Correlation functions and arctic curves of the four-vertex model in an external horizontal field: limit shapes of boxed plane partitions with fixed diagonal sums

The exactly solvable four-vertex model with fixed (scalar-product) boundary conditions has a determinantal partition function in an inhomogeneous external horizontal field (Bogoliubov and Malyshev, 2019), equivalent to the generating function of boxed plane partitions with fixed sums of their diagonals. The spatial structure of this field-deformed ensemble is, however, undetermined: its correlation functions have not been computed, the arctic curve (limit shape) — the boundary between frozen and disordered regions and its deformation with the field profile — is unknown, and the trace statistics for generic asymmetric boxes are unsettled. Only the uniform, zero-field case is resolved (Burenev, Colomo, Maroncelli and Pronko, 2023). The problem is to compute the correlation functions of the four-vertex model in the external horizontal field and to determine, by them or any equivalent method, the field-dependent arctic curve of the model — equivalently, the limit shape of boxed plane partitions with fixed (or diagonally weighted) sums of their diagonals — together with the asymmetric-box trace statistics.

Background

The six-vertex model of statistical mechanics places arrows on the edges of a square lattice so that two arrows point into and two out of every vertex; its name comes from the six local arrow configurations allowed by this ice rule. The four-vertex model is the subcase in which two of the six configurations are forbidden, i.e. their Boltzmann weights (the numbers weighting each local configuration in the partition function) are set to zero. Under the fixed, or scalar-product, boundary conditions used here — boundary arrows fixed so that the admissible configurations are nests of $N$ non-intersecting lattice paths on a grid of $2N$ vertical and $M+1$ horizontal lines, each path moving only east or north and never making two consecutive horizontal steps — the model is exactly solvable by the Quantum Inverse Scattering Method (an algebraic transfer-matrix technique) and is in one-to-one correspondence with boxed plane partitions: arrays $\pi_{km}$ of non-negative integers, strictly decreasing along rows and columns, fitting inside an $N\times N\times M$ box; equivalently, three-dimensional Young diagrams in a box, or lozenge tilings of a hexagon on the triangular lattice.

Bogoliubov and Malyshev (2019) solved this model in the presence of an inhomogeneous external horizontal field: a horizontal path step at height $\mu$ in column $s$ receives the weight $\exp(2h_s\mu)$, the field parameter $h_s$ being allowed to depend on the column $s$ (the associated potential grows linearly with the height; $h_s\equiv 0$ is the uniform model). They expressed the partition function as a determinant and, through the plane-partition correspondence, obtained the generating function $\sum_{\pi}q^{|\pi|}\prod_{s}a_{s}^{\mathrm{tr}_s\pi}$ of boxed (strict) plane partitions with fixed sums of their diagonals, $\mathrm{tr}_s\pi=\sum_{k+m=s}\pi_{km}$, where $a_s=e^{2h_s}$ and $|\pi|$ is the volume (total number of cubes); in the semi-infinite limit these formulas generalize the classical trace generating functions of Stanley (1973) and Gansner (1981) for plane partitions with fixed diagonal sums and unbounded parts.

In the scaling limit (lattice sizes growing at fixed proportions), random tiling and vertex models typically exhibit the arctic-curve, or limit-shape, phenomenon: a macroscopic boundary separates frozen (ordered) regions, where the configuration is deterministic, from a temperate, disordered region, with the boundary given by an explicit curve — the prototypes being the arctic circle of the Aztec diamond and the arctic curve of random boxed plane partitions (Cohn, Larsen and Propp, 1998). For vertex models such curves are obtained either by the Tangent Method (Colomo and Sportiello, 2016), which combines a boundary one-point correlation function with the entropy of a single escaping path, or by the Emptiness Formation Probability (EFP) method, which evaluates the probability that an entire spatial region is perfectly ordered. For the uniform four-vertex model with scalar-product boundary conditions, Burenev, Colomo, Maroncelli and Pronko (2023) determined the full arctic curve — a central disordered region bounded by four ferroelectric and two anti-ferroelectric frozen corners (ordered phases with, respectively, aligned and alternating local arrow patterns) — both by the Tangent Method and rigorously via EFP-type correlation functions evaluated as gap probabilities of a discrete log-gas (an ensemble of particles on lattice sites with logarithmic repulsion) built on the orthogonalizing measure of the Hahn polynomials, which are orthogonal polynomials on a finite arithmetic lattice; they also showed the boundary fluctuations are Tracy–Widom distributed (governed by the universal edge-fluctuation law of random matrix theory).

What is not known is the same structure in the presence of the field. The concluding section of Bogoliubov and Malyshev (2019) states that the problems of the trace statistic in the generic case of an asymmetric box and the arctic curves phenomena are connected with the calculation of the correlation functions of the four-vertex model in presence of the external horizontal field — and those correlation functions were not computed there, nor, as far as the literature shows, since. The field-deformed problem is equivalent to the limit shape of boxed plane partitions whose diagonal sums are fixed or diagonally weighted, a fixed-flux constraint that lies outside the standard uniform-tiling results.

Problem Statement

Determine the correlation structure and the limit-shape (arctic-curve) phase separation of the four-vertex model with fixed, scalar-product boundary conditions in the presence of the inhomogeneous external horizontal field — equivalently, the limit shape of boxed plane partitions with fixed (or diagonally weighted) sums of their diagonals. The single overarching objective is the scaling-limit spatial statistics of the field-deformed model; its required constituents, as posed by the source, are:

(1) Compute the correlation functions of the four-vertex model in the external horizontal field — for instance edge and boundary one-point functions, or emptiness- and anti-ferroelectric-formation probabilities — or an equivalent determinantal/variational characterization sufficient for a scaling-limit analysis. The determinantal partition function alone (Bogoliubov–Malyshev 2019) does not answer this.

(2) Derive the arctic curve in the scaling limit (lattice sizes $2N, M\to\infty$ at fixed proportions): the explicit boundary between the frozen (ferroelectric and anti-ferroelectric) and disordered regions as a function of the field profile $h_s$, equivalently of the per-diagonal weights $a_s=e^{2h_s}$, with all non-field weights (including the volume parameter $q$ of the generating function) held fixed, and including the homogeneous specialization $h_s\equiv h$; determine how the frozen regions of the zero-field model deform with the field profile.

(3) As the companion application the source explicitly ties to the same machinery, settle the trace statistics of boxed plane partitions in the generic asymmetric box (independent side lengths): the generating functions and their asymptotics, of which only special cases (symmetric boxes, semi-infinite limits, the high box) are known.

Part (1) is the route the source identifies, and parts (2)–(3) are its principal consequences; an answer may use any valid method (tangent method, EFP/log-gas asymptotics, free-fermion (determinantal) or variational dimer analysis) provided it yields the correlation functions or an equivalent characterization, the field-dependent arctic curve, and the generic asymmetric-box trace statistics of part (3). The zero-field case $h_s\equiv 0$ is already resolved by Burenev–Colomo–Maroncelli–Pronko (2023) and constitutes only the trivial specialization of the target.

References

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