pith. sign in

arxiv: 2304.08468 · v3 · submitted 2023-04-17 · 🧮 math.PR · math-ph· math.CO· math.MP

Large deviations for the 3D dimer model

Nishant Chandgotia , Scott Sheffield , Catherine Wolfram This is my paper

Pith reviewed 2026-05-24 09:06 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.COmath.MP
keywords dimer modellarge deviationsdivergence-free flowsspecific entropyGibbs measuresthree-dimensional latticeperfect matchingsvariational principle
0
0 comments X

The pith

Random flows from 3D dimer matchings converge to the unique flow maximizing integrated specific entropy under fixed boundary data.

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

The paper proves a large-deviation principle for the divergence-free flows induced by uniform random perfect matchings on fine 3D lattice approximations to a bounded region. As the mesh size tends to zero with controlled boundaries, these random flows converge in law to the deterministic flow that maximizes the integral of ent(g(x)) over the region, where ent(s) is the highest specific entropy attainable by any ergodic Gibbs measure with mean current s. The authors establish that ent is continuous and strictly concave on the interior of the octahedron of admissible currents, which forces uniqueness of the maximizer. The argument relies on Hall's theorem, the ergodic theorem, non-intersecting paths, and double-dimer cycle swaps rather than planar tools.

Core claim

The law of the random divergence-free flow induced by a uniform dimer matching on a fine discretization of a region R in R^3 converges to a delta mass at the flow g that maximizes the integral over R of ent(g(x)) dx, where ent(s) is the maximal specific entropy of an ergodic Gibbs measure with mean current s. Moreover, ent is continuous and strictly concave away from the edges of the octahedron of admissible currents, so the maximizer is unique under suitable boundary conditions. Large deviation principles quantify the exponential decay of probabilities for deviations from this g.

What carries the argument

The entropy function ent(s), which assigns to each admissible mean current s the supremum of specific entropies over all ergodic Gibbs measures with that mean current; it serves as both the rate function for the large deviations and the objective in the variational problem that selects the limiting flow.

If this is right

  • The limiting object is the unique solution of a strictly concave variational problem over divergence-free fields with prescribed boundary flux.
  • The probability of observing a flow whose average entropy is less than the maximum by a fixed positive amount decays exponentially in the volume.
  • Small changes in the boundary data produce small changes in the limiting flow because of strict concavity.
  • The same variational characterization holds for any bounded region whose boundary conditions admit at least one divergence-free extension.

Where Pith is reading between the lines

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

  • Numerical approximation of the limiting flow could proceed by discretizing the entropy-maximization problem once a computable surrogate for ent becomes available.
  • The result suggests that hydrodynamic limits for other three-dimensional matching models may likewise be governed by an entropy functional on currents.
  • Because ent is not given explicitly, its gradient or level sets might encode geometric constraints on admissible currents that are invisible from the octahedron alone.

Load-bearing premise

The maximal specific entropy for each interior mean current is achieved by at least one ergodic Gibbs measure, and the resulting function ent is strictly concave away from the edges of the octahedron.

What would settle it

A sequence of dimer configurations on successively finer meshes whose induced flows converge in probability to a divergence-free vector field other than the entropy maximizer for the same boundary data.

Figures

Figures reproduced from arXiv: 2304.08468 by Catherine Wolfram, Nishant Chandgotia, Scott Sheffield.

