Misc,
Introduction to a renormalisation group method
(2019)cite arxiv:1907.05474Comment: Book.
DOI: 10.1007/978-981-32-9593-3
Abstract
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.
