Ice model and eight-vertex model on the two-dimensional Sierpinski gasket

Abstract

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 $\frac{8}{7}\ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.

20 pages, 6 figures, 2 tables

References in corpus (2)

• Spanning trees on the Sierpinski gasket
• Dimer-monomer model on the Sierpinski gasket