An improvement of Liouville’s theorem for discrete harmonic functions [DELETED]

An improvement of Liouville’s theorem for discrete harmonic functions [DELETED]

Colloquium

Eugenia Malinnikova (Stanford University)

Wednesday, April 15, 2020 -
12:00pm to 1:00pm
Location: 
TBA

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.

Last updated: 2020/08/06 - 6:38am