Generic singularities of six-vertex limit shapes and the fluctuation processes at them

For the six-vertex model with fixed (domain-wall-type) boundary conditions, Palamarchuk and Reshetikhin (2010) posed two linked open problems that remain the core of the limit-shape theory for interacting (non-free-fermion) weights: (1) classify the generic singularities of limit shapes across the anisotropy regimes, in particular whether corner (angle-type) singularities, which are absent from the free-fermion dimer classification, appear generically in the ferroelectric regime $\Delta>1$; and (2) determine the scaling regimes and limiting stochastic processes governing fluctuations near limit-shape singularities, where the Airy and Pearcey processes of dimer models are conjectured analogues and genuinely new processes are expected at corners. All existing progress concerns degenerate limits (free fermion, five-vertex, stochastic weights, $c\to0$) or partial aspects (torus free energy for $\Delta<1$, boundary statistics, Monte-Carlo evidence at the ice point); the singularity classification and the corner fluctuation process are open.

Background

The six-vertex (ice-type) model places arrows on the edges of a square lattice subject to the ice rule — two arrows enter and two leave every vertex — leaving six local configurations with weights $a$, $b$, $c$, conventionally combined into the anisotropy parameter $\Delta=(a^2+b^2-c^2)/(2ab)$. A configuration can be encoded by a discrete height function whose graph is a stepped surface. Under domain-wall boundary conditions (DWBC: all arrows on the left and right boundaries point outward and all arrows on the top and bottom point inward, or vice versa), the Gibbs-random stepped surface concentrates in the thermodynamic limit on a deterministic limit shape: an arctic curve separates frozen regions, where the surface is flat, from a disordered (liquid) interior. The limit shape is characterized by a variational principle: it minimizes the integral of a surface tension $\sigma(\nabla h)$ over the domain, where $\sigma(H,V)$ is the Legendre transform of the free energy $f(H,V)$ as a function of two electric fields $H$, $V$ conjugate to the height slopes. For the six-vertex model, $f(H,V)$ is computed by the Bethe ansatz (a method that diagonalizes the row-to-row transfer matrix exactly by representing its eigenstates as superpositions of free quasiparticles) only non-rigorously: the derivation rests on unproven conjectures about the condensation of Bethe roots onto contours in the complex plane (Palamarchuk–Reshetikhin Appendix C, Conjectures C.1–C.3). The anisotropy separates regimes: disordered $|\Delta|<1$, ferroelectric $\Delta>1$, antiferroelectric $\Delta<-1$, with $\Delta=0$ the free-fermion point at which the model degenerates to a dimer (domino-tiling) model. At the free-fermion point the theory is complete: the variational principle is a theorem (Cohn–Kenyon–Propp) and the classification of Kenyon–Okounkov–Sheffield shows that frozen boundaries are smooth arcs with at most cusps, with local fluctuation processes given by the Airy process at smooth boundary points (the universal edge-scaling process with transverse scale $t^{2/3}$) and the Pearcey process at cusps (the process describing fluctuations where several caustic branches meet). For interacting weights ($\Delta\neq0$) much less is known. Bukman and Shore showed that the ferroelectric surface tension has a conical (corner-type) singularity and predicted a lens-shaped region of slopes admitting no pure (translation-invariant extremal) Gibbs state, with Kardar–Parisi–Zhang-type states on its boundary; these predictions were proved only in 2022 by Aggarwal, who also emphasized that essentially nothing is known about the conjecturally liquid states inside the lens. Within their (non-rigorous) variational framework, Palamarchuk and Reshetikhin identified tricritical points where two frozen phases and the disordered phase coexist, derived the Pokrovsky–Talapov $3/2$ law for the free energy at the antiferroelectric facet boundary and a $5/3$ law at a tricritical corner-type point, and observed that corners have no analogue in dimer models — motivating the conjecture that the interacting model exhibits genuinely new limit-shape singularities and fluctuation universality classes.

Problem Statement

For the six-vertex model with interacting weights ($\Delta\neq0$) on finite domains with fixed (domain-wall-type) boundary conditions, determine the local structure of limit-shape singularities, in the sense of Palamarchuk and Reshetikhin's open-problem list, in two linked parts that jointly constitute one objective — the geometry and the fluctuations of singular limit-shape points: (1) Classify the generic singularities of limit shapes: beyond the smooth arcs and cusps of the free-fermion classification, establish which singularity types occur generically — in particular whether for $\Delta>1$ corner (angle-type) singularities appear generically — and determine how the generic singularity type depends on the vertex weights and on the boundary data. The $\Delta>1$ corner question is the focal case of this classification; confirming or refuting it does not by itself complete the classification. (2) Find the scaling regimes and the limiting stochastic processes for fluctuations near the singularities of limit shapes: identify the fluctuation scale (powers of the lattice size $N$) and the limit process at smooth frozen boundaries (conjecturally the Airy process), at cusps (conjecturally the Pearcey process), and at corner singularities, where dimer models offer no analogue and the limiting process is expected to be new. The problem refers to the interacting model in its natural generality; results obtained only at the free-fermion point, in the stochastic-weight subfamily, in the $c\to0$ degeneration, or for the torus free energy do not by themselves complete it. Adjacent threads from the same source list — the role of integrability in limit-shape formation, full Liouville integrability of the Euler–Lagrange equations, rigorization of the Bethe-ansatz derivation of $f(H,V)$ (Conjectures C.1–C.3), and higher-rank generalizations — are out of scope for this problem.

References

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