Figure 1
Figure 1. Figure 1: Tiling of an Aztec diamond and bipartite coloring of squares in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The four brickwork patterns in two dimensions. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example of a fixed boundary region Rn ⊂ R for the LDP in two dimensions. Cohn, Kenyon and Propp showed that as the mesh size tends to zero (and the rescaled boundary heights converge to some function on ∂R) the random height function converges in probability to the unique continuum function u that (given the boundary values) minimizes the integral Z σ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A dimer tiling of the 10 × 10 × 10 cube and the bipartite coloring of the cubes in Z 3 . The figure below represents a random tiling τ of a region R called the Aztec pyramid (formed by stacking Aztec diamonds of width 2, 4, 6, . . . , 36). Next to it is again the underlying black￾and-white checkerboard coloring. Recall that (due to the reference flow) the divergence-free flow fτ sends a 1/6 unit of current… view at source ↗
Figure 5
Figure 5. Figure 5: A dimer tiling of an Aztec pyramid and the bipartite coloring of the cubes in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A tiling of a larger pyramid of Aztec diamonds, from the side and from below. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Tiling of an Aztec octahedron [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Tiling of an Aztec prism. A large deviation principle is a result about a sequence of probability measures (ρn)n≥1 which quantifies the probability of rare events at an exponential scale as n → ∞. More precisely, a sequence of probability measures (ρn)n≥1 on a topological space (X, B) is said to satisfy a large deviation principle (LDP) with rate function I and speed vn if I : X → [0, ∞) is a lower semicon… view at source ↗
Figure 9
Figure 9. Figure 9: An example of a free-boundary tiling of R We denote the corresponding free-boundary tiling flows of R at scale n by T Fn(R). Note that T Fn(R) is a finite set for all n. There is a signed flux measure on ∂R (supported on the points where edges of 1 n Z 3 cross ∂R) that encodes the net amount of flow directed into R. Since fτ is divergence-free, the total flux through ∂R is zero. (See Definition 5.5.) If τ … view at source ↗
Figure 10
Figure 10. Figure 10: Schematic for the patching theorem. For the hard boundary large deviation principle, we also prove a generalized patching theorem (Theorem 8.6.2), which says roughly that two tilings can be patched on a general annular region R \ R′ if they have the same boundary value b on ∂R and the inner one approximates a flexible flow g ∈ AF(R, b). Proving the patching theorems will be one of the more challenging asp… view at source ↗
Figure 11
Figure 11. Figure 11: A local move or flip in 2D. It turns out that the analogous statement is false in 3D. In fact, as we will explain in Section 3, there is no collection of local moves for which the analogous statement would be true in 3D. In 3D, one can construct (for any K > 0) a tiling τ of Z 3 that is 1. non-frozen — i.e., there exists a tiling τ ′ ̸= τ that disagrees with τ on finitely many edges. 2. locally frozen to … view at source ↗
Figure 12
Figure 12. Figure 12: 2D non-intersecting paths. There is an obvious bijection between non-intersecting path ensembles (as shown above) and dimer tilings (which is one way to deduce the integrability of the dimer model in two dimensions). Applying local moves corresponds to shifting these paths up and down locally. One can analogously superimpose a red three-dimensional matching with a black brickwork matching, to obtain an en… view at source ↗
Figure 13
Figure 13. Figure 13: Two examples of 3D non-intersecting paths with the same endpoints. [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A flip, a trit and a flip-rigid configuration called a hopfion. It remains open whether it is possible to connect any tiling of a rectangular box to any other using both the flip and trit moves shown above. It is still possible in 3D to generate random tilings of finite regions using Glauber dynamics (using an update algorithm that allows for the tiling to be modified along cycles of arbitrary length, see… view at source ↗
Figure 15
Figure 15. Figure 15: A local move or flip in two dimensions [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: (1) an example of a sequence of local moves transforming one tiling into another [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: A flip, a trit, and a flip-rigid configuration called a [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Parts of five strips drawn on the dual graph (left) and as a tiling (right) [PITH_FULL_IMAGE:figures/full_fig_p034_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (1) Cubes from a slab of Z 3 visible from above the slab in a dimer tiling τ of Z 3 , (2) tiles from τ drawn on the hexagonal lattice as edges colored pink,blue and orange, (3) the same tiles drawn as lozenges obtained by taking the Voronoi cells containing these edges, and (4) a key giving the translation between lozenge tiles and 3D dimer bricks. observations follow from this discussion: • All ergodic G… view at source ↗
Figure 20
Figure 20. Figure 20: Three adjacent cubes in C2c, with intersection with Lc in orange. For each C ∈ C2c, C ∩ Lc is a hexagon, hence the faces of Lc are hexagons. By observation we see that any two adjacent hexagons meet in an edge, any three adjacent hexagons meet at a vertex, and there are no collections of > 3 adjacent hexagons. Hence Lc is a copy of the hexagonal lattice. Finally [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Two dimensional schematic for patching. Equivalently, we want to show that this annular region can be tiled by dimers exactly so that it agrees with τ1 on one boundary and with τ2 on the other boundary. To do this, we need a condition to show that a region is tileable. A general condition for tileability, which works in any dimension, is given by the classical Hall’s matching theorem ([Hal35], stated here… view at source ↗
Figure 22
Figure 22. Figure 22: A region that is balanced (i.e. the number of black squares is equal to the number [PITH_FULL_IMAGE:figures/full_fig_p065_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Here is an example of a region of the form “ [PITH_FULL_IMAGE:figures/full_fig_p068_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: A potential counterexample set U ⊂ A. The red set U ⊂ A in [PITH_FULL_IMAGE:figures/full_fig_p070_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The middle (blue), thin (orange), and corner (green) regions of [PITH_FULL_IMAGE:figures/full_fig_p071_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: We can define U ′ to be the intersection of U with the middle region from [PITH_FULL_IMAGE:figures/full_fig_p071_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: The region U ′′ (blue) and the tiles from τ that intersect U ′ (green). In order to prove that U itself has more black than white squares, we will divide the rest of U into multiple pieces to treat separately, depicted in [PITH_FULL_IMAGE:figures/full_fig_p072_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Depiction of all the regions that we divide [PITH_FULL_IMAGE:figures/full_fig_p073_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: An example of tilings τ1, τ2, the loops in (τ1, τ2), and chain swapped tilings τ ′ 1 , τ ′ 2 . For the rest of this section, we study whether or not certain properties (ergodicity, the Gibbs property) are preserved under chain swapping, and how certain quantities (entropy, mean current) transform under chain swapping. The first result is that chain swapping preserves ergodicity. Proposition 7.4.2. If µ is… view at source ↗
Figure 30
Figure 30. Figure 30: Example of an infinite path ℓ ⊂ (τ1, τ2) hitting Bn, with first entrance, last exit, and left and right rays labeled. same direction.) Without loss of generality (by translating and possibly changing the choice of face F), there exists c ∈ (0, 1) such that given δ > 0, for n large enough |AP ∩ Sfirst(Bn)| > cn2 (52) with ν-probability 1 − δ. Given x ∈ AP ∩ Sfirst(Bn), there exists a unique infinite path ℓ… view at source ↗
Figure 31
Figure 31. Figure 31: In all three pictures, the transparent cube is [PITH_FULL_IMAGE:figures/full_fig_p102_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Two examples corresponding to Case 2. In this case the infinite path does not intersect T, so this can happen either if the final exit point y ∈ ∂Bn \ F (left) or if the final exit point y ∈ F, but the right ray ℓ+(y) crosses P again outside Bn (right). γ(x) x ∈ Zn y ℓ+(y) ℓ−(x) T T [PITH_FULL_IMAGE:figures/full_fig_p103_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Corresponding to Case 3, if γ(x) ∪ ℓ+(y) never crosses P \ F, then the resulting infinite path cannot have well-defined slope. 3. Remaining paths forced to have no well-defined slope: if x ∈ Zn is not in Case 1 or Case 2, then the path ℓ := ℓ−(x) ∪ γ(x) ∪ ℓ+(y) does not intersect T and does not cross P \ F ◦ at any time after going through x. This implies that ℓ−(x) and ℓ+(y) are contained in the same P h… view at source ↗
Figure 34
Figure 34. Figure 34: A cube cut into one regular tetrahedron and four right-angled tetrahedra. The [PITH_FULL_IMAGE:figures/full_fig_p119_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Above is an example of v = (v1, v2, v3) and its relationship to w(v) = w. backwards in k to choose the tile type for τv on all other slabs Lk = C2k ∪ C2k+1. The reference tiling τ1 consists of all −η1 tiles, which connect C2k to C2k−1. Subtracting τ1, the tiles in −τ1 connect C2k−1 to C2k, meaning that they connected the “odd” half of Lk−1 to the “even” half of Lk. Hence in the double dimer tiling (τv, τ1… view at source ↗
Figure 36
Figure 36. Figure 36: On the left is a face of one tetrahedron. The segments are the ends of channels, [PITH_FULL_IMAGE:figures/full_fig_p126_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: 2D schematic picture for the proof of the generalized patching theorem. [PITH_FULL_IMAGE:figures/full_fig_p132_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Flip and trit. Both the flip and the trit amount to finding a cycle in Z 3 (of length 4 or 6 respectively) that alternates between membership and non-membership in τ , and then swapping the members and non-members. Generally, a cycle swap is a swap of an alternating cycle of length k, and a k-swap is a cycle swap for which the cycle has length k. It is clear that any two perfect matchings of the same regi… view at source ↗
Figure 39
Figure 39. Figure 39: Two tilings of the slanted cylinder. The left and right edges are glued. [PITH_FULL_IMAGE:figures/full_fig_p146_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: Front and back sides of the surface of a cube, with five “stripes” wrapping around [PITH_FULL_IMAGE:figures/full_fig_p147_40.png] view at source ↗
read the original abstract

