Critical probabilities and convergence time of Percolation Probabilistic Cellular Automata

Abstract

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by $\mathcal{U}(x)$ the neighbourhood of site $x$, the transition probability is $T({\eta }_x=1|{\eta }_{\mathcal{U}(x)})=0$ if ${\eta }_{\mathcal{U}(x)}=0$ or $p$ otherwise, $\forall x\in \mathbb{Z}$. For any $\mathcal{U}$ there exists a non-trivial critical probability $p_c(\mathcal{U})$ that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of $p_c(\mathcal{U})$ and provides lower bounds for $p_c(\mathcal{U})$. Furthermore, by using dynamic renormalization techniques, 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 provides a partial answer to an open problem in Toom et al. (1990, 1994).

50 pages, 19 Figures

References in corpus (2)

• A Monte Carlo investigation of the critical behavior of Stavskaya's probabilistic cellular automaton
• Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

Cited by in corpus (3)

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• Phase transitions in random mixtures of elementary cellular automata
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