### Abstract

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q ^{-47} (resp. q ^{-46}). Finally, we compute chromatic roots for strips of widths 12 with free boundary conditions and locate roughly the limiting curves.

Original language | English (US) |
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Pages (from-to) | 279-373 |

Number of pages | 95 |

Journal | Journal of Statistical Physics |

Volume | 135 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2009 |

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### Keywords

- Antiferromagnetic Potts model
- Beraha-Kahane-Weiss theorem
- Chromatic polynomial
- Chromatic root
- Finite-lattice method
- Fortuin-Kasteleyn representation
- Large-q expansion
- One-dimensional polymer model
- Square lattice
- Transfer matrix

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*135*(2), 279-373. https://doi.org/10.1007/s10955-009-9725-1

**Transfer matrices and partition-function zeros for zntiferromagnetic potts models : VV. further results for the square-lattice chromatic polynomial.** / Salas, Jesús; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 135, no. 2, pp. 279-373. https://doi.org/10.1007/s10955-009-9725-1

}

TY - JOUR

T1 - Transfer matrices and partition-function zeros for zntiferromagnetic potts models

T2 - VV. further results for the square-lattice chromatic polynomial

AU - Salas, Jesús

AU - Sokal, Alan D.

PY - 2009/4

Y1 - 2009/4

N2 - We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q -47 (resp. q -46). Finally, we compute chromatic roots for strips of widths 12 with free boundary conditions and locate roughly the limiting curves.

AB - We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q -47 (resp. q -46). Finally, we compute chromatic roots for strips of widths 12 with free boundary conditions and locate roughly the limiting curves.

KW - Antiferromagnetic Potts model

KW - Beraha-Kahane-Weiss theorem

KW - Chromatic polynomial

KW - Chromatic root

KW - Finite-lattice method

KW - Fortuin-Kasteleyn representation

KW - Large-q expansion

KW - One-dimensional polymer model

KW - Square lattice

KW - Transfer matrix

UR - http://www.scopus.com/inward/record.url?scp=67349158736&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349158736&partnerID=8YFLogxK

U2 - 10.1007/s10955-009-9725-1

DO - 10.1007/s10955-009-9725-1

M3 - Article

VL - 135

SP - 279

EP - 373

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -