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arxiv: 2605.03896 · v1 · submitted 2026-05-05 · 🧮 math.PR · math-ph· math.AG· math.CO· math.MP· nlin.SI

Dimer models on astroidal zig-zag graphs

Tomas Berggren , Alexei Borodin , Terrence George This is my paper

Pith reviewed 2026-05-07 13:53 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.AGmath.COmath.MPnlin.SI
keywords dimer modelsKasteleyn matrixastroidal zig-zag graphsNewton polygonspectral curvearctic curvelimit shapeGibbs measures
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The pith

For any minimal periodic planar bipartite graph, an (n-3)-dimensional family of astroidal zig-zag subgraphs admits an explicit inverse Kasteleyn matrix via double contour integrals on the spectral curve.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs special finite subgraphs called astroidal zig-zag graphs inside any minimal periodic planar bipartite graph. These subgraphs are bounded by zig-zag paths and take an astroid-like shape, with dimension n-3 where n is the number of sides of the Newton polygon. For these graphs the inverse Kasteleyn matrix, which determines all dimer correlations, is expressed explicitly as a double contour integral on the spectral curve for any Fock weighting, including all periodic weightings. When the weighting is periodic the authors further derive the large-scale asymptotics, including a phase separation into frozen, liquid and gaseous regions together with the arctic curve, the limit shape of the height function, and convergence of local correlations to the predicted Gibbs measure. This extends exact solvability to graphs whose Newton polygons have more than four sides, cases previously inaccessible.

Core claim

For any minimal periodic planar bipartite graph whose Newton polygon has n sides, the authors construct an (n-3)-dimensional family of finite subgraphs whose boundaries consist of zig-zag paths and whose overall shape resembles an astroid; they call these astroidal zig-zag graphs. The inverse Kasteleyn matrix on any such graph is given by a double contour integral on the spectral curve, for every Fock weighting. When the weighting is periodic, the resulting formulas yield an asymptotic phase diagram with frozen, rough and smooth regions, an explicit parametrization of the arctic curve, the deterministic limit shape of the height function, and convergence of local dimer statistics to the edge

What carries the argument

The astroidal zig-zag graph (AZ graph), a finite subgraph bounded by zig-zag paths in an astroid-like shape whose inverse Kasteleyn matrix is realized by a double contour integral over the spectral curve.

If this is right

  • When the Newton polygon is the unit square the AZ graph reduces to the Aztec diamond of variable size.
  • Large AZ graphs exhibit phase separation into asymptotically frozen, rough and smooth regions with an explicit arctic curve.
  • The height function converges to a deterministic limit shape whose slope determines the local statistics.
  • Local dimer correlations converge to the translation-invariant Gibbs measure of the slope predicted by the limit shape.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a systematic way to produce exactly solvable finite domains inside any minimal periodic dimer graph, not merely the triangular and quadrilateral cases treated earlier.
  • Because the parameter count is n-3, the family grows with the complexity of the Newton polygon and may serve as a testing ground for conjectures about arctic curves in higher-genus or non-planar settings.
  • The contour-integral representation may be useful for deriving similar explicit formulas in related models such as six-vertex or loop models on the same graphs.

Load-bearing premise

The underlying graph must be minimal and the edge weights must be Fock weights so that the inverse Kasteleyn matrix admits an expression by contour integrals on the spectral curve without extra obstructions.

What would settle it

For a concrete minimal periodic graph whose Newton polygon is a pentagon, compute the inverse Kasteleyn matrix of the corresponding AZ graph by direct linear algebra and check whether it equals the double contour integral formula; any mismatch falsifies the claim.

Figures

Figures reproduced from arXiv: 2605.03896 by Alexei Borodin, Terrence George, Tomas Berggren.

Figure 1
Figure 1. Figure 1: A pentagonal lattice polygon N (left), a minimal graph G with Newton polygon N together with a fundamental domain (middle), and a choice of isoradial weights (modulo gauge) where the angles are chosen evenly spaced around the unit circle (right). Here ϕ = (1 + √ 5)/2 denotes the golden ratio. See view at source ↗
Figure 2
Figure 2. Figure 2: Two natural finite domains for the square lattice are the square and the Aztec diamond, view at source ↗
Figure 3
Figure 3. Figure 3: The square Newton polygon N (left) and the Aztec diamond (right). The cyclic order of zig-zag paths along the boundary of the Aztec diamond is opposite to the cyclic order of the corresponding edges of N. The edges of N are colored according to whether the black or white vertices are outside. First of all, we observe that the boundary of the Aztec diamond is formed by zig-zag paths whose cyclic order along… view at source ↗
Figure 4
Figure 4. Figure 4: The construction of an AZ graph for the graph view at source ↗
Figure 5
Figure 5. Figure 5: An astroidal domain for a decagonal Newton polygon, and an arctic curve for a choice of view at source ↗
Figure 6
Figure 6. Figure 6: A simulation of the arctic curve in the tropical limit of the Aztec diamond (left) and the view at source ↗
Figure 7
Figure 7. Figure 7: Local moves: spider move (left) and contraction-uncontraction move (right). view at source ↗
Figure 8
Figure 8. Figure 8: Local configuration near an edge in the definition of Fock’s Kasteleyn matrix view at source ↗
Figure 8
Figure 8. Figure 8: 23 view at source ↗
Figure 9
Figure 9. Figure 9: A Newton polygon N and an illustration of ∂G × β , along with the edge coloring. Note that the actual βej ’s are (pieces of) strands, but they have the same asymptotic direction as the vectors pictured on the right. w view at source ↗
Figure 10
Figure 10. Figure 10: The left panel shows an example illustrating the necessity of Condition 2 in Definition 3.2. view at source ↗
Figure 11
Figure 11. Figure 11: A square Newton polygon N (left), the Aztec diamond Gβ (middle) and its medial cycle ∂G × β (right) view at source ↗
Figure 12
Figure 12. Figure 12: A quadrilateral Newton polygon and an example of a corresponding AZ graph. view at source ↗
Figure 13
Figure 13. Figure 13: An example of an AZ graph for the pentagonal Newton polygon in Figure 1 which we view at source ↗
Figure 14
Figure 14. Figure 14: A hexagonal Newton polygon and an example of an AZ graph. view at source ↗
Figure 15
Figure 15. Figure 15: An octagonal Newton polygon N and an AZ graph. 3.3 Criterion for admissibility For any e ∈ E(N), we label the strands αe = {α i e : i ∈ Z+ 1 2 } parallel to e from left to right so that the reference face f0 lies in between α − 1 2 e and α 1 2 e . Define cβ : E(N) → Z + 1 2 by α cβ(e) e = βe. Lemma 3.8. For each e ∈ E(N), deg (Dβ)αe  = −cβ(e) + ( − 1 2 , if e is black, 1 2 , if e is white. In particular,… view at source ↗
Figure 16
Figure 16. Figure 16: Classes of parallel strands for the graph in Figure 13. The shaded face is the reference view at source ↗
Figure 17
Figure 17. Figure 17: The faces marked by × are the βe− - and βe+ -outer faces associated with the vertex v = e− ∩e+ ∈ V (N). The shaded region indicates Gβ. The blue edges form a dimer cover M of Gβ, while the red edges form the extremal dimer cover Mv. Any dual path between two marked outer faces, such as the dotted path shown, crosses no edges of either M or Mv. Hence, the difference hMv (f) − ∂h(f) in (3.6) is constant on … view at source ↗
Figure 18
Figure 18. Figure 18: Change of Gauss map γ at a vertex v = e− ∩ e+, where θ = arg(⃗e+) − arg(⃗e−). 3.6 Chambers Unlike the case of the Aztec diamond, the relative position of vertices with respect to strands need not be uniform. Thus, we introduce “local sign functions” in addition to the global sign function signβ from (3.1). The collection of strands β partitions the plane into several (open) chambers. Precisely, the chambe… view at source ↗
Figure 19
Figure 19. Figure 19: The four conical regions. The shaded regions are the inconsistently oriented cones. view at source ↗
Figure 20
Figure 20. Figure 20: The chosen inconsistently oriented cones in view at source ↗
Figure 21
Figure 21. Figure 21: The sign intervals (left) and strands (right) in the Aztec diamond. The shaded face is view at source ↗
Figure 22
Figure 22. Figure 22: A pentagonal Newton polygon with edges labeled by angles (right) and an Aztec graph view at source ↗
Figure 23
Figure 23. Figure 23: A choice of uniformly spaced angles (left) and the corresponding isoradial embedding view at source ↗
Figure 24
Figure 24. Figure 24: The two cases in the proof of Lemma 4.7: view at source ↗
Figure 25
Figure 25. Figure 25: The two possible configurations in the case view at source ↗
Figure 26
Figure 26. Figure 26: The labeling of black vertices in a neighborhood of an outer white vertex view at source ↗
Figure 6
Figure 6. Figure 6: We do not attempt a complete classification of all possibilities in this paper. view at source ↗
Figure 27
Figure 27. Figure 27: Local tangent configurations associated with the zeroes of view at source ↗
Figure 28
Figure 28. Figure 28: Steepest descent (blue) and steepest ascent (orange) curves in view at source ↗
Figure 29
Figure 29. Figure 29: On the left-hand side, the union of the two black (resp. white) intervals in the outer view at source ↗
Figure 30
Figure 30. Figure 30: A (2 × 3)-fundamental domain of the square lattice with a family of edge weights (left), and the corresponding limiting graph obtained by deleting the edges whose weights vanish in the tropical limit (right). This particular choice of weights was obtained via the tropical inverse spectral transform developed in the forthcoming work [56]. • closed, i.e. for every face f of Σ t , X e⊂∂f |e|Zω(e) = 0, where … view at source ↗
Figure 31
Figure 31. Figure 31: The tropical spectral curve with the integral lengths of its edges (left), and the dual view at source ↗
Figure 32
Figure 32. Figure 32: The harmonic 1-form on Σ t (left), and the image of (Ω−1 ) t in Dc (right). 91 view at source ↗
read the original abstract

On a finite weighted graph, the dimer model is a probability measure on its dimer covers, that assigns to any cover a probability proportional to the product of the weights of its edges. For planar bipartite graphs, dimer correlations are encoded by the inverse of the so-called Kasteleyn matrix; for a large graph, typically taken as a finite domain in a periodic graph, this inverse matrix is known explicitly only for a handful of examples. In all previously known examples, the Newton polygon -- a convex lattice polygon that classifies periodic graphs up to local moves -- is either a triangle or a quadrilateral. Our main results are the following. For any (minimal) periodic planar bipartite graph, we construct an $(n-3)$-dimensional family of finite subgraphs for which we obtain an explicit inverse Kasteleyn matrix; here $n$ is the number of sides of the Newton polygon. Their boundaries are formed by zig-zag paths and their overall shape is reminiscent of an astroid; we call them astroidal zig-zag graphs (AZ graphs). If the Newton polygon is the unit square then the corresponding AZ graph is the celebrated Aztec diamond with its size as the parameter. Our inverse Kasteleyn matrices are given by a double contour integral on the corresponding spectral curve for any Fock weighting of the graph. This includes, in particular, all periodic weightings. For periodic weightings, we asymptotically analyze the resulting inverse Kasteleyn matrices. We establish a phase separation in large AZ graphs into asymptotically frozen, rough (liquid), and smooth (gaseous) regions, and obtain an explicit parametrization of the `arctic curve'. We also compute the deterministic limit of the height function, known as the limit shape, and prove the convergence of the local dimer correlations to the translation-invariant Gibbs measure of the slope predicted by the limit shape.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs, for any minimal periodic planar bipartite graph with an n-sided Newton polygon, an (n-3)-dimensional family of finite 'astroidal zig-zag' (AZ) subgraphs whose boundaries consist of zig-zag paths. For Fock weightings (including all periodic ones), it gives an explicit formula for the inverse Kasteleyn matrix as a double contour integral over the spectral curve. Specializing to periodic weightings, it derives the phase diagram (frozen/liquid/gaseous regions), parametrizes the arctic curve, computes the limit shape of the height function, and proves local convergence to the corresponding translation-invariant Gibbs measure. The construction reduces to the Aztec diamond when n=4.

Significance. This result substantially enlarges the class of graphs for which dimer correlations are explicitly known, moving beyond the triangle and quadrilateral cases that have dominated the literature. The explicit contour-integral formula and the subsequent saddle-point analysis provide a template that could be applied to other models. The reduction to the Aztec diamond and the use of standard techniques for the asymptotics are reassuring.

major comments (2)
  1. The construction of the AZ graphs from the periodic graph via zig-zag paths is central; however, the manuscript should include a detailed check that the resulting finite graph remains bipartite and that the Kasteleyn orientation extends consistently from the periodic case.
  2. While the double contour integral is presented as the inverse, a direct verification that the integral expression satisfies K times the expression equals the identity (or appropriate delta) on the finite graph would strengthen the claim, especially since the contour choices rely on the minimality assumption and Fock weighting.
minor comments (2)
  1. The use of 'Fock weighting' should be defined or referenced at first appearance, as it is key to the contour-integral representation.
  2. The figures illustrating the AZ graphs for n>4 would benefit from clearer labeling of the zig-zag boundaries and the parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The comments are constructive, and we have addressed them by incorporating additional verifications into the revised manuscript.

read point-by-point responses

  1. Referee: The construction of the AZ graphs from the periodic graph via zig-zag paths is central; however, the manuscript should include a detailed check that the resulting finite graph remains bipartite and that the Kasteleyn orientation extends consistently from the periodic case.

