Clusters in middle-phase percolation on hyperbolic plane

Details Clusters in middle-phase percolation on hyperbolic plane Clusters in middle-phase percolation on hyperbolic plane

2011-03-31

arxiv.org/abs/1104.0027v1

Mathematics

Probability

13 pages, 9 figures. To appear in Banach Center Publications in Proceedings of the "13th Workshop: Non-Commutative Harmonic An

Scientific paper

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.

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