Mean time of archipelagos in $1D$ probabilistic cellular automata has phases

Abstract

We study a non-ergodic one-dimensional probabilistic cellular automata, where each component can assume the states $\+$ and $\-.$ We obtained the limit distribution for a set of measures on $\{\+,\-\}^\Z.$ Also, we show that for certain parameters of our process the mean time of convergence can be finite or infinity. When it is finite we have showed that the upper bound is function of the initial distribution.