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Thin-ended clusters in percolation in $\mathbb{H}^d$

Part of: Equilibrium statistical mechanics Special aspects of infinite or finite groups Special processes

Published online by Cambridge University Press:  10 March 2023

Jan Czajkowski Open the ORCID record for Jan Czajkowski [Opens in a new window]
Jan Czajkowski*
Affiliation:
Wrocław University of Science and Technology
*
*Postal address: Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland. Email address: [email protected]
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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.

Keywords

Bernoulli percolationhyperbolic spacecluster boundaryexponential decay

MSC classification

Primary: 82B43: Percolation
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 20F55: Reflection and Coxeter groups
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 581 - 610
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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