Open problems in the rigorous free-energy computation of the six-vertex model: analyticity of condensed Bethe solutions at all densities, optimal half-filling asymptotics, asymmetric weights, and full-range rotational invariance

Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia (Comm. Math. Phys. 2022) gave a fully rigorous computation of the free energy of the symmetric six-vertex model on the torus for anisotropy $\Delta<1$, through existence, condensation, and analyticity of solutions of the discrete Bethe equations and identification of the Bethe-ansatz eigenvalue as the Perron-Frobenius eigenvalue of the transfer matrix. Their methods leave four issues open, stated at four separate places in the paper (the inability remark after Theorem 4, the nonlinear-integral-equation barrier of Section 1.3, the future-paper deferral of Section 1.1, and the unproven rotational-invariance conjecture of Section 1.2): (i) existence of an analytic family of condensed Bethe solutions for arbitrary arrow density $n\le N/2$ over the whole intervals $(-\infty,-1)$ and $(-1,1)$, linked to the expected infinite-order transition at $\Delta=-1$; (ii) optimal higher-order asymptotics for linear statistics of Bethe roots at exact half filling $n/N=1/2$ for $-1\le\Delta<1$, where nonlinear-integral-equation techniques stop at $o(1)$; (iii) extension of the paper's results to fully asymmetric vertex weights, which the source defers to a future paper; and (iv) a rigorous proof that the scaling limit of the height function is rotation invariant for every $c\in[1,2]$, beyond the range $\sqrt3\le c\le2$ handled by the companion work. The sub-range $c\in[\sqrt3,2]$ is settled up to Gaussian-free-field convergence (2026), leaving $c\in[1,\sqrt3)$, equivalently $-1/2<\Delta\le1/2$, as the open core.

Background

The six-vertex model (Pauling's ice-type model) assigns an orientation to every edge of the square lattice subject to the ice rule: at each vertex exactly two arrows point in and two point out, giving six allowed local configurations with weights $a_1,a_2,b_1,b_2,c_1,c_2$. On the $N\times M$ torus with periodic boundary conditions and exactly $n$ up arrows per row, the partition function is governed by the transfer matrix $V_N$ (the operator adding one row), a $2^N$-dimensional positive matrix whose largest eigenvalue determines the free energy per site in the thermodynamic limit; by the Perron-Frobenius theorem this largest eigenvalue is unique with a strictly positive eigenvector. The Bethe ansatz (Lieb's 1967 solution of the ice-type phases) diagonalizes $V_N$: eigenvectors are labelled by roots $\lambda_1,\dots,\lambda_n$ of the discrete Bethe equations, and the choice corresponding to the ground state — conjectured by Hulthén and proved by Yang and Yang — gives the Perron-Frobenius eigenvalue. Throughout, the symmetric case $a_1=a_2=a$, $b_1=b_2=b$, $c_1=c_2=c$ is considered, with anisotropy parameter $\Delta=(a^2+b^2-c^2)/(2ab)$: $\Delta>1$ is the ferroelectric phase, $\Delta<-1$ the anti-ferroelectric phase, and $-1\le\Delta<1$ a massless (disordered) regime, separated by an expected infinite-order, Kosterlitz-Thouless-type transition at $\Delta=-1$.

Duminil-Copin, Kozlowski, Krachun, Manolescu and Tikhonovskaia (2022) proved, for all $n\le N/2$ and $\Delta<1$: existence and uniqueness of condensed symmetric strictly ordered solutions of the discrete Bethe equations (condensation meaning that the empirical measure of the roots converges to the solution $\rho(\cdot|q)$ of a linear integral equation, the continuum Bethe equation); analyticity of the solution family in $\Delta$ on sub-intervals (their Theorem 4: $(\Delta_0,1)$ for $\Delta_0>-1$ with $n\le N/2-C_0(\Delta_0)$, and $(-\infty,\Delta_0)$ for $\Delta_0<-1$ with all $n\le N/2$); a family analytic on the whole interval $(-1,1)$ for densities $n\le N/2-C_1(\log N)^2$ (their Theorem 6); identification of the resulting eigenvalue as the Perron-Frobenius eigenvalue; and, as consequences, a short rigorous computation of the free energy and asymptotic expansions of the partition functions as the density $m=n/N$ tends to $1/2$, with error $O(1/N)$ for $-1<\Delta<1$ and $O((\log N)/N)$ at $\Delta=-1$. The half-filling limit is the physically critical input: it underpins localization/delocalization results for the six-vertex height function and rotational-invariance results for critical random-cluster models. The sharpest previous technique, rigorous nonlinear integral equations (NLIE) for the counting function of the Bethe roots (an auxiliary monotone function whose derivative encodes the local density of roots), developed by Kozlowski (2017) — a rigorous form of the thermodynamic Bethe ansatz which yields complete $1/N$-expansions for $\Delta<-1$ at all $m\in[0,1/2]$ and for $-1\le\Delta<1$ at $m<1/2$ — provably cannot reach beyond $o(1)$ for linear statistics $\sum_j f(\lambda_j)-\int f(\rho)\,d\rho$ at $m=1/2$, $-1\le\Delta<1$, essentially because the Fermi boundary $Q(1/2)$ is infinite for $|\Delta|\le1$, so the relevant compactness is lost.

