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arXiv · 1712.07902

A discrete harmonic function bounded on a large portion of $\mathbb{Z}^2$ is constant

Lev Buhovsky·Alexander Logunov·Eugenia Malinnikova·Mikhail Sodin
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Abstract

An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at the origin $|u|\le 1$ on a $(1-\varepsilon)$ portion of $Q$ then $u$ is constant.

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Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova, Mikhail Sodin. 2017-12-21. A discrete harmonic function bounded on a large portion of $\mathbb{Z}^2$ is constant. https://arxiv.org/abs/1712.07902

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