The Boundary of a Square Tiling of a Graph coincides with the Poisson Boundary

Abstract

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.