Introduction to a Renormalisation Group Method

This book provides an introduction to a renormalisation group method in the spirit of that of Wilson. It starts with a concise overview of the theory of critical phenomena and the introduction of several tools required in the renormalisation group approach, including Gaussian integration and finite range decomposition. The bulk of the book consists of an analysis of the 4-dimensional hierarchical $|\varphi|^4$ model. This includes definition of the renormalisation group map, perturbation theory, development of a general family of norms that we use to analysis the renormalisation group map, and of a stability analysis of the dynamical system that the renormalisation group map defines. The analysis of the hierarchical $|\varphi|^4$ model is presented in a pedagodical way and set up in such that the extension to the more complicated Euclidean nearest-neighbour setting is very much parallel. This extension is outlined in an appendix with precise pointers to the literature where it is carried out in detail. Finally, we provide a short introduction to supersymmetry and the supersymmetric representation of self-avoiding walks. The renormalisation group method can be extended to this supersymmetric setting and has been used to study the 4-dimensional weakly self-avoiding walk.