Self-avoiding walks on Sierpinski lattices in two and three dimensions

The scaling properties of linear polymers on deterministic fractal structures, modeled by self-avoiding random walks (SAW) on Sierpinski lattices in two and three dimensions, are studied. To this end, all possible SAW configurations of N steps are enumerated exactly and averages over suitable sets of starting lattice points for the walks are performed to extract the mean quantities of interest reliably. We determine the critical exponent describing the mean end-to-end chemical distance (-)l(N) after N steps and the corresponding distribution function, P(S)(l,N). A des Cloizeaux-type relation between the exponent characterizing the asymptotic shape of the distribution, for l-->0 and N--> infinity, and the one describing the total number of SAW of N steps is suggested and supported by numerical results. These results are confronted with those obtained recently on the backbone of the incipient percolation cluster, where the corresponding exponents are very well described by a generalized des Cloizeaux relation valid for statistically self-similar structures.