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Graf, Robert (2014): Forest-fire models and their critical limits. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik



Forest-fire processes were first introduced in the physics literature as a toy model for self-organized criticality. The term self-organized criticality describes interacting particle systems which are governed by local interactions and are inherently driven towards a perpetual critical state. As in equilibrium statistical physics, the critical state is characterized by long-range correlations, power laws, fractal structures and self-similarity. We study several different forest-fire models, whose common features are the following: All models are continuous-time processes on the vertices of some graph. Every vertex can be "vacant" or "occupied by a tree". We start with some initial configuration. Then the process is governed by two competing random mechanisms: On the one hand, vertices become occupied according to rate 1 Poisson processes, independently of one another. On the other hand, occupied clusters are "set on fire" according to some predefined rule. In this case the entire cluster is instantaneously destroyed, i.e. all of its vertices become vacant. The self-organized critical behaviour of forest-fire models can only occur on infinite graphs such as planar lattices or infinite trees. However, in all relevant versions of forest-fire models, the destruction mechanism is a priori only well-defined for finite graphs. For this reason, one starts with a forest-fire model on finite subsets of an infinite graph and then takes the limit along increasing sequences of finite subsets to obtain a new forest-fire model on the infinite graph. In this thesis, we perform this kind of limit for two classes of forest-fire models and investigate the resulting limit processes.