On the Number of Self-Avoiding Walks

Let χn be the number of self-avoiding walks on the integral points in Euclidean d space and γn the number of n-stepped self-avoiding polygons. It is shown that χn+2/χn−β2 and γ2n+3/γ2n+1−β2 tend to zero as n → ∞ where β=limn→∞χn1/n. Asymptotic estimates for these differences are given. β is also characterized as the unique positive root of Σ1∞λkx−k = 1 where the λk are the number of certain k-stepped self-avoiding walks.