Open Access
2022 Subcritical bootstrap percolation via Toom contours
Ivailo Hartarsky, Réka Szabó
Author Affiliations +
Electron. Commun. Probab. 27: 1-13 (2022). DOI: 10.1214/22-ECP496


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.

Funding Statement

Supported by the ERC Starting Grant 680275 “MALIG”.


We thank Cristina Toninelli for helpful and stimulating discussions. We thank Rob Morris for information regarding [2].


Download Citation

Ivailo Hartarsky. Réka Szabó. "Subcritical bootstrap percolation via Toom contours." Electron. Commun. Probab. 27 1 - 13, 2022.


Received: 25 April 2022; Accepted: 29 October 2022; Published: 2022
First available in Project Euclid: 15 November 2022

arXiv: 2203.16366
MathSciNet: MR4510850
zbMATH: 1515.60313
Digital Object Identifier: 10.1214/22-ECP496

Primary: 60K35
Secondary: 60C05 , 82C20

Keywords: Bootstrap percolation , critical probability , North-East-Center majority , stability threshold , Toom contour , Toom rule

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