Year: 2009
Type: article
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≤r≤d 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.
Source: The Annals of Probability
DOI: 10.1214/08-aop433
Authors: József Balogh, Béla Bollobás, Robert Morris
Institutions: University of Cambridge, University of Memphis
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2276192920