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arXiv · 2309.13728

Unique continuation on planar graphs

Ahmed Bou-Rabee·William Cooperman·Shirshendu Ganguly
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Abstract

We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017).

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Ahmed Bou-Rabee, William Cooperman, Shirshendu Ganguly. 2023-09-24. Unique continuation on planar graphs. https://doi.org/10.19086/da.144015

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