### Abstract

We study the chromatic polynomial P _{G}(q) for m ≤ n triangular-lattice strips of widths m ≤ 12 _{P}, 9 _{F} (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin-Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m, n → ∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.

Original language | English (US) |
---|---|

Pages (from-to) | 921-1017 |

Number of pages | 97 |

Journal | Journal of Statistical Physics |

Volume | 112 |

Issue number | 5-6 |

DOIs | |

State | Published - Sep 2003 |

### Fingerprint

### Keywords

- Antiferromagnetic Potts model
- Beraha numbers
- Beraha-Kahane-Weiss theorem
- Chromatic polynomial
- Chromatic root
- Fortuin-Kasteleyn representation
- Transfer matrix
- Triangular lattice

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*112*(5-6), 921-1017. https://doi.org/10.1023/A:1024611424456

**Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. III. Triangular-Lattice Chromatic Polynomial.** / Jacobsen, Jesper Lykke; Salas, Jesús; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 112, no. 5-6, pp. 921-1017. https://doi.org/10.1023/A:1024611424456

}

TY - JOUR

T1 - Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. III. Triangular-Lattice Chromatic Polynomial

AU - Jacobsen, Jesper Lykke

AU - Salas, Jesús

AU - Sokal, Alan D.

PY - 2003/9

Y1 - 2003/9

N2 - We study the chromatic polynomial P G(q) for m ≤ n triangular-lattice strips of widths m ≤ 12 P, 9 F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin-Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m, n → ∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.

AB - We study the chromatic polynomial P G(q) for m ≤ n triangular-lattice strips of widths m ≤ 12 P, 9 F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin-Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m, n → ∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.

KW - Antiferromagnetic Potts model

KW - Beraha numbers

KW - Beraha-Kahane-Weiss theorem

KW - Chromatic polynomial

KW - Chromatic root

KW - Fortuin-Kasteleyn representation

KW - Transfer matrix

KW - Triangular lattice

UR - http://www.scopus.com/inward/record.url?scp=0038167272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038167272&partnerID=8YFLogxK

U2 - 10.1023/A:1024611424456

DO - 10.1023/A:1024611424456

M3 - Article

VL - 112

SP - 921

EP - 1017

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -