Percolation on 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 pc ∈ [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 pu ∈ [pc, 1] such that the number of infinite clusters is a.s. ∞ for p ∈ (pc, pu), and a.s. one for p > pu. 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 ∈ (pc, pu), we also prove that each infinite cluster has uncountably many ends. For p > pc 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 p1 < p2 in (pc, pu), every infinite cluster at level p2 contains infinitely many infinite clusters at level p1. We also show that any Cartesian product G of d infinite connected graphs of bounded degree satisfies pu(G) ≤ pc(Z ). Research supported by a grant from the Swedish Natural Science Research Council Research partially supported by NSF grant # DMS-9803597 Research partially supported by NSF grant # DMS-9703814