Bootstrap percolation in three dimensions

József Balogh, Béla Bollobás, Robert Morris

Research output: Contribution to journalArticlepeer-review


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 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. 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.

Original languageEnglish (US)
Pages (from-to)1329-1380
Number of pages52
JournalAnnals of Probability
Issue number4
StatePublished - Jul 2009


  • Bootstrap percolation
  • Sharp threshold

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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