A Stationary Planar Random Graph with Singular Stationary Dual: Dyadic Lattice Graphs

TL;DR

This paper constructs examples of stationary planar random graphs with duals that have singular stationary measures, revealing limitations of planar duality in stationary random graph models and analyzing harmonic measure singularity.

Contribution

It provides the first examples of stationary graphs with duals having singular measures, demonstrating a fundamental limitation in planar duality for stationary graphs.

Findings

Dual graphs can have singular stationary measures.

Planar duality does not preserve stationarity.

Harmonic measure on dyadic lattice graphs is singular.

Abstract

Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a singular stationary measure. This answers a question of Curien and shows behaviour different from the unimodular case. The consequence is that planar duality does not combine well with stationary random graphs. We also study harmonic measure on dyadic lattice graphs and show its singularity.