  Answering a question of Benjamini & Schramm [8], 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, namely as the
boundary of a tiling of a cylinder by squares. This implies a conjecture of
Northshield [34] of similar flavour. For our proof we introduce a general
criterion for identifying the Poisson boundary of a stochastic process that
might have further applications.
