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An improvement of the Liouville theorem for discrete harmonic functions on$\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists apositive constant $\varepsilon$ such that if $u$ is discrete harmonic on$\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at theorigin $|u|\le 1$ on a $(1-\varepsilon)$ portion of $Q$ then $u$ is constant.

A discrete harmonic function bounded on a large portion of $\mathbb{Z}^2$ is constant