Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990 ) and Benjamini and Schramm (1996), we show that the property of having (al most surely) a unique infinite open cluster is increasing in p. Moreover, i n the standard coupling of the percolation models for all parameters, a.s. for all p(2) > p(1) > p(c), each infinite p(2)-cluster contains an infinite pr-cluster; this yields an extension of Alexander's (1995) "simultaneous u niqueness" theorem. As a corollary, we obtain that the probability theta(v) (p) that a given vertex v belongs to an infinite cluster is depends contin uously on p throughout the supercritical phase p > p(c). All our results ex tend to quasi-transitive infinite graphs with a unimodular automorphism gro up.