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One-Dimensional General Forest Fire Processes
Xavier Bressaud, Université Paul Sabatier, Toulouse, France, and Nicolas Fournier, Université Paris-Est, Créteil, France
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2013; 138 pp; softcover
Number: 132
ISBN-13: 978-2-85629-765-0
List Price: US$48
Member Price: US$38.40
Order Code: SMFMEM/132
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The authors consider the one-dimensional generalized forest fire process: at each site of \(\mathbb{Z}\), seeds and matches fall according to i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model.

According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently previous results where seeds and matches were assumed to fall according to Poisson processes.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians.

Table of Contents

  • Introduction
  • Notation and results
  • Proofs
  • Numerical simulations
  • Appendix
  • Bibliography
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