The uniform dimer model (perfect matchings) on induced subgraphs of the cubic lattice Z3\mathbb{Z}^3 now has a law of large numbers: Chandgotia, Sheffield and Wolfram proved that the divergence-free flow induced by a uniformly random matching of a large region converges to the deterministic flow maximizing the integral of an entropy function ent⁡\operatorname{ent}, together with a large deviation principle. What happens at the fluctuation scale around this limit shape is unknown. The source poses the problem (Problems 9.0.5 and 9.0.6) and conjectures the answer: as the mesh tends to zero, the fine-mesh fluctuations of the tiling flow should converge to the Gaussian random generalized flow obtained by orthogonally projecting vector-valued white noise onto the space of divergence-free fields, the three-dimensional analog of Kenyon's theorem that planar domino-tiling height functions converge to the Gaussian free field. This problem asks for a proof or refutation of that conjecture, or an equivalent identification of the fluctuation limit, for uniformly random matchings of a simple region such as a cube or torus, or in the Z3\mathbb{Z}^3 Gibbs-measure setting.

Background

A dimer configuration, also called a perfect matching, of a graph is a set of edges that covers every vertex exactly once. On the square lattice Z2\mathbb{Z}^2, uniformly random dimer configurations are the classical model of random domino tilings, and the planar model is exactly solvable: Kasteleyn's theorem expresses the partition function of any planar graph as the Pfaffian of a suitably signed adjacency matrix (the Pfaffian of a skew-symmetric matrix is a signed sum over perfect matchings whose square is the determinant), local statistics form a determinantal point process, that is, a random point process whose correlation functions are determinants of a fixed kernel, Cohn, Kenyon and Propp proved a variational principle stating that the normalized tiling of a large region converges to the limit shape maximizing the integral of an explicitly computable entropy (surface tension) function, and Kenyon proved that the height function, the integer-valued function on faces whose gradient records the local tiling orientation, converges in law to the Gaussian free field (the Gaussian field with covariance given by the Green's function). Kenyon, Okounkov and Sheffield extended the limit-shape theory to periodically weighted planar dimers. Passing to the flow representation makes Kenyon's fluctuation theorem concrete: the tiling flow, the dual of the height gradient, converges to the Gaussian field obtained by orthogonally projecting vector-valued white noise onto the space of divergence-free vector fields.

None of this two-dimensional machinery survives in three dimensions. By a theorem of Little, a bipartite graph admits Pfaffian (Kasteleyn) weights, meaning edge weights that make the dimer partition function computable as a Pfaffian, only if it does not contain K3,3K_{3,3}, the complete bipartite graph on three and three vertices, in the subdivision sense: that is, the graph cannot be reduced to one having K3,3K_{3,3} as a subgraph by contracting a collection of disjoint paths, each with an even number of vertices, to single edges. The cubic lattice Z3\mathbb{Z}^3 contains K3,3K_{3,3} in this sense and therefore admits no Kasteleyn (Pfaffian) orientation, so there is no determinantal structure. Chandgotia, Sheffield and Wolfram nevertheless built a large deviation theory for uniform dimers on induced subgraphs of Z3\mathbb{Z}^3 by working with flows. A matching sends one unit of current from each even vertex (vertex with even coordinate sum, under the bipartition of Z3\mathbb{Z}^3) to its matched odd neighbor; subtracting the constant reference flow r≡1/6r \equiv 1/6 on edges oriented from even to odd vertices produces a divergence-free tiling flow fτf_\tau taking the value +5/6+5/6 on edges occupied by a dimer and −1/6-1/6 on unoccupied edges. An ergodic Gibbs measure is a probability measure on tilings of all of Z3\mathbb{Z}^3, invariant and ergodic under translations by vectors with even coordinate sum, whose conditional law in any finite region, given the tiling outside, is uniform over the extensions; its mean current ss lies in the octahedron O={s:∣s1∣+∣s2∣+∣s3∣≤1}\mathcal{O}=\{s:|s_1|+|s_2|+|s_3|\le 1\}, and ent⁡(s)\operatorname{ent}(s) is the maximal entropy per site achieved by such a measure with mean current ss. The main theorem of the paper is a 3D variational principle: under boundary conditions converging to bb on ∂R\partial R, the tiling flow of a uniform random matching of the mesh-nn approximation of a bounded region R⊂R3R\subset\mathbb{R}^3 converges in probability to the divergence-free flow gg maximizing ∫Rent⁡(g(x)) dx\int_R \operatorname{ent}(g(x))\,dx, and an associated large deviation principle holds; flows are viewed as vector-valued signed measures with the weak (Wasserstein) topology, and their weak limits are called asymptotic flows.

This settles the limit shape but says nothing about fluctuations. On the physics side, Huse, Krauth, Moessner and Sondhi argued, via a gauge-field generalization of the height representation together with large-scale Monte Carlo simulation, that the uniform dimer model on the bipartite cubic lattice is a critical Coulomb phase with algebraic dipolar correlations; the stationary Gaussian field obtained by projecting vector-valued white noise onto divergence-free fields (the Leray projection) has exactly this dipolar r−3r^{-3} correlation decay in three dimensions. Motivated by the planar theorem and by this physics picture, Chandgotia, Sheffield and Wolfram pose the fluctuation problem and conjecture that the same white-noise projection limit holds in any dimension.

Problem Statement

Characterize the typical fluctuations of the tiling flow of a uniformly random perfect matching in three dimensions. In the setting of Chandgotia-Sheffield-Wolfram, the tiling flow of a uniform random matching of the mesh-nn discretization of a region R⊂R3R\subset\mathbb{R}^3 converges in probability, in the weak topology on vector-valued signed measures, to the deterministic divergence-free flow gg maximizing ∫Rent⁡(g(x)) dx\int_R \operatorname{ent}(g(x))\,dx subject to the boundary data. The problem is to determine the law of the fluctuations of the random flow around gg as the mesh tends to zero. Specifically, prove or refute the following conjecture (Problems 9.0.5 and 9.0.6 of the source): for uniformly random perfect matchings of a simple region such as a cube, or on a torus, or in the Z3\mathbb{Z}^3 Gibbs-measure setting, the discrete divergence-free flows obtained from the matchings converge, in the fine-mesh limit and under the rescaling for which the two-dimensional analogue converges, to the Gaussian random generalized flow obtained by orthogonally projecting vector-valued white noise onto the space of divergence-free flows, this being the three-dimensional counterpart of the consequence of Kenyon's theorem that planar tiling flows converge to the dual of the gradient of the Gaussian free field. A complete answer either establishes this convergence (including the covariance structure of the limit field), or identifies a different limiting fluctuation field, or proves that no such limit exists in a stated natural topology, in at least one of the settings named in the source. The source formulates Problem 9.0.6 for all dimensions d≥3d\ge 3; this problem restricts to the case d=3d=3, which is the setting of the source's large deviation theory and of the Coulomb-phase physics prediction, and states that restriction openly.

References

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