Scaling of Hamiltonian walks on fractal lattices

Original · EN

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 Zₙ, for the lattice with N≫ 1 sites, varies as ωⁿ Nγ. We explicitly calculate 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 μⁿ¹/ᵈᶠ, where df 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.