TY - JOUR

T1 - Scaling limit and critical exponents for two-dimensional bootstrap percolation

AU - Camia, Federico

N1 - Funding Information:
The author thanks M. Isopi for starting his interest in the model considered here and for pointing at ref. 7, and an anonymous referee for helpful comments on the presentation of the paper. Interesting and helpful discussions with L. R. Fontes, F. den Hollander, M. Isopi, A. Sakai, V. Sidoravicius and A. C. D. van Enter are also acknowledged. Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740.
Publisher Copyright:
© 2005 Springer Science+Business Media, Inc.

PY - 2005

Y1 - 2005

N2 - Consider a cellular automaton with state space {0, 1}Z2 where the initial configuration ω0 is chosen according to a Bernoulli product measure, 1’s are stable, and 0’s become 1’s if they are surrounded by at least three neighboring 1’s. In this paper we show that the configuration ωn at time n converges exponentially fast to a final configuration. (Formula Presented.), and that the limiting measure corresponding to (Formula Presented.) is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of Z2 (i.e. for independent ∗-percolation on Z2), 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.

AB - Consider a cellular automaton with state space {0, 1}Z2 where the initial configuration ω0 is chosen according to a Bernoulli product measure, 1’s are stable, and 0’s become 1’s if they are surrounded by at least three neighboring 1’s. In this paper we show that the configuration ωn at time n converges exponentially fast to a final configuration. (Formula Presented.), and that the limiting measure corresponding to (Formula Presented.) is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents β, η, ν and γ, and of the continuum scaling limit of crossing probabilities for independent site percolation on the close-packed version of Z2 (i.e. for independent ∗-percolation on Z2), 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.

KW - Bootstrap percolation

KW - Critical exponents

KW - Scaling limit

KW - Universality

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U2 - 10.1007/s10955-004-8778-4

DO - 10.1007/s10955-004-8778-4

M3 - Article

AN - SCOPUS:53349162898

VL - 118

SP - 85

EP - 101

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -