Scaling of Hamiltonian walks on fractal lattices

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 three-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, the number of open HWs ZN, for the lattice with N1 sites, varies as ωN Nγ. We explicitly calculate the exponent γ 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: ZN ∼ ωN μ N1 df, where df is the 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. © 2007 The American Physical Society.