Limits of harmonic functions on $\mathbb{Z}^d$

TL;DR

This paper presents an example demonstrating that pointwise limits of positive harmonic functions on integer lattices can be non-harmonic, challenging assumptions about their convergence properties.

Contribution

It provides the first known example of positive harmonic functions on $ abla^d$ whose pointwise limit is not harmonic, revealing limitations in harmonic function convergence.

Findings

Positive harmonic functions can converge pointwise to non-harmonic functions.

The example applies to dimensions $d \, \geq \, 2$.

Convergence behavior of harmonic functions is more complex than previously understood.

Abstract

We give an example of a sequence of positive harmonic functions on $\mathbb{Z}^d$, $d\geq 2$, that converges pointwise to a non-harmonic function.