Unique continuation on planar graphs

TL;DR

This paper proves that bounded discrete harmonic functions on large parts of periodic planar graphs are constant, using a new unique continuation result and geometric properties of level sets, with implications for classical Liouville theorems.

Contribution

It introduces a novel unique continuation theorem for weighted graph Laplacians on planar graphs, providing a geometric proof of the Liouville theorem in the square lattice case.

Findings

Bounded harmonic functions are constant on large portions of periodic planar graphs.

A new unique continuation principle for weighted graph Laplacians is established.

Provides a geometric proof of the Liouville theorem for the square lattice.

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).