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

TL;DR

This paper models compact polymers on three-dimensional fractal lattices using Hamiltonian walks, revealing how their enumeration scales with lattice size and how fractal structure influences these scaling laws.

Contribution

It introduces an exact recursive method to enumerate Hamiltonian walks on 3D fractal lattices and derives their asymptotic behavior, highlighting differences from homogeneous lattices.

Findings

Number of Hamiltonian walks scales as Z_N ~ ω^N μ^{N^σ}.

The connectivity constant ω depends on the fractal parameter b.

Exponent σ is determined solely by the fractal parameter b.

Abstract

We study Hamiltonian walks (HWs) on the family of three--dimensional modified Sierpinski gasket fractals, as a model for compact polymers in nonhomogeneous media in three dimensions. Each member of this fractal family is labeled with an integer $b\geq 2$. We apply an exact recursive method which allows for explicit enumeration of extremely long Hamiltonian walks of different types: closed and open, with end-points anywhere in the lattice, or with one or both ends fixed at the corner sites, as well as some Hamiltonian conformations consisting of two or three strands. Analyzing large sets of data obtained for $b=2,3$ and 4, we find that numbers $Z_N$ of Hamiltonian walks, on fractal lattice with $N$ sites, for $N\gg 1$ behave as $Z_N\sim \omega^N \mu^{N^\sigma}$. The leading term $\omega^N$ is characterized by the value of the connectivity constant $\omega>1$, which depends on $b$, but not on the type of HW. In contrast to that, the stretched exponential term $\mu^{N^\sigma}$ depends on the type of HW through constant $\mu<1$, whereas exponent $\sigma$ is determined by $b$ alone. For larger $b$ values, using some general features of the applied recursive relations, without explicit enumeration of HWs, we argue that asymptotical behavior of $Z_N$ should be the same, with $\sigma=\ln 3/\ln[b(b+1)(b+2)/6]$, valid for all $b>2$. This differs from the formulae obtained recently for Hamiltonian walks on other fractal lattices, as well as from the formula expected for homogeneous lattices. We discuss the possible origin and implications of such result.