Infinite clusters in dependent automorphism invariant percolation on trees

We study dependent bond percolation on the homogeneous tree Tₙ of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters a.s. is either 0 or ∞. Furthermore, each infinite cluster a.s. has either 1, 2 or infinitely many topological ends, and infinite clusters with infinitely many topological ends have a.s. a branching number greater than 1. We also show that if the marginal probability that a single edge is open is at least $2/(n + 1)$, then the existence of infinite clusters has to have positive probability. Several concrete examples are considered.