In 2000, Cohn, Kenyon and Propp studied uniformly random perfect matchings of large induced subgraphs of $\mathbb Z^2$ (a.k.a. dimer configurations or domino tilings) and developed a large deviation theory for the associated height functions. We establish similar results for large induced subgraphs of $\mathbb Z^3$. To formulate these results, recall that a perfect matching on a bipartite graph induces a flow that sends one unit of current from each even vertex to its odd partner. One can then subtract a "reference flow'' to obtain a divergence-free flow. We show that the flow induced by a uniformly random dimer configuration converges in law (when boundary conditions on a bounded $R \subset \mathbb R^3$ are controlled and the mesh size tends to zero) to the deterministic divergence-free flow $g$ on $R$ that maximizes $$\int_{R} \text{ent}(g(x)) \,dx$$ given the boundary data, where $\text{ent}(s)$ is the maximal specific entropy obtained by an ergodic Gibbs measure with mean current $s$. The function $\text{ent}$ is not known explicitly, but we prove that it is continuous and {\em strictly concave} on the octahedron $\mathcal O$ of possible mean currents (except on the edges of $\mathcal O$) which implies (under reasonable boundary conditions) that the maximizer is uniquely determined. We further establish two versions of a large deviation principle, using the integral above to quantify how exponentially unlikely the discrete random flows are to approximate other deterministic flows. The planar dimer model is mathematically rich and well-studied, but many of the most powerful tools do not seem readily adaptable to higher dimensions. Our analysis begins with a smaller set of tools, which include Hall's matching theorem, the ergodic theorem, non-intersecting-lattice-path formulations, and double-dimer cycle swaps.

