Indexing metadata

The Cluster Size Distribution for a Forest-Fire Process on $Z$


 
Dublin Core PKP Metadata Items Metadata for this Document
 
1. Title Title of document The Cluster Size Distribution for a Forest-Fire Process on $Z$
 
2. Creator Author's name, affiliation, country Rachel M Brouwer; CWI
 
2. Creator Author's name, affiliation, country Juho Pennanen; University of Helsinki
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) forest-fires, self-organised criticality, cluster size distribution
 
3. Subject Subject classification 60G18
 
4. Description Abstract

Consider the following forest-fire model where trees are located on sites of $\mathbb{Z}$. A site can be vacant or be occupied by a tree. Each vacant site becomes occupied at rate $1$, independently of the other sites. Each site is hit by lightning with rate $\lambda$, which burns down the occupied cluster of that site instantaneously. As $\lambda \downarrow 0$ this process is believed to display self-organised critical behaviour.

This paper is mainly concerned with the cluster size distribution in steady-state. Drossel, Clar and Schwabl (1993) claimed that the cluster size distribution has a certain power law behaviour which holds for cluster sizes that are not too large compared to some explicit cluster size $s_{max}$. The latter can be written in terms of $\lambda$ approximately as $s_{max}\ln(s_{max}) = 1/ \lambda$. However, Van den Berg and Jarai (2005) showed that this claim is not correct for cluster sizes of order $s_{max}$, which left the question for which cluster sizes the power law behaviour does hold. Our main result is a rigorous proof of the power law behaviour up to cluster sizes of the order $s_{max}^{1/3}$. Further, it proves the existence of a stationary translation invariant distribution, which was always assumed but never shown rigorously in the literature.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2006-12-04
 
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/369
 
10. Identifier Digital Object Identifier 10.1214/EJP.v11-369
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 11
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
15. Rights Copyright and permissions The Electronic Journal of Probability applies the Creative Commons Attribution License (CCAL) to all articles we publish in this journal. Under the CCAL, authors retain ownership of the copyright for their article, but authors allow anyone to download, reuse, reprint, modify, distribute, and/or copy articles published in EJP, so long as the original authors and source are credited. This broad license was developed to facilitate open access to, and free use of, original works of all types. Applying this standard license to your work will ensure your right to make your work freely and openly available.

Summary of the Creative Commons Attribution License

You are free
  • to copy, distribute, display, and perform the work
  • to make derivative works
  • to make commercial use of the work
under the following condition of Attribution: others must attribute the work if displayed on the web or stored in any electronic archive by making a link back to the website of EJP via its Digital Object Identifier (DOI), or if published in other media by acknowledging prior publication in this Journal with a precise citation including the DOI. For any further reuse or distribution, the same terms apply. Any of these conditions can be waived by permission of the Corresponding Author.