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

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 ε 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 $(1-\mathit{\epsilon })$ portion of Q, then u is constant.