Asymptotics of one dimensional forest fire process with non instantaneous propagation

TL;DR

This paper analyzes the asymptotic behavior of a one-dimensional forest fire model with non-instantaneous fire propagation, identifying three classes of scaling limits as ignition rate diminishes and propagation rate increases.

Contribution

It characterizes the limiting behaviors of the forest fire process under different asymptotic regimes of ignition and propagation rates.

Findings

Three classes of scaling limits identified

Asymptotic regimes depend on the rates of ignition and propagation

Provides a comprehensive understanding of the process dynamics in different limits

Abstract

Consider the following forest fire model where the possible locations of trees are the sites of $\mathbb{Z}$. Each site has three possible states: 'vacant', 'occupied' or 'burning'. Vacant sites become occupied at rate $1$. At each site, ignition (by lightning) occurs at rate $\lambda$. When a site is ignited, a fire starts and propagates to neighbors at rate $\pi$. We study the asymptotic behavior of this process as $\lambda\to0$ and $\pi\to\infty$. We show that there are three possible classes of scaling limits, according to the regime in which $\lambda\to0$ and $\pi\to\infty$.