Undetermined multiplicative constants in the large-$N$ asymptotics of the six-vertex model with domain wall boundary conditions

On the critical line between the disordered and antiferroelectric phases of the six-vertex model with domain wall boundary conditions (weights $a=1-x$, $b=1+x$, $c=2$, $|x|<1$, i.e. $a+b=c$), the partition function satisfies the rigorous large-$N$ asymptotics $Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$ with $F=\frac{\pi(1-x^2)}{2\cos\frac{\pi x}{2}}$ and $C=C_0(\cos\frac{\pi x}{2})^{1/12}$: the $x$-dependence of $C$ is fixed by the Toda equation, but the universal constant $C_0>0$ is undetermined — no closed form and no numerical value is known. The same constant-factor problem is open for the companion scale constants $C_0(\gamma)$ (disordered region) and $C(\gamma)$ (antiferroelectric region), and the higher-order expansion coefficients (such as $c_1(x)$ in the expansion of the orthogonal-polynomial normalizing constants $h_N$) are uncomputed. The problem is the rigorous determination of these constants and coefficients, completing the multiplicative completion of the exact-asymptotics program for the six-vertex model with domain wall boundary conditions; of the phase-region constants only the ferroelectric one (explicit) and the ferroelectric–disordered line constant (computed in 2014) are known. None of these constants or coefficients has been determined in the literature.

Background

The six-vertex model is a two-dimensional lattice model of statistical mechanics in which arrows are placed on the edges of an $N\times N$ square lattice subject to the ice rule — at every vertex exactly two arrows point in and two point out — leaving six allowed vertex configurations, grouped into three classes with Boltzmann weights $a$, $b$, $c$. With domain wall boundary conditions (DWBC), all boundary arrows on the left and top edges point into the square and all on the right and bottom edges point out; this is the natural boundary for the Izergin–Korepin formula, which expresses the partition function $Z_N$ as a closed finite determinant. The bulk phase structure is governed by the anisotropy parameter $\Delta=(a^2+b^2-c^2)/(2ab)$: the model is antiferroelectric for $a+b<c$, disordered for $|a-b|<c<a+b$, and ferroelectric for $|a-b|>c$.

Zinn-Justin showed that the DWBC partition function can be written as a Hankel determinant (a determinant of moments $\int z^{j+k}w(z)\,dz$ of a weight function $w$), so that $Z_N$ reduces to a tau-function (a normalization-dependent generating function) of a system of orthogonal polynomials. This opened the way to rigorous asymptotics via the Deift–Zhou nonlinear steepest descent method — the asymptotic solution of the Riemann–Hilbert problem (a boundary-value problem for analytic matrix-valued functions with prescribed jumps) encoding the orthogonal polynomials — combined with the Toda equation, a difference equation relating the tau-functions at lattice sizes $N-1$, $N$ and $N+1$, which constrains the dependence of multiplicative constants on the weights. By this route Bleher and Fokin (disordered region) and Bleher and Liechty (ferroelectric, antiferroelectric, and the ferroelectric–disordered critical line) proved the large-$N$ asymptotics of $Z_N$ in every phase region, confirming conjectures of Zinn-Justin and making rigorous the physics results of Bogoliubov, Kitaev and Zvonarev.

On the critical line between the disordered and antiferroelectric phases one has $a+b=c$, i.e. $\Delta=-1$; parameterizing $a=1-x$, $b=1+x$, $c=2$ with $|x|<1$, this line contains the transition point of the F-model (the antiferroelectric six-vertex model with $a=b$, at $a=b=1$, $c=2$), which undergoes an infinite-order, Kosterlitz–Thouless-type phase transition. Bleher and Bothner proved that, as $N\to\infty$,

