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

TL;DR

This paper calculates the number of configurations and entropy for ice and eight-vertex models on the two-dimensional Sierpinski gasket and its generalizations, providing exact and bounded results with high precision.

Contribution

It derives exact formulas and bounds for the configuration counts and entropies of ice and eight-vertex models on Sierpinski gaskets and their generalizations, advancing understanding of these models on fractal lattices.

Findings

Exact configuration counts for models on Sierpinski gasket.

Precise entropy calculations with over 100 significant figures.

Generalized results and conjectures for arbitrary gasket parameters.

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 $\frac87 \ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.