Asymptotics of one-dimensional forest fire processes

TL;DR

This paper analyzes the asymptotic behavior of a one-dimensional forest fire process as the fire rate approaches zero, revealing a limiting process with explicit construction and deriving cluster-size distribution estimates.

Contribution

It introduces a precise limiting process for the forest fire model as fire rate tends to zero, including a detailed description and simulation method.

Findings

The limiting process is explicitly constructed and can be simulated.

Asymptotic estimates for cluster-size distribution are provided.

The process exhibits a simple, well-defined behavior in the limit.

Abstract

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.