Clusters in middle-phase percolation on hyperbolic plane

Volume 96 / 2011

Jan Czajkowski Banach Center Publications 96 (2011), 99-113 MSC: Primary: 60K35; Secondary: 20H10, 82B43. DOI: 10.4064/bc96-0-6


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].


  • Jan CzajkowskiMathematical Institute, University of Wrocław
    pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

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