The asymptotic behaviour of selfavoiding walks and returns on a lattice

New data for the number of selfavoiding walks and selfavoiding returns to the origin on two and three dimensional lattices are presented and studied numerically by the ratio method. Estimates for the critical attrition and critical indices are given. For a loose-packed lattice the selfavoiding walk generating function appears to have a singularity on the negative real axis. This singularity is at the same distance from the origin as the physical singularity, and is found to be cusp-like with an exponent of 1/2 in two dimensions and 3/4 in three dimensions. This behaviour enables a close analogy to be drawn between the behaviour of the Ising model high temperature susceptibility and the walk generating function.