Bootstrap percolation in high dimensions

Abstract

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A ⊂ 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 → ∞. The main question is to determine the critical probability pc([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 ≫ log n. The bootstrap process has been extensively studied on [n]d when d is a fixed constant and 2 ≤ r ≤ d, and in these cases pc([n]d, r) has recently been determined up to a factor of 1 + o(1) as n → ∞. At the other end of the scale, Balogh and Bollobás determined pc([2]d, 2) up to a constant factor, and Balogh, Bollobás and Morris determined pc([n]d, d) asymptotically if d ≥ (log log n)2+ε, and gave much sharper bounds for the hypercube. Here we prove the following result. Let λ be the smallest positive root of the equation Σ∞k=0(-1)kλk/2k2-kK! = 0, so λ ≈ 1.166. Then 16λ/d2(1 + logd/√d)2 -2√d ≤ pc([2]d, 2) ≤ 16λ/d2 (1 + 5(log d)2/√d) 2-2√d if d is sufficiently large, and moreover equation presented as d → ∞, for every function n = n(d) with d ≫ log n. © 2010 Cambridge University Press.

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Combinatorics Probability and Computing

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