Scaling theory for the statistics of self-avoiding walks on random lattices

Abstract

The authors study self-avoiding walks (SAW) on randomly diluted (quenched) lattices with direct configurational averaging over the moments of the SAW distribution function. A scaling function representation of R_N, the average end-to-end distance of N-step walks, is studied here both for SAW on (a) the infinite percolation cluster and (b) any cluster. They have shown that, at the percolation threshold nu^P= nu ^P(1-beta_P/2 nu_P), where beta _P and nu_P are the percolation order parameter and correlation length exponents respectively. The authors also propose a scaling function representation for the total number of N-step SAW configuration G_N( approximately mu ^NN^{gamma −1}) for infinite cluster averaging, which gives gamma ^P=gamma +d(nu^P-nu ). For all cluster averaging gamma will remain unchanged.