Electronic Journal of Probability

The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions

Alexander Holroyd

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Abstract

In the modified bootstrap percolation model, sites in the cube $\{1,\ldots,L\}^d$ are initially declared active independently with probability $p$. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the $d$ dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all $d\geq 2$ we prove that as $L\to\infty$ and $p\to 0$ simultaneously, this probability converges to $1$ if $L\geq\exp \cdots \exp \frac{\lambda+\epsilon}{p}$, and converges to $0$ if $L\leq\exp \cdots \exp \frac{\lambda-\epsilon}{p}$, for any $\epsilon>0$. Here the exponential function is iterated $d-1$ times, and the threshold $\lambda$ equals $\pi^2/6$ for all $d$.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 17, 418-433.

Dates
Accepted: 6 June 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730552

Digital Object Identifier
doi:10.1214/EJP.v11-326

Mathematical Reviews number (MathSciNet)
MR2223042

Zentralblatt MATH identifier
1112.60080

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 metastability finite-size scaling

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

Citation

Holroyd, Alexander. The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions. Electron. J. Probab. 11 (2006), paper no. 17, 418--433. doi:10.1214/EJP.v11-326. https://projecteuclid.org/euclid.ejp/1464730552


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