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Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. (English) Zbl 0948.60098

Bramson, Maury (ed.) et al., Perplexing problems in probability. Festschrift in honor of Harry Kesten. Boston: Birkhäuser. Prog. Probab. 44, 69-90 (1999).
Summary: Consider i.i.d. percolation with retention parameter \(p\) on an infinite graph \(G\). There is a well known critical parameter \(p_c\in [0, 1]\) for the existence of infinite open clusters. Recently, it has been shown that when \(G\) is quasi-transitive, there is another critical value \(p_u\in[p_c,1]\) such that the number of infinite clusters is a.s. \(\infty\) for \(p\in (p_c, p_u)\), 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 \(p\in [0,1]\). Simultaneously for all \(p\in (p_c, p_u)\), 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 \(p\) 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 \(p_u(G)\leq p_c(\mathbb{Z}^d)\).
For the entire collection see [Zbl 0919.00093].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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