On the Poisson Follower Model

This dissertation presents studies of dynamics over the Poisson point process. In particular, we study a special case of Hegselmann-Krause Dynamics [1] over ℝ². Chapter 1 is a brief introduction to the thesis and its structure. Chapter 2 introduces the notation, the definitions and examples of phenomena of interest. In Chapter 3, we go deeper in analyzing the phenomena described by calculating frequency of these phenomena. A system of quadratic inequalities will be introduced to allow one to calculate the probabilities of the events pertaining to this dynamics using methods from integral geometry. Chapter 4 uses percolation arguments to prove the absence of percolation at step 1. In Chapter 5, we provide geometric results of independent interest pertaining to the Follower Dynamics. In Chapter 6, we discuss the limiting behavior of this process and include some more simulations. In Chapter 7 we propose future steps and discuss more general Hegselmann-Krause Dynamics.