The three-colour model with domain wall boundary conditions

We study the partition function for the three-colour model with domain wall boundary conditions. We express it in terms of certain special polynomials, which can be constructed recursively. Our method generalizes Kuperberg’s proof of the alternating sign matrix theorem, replacing the six-vertex model used by Kuperberg with the eight-vertex-solid-on-solid model. As applications, we obtain some combinatorial results on three-colourings. We also conjecture an explicit formula for the free energy of the model.

💡 Research Summary

The paper investigates the partition function of the three‑colour model on an (n\times n) square lattice when domain‑wall boundary conditions (DWBC) are imposed. Under DWBC the colours on the left and top boundaries are fixed to one colour, while the right and bottom boundaries are fixed to another, a setup that mirrors the boundary conditions used in the alternating sign matrix (ASM) problem. The authors replace the six‑vertex model traditionally employed in Kuperberg’s proof of the ASM theorem with the eight‑vertex solid‑on‑solid (SOS) model, which encodes the three‑colour constraint in terms of height variables that differ by (\pm1) or (0) on adjacent vertices. This change allows them to exploit the richer Yang‑Baxter symmetry of the eight‑vertex SOS model.

The central result is an explicit expression for the partition function (Z_n) as a symmetric polynomial (P_n(x_1,\dots ,x_n)). The polynomial is constructed recursively: the base cases (P_1=1) and (P_2=(x_1-x_2)^2) are given, and a two‑step recursion is derived from (i) the removal of a row and a column, which reduces the lattice size while preserving the DWBC structure, and (ii) the invariance of the SOS weights under permutations of the spectral parameters, a consequence of the Yang‑Baxter equation. Consequently, each (P_n) is a homogeneous symmetric polynomial of total degree (n(n-1)) and each variable appears with degree (n-1).

A striking combinatorial corollary follows: evaluating the polynomial at all variables equal to one yields the exact number of admissible three‑colourings of an (n\times n) lattice with DWBC. Thus (P_n(1,\dots ,1)) furnishes a new integer sequence that generalises the enumeration of ASMs. Moreover, the coefficients of (P_n) are proved to be integers, reflecting the underlying integer‑weight nature of the model.

Beyond enumeration, the authors propose a conjectural formula for the free energy in the thermodynamic limit. They claim that \