Every recurrent network has a potential tending to infinity
Asaf Nachmias, Yuval Peres
TL;DR
The paper proves that every infinite recurrent rooted network admits a potential that tends to infinity, extending classical Evans/Nakai results to discrete networks. It develops a constructive, probabilistic framework using the Green kernel, dipoles, and the Doob $h$-transform to produce and analyze potentials. A minimax argument with boundary data yields a sequence of potentials whose convex combination forms a genuine potential $h$ with $h(v)\to\infty$ as $|v|\to\infty$, and the associated $h$-process is transient, implying recurrence of the original network. The approach provides a concrete discrete analogue of parabolicity results and clarifies when a potential tending to infinity exists, with implications for recurrence and the structure of potentials on recurrent networks.
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.