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Proof of a phase transition in probabilistic cellular automata

Abstract : Cellular automata are a model of parallel computing. It is well known that simple deterministic cellular automata may exhibit complex behaviors such as Turing universality [3,13] but only few results are known about complex behaviors of probabilistic cellular automata. Several studies have focused on a specific probabilistic dynamics: α-asynchronism where at each time step each cell has a probability α to be updated. Experimental studies [5] followed by mathematical analysis [2,4,7,8] have permitted to exhibit simple rules with interesting behaviors. Among these behaviors, most of these studies conjectured that some cellular automata exhibit a polynomial/exponential phase transition on their convergence time, i.e. the time to reach a stable configuration. The study of these phase transitions is crucial to understand the behaviors which appear at low synchronicity. A first analysis [14] proved the existence of the exponential phase in cellular automaton FLIP-IF-NOT-ALL-EQUAL but failed to prove the existence of the polynomial phase. In this paper, we prove the existence of a polynomial/exponential phase transition in a cellular automaton called FLIP-IF-NOT-ALL-0.
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Submitted on : Wednesday, September 25, 2013 - 11:09:54 AM
Last modification on : Tuesday, June 30, 2020 - 11:56:08 AM

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Damien Regnault. Proof of a phase transition in probabilistic cellular automata. 17th International Conference on Developments in Language Theory (DLT 2013), Jun 2013, Marne-la-Vallee, France. pp.433--444, ⟨10.1007/978-3-642-38771-5_38⟩. ⟨hal-00865846⟩

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