### Abstract

We derive the tightest known bounds on η = 2^{ν}, where ν is the growth rate of the logarithm of the number of independent sets on a hexagonal lattice. To obtain these bounds, we generalize a method proposed by Calkin and Wilf. Their original strategy cannot immediately be used to derive bounds for η, due to the difference in symmetry between square and hexagonal lattices, so we propose a modified method and an algorithm to derive rigorous bounds on η. In particular, we prove that 1.546440708536001 ≤ η ≤ 1.5513, which improves upon the best known bounds of 1.5463 ≤ η ≤ 1.5527 given by Nagy and Zeger. Our lower bound matches the numerical estimate of Baxter up to 9 digits after the decimal point, and our upper bound can be further improved by following our method.

Original language | English (US) |
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Title of host publication | 2017 IEEE International Symposium on Information Theory, ISIT 2017 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 2910-2914 |

Number of pages | 5 |

ISBN (Electronic) | 9781509040964 |

DOIs | |

State | Published - Aug 9 2017 |

Externally published | Yes |

Event | 2017 IEEE International Symposium on Information Theory, ISIT 2017 - Aachen, Germany Duration: Jun 25 2017 → Jun 30 2017 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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ISSN (Print) | 2157-8095 |

### Other

Other | 2017 IEEE International Symposium on Information Theory, ISIT 2017 |
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Country | Germany |

City | Aachen |

Period | 6/25/17 → 6/30/17 |

### Keywords

- Hexagonal lattice
- Independent set
- Transfer matrix

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## Cite this

*2017 IEEE International Symposium on Information Theory, ISIT 2017*(pp. 2910-2914). [8007062] (IEEE International Symposium on Information Theory - Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2017.8007062