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From PCA's to equilibrium systems and back

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Abstract

Stationary measures for probabilistic cellular automata (PCA's) ind dimensions give rise to space-time histories whose statistics may naturally be described by Gibbs states ind+1 dimensions for an interaction energy ℋ obtained from the PCA. In this note we study the converse question: Do all Gibbs states for this ℋ correspond to statistical space-time histories for the PCA? Our main result states that the answer is yes, at least for translation invariant or periodic Gibbs states. Thus ergodicity questions for PCA's can, at least partially, be formulated as questions of uniqueness of Gibbs states.

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Author notes

  1. Roelof Kuik

    Present address: Erasmus University, Faculteit Bedijfskunde Rotterdam, The Netherlands

Authors and Affiliations

  1. Department of Mathematics and Physics, Rutgers University, 08903, New Brunswick, NJ, USA

    Sheldon Goldstein, Roelof Kuik, Joel L. Lebowitz & Christian Maes

  2. Aspirant N.F.W.O., Instituut voor Theoretische Fysika, K.U. Leuven, Celestijnenlaan 200 D, B-3030, Leuven, Belgium

    Christian Maes

Authors
  1. Sheldon Goldstein
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  2. Roelof Kuik
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  3. Joel L. Lebowitz
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  4. Christian Maes
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Additional information

Communicated by A. Jaffe

Dedicated to Roland Dobrushin

Research supported by NSF Grants DMR-86-12369 and DMS-85-12505

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Goldstein, S., Kuik, R., Lebowitz, J.L. et al. From PCA's to equilibrium systems and back. Commun.Math. Phys. 125, 71–79 (1989). https://doi.org/10.1007/BF01217769

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  • DOI: https://doi.org/10.1007/BF01217769

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