Monotonicity of uniqueness for percolation on Cayley graphs: All infinite cluster are born simultaneously

Abstract

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 (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p2 > p1 > pc, each infinite p2-cluster contains an infinite p1-cluster; this yields an extension of Alexander's (1995) "simultaneous uniqueness" theorem. As a corollary, we obtain that the probability θυ (p) that a given vertex υ belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p > pc. All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.