Asymptotics of one-dimensional forest fire processes

Details Asymptotics of one-dimensional forest fire processes Asymptotics of one-dimensional forest fire processes

2008-12-05

arxiv.org/abs/0812.1099v2

Annals of Probability 2010, Vol. 38, No. 5, 1783-1816

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP524 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP524

We consider the so-called one-dimensional forest fire process. At each site of $\mathbb{Z}$, a tree appears at rate $1$. At each site of $\mathbb{Z}$, a fire starts at rate ${\lambda}>0$, immediately destroying the whole corresponding connected component of trees. We show that when ${\lambda}$ 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 $\log(1/{\lambda})$ and of compressing space by a factor ${\lambda}\log(1/{\lambda})$. 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 ${\lambda}\to0$) for the cluster-size distribution of the forest fire process.

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