We consider the simple random walk on two types of tilings of the hyperbolic plane. The first by 2π⁄q-angled regular polygons, and the second by the Voronoi tiling associated to a random discrete set of the hyperbolic plane, the Poisson point process. In the second case, we assume that there are on average λ points per unit area.
In both cases the random walk (almost surely) escapes to infinity with positive speed, and thus converges to a point on the circle. The distribution of this limit point is called the harmonic measure of the walk.
I will show that the Hausdorff dimension of the harmonic measure is strictly smaller than 1, for q sufficiently large in the Fuchsian case, and for λ sufficiently small in the Poisson case. In particular, the harmonic measure is singular with respect to the Lebesgue measure on the circle in these two cases.
The proof is based on a Furstenberg type formula for the speed together with an upper bound for the Hausdorff dimension by the ratio between the entropy and the speed of the walk.
This is joint work with P. Lessa and E. Paquette.