A model of compact polymers on a family of three-dimensional fractal lattices

Abstract

We study Hamiltonian walks (HWs) on the family of three-dimensional modified Sierpinskigasket fractals, as a model for compact polymers in nonhomogeneous media inthree dimensions. Each member of this fractal family is labeled with an integerb ≥ 2. We apply an exact recursive method which allows for explicit enumeration of extremelylong Hamiltonian walks of different types: closed and open, with end-points anywhere inthe lattice, or with one or both ends fixed at the corner sites, as well as some Hamiltonianconformations consisting of two or three strands. Analyzing large sets of data obtained forb = 2, 3 and 4, we findthat numbers ZN of Hamiltonian walks, on fractal lattice withN sites, for behave as ZN ∼ ωNμNσ. Theleading term ωN is characterized by the value of the connectivity constantω > 1, whichdepends on b, but not on the type of HW. In contrast to that, the stretched exponential termμNσ depends on the type of HW through the constantμ < 1, whereas theexponent σ isdetermined by b alone. For larger b values, using some general features of the applied recursive relations, withoutexplicit enumeration of HWs, we argue that the asymptotical behavior ofZN should be thesame, with σ = ln3/ln[b(b + 1)(b + 2)/6],valid for all b > 2. This differs from the formulae obtained recently for Hamiltonian walks on other fractallattices, as well as from the formula expected for homogeneous lattices. We discuss thepossible origins and implications of such a result.