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finitary coding for the sub-critical Ising model with finite expected coding volume

ELECTRONIC JOURNAL OF PROBABILITY 2020年第none期25卷 1-27页

作者： Spinka, Yinon Univ British Columbia Vancouver BC Canada

It has been shown by van den Berg and Steif [5] that the sub-critical Ising model on Z(d) is a finitary factor of a finite-valued i.i.d. process. We strengthen this by showing that the factor map can be made to have finite expected coding volume (in fact, stretched-exponential tails), answering a question of van den Berg and Steif. The result holds at any temperature above the critical temperature. An analogous result holds for Markov random fields satisfying a high-noise assumption and for proper colorings with a large number of colors.

关键词： Ising model finitary coding finite expected coding volume

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finitary codings for the random-cluster model and other infinite-range monotone models

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ELECTRONIC JOURNAL OF PROBABILITY 2022年第none期27卷 1-32页

作者： Harel, Matan Spinka, Yinon Northeastern Univ Boston MA 02115 USA Univ British Columbia Vancouver BC Canada

A random field X = (X-v)(v is an element of G) on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as X = phi(Y) for some i.i.d. process Y = (Y-v)(v is an element of G) and equivariant map phi. Such a map, also called a coding, is finitary if, for every vertex v is an element of G, there exists a finite (but random) set U subset of G such that X-v is determined by {Y-u}(u is an element of U). We construct a coding for the random-cluster model on G, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Haggstrom-Jonasson-Lyons [18]. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model. Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications - the loop O(n) model and long-range Ising models. In the case of G = Z(d), we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process Y. To do this, we extend a mixing-time result of Martinelli-Olivieri [22] to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.

关键词： factor of iid finitary coding quasi-transitive graph random-cluster model monotone specification coupling from the past

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On the existence and nonexistence of finitary codings for a class of random fields

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ANNALS OF PROBABILITY 1999年第3期27卷 1501-1522页

作者： van den Berg, J Steif, JE CWI Dept PNA NL-1098 SJ Amsterdam Netherlands Chalmers Univ Technol Dept Math S-41296 Gothenburg Sweden

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 greater than or equal to 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.

关键词： ising model random fields phase transitions finitary coding

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Generating stochastic processes based on the finitary interval algorithm

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2008年第9期E91A卷 2482-2488页

作者： Fujisaki, Hiroshi Kanazawa Univ Grad Sch Nat Sci & Technol Kanazawa Ishikawa 9201192 Japan

We point out that the interval algorithm can be expressed in the form of a shift on the sequence space. Then we clarify that, by using a Bernoulli process, the interval algorithm can generate only a block of Markov chains or a sequence of independent blocks of Markov chains but not a stationary Markov process. By virtue of the finitary coding constructed by Hamachi and Keane, we obtain the procedure, called the finitary interval algorithm, to generate a Markov process by using the interval algorithm. The finitary interval algorithm also gives maps, defined almost everywhere, which transform a Markov measure to a Bernoulli measure.

关键词： random number generation interval algorithm induced transformation Bernoulli shift Markov process finitary coding

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Markov chains with exponential return times are finitary

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ERGODIC THEORY AND DYNAMICAL SYSTEMS 2021年第10期41卷 2918-2926页

作者： Angel, Omer Spinka, Yinon Univ British Columbia Dept Math 1984 Math Rd Vancouver BC V6T 1Z2 Canada

Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of Z is a finitary factor of an i.i.d. process.

关键词： random dynamics symbolic dynamics factor of iid finitary coding Markov chain

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