Self-Avoiding Walk and Supersymmetry

Following a brief discussion of the critical behaviour of the standard self-avoiding walk, we introduce the continuous-time weakly self-avoiding walk (also called the lattice Edwards model). We derive the BFS-Dynkin isomorphism which provides a random walk representation for spin systems. We introduce an anti-commuting fermion field represented by differential 1-forms, and explain an important connection with supersymmetry. We prove the localisation theorem, and use it to derive a representation of the weakly self-avoiding walk in terms of a supersymmetric spin system. For the 4-dimensional weakly self-avoiding walk, the renormalisation group method developed in this book has been extended to analyse the supersymmetric spin system and thereby yield results concerning the critical behaviour of the weakly self-avoiding walk.