Hutchcroft, Tom and Peres, Yuval (2017) Boundaries of planar graphs: a unified approach. Electronic Journal of Probability, 22 . Art. No. 100. ISSN 1083-6489. doi:10.1214/17-ejp116. https://resolver.caltech.edu/CaltechAUTHORS:20210924-183504452
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Abstract
We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos [9] and Angel, Barlow, Gurel-Gurevich and Nachmias [2] respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs. We also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.
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| Additional Information: | © 2017 The Author(s). Creative Commons Attribution 4.0 International License. Submitted to EJP on August 13, 2016, final version accepted on October 9, 2017. This work was carried out while TH was an intern at Microsoft Research, Redmond. We thank Russ Lyons and Asaf Nachmias for helpful discussions, and thank the anonymous referee for their careful reading of the paper. We thank Itai Benjamini for granting us permission to include the square tiling of Figure 1, which originally appeared in [5]. The circle packing in Figure 1 was created using Ken Stephenson’s CirclePack software. | |||||||||
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| Subject Keywords: | Circle packing, Harmonic functions, Martin boundary, Planar graphs, Poisson boundary, Random walk, Rough isometry, square tiling | |||||||||
| Classification Code: | AMS MSC 2010: 05C81 | |||||||||
| DOI: | 10.1214/17-ejp116 | |||||||||
| Record Number: | CaltechAUTHORS:20210924-183504452 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210924-183504452 | |||||||||
| Official Citation: | Tom Hutchcroft. Yuval Peres. "Boundaries of planar graphs: a unified approach." Electron. J. Probab. 22 1 - 20, 2017. https://doi.org/10.1214/17-EJP116 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 111023 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 27 Sep 2021 21:25 | |||||||||
| Last Modified: | 27 Sep 2021 21:25 |
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