k-independent percolation on trees

Pierre Mathieu and Christoph Temmel

Stochastic Processes and their Applications, 2012, vol. 122, issue 3, 1129-1153

Abstract: 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.

Keywords: k-independent; k-dependent; Tree percolation; Critical value; Percolation kernel; Second moment method; Shearer’s measure (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:122:y:2012:i:3:p:1129-1153

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DOI: 10.1016/j.spa.2011.10.014

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