## Abstract

We present the numbers of ice model configurations (with Boltzmann factors equal to one) I(n) on the two-dimensional Sierpinski gasket SG(n) at stage n. The upper and lower bounds for the entropy per site, defined as lim _{v→}lnI(n)/v, where v is the number of vertices on SG(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^{Gb}(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 SG(n) and the number of generalized vertex models ^{Eb}(n) on S^{Gb}(n), and obtain exactly ^{Eb}(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^{Eb}(n)/v=2(b+1)b+4ln2.

Original language | English |
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Pages (from-to) | 1776-1787 |

Number of pages | 12 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 392 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2013 Apr 15 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics