Comptes Rendus
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Every recurrent network has a potential tending to infinity
[Tout réseau récurrent a un potentiel tendant vers l’infini]
Asaf Nachmias1 ; Yuval Peres2
1 Department of Mathematical Sciences, Tel Aviv University, Israel
2 Beijing Institute of Mathematical Sciences and Applications, Beijing, China
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 793-798.

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.

Un réseau enraciné est un graphe G localement fini et connexe, doté de conductances sur les arêtes et d’un sommet distingué o. Un potentiel est une fonction sur les sommets de G qui est positive, s’annule en o, a un laplacien 1 en o et est harmonique en tous les autres sommets. Nous prouvons que tout réseau enraciné, récurrent et infini admet un potentiel tendant vers l’infini. Il s’agit d’un analogue de théorèmes classiques de Evans et Nakai dans le cadre des domaines euclidiens et des surfaces riemanniennes.

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DOI : 10.5802/crmath.758

Asaf Nachmias 1 ; Yuval Peres 2

1 Department of Mathematical Sciences, Tel Aviv University, Israel
2 Beijing Institute of Mathematical Sciences and Applications, Beijing, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Every recurrent network has a potential tending to infinity},
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Asaf Nachmias; Yuval Peres. Every recurrent network has a potential tending to infinity. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 793-798. doi : 10.5802/crmath.758. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.758/

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