Self-avoiding walks on self-similar structures: finite versus infinite ramification

Abstract

Self-avoiding walks (SAWs) generated by exact enumeration techniques are studied on the Sierpinski carpet (d = 2) and on the Sierpinski sponge (d = 3) (also called Sierpinski square lattices). A detailed comparison of the results for SAWs on these infinitely ramified fractals to SAWs on finitely ramified Sierpinski gaskets (Sierpinski triangular lattices), on regular lattices, and on the incipient percolation cluster is done, providing insight into the behaviour of SAWs on ordered and disordered structures. The SAWs on Sierpinski square lattices are found to display a kind of intermediate behaviour, sharing aspects of both SAWs on ordered and on fractal structures. As a consequence, a des Cloizeaux relation does not seem to hold for this structure, as opposed to its validity for SAWs on regular lattices, on Sierpinski triangular lattices and on the incipient percolation cluster.