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The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions


 
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1. Title Title of document The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions
 
2. Creator Author's name, affiliation, country Alexander E Holroyd; Department of Mathematics, University of British Columbia
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) bootstrap percolation; cellular automaton; metastability; finite-size scaling
 
3. Subject Subject classification 60K35; 82B43
 
4. Description Abstract In the modified bootstrap percolation model, sites in the cube $\{1,\ldots,L\}^d$ are initially declared active independently with probability $p$. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the $d$ dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all $d\geq 2$ we prove that as $L\to\infty$ and $p\to 0$ simultaneously, this probability converges to $1$ if $L\geq\exp \cdots \exp \frac{\lambda+\epsilon}{p}$, and converges to $0$ if $L\leq\exp \cdots \exp \frac{\lambda-\epsilon}{p}$, for any $\epsilon>0$. Here the exponential function is iterated $d-1$ times, and the threshold $\lambda$ equals $\pi^2/6$ for all $d$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Funded in part by an NSERC (Canada) Discovery Grant, and by MSRI (Berkeley USA)
 
7. Date (YYYY-MM-DD) 2006-06-06
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/326
 
10. Identifier Digital Object Identifier 10.1214/EJP.v11-326
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 11
 
12. Language English=en
 
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