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Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously
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  • Published: February 1999

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

  • Olle Häggström1 &
  • Yuval Peres2 

Probability Theory and Related Fields volume 113, pages 273–285 (1999)Cite this article

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  • 47 Citations

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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.

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Authors and Affiliations

  1. Mathematical Statistics, Chalmers University of Technology, S-412 96 Göteborg, Sweden (e-mail: olleh@math.chalmers.se) , , , , , , SE

    Olle Häggström

  2. Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860 USA (e-mail: peres@stat.berkeley.edu), , , , , , US

    Yuval Peres

Authors
  1. Olle Häggström
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  2. Yuval Peres
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Received: 22 December 1997 / Revised version: 1 July 1998

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Häggström, O., Peres, Y. Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab Theory Relat Fields 113, 273–285 (1999). https://doi.org/10.1007/s004400050208

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  • Issue Date: February 1999

  • DOI: https://doi.org/10.1007/s004400050208

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  • Mathematics Subject Classification (1991): 60K35
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