Explicit closed-form evaluation of the partial partition functions of the three-coloring model with domain wall boundary conditions in the general elliptic case

Razumov and Stroganov (2008) proved that the partial partition functions of the three-coloring statistical model with domain wall boundary conditions satisfy three-term theta-functional equations, and solved them in a nontrivial trigonometric limit, obtaining a determinant representation built from two alternating-power determinants with explicit scalar prefactors. They closed by posing, as a challenging open problem, the corresponding explicit expression for the general elliptic case (generic nome $p\neq0$). The subsequent literature establishes that the total elliptic domain-wall partition function of the parent eight-vertex-solid-on-solid model is known (Rosengren's Izergin-Korepin-type identity; Galléas's single-determinant representations), and the homogeneous three-colour partition function is known through recursive polynomials and Hankel determinants, but the three $\mathbb{Z}_3$-color sectors individually have no closed form at generic elliptic parameters, and the constructive solution of the elliptic functional equations remains unavailable. The problem remains open, with a sharp surviving core.

Background

The three-coloring model of Baxter (1970) is a lattice statistical model whose states are colorings of the faces of a square grid with three colors such that adjacent faces always carry different colors; Baxter computed its free energy on the torus by relating it to the zero-field eight-vertex model. Razumov and Stroganov (2008-2009) initiated the study of this model under domain wall boundary conditions (DWBC), the boundary setup in which the six-vertex model is exactly solved by the Izergin-Korepin determinant (a closed determinant formula for the inhomogeneous six-vertex model partition function on an $n\times n$ square with all boundary arrows pointing inward). For the three-coloring model, DWBC means that walking anticlockwise along the boundary the color changes by $+1$ from face to face along the vertical sides and by $-1$ along the horizontal sides; the admissible states of an $(n+1)\times(n+1)$ grid of faces are in bijection with alternating sign matrices (square matrices with entries $0,\pm1$ whose nonzero entries alternate in sign and sum to one in each row and column). With Stroganov's integrable weights, the Boltzmann weight of each of the six local vertex configurations is built from Jacobi theta functions of nome $p=e^{i\pi\tau}$ (with $\mathrm{Im}\,\tau>0$, so $p=0$ is the trigonometric limit), and depends on the spectral parameter $\chi_i-\psi_j$ attached to the crossing of horizontal line $i$ and vertical line $j$, on a fixed parameter $\lambda$, and on a color index $r\in\mathbb{Z}_3$. The total partition function decomposes over the residual $\mathbb{Z}_3$ symmetry as $Z_n=\sum_{r\in\mathbb{Z}_3}Z_n^r$, where the partial partition function $Z_n^r$ sums the states of one boundary-color sector. Part I of the series proved that the normalized functions $F_n^r$ (the sectors divided by explicit theta-function factors) satisfy three-term theta-functional equations such as $\sum_{s\in\mathbb{Z}_3}F_n^{r+s}(\chi_1,\ldots,\chi_k+2\pi s/3,\ldots,\chi_n;\{\psi\})=0$, together with the analogous equations in the $\psi$ variables. Part II solved these equations in a nontrivial trigonometric limit ($p\to0$ after the shift $\lambda\to\lambda+\pi\tau/2$): after a discrete Fourier transform in the sector index, the transformants are proportional to two alternating-power determinants $\mathcal{P}_n$ and $\mathcal{Q}_n$ in the $2n$ interlaced variables $u_1,\ldots,u_{2n}$, and each partial partition function is an explicit linear combination of $\mathcal{P}_n$ and $\mathcal{Q}_n$ with scalar prefactors $A_n^r,B_n^r,C_n^r$ fixed by recursion relations. In the naive trigonometric limit each sector $Z_n^r$ separately coincides with the six-vertex DWBC partition function at crossing parameter $2\pi/3$, that is, with the Izergin-Korepin determinant at a third root of unity (Stroganov). The model is a special, root-of-unity-anisotropy case of Baxter's eight-vertex-solid-on-solid (8VSOS) model, an ice-type model whose elliptic weights satisfy the dynamical Yang-Baxter equation (a Yang-Baxter equation with an extra shift parameter, the dynamical parameter $\lambda$); Rosengren obtained an Izergin-Korepin-type identity, a sum of Frobenius-type determinants, for the total 8VSOS DWBC partition function at generic elliptic parameters, and Galléas later produced continuous families of single-determinant representations of that total partition function. All of these results concern the total partition function; they do not resolve the three color sectors individually, and no elliptic analogue of the $\mathcal{P}_n/\mathcal{Q}_n$ sector formulas is known.

Problem Statement

Find an explicit closed-form expression for the partial partition functions $Z_n^r(\{\chi\};\{\psi\})$ (equivalently, for the normalized functions $F_n^r$) of the three-coloring statistical model with domain wall boundary conditions and Stroganov's elliptic vertex weights, in the general elliptic case of generic nome $p\neq0$, generic fixed parameter $\lambda$, and generic inhomogeneous spectral parameters $\chi_i,\psi_j$; this is the problem posed in the closing sentence of Razumov and Stroganov, Part II. The expression is the one 'corresponding' to the trigonometric result in the following determinate sense: (a) it must be given for every lattice size $n$ and every sector $r\in\mathbb{Z}_3$ individually, with the dependence on $\lambda$ and the nome $p$ explicit; (b) in the degeneration $p\to0$ (after the substitution $\lambda\to\lambda+\pi\tau/2$) it must reduce to the trigonometric representation of Part II, that is, to linear combinations of the determinants $\mathcal{P}_n$ and $\mathcal{Q}_n$ with the scalar prefactors $A_n^r,B_n^r,C_n^r$; and (c) in the naive trigonometric limit each sector must reproduce the Izergin-Korepin determinant at a third root of unity. An equivalent and equally acceptable route is a constructive solution of the elliptic three-term theta-functional equations satisfied by $F_n^r$ (Part I), from which such closed forms are extracted. Determinants, pfaffians, elliptic hypergeometric sums, or any other explicit special-function form are admissible, provided each color sector is obtained individually and the sum $\sum_{r\in\mathbb{Z}_3}Z_n^r$ is consistent with the known 8VSOS domain-wall partition function identities.

References

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