×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adams, S. (1992). Fölner Independence and the amenable Ising model. Ergodic Theory Dynam. Systems 12 633-657. · Zbl 0787.58025
[2] Aizenman, M., Barsky, D. J. and Fernandez, R. (1987). The phase transition in a general class of Ising-type models is sharp. J. Statist Phys. 47 343-374.
[3] Aizenman, M. and Fernandez, R. (1986). On the critical behavior of the magnetization in high-dimensional Ising models. J. Statist Phys. 44 393-454. · Zbl 0629.60106
[4] Akcoglu, M. A., del Junco, A. and Rahe, M. (1979). Finitary codes between Markov processes.Wahrsch. Verw. Gebiete 47 305-314. · Zbl 0403.28017
[5] Blum, J. R. and Hanson, D. (1963). On the isomorphism problem for Bernoulli schemes. Bull. Amer. Math. Soc. 69 221-223. · Zbl 0121.13601
[6] Burton, R. M. and Steif, J. E. (1994). Nonuniqueness of measures of maximal entropy for subshifts of finite type. Ergodic Theory Dynam. Systems 14 213-235. · Zbl 0807.58023
[7] Burton, R. M. and Steif, J. E. (1995). New results on measures of maximal entropy, Israel J. Math. 89 275-300. · Zbl 0826.28009
[8] Burton, R. M. and Steif, J. E. (1996). Some 2-d symbolic dynamical systems: entropy and mixing. In Ergodic Theory of Zd Actions (M. Pollicott and K. Schmidt, eds.) 297-305. Cambridge Univ. Press. · Zbl 0852.58029
[9] Csiszar, I. and Korner, J. (1981). Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, New York.
[10] del Junco, A. (1980). Finitary coding of Markov random fields.Wahrsch. Verw. Gebiete 52 193-202. · Zbl 0411.60055
[11] den Hollander, F. and Steif, J. E. (1997). On K-automorphisms, Bernoulli shifts and Markov random fields. Ergodic Theory Dynam. Systems 17 405-415. · Zbl 0949.60501
[12] den Hollander, F. and Steif, J. E. (1998). On the equivalence of certain ergodic properties for Gibbs states. Ergodic Theory Dynam. Systems. · Zbl 0968.60046
[13] Dobrushin, R. L. (1972). Gibbs states describing coexistence of phases for a threedimensional Ising model. Theory Probab. Appl. 17 582-600. · Zbl 0275.60119
[14] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York. · Zbl 0566.60097
[15] F öllmer, H. and Orey, S. (1988). Large deviations for the empirical field of a Gibbs measure. Ann. Probab. 16 961-977. · Zbl 0648.60028
[16] Friedman, N. A. and Ornstein, D. (1971). On isomorphism of weak Bernoulli transformations. Adv. Math. 5 365-394. · Zbl 0203.05801
[17] Georgii, H. (1988). Gibbs Measures and Phase Transitions. de Gruyter, New York. · Zbl 0657.60122
[18] Higuchi, Y. (1993). Coexistence of infinite -clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97 1-33. · Zbl 0794.60102
[19] Hoffman, C. (1999). A Markov random field which is K but not Bernoulli. Israel J. Math. · Zbl 0939.60036
[20] Katznelson, Y. and Weiss, B. (1972). Commuting measure-preserving transformations. Israel J. Math. 12 161-173. · Zbl 0239.28014
[21] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26 352- 371. · Zbl 0357.94012
[22] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. Math. 109 397-406. JSTOR: · Zbl 0405.28017
[23] Keane, M. and Smorodinsky, M. (1979). Finitary isomorphisms of irreducible Markov shifts. Israel J. Math. 34 281-286. · Zbl 0431.28015
[24] Lieb, E. H. (1980). A refinement of Simon’s correlation inequality. Comm. Math. Phys. 77 127-135.
[25] Liggett, T. M. (1985). Interacting Particle Systems. Springer, Berlin. · Zbl 0559.60078
[26] Lind, (1982). Ergodic group automorphisms are exponentially recurrent. Israel J. Math. 41 313-320. · Zbl 0495.28018
[27] Maes, C. and Shlosman, S. B. (1993). When is an interacting particle system ergodic? Comm. Math. Phys. 151 447-466. · Zbl 0765.60102
[28] Martinelli, F. and Olivieri, E. (1994). Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys. 161 447-486. · Zbl 0793.60110
[29] Marton, K. (1998). The blowing-up property for processes with memory. Unpublished manuscript. · Zbl 0927.60050
[30] Marton, K. and Shields, P. (1994). The positive-divergence and blowing-up properties. Israel J. Math. 86 331-348. · Zbl 0797.60044
[31] Meshalkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk. SSSR 128 41-44. · Zbl 0099.12301
[32] Monroy, G. and Russo, B. (1975). A family of codes between some Markov and Bernoulli schemes. Comm. Math. Phys. 43 155-159. · Zbl 0349.60070
[33] Ornstein, D. S. (1974). Ergodic Theory, Randomness and Dynamical Systems. Yale Univ. Press. · Zbl 0296.28016
[34] Ornstein, D. S. and Weiss, B. (1973). Zd-actions and the Ising model. Unpublished manuscript.
[35] Ornstein, D. S. and Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 1-141. · Zbl 0637.28015
[36] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223-252. · Zbl 0859.60067
[37] Schmidt, K. (1995). Dynamical Systems of Algebraic Origin. Birkhäuser, Boston. · Zbl 0833.28001
[38] Simon, B. (1980). Correlation inequalities and the decay of correlations in ferromagnets. Comm. Math. Phys. 77 111-126.
[39] Slawny, J. (1981). Ergodic properties of equilibrium states. Comm. Math. Phys. 80 477-483.
[40] Steif, J. E. (1988). The ergodic structure of interacting particle systems. Ph.D. dissertation, Stanford Univ.
[41] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publications I.H.E.S. 81 73-205. · Zbl 0864.60013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.