An improvement of Liouville theorem for discrete harmonic functions

Abstract: The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. At the same time for any angle on the plane, there exist non-constant harmonic functions that are bounded outside the angle. The situation is different for discrete harmonic functions on Z^2. We show that the following improved version of the Liouville theorem holds. If a discrete harmonic function is bounded on 99% of the plane then it is constant. It is a report on a joint work (in progress) with L. Buhovsky, A. Logunov and M. Sodin.