Transfer matrices and partition-function zeros for zntiferromagnetic potts models

VV. further results for the square-lattice chromatic polynomial

Jesús Salas, Alan D. Sokal

    Research output: Contribution to journalArticle

    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 languageEnglish (US)
    Pages (from-to)279-373
    Number of pages95
    JournalJournal of Statistical Physics
    Volume135
    Issue number2
    DOIs
    StatePublished - Apr 2009

    Fingerprint

    Chromatic Polynomial
    free boundaries
    Matrix Function
    Potts Model
    Transfer Matrix
    Square Lattice
    Partition Function
    Transfer Function
    partitions
    polynomials
    boundary conditions
    Boundary conditions
    Antiferromagnet
    Zero
    Lattice Model
    Free Boundary
    entry
    Strip
    Free Energy
    strip

    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

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

    In: Journal of Statistical Physics, Vol. 135, No. 2, 04.2009, p. 279-373.

    Research output: Contribution to journalArticle

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