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Sharp thresholds in Bootstrap Percolation
Jozef Balogh, Bela Bollobas

Citation
Jozef Balogh, Bela Bollobas. "Sharp thresholds in Bootstrap Percolation". Physica A: Statistical Mechanics and its Applications, 326(3-4):305-312, August 2003.

Abstract
In the standard bootstrap percolation on the d-dimensional grid, in the initial position each of the n^d sites is occupied with probability p and empty with probability 1 - p, independently of the state of every other site. Once a site is occupied, it remains occupied for ever, while an empty site becomes occupied if at least two of its neighbors are occupied. If at the end of the process every site is occupied, we say that the (initial) configuration percolates. By making use of a theorem of Friedgut and Kalai (Proc. Amer. Math. Soc. 124 (1996) 2993), we shall show that the threshold function of the percolation is sharp. We shall prove similar results for three other models of bootstrap percolation as well.

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Citation formats  
  • HTML
    Jozef Balogh, Bela Bollobas. <a
    href="http://chess.eecs.berkeley.edu/pubs/720.html"
    >Sharp thresholds in Bootstrap Percolation</a>,
    <i>Physica A: Statistical Mechanics and its
    Applications</i>, 326(3-4):305-312, August 2003.
  • Plain text
    Jozef Balogh, Bela Bollobas. "Sharp thresholds in
    Bootstrap Percolation". <i>Physica A: Statistical
    Mechanics and its Applications</i>, 326(3-4):305-312,
    August 2003.
  • BibTeX
    @article{BaloghBollobas03_SharpThresholdsInBootstrapPercolation,
        author = {Jozef Balogh and Bela Bollobas},
        title = {Sharp thresholds in Bootstrap Percolation},
        journal = {Physica A: Statistical Mechanics and its
                  Applications},
        volume = {326},
        number = {3-4},
        pages = {305-312},
        month = {August},
        year = {2003},
        abstract = {In the standard bootstrap percolation on the
                  d-dimensional grid, in the initial position each
                  of the n^d sites is occupied with probability p
                  and empty with probability 1 - p, independently of
                  the state of every other site. Once a site is
                  occupied, it remains occupied for ever, while an
                  empty site becomes occupied if at least two of its
                  neighbors are occupied. If at the end of the
                  process every site is occupied, we say that the
                  (initial) configuration percolates. By making use
                  of a theorem of Friedgut and Kalai (Proc. Amer.
                  Math. Soc. 124 (1996) 2993), we shall show that
                  the threshold function of the percolation is
                  sharp. We shall prove similar results for three
                  other models of bootstrap percolation as well. },
        URL = {http://chess.eecs.berkeley.edu/pubs/720.html}
    }
    

Posted by Christopher Brooks on 4 Nov 2010.
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