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

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 $\mathcal{L}_φh=0$. The potential $φ$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $φ$. Even if $φ$ 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.

💡 Research Summary

This paper establishes a fundamental convergence result in discrete potential theory, demonstrating that harmonic functions defined on Tutte’s harmonic embeddings of planar weighted graphs converge to solutions of the linearized Monge–Ampère equation as the mesh size of the embedding tends to zero.

The authors consider a sequence (Γ_δ){δ→0} of Tutte embeddings (also known as barycentric or harmonic embeddings) of finite planar graphs into a complex domain U. Each edge carries a conductance weight c{vv’}>0, and the embedding is harmonic in the sense that each interior vertex is positioned at the weighted barycenter of its neighbors. From this embedding, a dual embedding H*_δ is constructed via the relation (1.1). A central object is the Maxwell–Cremona potential Φ_δ, a piecewise linear convex function defined on U whose gradient (in a distributional sense) is essentially the piecewise constant function Ψ_δ taking the value H*_δ(v*) on each face corresponding to the dual vertex v*.

The main geometric assumption is that these potentials satisfy a uniform convexity condition (CONV) at scales larger than δ. This condition, equivalent to a Lipschitz condition (LIP) on Ψ_δ, ensures that as δ→0, the potentials Φ_δ converge uniformly to a limit function φ which is uniformly convex and C^{1,1} smooth. The authors prove in Theorem 1 that this geometric condition is equivalent to a probabilistic property (RW) of the continuous-time random walk on the graph Γ_δ with the given conductances, which requires uniform ellipticity and comparability of the invariant measure to the Lebesgue measure starting from scale δ.

The limit potential φ defines a non-standard complex structure. The associated linearized Monge–Ampère operator is L_φ h = -div(A_φ ∇ h), where the matrix A_φ is the cofactor matrix of the Hessian of φ (see (1.8)). This elliptic operator generalizes the standard Laplacian, which corresponds to the special case φ(w)=|w|^2/2.

The paper’s main convergence results are Theorems 2 and 3. Theorem 2 states that solutions to the discrete Dirichlet problem on the subgraph Ω_δ (with boundary values given by a continuous function g) converge uniformly to the solution of the continuous Dirichlet problem for L_φ in Ω. Furthermore, appropriately chosen discrete harmonic conjugates converge to an A_φ-harmonic conjugate of the limit. Theorem 3 proves the analogous convergence for Green’s functions uniformly away from the diagonal.

The proof strategy first handles the case where the limit potential φ is C^3 smooth, leveraging classical PDE estimates and probabilistic techniques related to the random walk. The general C^{1,1} case is then addressed by approximating φ with smooth, uniformly convex potentials and controlling the approximation errors using the stability properties of both the discrete and continuous problems.

This work significantly generalizes prior convergence results for discrete harmonic functions, which were largely confined to orthodiagonal tilings (a special case where φ is quadratic). By allowing φ to be any uniformly convex function, the framework can accommodate highly irregular graph embeddings that arise naturally in contexts like the study of two-dimensional statistical physics models on random planar maps and Liouville Quantum Gravity. The final remarks about harmonic functions in a “different complex structure” hint at the potential application of these results to understanding scaling limits in such random geometric settings.