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.
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 . The circle packing in Figure 1 was created using Ken Stephenson's CirclePack software.
Published - 17-EJP116.pdf
Submitted - 1508.03923.pdf