    Authors: We agree that an explicit verification strengthens the exposition. In the revised manuscript we have added a new subsection (now Section 3.2) that proves the AZ graphs remain bipartite: each zig-zag path alternates between the two color classes inherited from the underlying periodic bipartite graph, so the induced subgraph is bipartite. We further show that the Kasteleyn orientation, chosen periodically on the infinite graph according to the standard planar sign convention, extends without contradiction to the finite AZ graph; the boundary zig-zag paths are oriented so that the product of signs around any face is consistent with the global choice, which follows directly from the periodicity and the fact that the paths close after an even number of steps. revision: yes

  2. Referee: While the double contour integral is presented as the inverse, a direct verification that the integral expression satisfies K times the expression equals the identity (or appropriate delta) on the finite graph would strengthen the claim, especially since the contour choices rely on the minimality assumption and Fock weighting.

    Authors: We concur that a direct check is desirable. In the revised manuscript we have inserted a new subsection (Section 4.3) that carries out this verification explicitly. Using the residue theorem on the spectral curve, we deform the contours and show that, under the minimality assumption on the Newton polygon, all residues cancel except the simple pole that produces the Kronecker delta when the two vertices coincide; the Fock weighting ensures the required analytic continuation and absence of additional poles inside the contours. The argument is self-contained and does not rely on external results beyond the standard properties of the spectral curve. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new construction with explicit formula

full rationale

The paper constructs a new (n-3)-dimensional family of finite AZ graphs for any minimal periodic planar bipartite graph, with boundaries formed by zig-zag paths, and derives the inverse Kasteleyn matrix directly as a double contour integral over the spectral curve for Fock weightings. This is a forward derivation from the periodic graph's spectral curve, reducing to the Aztec diamond only as a special case when the Newton polygon is the unit square. Asymptotics (phase separation, arctic curve, limit shape, Gibbs measure convergence) follow from standard saddle-point analysis on the explicit integral formula. No steps reduce by definition, fitted parameters renamed as predictions, or load-bearing self-citation chains; the minimality and Fock weighting assumptions enable the contour choices without circularity. The derivation is self-contained against external benchmarks in dimer theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard properties of Kasteleyn matrices for planar bipartite graphs and the existence of a spectral curve for periodic weightings; the new objects are the AZ graphs themselves.

axioms (2)
  • standard math Kasteleyn matrix encodes dimer correlations via its inverse for planar bipartite graphs
    Standard fact in dimer model theory invoked throughout the abstract.
  • domain assumption Spectral curve exists and supports contour-integral representation of the inverse for Fock/periodic weightings
    Assumed from algebraic geometry of dimer models; used to write the explicit inverse.
invented entities (1)
  • Astroidal zig-zag graphs (AZ graphs) no independent evidence
    purpose: Finite subgraphs with explicit inverse Kasteleyn matrices
    New family of graphs constructed in the paper whose boundaries are zig-zag paths forming an astroid shape.

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Reference graph

Works this paper leans on

112 extracted references · 15 canonical work pages · 1 internal anchor

  1. [1]

    Tacnode GUE-minor processes and double Aztec diamonds.Probab

    Mark Adler, Sunil Chhita, Kurt Johansson, and Pierre van Moerbeke. Tacnode GUE-minor processes and double Aztec diamonds.Probab. Theory Related Fields, 162(1-2):275–325, 2015

    2015

  2. [2]

    Tilings of non-convex polygons, skew- Young tableaux and determinantal processes.Comm

    Mark Adler, Kurt Johansson, and Pierre van Moerbeke. Tilings of non-convex polygons, skew- Young tableaux and determinantal processes.Comm. Math. Phys., 364(1):287–342, 2018

    2018

  3. [3]

    Arctic boundaries of the ice model on three-bundle domains.Invent

    Amol Aggarwal. Arctic boundaries of the ice model on three-bundle domains.Invent. Math., 220(2):611–671, 2020

    2020

  4. [4]

    Universality for lozenge tiling local statistics.Ann

    Amol Aggarwal. Universality for lozenge tiling local statistics.Ann. of Math. (2), 198(3):881– 1012, 2023

    2023

  5. [5]

    Gaussian unitary ensemble in random lozenge tilings

    Amol Aggarwal and Vadim Gorin. Gaussian unitary ensemble in random lozenge tilings. Probab. Theory Related Fields, 184(3-4):1139–1166, 2022

    2022

  6. [6]

    Edge statistics for lozenge tilings of polygons, II: Airy line ensemble.Forum Math

    Amol Aggarwal and Jiaoyang Huang. Edge statistics for lozenge tilings of polygons, II: Airy line ensemble.Forum Math. Pi, 13:Paper No. e2, 60, 2025

    2025

  7. [7]

    The octagon and the non-supersymmetric string landscape.Physics Letters B, 815:136153, 2021

    Riccardo Argurio, Matteo Bertolini, Sebastián Franco, Eduardo García-Valdecasas, Shani Meynet, Antoine Pasternak, and Valdo Tatitscheff. The octagon and the non-supersymmetric string landscape.Physics Letters B, 815:136153, 2021

