Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Critical Scaling in Particle Systems and Random Graphs

Goriachkin, Vasilii LU orcid (2023) In Doctoral Theses in Mathematical Sciences 2023(5).
Abstract
The purpose of this thesis is to study the behavior of macro-systems through their micro-parameters. In particular, we are interested in finding critical scaling in various models.
Paper I investigates the influence of discrete-time collisions on particle dynamics. By analyzing two models — one involving external forces and friction, and another incorporating collisions with lighter particles — a nuanced understanding of particle trajectories emerges. Conditions for the equivalence of these models are established, encompassing both deterministic and stochastic collision scenarios.
Paper II focuses on scaling properties within critical geometric random graphs on a 2-dimensional torus. This is an example of an inhomogeneous random... (More)
The purpose of this thesis is to study the behavior of macro-systems through their micro-parameters. In particular, we are interested in finding critical scaling in various models.
Paper I investigates the influence of discrete-time collisions on particle dynamics. By analyzing two models — one involving external forces and friction, and another incorporating collisions with lighter particles — a nuanced understanding of particle trajectories emerges. Conditions for the equivalence of these models are established, encompassing both deterministic and stochastic collision scenarios.
Paper II focuses on scaling properties within critical geometric random graphs on a 2-dimensional torus. This is an example of an inhomogeneous random graph that is not of rank 1. Drawing parallels with classic Erdős-Rényi graphs, the study unveils scaling patterns of the size of the largest connected component and its diffusion approximation.
In Paper III and Paper IV, we examine axon tree growth models in dimensions 2 and 3. We uncover the relationship between the probability of neuron connections and micro-level growth parameters. Notably, we demonstrate that connection probabilities do not strictly decrease exponentially or polynomially with the distance between neurons. While finding the critical scaling for the connection probability over time (determined by distance) remains challenging, the insights from Papers III and IV will aid in addressing this issue. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Associate Professor Tykesson, Johan, Chalmers University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
dissipative force, particles collisions, random graphs, random walks, martingales, geometric random graphs, neural networks, critical scaling, diffusion approximation
in
Doctoral Theses in Mathematical Sciences
volume
2023
issue
5
pages
171 pages
publisher
Lund University
defense location
Matematikhuset, Hörmandersalen
defense date
2023-11-02 13:00:00
ISSN
1404-0034
1404-0034
ISBN
978-91-8039-820-6
978-91-8039-819-0
language
English
LU publication?
yes
id
cf265749-ff01-4b6c-9814-57cca53c0540
date added to LUP
2023-10-03 16:53:34
date last changed
2024-02-13 11:31:58
@phdthesis{cf265749-ff01-4b6c-9814-57cca53c0540,
  abstract     = {{The purpose of this thesis is to study the behavior of macro-systems through their micro-parameters. In particular, we are interested in finding critical scaling in various models.<br/>Paper I investigates the influence of discrete-time collisions on particle dynamics. By analyzing two models — one involving external forces and friction, and another incorporating collisions with lighter particles — a nuanced understanding of particle trajectories emerges. Conditions for the equivalence of these models are established, encompassing both deterministic and stochastic collision scenarios.<br/>Paper II focuses on scaling properties within critical geometric random graphs on a 2-dimensional torus. This is an example of an inhomogeneous random graph that is not of rank 1. Drawing parallels with classic Erdős-Rényi graphs, the study unveils scaling patterns of the size of the largest connected component and its diffusion approximation.<br/>In Paper III and Paper IV, we examine axon tree growth models in dimensions 2 and 3. We uncover the relationship between the probability of neuron connections and micro-level growth parameters. Notably, we demonstrate that connection probabilities do not strictly decrease exponentially or polynomially with the distance between neurons. While finding the critical scaling for the connection probability over time (determined by distance) remains challenging, the insights from Papers III and IV will aid in addressing this issue.}},
  author       = {{Goriachkin, Vasilii}},
  isbn         = {{978-91-8039-820-6}},
  issn         = {{1404-0034}},
  keywords     = {{dissipative force; particles collisions; random graphs; random walks; martingales; geometric random graphs; neural networks; critical scaling; diffusion approximation}},
  language     = {{eng}},
  month        = {{10}},
  number       = {{5}},
  publisher    = {{Lund University}},
  school       = {{Lund University}},
  series       = {{Doctoral Theses in Mathematical Sciences}},
  title        = {{Critical Scaling in Particle Systems and Random Graphs}},
  url          = {{https://lup.lub.lu.se/search/files/160172109/Thesis_Vasilii_Goriachkin_without_papers.pdf}},
  volume       = {{2023}},
  year         = {{2023}},
}