Clusters in middle-phase percolation on hyperbolic plane (Q2905541)

\(p\)-Bernoulli bond percolation is considered on transitive nonamenable planar graphs with one end (vertex transitive tiling graphs in the hyperbolic plane \(H^2\)), and on their duals. It is known [\textit{I. Benjamini} and \textit{O. Schramm}, J. Am. Math. Soc. 14, No. 2, 487--507 (2001; Zbl 1037.82018)] that, in this case, we have three essential phases of percolation whose tresholds end and/or begin at the values of the critical and unification probabilities \(0<p_c<p_u<1\). It is proven that, in the middle phase \((p_c,p_u)\subset [0,1]\), almost surely all infinite clusters have one-point boundaries in \(\partial H^2\).NEWLINENEWLINEFor the entire collection see [Zbl 1248.46002].