Contacts in self-avoiding walks and polygons

We prove several results concerning the numbers of n-edge self-avoiding pol ygons and walks in the lattice Z(d) which had previously been conjectured o n the basis of numerical results. If the number df n-edge self-avoiding pol ygons (walks) with k contacts is p(n)(k) (c(n)(k)) then we prove that kappa (0) equivalent to lim(n --> infinity) n(-1) log p(n) (k) = lim(n --> infin ity) n(-1) logc(n)(k) exists for all fixed k and is independent of k. For p olygons in Z(2), we prove that there exist two positive functions B-1 and B -2, independent of n but depending on k, such that B(1)n(k) p(n)(0) less th an or equal to p(n) (k) less than or equal to B(2)n(k) p(n) (0) for fixed k and n large. Also, provided the limit exists, we prove that 0 < lim(n --> infinity) <k > (n)/n < 1. In addition, we consider the number of polygons w ith a density of contacts, i.e. k = alphan, and show that the corresponding connective constant, K(alpha), exists and is a concave function of alpha. For d = 2, we prove that lim(alpha -->0+) K(alpha) = K-0 and the right deri vative of kappa(alpha) at alpha = 0 is infinite.