**
Below is the ascii version of the abstract for 98-724.
The html version should be ready soon.**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