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

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 N N gamma -1 ) for infinite cluster averaging, which gives gamma P = gamma +d( nu P - nu ). For all cluster averaging gamma will remain unchanged.