Selected Topics in Continuum Percolation : Phase Transitions, Cover Times and Random Fractals
Sammanfattning: This thesis consists of an introduction and three research papers. The subject is probability theory and in particular concerns the topics of percolation, cover times and random fractals.Paper I deals with the Poisson Boolean model in locally compact Polish metric spaces. We prove that if a metric space M1 is mm-quasi-isometric to another metric space M2 and the Poisson Boolean model in M1 features one of the following percolation properties: it has a subcritical phase or it has a supercritical phase, then respectively so does the Poisson Boolean model in M2. In particular, if the process in M1 undergoes a phase transition, then so does the process in M2. We use these results to study phase transitions in a large family of metric spaces, including Riemannian manifolds, Gromov spaces and Caley graphs.In Paper II we study the distribution of the time it takes for a Poisson process of cylinders to cover a bounded subset of d-dimensional Euclidean space. The Poisson process of cylinders is invariant under rotations, reﬂections and translations. Furthermore, we add a time component, so that one can imagine that the cylinders are “raining from the sky” at unit rate. We show that the cover times of a sequence of discrete and well separated sets converge to a Gumbel distribution as the cardinality of the sets grows. For sequences of sets with positive box dimension, we determine the correct speed at which the cover times of the sets An grows.In Paper III we consider a semi-scale invariant version of the Poisson cylinder model. This model induces a random fractal set in the vacant region of the process. We establish an existence phase transition for dimensions d ≥ 2 and a connectivity phase transition for dimensions d ≥ 4. An important step when analysing the connectivity phase transition is to consider the restriction of the process onto subspaces. We show that this restriction induces a fractal ellipsoid model in the corresponding subspace. We then present a detailed description of this induced ellipsoid model. Moreover, the almost sure Hausdorff dimension of the fractal set is also determined.
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