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Transactions of the American Mathematical Society

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The second term for two-neighbour bootstrap percolation in two dimensions


Authors: Ivailo Hartarsky and Robert Morris
Journal: Trans. Amer. Math. Soc. 372 (2019), 6465-6505
MSC (2010): Primary 60C05; Secondary 60K35, 82B20
DOI: https://doi.org/10.1090/tran/7828
Published electronically: August 1, 2019
MathSciNet review: 4024528
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Abstract: In the $ r$-neighbour bootstrap process on a graph $ G$, vertices are infected (in each time step) if they have at least $ r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism and kinetically constrained spin models of the liquid-glass transition, the most extensively studied case is the two-neighbour bootstrap process on the two-dimensional grid $ [n]^2$. Around 15 years ago, in a major breakthrough, Holroyd determined the sharp threshold for percolation in this model, and his bounds were subsequently sharpened further by Gravner and Holroyd, and by Gravner, Holroyd, and Morris.

In this paper we strengthen the lower bound of Gravner, Holroyd, and Morris by proving that the critical probability $ p_c\big ( [n]^2,2 \big )$ for percolation in the two-neighbour model on $ [n]^2$ satisfies

$\displaystyle p_c\big ( [n]^2,2 \big ) = \frac {\pi ^2}{18\log n} - \frac {\Theta (1)}{(\log n)^{3/2}}\,.$

The proof of this result requires a very precise understanding of the typical growth of a critical droplet and involves a number of technical innovations. We expect these to have other applications, for example, to the study of more general two-dimensional cellular automata and to the $ r$-neighbour process in higher dimensions.

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Additional Information

Ivailo Hartarsky
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, CNRS, PSL Research University, 45 rue d’Ulm, Paris, France
Email: ivailo.hartarsky@ens.fr

Robert Morris
Affiliation: IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, Brazil
Email: rob@impa.br

DOI: https://doi.org/10.1090/tran/7828
Keywords: Bootstrap percolation, critical probability, finite-size scaling, sharp threshold
Received by editor(s): July 2, 2018
Received by editor(s) in revised form: January 14, 2019
Published electronically: August 1, 2019
Additional Notes: Both authors were partially supported by ERC Starting Grant 680275 MALIG
The research of the second author was also partially supported by CNPq (Proc. 303275/2013-8), by FAPERJ (Proc. 201.598/2014), and by JSPS
Article copyright: © Copyright 2019 American Mathematical Society