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A sharper threshold for bootstrap percolation in two dimensions
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  • Open Access
  • Published: 31 December 2010

A sharper threshold for bootstrap percolation in two dimensions

  • Janko Gravner1,
  • Alexander E. Holroyd2,3 &
  • Robert Morris4 

Probability Theory and Related Fields volume 153, pages 1–23 (2012)Cite this article

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  • 32 Citations

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Abstract

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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Open Access

This article is 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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Authors and Affiliations

  1. Mathematics Department, University of California, Davis, CA, 95616, USA

    Janko Gravner

  2. Microsoft Research, 1 Microsoft Way, Redmond, WA, 98052, USA

    Alexander E. Holroyd

  3. University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Alexander E. Holroyd

  4. IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, Brazil

    Robert Morris

Authors
  1. Janko Gravner
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  2. Alexander E. Holroyd
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  3. Robert Morris
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Corresponding author

Correspondence to Robert Morris.

Additional information

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 (https://creativecommons.org/licenses/by-nc/2.0), 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). https://doi.org/10.1007/s00440-010-0338-z

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  • Received: 20 February 2010

  • Revised: 26 October 2010

  • Published: 31 December 2010

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-010-0338-z

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Mathematics Subject Classification (2000)

  • 60C05
  • 60K35
  • 82B20
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