On the Number of Self-Avoiding Walks. II

Abstract

Let χ n (d) [resp. γ2n−1(d)] be the number of self‐avoiding walks (polygons) of n (2n − 1) steps on the integral points in d dimensions. It is known that β d = lim n→∞ [χ n (d)] 1/n = lim n→∞ [γ 2n−1 (d)] 1/2n−1 In this paper β d is compared with β d,2r = lim n→∞ [χ n,2r (d)] 1/n where χ n,2r (d) is the number of n‐step walks on the integral points in d dimensions with no loops of 2r steps or less. In other words the walks counted in χ n,2r (d) may visit the same point more than once as long as there are more than 2r steps between consecutive visits. It turns out that β d,2r −β d =O(d −r )(d→∞) and it follows in particular that 1 β d =2d−1−1/2d+O(1/d 2 )(d→∞) It is also shown that, for suitable constants α6 = α6(d) and α7 = α7(d), χ n (d)≤β n exp [α 6 n 2/(d+2) log n] and β 2n−1 exp [−α 7 n 2/(d+1) log n]≤γ 2n−1 (d) .