Scaling of Hamiltonian walks on fractal lattices

Details Scaling of Hamiltonian walks on fractal lattices Scaling of Hamiltonian walks on fractal lattices

2006-12-19

arxiv.org/abs/cond-mat/0612482v2

Phys.Rev.E 76, 011107 (2007)

Physics

Condensed Matter

Statistical Mechanics

19 pages, 13 figures, RevTex4; extended Introduction, several references added; one figure added in section II; corrected typo

Scientific paper

10.1103/PhysRevE.76.011107

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.

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