On a class of probabilistic cellular automata with size-3 neighbourhood and their applications in percolation games

2022/08/24 by Dhruv Bhasin, Sayar Karmakar, Bhasin, Dhruv +5 · 1 citation
Computer Science · Mathematics · #05C57 #37A25 #37B15 #68Q80 #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2208.11670

openalex publication_date 2022/08/24 · openalex created_date 2022/08/26 · openalex updated_date 2026/07/28

Abstract

Different versions of percolation games on ℤ², with parameters p and q that indicate, respectively, the probability with which a site in ℤ² is labeled a trap and the probability with which it is labeled a target, are shown to have probability 0 of culminating in draws when p+q > 0. We show that, for fixed p and q, the probability of draw in each of these games is 0 if and only if a certain 1-dimensional probabilistic cellular automaton (PCA) F_p,q with a size-3 neighbourhood is ergodic. This allows us to conclude that F_p,q is ergodic whenever p+q > 0, thereby rigorously establishing ergodicity for a considerable class of PCAs.

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