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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
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