Bootstrap percolation in three dimensions

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 p_c(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.