Potentials in recurrent networks: a survey

Abstract

A nonnegative function on the vertices of an infinite graph G which vanishes at a distinguished vertex o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We survey basic properties of potentials in recurrent networks. In particular, we show that potentials are Lipschitz with respect to the effective resistance metric, and if the potential is unique, then there is a determinantal formula for the harmonic measures from infinity. We also infer from the von Neumann minimax theorem that there always exists a potential tending to infinity.

Some material in this survey article is drawn from the authors' recent note arXiv:2502.17285