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Thin-ended clusters in percolation in
$\mathbb{H}^d$
Published online by Cambridge University Press: 10 March 2023
Abstract
Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space
$\mathbb{H}^d$
in such a way that it admits a transitive action by isometries of
$\mathbb{H}^d$
. Let
$p_{\text{a}}$
be the supremum of all percolation parameters such that no point at infinity of
$\mathbb{H}^d$
lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter
$p < p_{\text{a}}$
, almost surely every percolation cluster is thin-ended, i.e. has only one-point boundaries of ends.
MSC classification
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- Original Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust
References
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