Process of a Random Planar Quadrangulation
(2024) In Master’s Theses in Mathematical Sciences MASM02 20241Mathematical Statistics
- Abstract
- A process of a random graph, that grows in a box [0,N]^2 ⊂ R^2 and yields a quadrangulation when it stops growing, is investigated in this work. The random components in this model come from the Poisson distributed number of points, the position of those, and the either horizontal or vertical direction in which lines grow from these points. The model finds inspiration in a cellular automata model by Ekström and Turova (2022), which is used to model neuronal activity in the brain. This work provides results on the quadrangulation model, including tail bounds for the distribution of the growth of lines, bounds for the expected values of a model characteristic and the covariance structures of the behaviour around points depending on their... (More)
- A process of a random graph, that grows in a box [0,N]^2 ⊂ R^2 and yields a quadrangulation when it stops growing, is investigated in this work. The random components in this model come from the Poisson distributed number of points, the position of those, and the either horizontal or vertical direction in which lines grow from these points. The model finds inspiration in a cellular automata model by Ekström and Turova (2022), which is used to model neuronal activity in the brain. This work provides results on the quadrangulation model, including tail bounds for the distribution of the growth of lines, bounds for the expected values of a model characteristic and the covariance structures of the behaviour around points depending on their distance. It is also set into relation with work on Gilbert tessellation. (Less)
- Popular Abstract
- There are various stochastic models that aim to capture random spatial patterns appearing in nature. Examples of that include the spread of disease in an epidemic, the growth of crystals, foam structures, and many more. The model presented in this work is based on a model that, in a very simplified way, aims to model neuronal activity in the brain and is presented by Ekström and Turova (2022).
Our model is defined on a finite square in the two-dimensional real plane. It results in a partition of the square into rectangles, which we call random quadrangulation. The rectangles are formed by lines growing horizontally or vertically from the points of a Poisson process. This work provides results that characterise the probabilistic... (More) - There are various stochastic models that aim to capture random spatial patterns appearing in nature. Examples of that include the spread of disease in an epidemic, the growth of crystals, foam structures, and many more. The model presented in this work is based on a model that, in a very simplified way, aims to model neuronal activity in the brain and is presented by Ekström and Turova (2022).
Our model is defined on a finite square in the two-dimensional real plane. It results in a partition of the square into rectangles, which we call random quadrangulation. The rectangles are formed by lines growing horizontally or vertically from the points of a Poisson process. This work provides results that characterise the probabilistic properties of this model.
We derive tail bounds on the probability distribution of the lengths of the lines growing from the points. This is used to derive a bound on the covariance of events related to the line lengths from two different points depending on their distance. Our results are also set into relation with works on Gilbert tessellation. The findings in our work are the only ones, to the best knowledge of the author, that hold exactly for this model and are not only approximations. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9163014
- author
- Ewers, Emily LU
- supervisor
- organization
- course
- MASM02 20241
- year
- 2024
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- random quadrangulation, random graph, tessellation, rectangular tessellation, Gilbert tessellation, Poisson process, exponential decay, tail bounds
- publication/series
- Master’s Theses in Mathematical Sciences
- report number
- LUNFMS-3126-2024
- ISSN
- 1404-6342
- other publication id
- 2024:E21
- language
- English
- id
- 9163014
- date added to LUP
- 2024-06-19 15:23:01
- date last changed
- 2024-06-19 15:23:01
@misc{9163014,
abstract = {{A process of a random graph, that grows in a box [0,N]^2 ⊂ R^2 and yields a quadrangulation when it stops growing, is investigated in this work. The random components in this model come from the Poisson distributed number of points, the position of those, and the either horizontal or vertical direction in which lines grow from these points. The model finds inspiration in a cellular automata model by Ekström and Turova (2022), which is used to model neuronal activity in the brain. This work provides results on the quadrangulation model, including tail bounds for the distribution of the growth of lines, bounds for the expected values of a model characteristic and the covariance structures of the behaviour around points depending on their distance. It is also set into relation with work on Gilbert tessellation.}},
author = {{Ewers, Emily}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master’s Theses in Mathematical Sciences}},
title = {{Process of a Random Planar Quadrangulation}},
year = {{2024}},
}