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.

Electronic downloads

• http://dx.doi.org/10.1016/S0378-4371(03)00364-9

• 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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