“Generalized des Cloizeaux” exponent for self-avoiding walks on the incipient percolation cluster

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function [formula presented] for the end-to-end distance r of an N step SAW behaves as a power law for [formula presented] In this work, we determine the corresponding exponent using scaling arguments, and show that our suggested “generalized des Cloizeaux” expression for the exponent is in excellent agreement with exact enumeration results in two and three dimensions.