The height function $h$ maps a six-vertex arrow configuration to an integer-valued function on the faces of the lattice (height increases by one to the left of each arrow). For $a=b=1$ it is localized (bounded variance of increments) when $c>2$ and delocalized (logarithmically growing variance) when $1\le c\le2$, i.e. $-1\le\Delta\le1/2$. In the delocalized regime $h$ is expected to converge, after diffusive rescaling, to a multiple of the Gaussian free field (GFF), the universal massless Gaussian scaling limit; since the GFF is rotation invariant, this predicts rotational invariance of all large-scale correlations of $h$ for every $c\in[1,2]$, and its failure for $c>2$. The companion paper (Duminil-Copin, Kozlowski, Krachun, Manolescu, Oulamara) proved large-scale rotational invariance for the critical random-cluster model (Fortuin-Kasteleyn percolation with cluster weight $q\in[1,4]$) and, via the Baxter-Kelland-Wu correspondence (the exact solvability identification between the six-vertex and random-cluster models) — which matches that range of $q$ to $c\in[\sqrt3,2]$ — combined with the FKG lattice condition (positive association), for the six-vertex height function on $\sqrt3\le c\le2$. The transfer breaks down below $c=\sqrt3$, where no FK representation with positive association is available. In 2026 the same circle settled convergence of the zero-slope (zero average macroscopic slope, i.e. exactly balanced arrow densities) height function to the GFF for $c\in[\sqrt3,2]$ (including anisotropic $a\ne b$), explicitly flagging rotational invariance on the remaining range as the one missing input.

Problem Statement

Resolve the following open issues left by the rigorous Bethe-ansatz computation of the six-vertex model's free energy (Duminil-Copin, Kozlowski, Krachun, Manolescu, Tikhonovskaia, Comm. Math. Phys. 2022), stated by the source at four separate places; each component (A)-(D) is an independent open target, and each is accepted separately: a complete resolution of any one of them suffices, and the four need not all be resolved.

(A) Analyticity at all densities: prove (or disprove) that there exists $N_0$ such that for every $N\ge N_0$ and every arrow density $n\le N/2$ there is a family of condensed symmetric strictly ordered solutions of the discrete Bethe equations which is analytic in $\Delta$ throughout each of the whole intervals $(-\infty,-1)$ and $(-1,1)$; that is, close the two gaps of the source's Theorem 4 — the behavior as $\Delta\to-1^-$ at fixed $n\le N/2$, and the case of $n$ within $O((\log N)^2)$ of $N/2$ for $-1<\Delta<1$, in view of the source's Theorem 6, which already provides a family analytic on the whole interval $(-1,1)$ for $n\le N/2-C_1(\log N)^2$ — and clarify the relation to the expected infinite-order phase transition of the model at $\Delta=-1$.

(B) Optimal asymptotics at half filling: develop techniques that improve and unify the control of linear statistics of the counting measure of the Bethe roots, $\sum_{j=1}^n f(\lambda^{(n)}_j)-\int_{-Q}^{Q} f(\mu)\,\rho(\mu|Q)\,d\mu$ with $Q=Q(n/N)$ the Fermi boundary (the half-width of the support of the limiting root density $\rho(\cdot|Q)$, the solution of the continuum Bethe equation, fixed by the density $m=n/N$), at exact half filling $m=n/N=1/2$ for $-1\le\Delta<1$, beyond the currently available $O(1/N)$ (for $-1<\Delta<1$) and $O((\log N)/N)$ (at $\Delta=-1$) bounds, to the full all-order asymptotic expansion in $N$ that the nonlinear-integral-equation method provides for each fixed $\Delta<-1$ with $m\in[0,1/2]$ and for each fixed $-1\le\Delta<1$ with $m\in[0,1/2)$.

(C) Asymmetric weights: establish which of the paper's results — existence and condensation of Bethe roots, identification of the Bethe-ansatz eigenvalue as the Perron-Frobenius eigenvalue of the transfer matrix, and the resulting rigorous free-energy computation — extend to the asymmetric six-vertex model with general weights $a_1,a_2,b_1,b_2,c_1,c_2\ge0$ ($a_1\ne a_2$, $b_1\ne b_2$, $c_1\ne c_2$ in general), which the source defers to a future paper ("Some of the results of this paper may extend to the asymmetric case and will be the object of a future paper", Section 1.1).

(D) Full-range rotational invariance: prove (or disprove) that the scaling limit of the height function of the six-vertex model with $a=b=1$ is rotation invariant for every $c\in[1,2]$ (equivalently $\Delta\in[-1,1/2]$). The sub-range $c\in[\sqrt3,2]$ is settled; the open core is $1\le c<\sqrt3$, with the conjecture extending to all $c\in[1,2]$ and failing for $c>2$ where the height function localizes.

References

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