Abstract
Different versions of percolation games on , with parameters p and q that indicate, respectively, the probability with which a site in 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 . 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) with a size-3 neighborhood is ergodic. This allows us to conclude that is ergodic whenever , thereby rigorously establishing ergodicity for a considerable class of PCAs that tie in closely with important topics such as the enumeration of directed animals, broadcasting of information on directed infinite lattices, examining reliability of computations against the presence of noise etc. The key to our proof is the technique of weight functions. We include extensive discussions on game theoretic PCAs to which this technique may be applicable to establish ergodicity, and on percolation games to which this technique may be applicable to explore the ‘regimes’ (depending on the underlying parameter(s), such as in our case) in which the probabilities of draw are 0.
Funding Statement
For the duration of this project, Dhruv Bhasin has been supported by Grant No. 0203/2/2021/RD-II/3033 awarded by the National Board for Higher Mathematics (NBHM), Sayar Karmakar has been supported by Grant No. NSF DMS 2124222 awarded by the National Science Foundation (NSF), and Moumanti Podder has been supported by Grant No. SERB CRG/2021/006785 awarded by the Science and Engineering Research Board (SERB). In addition to these funders, the authors express their gratitude towards their respective institutes / university for supporting them in myriad ways and for fostering wonderful environments conducive to collaborations in research.
Acknowledgments
We are thankful to the editor and two anonymous referee for their useful feedbacks that certainly helped improve the paper.
Citation
Dhruv Bhasin. Sayar Karmakar. Moumanti Podder. Souvik Roy. "On a class of PCA with size-3 neighborhood and their applications in percolation games." Electron. J. Probab. 28 1 - 60, 2023. https://doi.org/10.1214/23-EJP1046
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