The ratio set of the harmonic measure of a random walk on a hyperbolic group

We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III_0.