Potentials in recurrent networks: a survey

TL;DR

This survey explores the properties of potentials in recurrent networks, highlighting their Lipschitz continuity, determinantal formulas for harmonic measures, and the existence of potentials tending to infinity, based on graph theory and network analysis.

Contribution

It provides a comprehensive overview of potentials in recurrent networks, including new insights into their Lipschitz properties and formulas for harmonic measures.

Findings

Potentials are Lipschitz continuous with respect to the effective resistance metric.

Existence of a determinantal formula for harmonic measures from infinity.

There always exists a potential tending to infinity, proven via von Neumann minimax theorem.

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.