Contacts in self-avoiding walks and polygons

Abstract

We prove several results concerning the numbers of n-edge self-avoiding polygons and walks in the lattice Zd which had previously been conjectured on the basis of numerical results. If the number of n-edge self-avoiding polygons (walks) with k contacts is pn(k) (cn(k)) then we prove that κ0 ≡ limn→∞ n-1 log pn(k) = limn→∞ n-1 log cn(k) exists for all fixed k and is independent of k. For polygons in Z2, we prove that there exist two positive functions B1 and B2, independent of n but depending on k, such that B1nkpn(0) ≤ pn(k) ≤ B2nkpn(0) for fixed k and n large. Also, provided the limit exists, we prove that 0 < limn→∞ kn/n < 1. In addition, we consider the number of polygons with a density of contacts, i.e. k = αn, and show that the corresponding connective constant, κ(α), exists and is a concave function of α. For d = 2, we prove that limα→0+ κ(α) = κ0 and the right derivative of κ(α) at α = 0 is infinite.