Abstract
The distribution of end-to-end lengths of an n step self-avoiding walk has been calculated for walks of up to 10 steps on the face-centred cubic lattice and up to 12 steps on the triangular lattice. The data on both lattices have been extended to higher step lengths for walks with short end-to-end lengths. The regions of short end-to-end lengths and long end-to-end lengths have been analysed separately. The results show that the region of long end-to-end lengths behaves predictably in contrast to the region of short end-to-end lengths. A new analytical form for the distribution close to the origin is suggested. The results throw considerable light on the scaling laws first introduced to explain critical phenomena. It is suggested that the strong-scaling hypothesis is untenable in both two and three dimensions. A new characteristic length is introduced and its magnitude is calculated.