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A sharper threshold for bootstrap percolation in two dimensions

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Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability p c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p c ~ π 2/(18 log n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is −(log n)−3/2+o(1), and moreover determining it up to a poly(log log n)-factor. The exponent −3/2 corrects numerical predictions from the physics literature.


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Correspondence to Robert Morris.

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Supported by NSF grant DMS 0204376 and the Republic of Slovenia Ministry of Science program P1-285 (JG); NSERC and Microsoft Research (AEH); a JSPS Fellowship and a Research Fellowship from Murray Edwards College, Cambridge (RM).

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Gravner, J., Holroyd, A.E. & Morris, R. A sharper threshold for bootstrap percolation in two dimensions. Probab. Theory Relat. Fields 153, 1–23 (2012).

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Mathematics Subject Classification (2000)

  • 60C05
  • 60K35
  • 82B20