Lectures on Self-Avoiding Walks

TL;DR

This paper provides a comprehensive overview of rigorous results on self-avoiding walks, including bounds, critical behavior, and advanced mathematical techniques like lace expansion and renormalisation group analysis.

Contribution

It offers detailed proofs of key results, such as the connective constant on the hexagonal lattice, and discusses advanced methods for understanding self-avoiding walks in various dimensions.

Findings

Proof that the connective constant on the hexagonal lattice equals √(2+√2)

Discussion of the lace expansion and its application in dimensions >4

Development of functional integral representations and renormalisation group analysis in dimension 4

Abstract

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals $\sqrt{2+\sqrt{2}}$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions $d>4$ is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.