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

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arxiv 1712.07902 v1 pith:5NZMMZVC submitted 2017-12-21 math.CA math.AP

Lev Buhovsky , Alexander Logunov , Eugenia Malinnikova , Mikhail Sodin This is my paper · ORCID
classification math.CAmath.AP
keywords constantdiscreteharmonicmathbblargeportionvarepsilonbounded
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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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On Support Cardinality for the Discrete Schr\"odinger Equation

    math-ph 2026-06 unverdicted novelty 6.0 of 10

    Minimal support cardinality S_d(N) of nontrivial origin-nonzero Dirichlet solutions is nondecreasing in d, so S_4(N) ≳ N²/log N, matching even-dimensional constructions up to a log factor.

  2. A Liouville principle for the random conductance model under degenerate conditions

    math.AP 2019-08 conditional novelty 6.0 of 10

    For stationary ergodic conductances on Z^d with a (p,q)-moment condition and reflection invariance, the space of harmonic functions growing slower than |x|^(1+alpha) has dimension d+1.

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