Asymptotic Behaviour of Probabilistic Cellular Automata

Abstract

The automaton iteratively evolves from one configuration to another using local transition rules based on the neighborhood of each cell. The aim of this project is to study the notions of ergodicity and uniform ergodicity in probabilistic cellular automata. Before studying the notion of ergodicity in probabilistic cellular automata, I will start by studying the notion of ergodicity in finite-state Markov chains. The reason behind doing this is that finite- state Markov chains are simpler than probabilistic cellular automata. I will then introduce the notion of a probabilistic cellular automaton. Here, I distinguish between three classes of probabilistic cellular automata: the fully deterministic ones, the fully probabilistic ones, and the rest. Afterwards, I will present the proof of the equivalence between ergodicity and uniform ergodicity in two special cases of PCA which are: fully probabilistic and fully deterministic. But first I will give the needed background to get the result for both cases.