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

0 major / 2 minor

Summary. The paper establishes large deviation principles and a law of large numbers for the divergence-free flows induced by uniformly random perfect matchings (dimer configurations) on large induced subgraphs of Z^3. Under controlled boundary conditions and as the mesh size tends to zero, these random flows converge in law to the deterministic divergence-free flow g that maximizes the integral over R of ent(g(x)) dx, where ent(s) is the maximal specific entropy of an ergodic Gibbs measure with mean current s. The function ent is proved continuous and strictly concave on the interior of the octahedron O of admissible mean currents (except the edges), implying uniqueness of the maximizer; two versions of the LDP are derived from this variational principle. The proofs rely on Hall's theorem, the ergodic theorem, non-intersecting lattice paths, and double-dimer cycle swaps.

Significance. If the central claims hold, the work provides the first large-deviation and hydrodynamic-limit results for the three-dimensional dimer model, extending the 2000 Cohn-Kenyon-Propp theory from Z^2. The strict concavity of ent on int(O) is a key technical contribution that guarantees uniqueness without explicit knowledge of ent, and the use of ergodic Gibbs measures together with matching and cycle-swap arguments supplies a workable toolkit for higher-dimensional tilings where planar techniques fail. The results are falsifiable via the stated variational characterization and supply a concrete entropy functional for future numerical or asymptotic checks.

minor comments (2)
  1. [Abstract / Introduction] The abstract and introduction refer to 'post-hoc boundary control' and 'reasonable boundary conditions,' but the precise formulation of admissible boundary data (e.g., how the mesh-size limit interacts with the prescribed divergence-free boundary flow) is not visible in the high-level statements; a dedicated subsection or theorem statement spelling out the admissible class would improve readability.
  2. [§2 (preliminaries)] The reference flow subtracted to obtain the divergence-free current is mentioned but its explicit construction on the discrete graph (and its convergence to a continuum reference) is not detailed in the provided overview; adding a short paragraph or diagram in §2 would clarify the normalization step.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs the entropy ent(s) externally as the maximal specific entropy of ergodic Gibbs measures with mean current s, then proves its continuity and strict concavity on the interior of the octahedron O (except edges) via Hall's theorem, the ergodic theorem, non-intersecting paths, and double-dimer swaps. The LDP and convergence to the unique maximizer of ∫ ent(g(x)) dx are derived from these independently established properties and boundary data, without any reduction of the maximizer to a fitted quantity, self-citation chain, or definitional equivalence. The reference to Cohn-Kenyon-Propp is to independent 2D prior work and is not load-bearing for the 3D results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of ergodic Gibbs measures realizing the maximal entropy ent(s) and on the applicability of the ergodic theorem to dimer configurations; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption An ergodic Gibbs measure with given mean current s achieves the maximal specific entropy ent(s)
    Used to define the limiting variational problem; invoked when the entropy functional is introduced.
  • standard math The ergodic theorem applies to the sequence of finite-volume dimer measures
    Supports passage to the continuum limit and definition of ent.

