Every recurrent network has a potential tending to infinity

Asaf Nachmias; Yuval Peres

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arXiv · 2502.17285

Every recurrent network has a potential tending to infinity

Asaf Nachmias·Yuval Peres

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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.

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Asaf Nachmias, Yuval Peres. 2025-02-24. Every recurrent network has a potential tending to infinity. https://arxiv.org/abs/2502.17285

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