Critical Probabilities and Convergence Time of Stavskaya's Probabilistic Cellular Automata
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Submission date: 20. Mar. 2014
published in: Journal of statistical physics, 159 (2015) 4, p. 853-892
DOI number (of the published article): 10.1007/s10955-015-1199-8
with the following different title: Critical probabilities and convergence time of percolation probabilistic cellular automata
MSC-Numbers: 82C27, 82C22, 82C20
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We consider a class of probabilistic cellular automata undergoing a phase transition with an absorbing-state. Denoting by 𝒰(s) the neighbourhood of the site s, the transition probability is T(ηs = 1|η𝒰(s)) = 0 ifη𝒰(s) = 0 or p otherwise, ∀s ∈ ℤ. For any 𝒰 there exists a non-trivial critical probability pc(𝒰) which separates a phase with an absorbing-state from a ﬂuctuating phase. We study how the neighbourhood aﬀects the value of pc(𝒰) and we provide lower bounds for pc(𝒰). Furthermore, using techniques of dynamic renormalization, we prove that the expected convergence time of the processes on a ﬁnite space with periodic boundaries grows exponentially (resp. logarithmically) with the system size if p > pc (resp. p < pc). This appears as an open problem in Toom et al. (2004, 1995, 1990).