Temperature-dependent structural behavior of self-avoiding walks on three-dimensional Sierpinski sponges

Abstract

The statistics of self-avoiding random walks (SAWs), consisting of up to N=1280 steps, on deterministic fractal structures with infinite ramification, modeled by Sierpinski cubic lattices, in the presence of a finite temperature is investigated as a model for polymers absorbed on a disordered medium. Thereby, the three-dimensional Sierpinski sponge is defined by two types of sites with energy 0 and ϵ>0 , respectively, yielding a deterministic fractal energy landscape. The probability distribution function of the end-to-end distance of SAWs is obtained and its scaling behavior studied. In the limiting case of temperature T → ∞, the known behavior of SAWs on regular cubic lattices is recovered, while for T → 0 the resulting scaling exponents are confronted with previous calculations for much shorter linear chains based on the exact enumeration technique. For finite temperatures, the structural behavior of SAWs in three dimensions is compared to its two-dimensional counterpart and found to be intermediate between the two limiting cases (T → 0 and T → ∞, respectively), where the characteristic exponents, however, display a nontrivial dependence on temperature.