Clusters in middle-phase percolation on hyperbolic plane

I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i. e. 0 < pc(G) < pu(G) < 1, where pc is the critical probability and pu – the unification probability. I prove that in the middle phase a. s. all the ends of all the infinite clusters have one-point boundary in ∂H2. This result is similar to some results in [Lal].