Critical Scaling in Particle Systems and Random Graphs

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 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.