Shape of a Self-Avoiding Walk or Polymer Chain

If pn(r) is the probability density of self-avoiding walks of m steps which reach the point r it is proved rigorously that the generating function P(θ,r)= ∑ n=1∞pn(r)exp(−nθ)decays exponentially with r. This result is used to derive restrictions on the form of the distribution pn(r). In particular it is argued that if, for large n the distribution in d dimensions approaches a limiting shape, Rn−dF(r/Rn), where the scaling length Rn, which measures the mean end-to-end distance, varies as r0nv, then the shape factor has the form F(y)=A(y)exp(−yδ),where δ=1/(1—v) and where A(y) does not vary exponentially fast for large y. Accepting the values v2=¾ and v3=⅗ for d=2 and 3, as suggested originally by Flory and since supported by numerical and theoretical calculations, yields δ2=4 and δ3=2½ so that F(y) has the form conjectured recently by Domb et al. on the basis of the numerical analysis of finite lattice chains.