Asymptotics of one dimensional forest fire process with non instantaneous propagation

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 $\pi\to\infty$.