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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.

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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
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URLURL TypeDescription Paper
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 (, 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.
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Microsoft ResearchUNSPECIFIED
Subject Keywords:Circle packing · Planar graphs · Harmonic functions · Dirichlet space · Electrical networks
Issue or Number:3
Record Number:CaltechAUTHORS:20210922-193309575
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Official Citation:Hutchcroft, T. Harmonic Dirichlet Functions on Planar Graphs. Discrete Comput Geom 61, 479–506 (2019).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111007
Deposited By: George Porter
Deposited On:27 Sep 2021 18:42
Last Modified:27 Sep 2021 18:42

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