Asymptotics of one-dimensional forest fire processes

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.