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 | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.