A note on unique continuation for discrete harmonic functions

TL;DR

This paper explores the properties of discrete harmonic functions, providing a quantitative perspective on their unique continuation and establishing a three balls theorem analogue for these functions.

Contribution

It introduces a quantitative version of the unique continuation property and a three balls theorem for discrete harmonic functions, bridging discrete and continuous cases.

Findings

Discrete harmonic functions can vanish on large cubes without being identically zero.

A three balls theorem analogue is established for discrete harmonic functions.

Discrete harmonic functions resemble continuous ones on large scales.

Abstract

In this note we give a quantitative version of the following simple observation: a discrete harmonic function on the lattice may vanish at each point of a large cube without being zero identically, at the same time there is a version of three balls theorem for discrete harmonic functions since they resemble continuous ones on large scales.