Shape of self-avoiding walk or polymer chain

If p n (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≫R n , that pn(r)∼R n -d (r/R n ) t exp{-(r/R n ) l/(l-ν) } where R n is a scaling length which varies as n ν , 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 p n (r) ∼ R n -d (r/R n ) g for large n and r≫R n 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.