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

TL;DR

This paper presents new structural insights and explicit formulas for transfer matrices of square-lattice Potts models, along with high-order large-q expansions and chromatic root computations for specific strip widths.

Contribution

It provides explicit transfer matrix expressions, high-order free energy expansions, and chromatic root analyses for square-lattice Potts models at zero temperature.

Findings

Explicit transfer matrix diagonal entries for arbitrary widths.

Large-q expansion of free energies up to order q^{-47}.

Chromatic roots and limiting curves for strips of widths 9 to 12.

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.