Kenyon-Okounkov Gaussian free field convergence for uniformly weighted generalized tower graphs: upgrading gradient convergence to the full height field

For uniformly weighted generalized tower graphs - domino-tiling graphs built from up-down paths approximating an arbitrary admissible boundary profile f, with the Aztec diamond (f identically 1) and the Borodin-Ferrari tower graph as special cases - Keating and Vu (2025) constructed perfect t-embeddings and proved that the gradients of the height-function k-point correlation functions converge to those of the Gaussian free field. This is strictly weaker than the Kenyon-Okounkov prediction that the height field itself converges to the Gaussian free field in the intrinsic conformal structure. The problem asks for the missing step of the program: prove full field-level convergence of the centered height fluctuations to the Gaussian free field in the Kenyon-Okounkov conformal structure for this family, or rigorously identify any additional limiting component - which amounts to closing the explicit boundary-control gap of the t-embedding framework. This step is open.

Background

Dimer model and height function. A dimer configuration (perfect matching) of a finite planar bipartite graph $G=(B,W,E)$ is a set of edges covering every vertex exactly once; for subgraphs of the square grid these are in bijection with domino tilings. Under the uniform measure on dimer configurations, one defines the Thurston height function $h$: an integer-valued function on the faces of $G$ whose value changes by 1 or 3 (in absolute value) across each edge according to the dimer occupancy. Height functions are the central observable of the model: they linearize the local dimer statistics and exhibit macroscopic structure.

Limit shapes and the arctic curve. For large tilings of a fixed domain, the rescaled height function concentrates around a deterministic limit shape, and the domain splits into frozen regions (height asymptotically deterministic, tiled periodically) and liquid regions (both types of edges present), separated by sharp arctic curves. Kenyon, Okounkov and Sheffield characterized limit shapes variationally through the amoeba of the dimer spectral curve (for periodically weighted graphs, the algebraic curve parameterizing the model, projected by logarithmic coordinates), and Kenyon and Okounkov showed the interface dynamics is governed by the complex Burgers equation, a first-order nonlinear PDE for the limit-shape gradient.

Gaussian free field and the Kenyon-Okounkov (KO) conjecture. The Gaussian free field (GFF) is the centered Gaussian generalized function whose covariance is the Green's function of the Laplacian on a domain, with respect to a given conformal structure. The KO conjecture predicts that in the liquid region, the fluctuations of the height function around its limit shape, centered and rescaled by $\sqrt{\pi}$, converge as random generalized functions (modulo additive constants) to a GFF with respect to a non-trivial complex conformal structure intrinsic to the model, determined by the spectral curve via the critical point map. This conformal structure does not coincide with the Euclidean one and is generally only implicit, which is what makes the conjecture difficult. It is confirmed for Temperleyan domains (graphs coupled to spanning trees by Temperley's bijection; Kenyon, in the sense of moments), polygonal lozenge domains (Petrov), rectangular Aztec diamonds (Bufetov-Knizel), and piecewise-periodic rail-yard graphs (a rail-yard graph is a half-plane dimer graph assembled from a sequence of columns of two lattice types; Li-Vuletic) - all settings with integrable structure, namely the Schur process (a determinantal point process describing random interlacing partitions) or orthogonal-polynomial representations. The general case is a central open problem of random-tiling theory. In multiply connected liquid regions the expected limit is refined: a GFF plus an independent discrete-Gaussian (compactified free field) harmonic component, as proved in the doubly periodic Aztec case by Berggren-Nicoletti and in doubly connected Temperleyan cylinders by Chelkak-Deiman.

t-embeddings and the gradient-vs-field gap. A t-embedding of a planar bipartite graph is an embedding of its dual graph whose white faces are tangential polygons sharing an incircle (a generalization of isoradial embeddings, in which every face is inscribed in a circle of a fixed common radius; introduced by Kenyon-Lam-Ramassamy-Russkikh); a perfect t-embedding additionally requires the complementary black faces to be tangential. Perfect t-embeddings come with an origami map, a piecewise-linear map recording the turning of the edges, whose image is a surface in Minkowski space $\mathbb{R}^{2,1}$. Under the framework assumptions LIP($\kappa,\delta$) (a Lipschitz-type bound on the slopes of the embedding) and EXP-FAT($\delta$) (a lower bound on the size ratio of adjacent faces), together with the hypothesis that the origami maps converge to a Lorentz-minimal surface (for which the rigidity of the convex hull of the origami image, verified for Aztec diamonds and tower graphs by Berggren-Nicoletti-Russkikh, is the standard sufficient condition), Chelkak-Laslier-Russkikh proved that the gradients of the k-point correlation functions of the height function converge to those of the GFF in the conformal structure induced by that surface. However, convergence of gradients does not imply convergence of the height field itself: upgrading requires uniform bounds on the coupling functions - the entries of the inverse Kasteleyn operator, the signed weighted adjacency matrix between white and black vertices whose determinant counts dimer configurations - when both arguments approach the boundary simultaneously, and Chelkak-Laslier-Russkikh explicitly state they are not aware of such an argument; their theorem does not exclude, for example, the height field converging to the GFF plus an independent global random variable. This is the precise technical gap between the t-embedding framework and the KO conjecture.

Generalized tower graphs. Fix an admissible profile $f:[-1,x^*]\to\mathbb{R}$: $f(-1)=1$, $|f'(x)|\le 1$ on $[-1,x^*]$ (so $f$ is 1-Lipschitz), and $x^*$ is the smallest point with $f(x^*)=x^*$. For each $n$, the generalized tower graph $G_n^f$ is obtained by stacking Aztec-diamond blocks of varying widths and applying the shuffling algorithm (Propp's local update rule that grows the graph while tracking the dimer model) along up-down paths - lattice paths with steps $\pm 1$ - that approximate $n f$; the Aztec diamond corresponds to $f$ identically 1 and the tower graph of Borodin-Ferrari to $f(x)=x/3+4/3$. These graphs are a special case of rail-yard graphs, but without periodicity assumptions. Keating and Vu constructed perfect t-embeddings of the reduced graphs (obtained by contracting all degree-two vertices) via an explicit connection to the Aztec diamond, proved convergence of the corresponding origami maps to a maximal surface, verified all framework assumptions with $\delta_n=\log(n)/n$, computed the arctic curves $x^2+y^2=\frac{1}{2}f(x+y)^2$, and deduced gradient convergence of height correlations for arbitrary admissible $f$ - while noting this remains weaker than the KO conjecture.

Two-dimensional initial data. Keating-Vu Remark 3.7 describes a natural generalization in which the boundary data is encoded by a two-dimensional up-down surface $H:\mathbb{Z}^2\to\mathbb{Z}$ instead of a path; the local mixing moves are then flips of octahedron-recurrence type (the octahedron recurrence is the local linear recurrence on $\mathbb{Z}^3$ governing lozenge-tiling partition functions), and the family contains the $(r,s,t)$-graphs (pinecone graphs) of Di Francesco-Vu, whose arctic curves are known but whose fluctuations are not. The paper restricts to path data for simplicity, asserting that its Section 5 asymptotic analysis of the technical assumptions would apply equally well in the two-dimensional case, but the construction of the graphs, their perfect t-embeddings, and the fluctuation theorems are not carried out there. This two-dimensional direction is background context, not part of the problem asked below.

Problem Statement

Shared objective: complete the Kenyon-Okounkov Gaussian-fluctuation picture for uniformly weighted tower-type dimer graphs by upgrading the established gradient convergence to convergence of the full height field.

Setup. Let $f:[-1,x^*]\to\mathbb{R}$ be an admissible profile ($f(-1)=1$, $|f'|\le 1$ on $[-1,x^*]$ so that $f$ is 1-Lipschitz, and $x^*$ the smallest point with $f(x^*)=x^*$), and let $G_n^f$ be the associated generalized tower graph with uniform dimer weights, with centered height function $h_n$ under the uniform measure on perfect matchings. Keating-Vu construct perfect t-embeddings $\mathcal{T}_n$ of the reduced graphs (obtained from $G_n^f$ by contracting all degree-two vertices); the liquid region $\Omega_f$, with boundary curve $x^2+y^2=\frac{1}{2}f(x+y)^2$ in macroscopic coordinates, carries the limiting complex structure $\omega_f$ induced by the limiting maximal (Lorentz-minimal) surface, equivalently by the spectral curve via the critical point map.

Goal - from gradients to the field. Prove that, for arbitrary admissible profiles $f$, the centered height functions $h_n$ converge, as random generalized functions on $\Omega_f$ modulo deterministic additive constants (for instance in the sense of joint moments at any finite collection of faces, or in distribution against smooth compactly supported test functions), to the Gaussian free field on $(\Omega_f,\omega_f)$. In particular, either rule out or rigorously identify additional limiting components, such as an independent global random variable or a discrete-Gaussian / compactified-field component of the type that appears in multiply connected settings. A natural route, within the t-embedding framework, is to upgrade the gradient-convergence theorem for this family by closing its boundary-control gap - uniform bounds on the inverse-Kasteleyn coupling functions in the regime where both arguments approach the boundary simultaneously - although other approaches are not excluded. A solution that proves convergence to a rigorously and fully characterized different limit (for example, the GFF plus a specified independent component) is an acceptable answer, provided the limit is completely identified for the whole family of admissible profiles.

