Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 100, 20 pp.
Boundaries of planar graphs: a unified approach
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  and Angel, Barlow, Gurel-Gurevich and Nachmias  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.
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
Permanent link to this document
Digital Object Identifier
Primary: 05C81: Random walks on graphs
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