98-724 Olle H\"aggstr\"om, Yuval Peres and Roberto H. Schonmann

Percolation on Transitive Graphs as a Coalescent Process: Relentless Merging Followed by Simultaneous Uniqueness (278K, postcript) Nov 19, 98

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Abstract. 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 $p_2$ 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 $p_u(G) \leq p_c(\Z^d)$.

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