A discrete harmonic function bounded on a large portion of Z2 is constant
Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova, Mikhail Sodin · Duke Mathematical Journal · 2022
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.