The problem is solved when this field-level convergence - or a rigorously and fully characterized alternative limiting behavior - is established for arbitrary admissible profiles, at the standard of rigor of the cited literature.

References

1. D. Keating and H. T. Vu, "Perfect t-Embeddings of Uniformly Weighted Generalized Tower Graphs", arXiv:2509.18791, 2025. DOI: 10.48550/arXiv.2509.18791. https://arxiv.org/abs/2509.18791
2. D. Chelkak, B. Laslier and M. Russkikh, "Dimer model and holomorphic functions on t-embeddings of planar graphs", Proceedings of the London Mathematical Society (3) 126(5):1656-1739, 2023. DOI: 10.1112/plms.12516. https://arxiv.org/abs/2001.11871
3. D. Chelkak, B. Laslier and M. Russkikh, "Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces", arXiv:2109.06272, 2021. https://arxiv.org/abs/2109.06272
4. T. Berggren, M. Nicoletti and M. Russkikh, "Perfect t-embeddings of uniformly weighted Aztec diamonds and tower graphs", International Mathematics Research Notices 2024(7):5963-6007, 2024. DOI: 10.1093/imrn/rnad299. https://arxiv.org/abs/2303.10045
5. T. Berggren, M. Nicoletti and M. Russkikh, "Perfect t-embeddings of doubly periodic Aztec diamonds", arXiv:2508.04938, 2025. https://arxiv.org/abs/2508.04938
6. T. Berggren, M. Nicoletti and M. Russkikh, "Perfect t-embeddings and Lozenge Tilings" (perfect t-embeddings for graphs with outer face of degree greater than 4), arXiv:2408.05441, 2024. https://arxiv.org/abs/2408.05441
7. T. Berggren and M. Russkikh, "Perfect t-embeddings and the octahedron equation of the two-periodic Aztec diamond", Annales de l'Institut Henri Poincare D, 2025. DOI: 10.4171/aihpd/220. https://arxiv.org/abs/2508.06697
8. T. Berggren and M. Nicoletti, "Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models", arXiv:2502.07241, 2025. https://arxiv.org/abs/2502.07241
9. M. Basok, "Kenyon's identities for the height function and compactified free field in the dimer model", arXiv:2511.03804, 2025. https://arxiv.org/abs/2511.03804
10. D. Chelkak and Z. Deiman, "Domino tilings of black-and-white Temperleyan cylinders", arXiv:2601.13332, 2026. https://arxiv.org/abs/2601.13332
11. R. Kenyon and A. Okounkov, "Limit shapes and the complex Burgers equation", Acta Mathematica 199(2):263-302, 2007. DOI: 10.1007/s11511-007-0021-0. https://arxiv.org/abs/math-ph/0507007
12. R. Kenyon, A. Okounkov and S. Sheffield, "Dimers and amoebae", Annals of Mathematics 163(3):1019-1056, 2006. DOI: 10.4007/annals.2006.163.1019. https://doi.org/10.4007/annals.2006.163.1019
13. R. Kenyon, "Dominos and the Gaussian Free Field", The Annals of Probability 29(3):1128-1137, 2001. DOI: 10.1214/aop/1015345599. https://doi.org/10.1214/aop/1015345599
14. R. Kenyon, W. Y. Lam, S. Ramassamy and M. Russkikh, "Dimers and circle patterns", Annales scientifiques de l'Ecole Normale Superieure 55(3):865-903, 2022. DOI: 10.24033/asens.2507. https://arxiv.org/abs/1810.05616
15. L. Petrov, "Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field", The Annals of Probability 43(1):1-43, 2015. DOI: 10.1214/12-AOP823. https://arxiv.org/abs/1206.5123
16. A. Bufetov and A. Knizel, "Asymptotics of random domino tilings of rectangular Aztec diamonds", Annales de l'Institut Henri Poincare, Probabilites et Statistiques 54(3):1250-1290, 2018. DOI: 10.1214/17-AIHP838. https://arxiv.org/abs/1604.01491
17. Z. Li and M. Vuletic, "Asymptotics of pure dimer coverings on rail-yard graphs", Forum of Mathematics, Sigma 11:e94, 2023. DOI: 10.1017/fms.2023.90. https://arxiv.org/abs/2110.11393
18. P. Di Francesco and H. T. Vu, "Arctic curves of the T-system with slanted initial data", Journal of Physics A: Mathematical and Theoretical 57(33):335201, 2024. DOI: 10.1088/1751-8121/ad65a5. https://arxiv.org/abs/2403.02479
19. D. Chelkak and S. Ramassamy, "Fluctuations in the Aztec diamonds via a space-like maximal surface in Minkowski 3-space", Confluentes Mathematici 16:1-17, 2024. DOI: 10.5802/cml.94. https://arxiv.org/abs/2002.07540