| Résume | Tutte’s harmonic embeddings of planar graphs appear in various contexts including combinatorics, electrostatics, discrete geometry and more. Recent studies of 2d lattice models in the context of conformal field theory suggest yet another angle from which Tutte’s embeddings can be viewed as "discrete uniformizing maps": given an abstract planar graph with positive edge weights, we can hope that such embeddings encode the ”discrete conformal structure” induced by the simple random walk on this graph. This poses the following structural problem: given a sequence of Tutte's embeddings of planar graphs, describe the diffusion that appears in the limit of the underlying simple random walks. It turns out that linearized Monge-Ampère equations provide a convenient language for solving this problem. The link between Tutte's embeddings and linearized Monge-Ampère equations is built using the so-called Maxwell-Cremona lifts of the former which allows to associate a convex potential with every embedding. We will discuss the convergence of Green’s functions and solutions of Dirichlet problems on Tutte's embeddings under the convergence of potentials, and how the discrete complex analysis on t-embeddings can be used as a tool to prove it and to study the equation in the limit.
Based on a joint work with Dmitry Chelkak, Benoit Laslier and Marianna Russkikh.
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