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arXiv · 2610.39242

The Liouville property for geometric graphs in the plane

Kuijun Liu·Feng Luo·Yanwen Luo·Zhenghao Rao
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Abstract

We show that every bounded discrete harmonic function with respect to the cotangent Laplacian on a Delaunay triangulation of the Euclidean plane is constant; equivalently, the cotangent-weighted triangulation has the Liouville property. Our main theorem establishes this property for a more general class of connected weighted geometric graphs in the plane. Furthermore, we show that a planar Delaunay triangulation with cotangent weights could be transient but is recurrent if edge lengths are uniformly bounded.

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Kuijun Liu, Feng Luo, Yanwen Luo, Zhenghao Rao. 2026-10-29. The Liouville property for geometric graphs in the plane. https://arxiv.org/abs/2610.39242

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