Scaling of Hamiltonian walks on fractal lattices

TL;DR

This paper analyzes the asymptotic behavior of Hamiltonian walks on various fractal lattices, revealing different scaling forms depending on lattice type and providing explicit exponents, with implications for understanding real compact polymers.

Contribution

It introduces exact recursive techniques to determine the scaling forms and exponents of Hamiltonian walks on multiple fractal lattices, including new forms for even n-simplex fractals.

Findings

Number of open HWs on certain fractals scales as $ au^N N^eta$ for large N.

Explicit calculation of scaling exponents for GM, MSG, and n-simplex fractals.

Different scaling behavior observed for even n-simplex fractals, involving a $ ext{N}^{1/d_f}$ term.

Abstract

We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e. self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on 3-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG) and n-simplex fractal families. For GM, MSG and n-simplex lattices with odd values of n, number of open HWs $Z_N$, for the lattice with $N\gg 1$ sites, varies as $\omega^N N^\gamma$. We explicitly calculate exponent $\gamma$ for several members of GM and MSG families, as well as for n-simplices with n=3,5, and 7. For n-simplex fractals with even n we find different scaling form: $Z_N\sim \omega^N \mu^{N^{1/d_f}}$, where $d_f$ is fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.