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Bootstrap Percolation in High Dimensions

  • DOI:
  • Published online: 12 August 2010

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices AV(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 ⩽ rd, 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 so λ ≈ 1.166. Then if d is sufficiently large, and moreover as d → ∞, for every function n = n(d) with d ≫ log n.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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