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The number of infinite clusters in dynamical percolation
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  • Published: June 1998

The number of infinite clusters in dynamical percolation

  • Yuval Peres1 &
  • Jeffrey E. Steif3 

Probability Theory and Related Fields volume 111, pages 141–165 (1998)Cite this article

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  • 24 Citations

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Summary.

Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. Häggström, we found conditions for existence of infinite clusters at exceptional times. Here we show that for ℤd, with p>p c , a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p c , a.s. there are infinitely many infinite clusters at all times. At the critical value p=p c , the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T k of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T k. The proof of this capacity criterion is based on a new kernel truncation technique.

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Authors and Affiliations

  1. Department of Statistics, University of California at Berkeley, Berkeley, CA 94720, USA. e-mail: peres@stat.berkeley.edu, , , , , , US

    Yuval Peres

  2. Department of Mathematical Statistics, Chalmers University of Technology, S-41296 Gothenburg, Sweden. e-mail: steif@math.chalmers.se, , , , , , SE

    Jeffrey E. Steif

Authors
  1. Yuval Peres
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  2. Jeffrey E. Steif
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Received: 5 May 1997 / In revised form: 24 November 1997

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Peres, Y., Steif, J. The number of infinite clusters in dynamical percolation. Probab Theory Relat Fields 111, 141–165 (1998). https://doi.org/10.1007/s004400050165

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  • Issue Date: June 1998

  • DOI: https://doi.org/10.1007/s004400050165

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  • Mathematics Subject Classification (1991): primary 60K35
  • secondary 31C15 60J45
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