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

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.