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Article Dans Une Revue Stochastic Processes and their Applications Année : 2012

k-independent percolation on trees

Résumé

Consider the class of k-independent bond, respectively site, percolations with parameter p on an infinite tree T. We derive tight bounds on p for both a.s. percolation and a.s. 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 Bollob\`as & Balister for 1-independent bond percolations. Central to our argumentation are moment method bounds \`a la Lyons supplemented by explicit percolation models \`a la Bollob\`as & Balister. An indispensable tool is the minimality and explicit construction of Shearer's measure on the k-fuzz of Z.

Dates et versions

hal-01271014 , version 1 (08-02-2016)

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Pierre Mathieu, Christoph Temmel. k-independent percolation on trees. Stochastic Processes and their Applications, 2012, 122 (3), pp.1129-1153. ⟨10.1016/j.spa.2011.10.014⟩. ⟨hal-01271014⟩
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