In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom’s North-East-Center majority rule cellular automaton.
Supported by the ERC Starting Grant 680275 “MALIG”.
We thank Cristina Toninelli for helpful and stimulating discussions. We thank Rob Morris for information regarding .
"Subcritical bootstrap percolation via Toom contours." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP496