Work

Every recurrent network has a potential tending to infinity

Year: 2025

Type: article

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.

Source: Comptes Rendus Mathématique

DOI: 10.5802/crmath.758

Authors: Asaf Nachmias, Yuval Peres

Institutions: Tel Aviv University, Beijing Institute of Mathematical Sciences and Applications

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Israel Science Foundation, H2020 European Research Council

Awards: G2004740150, Universality of random planar maps and trees