Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

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.

111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phys

References in corpus (5)

• Phase diagram of the chromatic polynomial on a torus
• Partition function zeros at first-order phase transitions: A general analysis
• Partition function zeros at first-order phase transitions: Pirogov-Sinai theory
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
• Phase transition in the 3-state Potts antiferromagnet on the diced lattice

Cited by in corpus (8)

• Logarithmic Minimal Models
• Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
• Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions
• Phase diagram of the triangular-lattice Potts antiferromagnet
• A generalized Beraha conjecture for non-planar graphs
• Potts model on recursive lattices: some new exact results
• The phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet
• Parallel family trees for transfer matrices in the Potts model