A discrete harmonic function bounded on a large portion of ${\mathbb{Z}}^2$ 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 $\epsilon$ 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 $(1-\epsilon )$ portion of $Q$ then $u$ is constant.