Clusters in middle-phase percolation on hyperbolic plane

Abstract

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 (Benjamini and Schramm) that in such a graph G we have three essential phases of percolation, i. e. 0 < p_c(G) < p_u(G) < 1, where p_c is the critical probability and p_u - 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 the boundary of H^2. This result is similar to some results of Lalley.