$Z_N = C\,F^{N^2}\,N^{1/12}\left(1+O(N^{-1})\right),\qquad F=\frac{\pi(1-x^2)}{2\cos\frac{\pi x}{2}},\qquad C=C_0\left(\cos\frac{\pi x}{2}\right)^{1/12},$

with the error uniform on compact subsets of $|x|<1$; the finite-size exponent $1/12$ characterizes the transition. The $x$-dependence of $C$ follows from the Toda equation, but the overall scale — the universal constant $C_0>0$ — is not determined by the method: neither a closed-form expression nor a numerical value is known. The situation is the same for the companion asymptotics in the disordered region, $Z_N=CF^{N^2}N^{\kappa}(1+O(N^{-\varepsilon}))$ with $\kappa=\frac{1}{12}-\frac{2\gamma^2}{3\pi(\pi-2\gamma)}$ and $C=C_0(\gamma)(\cos\frac{\pi t}{2\gamma})^{\kappa}$, where $\gamma,t$ parametrize the disordered weights $a=\sin(\gamma-t)$, $b=\sin(\gamma+t)$, $c=\sin(2\gamma)$ with $0\le|t|<\gamma<\pi/2$, and in the antiferroelectric region, $Z_N=CF^{N^2}\vartheta_4(N\omega)(1+O(N^{-1}))$ with $\vartheta_4$ a Jacobi theta function, where the constants $C_0(\gamma)$ and $C(\gamma)$ are likewise known only structurally. By contrast, in the ferroelectric region the constant is explicit, $C=1-e^{-4\gamma}$, and on the ferroelectric–disordered critical line Bleher and Bothner (2014) did compute the constant, $C=e^{c}(\alpha-1)^{1/4}$ with $c=0.4616571\ldots$, by evaluating a double-scaling limit (in which $N$ and the line parameter $\alpha$ tend jointly to infinity) and combining it with the structural form imposed by the Toda equation. That 2014 paper states plainly that the antiferroelectric, disordered, and disordered–antiferroelectric constants are not known, and that the calculation of constant factors in this model is "a notoriously difficult problem". The 2012 source analysis also stops short of the higher-order expansion: in the asymptotics of the orthogonal-polynomial normalizing constants $h_N$ (the squared norms of the polynomials) on the critical line,

$h_N=\frac{\pi}{2}\left(\frac{2\pi N}{\cos\frac{\pi x}{2}}\right)^{2N+1}\frac{e^{-2N}}{16^{N}}\left[1+\frac{1}{4N}+\frac{(-1)^{N}}{2}\cos\left(\pi x\left(N+\frac{1}{2}\right)\right)A(N)+O(N^{-2})\right],$

the small oscillatory function $A(N)=A(N;x)$ is expanded as

$A(N)=\frac{1}{N(\ln N)^{2}}\left\{1-\frac{2\ln\ln N}{\ln N}+\frac{c_0(x)}{\ln N}+O\left(\left(\frac{\ln\ln N}{\ln N}\right)^{2}\right)\right\},$

and the full expansion of $h_N$ contains, in addition, a separate non-oscillatory term $\frac{c_1(x)}{N^{2}}$. The coefficient $c_0(x)=1-2\gamma_{\rm E}-4\ln 2-2\ln\cos\frac{\pi x}{2}$ (with Euler's constant $\gamma_{\rm E}$) of the $1/\ln N$ order inside the oscillatory bracket is computed, but $c_1(x)$ is not, and $A(N)$ is derived only to the orders displayed.

Problem Statement

Rigorously determine the undetermined multiplicative constants — and the missing higher-order coefficients — in the large-$N$ asymptotics of the six-vertex model with domain wall boundary conditions. These form one acknowledged constant-factor problem for the same partition function $Z_N$ (given by the Izergin–Korepin determinant), with tiers (1) and (2) constituting the multiplicative completion of the $Z_N$ asymptotics across the phase diagram and tier (3) the companion higher-order expansion on the critical line; a complete answer requires all three tiers, and a rigorous determination of any single tier is an accepted partial solution:

(1) The universal constant $C_0>0$ on the critical line between the disordered and antiferroelectric phases (weights $a=1-x$, $b=1+x$, $c=2$, $|x|<1$), which enters

$Z_N=C_0\left(\cos\frac{\pi x}{2}\right)^{1/12}\left[\frac{\pi(1-x^2)}{2\cos\frac{\pi x}{2}}\right]^{N^2}N^{1/12}\left(1+O(N^{-1})\right).$

Determine its closed-form expression, or at minimum a rigorous numerical determination with controlled error bounds, under the same normalization as the source theorem (the Izergin–Korepin determinant with these weights).

