Open Access
2017 Boundaries of planar graphs: a unified approach
Tom Hutchcroft, Yuval Peres
Electron. J. Probab. 22: 1-20 (2017). DOI: 10.1214/17-EJP116


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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Tom Hutchcroft. Yuval Peres. "Boundaries of planar graphs: a unified approach." Electron. J. Probab. 22 1 - 20, 2017.


Received: 13 August 2016; Accepted: 9 October 2017; Published: 2017
First available in Project Euclid: 25 November 2017

zbMATH: 1378.05189
MathSciNet: MR3733658
Digital Object Identifier: 10.1214/17-EJP116

Primary: 05C81

Keywords: Circle packing , Harmonic functions , Martin boundary , Planar graphs , Poisson boundary , Random walk , Rough isometry , square tiling

Vol.22 • 2017
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