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A general perturbation series is developed for a random walk with a repulsive interaction w delta ij between the ith and jth steps using the generating function for returns to the origin. The interaction w is equal to -1+exp(- beta J), so that for a lattice model, beta to infinity (w to -1) corresponds to a self avoiding wall. Expansions are obtained in powers of w for cN(w) the total number of walks of N steps, uN(w) the number of walks of N steps terminating at the origin and (RN2(w)) the mean square length of a walk of N steps. The analytic behaviour of the corresponding generating functions is examined, and it is suggested that they are analogous to Ising model expansions about a singular point which is associated with a change of exponent. Hence it is conjectured using the 'smoothness postulate' of Griffiths (1970), that a change of exponent occurs at w=0 but for no other value of w. A distinction is drawn between ladder and non-ladder contributions, following Chikahisa (1970). Using a method of Brout (1961), it is found that all ladder diagrams can be summed, and the result conforms to the conjectured pattern of behaviour. However non-ladder contributions give rise to logarithmic terms as envisaged by Chikahisa. It is also concluded that the two-parameter theory which assumes that (RN2(w))/N is a function of wN1/2 is an approximation of limited validity.
Domb et al. (Wed,) studied this question.