Bounding the number of self-avoiding walks: Hammersley.Welsh with polygon insertion

Let cn=cn(d) denote the number of self-avoiding walks of length n starting at the origin in the Euclidean nearest-neighbour lattice Zd. Let .=limnc1/nn denote the connective constant of Zd. 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 cn.exp(Cn1/2).n for all n.N. While it is anticipated that cn..n has a power-law growth in n, the best-known upper bound in dimension two has remained of the form n1/2 inside the exponential. The natural first improvement to demand for a given planar lattice is a bound of the form cn.exp(Cn1/2..).n , where . 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 .>0 in each case. For the hexagonal lattice H, the bound is proved for all n.N; while for the Euclidean lattice Z2, it is proved for a set of n.N of limit supremum density equal to one. A power-law upper bound on cn..n for H is also proved, contingent on a nonquantitative assertion concerning this lattice.s connective constant.