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

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.

29 pages