## Abstract

By bootstrap percolation we mean the following deterministic process on a graph G. Given a set A of vertices "infected" at time 0, new vertices are subsequently infected, at each time step, if they have at least r ∈ N previously infected neighbors. When the set A is chosen at random, the main aim is to determine the critical probability p_{c}(G, r) at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the d-dimensional grid [n]^{d}: with 2 ≤ r ≤ d fixed, it was proved by Cerf and Cirillo (for d = r = 3), and by Cerf and Manzo (in general), that where log_{(r)} is an r-times iterated logarithm. However, the exact threshold function is only known in the case d = r = 2, where it was shown by Holroyd to be. In this paper we shall determine the exact threshold in the crucial case d = r = 3, and lay the groundwork for solving the problem for all fixed d and r.

Original language | English (US) |
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Pages (from-to) | 1329-1380 |

Number of pages | 52 |

Journal | Annals of Probability |

Volume | 37 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2009 |

## Keywords

- Bootstrap percolation
- Sharp threshold

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty