### Abstract

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P_{G}(q) for m×n rectangular subsets of the square lattice, with m≤8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers B_{2}, B_{3}, B_{4}, A_{5} are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B_{10}) cannot be a chromatic root of any graph.

Original language | English (US) |
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Pages (from-to) | 609-699 |

Number of pages | 91 |

Journal | Journal of Statistical Physics |

Volume | 104 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 2001 |

### Keywords

- Antiferromagnetic Potts model
- Beraha numbers
- Beraha-Kahane-Weiss theorem
- Chromatic polynomial
- Chromatic root
- Fortuin-Kasteleyn representation
- Square lattice
- Temperley-Lieb algebra
- Transfer matrix

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Statistical Physics*,

*104*(3-4), 609-699. https://doi.org/10.1023/A:1010376605067