Shape of self-avoiding walk or polymer chain

Abstract

If pn(r) is the probability that a self-avoiding walk of n steps reaches a distance r from the origin, then it is shown, for large n and r≫Rn, that pn(r)∼Rn-d(r/Rn)t exp{-(r/Rn)l/(l-ν)} where Rn is a scaling length which varies as nν, and d is the dimensionality. Furthermore, the index t is related to d, ν, and a further index y which des- cribes the asymptotic behaviour of the total number of self-avoiding walks. We have also shown, on the assumption that pn(r) ∼ Rn-d(r/Rn)g for large n and r≫Rn that the index g can be related to d, v, y, and an index α which describes the asymptotic behaviour of the total number of self-avoiding walks which return to the origin.