Publication: Transfer matrices and partition-function zeros for antiferromagnetic Potts models. V. Further results for the square-lattice chromatic polynomial
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2009-04
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Springer
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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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Chromatic polynomial, Chromatic root, Antiferromagnetic Potts model, Square lattice, Transfer matrix, Fortuin-Kasteleyn representation, Beraha-Kahane-Weiss theorem, Large-q expansion, One-dimensional polymer model, Finite lattice method
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Salas, J. & Sokal, A. D. (2009). Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models V. Further Results for the Square-Lattice Chromatic Polynomial. Journal of Statistical Physics, 135(2), pp. 279–373.