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

TL;DR

This paper demonstrates that the Poisson boundary of certain planar graphs can be geometrically represented as a circle boundary of a square tiling, confirming a related conjecture and introducing a new criterion for identifying Poisson boundaries.

Contribution

It proves a geometric realization of the Poisson boundary for planar, uniquely absorbing graphs and introduces a general criterion for identifying Poisson boundaries.

Findings

Poisson boundary coincides with the boundary of a square tiling

Confirms a conjecture of Northshield

Provides a new criterion for Poisson boundary identification

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.