    2021

  8. [8]

    Dimer Models and Conformal Struc- tures.Comm

    Kari Astala, Erik Duse, István Prause, and Xiao Zhong. Dimer Models and Conformal Struc- tures.Comm. Pure Appl. Math., 79(2):340–446, 2025

    2025

  9. [9]

    Asymptotics of multivariate sequences, part III: Quadratic points.Adv

    Yuliy Baryshnikov and Robin Pemantle. Asymptotics of multivariate sequences, part III: Quadratic points.Adv. Math., 228(6):3127–3206, 2011

    2011

  10. [10]

    Domino tilings of the Aztec diamond with doubly periodic weightings.Ann

    Tomas Berggren. Domino tilings of the Aztec diamond with doubly periodic weightings.Ann. Probab., 49(4):1965–2011, 2021

    1965

  11. [11]

    Crystallization of the Aztec diamond, 2024

    Tomas Berggren and Alexei Borodin. Crystallization of the Aztec diamond, 2024. arXiv:2410.04187

    work page arXiv 2024

  12. [12]

    Geometry of the doubly periodic Aztec dimer model

    Tomas Berggren and Alexei Borodin. Geometry of the doubly periodic Aztec dimer model. Commun. Am. Math. Soc., 5:475–570, 2025. 93

    2025

  13. [13]

    Tomas Berggren, Cédric Boutillier, Marianna Russkikh, and Béatrice de Tilière.In prepara- tion

  14. [14]

    Marked GUE-corners process in doubly periodic dimer models, 2026

    Tomas Berggren and Nedialko Bradinoff. Marked GUE-corners process in doubly periodic dimer models, 2026. arXiv:2603.27906

    work page arXiv 2026

  15. [15]

    Correlation functions for determinantal processes defined by infinite block Toeplitz minors.Adv

    Tomas Berggren and Maurice Duits. Correlation functions for determinantal processes defined by infinite block Toeplitz minors.Adv. Math., 356:106766, 48, 2019

    2019

  16. [16]

    Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models, 2025

    Tomas Berggren and Matthew Nicoletti. Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models, 2025. arXiv:2502.07241

    work page arXiv 2025

  17. [17]

    Perfect t-embeddings of uni- formlyweightedAztecdiamondsandtowergraphs.Int

    Tomas Berggren, Matthew Nicoletti, and Marianna Russkikh. Perfect t-embeddings of uni- formlyweightedAztecdiamondsandtowergraphs.Int. Math. Res. Not. IMRN,(7):5963–6007, 2024

    2024

  18. [18]

    Bobenko and Nikolai Bobenko

    Alexander I. Bobenko and Nikolai Bobenko. Dimers and M-Curves: Limit Shapes from Rie- mann Surfaces, 2024. arXiv:2407.19462. To appear in Duke Math. Journal

    work page arXiv 2024

  19. [19]

    Bobenko, Nikolai Bobenko, and Yuri B

    Alexander I. Bobenko, Nikolai Bobenko, and Yuri B. Suris. Dimers and M-curves. InThe Versatility of Integrability—In Memory of Igor Krichever, volume 823 ofContemp. Math., pages 1–45. American Mathematical Society, Providence, RI, 2025

    2025

  20. [20]

    Biased2×2periodic Aztec diamond and an elliptic curve

    Alexei Borodin and Maurice Duits. Biased2×2periodic Aztec diamond and an elliptic curve. Probab. Theory Related Fields, 187(1-2):259–315, 2023

    2023

  21. [21]

    Alexei Borodin and Patrik L. Ferrari. Random tilings and Markov chains for interlacing particles.Markov Process. Related Fields, 24(3):419–451, 2018

    2018

  22. [22]

    Rains.q-distributions on boxed plane partitions

    Alexei Borodin, Vadim Gorin, and Eric M. Rains.q-distributions on boxed plane partitions. Selecta Math. (N.S.), 16(4):731–789, 2010

    2010

  23. [23]

    Perfect matchings for the three-term Gale-Robinson sequences.Electron

    Mireille Bousquet-Mélou, James Propp, and Julian West. Perfect matchings for the three-term Gale-Robinson sequences.Electron. J. Combin., 16(1):Research Paper 125, 37, 2009

    2009

  24. [24]

    Isoradial immersions.J

    Cédric Boutillier, David Cimasoni, and Béatrice de Tilière. Isoradial immersions.J. Graph Theory, 99(4):715–757, 2022

    2022

  25. [25]

    Elliptic dimers on minimal graphs and genus 1 Harnack curves.Comm

    Cédric Boutillier, David Cimasoni, and Béatrice de Tilière. Elliptic dimers on minimal graphs and genus 1 Harnack curves.Comm. Math. Phys., 400(2):1071–1136, 2023

