Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously |
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Authors: | Olle Häggström Yuval Peres |
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Institution: | Mathematical Statistics, Chalmers University of Technology, S-412 96 G?teborg, Sweden (e-mail: olleh@math.chalmers.se), SE Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860 USA (e-mail: peres@stat.berkeley.edu), US
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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.
Received: 22 December 1997 / Revised version: 1 July 1998 |
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Keywords: | Mathematics Subject Classification (1991): 60K35 |
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