### Abstract

Consider a cellular automaton with state space {0,1} 2 where the initial configuration _0 is chosen according to a Bernoulli product measure, 1s are stable, and 0s become 1s if they are surrounded by at least three neighboring 1s. In this paper we show that the configuration _n at time n converges exponentially fast to a final configuration , and that the limiting measure corresponding to is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents , , and , and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of 2 (i.e. for independent *-percolation on ), we prove that the bootstrapped percolation model has the same scaling limit and critical exponents.This type of bootstrap percolation can be seen as a paradigm for a class of cellular automata whose evolution is given, at each time step, by a monotonic and nonessential enhancement [Aizenman and Grimmett, J. Stat. Phys. 63: 817--835 (1991); Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999)

Original language | English |
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Pages (from-to) | 85-101 |

Journal | Journal of Statistical Physics |

Volume | 118 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2005 |

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## Cite this

Camia, F. (2005). Scaling limit and critical exponents for two-dimensional bootstrap percolation.

*Journal of Statistical Physics*,*118*(1-2), 85-101. https://doi.org/10.1007/s10955-004-8778-4