Weighting of graphs for the Ising and classical vector models

Abstract

A simplified procedure is developed for calculating the weights of configurations which enter the high temperature expansions for the Ising and classical vector models. For ladder structures the weights can be derived from the partition function of a simple polygon by replacing a single interaction by a pair of parallel interactions successively along suitably chosen lengths of bonds. For nonladder structures a number of relations can be obtained by allowing chosen interactions to become infinite so that the nodes which they connect coalesce. The method can be used in d dimensions and results check with the calculations of Joyce (1967) for d=3 in terms of 3-j and 6-j symbols.