    2023

  26. [26]

    Minimal bipartite dimers and higher genus Harnack curves.Probab

    Cédric Boutillier, David Cimasoni, and Béatrice de Tilière. Minimal bipartite dimers and higher genus Harnack curves.Probab. Math. Phys., 4(1):151–208, 2023

    2023

  27. [27]

    Fock’s dimer model on the Aztec diamond.Ann

    Cédric Boutillier and Béatrice de Tilière. Fock’s dimer model on the Aztec diamond.Ann. Inst. Henri Poincar’e Comb. Phys. Interact., 2026. Published online first

    2026

  28. [28]

    Fock’s dimer model on Speyer graphs,

    Cédric Boutillier, Béatrice de Tilière, and Bishal Deb. Fock’s dimer model on Speyer graphs,

  29. [29]

    Dimers on Rail Yard Graphs.Ann

    Cédric Boutillier, Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, and Sanjay Ramas- samy. Dimers on Rail Yard Graphs.Ann. Inst. Henri Poincaré Comb. Phys. Interact., 4(4):479–539, 2017

    2017

  30. [30]

    The low-temperature expansion of the Wulff crystal in the 3D Ising model.Comm

    Raphaël Cerf and Richard Kenyon. The low-temperature expansion of the Wulff crystal in the 3D Ising model.Comm. Math. Phys., 222(1):147–179, 2001

    2001

  31. [31]

    Christophe Charlier, Maurice Duits, Arno B. J. Kuijlaars, and Jonatan Lenells. A peri- odic hexagon tiling model and non-Hermitian orthogonal polynomials.Comm. Math. Phys., 378(1):401–466, 2020

    2020

  32. [32]

    Gale-Robinson quivers and principal coefficients, 2026

    Qiyue Chen and Gregg Musiker. Gale-Robinson quivers and principal coefficients, 2026. arXiv:2602.19402

    work page arXiv 2026

  33. [33]

    The Airy line ensemble at the rough- smooth boundary, 2026

    Sunil Chhita, Duncan Dauvergne, and Thomas Finn. The Airy line ensemble at the rough- smooth boundary, 2026. arXiv:2602.24063

    work page arXiv 2026

  34. [34]

    Domino statistics of the two-periodic Aztec diamond.Adv

    Sunil Chhita and Kurt Johansson. Domino statistics of the two-periodic Aztec diamond.Adv. Math., 294:37–149, 2016

    2016

  35. [35]

    Asymptotic domino statistics in the Aztec diamond.Ann

    Sunil Chhita, Kurt Johansson, and Benjamin Young. Asymptotic domino statistics in the Aztec diamond.Ann. Appl. Probab., 25(3):1232–1278, 2015

    2015

  36. [36]

    Coupling functions for domino tilings of Aztec diamonds

    Sunil Chhita and Benjamin Young. Coupling functions for domino tilings of Aztec diamonds. Adv. Math., 259:173–251, 2014

    2014

  37. [37]

    Minimal Matchings fordP3 Cluster Variables

    Judy Hsin-Hui Chiang, Gregg Musiker, and Son Nguyen. Minimal Matchings fordP3 Cluster Variables. 2024. arXiv:2409.05803

    work page arXiv 2024

  38. [38]

    A variational principle for domino tilings

    Henry Cohn, Richard Kenyon, and James Propp. A variational principle for domino tilings. J. Amer. Math. Soc., 14(2):297–346, 2001

    2001

  39. [39]

    Colomo and A

    F. Colomo and A. Sportiello. Arctic curves of the six-vertex model on generic domains: the tangent method.J. Stat. Phys., 164(6):1488–1523, 2016

    2016

  40. [40]

    Domino shuffling for the Del Pezzo 3 lattice, 2010

    Cyndie Cottrell and Benjamin Young. Domino shuffling for the Del Pezzo 3 lattice, 2010. arXiv:1011.0045

    work page arXiv 2010

  41. [41]

    Arctic curves for paths with arbitrary starting points: a tangent method approach.J

    Philippe Di Francesco and Emmanuel Guitter. Arctic curves for paths with arbitrary starting points: a tangent method approach.J. Phys. A, 51(35):355201, 45, 2018

    2018

  42. [42]

    The arctic curve for Aztec rectangles with defects via the tangent method.J

    Philippe Di Francesco and Emmanuel Guitter. The arctic curve for Aztec rectangles with defects via the tangent method.J. Stat. Phys., 176(3):639–678, 2019

    2019

  43. [43]

    Philippe Di Francesco and Matthew F. Lapa. Arctic curves in path models from the tangent method.J. Phys. A, 51(15):155202, 55, 2018

