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Dürre, Florian Maximilian (2009): Self-organized critical phenomena: Forest fire and sandpile models. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics
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Abstract

The concept of self-organized criticality was proposed as an explanation for the occurrence of fractal structures in diverse natural phenomena. Roughly speaking the idea behind self-organized criticality is that a dynamic drives a system towards a stationary state that is characterized by power law correlations in space and time. We study two of the most famous models that were introduced as models exhibiting self-organized criticality. The first of them is the forest fire model. In a forest fire model each site (vertex) of a graph is either vacant or occupied by a tree. Vacant sites get occupied according to independent rate 1 Poisson processes. Independently, at each sites ignition (by lightning) occurs according to independent Poisson processes that have rate Lambda>0. When a site is ignited its whole cluster of occupied sites becomes vacant instantaneously. It is known that infinite volume forest fire processes exist for all ignition rates Lambda>0. The proof of existence is rather abstract, and does not imply uniqueness. Nor does the construction answer the question whether infinite volume forest fire processes are measurable with respect to their driving Poisson processes. Motivated by these questions, we show the almost sure infinite volume convergence for forest fire models with respect to their driving Poisson processes. Our proof is quite general and covers all graphs with bounded vertex, all positive ignition rates Lambda>0, and a quite large set of initial configurations. One of the main ingredients of the proof is an estimate for the decay of the cluster size distribution in a forest fire model. For Gamma>0, we study the probability that the cluster at site x and time t>=Gamma is larger than m, conditioned on the configuration of some further clusters at time t. We show that as m tends to infinity, this conditional probability decays to zero. The convergence is uniform in the choice of the site x, the time t, and the configuration of the further clusters we condition on. Being a consequence of almost sure infinite volume convergence, we obtain uniqueness and measurability with respect to the driving Poisson processes, and the Markov property. The second model in focus is the Abelian sandpile model. Let Lambda be a finite subset of the two-dimensional integer lattice. We consider the following sandpile model on Lambda: each vertex in Lambda contains a sandpile with a height between one and four sand grains. At discrete times, we choose a site v in Lambda randomly and add a sand grain at the site v. If after adding the sand grain the height at the site v is strictly larger than four, then the site topples. That is, four sand grains leave the site v, and each distance-one-neighbour of v gets one of these grains. If after toppling the site v there are other sites with a height strictly larger than four, we continue by toppling these sites until we obtain a configuration where all sites have a height between one and four. We study the scaling limit for the height one field in such a sandpile model. More precisely, we identify the scaling limit for the covariance of having height one at two macroscopically distant sites. We show that this scaling limit is conformally covariant. Furthermore, we show a central limit theorem for the sandpile height one field. Our results are based on a representation of the height one joint intensities that is close to a block-determinantal structure.