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July 2009 Bootstrap percolation in three dimensions
József Balogh, Béla Bollobás, Robert Morris
Ann. Probab. 37(4): 1329-1380 (July 2009). DOI: 10.1214/08-AOP433

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 pc(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≤rd fixed, it was proved by Cerf and Cirillo (for d=r=3), and by Cerf and Manzo (in general), that $$p_{c}([n]^{d},r)=\Theta\biggl(\frac{1}{\log_{(r-1)}n}\biggr)^{d-r+1}$$ 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 $(1+o(1))\frac{\pi^{2}}{18\log n}$. 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.

Citation

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József Balogh. Béla Bollobás. Robert Morris. "Bootstrap percolation in three dimensions." Ann. Probab. 37 (4) 1329 - 1380, July 2009. https://doi.org/10.1214/08-AOP433

Information

Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1187.60082
MathSciNet: MR2546747
Digital Object Identifier: 10.1214/08-AOP433

Subjects:
Primary: 60C05, 60K35

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 4 • July 2009
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