The Cluster Size Distribution for a Forest-Fire Process on $Z$

Abstract

Consider the following forest-fire model where trees are located on sites of $\mathbb{Z}$. A site can be vacant or be occupied by a tree. Each vacant site becomes occupied at rate $1$, independently of the other sites. Each site is hit by lightning with rate $\lambda$, which burns down the occupied cluster of that site instantaneously. As $\lambda \downarrow 0$ this process is believed to display self-organised critical behaviour.

This paper is mainly concerned with the cluster size distribution in steady-state. Drossel, Clar and Schwabl (1993) claimed that the cluster size distribution has a certain power law behaviour which holds for cluster sizes that are not too large compared to some explicit cluster size $s_{max}$. The latter can be written in terms of $\lambda$ approximately as $s_{max}\ln(s_{max}) = 1/ \lambda$. However, Van den Berg and Jarai (2005) showed that this claim is not correct for cluster sizes of order $s_{max}$, which left the question for which cluster sizes the power law behaviour does hold. Our main result is a rigorous proof of the power law behaviour up to cluster sizes of the order $s_{max}^{1/3}$. Further, it proves the existence of a stationary translation invariant distribution, which was always assumed but never shown rigorously in the literature.