Abstract

We characterize the locally finite networks admitting non-constant harmonic functions of finite energy. Our characterization unifies the necessary existence criteria of Thomassen (J Comb Theory, Ser B 49:87–102, 1990) and of Lyons and Peres (2011) with the sufficient criterion of Soardi (1991). We also extend a necessary existence criterion for non-elusive non-constant harmonic functions of finite energy due to Georgakopoulos (J Lond Math Soc, 2010).

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