Self-avoiding walks and polygons -- an overview

Abstract

This is a rather personal review of the problem of self-avoiding walks and polygons. After defining the problem, and outlining what is known rigorously and what is merely conjectured, I highlight the major outstanding problems. I then give several applications in which the I have been involved. These include a study of surface adsorption of polymers, counting possible paths in a telecommunication network, hitting probabilities of SAWs in a rectangle, and the modelling of biological experiments on polymers. I hope to show that SAWs are not only of intrinsic mathematical interest, but also have many interesting and useful applications.

This is a slightly updated and expanded version, with minor corrections, of an article that first appeared in the October 2012 issue of the Asia Pacific Newsletter, accompanying an interview with the author. 19 pages, 9 figures

References in corpus (7)

• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions
• Efficient implementation of the pivot algorithm for self-avoiding walks
• Lectures on Self-Avoiding Walks
• Scaling behaviour of two-dimensional polygon models
• The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$
• Self-avoiding walks in a rectangle

Cited by in corpus (2)

• On self-avoiding polygons and walks: counting, joining and closing
• Area-width scaling in generalised Motzkin paths