(CANCELLED) 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.