The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

We present the numbers of ice model configurations (with Boltzmann factors equal to one) I ( n ) on the two-dimensional Sierpinski gasket S G ( n ) at stage n . The upper and lower bounds for the entropy per site, defined as lim v → ∞ ln I ( n ) / v , where v is the number of vertices on S G ( n ) , 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 accuracy. The corresponding result of the ice model on the generalized two-dimensional Sierpinski gasket S G b ( n ) with b = 3 is also obtained, and the general upper and lower bounds for the entropy per site for arbitrary b are conjectured. We also consider the number of eight-vertex model configurations on S G ( n ) and the number of generalized vertex models E b ( n ) on S G b ( n ) , and obtain exactly E b ( n ) = 2 { 2 ( b + 1 ) [ b ( b + 1 ) / 2 ] n + b + 4 } / ( b + 2 ) . It follows that the entropy per site is lim v → ∞ ln E b ( n ) / v = 2 ( b + 1 ) b + 4 ln 2 .