Critical probabilities and convergence time of percolation probabilistic cellular automata

This paper considers a class of probabilistic cellular automata undergoing a phase transition with an absorbing state. Denoting by U(x) the neighbourhood of site (Formula Presented), the transition probability is (Formula Presented) otherwise, (Formula Presented). For any (Formula Presented) there exists a non-trivial critical probability that separates a phase with an absorbing state from a fluctuating phase. This paper studies how the neighbourhood affects the value of (Formula Presented) and provides lower bounds for (Formula Presented). 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 (Formula Presented). This provides a partial answer to an open problem in Toom et al. (Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, pp. 1–182. Manchester University Press, Manchester, 1990; Topics in Contemporary Probability and its Applications, pp. 117–157. CRC Press, Boca Raton, 1995).