Probability Theory and Related Fields

, Volume 134, Issue 4, pp 624–648 | Cite as

Bootstrap percolation on the hypercube

  • József BaloghEmail author
  • Béla Bollobás


In the bootstrap percolation on the n-dimensional hypercube, in the initial position each of the 2 n 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 Open image in new window the probability of spanning tends to 1 as n→∞, while for Open image in new window the probability tends to 0. Furthermore, we shall show that for each n the transition has a sharp threshold function.


Stochastic Process Probability Theory Initial Position Mathematical Biology Threshold Function 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Ohio State UniversityUSA
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  3. 3.Trinity CollegeCambridgeUK

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