Harmonic functions on Tutte embeddings and linearized Monge-Amp\`ere equation

Beno\^it Laslier; Dmitry Chelkak; Marianna Russkikh; Mikhail Basok

arxiv: 2511.06587 · v3 · pith:IGQD6HCKnew · submitted 2025-11-10 · 🧮 math-ph · math.MP· math.PR

Harmonic functions on Tutte embeddings and linearized Monge-Amp\`ere equation

Mikhail Basok , Dmitry Chelkak , Beno\^it Laslier , Marianna Russkikh This is my paper

Pith reviewed 2026-05-21 20:14 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR

keywords Tutte embeddingsharmonic functionslinearized Monge-Ampère equationMaxwell-Cremona potentialsDirichlet problemsGreen's functionsdiscrete-to-continuous convergenceuniform convexity

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The pith

Solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings converge to those of the linearized Monge-Ampère equation under uniform convexity of the limiting potential.

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

The paper shows that discrete harmonic functions defined via Tutte embeddings of graphs converge to continuous solutions of the PDE L_φ h = 0 when the associated piecewise linear Maxwell-Cremona potentials converge to a continuous uniformly convex potential φ. This convergence applies to both Dirichlet problem solutions and Green's functions, and the uniform convexity of φ guarantees uniform ellipticity of the operator. The result extends earlier work on regular orthodiagonal tilings to more irregular setups and also examines limits that are harmonic with respect to a modified complex structure. The motivation is to handle 2d lattice models on irregular graphs by relating them to a continuous linear PDE.

Core claim

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge-Ampère equation L_φ h=0. We assume that piecewise linear Maxwell-Cremona potentials associated with the embeddings converge to a continuous potential φ and the only assumption that we use is the uniform convexity of φ or, equivalently, the uniform ellipticity of the operator L_φ. Even if φ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. We also study the situation in which the limits are harmonic in a different complex structure.

What carries the argument

The linearized Monge-Ampère operator L_φ (or mathcal{L}_varphi), whose uniform ellipticity is equivalent to uniform convexity of the limiting potential φ, acting on the discrete harmonic functions coming from Tutte embeddings whose Maxwell-Cremona potentials converge to φ.

If this is right

Discrete harmonic functions on the embeddings approximate the continuous solutions to the linearized equation.
Green's functions on the discrete side converge to their continuous counterparts.
The framework applies even when the limiting potential is quadratic, covering and extending orthodiagonal tiling cases.
Limits can satisfy harmonicity with respect to an altered complex structure instead of the standard one.

Where Pith is reading between the lines

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

The convergence may let researchers import analytic tools from the continuous PDE to study statistical mechanics on irregular or random lattices.
Numerical schemes based on Tutte embeddings could be used to approximate solutions of the linearized Monge-Ampère equation on domains with complicated boundaries.
Similar convergence statements might hold for other linear elliptic operators obtained by linearizing nonlinear geometric PDEs around convex potentials.

Load-bearing premise

The limiting potential φ must be uniformly convex so that the operator remains uniformly elliptic.

What would settle it

A sequence of Tutte embeddings whose Maxwell-Cremona potentials converge uniformly to a convex but not uniformly convex φ, yet the discrete Dirichlet solutions or Green's functions fail to converge to solutions of L_φ h=0.

Figures

Figures reproduced from arXiv: 2511.06587 by Beno\^it Laslier, Dmitry Chelkak, Marianna Russkikh, Mikhail Basok.

**Figure 1.** Figure 1: From left to right: an example of a harmonic embedding H of a weighted graph Γ, dual harmonic embedding H∗ of Γ∗ (with modified outer vertex), and the superposition graph Γ ∪ Γ ∗ . Note that we do not fix the embedding of Γ ∪ Γ ∗ into C. space R 3 . The second is called a t-surface: this is a polygonal surface Θδ embedded into R 2,2 ∼= C 1,1 , which is obtained as the lift to this Minkowski space of the t-… view at source ↗

**Figure 2.** Figure 2: From left to right: an example of a harmonic embedding H of a weighted graph Γ, the corner graph V of Γ, and the t-embedding of V constructed out of H. Note that there is a bijection between the edges of V and the edges of Γ ∪ Γ ∗ . Also, there is a bijection between faces of V and all vertices of Γ ∪ Γ ∗ except the black boundary ones. The latter correspondence allows us to color the faces of V in black a… view at source ↗

read the original abstract

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Amp\`ere equation $\mathcal{L}_\varphi h=0$. More precisely, we assume that piecewise linear Maxwell--Cremona potentials associated with the embeddings converge to a continuous potential $\varphi$ and the only assumption that we use is the uniform convexity of $\varphi$ or, equivalently, the uniform ellipticity of the operator $\mathcal{L}_\varphi$. Even if $\varphi$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

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.

