Lagrangian theory for a self-avoiding random chain

The Lagrangian theory of random chains with excluded volume is used to study ZN(→r), the number of chains with N links, starting from the origin and arriving at a point →r. Its asymptotic expression (N→∞) is ZN(r)N^γ-1-νd F(rN^-ν), where γ and ν are critical indices. The short- and long-range behaviors of $F(x)$ are calculated in terms of γ and ν. In particular, it is shown that for x1, we have F(x)Fx^θ with F=const and θ=(γ-1)ν.