# zbMATH — the first resource for mathematics

The cluster size distribution for a forest-fire process on $$\mathbb Z$$. (English) Zbl 1128.60080
Summary: 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. B. Drossel, S. Clar and F. Schwabl [Exact results for the one-dimensional self-organized critical forest-fire model, Phys. Rev. Lett. 71, No. 23, 3739–3742 (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 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.

##### MSC:
 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.) 60G18 Self-similar stochastic processes
##### Keywords:
self-organized criticality; cluster size distribution
Full Text: