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From equilibrium spin models to probabilistic cellular automata

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Abstract

The general equivalence betweenD-dimensional probabilistic cellular automata (PCA) and (D+1)-dimensional equilibrium spin models satisfying a “disorder condition” is first described in a pedagogical way and then used to analyze the phase diagrams, the critical behavior, and the universality classes of some automata. Diagrammatic representations of time-dependent correlation functions of PCA are introduced. Two important classes of PCA are singled out for which these correlation functions simplify: (1) “Quasi-Hamiltonian” automata, which have a current-carrying steady state, and for which some correlation functions are those of aD-dimensional static model. PCA satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes. (2) “Linear” (and more generally “affine”) PCA for which the diagrammatics reduces to a random walk problem closely related to (D+1)-dimensional directed SAWs: both problems display a critical behavior with mean-field exponents in any dimension. The correlation length and effective velocity of propagation of excitations can be calculated for affine PCA, as is shown on an explicitD=1 example. We conclude with some remarks on nonlinear PCA, for which the diagrammatics is related to reaction-diffusion processes, and which belong in some cases to the universality class of Reggeon field theory.

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  1. Pierre Le Doussal

    Present address: Sloane Laboratory, Physics Department, Yale University, New Haven, Connecticut

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  1. Laboratoire de Physique Théorique de l'École Normale Supérieure (Laboratoire propre du Centre National de la Recherche Scientifique, associé à l'École Normale Supérieure et à l'Université de Paris-Sud), France

    Antoine Georges & Pierre Le Doussal

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  1. Antoine Georges
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Georges, A., Le Doussal, P. From equilibrium spin models to probabilistic cellular automata. J Stat Phys 54, 1011–1064 (1989). https://doi.org/10.1007/BF01019786

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  • DOI: https://doi.org/10.1007/BF01019786

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