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Statistical mechanics of probabilistic cellular automata

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Abstract

We investigate the behavior of discrete-time probabilistic cellular automata (PCA), which are Markov processes on spin configurations on ad-dimensional lattice, from a rigorous statistical mechanics point of view. In particular, we exploit, whenever possible, the correspondence between stationary measures on the space-time histories of PCAs on ℤd and translation-invariant Gibbs states for a related Hamiltonian on ℤ(d+1). This leads to a simple large-deviation formula for the space-time histories of the PCA and a proof that in a high-temperature regime the stationary states of the PCA are Gibbsian. We also obtain results about entropy, fluctuations, and correlation inequalities, and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high-noise regime. We discuss phase transitions in the low-noise (or low-temperature) regime and review Toom's proof of nonergodicity of a certain class of PCAs.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, New Jersey

    Joel L. Lebowitz, Christian Maes & Eugene R. Speer

  2. Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France

    Joel L. Lebowitz & Christian Maes

  3. Department of Physics, Rutgers University, New Brunswick, New Jersey

    Joel L. Lebowitz

  4. Aangesteld Navorser NFWO, Instituut voor Theoretische Fysica, K. U. Leuven, Belgium

    Christian Maes

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  1. Joel L. Lebowitz
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  2. Christian Maes
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Lebowitz, J.L., Maes, C. & Speer, E.R. Statistical mechanics of probabilistic cellular automata. J Stat Phys 59, 117–170 (1990). https://doi.org/10.1007/BF01015566

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