Every planar graph with the Liouville property is amenable

Johannes Carmesin; Agelos Georgakopoulos

doi:10.1002/rsa.20936

Every planar graph with the Liouville property is amenable

Johannes Carmesin^*, Agelos Georgakopoulos

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic functions.

Original language	English
Pages (from-to)	706-729
Number of pages	24
Journal	Random Structures and Algorithms
Volume	57
Issue number	3
Early online date	7 Jun 2020
DOIs	https://doi.org/10.1002/rsa.20936
Publication status	Published - Oct 2020

Bibliographical note

Publisher Copyright:
© 2020 Crown copyright. Random Structures and Algorithms published by Wiley Periodicals, LLC. This article is published with the permission of the Controller of HMSO and the Queen’s Printer for Scotland.

Keywords

circle packing
electrical network
harmonic function
infinitegraph
non-amenable
planar graphs
recurrent
transient
infinite graph

ASJC Scopus subject areas

Software
Applied Mathematics
General Mathematics
Computer Graphics and Computer-Aided Design

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10.1002/rsa.20936Licence: Creative Commons: Attribution (CC BY)

CarmesinJ2020EveryFinal published version, 1.21 MBLicence: Creative Commons: Attribution (CC BY)

Cite this

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Every planar graph with the Liouville property is amenable

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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