A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy

TL;DR

This paper provides a comprehensive characterization of locally finite networks that admit non-constant harmonic functions with finite energy, unifying previous criteria and extending existing results.

Contribution

It unifies and extends criteria for the existence of non-constant harmonic functions of finite energy on locally finite networks.

Findings

Unified necessary and sufficient criteria for harmonic functions of finite energy

Extended existence criteria for non-elusive harmonic functions

Connected previous theoretical results into a comprehensive framework

Abstract

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen and of Lyons and Peres with the sufficient criterion of Soardi. We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos.