Counting Theorem for the Generating Function of Self-Avoiding Walks

A generating function Gn(α, β, γ) is defined to contain the information which gives the number of self-avoiding walks which reach the point (I, J, K) after n jumps. Hence, the generating function yields the distribution function and the mean-square end-to-end distance of self-avoiding walks. One way of obtaining Gn is by direct computer enumeration. However, this method becomes exceedingly expensive beyond a certain value of n. In this paper we formulate a recursion relation for the generating function in terms of a restricted subclass of configurations. Since only small numbers of these configurations contribute to Gn for given n, the formulation enables one to obtain Gn for very large n with considerably less computer time.