Universität zu Köln

Korfhage, Marilyn (2025). Percolation in the Poisson Boolean model with convex grains. PhD thesis, Universität zu Köln.

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Abstract

We study the Poisson Boolean model where the grains are random convex bodies with a rotation-invariant distribution, i.e. we have a Poisson Boolean model where the locations of the vertices are distributed in d-dimensional Euclidean space according to a Poisson process. The vertices are than given by i.i.d. copies of a random convex body attached to the individual locations. The random convex bodies are called the grains of the vertices. We say that a grain distribution is dense if the union of the grains covers the entire space and robust if the union of the grains has an unbounded connected component irrespective of the intensity of the underlying Poisson process. If the grains are balls of random radius, then density and robustness are equivalent, but in general this is not the case. In this work we are dealing with the Poisson Boolean model where convex bodies are rotation-invariant distributed and have regularly varying diameters with negative indices. We show in this model that in any dimension bigger or equal two, there are parameter regimes for the indices such that the grain distribution is robust but not dense. We give on the one hand some universal criteria for density and robustness of a grain distribution and on the other hand also some special criteria on the grain distribution being robust. In addition to that we prove non-robustness for a generalisation of a 2-dimensional ellipses model. We also investigate the chemical distance in the parameter regime for the universal criteria for robustness without having density. We prove that the model is ultrasmall under these conditions, i.e. die chemical distance of two far apart vertices grows like double logarithmic of the Euclidean distance of the locations of them. Moreover, a surprising constant c appears in this asymptotic behavior, which depends only on the model parameters.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Korfhage, Marilyn korfhage.marilyn@gmail.com UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-787080
Date:	2025
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language spatial random graphs, Poisson Boolean model, robustness, convex grains, chemical distance, ultrasmallness UNSPECIFIED
Date of oral exam:	24 July 2025
Referee:	Name Academic Title Mörters, Peter Prof. Dr. Gusakova, Anna JProf. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/78708