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Asymptotics of one-dimensional forest fire processes. (English) Zbl 1205.60167
Two families of independent Poisson processes, one with rate $$1$$ another with rate $$\lambda$$, are combined to model the so-called one-dimensional forest fire process (FFP). It is known that, from the point of view of self-organized criticality, an interesting behavior appears in the asymptotic regime $$\lambda \rightarrow 0$$. The asymptotic behavior of such $$\lambda$$-FFP model as $$\lambda \rightarrow 0$$ has been studied heuristically and numerically. However, mathematically rigorous results are scarse. In this paper a limit theorem is derived rigorusly, showing that the $$\lambda$$-FFP converges (under suitable rescaling) to some limit forest fire process. Its dynamics is precisely described. The limiting process is unique, can be built by means of a graphical construction and is amenable to reliable simulations. Asymptotic estimates are derived for the cluster-size distribution of the forest fire process.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 82C27 Dynamic critical phenomena in statistical mechanics
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