Transfer matrices and partition-function zeros for antiferromagnetic Potts models. V. Further results for the square-lattice chromatic polynomial Articles uri icon

publication date

  • April 2009

start page

  • 279

end page

  • 373


  • 2


  • 135

International Standard Serial Number (ISSN)

  • 0022-4715

Electronic International Standard Serial Number (EISSN)

  • 1572-9613


  • 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


  • 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