BOUNDING THE NUMBER OF SELF-AVOIDING WALKS: HAMMERSLEY-WELSH WITH POLYGON INSERTION

Let c(n) = c(n)(d) denote the number of self-avoiding walks of length n starting at the origin in the Euclidean nearest-neighbour lattice Z(d). Let mu, = lim(n) c(n)(1/n) denote the connective constant of Z(d). In 1962, Hammersley and Welsh (Quart. J. Math. Oxford Ser (2) 13 (1962) 108-110) proved that, for each d >= 2, there exists a constant C > 0 such that c(n) <= exp(C-n(1/2))mu(n) for all n is an element of N. While it is anticipated that c(n)mu(-n). has a power-law growth in n, the best-known upper bound in dimension two has remained of the form n(1/2 )inside the exponential. The natural first improvement to demand for a given planar lattice is a bound of the form c(n) <= exp(C-n(1/2-epsilon))mu(n), where mu denotes the connective constant of the lattice in question. We derive a bound of this form for two such lattices, for an explicit choice of epsilon > 0 in each case. For the hexagonal lattice IR, the bound is proved for all n is an element of N; while for the Euclidean lattice Z(2), it is proved for a set of n is an element of N of limit supremum density equal to one. A power-law upper bound on c(n)mu(-n) for H is also proved, contingent on a nonquantitative assertion concerning this lattice's connective constant.