Applications of Random Walks in Science (3rd year project)

Random walks are a central object in probability theory, and they come in a variety of flavours. The most classical are stochastic processes that arise from sums of independent random variables — e.g., simple random walk on the integers, or on a d-dimensional integer lattice, which you may have seen in a probability module. These random walks are on discrete spaces. Random walks can also take values in continuous space — e.g., you can think of the central limit theorem as describing the long-time behaviour of a random walk on the real numbers.

This project concerns the study of random walks in some of their many guises, particularly in connection with physics, computer science, and chemistry. Random walks can be used in algorithm design; for modeling polymers; for explaining wetting and crystal growth phenomena. In these applications it is something necessary to relax the assumption of independent increments, and this leads to a rich variety of mathematical phenomena.

This project will begin by reviewing some classical theory about random walks and renewal structures, along with some indications of how they arise in applications. There will then be some choice about what topics are investigated further in individual projects, with possibilities including:

• Algorithmic applications of random walks (sampling spanning trees; estimating volumes of convex bodies; Markov Chain Monte Carlo algorithms)

• Mathematical polymer chemistry (self-avoiding walks; Poland-Scheraga model of DNA denaturation)

• Scaling limits of random walks (Donsker’s theorem; central and local central limit theorems)

• Wetting and melting phenomena (non-intersecting random walks; renewal theory)

• Electrical Networks and random walks.

Prerequisites: Probability II.

Resources:

• Random Walk: A Modern Introduction by G. Lawler and V. Limic

• Probability on Trees and Networks by R. Lyons and Y. Peres

• Walks, Walls, Wetting and Melting by M. Fisher.

• Further resources may be suggested once the project is underway.

Supervisors: Michaelmas term will be supervised by Prof. Tyler Helmuth, and Epiphany term by Prof. Ostap Hryniv.