## Geometrical and percolative properties of spatially correlated models

##### Abstract

This thesis consists of four papers dealing with phase transitions in various
models of continuum percolation. These models exhibit complicated dependencies
and are generated by different Poisson processes. For each such process
there is a parameter, known as the intensity, governing its behavior. By varying
the value of this parameter, the geometrical and topological properties of these
models may undergo dramatic and rapid changes. This phenomenon is called
a phase transition and the value at which the change occur is called a critical
value.
In Paper I, we study the topic of visibility in the vacant set of the Brownian
interlacements in Euclidean space and the Brownian excursions process in the
unit disc. For the vacant set of the Brownian interlacements we obtain upper
and lower bounds of the probability of having visibility in some direction to a
distance r in terms of the probability of having visibility in a fixed direction
of distance r. For the vacant set of the Brownian excursions we prove a phase
transition in terms of visibility to infinity (with respect to the hyperbolic metric).
We also determine the critical value and show that at the critical value there is
no visibility to infinity.
In Paper II we compute the critical value for percolation in the vacant set of
the Brownian excursions process. We also show that the Brownian excursions
process is a hyperbolic analogue of the Brownian interlacements. In Paper III, we study the vacant set of a semi scale invariant version of the
Poisson cylinder model. In this model it turns out that the vacant set is a fractal.
We determine the critical value for the so-called existence phase transition and
what happens at the critical value. We also compute the Hausdorff dimension of
the fractal whenever it exists. Furthermore, we prove that the fractal exhibits a
nontrivial connectivity phase transition for dimensions four and greater and that the fractal is totally
disconnected for dimension two. In the three dimensional case we prove a partial result showing that the fractal restricted to a plane is totally disconnected with probability one. In Paper IV we study a continuum percolation process, the random ellipsoid
model, generated by taking the intersection of a Poisson cylinder model in d dimensions and a subspace of dimension k. For k between 2 and d-2, we show that there is a non-trivial phase transition concerning the expected number of ellipsoids in the cluster of the origin. When k=d-1 this critical value is zero. We compare these results with results for the classical Poisson Boolean model.

##### Parts of work

Visibility in the vacant set of the Brownian
interlacements and the Brownian excursion process. ::doi::10.30757/ALEA.v16-36 Percolation of the vacant set of the Brownian
excursions process. Manuscript The fractal cylinder process: existence and
connectivity phase transitio. Submitted Properties of a random ellipsoid model. Submitted

##### Degree

Doctor of Philosophy

##### University

Göteborgs universitet. Naturvetenskapliga fakulteten

##### Institution

Department of Mathematical Sciences ; Institutionen för matematiska vetenskaper

##### Disputation

10:15 Hörsal Pascal, Matematiska vetenskaper, Chalmers Tvärgata 3, Göteborg. https://chalmers.zoom.us/j/631607089

##### Date of defence

2020-04-17

olofel@chalmers.se

##### Date

2020-03-10##### Author

Hallqvist Elias, Karl Olof

##### Keywords

continuum percolation,

brownian excursions

brownian interlacements

poisson cylinder model

fractal percolation

##### Publication type

Doctoral thesis

##### ISBN

978-91-7833-872-6

978-91-7833-873-3

##### Language

eng