(2) The companion scale constants of the same problem: the function $C_0(\gamma)>0$ in the disordered-region asymptotics $Z_N=C_0(\gamma)\left(\cos\frac{\pi t}{2\gamma}\right)^{\kappa}F^{N^2}N^{\kappa}(1+O(N^{-\varepsilon}))$ (disordered weights $a=\sin(\gamma-t)$, $b=\sin(\gamma+t)$, $c=\sin(2\gamma)$, $0\le|t|<\gamma<\pi/2$, some $\varepsilon>0$, $\kappa=\frac{1}{12}-\frac{2\gamma^2}{3\pi(\pi-2\gamma)}$, and $F=\frac{\pi(\sin(\gamma-t)\sin(\gamma+t))}{2\gamma\cos\frac{\pi t}{2\gamma}}$; note that $C_0(\gamma)$ is a function of $\gamma$ — a distinct object from the universal constant $C_0$ of tier (1) despite the shared base symbol) and the constant $C(\gamma)>0$ in the antiferroelectric-region asymptotics $Z_N=C(\gamma)F^{N^2}\vartheta_4(N\omega)(1+O(N^{-1}))$ (antiferroelectric weights $a=\sinh(\gamma-t)$, $b=\sinh(\gamma+t)$, $c=\sinh(2\gamma)$, $0\le|t|<\gamma$, with $\omega=\frac{\pi}{2}\left(1+\frac{t}{\gamma}\right)$, $F=\frac{\pi(\sinh(\gamma-t)\sinh(\gamma+t))\vartheta_1'(0)}{2\gamma\vartheta_1(\omega)}$, and $\vartheta_1(z)=\vartheta_1(z|q)$, $\vartheta_4(z)=\vartheta_4(z|q)$ the Jacobi theta functions with elliptic nome $q=e^{-\pi^2/(2\gamma)}$; $C(\gamma)$ does not depend on $t$), each currently known only structurally.

(3) The higher-order coefficients not computed in the source analysis, needed for the higher-order expansion on the critical line: the coefficient function $c_1(x)$ multiplying the separate non-oscillatory $\frac{1}{N^2}$ term in the $h_N$ expansion displayed in the Background (of which only $c_0(x)$, the coefficient of the $1/\ln N$ order inside the oscillatory bracket, is known), and the further terms of the $1/\ln N$-type expansion of the small oscillatory function $A(N;x)$ beyond those displayed in the Background, up to and including the order $\left(\frac{\ln\ln N}{\ln N}\right)^{2}$.

The completed instances of the same program — the explicit ferroelectric constant and the ferroelectric–disordered line constant computed by Bleher and Bothner in 2014 — show what a solution looks like; a full answer closes the program, and progress on any tier is meaningful partial progress toward the same objective.

