One-point boundaries of ends of clusters in percolation in $\mathbb H^d$

Jan Czajkowski

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arXiv · 1804.05948

One-point boundaries of ends of clusters in percolation in $\mathbb H^d$

Jan Czajkowski

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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 $p_0$ be the supremum of such percolation parameters 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 $p < p_0$, a.s. every percolation cluster has only one-point boundaries of ends.

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Jan Czajkowski. 2018-04-16. One-point boundaries of ends of clusters in percolation in $\mathbb H^d$. https://arxiv.org/abs/1804.05948

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