Project: Self-avoiding Walk (Level 3)

A linear polymer is a string of molecules strung together in a chain. They are studied in physics, chemistry, and beyond, and part of this research involves the creation of mathematical models which can be used to derive predictions and compare the results with experiment. A fascinating mathematical object was born from this line of research: the study of self-avoiding walks. These are like the random walks you may be familiar with from probability modules, but with the additional rule that the walk cannot revisit places it has been in the past. This constraint reflects the fact that a polymer chain is made of individual molecules that cannot overlap.

Basic questions about self-avoiding walk turn out to be a rich source of interesting mathematics. For example, it is a fundamental challenge to even enumerate how many self-avoiding walks of length N there are on an infinite lattice. To get a feeling for this, you might like to try counting how many self-avoiding walks of length 3, 4, 5 and 6 there are on a checkboard. Other questions about self-avoiding walk concern it's probabilistic and statistical properties. What does a random self-avoiding walk of length N look like? This is the type of question that can be asked of a linear polymer chain in a lab. Do self-avoiding walks tend to knot themselves up? What do the patterns along them look like?

This project will initially involve learning some basic tools that have been developed for studying self-avoiding walks and fundamental results like the Hammersley--Welsh bound. You will then be able to explore more advanced topics depending on your interests.

Prerequisites: Probability II is advised, but combinatorially-minded projects are possible without this.

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