Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harmonic functions on Tutte's embeddings and linearized Monge-Ampère equation

Mikhail Basok, Dmitry Chelkak, Benoît Laslier, Marianna Russkikh

TL;DR

This work establishes that discrete harmonic problems on Tutte embeddings of irregular planar graphs converge to continuum solutions of the linearized Monge–Ampère equation $\mathcal{L}_\varphi h=0$ under uniformly convex potentials $\varphi$. By developing the t-embedding framework and analyzing t-surfaces in Minkowski space $\mathbb{R}^{2,2}$, the authors connect discrete Dirichlet problems and Green’s functions to weak and strong solutions of $\mathcal{L}_\varphi$, with convergence shown first for $\varphi\in C^3$ and then extended to $\varphi\in C^{1,1}$. A key part of the approach is the introduction of Lip$(\kappa,\delta)$-type regularity for the t-surface and its relation to discrete holomorphicity via t-holomorphic functions, which yield ellipticity and Hölder bounds for the associated random walks. The paper also characterizes when a conformal coordinate $\zeta$ exists so that $\mathcal{L}_\varphi$ corresponds to a Laplacian, tying geometric maximal surface theory in $\mathbb{R}^{2,2}$ to PDE convergence. These results generalize prior orthodiagonal-tile analyses and have potential implications for 2D lattice models on irregular graphs and discrete conformal geometry in Liouville quantum gravity contexts, including both convergence of observables and the structure of the limiting diffusion.

Abstract

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation $\mathcal{L}_\varphi h=0$. The potential $\varphi$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $\varphi$. 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.

Harmonic functions on Tutte's embeddings and linearized Monge-Ampère equation

TL;DR

This work establishes that discrete harmonic problems on Tutte embeddings of irregular planar graphs converge to continuum solutions of the linearized Monge–Ampère equation under uniformly convex potentials . By developing the t-embedding framework and analyzing t-surfaces in Minkowski space , the authors connect discrete Dirichlet problems and Green’s functions to weak and strong solutions of , with convergence shown first for and then extended to . A key part of the approach is the introduction of Lip-type regularity for the t-surface and its relation to discrete holomorphicity via t-holomorphic functions, which yield ellipticity and Hölder bounds for the associated random walks. The paper also characterizes when a conformal coordinate exists so that corresponds to a Laplacian, tying geometric maximal surface theory in to PDE convergence. These results generalize prior orthodiagonal-tile analyses and have potential implications for 2D lattice models on irregular graphs and discrete conformal geometry in Liouville quantum gravity contexts, including both convergence of observables and the structure of the limiting diffusion.

Abstract

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation . The potential appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of . Even if 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.

Paper Structure

This paper contains 20 sections, 33 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Motivation
  3. Tutte's embeddings $\Gamma_\delta$, potentials $\Phi_\delta$, and random walks on $\Gamma_\delta$
  4. Main convergence results
  5. Convergence to harmonic functions in a non-trivial metric and t-surfaces in ${\mathbb{R}}^{2,2}$
  6. Organization of the paper
  7. T-surface of a harmonic embedding
  8. Definition of the t-surface
  9. Equivalence of property \ref{['prty:LIP']} and property $\mathrm{Lip}(\kappa,\delta)$ of the t-surface $\Theta$.
  10. Corollaries of property $\mathrm{Lip}(\kappa,\delta)$ for random walks and harmonic functions on $\Gamma$.
  11. T-holomorphic functions
  12. Elliptic equations and closed 1-forms in 2D
  13. Convergence of discrete harmonic functions on $\Gamma_\delta$ as $\delta\to 0$
  14. Oscillations estimate via the Dirichlet energy
  15. Discrete Caccioppoli estimate
  16. ...and 5 more sections

Key Result

Theorem 1

(i) Properties prty:CONV and/or prty:LIP of Tutte's embeddings $\Gamma_\delta$ on a domain $U\subset{\mathbb{C}}$ imply property prty:RW of the random walks on $\Gamma_\delta$ on each subdomain $U'\Subset U$. (ii) Vice versa, property prty:RW on $U'$ implies properties prty:CONV,prty:LIP on each sub

Figures (2)

  • Figure 1: From left to right: an example of a harmonic embedding ${\CMcal{H}}$ of a weighted graph $\Gamma$, dual harmonic embedding ${\CMcal{H}}^\ast$ of $\Gamma^\ast$ (with modified outer vertex), and the superposition graph $\Gamma\cup \Gamma^\ast$. Note that we do not fix the embedding of $\Gamma\cup\Gamma^\ast$ into ${\mathbb{C}}$.
  • Figure 2: From left to right: an example of a harmonic embedding ${\CMcal{H}}$ of a weighted graph $\Gamma$, the corner graph ${\CMcal{V}}$ of $\Gamma$, and the t-embedding of ${\CMcal{V}}$ constructed out of ${\CMcal{H}}$.

Theorems & Definitions (82)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1
  • Theorem 2
  • Remark 1.3
  • Theorem 3
  • Remark 1.4
  • Remark 1.5
  • Theorem 4
  • Remark 1.6
  • ...and 72 more