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Abstract

I consider $p$-Bernoulli bond percolation 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. \[ 0<{p_\mathrm c}(G)<{p_\mathrm u}(G)<1, \] where ${p_\mathrm c}$ is the critical probability and ${p_\mathrm u}$—the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in $\partial{\mathbb{H}^2}$. This result is similar to some results in [Lal].