Percolation transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness

Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasi-transitive, there is another critical value p u ∈ [p c , 1] such that the number of infinite clusters is a.s. ∞ for p ∈ (p c , p u ), and a.s. one for p > p u . We prove a simultaneous version of this result in the canonical coupling of the percolation processes for all p ∈ [0, 1]. Simultaneously for all p ∈ (p c , p u ), we also prove that each infinite cluster has uncountably many ends. For p > p c we prove that all infinite clusters are indistinguishable by robust properties. Under the additional assumption that G is unimodular, we prove that a.s. for all p 1 < p 2 in (p c , p u ), every infinite cluster at level p 2 contains infinitely many infinite clusters at level p 1. We also show that any Cartesian product G of d infinite connected graphs of bounded degree satisfies p u (G) ≤ p c (Z d ).