Analysis & Applied Math

Consider a sequence of Tutte's barycentric embedding of planar graphs with 'mesh size' tending to zero. How to describe the scaling limit of random walks/discrete harmonic functions on these graphs and in which generality does such a convergence result hold? It is not hard to guess that the linearized Monge-Ampère equation should appear in the limit and we prove the convergence using only the assumption that piecewise linear Maxwell-Cremona potentials associated with the embeddings converge to a uniformly convex potential. This significantly generalizes previously known results on the convergence of discrete harmonic functions on orthodiagonal tilings even in the situation when this potential is quadratic.

Based upon a joint work with Mikhail Basok, Benoît Laslier, and Marianna Russkikh arXiv:2511.06587.