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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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