## 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.

### 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

### Keywords:

random field; finitary coding; Ising model; phase transition
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### References:

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