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Bootstrap Percolation, Probabilistic Cellular Automata and Sharpness

Abstract

We establish new connections between percolation, bootstrap percolation, probabilistic cellular automata and deterministic ones. Surprisingly, by juggling with these in various directions, we effortlessly obtain a number of new results in these fields. In particular, we prove the sharpness of the phase transition of attractive absorbing probabilistic cellular automata, a class of bootstrap percolation models and kinetically constrained models. We further show how to recover a classical result of Toom on the stability of cellular automata w.r.t. noise and, inversely, how to deduce new results in bootstrap percolation universality from his work.

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Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. More generally, PCA are defined identically by a rates measure supported not only on \(\mathscr {U} \), but on the entire power set of \({\text{\O}mega }_R\). However, attractiveness is essential for everything we will say, so we restrict directly to the relevant setting. See e.g. [27] for problems arising immediately without this assumption.

  2. This property is sometimes called freezing to distinguish from attractiveness, which is also a type of monotonicity.

  3. There may be issues defining this if \(\chi \) has infinite support. We will only consider finitely supported \(\chi \) measures in this work.

  4. We refer the reader to [20] for a related correspondence between PCA and equilibrium statistical mechanics models.

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Acknowledgements

We thank Irène Marcovici, Jan Swart, Réka Szabó, Siamak Taati and Cristina Toninelli for enlightening discussions. We also thank Réka for bringing important references to our attention. We thank the annonymous referees for helpful remarks on the presentation.

Funding

This work is supported by European Research Council Starting Grant 680275 “MALIG.”

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Correspondence to Ivailo Hartarsky.

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Communicated by Giulio Biroli.

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Hartarsky, I. Bootstrap Percolation, Probabilistic Cellular Automata and Sharpness. J Stat Phys 187, 21 (2022). https://doi.org/10.1007/s10955-022-02922-6

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  • DOI: https://doi.org/10.1007/s10955-022-02922-6

Keywords

  • Probabilistic cellular automata
  • Bootstrap percolation
  • Kinetically constrained models
  • Sharp phase transition
  • Stability

Mathematics Subject Classification

  • Primary 60K35
  • Secondary 37B15
  • 60C05
  • 82B43
  • 82C20