Distribution Function for Self-Avoiding Walks

Abstract

In this paper the asymptotic behavior of the distribution function for self‐avoiding walks is derived from the generating function G n (θ) , where n is the number of steps and θ is a variable of the function. The asymptotic form of G n (θ) for large n is determined as follows: First, from the subadditive property of ln G n (θ) we prove rigorously the existence of κ(θ) = lim n→∞ (1 / n) ln G n (θ) and determine its upper and lower bounds. Second, from the enumeration data for small n we determine the form of G n (θ) exp [− nκ(θ)] for large n . The relation between δ and ν , defined, respectively, by F n (x) ∼ exp [− (x / x n ) δ ] and x n ∼ n ν , are derived from the distribution function thus obtained.