### Abstract

Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied here at a rigorous level. We find that for an initial configuration obtained by placing particles independently with probability p«1, at sites of Z^{d}(d≥2), the density of the ‘bootstrapped’ (final) configurations in the sequence of cubes (−L/2 , L/2)^{d} typically undergoes an abrupt transition, as L is increased, from being close to 0 to the value 1. With L fixed at a large value, the mean finaldensity as a function of p changes from 0 to 1 around a value which varies only slowly with L— pertinent parameter beingλ=p^{1/(d−1)} In L. The driving mechanism is the capture of a ‘critical droplet’. This behaviour is analogous to the decay of a metastable statenear a first-order phase transition, for which the present analysis offers some suggestive ideas.

Original language | English (US) |
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Pages (from-to) | 3801-3813 |

Number of pages | 13 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 21 |

Issue number | 19 |

DOIs | |

State | Published - Oct 7 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)

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## Cite this

*Journal of Physics A: Mathematical and General*,

*21*(19), 3801-3813. https://doi.org/10.1088/0305-4470/21/19/017