Electronic Journal of Probability

Local Bootstrap Percolation

Janko Gravner and Alexander Holroyd

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 14, 385-399.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819475

Digital Object Identifier
doi:10.1214/EJP.v14-607

Mathematical Reviews number (MathSciNet)
MR2480546

Zentralblatt MATH identifier
1190.60094

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
bootstrap percolation cellular automaton crossover finite-size scaling metastability

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Gravner, Janko; Holroyd, Alexander. Local Bootstrap Percolation. Electron. J. Probab. 14 (2009), paper no. 14, 385--399. doi:10.1214/EJP.v14-607. https://projecteuclid.org/euclid.ejp/1464819475


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