Summer 2024

Mentored Advanced Project (MAP) at Grinnell College.

Students: Owen Gould, Sam Hall, Eloise McSurdy, Jianna Pollack.

Exploring Random Walks.

Expository Article

In this project, students studied the probability techniques and computational simulations of random walks. Random walks are fundamental stochastic processes used to model various phenomena in diverse fields such as physics, biology, finance, and computer science. This undergraduate research project aims to investigate random walk dynamics and properties, focusing on theoretical analysis and computational simulations. The project began with an introduction to the basics of random walks, including simple one-dimensional walks and their extensions to higher dimensions. Students will explore the concept of discrete and continuous random walks and understand how different types of steps (e.g., symmetric, and asymmetric) influence the behavior of the walk. Next, the research explored more advanced topics such as random walks in random environments (RWRE) and branching random walks. Students read the recent developments in this field, including research papers. They studied the concepts of recurrence, transience, and convergence to stationary distributions, gaining insight into the long-term behavior of such random processes. Furthermore, the project involved using programming languages like Python to simulate random walks to model animal movement. In particular, using GPS telemetry data to optimize parameters, students constructed a RWRE model to predict and study elephant movement in Kruger National Park.