can be obtained as finitary codings by finite-valued i.i.d. random fields? The authors show that the stationary distributions of a monotone exponentially ergodic probabilistic cellular automaton does admit such a coding. On the other hand, for the Markov random field in the phase transition regime such coding does not exist. Applying the obtained results to the ferromagnetic Ising model in \({\mathbb Z}^d\) (at zero external field), the authors deduce that the corresponding plus state admits a finitary coding by a finite-valued i.i.d. random field only in the high-temperature (i.e. uniqueness) region.

Reviewer: Ostap Hryniv (Lausanne)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

28Dxx | Measure-theoretic ergodic theory |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

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\textit{J. van den Berg} and \textit{J. E. Steif}, Ann. Probab. 27, No. 3, 1501--1522 (1999; Zbl 0968.60091)

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