A discrete harmonic function bounded on a large portion of Z2 is constant

2022/03/30 by Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems

paper · doi:10.1215/00127094-2021-0037

Abstract

An improvement of the Liouville theorem for discrete harmonic functions on Z2 is obtained. More precisely, we prove that there exists a positive constant ε such that if u is discrete harmonic on Z2 and for each sufficiently large square Q centered at the origin |u|≤1 on a (1−ε) portion of Q, then u is constant.

Citations

• On localization in the continuous Anderson-Bernoulli model in higher dimension
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

Cited by

• Random Schrödinger operator with singular potentials
• On localization for the alloy-type Anderson-Bernoulli model with long-range hopping

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