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One-dimensional general forest fire processes. (English) Zbl 1297.60063
In the classical forest-fire model on a graph, every vertex of the graph is either vacant or occupied by a tree, and the time evolution is controlled by seeds and matches which are falling on vertices according to independent Poisson point processes with rates $$1$$ and $$\lambda$$, respectively. If a seed falls on a vacant vertex, it immediately grows into a tree making the vertex occupied. If a match falls on a tree, the connected component (forest) of the tree is instantaneously burned making all the vertices of the forest vacant.
In their previous work [Ann. Probab. 38, No. 5, 1783–1816 (2010; Zbl 1205.60167)], the authors considered the forest-fire model on $$\mathbb Z$$ in the regime $$\lambda\to 0$$. They proved that under suitable rescaling of time and space, the forest-fire process converges to a limiting process, which is defined as a unique solution to a stochastic differential equation and admits a graphical representation.
In the current paper, the authors study scaling limits of generalized forest-fire processes on $$\mathbb Z$$ governed by independent stationary renewal processes of seeds and matches. Under some general assumptions on the renewal processes, they prove convergence (as $$\lambda\to 0$$) of the forest-fire process to a limiting process. Depending on the renewal process of the seeds, four qualitatively different types of limiting processes can arise, each is defined as a unique solution to a stochastic differential equation and admits a graphical representation. One of the types coincided with the limiting process for the classical forest-fire model obtained in [loc. cit.], proving certain universality with respect to the processes of seeds and matches. Asymptotic bounds on the distribution of the forest size at $$0$$ are obtained in all four cases, implying, in particular, the absence of self-organized criticality.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 92D40 Ecology
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