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