Boundaries of Planar Graphs: A Unified Approach

TL;DR

This paper provides a unified, robust proof that the Poisson boundary of planar graphs matches their geometric boundaries, extending to graphs roughly isometric to planar graphs, and relates square tiling boundaries to Martin boundaries.

Contribution

It offers a new unified proof connecting various boundaries of planar graphs and extends results to graphs roughly isometric to planar graphs, also relating square tiling and Martin boundaries.

Findings

Poisson boundary coincides with circle packing boundary

Poisson boundary matches square tiling boundary for planar graphs

Square tiling boundary equals Martin boundary in bounded degree triangulations

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 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.