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

TL;DR

This paper investigates the long-term behavior of a one-dimensional probabilistic cellular automaton, revealing conditions under which the mean convergence time is finite or infinite and how it depends on initial states.

Contribution

It provides the first analysis of phase behavior and convergence times in a non-ergodic 1D probabilistic cellular automaton with detailed limit distribution results.

Findings

Mean convergence time can be finite or infinite depending on parameters.

Upper bounds of convergence time depend on initial distribution.

Identifies phase transitions in the automaton's behavior.

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.