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Selected Topics in Continuum Percolation: Phase Transitions, Cover Times and Random Fractals
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2019. , p. 54
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 117
Keywords [en]
Poisson point process, Percolation, Boolean model, Quasi-isometries, Cover times, Poisson cylinder process, Ellipsoid process, Phase transition, Random fractals
National Category
Probability Theory and Statistics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-392552ISBN: 978-91-506-2787-9 (print)OAI: oai:DiVA.org:uu-392552DiVA, id: diva2:1348901
Public defence
2019-10-24, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2019-10-01 Created: 2019-09-05 Last updated: 2019-10-01
List of papers
1. Invariance Under Quasi-isometries of Subcritical and Supercritical Behavior in the Boolean Model of Percolation
Open this publication in new window or tab >>Invariance Under Quasi-isometries of Subcritical and Supercritical Behavior in the Boolean Model of Percolation
2016 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 162, no 3, p. 685-700Article in journal (Refereed) Published
Abstract [en]

In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. More precisely, we prove that if a metric space M is mm-quasi-isometric to another metric space N and the Poisson Boolean model in M exhibits any of the following: (a) a subcritical phase; (b) a supercritical phase; or (c) a phase transition, then respectively so does the Poisson Boolean model of percolation in N. Then we use these results in order to understand the phase transition phenomenon in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

Keywords
Poisson point process, Percolation, Boolean model, Quasi-isometries
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-282329 (URN)10.1007/s10955-015-1422-7 (DOI)000371086600006 ()
Available from: 2016-04-05 Created: 2016-04-05 Last updated: 2019-09-05Bibliographically approved
2. Random cover times using the Poisson cylinder process.
Open this publication in new window or tab >>Random cover times using the Poisson cylinder process.
(English)In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436Article in journal (Refereed) Accepted
Abstract [en]

In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a subset A of the d-dimensional Euclidean space. This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are “raining from the sky” at unit rate. Our main results concerns the asymptotic of this cover time as the set A grows. If the set A is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead A has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.

Keywords
Cover times, Poisson cylinder process
National Category
Probability Theory and Statistics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-392502 (URN)
Available from: 2019-09-05 Created: 2019-09-05 Last updated: 2019-09-05
3. The fractal cylinder process: existence and connectivity phase transition
Open this publication in new window or tab >>The fractal cylinder process: existence and connectivity phase transition
(English)In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737Article in journal (Refereed) Submitted
Abstract [en]

We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension d ≥ 2, and a connectivityphase transition whenever d ≥ 4. We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.

A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.

In addition we also determine the almost sure Hausdorff dimension of the fractal set.

National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-392504 (URN)
Available from: 2019-09-05 Created: 2019-09-05 Last updated: 2019-09-05

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