# An improvement of Liouville's theorem for discrete harmonic functions

An improvement of Liouville's theorem for discrete harmonic functions

The classical Liouville theorem tells us that a bounded harmonic

function on the plane is a constant. At the same time for any

(arbitrarily small) angle on the plane there exist non-constant

harmonic functions that are bounded everywhere outside this angle.

The situation is completely different for discrete harmonic functions

on the standard square lattices. The following strong version of the

Liouville theorem holds on the two-dimensional lattice. If a discrete

harmonic function is bounded on 99% of the lattice then it is constant.

A simple counter-example shows that in higher dimensions such

improvement is no longer true.

The talk is based on a joint work with L. Buhovsky, A. Logunov and M.

Sodin.