Violation of the des Cloizeaux relation for self-avoiding walks on Sierpinski square lattices

The statistics of self-avoiding walks (SAWs) on deterministic fractal structures with infinite ramification, modeled by Sierpinski square lattices, is revisited in two and three dimensions using the reptation algorithm. The probability distribution function of the end-to-end distance of SAWs, consisting of up to 400 steps, is obtained and its scaling behavior at small distances is studied. The resulting scaling exponents are confronted with previous calculations for much shorter linear chains (20 to 30 steps) based on the exact enumeration (EE) technique. The present results coincide with the EE values in two dimensions, but differ slightly in three dimensions. A possible explanation for this discrepancy is discussed. Despite this, the violation of the so-called des Cloizeaux relation, a renormalization result that holds on regular lattices and on deterministic fractal structures with finite ramification, is confirmed numerically. © 2006 The American Physical Society.