Bootstrap percolation in high dimensions

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected neighbours. The set A is said to percolate if eventually all vertices are infected. Our aim is to understand this process on the grid, [n]^d, for arbitrary functions n = n(t), d = d(t) and r = r(t), as t -> infinity. The main question is to determine the critical probability p_c([n]^d,r) at which percolation becomes likely, and to give bounds on the size of the critical window. In this paper we study this problem when r = 2, for all functions n and d satisfying d \gg log n. The bootstrap process has been extensively studied on [n]^d when d is a fixed constant and 2 \leq r \leq d, and in these cases p_c([n]^d,r) has recently been determined up to a factor of 1 + o(1) as n -> infinity. At the other end of the scale, Balogh and Bollobas determined p_c([2]^d,2) up to a constant factor, and Balogh, Bollobas and Morris determined p_c([n]^d,d) asymptotically if d > (log log n)^{2+\eps}, and gave much sharper bounds for the hypercube. Here we prove the following result: let \lambda be the smallest positive root of the equation \sum_{k=0}^\infty (-1)^k \lambda^k / (2^{k^2-k} k!) = 0, so \lambda \approx 1.166. Then (16\lambda / d^2) (1 + (log d / \sqrt{d})) 2^{-2\sqrt{d}} < p_c([2]^d,2) < (16\lambda / d^2) (1 + (5(log d)^2 / \sqrt{d})) 2^{-2\sqrt{d}} if d is sufficiently large, and moreover we determine a sharp threshold for the critical probability p_c([n]^d,2) for every function n = n(d) with d \gg log n.Comment: 51 pages, revised versio

Extremal bounds for bootstrap percolation in the hypercube

The r-neighbour bootstrap percolation process on a graph G starts with an initial set A0 of “infected” vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of G eventually becomes infected, then we say that A0 percolates. We prove a conjecture of Balogh and Bollob ́as which says that, for fixed r and d →∞ , every percolating set in the d -dimensional hypercube has cardinality at least 1+ o (1) / r ( d r − 1 ). We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve on the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when r = 3, we prove that the minimum cardinality of a percolating set in the d -dimensional hypercube is ⌈ d (d +3) / 6 ⌉ + 1 for all d ≥ 3