Exact analysis of the self-avoiding random walks on two infinite families of fractals

Abstract

Abstract We introduce two infinite families of fractals that we name the Φ family and the Koch family, according to their first members, which are the plane-filling Φ lattice and the Koch fractal curve, respectively. The fractal dimension d f of Φ family varies from 2 to 1 (and from 1.465 to 1, in the case of the Koch family) when the fractal enumerator b (an odd integer) varies from 3 to ∞. We have calculated exactly the critical exponents of the self-avoiding random walks (SAWs) on these fractals. Our results render it possible to perform a complete and exact study of the fractal to Euclidean crossover, which, in this case, occurs when b →∞. It turns out that all critical exponents, when d f →1 ( b →∞), tend to the corresponding Euclidean values with a unique correction term of the type constant/ln( b ).