Effects of the boundaries on the scaling form of Hamiltonian walks on fractal lattices

Abstract

Hamiltonian walks (HWs) on a lattice are random walks that visit each lattice site exactly once. They are commonly used to model compact polymer conformations. The scaling form for the number of HWs, on translationary invariant lattices, consists of the leading exponential factor with the power law and stretched exponential factor as corrections. The stretched exponential factor, with the exponent σ that depends on the lattice dimension only, is caused and determined by the boundary sites of the lattice and corresponds to the surface tension effects of the compact globule. On fractal lattices, on the contrary, the existence of the stretched exponential factor in the scaling form of HWs is not so straightforward, and such a correspondence cannot be drawn equivalently. In this paper, we reinvestigate the appearance of the stretched exponential factor in the scaling form of HWs on fractal lattices and consider the effects of some kind of 'boundary' condition on it. In particular, in the case of 4-simplex lattice, we explicitly show that the introduction of only two extra links between the corner vertices of the largest generator, leads to complete disappearance of the stretched exponential factor. We also discuss impact of the boundaries on the scaling form of HWs on other fractal lattices.