Desk Editor's Note private letter to a colleague

The paper proves convergence of discrete Dirichlet solutions and Green's functions on Tutte embeddings to the linearized Monge-Ampère equation under the assumption that the associated Maxwell-Cremona potentials converge to a uniformly convex limit.

read the letter

The main point is that under the assumption that the piecewise linear Maxwell-Cremona potentials converge to a continuous uniformly convex potential, the discrete harmonic functions on these Tutte embeddings converge to the solutions of the linearized Monge-Ampère equation. This includes both Dirichlet problems and Green's functions. The paper makes this work even when the limit potential is quadratic, which broadens the known results from orthodiagonal tilings to more general irregular embeddings. They do a good job laying out a framework that could apply to 2D lattice models on graphs that aren't nicely symmetric. The uniform ellipticity condition is the key assumption, and they are upfront about it rather than hiding extra requirements. The part about limits being harmonic in a different complex structure is a nice addition for potential applications. The argument seems sound because it builds on the potential convergence and then applies standard elliptic theory in the limit. There is no sign of the kind of circular reasoning where the result depends on fitted parameters from the same setup. The math looks like a legitimate extension rather than a restatement. One soft spot is that the convergence of the potentials themselves is taken as given, so for concrete embeddings this might need separate verification. That's expected for this kind of conditional result, but it limits how plug-and-play the theorem is. Also, since the full proof details aren't in the abstract, the error estimates and boundary handling would need checking in the manuscript, but the stress test indicates no internal contradictions. This paper is for specialists in discrete complex analysis and statistical mechanics who want to move beyond regular lattices. Anyone working on convergence theorems for discrete operators on irregular graphs would get something out of it. It deserves serious peer review because the generalization is non-routine and the setup is honest about its assumptions. I would recommend engaging with the work and sending it out for refereeing.

Referee Report

1 major / 2 minor

Summary. The paper proves convergence of solutions to Dirichlet problems and Green's functions for discrete harmonic functions on Tutte embeddings to the corresponding objects for the linearized Monge-Ampère equation L_φ h = 0. The proof assumes that the piecewise-linear Maxwell-Cremona potentials associated with the embeddings converge to a continuous potential φ, with the sole additional hypothesis being uniform convexity of φ (equivalently, uniform ellipticity of L_φ). The result is stated to hold even when φ is quadratic, recovering and generalizing known cases for orthodiagonal tilings, and the authors also consider limits that are harmonic with respect to a different complex structure.

Significance. If the convergence holds under the stated hypotheses, the result provides a useful extension of discrete harmonic analysis from regular orthodiagonal meshes to irregular graphs arising from Tutte embeddings. This could support analysis of 2d lattice models on non-uniform discretizations. The explicit isolation of the uniform ellipticity assumption and the recovery of quadratic cases are positive features; the conditional character of the theorem is clearly articulated.

major comments (1)

The manuscript states that the proof relies on convergence of the piecewise-linear potentials to a continuous uniformly convex φ, but the error estimates controlling the passage from discrete to continuous Dirichlet solutions and Green's functions are not detailed in a way that makes the dependence on boundary data fully transparent. A concrete bound relating the discrete and continuous solutions near the boundary would strengthen the claim.

minor comments (2)

Notation for the linearized operator L_φ is introduced in the abstract but would benefit from an explicit definition in the first section, including the precise form of the coefficients in terms of second derivatives of φ.
The discussion of the alternative complex structure in the final section would be clearer if the change of coordinates or the modified ellipticity condition were written out explicitly rather than left as a reference to prior work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive suggestion regarding error estimates. We address the major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

Referee: The manuscript states that the proof relies on convergence of the piecewise-linear potentials to a continuous uniformly convex φ, but the error estimates controlling the passage from discrete to continuous Dirichlet solutions and Green's functions are not detailed in a way that makes the dependence on boundary data fully transparent. A concrete bound relating the discrete and continuous solutions near the boundary would strengthen the claim.

Authors: We appreciate the referee's observation. The convergence arguments in Sections 3 and 4 rely on the uniform ellipticity of L_φ to apply discrete maximum principles and barrier constructions that control the difference between discrete and continuous solutions, with the boundary data entering through the assumed uniform convergence of the potentials on the boundary. While these controls are present, we agree that the dependence is not stated as explicitly as it could be. In the revision we will add a short lemma (or remark) deriving an explicit bound of the form |h_disc - h_cont| ≤ C(φ,∂Ω)·(sup|φ_n - φ| + dist(x,∂Ω)^α) near the boundary, where the constants depend only on the uniform convexity modulus and the boundary data. This will be placed after the statement of the main convergence theorem and will not change any hypotheses or conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated assumptions

full rationale

The paper explicitly assumes convergence of piecewise-linear Maxwell-Cremona potentials to a continuous uniformly convex φ (equivalently, uniform ellipticity of L_φ) and proves convergence of discrete Dirichlet solutions and Green's functions to the continuous linearized Monge-Ampère objects under this hypothesis. This assumption is external to the derivation and not obtained by fitting or self-definition within the paper. The result generalizes known orthodiagonal cases without reducing any central step to a prior self-citation chain or renaming of inputs. The argument is conditional and does not claim to derive the potential convergence itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the modeling assumption that discrete potentials converge to a uniformly convex continuous limit; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Uniform convexity of the limiting potential φ is equivalent to uniform ellipticity of L_φ
Invoked as the sole assumption enabling the convergence proof.

pith-pipeline@v0.9.0 · 5672 in / 1147 out tokens · 33119 ms · 2026-05-21T20:14:25.072116+00:00 · methodology

discussion (0)

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We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge–Ampère equation L_φ h=0. ... the only assumption that we use is the uniform convexity of φ or, equivalently, the uniform ellipticity of the operator L_φ.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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