# Selected Topics in Continuum Percolation: Phase Transitions, Cover Times and Random Fractals

- Date: –15:00
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 Häggsalen
- Doctoral student: Filipe Mussini
- Organiser: Matematiska institutionen
- Contact person: Erik Broman
- Disputation

Abstract: 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 *M*1 is mm-quasi-isometric to another metric space *M*2 and the Poisson Boolean model in *M*1 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 *M*2. In particular, if the process in *M*1 undergoes a phase transition, then so does the process in *M*2. 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 *A*n 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.