Critical Probabilities and Convergence Time of Stavskaya's Probabilistic Cellular Automata

Lorenzo Taggi

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Submission date: 20. Mar. 2014
Pages: 33
published in: Journal of statistical physics, 159 (2015) 4, p. 853-892 
DOI number (of the published article): 10.1007/s10955-015-1199-8
Bibtex
with the following different title: Critical probabilities and convergence time of percolation probabilistic cellular automata
MSC-Numbers: 82C27, 82C22, 82C20
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Abstract:
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(η_s = 1|η_𝒰(s)) = 0 ifη_𝒰(s) = 0 or p otherwise, ∀s ∈ ℤ. For any 𝒰 there exists a non-trivial critical probability p_c(𝒰) which separates a phase with an absorbing-state from a fluctuating phase. We study how the neighbourhood affects the value of p_c(𝒰) and we provide lower bounds for p_c(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 > p_c (resp. p < p_c). This appears as an open problem in Toom et al. (2004, 1995, 1990).