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k-independent percolation on trees


  • Mathieu, Pierre
  • Temmel, Christoph


Consider the class of k-independent bond or site percolations with parameter p on a tree T. We derive tight bounds on p for both almost sure percolation and almost sure nonpercolation. The bounds are continuous functions of k and the branching number of T. This extends previous results by Lyons for the independent case (k=0) and by Balister & Bollobás for 1-independent bond percolations. Central to our argumentation are moment method bounds à la Lyons supplemented by explicit percolation models à la Balister & Bollobás. An indispensable tool is the minimality and explicit construction of Shearer’s measure on the k-fuzz of Z.

Suggested Citation

  • Mathieu, Pierre & Temmel, Christoph, 2012. "k-independent percolation on trees," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1129-1153.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:3:p:1129-1153
    DOI: 10.1016/

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