### Abstract

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m×n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B∞(sq) for this model with ordinary (e. g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.

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

Pages (from-to) | 1028-1122 |

Number of pages | 95 |

Journal | Journal of Statistical Physics |

Volume | 144 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2011 |

### Fingerprint

### Keywords

- Beraha-Kahane-Weiss theorem
- Chromatic polynomial
- Chromatic roots
- Extra-vertex boundary conditions
- Planar graph
- Potts model
- Square lattice
- Transfer matrix
- Tutte polynomial

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*144*(5), 1028-1122. https://doi.org/10.1007/s10955-011-0292-x

**Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions.** / Salas, Jesús; Sokal, Alan D.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 144, no. 5, pp. 1028-1122. https://doi.org/10.1007/s10955-011-0292-x

}

TY - JOUR

T1 - Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models VI. Square Lattice with Extra-Vertex Boundary Conditions

AU - Salas, Jesús

AU - Sokal, Alan D.

PY - 2011/9

Y1 - 2011/9

N2 - We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m×n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B∞(sq) for this model with ordinary (e. g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.

AB - We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the boundary conditions that are obtained from an m×n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B∞(sq) for this model with ordinary (e. g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.

KW - Beraha-Kahane-Weiss theorem

KW - Chromatic polynomial

KW - Chromatic roots

KW - Extra-vertex boundary conditions

KW - Planar graph

KW - Potts model

KW - Square lattice

KW - Transfer matrix

KW - Tutte polynomial

UR - http://www.scopus.com/inward/record.url?scp=80052735319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052735319&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0292-x

DO - 10.1007/s10955-011-0292-x

M3 - Article

VL - 144

SP - 1028

EP - 1122

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5

ER -