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Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. (English) Zbl 1130.60091

The forest-fire model on the square lattice \(\mathbb{Z}^d, d>1\) is considered. Each site of it is either vacant or occupied by a tree. Vacant sites become occupied according to independent rate \(1\) Poisson processes, the growth processes. Independently at each site lightning strikes according to independent rate \(\lambda\) Poisson ignition processes, \(\lambda>0\) is the parameter of the model. When an occupied site is hit by ignition, its entire occupied cluster burns down, that is, becomes vacant instantaneously. This article studies whether a forest-fire process with given parameter that starts with all sites vacant is unique. Under the assumption on the decay of the cluster size distribution of a process it is shown that it dominates the forest-fire process, the forest-fire process with given parameter and the vacant initial configuration is unique, adapted to the filtration generated by its driving growth and ignition process, and can be constructed in a direct way. This assumption is satisfied provided that \(\lambda\) is big enough.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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