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The \(T,T^{-1}\)-process, finitary codings and weak Bernoulli. (English) Zbl 1022.60089

The \(T,T^{-1}\)-process in \(\mathbb Z^d\) (\(d\geq 1\)), associated with a stationary process \(\{X_i\}_{i\in Z}\) taking values in \(\mathbb Z^d\), is the stationary process \(\{Z_i\}_{i\in \mathbb Z}\) where \(Z_i=(X_i,C_{S_i})\), \(\{S_i\}_{i\in \mathbb Z}\) is the random walk on \(\mathbb Z^d\) corresponding to \(\{X_i\}_{i\in Z}\), and \(\{C_z\}_{z\in \mathbb Z^d}\) is an i.i.d. sequence of random variables taking values \(0\) and \(1\) with probability \(1/2.\) \(T,T^{-1}\)-processes were first considered by S. A. Kalikow for \(d=1\), and \(\{S_i\}_{i\in \mathbb Z}\) the simple symmetric random walk on \(\mathbb Z\) [Ann. Math., II. Ser. 115, 393-409 (1982; Zbl 0523.28018)], where he showed that the process \(\{Z_i\}_{i\in \mathbb Z}\) is not Bernoulli. In this paper the author gives an elementary proof that for any \(d\geq 1\), the process \(\{C_z\}_{z\in \mathbb Z^d}\), associated with an i.i.d. process \(\{X_i\}_{i\in Z}\) with mean zero, is not a finitary factor of any i.i.d process. This result implies that the corresponding \(T,T^{-1}\)-process is not finitarily isomorphic to an i.i.d. process. In the second half of the paper, different types of the weak Bernoulli property of finitary factors of i.i.d. processes indexed by \(\mathbb Z^d\) are established. As a result the weak Bernoulli property of the Ising model in the subcritical regime is obtained.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
28D05 Measure-preserving transformations
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

Citations:

Zbl 0523.28018
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References:

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