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.
"Boundaries of planar graphs: a unified approach." Electron. J. Probab. 22 1 - 20, 2017. https://doi.org/10.1214/17-EJP116