@article{EJP326,
author = {Alexander Holroyd},
title = {The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {11},
year = {2006},
keywords = {bootstrap percolation; cellular automaton; metastability; finite-size scaling},
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 $d$.},
pages = {no. 17, 418-433},
issn = {1083-6489},
doi = {10.1214/EJP.v11-326},
url = {http://ejp.ejpecp.org/article/view/326}}