Open Access
2009 Local Bootstrap Percolation
Janko Gravner, Alexander Holroyd
Author Affiliations +
Electron. J. Probab. 14: 385-399 (2009). DOI: 10.1214/EJP.v14-607


We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of $\mathbb{Z}^2$ are independently occupied with small probability $p$, otherwise empty. Subsequently, an empty site becomes active by contact with two or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability $\exp [\alpha(p)/p]$, where $\alpha(p)$ is between $-\pi^2/9+c\sqrt p$ and $-\pi^2/9+C\sqrt p(\log p^{-1})^3$. This corrects previous numerical predictions for the scaling of the correction term.


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Janko Gravner. Alexander Holroyd. "Local Bootstrap Percolation." Electron. J. Probab. 14 385 - 399, 2009.


Accepted: 9 February 2009; Published: 2009
First available in Project Euclid: 1 June 2016

zbMATH: 1190.60094
MathSciNet: MR2480546
Digital Object Identifier: 10.1214/EJP.v14-607

Primary: 60K35
Secondary: 82B43

Keywords: Bootstrap percolation , cellular automaton , crossover , finite-size scaling , metastability

Vol.14 • 2009
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