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The hard-core probabilistic cellular automaton has attracted a renewed interest in the last few years, thanks to its connection with the study of a combinatorial game on percolation configurations. We provide an alternative proof for the ergodicity of this PCA for a neighbourhood of size 2 and 3, using the notion of decorrelated islands introduced by Casse in 2023, together with some new ideas. This shortens the previous proofs and provides a more intuitive and unified approach.

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Ergodicity of the Hard-Core PCA with a Random Walk Method

  • 会议论文

  • 发表日期：

    2026年

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  • Jérôme Casse,
  • Irène Marcovici,
  • Maxence Poutrel

摘要

The hard-core probabilistic cellular automaton has attracted a renewed interest in the last few years, thanks to its connection with the study of a combinatorial game on percolation configurations. We provide an alternative proof for the ergodicity of this PCA for a neighbourhood of size 2 and 3, using the notion of decorrelated islands introduced by Casse in 2023, together with some new ideas. This shortens the previous proofs and provides a more intuitive and unified approach.