Every recurrent network has a potential tending to infinity

TL;DR

This paper proves that in any infinite recurrent rooted network, there exists a potential function that tends to infinity, extending classical results from Euclidean and Riemannian geometries to network theory.

Contribution

It establishes the existence of unbounded potentials in infinite recurrent networks, bridging classical potential theory with graph-based network analysis.

Findings

Every infinite recurrent rooted network admits a potential tending to infinity.

The result extends classical potential theory to the setting of infinite networks.

Provides a new perspective on the behavior of potentials in complex network structures.

Abstract

A rooted network consists of a connected, locally finite graph G, equipped with edge conductances and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and Nakai in the settings of Euclidean domains and Riemannian surfaces.