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Harmonic Dirichlet functions on planar graphs

Hutchcroft, Tom (2019) Harmonic Dirichlet functions on planar graphs. Discrete & Computational Geometry, 61 (3). pp. 479-506. ISSN 0179-5376. doi:10.1007/s00454-019-00057-2. https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309575

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Abstract

Benjamini and Schramm (Invent Math 126(3):565–587, 1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation T and every circle packing of T in a domain D, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on T and the space of harmonic Dirichlet functions on D.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00454-019-00057-2DOIArticle
https://arxiv.org/abs/1707.07751arXivDiscussion Paper
ORCID:
AuthorORCID
Hutchcroft, Tom0000-0003-0061-593X
Additional Information:© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 26 July 2017 / Revised: 24 October 2018 / Accepted: 7 January 2019 / Published online: 30 January 2019. The author was supported by a Microsoft Research PhD Fellowship. We thank the anonymous referees for their comments and corrections.
Funders:
Funding AgencyGrant Number
Microsoft ResearchUNSPECIFIED
Subject Keywords:Circle packing · Planar graphs · Harmonic functions · Dirichlet space · Electrical networks
Issue or Number:3
DOI:10.1007/s00454-019-00057-2
Record Number:CaltechAUTHORS:20210922-193309575
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309575
Official Citation:Hutchcroft, T. Harmonic Dirichlet Functions on Planar Graphs. Discrete Comput Geom 61, 479–506 (2019). https://doi.org/10.1007/s00454-019-00057-2
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111007
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:27 Sep 2021 18:42
Last Modified:27 Sep 2021 18:42

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