The Annals of Probability
- Ann. Probab.
- Volume 38, Number 5 (2010), 1783-1816.
Asymptotics of one-dimensional forest fire processes
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.
Ann. Probab., Volume 38, Number 5 (2010), 1783-1816.
First available in Project Euclid: 17 August 2010
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Bressaud, Xavier; Fournier, Nicolas. Asymptotics of one-dimensional forest fire processes. Ann. Probab. 38 (2010), no. 5, 1783--1816. doi:10.1214/09-AOP524. https://projecteuclid.org/euclid.aop/1282053772