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