The sharp threshold for bootstrap percolation in all dimensions

József Balogh, Béla Bollobás, Hugo Duminil-Copin, Robert Morris

Research output: Contribution to journalArticlepeer-review


In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid [n]d. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair d, for some constant, and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine.

Original languageEnglish (US)
Pages (from-to)2667-2701
Number of pages35
JournalTransactions of the American Mathematical Society
Issue number5
StatePublished - 2012

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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