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September 2010 Asymptotics of one-dimensional forest fire processes
Xavier Bressaud, Nicolas Fournier
Ann. Probab. 38(5): 1783-1816 (September 2010). DOI: 10.1214/09-AOP524

Abstract

We consider the so-called one-dimensional forest fire process. At each site of ℤ, a tree appears at rate 1. At each site of ℤ, a fire starts at rate λ>0, immediately destroying the whole corresponding connected component of trees. We show that when λ is made to tend to 0 with an appropriate normalization, the forest fire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor log(1/λ) and of compressing space by a factor λ log(1/λ). The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when λ→0) for the cluster-size distribution of the forest fire process.

Citation

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Xavier Bressaud. Nicolas Fournier. "Asymptotics of one-dimensional forest fire processes." Ann. Probab. 38 (5) 1783 - 1816, September 2010. https://doi.org/10.1214/09-AOP524

Information

Published: September 2010
First available in Project Euclid: 17 August 2010

zbMATH: 1205.60167
MathSciNet: MR2722786
Digital Object Identifier: 10.1214/09-AOP524

Subjects:
Primary: 60K35, 82C22

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 5 • September 2010
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