Self-destructive percolation as a limit of forest-fire models on regular rooted trees
- Robert Graf
- Random Struct. Algorithms
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.