## Electronic Journal of Probability

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

Alexander Holroyd

#### 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

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

Rights

#### 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

#### References

• Adler J.; Stauffer, D.; Aharony, A. Comparison of bootstrap percolation models. J. Phys. A 22 (1989), L297–L301.
• Aizenman, M.; Lebowitz, J. L. Metastability effects in bootstrap percolation. J. Phys. A 21 (1988), no. 19, 3801–3813.
• Balogh, J.; Bollobas, B. Sharp thresholds in bootstrap percolation. Physica A, 326 (2003), 305–312.
• Cerf, Raphaël; Cirillo, Emilio N. M. Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27 (1999), no. 4, 1837–1850.
• Cerf, R.; Manzo, F.. The threshold regime of finite volume bootstrap percolation. Stochastic Process. Appl. 101 (2002), no. 1, 69–82.
• Gregorio, P. D.; Lawlor, A.; Bradley, P.; Dawson, K. A. Clarification of the bootstrap percolation paradox. Phys. Rev. Lett., 93 (2004), no. 2, 025501
• Grimmett, Geoffrey. Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
• Holroyd, Alexander E. Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 (2003), no. 2, 195–224.
• Holroyd, Alexander E.; Liggett, Thomas M.; Romik, Dan. Integrals, partitions, and cellular automata. Trans. Amer. Math. Soc. 356 (2004), no. 8, 3349–3368 (electronic).
• Schonmann, Roberto H. On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 (1992), no. 1, 174–193.
• van Enter, Aernout C. D. Proof of Straley's argument for bootstrap percolation. J. Statist. Phys. 48 (1987), no. 3-4, 943–945.