The bounded 15-vertex model (K. Eloranta, 2018) is a blank-edge vertex model on the square lattice that relaxes the six-vertex (ice) rule by allowing unoriented ('blank') edges in a rotationally asymmetric pattern, interpolating between the six- and 19-vertex models. On a square domain with partially oriented boundary, evolved by a two-action probabilistic cellular automaton, it conserves the total number of blank edges and produces limit shapes ranging from a nearly isotropic disk (K-type, asymptotically domain-wall boundaries) to multi-blob, necklace-like equilibria (T-type and more complex boundaries). The source paper poses but does not solve the macroscopic theory of these equilibria: the infinite-time fate of persistent small blank structures, the conjecture that K-type boundaries converge to the six-vertex limit shape at uniform weights (the alternating-sign-matrix limit shape), the classification of limit shapes across the hierarchy of non-DWBC boundary classes, and how the conserved blank sector and blank-ribbon geometry determine the equilibrium.

Background

In a vertex model of statistical mechanics, the degrees of freedom are arrow orientations on the edges of a lattice, and a local vertex rule restricts the allowed arrow patterns around each vertex. The six-vertex (ice) model on the square lattice requires exactly two arrows in and two out at every vertex. Under domain-wall boundary conditions (DWBC) — arrows all pointing inward on two adjacent sides of a square domain and all outward on the other two, alternating as one goes around — the model is exactly solvable and exhibits phase separation in the thermodynamic limit: a disordered ('temperate') central region is separated from four frozen corners by a smooth curve, the arctic curve or limit shape. F. Colomo and A. G. Pronko (J. Stat. Phys. 138, 2010) computed this curve for the whole disordered regime; at uniform weights a=b=ca=b=c (the point Δ=1/2\Delta = 1/2) the DWBC six-vertex model is equivalent, via the standard bijection, to uniformly random alternating sign matrices (square matrices with entries 0,±10, \pm 1, each row and column summing to 1 with nonzero entries alternating in sign), and the arctic curve is the ASM limit shape, a nearly circular but non-circular algebraic curve whose north-west portion is (2x−1)2+(2y−1)2−4xy=1(2x-1)^2 + (2y-1)^2 - 4xy = 1.

Eloranta's 'bounded' vertex-model series — Diamond Ice (J. Stat. Phys. 96, 1999), the bounded eight-vertex model (Theor. Comput. Sci., 2003), the bounded 19-vertex model (arXiv:1710.03609), and the bounded 15-vertex model (arXiv:1807.07567) — studies vertex models in finite domains with partially oriented boundary through explicit dynamics. The 19-vertex rule requires an equal number of incoming and outgoing arrows at each vertex while allowing unoriented ('blank') edges; enumerating the legal arrow patterns gives 19 vertex configurations. The 15-vertex rule is its rotationally asymmetric reduction: it keeps the six ice vertices, the fully blank vertex, and only part of the rotations of the remaining blank-containing ('L-type') configurations, for 15 configurations in total. The model is treated as a purely combinatorial object, without vertex weights. Two elementary facts delimit it: an unoriented interior edge forces an unoriented path to the boundary (Proposition 1 of the source), so under DWBC the 15-vertex and six-vertex configuration sets coincide (Corollary 1) — genuinely new behavior requires boundary conditions with blank boundary edges; and the number of blank edges is conserved for a fixed boundary condition (Proposition 2), so the configuration space splits into disconnected blank-number sectors.

The dynamics is a probabilistic cellular automaton (PCA — a discrete-time Markov chain updating sites in parallel rather than one at a time) built from two local moves: action I, a corner flip that creates or transports a minimal blank path, and action II, the reversal of a unidirectional loop, which is the standard generating move of the six-vertex model. The moves are applied alternately on the even and odd sublattices of a checkerboard decomposition of a diamond domain containing the square. The height function — an integer function on the dual lattice that increases or decreases by one when crossing a left- or right-pointing arrow and stays constant across a blank edge — turns each configuration into a discrete surface, and the boundary condition into a 'roof' over the diamond. Since a straight blank path across the domain cannot branch (Corollary 2 of the source), blank 'ribbons' behave much like elastic bands attached to the boundary.

Simulations in the source paper sweep a hierarchy of boundary conditions of increasing distance from DWBC, obtained by planing the ridge of the DWBC height roof. K-type boundaries introduce two neighboring blank zig-zag ribbons running parallel to a diagonal; projected onto the inscribed square the boundary differs from DWBC by only two doublets of blank edges at opposite corners, so the fraction of boundary edges that are blank vanishes as the lattice size grows ('asymptotically DWBC'). The equilibrium is a nearly isotropic disk-shaped temperate domain — surprising, given the rule's strong microscopic asymmetry — with the blank ribbons pushed to its rim. T-type boundaries add further blank zig-zag 'terraces' on the slopes of the roof; here the blank zig-zags enter the square, no fixed boundary condition holds on it, and the natural domain is the diamond itself. With several zig-zags initially bundled at the top, the equilibrium becomes multi-blob, with small blank disks persisting for very long times; one scenario suggested in the source is a 'necklace' of disks governed by a surface tension of the height surfaces, which is however not defined there.

