Electronic Journal of Probability

Boundaries of planar graphs: a unified approach

Tom Hutchcroft and Yuval Peres

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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.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 100, 20 pp.

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

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Digital Object Identifier

Primary: 05C81: Random walks on graphs

planar graphs harmonic functions Poisson boundary Martin boundary random walk circle packing square tiling rough isometry

Creative Commons Attribution 4.0 International License.


Hutchcroft, Tom; Peres, Yuval. Boundaries of planar graphs: a unified approach. Electron. J. Probab. 22 (2017), paper no. 100, 20 pp. doi:10.1214/17-EJP116. https://projecteuclid.org/euclid.ejp/1511578855

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