Centrum Voor Wiskunde En Informatica on the Existence and Non-existence of Finitary Codings for a Class of Random Fields

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 Z d , d ≥ 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.

[1]  Robert M. Burton,et al.  New results on measures of maximal entropy , 1995 .

[2]  D. Ornstein,et al.  On isomorphism of weak Bernoulli transformations , 1970 .

[3]  Meir Smorodinsky,et al.  Finitary isomorphisms of irreducible Markov shifts , 1979 .

[4]  F. Martinelli,et al.  Approach to equilibrium of Glauber dynamics in the one phase region , 1994 .

[5]  Klaus Schmidt.  Dynamical Systems of Algebraic Origin , 1995 .

[6]  Michael Aizenman,et al.  On the critical behavior of the magnetization in high-dimensional Ising models , 1986 .

[7]  D. Ornstein.  Ergodic theory, randomness, and dynamical systems , 1974 .

[8]  J. Steif,et al.  Ergodic Theory of ℤ d Actions: Some 2-d symbolic dynamical systems: Entropy and mixing , 1996 .

[9]  On the Isomorphism Problem for Bernoulli Schemes , 1975 .

[10]  Finitary codes between Markov processes , 1979 .

[11]  Andrés del Junco.  Finitary coding of Markov random fields , 1980 .

[12]  Benjamin Weiss,et al.  Entropy and isomorphism theorems for actions of amenable groups , 1987 .

[13]  L. Russo,et al.  A family of codes between some Markov and Bernoulli schemes , 1975 .

[14]  R. L. Dobrushin,et al.  Gibbs State Describing Coexistence of Phases for a Three-Dimensional Ising Model , 1973 .

[15]  Barry Simon,et al.  Correlation inequalities and the decay of correlations in ferromagnets , 1980 .

[16]  Steven Orey,et al.  Large Deviations for the Empirical Field of a Gibbs Measure , 1988 .

[17]  M. Aizenman,et al.  The phase transition in a general class of Ising-type models is sharp , 1987 .

[18]  J. Slawny.  Ergodic properties of equilibrium states , 1981 .

[19]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[20]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[21]  Paul C. Shields,et al.  The positive-divergence and blowing-up properties , 1994 .

[22]  J. Steif,et al.  On K-automorphisms, Bernoulli shifts and Markov random fields , 1997, Ergodic Theory and Dynamical Systems.

[23]  D. Lind.  Ergodic group automorphisms are exponentially recurrent , 1982 .

[24]  J. Steif,et al.  On the equivalence of certain ergodic properties for Gibbs states , 2000, Ergodic Theory and Dynamical Systems.

[25]  Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions , 1993 .

[26]  Christopher Hoffman.  A markov random field which isK but not Bernoulli , 1999 .

[27]  Meir Smorodinsky,et al.  Bernoulli schemes of the same entropy are finitarily isomorphic , 1979 .

[28]  E. Lieb.  A refinement of Simon's correlation inequality , 1980 .

[29]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[30]  David Bruce Wilson,et al.  Exact sampling with coupled Markov chains and applications to statistical mechanics , 1996, Random Struct. Algorithms.

[31]  Senya Shlosman,et al.  When is an interacting particle system ergodic? , 1993 .

[32]  J. Steif,et al.  Non-uniqueness of measures of maximal entropy for subshifts of finite type , 1994, Ergodic Theory and Dynamical Systems.

[33]  Michael Keane,et al.  A class of finitary codes , 1977 .

[34]  Benjamin Weiss,et al.  Commuting measure-preserving transformations , 1972 .

[35]  S. Adams.  Følner Independence and the amenable Ising model , 1992, Ergodic Theory and Dynamical Systems.

[36]  J. Propp,et al.  Exact sampling with coupled Markov chains and applications to statistical mechanics , 1996 .