Comptes Rendus
Research article - Probability theory
Every recurrent network has a potential tending to infinity
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.

Received:
Revised:
Accepted:
Published online:
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{CRMATH_2025__363_G8_793_0,
     author = {Asaf Nachmias and Yuval Peres},
     title = {Every recurrent network has a potential tending to infinity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {793--798},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.758},
     language = {en},
}
TY  - JOUR
AU  - Asaf Nachmias
AU  - Yuval Peres
TI  - Every recurrent network has a potential tending to infinity
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 793
EP  - 798
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.758
LA  - en
ID  - CRMATH_2025__363_G8_793_0
ER  - 
%0 Journal Article
%A Asaf Nachmias
%A Yuval Peres
%T Every recurrent network has a potential tending to infinity
%J Comptes Rendus. Mathématique
%D 2025
%P 793-798
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.758
%G en
%F CRMATH_2025__363_G8_793_0
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/

[1] Nathanaël Berestycki; Diederik van Engelenburg Harnack inequality and one-endedness of UST on reversible random graphs, Probab. Theory Relat. Fields, Volume 188 (2024) no. 1-2, pp. 487-548 | DOI | MR | Zbl

[2] Diederik van Engelenburg; Tom Hutchcroft The number of ends in the uniform spanning tree for recurrent unimodular random graphs, Ann. Probab., Volume 52 (2024) no. 6, pp. 2079-2103 | DOI | MR | Zbl

[3] Griffith C. Evans Potentials and positively infinite singularities of harmonic functions, Monatsh. Math. Phys., Volume 43 (1936) no. 1, pp. 419-424 | DOI | MR | Zbl

[4] Mitsuru Nakai On Evans potential, Proc. Japan Acad., Volume 38 (1962), pp. 624-629 | Zbl

[5] Maurice Sion On general minimax theorems, Pac. J. Math., Volume 8 (1958), pp. 171-176 | DOI | Zbl

[6] Frank Spitzer Principles of random walk, Graduate Texts in Mathematics, 34, Springer, 1976, xiii+408 pages | DOI | MR | Zbl

Cited by Sources:

Comments - Policy