  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.
