Abstract
Ergodicity of probabilistic cellular automata is a very important issue in the PCA theory. In particular, the question about the ergodicity of all PCA with neighbourhood of size two, binary alphabet and positive rates is still open. In this article, we do not try to improve this issue, but we show a new kind of proof (to the best knowledge of the author) about the ergodicity of some of those PCA, including also some CA with errors. The proof is based on the study of the boundaries of islands where the PCA is totally decorrelated from its initial condition. The behaviours of these boundaries are the ones of random walks.
Acknowledgments
I am very grateful to Irène Marcovici. Her knowledge on the subject confirms me that the idea developed in the paper should be written and published. Moreover, her attentive reading of the paper permits to clarify and improve it. I am also very grateful to the referee for its careful reading and remarks that improve this paper.
Citation
Jérôme Casse. "Ergodicity of some probabilistic cellular automata with binary alphabet via random walks." Electron. J. Probab. 28 1 - 17, 2023. https://doi.org/10.1214/23-EJP971
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