Term structure of series expansions for the Ising and classical vector models and dilute magnetism

A detailed investigation is undertaken of the relationship between the parameters characterizing the geometrical properties of self avoiding walks (Saw approximation) and those describing the critical behaviour of the Ising and classical vector models. There are strong indications that the statistical distribution of contacts is the same for chains and polygons, and this gives rise to the identity of critical temperature for specific heat and susceptibility. It is suggested that a particular set of configurations consisting of bridges across chains and polygons are sufficient to account for the critical exponents. Following a previous paper by Domb and Joyce (1972) on interactions in random walks a distinction is drawn between ladder and nonladder configurations. The difference in critical exponents for the different models is due to the difference in weighting of ladder and nonladder configurations. The same analysis applied to a dilute magnet leads to the conclusion that the critical exponents of specific heat and susceptibility are identical with those of an undiluted magnet. A series expansion is obtained for the Curie temperature of all the above models in terms of the statistics of contacts of various types in a self avoiding walk.