pith-pipeline@v0.9.0 · 5891 in / 1356 out tokens · 28048 ms · 2026-05-24T09:06:25.143759+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 5 internal anchors

  1. [1]

    Yang–mills for probabilists

    [Cha19] Sourav Chatterjee. Yang–mills for probabilists. In Probability and Analysis in Interacting Physical Systems: In Honor of SRS Varadhan, Berlin, August, 2016 , pages 1–16. Springer,

    work page 2016

  2. [2]

    Non- integrable dimers: universal fluctuations of tilted height profiles

    [GMT20] Alessandro Giuliani, Vieri Mastropietro, and Fabio Lucio Toninelli. Non- integrable dimers: universal fluctuations of tilted height profiles. Comm. Math. Phys., 377(3):1883–1959,

    work page 1959

  3. [3]

    Weakly non-planar dimers

    [GRT22] Alessandro Giuliani, Bruno Renzi, and Fabio Toninelli. Weakly non-planar dimers. arXiv preprint arXiv:2207.10428 ,

    work page arXiv

  4. [4]

    Local dimer dynamics in higher dimensions

    [HLT23] Ivailo Hartarsky, Lyuben Lichev, and Fabio Toninelli. Local dimer dynamics in higher dimensions. arXiv preprint arXiv:2304.10930 ,

    work page arXiv

  5. [5]

    Random Domino Tilings and the Arctic Circle Theorem

    [JPS98] William Jockusch, James Propp, and Peter Shor. Random domino tilings and the Arctic circle theorem. arXiv preprint arXiv:math/9801068 ,

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Lectures on Dimers

    [Ken09] Richard Kenyon. Lectures on dimers. aXiv preprint arXiv:0910.3129 ,

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    A variational principle for domino tilings of multiply con- nected domains

    [Kuc22] Nikolai Kuchumov. A variational principle for domino tilings of multiply con- nected domains. arXiv preprint arXiv:2110.06896 ,

    work page arXiv

  8. [8]

    [Mil15] Pedro H. Milet. Domino tilings of three-dimensional regions. PhD thesis, arXiv preprint arXiv:1503.04617,

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Domino tilings of three-dimensional regions: flips, trits and twists

    [MS] Pedro H. Milet and Nicolau C. Saldanha. Enumeration of tilings for the 4x4x4 box. http://mat.puc-rio.br/~nicolau/multiplex/example444.html. [MS14a] Pedro H. Milet and Nicolau C. Saldanha. Domino tilings of three-dimensional regions: flips, trits and twists. arXiv preprint arXiv:1410.7693 ,

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Twists for duplex regions

    [MS14b] Pedro H. Milet and Nicolau C. Saldanha. Twists for duplex regions. arXiv preprint arXiv:1411.1793,

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term

    [PRT19] Benedetto Piccoli, Francesco Rossi, and Magali Tournus. A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term. arXiv preprint arXiv:1910.05105 ,

    work page arXiv 1910

  12. [12]

    Macroscopic loops in the 3 d double- dimer model

    [QT22] Alexandra Quitmann and Lorenzo Taggi. Macroscopic loops in the 3 d double- dimer model. arXiv preprint arXiv:2206.08284 ,

    work page arXiv

  13. [13]

    Random three-dimensional tilings of Aztec oc- tahedra and tetrahedra: an extension of domino tilings

    [RY00] Dana Randall and Gary Yngve. Random three-dimensional tilings of Aztec oc- tahedra and tetrahedra: an extension of domino tilings. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 2000), pages 636–645. ACM, New York,

    work page 2000

  14. [14]

    In 2020 Fall Central Sec- tional Meeting. AMS,

    work page 2020