MiS Preprint Repository

We consider a class of probabilistic cellular automata undergoing a phase transition with an absorbing-state. Denoting by ${\mathcal{U}} (s)$ the neighbourhood of the site $s$, the transition probability is $T(\eta_s = 1 | \eta_{{\mathcal{U}}(s)}) = 0$ if $\eta_{{\mathcal{U}}(s)}= \mathbf{0}$ or $p$ otherwise, $\forall s \in \mathbb{Z}$. For any $\mathcal{U}$ there exists a non-trivial critical probability $p_c( {\mathcal{U}})$ which separates a phase with an absorbing-state from a fluctuating phase. We study how the neighbourhood affects the value of $p_c( {\mathcal{U}})$ and we provide lower bounds for $p_c( {\mathcal{U}})$. 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).