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Bootstrap percolation on the hypercube
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  • Published: 14 July 2005

Bootstrap percolation on the hypercube

  • József Balogh1 &
  • Béla Bollobás2,3 

Probability Theory and Related Fields volume 134, pages 624–648 (2006)Cite this article

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Abstract

In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2n sites is occupied with probability p and empty with probability 1−p, independently of the state of the other sites. Every occupied site remains occupied for ever, while an empty site becomes occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that the (initial) position spans the hypercube. We shall show that there are constants c 1,c 2>0 such that for the probability of spanning tends to 1 as n→∞, while for the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function.

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Authors and Affiliations

  1. Ohio State University, USA

    József Balogh

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Béla Bollobás

  3. Trinity College, Cambridge, CB2 1TQ, UK

    Béla Bollobás

Authors
  1. József Balogh
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  2. Béla Bollobás
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Corresponding author

Correspondence to József Balogh.

Additional information

J. Balogh: work was done while at The University of Memphis, USA

Research supported in part by NSF grant DMS0302804

Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900

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Cite this article

Balogh, J., Bollobás, B. Bootstrap percolation on the hypercube. Probab. Theory Relat. Fields 134, 624–648 (2006). https://doi.org/10.1007/s00440-005-0451-6

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  • Received: 18 December 2004

  • Revised: 19 April 2005

  • Published: 14 July 2005

  • Issue Date: April 2006

  • DOI: https://doi.org/10.1007/s00440-005-0451-6

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Keywords

  • Stochastic Process
  • Probability Theory
  • Initial Position
  • Mathematical Biology
  • Threshold Function
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