References

1. Pavel Bleher and Thomas Bothner, "Exact solution of the six-vertex model with domain wall boundary conditions. Critical line between disordered and antiferroelectric phases", Random Matrices: Theory and Applications 1 (2012) 1250012. DOI: 10.1142/S2010326312500128, https://doi.org/10.1142/S2010326312500128; preprint arXiv:1208.6276, https://arxiv.org/abs/1208.6276
2. Pavel Bleher and Thomas Bothner, "Calculation of the constant factor in the six-vertex model", Annales de l'Institut Henri Poincaré D 1 (2014) 363–427. DOI: 10.4171/AIHPD/11, https://doi.org/10.4171/AIHPD/11; preprint arXiv:1306.3510, https://arxiv.org/abs/1306.3510
3. Pavel M. Bleher and Vladimir V. Fokin, "Exact solution of the six-vertex model with domain wall boundary conditions. Disordered phase", Communications in Mathematical Physics 268 (2006) 223–284. DOI: 10.1007/s00220-006-0097-y, https://doi.org/10.1007/s00220-006-0097-y
4. Pavel Bleher and Karl Liechty, "Exact solution of the six-vertex model with domain wall boundary conditions. Antiferroelectric phase", Communications on Pure and Applied Mathematics 63 (2010) 779–829. DOI: 10.1002/cpa.20311, https://doi.org/10.1002/cpa.20311
5. Pavel Bleher and Karl Liechty, "Exact solution of the six-vertex model with domain wall boundary conditions. Ferroelectric phase", Communications in Mathematical Physics 286 (2009) 777–801. DOI: 10.1007/s00220-008-0709-9, https://doi.org/10.1007/s00220-008-0709-9
6. Pavel Bleher and Karl Liechty, "Exact solution of the six-vertex model with domain wall boundary conditions. Critical line between ferroelectric and disordered phases", Journal of Statistical Physics 134 (2009) 463–485. DOI: 10.1007/s10955-009-9688-2, https://doi.org/10.1007/s10955-009-9688-2
7. Pavel Bleher and Karl Liechty, "Domain wall six-vertex model with half-turn symmetry", Constructive Approximation 47 (2018) 141–162. DOI: 10.1007/s00365-017-9405-3, https://doi.org/10.1007/s00365-017-9405-3; preprint arXiv:1702.01190, https://arxiv.org/abs/1702.01190
8. P. Zinn-Justin, "Six-vertex model with domain wall boundary conditions and one-matrix model", Physical Review E 62 (2000) 3411–3418. DOI: 10.1103/PhysRevE.62.3411, https://doi.org/10.1103/PhysRevE.62.3411
9. N. M. Bogoliubov, A. V. Kitaev and M. B. Zvonarev, "Boundary polarization in the six-vertex model", Physical Review E 65 (2002) 026126. DOI: 10.1103/PhysRevE.65.026126, https://doi.org/10.1103/PhysRevE.65.026126
10. Alexander Glazman and Ron Peled, "On the transition between the disordered and antiferroelectric phases of the 6-vertex model", Electronic Journal of Probability 28 (2023) paper no. 98. DOI: 10.1214/23-EJP980, https://doi.org/10.1214/23-EJP980; preprint arXiv:1909.03436, https://arxiv.org/abs/1909.03436
11. Rick Keesman and Jules Lamers, "Numerical study of the F-model with domain-wall boundaries", Physical Review E 95 (2017) 052117. DOI: 10.1103/PhysRevE.95.052117, https://doi.org/10.1103/PhysRevE.95.052117; preprint arXiv:1702.05474, https://arxiv.org/abs/1702.05474
12. Vadim Gorin and Karl Liechty, "Boundary statistics for the six-vertex model with DWBC", Communications on Pure and Applied Mathematics 78 (2025) 1847–1948. DOI: 10.1002/cpa.22254, https://doi.org/10.1002/cpa.22254; preprint arXiv:2310.12735, https://arxiv.org/abs/2310.12735
13. Jan de Gier, Richard Kenyon, Michael Wheeler and Yuyang Zhou, "The asymmetric five vertex model on a rectangle", arXiv:2606.22004 (2026), https://arxiv.org/abs/2606.22004
14. Jean-Marie Stéphan, "Two dimensional inhomogeneous classical systems at criticality", arXiv:2608.02903 (2026), https://arxiv.org/abs/2608.02903
15. Kohei Motegi, "Factorization of rational six vertex model partition functions", Nuclear Physics B 1009 (2024) 116743. DOI: 10.1016/j.nuclphysb.2024.116743, https://doi.org/10.1016/j.nuclphysb.2024.116743; preprint arXiv:2410.03153, https://arxiv.org/abs/2410.03153