The boundary of a square tiling of a graph coincides with the Poisson boundary.

Answering a question of Benjamini and Schramm (Ann Probab 24(3):1219-1238, ), we show that the Poisson boundary of any planar, uniquely absorbing (e.g. one-ended and transient) graph with bounded degrees can be realised geometrically as a circle, which arises from a discrete version of Riemann's mapping theorem. This implies a conjecture of Northshield (Potential Anal 2(4):299-314, ). Some of our technique apply to the non-planar case and might have further applications. When the graph is also hyperbolic then, under mild conditions, we prove the equivalence of several boundary constructions. [ABSTRACT FROM AUTHOR]