  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.
