The Annals of Probability

Bootstrap percolation in three dimensions

József Balogh, Béla Bollobás, and Robert Morris

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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


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.

Article information

Ann. Probab., Volume 37, Number 4 (2009), 1329-1380.

First available in Project Euclid: 21 July 2009

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60C05: Combinatorial probability

Bootstrap percolation sharp threshold


Balogh, József; Bollobás, Béla; Morris, Robert. Bootstrap percolation in three dimensions. Ann. Probab. 37 (2009), no. 4, 1329--1380. doi:10.1214/08-AOP433.

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