Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2009-09-30
J. Math. Phys. 52, 023301 (2011)
Physics
Condensed Matter
Statistical Mechanics
33 pages, 21 figures
Scientific paper
We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{5^2 \times 7^2 \times 17^2}{2^{12} \times 3^5 \times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $\frac {\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac {7 \times 17}{2^4 \times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $\ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $\ell \log 2 / \log 3$ for $\ell>0$.
Chang Shu-Chiuan
Chen Lung-Chi
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