K-independent percolation on trees

Abstract

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 Bollobàs & Balister for 1-independent bond percolations. Central to our argumentation are moment method bounds à la Lyons supplemented by explicit percolation models à la Bollobàs & Balister. An indispensable tool is the minimality and explicit construction of Shearer's measure on the k-fuzz of Z.

28 pages, 4 figures

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