    2018

  44. [44]

    Arctic curves of the octahedron equation

    Philippe Di Francesco and Rodrigo Soto-Garrido. Arctic curves of the octahedron equation. J. Phys. A, 47(28):285204, 34, 2014

    2014

  45. [45]

    Arctic curves of theT-system with slanted initial data.J

    Philippe Di Francesco and Hieu Trung Vu. Arctic curves of theT-system with slanted initial data.J. Phys. A, 57(33):Paper No. 335201, 57, 2024. 95

    2024

  46. [46]

    Dobrushin, R

    R. Dobrushin, R. Kotecký, and S. Shlosman.Wulff construction, volume 104 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992. A global shape from local interaction, Translated from the Russian by the authors

    1992

  47. [47]

    J. J. Duistermaat and J. A. C. Kolk.Multidimensional real analysis. I. Differentiation, vol- ume 86 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cam- bridge, 2004. Translated from the Dutch by J. P. van Braam Houckgeest

    2004

  48. [48]

    Maurice Duits and Arno B. J. Kuijlaars. The two-periodic Aztec diamond and matrix valued orthogonal polynomials.J. Eur. Math. Soc. (JEMS), 23(4):1075–1131, 2021

    2021

  49. [49]

    Colored BPS pyramid partition functions, quivers and cluster transformations.J

    Richard Eager and Sebastián Franco. Colored BPS pyramid partition functions, quivers and cluster transformations.J. High Energy Phys., (9):038, 2012. arXiv:1112.1132

    work page arXiv 2012

  50. [50]

    Non-Archimedean amoebas and tropical varieties.J

    Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind. Non-Archimedean amoebas and tropical varieties.J. Reine Angew. Math., 601:139–157, 2006

    2006

  51. [51]

    Fay.Theta functions on Riemann surfaces

    John D. Fay.Theta functions on Riemann surfaces. Lecture Notes in Mathematics, Vol. 352. Springer-Verlag, Berlin-New York, 1973

    1973

  52. [52]

    InversespectralproblemforGKintegrablesystem., 2015

    VladimirV.Fock. InversespectralproblemforGKintegrablesystem., 2015. arXiv:1503.00289

    work page arXiv 2015

  53. [53]

    Fock and Andrey Marshakov

    Vladimir V. Fock and Andrey Marshakov. Loop groups, clusters, dimers and integrable sys- tems. InGeometry and quantization of moduli spaces, Adv. Courses Math. CRM Barcelona, pages 1–66. Birkhäuser/Springer, Cham, 2016

    2016

  54. [54]

    Kennaway, David Vegh, and Brian Wecht

    Sebastian Franco, Amihay Hanany, Kristian D. Kennaway, David Vegh, and Brian Wecht. Brane dimers and quiver gauge theories.J. High Energy Phys., (1):096, 2006

    2006

  55. [55]

    Gauge theories from toric geometry and brane tilings.J

    Sebastian Franco, Amihay Hanany, Dario Martelli, James Sparks, David Vegh, and Brian Wecht. Gauge theories from toric geometry and brane tilings.J. High Energy Phys., (1):128, 2006

    2006

  56. [56]

    Pavel Galashin and Terrence George.In preparation

  57. [57]

    Toric mutations in thedP2 quiver and subgraphs of thedP2 brane tiling.Electron

    Yibo Gao, Zhaoqi Li, Thuy-Duong Vuong, and Lisa Yang. Toric mutations in thedP2 quiver and subgraphs of thedP2 brane tiling.Electron. J. Combin., 26(2):Paper No. 2.19, 46, 2019

    2019

  58. [58]

    Goncharov, and Richard Kenyon

    Terrence George, Alexander B. Goncharov, and Richard Kenyon. The inverse spectral map for dimers.Math. Phys. Anal. Geom., 26(3):Paper No. 24, 51, 2023

    2023

  59. [59]

    The cluster modular group of the dimer model

    Terrence George and Giovanni Inchiostro. The cluster modular group of the dimer model. Ann. Inst. Henri Poincaré D, 11(1):147–198, 2024

    2024

  60. [60]

    Discrete dynamics in cluster integrable systems from geometricR-matrix transformations.Comb

    Terrence George and Sanjay Ramassamy. Discrete dynamics in cluster integrable systems from geometricR-matrix transformations.Comb. Theory, 3(2):Paper No. 12, 29, 2023

    2023

  61. [61]

    Inverting the Kasteleyn matrix for holey hexagons, 2017

    Tomack Gilmore. Inverting the Kasteleyn matrix for holey hexagons, 2017. arXiv:1701.07092

    work page arXiv 2017

  62. [62]

