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Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

V. Further Results for the Square-Lattice Chromatic Polynomial

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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 9≤m≤12 with free boundary conditions and locate roughly the limiting curves.

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Authors and Affiliations

  1. Instituto Gregorio Millán and Grupo de Modelización, Simulación Numérica y Matemática Industrial, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911, Leganés, Spain

    Jesús Salas

  2. Department of Physics, New York University, 4 Washington Place, New York, NY, 10003, USA

    Alan D. Sokal

  3. Department of Mathematics, University College London, London, WC1E 6BT, England, UK

    Alan D. Sokal

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  1. Jesús Salas
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  2. Alan D. Sokal
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Salas, J., Sokal, A.D. Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. J Stat Phys 135, 279–373 (2009). https://doi.org/10.1007/s10955-009-9725-1

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