The name '15-vertex model' also denotes an unrelated integrable spin-1 (three-state edge) vertex model, whose non-symmetric R-matrices were classified by R. S. Vieira (arXiv:1908.06932); Eloranta's blank-edge model is a distinct, non-integrable combinatorial object, and the two must not be conflated.

Problem Statement

Determine the macroscopic equilibrium theory — limit shapes and their metastability — of the bounded 15-vertex model (Eloranta's blank-edge 15-vertex rule on the square lattice, evolved by its two-action probabilistic cellular automaton) on a square/diamond domain for boundary conditions beyond DWBC. The source paper poses four linked questions that jointly constitute this objective; the relations among them are: (1) settles whether the observed multi-blob states are genuine equilibria or metastable transients, which any limit-shape statement must presuppose; (2) is the anchor case connecting the blank-edge model to the exactly solved six-vertex picture; (3) is the classification proper; and (4) identifies the state variables (conserved blank count, ribbon attachment geometry) on which the classification depends.

(1) Metastability and long-time fate of blank structures. In 'unclean' K-type evolutions a corner-like cluster of blank vertices persists, and in T-type evolutions with bundled blank zig-zags a multi-blob equilibrium of small blank disks persists, for very long simulation times. Determine whether such structures are always eventually removed by rare (large-deviation) fluctuations — and on what time scale as a function of lattice size — or whether they survive in the infinite-time or scaling limit, so that multi-blob equilibria are genuine rather than metastable.

(2) K-type scaling limit. Prove or decisively test the conjecture that for K-type (asymptotically DWBC) boundaries the contribution of the blank ribbons is negligible in the scaling limit, so that the temperate domain converges, as the lattice size N→∞N \to \infty, to the six-vertex limit shape at uniform weights — the limit shape of large uniformly random alternating sign matrices.

(3) Classification of limit shapes across non-DWBC boundary classes. For the hierarchy of boundary conditions beyond DWBC obtained by planing the height roof (T-type and more complex partially oriented boundaries, including bundled blank zig-zags), determine the limit shape / arctic curve as a function of the boundary data — in particular, which boundary configurations produce single-component temperate domains and which produce multi-component (multi-blob, necklace-like) ones.

(4) Dependence on conserved blank sectors and ribbon placement. Quantify how the conserved number of blank edges and the attachment geometry of the blank ribbons on the boundary determine the equilibrium geometry — for instance, whether an effective surface tension for the height surfaces exists and determines the observed equilibria, including the attachment-point separations that simulations indicate are the primary control parameter.

A satisfactory answer either provides a general principle from which the equilibria for each boundary class follow (for example a variational or surface-tension theory for blank-edge vertex models), or settles the four questions individually, in each case by rigorous mathematics or by systematically converged numerical evidence with quantified finite-size control. The questions must be answered for the model as defined in the source — the 15-vertex rule, the two PCA actions, and the height-roof boundary family — not for a substituted or restricted model class.

References

  1. Kari Eloranta, The bounded 15-vertex model, arXiv:1807.07567 (2018), https://arxiv.org/abs/1807.07567
  2. Kari Eloranta, The bounded 19-vertex model, arXiv:1710.03609 (2017), https://arxiv.org/abs/1710.03609
  3. Kari Eloranta, The bounded eight-vertex model, Theoretical Computer Science 303 (2003) 539-552, DOI 10.1016/S0304-3975(02)00505-4, https://doi.org/10.1016/S0304-3975(02)00505-4
  4. Kari Eloranta, Diamond Ice, Journal of Statistical Physics 96 (1999) 1091-1109, DOI 10.1023/A:1004644418182, https://doi.org/10.1023/A:1004644418182
  5. F. Colomo and A. G. Pronko, The arctic curve of the domain-wall six-vertex model, Journal of Statistical Physics 138 (2010) 662-700, DOI 10.1007/s10955-009-9902-2, https://doi.org/10.1007/s10955-009-9902-2
  6. Ivar Lyberg, Vladimir Korepin, G. A. P. Ribeiro, Jacopo Viti, Phase separation in the six-vertex model with a variety of boundary conditions, Journal of Mathematical Physics 59, 053301 (2018), DOI 10.1063/1.5018324, https://arxiv.org/abs/1711.07905
  7. Ivar Lyberg, Vladimir Korepin, Jacopo Viti, Fluctuation of the phase boundary in the six-vertex model with domain wall boundary conditions: a Monte Carlo study, Journal of Physics A: Mathematical and Theoretical 56, 495002 (2023), DOI 10.1088/1751-8121/ad0a43, https://arxiv.org/abs/2303.14669
  8. Zhao Zhang, Israel Klich, Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models, SciPost Physics 15, 044 (2023), DOI 10.21468/SciPostPhys.15.2.044, https://arxiv.org/abs/2210.03038
  9. R. S. Vieira, Fifteen-vertex models with non-symmetric R matrices, arXiv:1908.06932 (2019), https://arxiv.org/abs/1908.06932