ar X iv : 0 81 2 . 10 99 v 1 [ m at h . PR ] 5 D ec 2 00 8 ASYMPTOTICS OF ONE - DIMENSIONAL FOREST FIRE PROCESSES

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, destroying immediately the whole corresponding connected component of trees. We show that when making λ tend to 0, with a correct normalization, the forest-fire process tends to an uniquely defined process, of which we describe precisely the dynamics. 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.

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Cite this paper

@inproceedings{Bressaud2008arXI, title={ar X iv : 0 81 2 . 10 99 v 1 [ m at h . PR ] 5 D ec 2 00 8 ASYMPTOTICS OF ONE - DIMENSIONAL FOREST FIRE PROCESSES}, author={Xavier Bressaud and Nicolas Fournier}, year={2008} }