Boundaries of planar graphs: a unified approach

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.