Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters 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 p 2>p 1>p c , each infinite p 2-cluster contains an infinite p 1-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.