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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 14 open journal systems 


Local Bootstrap Percolation

Janko Gravner, University of California Davis
Alexander E. Holroyd, University of British Columbia, Microsoft Research


Abstract
We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions for the scaling of the correction term.


Full text: PDF

Pages: 385-399

Published on: February 9, 2009
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Electronic Journal of Probability. ISSN: 1083-6489