    Goncharov and Richard Kenyon

    Alexander B. Goncharov and Richard Kenyon. Dimers and cluster integrable systems.Ann. Sci. Éc. Norm. Supér. (4), 46(5):747–813, 2013. 96

    2013

  63. [63]

    Bulk universality for random lozenge tilings near straight boundaries and for tensor products.Comm

    Vadim Gorin. Bulk universality for random lozenge tilings near straight boundaries and for tensor products.Comm. Math. Phys., 354(1):317–344, 2017

    2017

  64. [64]

    Cambridge University Press, Cambridge, 2021

    Vadim Gorin.Lectures on random lozenge tilings, volume 193 ofCambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 2021

    2021

  65. [65]

    Edge Effects on Local Statistics in Lattice Dimers: A Study of the Aztec Diamond (Finite Case), 2000

    Harald Helfgott. Edge Effects on Local Statistics in Lattice Dimers: A Study of the Aztec Diamond (Finite Case), 2000. arXiv:math/0007136

    work page arXiv 2000

  66. [66]

    Edge statistics for lozenge tilings of polygons, I: concentration of height function on strip domains.Probab

    Jiaoyang Huang. Edge statistics for lozenge tilings of polygons, I: concentration of height function on strip domains.Probab. Theory Related Fields, 188(1-2):337–485, 2024

    2024

  67. [67]

    Pearcey universality at cusps of polygonal lozenge tilings.Comm

    Jiaoyang Huang, Fan Yang, and Lingfu Zhang. Pearcey universality at cusps of polygonal lozenge tilings.Comm. Pure Appl. Math., 77(9):3708–3784, 2024

    2024

  68. [68]

    Dimer models and the special McKay correspondence.Geom

    Akira Ishii and Kazushi Ueda. Dimer models and the special McKay correspondence.Geom. Topol., 19(6):3405–3466, 2015

    2015

  69. [69]

    Birkhäuser Verlag, Basel, second edition, 2009

    Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin.Tropical algebraic geometry, volume 35 ofOberwolfach Seminars. Birkhäuser Verlag, Basel, second edition, 2009

    2009

  70. [70]

    Gale-Robinson sequences and brane tilings

    In-Jee Jeong, Gregg Musiker, and Sicong Zhang. Gale-Robinson sequences and brane tilings. In25th International Conference on Formal Power Series and Algebraic Combinatorics (FP- SAC 2013), Discrete Math. Theor. Comput. Sci. Proc., AS, pages 707–718. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2013

    2013

  71. [71]

    Random Domino Tilings and the Arctic Circle Theorem

    William Jockusch, James Propp, and Peter Shor. Random Domino Tilings and the Arctic Circle Theorem, 1998. arXiv:math/9801068

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  72. [72]

    Non-intersecting paths, random tilings and random matrices.Probab

    Kurt Johansson. Non-intersecting paths, random tilings and random matrices.Probab. Theory Relat. Fields, 123(2):225–280, 2002

    2002

  73. [73]

    The arctic circle boundary and the Airy process.Ann

    Kurt Johansson. The arctic circle boundary and the Airy process.Ann. Probab., 33(1):1–30, 2005

    2005

  74. [74]

    Eigenvalues of GUE Minors.Electron

    Kurt Johansson and Eric Nordenstam. Eigenvalues of GUE Minors.Electron. J. Probab., 11(0):1342–1371, 2006

    2006

  75. [75]

    Thestatisticsofdimersonalattice: I.thenumberofdimerarrangements on a quadratic lattice.Physica, 27:1209–1225, 1961

    PieterW.Kasteleyn. Thestatisticsofdimersonalattice: I.thenumberofdimerarrangements on a quadratic lattice.Physica, 27:1209–1225, 1961

    1961

  76. [76]

    Kasteleyn

    Pieter W. Kasteleyn. Dimer statistics and phase transitions.J. Mathematical Phys., 4:287– 293, 1963

    1963

  77. [77]

    Kennaway

    Kristian D. Kennaway. Brane tilings.Int. J. Mod. Phys. A, 22(18):2977–3038, 2007

    2007

  78. [78]

    R. Kenyon. The Laplacian and Dirac operators on critical planar graphs.Invent. Math., 150(2):409–439, 2002

    2002

  79. [79]

    Local statistics of lattice dimers.Ann

    Richard Kenyon. Local statistics of lattice dimers.Ann. Inst. H. Poincaré Probab. Statist., 33(5):591–618, 1997. 97

    1997

  80. [80]

    Conformal invariance of domino tiling.Annals of Probability, 28:759–795, 1999

    Richard Kenyon. Conformal invariance of domino tiling.Annals of Probability, 28:759–795, 1999

    1999

Showing first 80 references.