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)

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