7 - Diagrammatic methods

Summary

This chapter is devoted to technicalities related to various expansions already encountered in volume 1, mostly those that derive from the original lattice formulation of the models, be it high or low temperature, strong coupling expansions and to some extent those arising in the guise of Feynman diagrams in the continuous framework. We shall not try to be exhaustive, but rather illustrative, relying on the reader's interest to investigate in greater depth some aspects inadequately treated. Nor shall we try to explore with great sophistication the vast domain of graph theory. There are, however, a number of common features, mostly of topological nature, which we would like to present as examples of the diversity of what looks at first sight like straightforward procedures.

General Techniques

Definitions and notations

Let a labelled graph G be a collection of v elements from a set of indices, and l pairs of such elements with possible duplications (i.e. multiple links). We shall also interchangeably use the word diagram instead of graph. This abstract object is represented by v points (vertices) and l links. Each vertex is labelled by its index.

The problem under consideration will define a set of admissible graphs, with a corresponding weight ω(G) (a real or complex number) according to a well-defined set of rules. We wish to find the sum of weights over all admissible graphs.