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تسجيل مستخدم جديدIce model and eight-vertex model on the two-dimensional Sierpinski gasket
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We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $lim_{v to infty} ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $ln 2$. For the ice model, the upper and lower bounds for the entropy per site $lim_{v to infty} ln I(n)/v$ are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 times 6^n +7)/5}$ and the entropy per site is equal to $frac87 ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.
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